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In physics, special relativity (SR, also known as the special theory of relativity or STR) is the generally accepted and experimentally well confirmed physical theory regarding the relationship between space and time. In Albert Einstein's original pedagogical treatment, it is based on two postulates:
- that the laws of physics are invariant (i.e. identical) in all inertial systems (non-accelerating frames of reference).
- that the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.
It was originally proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".^{[1]} The inconsistency of Newtonian mechanics with Maxwell’s equations of electromagnetism and the lack of experimental confirmation for a hypothesized luminiferous aether led to the development of special relativity, which corrects mechanics to handle situations involving motions nearing the speed of light. As of today, special relativity is the most accurate model of motion at any speed. Even so, the Newtonian mechanics model is still useful (due to its simplicity and high accuracy) as an approximation at small velocities relative to the speed of light.
Special relativity implies a wide range of consequences, which have been experimentally verified,^{[2]} including length contraction, time dilation, relativistic mass, mass–energy equivalence, a universal speed limit and relativity of simultaneity. It has replaced the conventional notion of an absolute universal time with the notion of a time that is dependent on reference frame and spatial position. Rather than an invariant time interval between two events, there is an invariant spacetime interval. Combined with other laws of physics, the two postulates of special relativity predict the equivalence of mass and energy, as expressed in the mass–energy equivalence formula E = mc^{2}, where c is the speed of light in a vacuum.^{[3]}^{[4]}
A defining feature of special relativity is the replacement of the Galilean transformations of Newtonian mechanics with the Lorentz transformations. Time and space cannot be defined separately from each other. Rather space and time are interwoven into a single continuum known as spacetime. Events that occur at the same time for one observer can occur at different times for another.
The theory is "special" in that it only applies in the special case where the curvature of spacetime due to gravity is negligible.^{[5]}^{[6]} In order to include gravity, Einstein formulated general relativity in 1915. Special relativity, contrary to some outdated descriptions, is capable of handling accelerated frames of reference.^{[7]}
As Galilean relativity is now considered an approximation of special relativity that is valid for low speeds, special relativity is considered an approximation of general relativity that is valid for weak gravitational fields, i.e. at a sufficiently small scale and in conditions of free fall. Whereas general relativity incorporates noneuclidean geometry in order to represent gravitational effects as the geometric curvature of spacetime, special relativity is restricted to the flat spacetime known as Minkowski space. A locally Lorentz-invariant frame that abides by special relativity can be defined at sufficiently small scales, even in curved spacetime.
Galileo Galilei had already postulated that there is no absolute and well-defined state of rest (no privileged reference frames), a principle now called Galileo's principle of relativity. Einstein extended this principle so that it accounted for the constant speed of light,^{[8]} a phenomenon that had been recently observed in the Michelson–Morley experiment. He also postulated that it holds for all the laws of physics, including both the laws of mechanics and of electrodynamics.^{[9]}
Contents
- 1 Postulates
- 2 Lack of an absolute reference frame
- 3 Reference frames, coordinates, and the Lorentz transformation
- 4 Consequences derived from the Lorentz transformation
- 5 Other consequences
- 6 Causality and prohibition of motion faster than light
- 7 Geometry of spacetime
- 8 Physics in spacetime
- 9 Relativity and unifying electromagnetism
- 10 Status
- 11 Theories of relativity and quantum mechanics
- 12 See also
- 13 References
- 14 External links
- 15 History
- 16 From classical mechanics to general relativity
- 17 Definition and basic applications
- 18 Consequences of Einstein's theory
- 19 Astrophysical applications
- 20 Advanced concepts
- 21 Relationship with quantum theory
- 22 Current status
- 23 See also
- 24 Notes
- 25 References
- 26 Further reading
- 27 External links
- 28 Fundamentals
- 29 M-theory
- 30 Black holes
- 31 AdS/CFT correspondence
- 32 Phenomenology
- 33 Connections to mathematics
- 34 History
- 35 Criticism
- 36 References
- 37 Further reading
- 38 External links
- 39 History
- 40 Mathematical formulations
- 41 Mathematically equivalent formulations of quantum mechanics
- 42 Interactions with other scientific theories
- 43 Philosophical implications
- 44 Applications
- 45 Examples
- 46 See also
- 47 Notes
- 48 References
- 49 Further reading
- 50 External links
- 51 Formulation in terms of electric and magnetic fields
- 52 Relationship between differential and integral formulations
- 53 Conceptual descriptions
- 54 Vacuum equations, electromagnetic waves and speed of light
- 55 Macroscopic formulation
- 56 Alternative formulations
- 57 Solutions
- 58 Limitations of the Maxwell equations as a theory of electromagnetism
- 59 Variations
- 60 See also
- 61 Notes
- 62 References
- 63 Historical publications
- 64 External links
- 65 History
- 66 Feynman's view of quantum electrodynamics
- 67 Mathematics
- 68 Renormalizability
- 69 Nonconvergence of series
- 70 See also
- 71 References
- 72 Further reading
- 73 External links
PostulatesEdit
“ | Reflections of this type made it clear to me as long ago as shortly after 1900, i.e., shortly after Planck's trailblazing work, that neither mechanics nor electrodynamics could (except in limiting cases) claim exact validity. Gradually I despaired of the possibility of discovering the true laws by means of constructive efforts based on known facts. The longer and the more desperately I tried, the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results... How, then, could such a universal principle be found? | ” |
— Albert Einstein: Autobiographical Notes^{[10]} |
Einstein discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of the (then) known laws of either mechanics or electrodynamics. These propositions were the constancy of the speed of light and the independence of physical laws (especially the constancy of the speed of light) from the choice of inertial system. In his initial presentation of special relativity in 1905 he expressed these postulates as:^{[1]}
- The Principle of Relativity – The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other.^{[1]}
- The Principle of Invariant Light Speed – "... light is always propagated in empty space with a definite velocity [speed] c which is independent of the state of motion of the emitting body" (from the preface).^{[1]} That is, light in vacuum propagates with the speed c (a fixed constant, independent of direction) in at least one system of inertial coordinates (the "stationary system"), regardless of the state of motion of the light source.
The derivation of special relativity depends not only on these two explicit postulates, but also on several tacit assumptions (made in almost all theories of physics), including the isotropy and homogeneity of space and the independence of measuring rods and clocks from their past history.^{[11]}
Following Einstein's original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations.^{[12]} However, the most common set of postulates remains those employed by Einstein in his original paper. A more mathematical statement of the Principle of Relativity made later by Einstein, which introduces the concept of simplicity not mentioned above is:
Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K.^{[13]}
Henri Poincaré provided the mathematical framework for relativity theory by proving that Lorentz transformations are a subset of his Poincaré group of symmetry transformations. Einstein later derived these transformations from his axioms.
Many of Einstein's papers present derivations of the Lorentz transformation based upon these two principles.^{[14]}
Einstein consistently based the derivation of Lorentz invariance (the essential core of special relativity) on just the two basic principles of relativity and light-speed invariance. He wrote:
The insight fundamental for the special theory of relativity is this: The assumptions relativity and light speed invariance are compatible if relations of a new type ("Lorentz transformation") are postulated for the conversion of coordinates and times of events... The universal principle of the special theory of relativity is contained in the postulate: The laws of physics are invariant with respect to Lorentz transformations (for the transition from one inertial system to any other arbitrarily chosen inertial system). This is a restricting principle for natural laws...^{[10]}
Thus many modern treatments of special relativity base it on the single postulate of universal Lorentz covariance, or, equivalently, on the single postulate of Minkowski spacetime.^{[15]}^{[16]}
From the principle of relativity alone without assuming the constancy of the speed of light (i.e. using the isotropy of space and the symmetry implied by the principle of special relativity) one can show that the spacetime transformations between inertial frames are either Euclidean, Galilean, or Lorentzian. In the Lorentzian case, one can then obtain relativistic interval conservation and a certain finite limiting speed. Experiments suggest that this speed is the speed of light in vacuum.^{[17]}^{[18]}
The constancy of the speed of light was motivated by Maxwell's theory of electromagnetism and the lack of evidence for the luminiferous ether. There is conflicting evidence on the extent to which Einstein was influenced by the null result of the Michelson–Morley experiment.^{[19]}^{[20]} In any case, the null result of the Michelson–Morley experiment helped the notion of the constancy of the speed of light gain widespread and rapid acceptance.
Lack of an absolute reference frameEdit
The principle of relativity, which states that there is no preferred inertial reference frame, dates back to Galileo, and was incorporated into Newtonian physics. However, in the late 19th century, the existence of electromagnetic waves led physicists to suggest that the universe was filled with a substance that they called "aether", which would act as the medium through which these waves, or vibrations travelled. The aether was thought to constitute an absolute reference frame against which speeds could be measured, and could be considered fixed and motionless. Aether supposedly possessed some wonderful properties: it was sufficiently elastic to support electromagnetic waves, and those waves could interact with matter, yet it offered no resistance to bodies passing through it. The results of various experiments, including the Michelson–Morley experiment, led to the theory of special relativity, by showing that there was no aether.^{[21]} Einstein's solution was to discard the notion of an aether and the absolute state of rest. In relativity, any reference frame moving with uniform motion will observe the same laws of physics. In particular, the speed of light in vacuum is always measured to be c, even when measured by multiple systems that are moving at different (but constant) velocities.
Reference frames, coordinates, and the Lorentz transformationEdit
Reference frames play a crucial role in relativity theory. The term reference frame as used here is an observational perspective in space which is not undergoing any change in motion (acceleration), from which a position can be measured along 3 spatial axes. In addition, a reference frame has the ability to determine measurements of the time of events using a 'clock' (any reference device with uniform periodicity).
An event is an occurrence that can be assigned a single unique time and location in space relative to a reference frame: it is a "point" in spacetime. Since the speed of light is constant in relativity in each and every reference frame, pulses of light can be used to unambiguously measure distances and refer back the times that events occurred to the clock, even though light takes time to reach the clock after the event has transpired.
For example, the explosion of a firecracker may be considered to be an "event". We can completely specify an event by its four spacetime coordinates: The time of occurrence and its 3-dimensional spatial location define a reference point. Let's call this reference frame S.
In relativity theory we often want to calculate the position of a point from a different reference point.
Suppose we have a second reference frame S′, whose spatial axes and clock exactly coincide with that of S at time zero, but it is moving at a constant velocity v with respect to S along the x-axis.
Since there is no absolute reference frame in relativity theory, a concept of 'moving' doesn't strictly exist, as everything is always moving with respect to some other reference frame. Instead, any two frames that move at the same speed in the same direction are said to be comoving. Therefore, S and S′ are not comoving.
Define the event to have spacetime coordinates (t,x,y,z) in system S and (t′,x′,y′,z′) in S′. Then the Lorentz transformation specifies that these coordinates are related in the following way:
where
is the Lorentz factor and c is the speed of light in vacuum, and the velocity v of S′ is parallel to the x-axis. The y and z coordinates are unaffected; only the x and t coordinates are transformed. These Lorentz transformations form a one-parameter group of linear mappings, that parameter being called rapidity.
There is nothing special about the x-axis, the transformation can apply to the y or z axes, or indeed in any direction, which can be done by directions parallel to the motion (which are warped by the γ factor) and perpendicular; see main article for details.
A quantity invariant under Lorentz transformations is known as a Lorentz scalar.
Writing the Lorentz transformation and its inverse in terms of coordinate differences, where for instance one event has coordinates (x_{1}, t_{1}) and (x′_{1}, t′_{1}), another event has coordinates (x_{2}, t_{2}) and (x′_{2}, t′_{2}), and the differences are defined as
we get
These effects are not merely appearances; they are explicitly related to our way of measuring time intervals between events which occur at the same place in a given coordinate system (called "co-local" events). These time intervals will be different in another coordinate system moving with respect to the first, unless the events are also simultaneous. Similarly, these effects also relate to our measured distances between separated but simultaneous events in a given coordinate system of choice. If these events are not co-local, but are separated by distance (space), they will not occur at the same spatial distance from each other when seen from another moving coordinate system. However, the spacetime interval will be the same for all observers. The underlying reality remains the same. Only our perspective changes.
Consequences derived from the Lorentz transformationEdit
The consequences of special relativity can be derived from the Lorentz transformation equations.^{[22]} These transformations, and hence special relativity, lead to different physical predictions than those of Newtonian mechanics when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything humans encounter that some of the effects predicted by relativity are initially counterintuitive.
Relativity of simultaneityEdit
Two events happening in two different locations that occur simultaneously in the reference frame of one inertial observer, may occur non-simultaneously in the reference frame of another inertial observer (lack of absolute simultaneity).
From the first equation of the Lorentz transformation in terms of coordinate differences
it is clear that two events that are simultaneous in frame S (satisfying Δt = 0), are not necessarily simultaneous in another inertial frame S′ (satisfying Δt′ = 0). Only if these events are additionally co-local in frame S (satisfying Δx = 0), will they be simultaneous in another frame S′.
Time dilationEdit
The time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers' reference frames (e.g., the twin paradox which concerns a twin who flies off in a spaceship traveling near the speed of light and returns to discover that his or her twin sibling has aged much more).
Suppose a clock is at rest in the unprimed system S. The location of the clock on two different ticks is then characterized by Δx = 0. To find the relation between the times between these ticks as measured in both systems, the first equation can be used to find:
- for events satisfying
This shows that the time (Δt′) between the two ticks as seen in the frame in which the clock is moving (S′), is longer than the time (Δt) between these ticks as measured in the rest frame of the clock (S). Time dilation explains a number of physical phenomena; for example, the lifetime of muons produced by cosmic rays impinging on the Earth's atmosphere is measured to be greater than the lifetimes of muons measured in the laboratory.^{[23]}
Length contractionEdit
The dimensions (e.g., length) of an object as measured by one observer may be smaller than the results of measurements of the same object made by another observer (e.g., the ladder paradox involves a long ladder traveling near the speed of light and being contained within a smaller garage).
Similarly, suppose a measuring rod is at rest and aligned along the x-axis in the unprimed system S. In this system, the length of this rod is written as Δx. To measure the length of this rod in the system S′, in which the clock is moving, the distances x′ to the end points of the rod must be measured simultaneously in that system S′. In other words, the measurement is characterized by Δt′ = 0, which can be combined with the fourth equation to find the relation between the lengths Δx and Δx′:
- for events satisfying
This shows that the length (Δx′) of the rod as measured in the frame in which it is moving (S′), is shorter than its length (Δx) in its own rest frame (S).
Composition of velocitiesEdit
Velocities (speeds) do not simply add. If the observer in S measures an object moving along the x axis at velocity u, then the observer in the S′ system, a frame of reference moving at velocity v in the x direction with respect to S, will measure the object moving with velocity u′ where (from the Lorentz transformations above):
The other frame S will measure:
Notice that if the object were moving at the speed of light in the S system (i.e. u = c), then it would also be moving at the speed of light in the S′ system. Also, if both u and v are small with respect to the speed of light, we will recover the intuitive Galilean transformation of velocities
The usual example given is that of a train (frame S′ above) traveling due east with a velocity v with respect to the tracks (frame S). A child inside the train throws a baseball due east with a velocity u′ with respect to the train. In nonrelativistic physics, an observer at rest on the tracks will measure the velocity of the baseball (due east) as u = u′ + v, while in special relativity this is no longer true; instead the velocity of the baseball (due east) is given by the second equation: u = (u′ + v)/(1 + u′v/c^{2}). Again, there is nothing special about the x or east directions. This formalism applies to any direction by considering parallel and perpendicular components of motion to the direction of relative velocity v, see main article for details.
Other consequencesEdit
Thomas rotationEdit
The orientation of an object (i.e. the alignment of its axes with the observer's axes) may be different for different observers. Unlike other relativistic effects, this effect becomes quite significant at fairly low velocities as can be seen in the spin of moving particles.
Equivalence of mass and energyEdit
As an object's speed approaches the speed of light from an observer's point of view, its relativistic mass increases thereby making it more and more difficult to accelerate it from within the observer's frame of reference.
The energy content of an object at rest with mass m equals mc^{2}. Conservation of energy implies that, in any reaction, a decrease of the sum of the masses of particles must be accompanied by an increase in kinetic energies of the particles after the reaction. Similarly, the mass of an object can be increased by taking in kinetic energies.
In addition to the papers referenced above—which give derivations of the Lorentz transformation and describe the foundations of special relativity—Einstein also wrote at least four papers giving heuristic arguments for the equivalence (and transmutability) of mass and energy, for E = mc^{2}.
Mass–energy equivalence is a consequence of special relativity. The energy and momentum, which are separate in Newtonian mechanics, form a four-vector in relativity, and this relates the time component (the energy) to the space components (the momentum) in a non-trivial way. For an object at rest, the energy–momentum four-vector is (E/c, 0, 0, 0): it has a time component which is the energy, and three space components which are zero. By changing frames with a Lorentz transformation in the x direction with a small value of the velocity v, the energy momentum four-vector becomes (E/c, Ev/c^{2}, 0, 0). The momentum is equal to the energy multiplied by the velocity divided by c^{2}. As such, the Newtonian mass of an object, which is the ratio of the momentum to the velocity for slow velocities, is equal to E/c^{2}.
The energy and momentum are properties of matter and radiation, and it is impossible to deduce that they form a four-vector just from the two basic postulates of special relativity by themselves, because these don't talk about matter or radiation, they only talk about space and time. The derivation therefore requires some additional physical reasoning. In his 1905 paper, Einstein used the additional principles that Newtonian mechanics should hold for slow velocities, so that there is one energy scalar and one three-vector momentum at slow velocities, and that the conservation law for energy and momentum is exactly true in relativity. Furthermore, he assumed that the energy of light is transformed by the same Doppler-shift factor as its frequency, which he had previously shown to be true based on Maxwell's equations.^{[1]} The first of Einstein's papers on this subject was "Does the Inertia of a Body Depend upon its Energy Content?" in 1905.^{[24]} Although Einstein's argument in this paper is nearly universally accepted by physicists as correct, even self-evident, many authors over the years have suggested that it is wrong.^{[25]} Other authors suggest that the argument was merely inconclusive because it relied on some implicit assumptions.^{[26]}
Einstein acknowledged the controversy over his derivation in his 1907 survey paper on special relativity. There he notes that it is problematic to rely on Maxwell's equations for the heuristic mass–energy argument. The argument in his 1905 paper can be carried out with the emission of any massless particles, but the Maxwell equations are implicitly used to make it obvious that the emission of light in particular can be achieved only by doing work. To emit electromagnetic waves, all you have to do is shake a charged particle, and this is clearly doing work, so that the emission is of energy.^{[27]}^{[28]}
How far can one travel from the Earth?Edit
Since one can not travel faster than light, one might conclude that a human can never travel farther from Earth than 40 light years if the traveller is active between the ages of 20 and 60. One would easily think that a traveller would never be able to reach more than the very few solar systems which exist within the limit of 20–40 light years from the earth. But that would be a mistaken conclusion. Because of time dilation, a hypothetical spaceship can travel thousands of light years during the pilot's 40 active years. If a spaceship could be built that accelerates at a constant 1 g, it will, after a little less than a year, be travelling at almost the speed of light as seen from Earth. This is described by:
where v(t) it the velocity at a time, t, a is the acceleration of 1g and t is the time as measured by people on Earth.^{[29]} Therefore, after 1 year of accelerating at 9.81m/s^{2}, the spaceship will be travelling at v = 0.77c relative to Earth. Time dilation will increase the travellers life span as seen from the reference frame of the Earth to 2.7 years, but his lifespan measured by a clock travelling with him will not change. During his journey, people on Earth will experience more time than he does. A 5-year round trip for him will take 6½ Earth years and cover a distance of over 6 light-years. A 20-year round trip for him (5 years accelerating, 5 decelerating, twice each) will land him back on Earth having travelled for 335 Earth years and a distance of 331 light years.^{[30]} A full 40-year trip at 1 g will appear on Earth to last 58,000 years and cover a distance of 55,000 light years. A 40-year trip at 1.1 g will take 148,000 Earth years and cover about 140,000 light years. A one-way 28 year (14 years accelerating, 14 decelerating as measured with the cosmonaut's clock) trip at 1 g acceleration could reach 2,000,000 light-years to the Andromeda Galaxy.^{[31]} This same time dilation is why a muon travelling close to c is observed to travel much further than c times its half-life (when at rest).^{[32]}
Causality and prohibition of motion faster than lightEdit
In diagram 2 the interval AB is 'time-like'; i.e., there is a frame of reference in which events A and B occur at the same location in space, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames. It is hypothetically possible for matter (or information) to travel from A to B, so there can be a causal relationship (with A the cause and B the effect).
The interval AC in the diagram is 'space-like'; i.e., there is a frame of reference in which events A and C occur simultaneously, separated only in space. There are also frames in which A precedes C (as shown) and frames in which C precedes A. If it were possible for a cause-and-effect relationship to exist between events A and C, then paradoxes of causality would result. For example, if A was the cause, and C the effect, then there would be frames of reference in which the effect preceded the cause. Although this in itself won't give rise to a paradox, one can show^{[33]}^{[34]} that faster than light signals can be sent back into one's own past. A causal paradox can then be constructed by sending the signal if and only if no signal was received previously.
Therefore, if causality is to be preserved, one of the consequences of special relativity is that no information signal or material object can travel faster than light in vacuum. However, some "things" can still move faster than light. For example, the location where the beam of a search light hits the bottom of a cloud can move faster than light when the search light is turned rapidly.^{[35]}
Even without considerations of causality, there are other strong reasons why faster-than-light travel is forbidden by special relativity. For example, if a constant force is applied to an object for a limitless amount of time, then integrating F = dp/dt gives a momentum that grows without bound, but this is simply because approaches infinity as approaches c. To an observer who is not accelerating, it appears as though the object's inertia is increasing, so as to produce a smaller acceleration in response to the same force. This behavior is observed in particle accelerators, where each charged particle is accelerated by the electromagnetic force.
Geometry of spacetimeEdit
Comparison between flat Euclidean space and Minkowski spaceEdit
Special relativity uses a 'flat' 4-dimensional Minkowski space – an example of a spacetime. Minkowski spacetime appears to be very similar to the standard 3-dimensional Euclidean space, but there is a crucial difference with respect to time.
In 3D space, the differential of distance (line element) ds is defined by
where dx = (dx_{1}, dx_{2}, dx_{3}) are the differentials of the three spatial dimensions. In Minkowski geometry, there is an extra dimension with coordinate X^{0} derived from time, such that the distance differential fulfills
where dX = (dX_{0}, dX_{1}, dX_{2}, dX_{3}) are the differentials of the four spacetime dimensions. This suggests a deep theoretical insight: special relativity is simply a rotational symmetry of our spacetime, analogous to the rotational symmetry of Euclidean space (see image right).^{[37]} Just as Euclidean space uses a Euclidean metric, so spacetime uses a Minkowski metric. Basically, special relativity can be stated as the invariance of any spacetime interval (that is the 4D distance between any two events) when viewed from any inertial reference frame. All equations and effects of special relativity can be derived from this rotational symmetry (the Poincaré group) of Minkowski spacetime.
The actual form of ds above depends on the metric and on the choices for the X^{0} coordinate. To make the time coordinate look like the space coordinates, it can be treated as imaginary: X_{0} = ict (this is called a Wick rotation). According to Misner, Thorne and Wheeler (1971, §2.3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) and to take X^{0} = ct, rather than a "disguised" Euclidean metric using ict as the time coordinate.
Some authors use X^{0} = t, with factors of c elsewhere to compensate; for instance, spatial coordinates are divided by c or factors of c^{±2} are included in the metric tensor.^{[38]} These numerous conventions can be superseded by using natural units where c = 1. Then space and time have equivalent units, and no factors of c appear anywhere.
3D spacetimeEdit
If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3D space
we see that the null geodesics lie along a dual-cone (see image right) defined by the equation;
or simply
which is the equation of a circle of radius c dt.
4D spacetimeEdit
If we extend this to three spatial dimensions, the null geodesics are the 4-dimensional cone:
so
This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars and say "The light from that star which I am receiving is X years old", we are looking down this line of sight: a null geodesic. We are looking at an event a distance away and a time d/c in the past. For this reason the null dual cone is also known as the 'light cone'. (The point in the lower left of the picture above right represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".)
The cone in the −t region is the information that the point is 'receiving', while the cone in the +t section is the information that the point is 'sending'.
The geometry of Minkowski space can be depicted using Minkowski diagrams, which are useful also in understanding many of the thought-experiments in special relativity.
Note that, in 4d spacetime, the concept of the center of mass becomes more complicated, see center of mass (relativistic).
Physics in spacetimeEdit
Transformations of physical quantities between reference framesEdit
Above, the Lorentz transformation for the time coordinate and three space coordinates illustrates that they are intertwined. This is true more generally: certain pairs of "timelike" and "spacelike" quantities naturally combine on equal footing under the same Lorentz transformation.
The Lorentz transformation in standard configuration above, i.e. for a boost in the x direction, can be recast into matrix form as follows:
In Newtonian mechanics, quantities which have magnitude and direction are mathematically described as 3d vectors in Euclidean space, and in general they are parametrized by time. In special relativity, this notion is extended by adding the appropriate timelike quantity to a spacelike vector quantity, and we have 4d vectors, or "four vectors", in Minkowski spacetime. The components of vectors are written using tensor index notation, as this has numerous advantages. The notation makes it clear the equations are manifestly covariant under the Poincaré group, thus bypassing the tedious calculations to check this fact. In constructing such equations, we often find that equations previously thought to be unrelated are, in fact, closely connected being part of the same tensor equation. Recognizing other physical quantities as tensors simplifies their transformation laws. Throughout, upper indices (superscripts) are contravariant indices rather than exponents except when they indicate a square (this is should be clear from the context), and lower indices (subscripts) are covariant indices. For simplicity and consistency with the earlier equations, Cartesian coordinates will be used.
The simplest example of a four-vector is the position of an event in spacetime, which constitutes a timelike component ct and spacelike component x = (x, y, z), in a contravariant position four vector with components:
where we define X^{0} = ct so that the time coordinate has the same dimension of distance as the other spatial dimensions; so that space and time are treated equally.^{[39]}^{[40]}^{[41]} Now the transformation of the contravariant components of the position 4-vector can be compactly written as:
where there is an implied summation on from 0 to 3, and is a matrix.
More generally, all contravariant components of a four-vector transform from one frame to another frame by a Lorentz transformation:
Examples of other 4-vectors include the four-velocity U^{μ}, defined as the derivative of the position 4-vector with respect to proper time:
where the Lorentz factor is:
The relativistic energy and relativistic momentum of an object are respectively the timelike and spacelike components of a contravariant four momentum vector:
where m is the invariant mass.
The four-acceleration is the proper time derivative of 4-velocity:
The transformation rules for three-dimensional velocities and accelerations are very awkward; even above in standard configuration the velocity equations are quite complicated owing to their non-linearity. On the other hand, the transformation of four-velocity and four-acceleration are simpler by means of the Lorentz transformation matrix.
The four-gradient of a scalar field φ transforms covariantly rather than contravariantly:
that is:
only in Cartesian coordinates. It's the covariant derivative which transforms in manifest covariance, in Cartesian coordinates this happens to reduce to the partial derivatives, but not in other coordinates.
More generally, the covariant components of a 4-vector transform according to the inverse Lorentz transformation:
where is the reciprocal matrix of .
The postulates of special relativity constrain the exact form the Lorentz transformation matrices take.
More generally, most physical quantities are best described as (components of) tensors. So to transform from one frame to another, we use the well-known tensor transformation law^{[42]}
where is the reciprocal matrix of . All tensors transform by this rule.
An example of a four dimensional second order antisymmetric tensor is the relativistic angular momentum, which has six components: three are the classical angular momentum, and the other three are related to the boost of the center of mass of the system. The derivative of the relativistic angular momentum with respect to proper time is the relativistic torque, also second order antisymmetric tensor.
The electromagnetic field tensor is another second order antisymmetric tensor field, with six components: three for the electric field and another three for the magnetic field. There is also the stress–energy tensor for the electromagnetic field, namely the electromagnetic stress–energy tensor.
MetricEdit
The metric tensor allows one to define the inner product of two vectors, which in turn allows one to assign a magnitude to the vector. Given the four-dimensional nature of spacetime the Minkowski metric η has components (valid in any inertial reference frame) which can be arranged in a 4 × 4 matrix:
which is equal to its reciprocal, , in those frames. Throughout we use the signs as above, different authors use different conventions – see Minkowski metric alternative signs.
The Poincaré group is the most general group of transformations which preserves the Minkowski metric:
and this is the physical symmetry underlying special relativity.
The metric can be used for raising and lowering indices on vectors and tensors. Invariants can be constructed using the metric, the inner product of a 4-vector T with another 4-vector S is:
Invariant means that it takes the same value in all inertial frames, because it is a scalar (0 rank tensor), and so no Λ appears in its trivial transformation. The magnitude of the 4-vector T is the positive square root of the inner product with itself:
One can extend this idea to tensors of higher order, for a second order tensor we can form the invariants:
similarly for higher order tensors. Invariant expressions, particularly inner products of 4-vectors with themselves, provide equations that are useful for calculations, because one doesn't need to perform Lorentz transformations to determine the invariants.
Relativistic kinematics and invarianceEdit
The coordinate differentials transform also contravariantly:
so the squared length of the differential of the position four-vector dX^{μ} constructed using
is an invariant. Notice that when the line element dX^{2} is negative that √−dX^{2} is the differential of proper time, while when dX^{2} is positive, √dX^{2} is differential of the proper distance.
The 4-velocity U^{μ} has an invariant form:
which means all velocity four-vectors have a magnitude of c. This is an expression of the fact that there is no such thing as being at coordinate rest in relativity: at the least, you are always moving forward through time. Differentiating the above equation by τ produces:
So in special relativity, the acceleration four-vector and the velocity four-vector are orthogonal.
Relativistic dynamics and invarianceEdit
The invariant magnitude of the momentum 4-vector generates the energy–momentum relation:
We can work out what this invariant is by first arguing that, since it is a scalar, it doesn't matter in which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero.
We see that the rest energy is an independent invariant. A rest energy can be calculated even for particles and systems in motion, by translating to a frame in which momentum is zero.
The rest energy is related to the mass according to the celebrated equation discussed above:
Note that the mass of systems measured in their center of momentum frame (where total momentum is zero) is given by the total energy of the system in this frame. It may not be equal to the sum of individual system masses measured in other frames.
To use Newton's third law of motion, both forces must be defined as the rate of change of momentum with respect to the same time coordinate. That is, it requires the 3D force defined above. Unfortunately, there is no tensor in 4D which contains the components of the 3D force vector among its components.
If a particle is not traveling at c, one can transform the 3D force from the particle's co-moving reference frame into the observer's reference frame. This yields a 4-vector called the four-force. It is the rate of change of the above energy momentum four-vector with respect to proper time. The covariant version of the four-force is:
In the rest frame of the object, the time component of the four force is zero unless the "invariant mass" of the object is changing (this requires a non-closed system in which energy/mass is being directly added or removed from the object) in which case it is the negative of that rate of change of mass, times c. In general, though, the components of the four force are not equal to the components of the three-force, because the three force is defined by the rate of change of momentum with respect to coordinate time, i.e. dp/dt while the four force is defined by the rate of change of momentum with respect to proper time, i.e. dp/dτ.
In a continuous medium, the 3D density of force combines with the density of power to form a covariant 4-vector. The spatial part is the result of dividing the force on a small cell (in 3-space) by the volume of that cell. The time component is −1/c times the power transferred to that cell divided by the volume of the cell. This will be used below in the section on electromagnetism.
Relativity and unifying electromagnetismEdit
Theoretical investigation in classical electromagnetism led to the discovery of wave propagation. Equations generalizing the electromagnetic effects found that finite propagation speed of the E and B fields required certain behaviors on charged particles. The general study of moving charges forms the Liénard–Wiechert potential, which is a step towards special relativity.
The Lorentz transformation of the electric field of a moving charge into a non-moving observer's reference frame results in the appearance of a mathematical term commonly called the magnetic field. Conversely, the magnetic field generated by a moving charge disappears and becomes a purely electrostatic field in a comoving frame of reference. Maxwell's equations are thus simply an empirical fit to special relativistic effects in a classical model of the Universe. As electric and magnetic fields are reference frame dependent and thus intertwined, one speaks of electromagnetic fields. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame.
Maxwell's equations in the 3D form are already consistent with the physical content of special relativity, although they are easier to manipulate in a manifestly covariant form, i.e. in the language of tensor calculus.^{[43]} See main links for more detail.
StatusEdit
Special relativity in its Minkowski spacetime is accurate only when the absolute value of the gravitational potential is much less than c^{2} in the region of interest.^{[44]} In a strong gravitational field, one must use general relativity. General relativity becomes special relativity at the limit of a weak field. At very small scales, such as at the Planck length and below, quantum effects must be taken into consideration resulting in quantum gravity. However, at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10^{−20})^{[45]} and thus accepted by the physics community. Experimental results which appear to contradict it are not reproducible and are thus widely believed to be due to experimental errors.
Special relativity is mathematically self-consistent, and it is an organic part of all modern physical theories, most notably quantum field theory, string theory, and general relativity (in the limiting case of negligible gravitational fields).
Newtonian mechanics mathematically follows from special relativity at small velocities (compared to the speed of light) – thus Newtonian mechanics can be considered as a special relativity of slow moving bodies. See classical mechanics for a more detailed discussion.
Several experiments predating Einstein's 1905 paper are now interpreted as evidence for relativity. Of these it is known Einstein was aware of the Fizeau experiment before 1905,^{[46]} and historians have concluded that Einstein was at least aware of the Michelson–Morley experiment as early as 1899 despite claims he made in his later years that it played no role in his development of the theory.^{[20]}
- The Fizeau experiment (1851, repeated by Michelson and Morley in 1886) measured the speed of light in moving media, with results that are consistent with relativistic addition of colinear velocities.
- The famous Michelson–Morley experiment (1881, 1887) gave further support to the postulate that detecting an absolute reference velocity was not achievable. It should be stated here that, contrary to many alternative claims, it said little about the invariance of the speed of light with respect to the source and observer's velocity, as both source and observer were travelling together at the same velocity at all times.
- The Trouton–Noble experiment (1903) showed that the torque on a capacitor is independent of position and inertial reference frame.
- The Experiments of Rayleigh and Brace (1902, 1904) showed that length contraction doesn't lead to birefringence for a co-moving observer, in accordance with the relativity principle.
Particle accelerators routinely accelerate and measure the properties of particles moving at near the speed of light, where their behavior is completely consistent with relativity theory and inconsistent with the earlier Newtonian mechanics. These machines would simply not work if they were not engineered according to relativistic principles. In addition, a considerable number of modern experiments have been conducted to test special relativity. Some examples:
- Tests of relativistic energy and momentum – testing the limiting speed of particles
- Ives–Stilwell experiment – testing relativistic Doppler effect and time dilation
- Time dilation of moving particles – relativistic effects on a fast-moving particle's half-life
- Kennedy–Thorndike experiment – time dilation in accordance with Lorentz transformations
- Hughes–Drever experiment – testing isotropy of space and mass
- Modern searches for Lorentz violation – various modern tests
- Experiments to test emission theory demonstrated that the speed of light is independent of the speed of the emitter.
- Experiments to test the aether drag hypothesis – no "aether flow obstruction".
Theories of relativity and quantum mechanicsEdit
Special relativity can be combined with quantum mechanics to form relativistic quantum mechanics. It is an unsolved problem in physics how general relativity and quantum mechanics can be unified; quantum gravity and a "theory of everything", which require such a unification, are active and ongoing areas in theoretical research.
The early Bohr–Sommerfeld atomic model explained the fine structure of alkali metal atoms using both special relativity and the preliminary knowledge on quantum mechanics of the time.^{[47]}
In 1928, Paul Dirac constructed an influential relativistic wave equation, now known as the Dirac equation in his honour,^{[48]} that is fully compatible both with special relativity and with the final version of quantum theory existing after 1926. This equation explained not only the intrinsic angular momentum of the electrons called spin, it also led to the prediction of the antiparticle of the electron (the positron),^{[48]}^{[49]} and fine structure could only be fully explained with special relativity. It was the first foundation of relativistic quantum mechanics. In non-relativistic quantum mechanics, spin is phenomenological and cannot be explained.
On the other hand, the existence of antiparticles leads to the conclusion that relativistic quantum mechanics is not enough for a more accurate and complete theory of particle interactions. Instead, a theory of particles interpreted as quantized fields, called quantum field theory, becomes necessary; in which particles can be created and destroyed throughout space and time.
See alsoEdit
- People: Hendrik Lorentz | Henri Poincaré | Albert Einstein | Max Planck | Hermann Minkowski | Max von Laue | Arnold Sommerfeld | Max Born | Gustav Herglotz | Richard C. Tolman
- Relativity: Theory of relativity | History of special relativity | Principle of relativity | General relativity | Frame of reference | Inertial frame of reference | Lorentz transformations | Bondi k-calculus | Einstein synchronisation | Rietdijk–Putnam argument | Special relativity (alternative formulations) | Criticism of relativity theory | Relativity priority dispute
- Physics: Newtonian Mechanics | spacetime | speed of light | simultaneity | center of mass (relativistic) | physical cosmology | Doppler effect | relativistic Euler equations | Aether drag hypothesis | Lorentz ether theory | Moving magnet and conductor problem | Shape waves | Relativistic heat conduction | Relativistic disk | Thomas precession | Born rigidity | Born coordinates
- Mathematics: Derivations of the Lorentz transformations | Minkowski space | four-vector | world line | light cone | Lorentz group | Poincaré group | geometry | tensors | split-complex number | Relativity in the APS formalism
- Philosophy: actualism | conventionalism | formalism
- Paradoxes: Twin paradox | Ehrenfest paradox | Ladder paradox | Bell's spaceship paradox | Velocity composition paradox
ReferencesEdit
- ^ ^{a} ^{b} ^{c} ^{d} ^{e} Albert Einstein (1905) "Zur Elektrodynamik bewegter Körper", Annalen der Physik 17: 891; English translation On the Electrodynamics of Moving Bodies by George Barker Jeffery and Wilfrid Perrett (1923); Another English translation On the Electrodynamics of Moving Bodies by Megh Nad Saha (1920).
- ^ Tom Roberts and Siegmar Schleif (October 2007). "What is the experimental basis of Special Relativity?". Usenet Physics FAQ. Retrieved 2008-09-17.
- ^ Albert Einstein (2001). Relativity: The Special and the General Theory (Reprint of 1920 translation by Robert W. Lawson ed.). Routledge. p. 48. ISBN 0-415-25384-5.
- ^ Richard Phillips Feynman (1998). Six Not-so-easy Pieces: Einstein's relativity, symmetry, and space–time (Reprint of 1995 ed.). Basic Books. p. 68. ISBN 0-201-32842-9.
- ^ Sean Carroll, Lecture Notes on General Relativity, ch. 1, "Special relativity and flat spacetime," http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll1.html
- ^ Wald, General Relativity, p. 60: "...the special theory of relativity asserts that spacetime is the manifold ℝ^{4} with a flat metric of Lorentz signature defined on it. Conversely, the entire content of special relativity ... is contained in this statement ..."
- ^ Rindler, W., 1969, Essential Relativity: Special, General, and Cosmological
- ^ Edwin F. Taylor and John Archibald Wheeler (1992). Spacetime Physics: Introduction to Special Relativity. W. H. Freeman. ISBN 0-7167-2327-1.
- ^ Wolfgang Rindler (1977). Essential Relativity. Birkhäuser. p. §1,11 p. 7. ISBN 3-540-07970-X.
- ^ ^{a} ^{b} Einstein, Autobiographical Notes, 1949.
- ^ Einstein, "Fundamental Ideas and Methods of the Theory of Relativity", 1920
- ^ For a survey of such derivations, see Lucas and Hodgson, Spacetime and Electromagnetism, 1990
- ^ Einstein, A., Lorentz, H. A., Minkowski, H., & Weyl, H. (1952). The Principle of Relativity: a collection of original memoirs on the special and general theory of relativity. Courier Dover Publications. p. 111. ISBN 0-486-60081-5.CS1 maint: multiple names: authors list (link)
- ^ Einstein, On the Relativity Principle and the Conclusions Drawn from It, 1907; "The Principle of Relativity and Its Consequences in Modern Physics", 1910; "The Theory of Relativity", 1911; Manuscript on the Special Theory of Relativity, 1912; Theory of Relativity, 1913; Einstein, Relativity, the Special and General Theory, 1916; The Principle Ideas of the Theory of Relativity, 1916; What Is The Theory of Relativity?, 1919; The Principle of Relativity (Princeton Lectures), 1921; Physics and Reality, 1936; The Theory of Relativity, 1949.
- ^ Das, A. (1993) The Special Theory of Relativity, A Mathematical Exposition, Springer, ISBN 0-387-94042-1.
- ^ Schutz, J. (1997) Independent Axioms for Minkowski Spacetime, Addison Wesley Longman Limited, ISBN 0-582-31760-6.
- ^ Yaakov Friedman (2004). Physical Applications of Homogeneous Balls. Progress in Mathematical Physics. 40. pp. 1–21. ISBN 0-8176-3339-1.
- ^ David Morin (2007) Introduction to Classical Mechanics, Cambridge University Press, Cambridge, chapter 11, Appendix I, ISBN 1-139-46837-5.
- ^ Michael Polanyi (1974) Personal Knowledge: Towards a Post-Critical Philosophy, ISBN 0-226-67288-3, footnote page 10–11: Einstein reports, via Dr N Balzas in response to Polanyi's query, that "The Michelson–Morley experiment had no role in the foundation of the theory." and "..the theory of relativity was not founded to explain its outcome at all." [1]
- ^ ^{a} ^{b} Jeroen van Dongen (2009). "On the role of the Michelson–Morley experiment: Einstein in Chicago" (PDF). Eprint arXiv:0908.1545. 0908: 1545. arXiv:0908.1545. Bibcode:2009arXiv0908.1545V.
- ^ Staley, Richard (2009), "Albert Michelson, the Velocity of Light, and the Ether Drift", Einstein's generation. The origins of the relativity revolution, Chicago: University of Chicago Press, ISBN 0-226-77057-5
- ^ Robert Resnick (1968). Introduction to special relativity. Wiley. pp. 62–63.
- ^ Daniel Kleppner and David Kolenkow (1973). An Introduction to Mechanics. pp. 468–70. ISBN 0-07-035048-5.
- ^ Does the inertia of a body depend upon its energy content? A. Einstein, Annalen der Physik. 18:639, 1905 (English translation by W. Perrett and G.B. Jeffery)
- ^ Max Jammer (1997). Concepts of Mass in Classical and Modern Physics. Courier Dover Publications. pp. 177–178. ISBN 0-486-29998-8.
- ^ John J. Stachel (2002). Einstein from B to Z. Springer. p. 221. ISBN 0-8176-4143-2.
- ^ On the Inertia of Energy Required by the Relativity Principle, A. Einstein, Annalen der Physik 23 (1907): 371–384
- ^ In a letter to Carl Seelig in 1955, Einstein wrote "I had already previously found that Maxwell's theory did not account for the micro-structure of radiation and could therefore have no general validity.", Einstein letter to Carl Seelig, 1955.
- ^ Baglio, Julien (26 May 2007). "Acceleration in special relativity: What is the meaning of "uniformly accelerated movement" ?" (PDF). Physics Department, ENS Cachan. Retrieved 22 January 2016.
- ^ Philip Gibbs and Don Koks. "The Relativistic Rocket". Retrieved 30 August 2012.
- ^ Philip Gibbs and Don Koks. "The Relativistic Rocket". Retrieved 13 October 2013.
- ^ The special theory of relativity shows that time and space are affected by motion. Library.thinkquest.org. Retrieved on 2013-04-24.
- ^ R. C. Tolman, The theory of the Relativity of Motion, (Berkeley 1917), p. 54
- ^ G. A. Benford, D. L. Book, and W. A. Newcomb (1970). "The Tachyonic Antitelephone". Physical Review D. 2 (2): 263. Bibcode:1970PhRvD...2..263B. doi:10.1103/PhysRevD.2.263.CS1 maint: multiple names: authors list (link)
- ^ Wesley C. Salmon (2006). Four Decades of Scientific Explanation. University of Pittsburgh. p. 107. ISBN 0-8229-5926-7., Section 3.7 page 107
- ^ J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 58. ISBN 0-7167-0344-0.CS1 maint: multiple names: authors list (link)
- ^ J.R. Forshaw, A.G. Smith (2009). Dynamics and Relativity. Wiley. p. 247. ISBN 978-0-470-01460-8.
- ^ R. Penrose (2007). The Road to Reality. Vintage books. ISBN 0-679-77631-1.
- ^ Jean-Bernard Zuber & Claude Itzykson, Quantum Field Theory, pg 5, ISBN 0-07-032071-3
- ^ Charles W. Misner, Kip S. Thorne & John A. Wheeler, Gravitation, pg 51, ISBN 0-7167-0344-0
- ^ George Sterman, An Introduction to Quantum Field Theory, pg 4 , ISBN 0-521-31132-2
- ^ Sean M. Carroll (2004). Spacetime and Geometry: An Introduction to General Relativity. Addison Wesley. p. 22. ISBN 0-8053-8732-3.
- ^ E. J. Post (1962). Formal Structure of Electromagnetics: General Covariance and Electromagnetics. Dover Publications Inc. ISBN 0-486-65427-3.
- ^ Øyvind Grøn and Sigbjørn Hervik (2007). Einstein's general theory of relativity: with modern applications in cosmology. Springer. p. 195. ISBN 0-387-69199-5. Extract of page 195 (with units where c=1)
- ^ The number of works is vast, see as example:
Sidney Coleman, Sheldon L. Glashow (1997). "Cosmic Ray and Neutrino Tests of Special Relativity". Phys. Lett. B405 (3–4): 249–252. arXiv:hep-ph/9703240. Bibcode:1997PhLB..405..249C. doi:10.1016/S0370-2693(97)00638-2.
An overview can be found on this page - ^ John D. Norton, John D. (2004). "Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905". Archive for History of Exact Sciences. 59: 45–105. Bibcode:2004AHES...59...45N. doi:10.1007/s00407-004-0085-6.
- ^ R. Resnick, R. Eisberg (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd ed.). John Wiley & Sons. pp. 114–116. ISBN 978-0-471-87373-0.
- ^ ^{a} ^{b} P.A.M. Dirac (1930). "A Theory of Electrons and Protons". Proceedings of the Royal Society. A126 (801): 360. Bibcode:1930RSPSA.126..360D. doi:10.1098/rspa.1930.0013. JSTOR 95359.
- ^ C.D. Anderson (1933). "The Positive Electron". Phys. Rev. 43 (6): 491–494. Bibcode:1933PhRv...43..491A. doi:10.1103/PhysRev.43.491.
TextbooksEdit
- Einstein, Albert (1920). Relativity: The Special and General Theory.
- Einstein, Albert (1996). The Meaning of Relativity. Fine Communications. ISBN 1-56731-136-9
- Logunov, Anatoly A. (2005) Henri Poincaré and the Relativity Theory (transl. from Russian by G. Pontocorvo and V. O. Soleviev, edited by V. A. Petrov) Nauka, Moscow.
- Charles Misner, Kip Thorne, and John Archibald Wheeler (1971) Gravitation. W. H. Freeman & Co. ISBN 0-7167-0334-3
- Post, E.J., 1997 (1962) Formal Structure of Electromagnetics: General Covariance and Electromagnetics. Dover Publications.
- Wolfgang Rindler (1991). Introduction to Special Relativity (2nd ed.), Oxford University Press. ISBN 978-0-19-853952-0; ISBN 0-19-853952-5
- Harvey R. Brown (2005). Physical relativity: space–time structure from a dynamical perspective, Oxford University Press, ISBN 0-19-927583-1; ISBN 978-0-19-927583-0
- Qadir, Asghar (1989). Relativity: An Introduction to the Special Theory. Singapore: World Scientific Publications. p. 128. ISBN 9971-5-0612-2.
- Silberstein, Ludwik (1914) The Theory of Relativity.
- Lawrence Sklar (1977). Space, Time and Spacetime. University of California Press. ISBN 0-520-03174-1.
- Lawrence Sklar (1992). Philosophy of Physics. Westview Press. ISBN 0-8133-0625-6.
- Taylor, Edwin, and John Archibald Wheeler (1992) Spacetime Physics (2nd ed.). W.H. Freeman & Co. ISBN 0-7167-2327-1
- Tipler, Paul, and Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman & Co. ISBN 0-7167-4345-0
Journal articlesEdit
- Alvager, T.; Farley, F. J. M.; Kjellman, J.; Wallin, L.; et al. (1964). "Test of the Second Postulate of Special Relativity in the GeV region". Physics Letters. 12 (3): 260. Bibcode:1964PhL....12..260A. doi:10.1016/0031-9163(64)91095-9.
- Darrigol, Olivier (2004). "The Mystery of the Poincaré–Einstein Connection". Isis. 95 (4): 614–26. doi:10.1086/430652. PMID 16011297.
- Wolf, Peter; Petit, Gerard (1997). "Satellite test of Special Relativity using the Global Positioning System". Physical Review A. 56 (6): 4405–09. Bibcode:1997PhRvA..56.4405W. doi:10.1103/PhysRevA.56.4405.
- Special Relativity Scholarpedia
- Special relativity: Kinematics Wolfgang Rindler, Scholarpedia, 6(2):8520. doi:10.4249/scholarpedia.8520
External linksEdit
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Original worksEdit
- Zur Elektrodynamik bewegter Körper Einstein's original work in German, Annalen der Physik, Bern 1905
- On the Electrodynamics of Moving Bodies English Translation as published in the 1923 book The Principle of Relativity.
Special relativity for a general audience (no mathematical knowledge required)Edit
- Einstein Light An award-winning, non-technical introduction (film clips and demonstrations) supported by dozens of pages of further explanations and animations, at levels with or without mathematics.
- Einstein Online Introduction to relativity theory, from the Max Planck Institute for Gravitational Physics.
- Audio: Cain/Gay (2006) – Astronomy Cast. Einstein's Theory of Special Relativity
Special relativity explained (using simple or more advanced mathematics)Edit
- Greg Egan's Foundations.
- The Hogg Notes on Special Relativity A good introduction to special relativity at the undergraduate level, using calculus.
- Relativity Calculator: Special Relativity – An algebraic and integral calculus derivation for E = mc^{2}.
- MathPages – Reflections on Relativity A complete online book on relativity with an extensive bibliography.
- Relativity An introduction to special relativity at the undergraduate level, without calculus.
- Relativity: the Special and General Theory at Project Gutenberg, by Albert Einstein
- Special Relativity Lecture Notes is a standard introduction to special relativity containing illustrative explanations based on drawings and spacetime diagrams from Virginia Polytechnic Institute and State University.
- Understanding Special Relativity The theory of special relativity in an easily understandable way.
- An Introduction to the Special Theory of Relativity (1964) by Robert Katz, "an introduction ... that is accessible to any student who has had an introduction to general physics and some slight acquaintance with the calculus" (130 pp; pdf format).
- Lecture Notes on Special Relativity by J D Cresser Department of Physics Macquarie University.
- SpecialRelativity.net - An overview with visualizations and minimal mathematics.
VisualizationEdit
- Raytracing Special Relativity Software visualizing several scenarios under the influence of special relativity.
- Real Time Relativity The Australian National University. Relativistic visual effects experienced through an interactive program.
- Spacetime travel A variety of visualizations of relativistic effects, from relativistic motion to black holes.
- Through Einstein's Eyes The Australian National University. Relativistic visual effects explained with movies and images.
- Warp Special Relativity Simulator A computer program to show the effects of traveling close to the speed of light.
- Animation clip on YouTube visualizing the Lorentz transformation.
- Original interactive FLASH Animations from John de Pillis illustrating Lorentz and Galilean frames, Train and Tunnel Paradox, the Twin Paradox, Wave Propagation, Clock Synchronization, etc.
General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915^{[1]} and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.
Some predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light. Examples of such differences include gravitational time dilation, gravitational lensing, the gravitational redshift of light, and the gravitational time delay. The predictions of general relativity have been confirmed in all observations and experiments to date. Although general relativity is not the only relativistic theory of gravity, it is the simplest theory that is consistent with experimental data. However, unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity.
Einstein's theory has important astrophysical implications. For example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars. There is ample evidence that the intense radiation emitted by certain kinds of astronomical objects is due to black holes; for example, microquasars and active galactic nuclei result from the presence of stellar black holes and black holes of a much more massive type, respectively. The bending of light by gravity can lead to the phenomenon of gravitational lensing, in which multiple images of the same distant astronomical object are visible in the sky. General relativity also predicts the existence of gravitational waves, which have since been observed directly by physics collaboration LIGO. In addition, general relativity is the basis of current cosmological models of a consistently expanding universe.
HistoryEdit
Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his new relativistic framework. In 1907, beginning with a simple thought experiment involving an observer in free fall, he embarked on what would be an eight-year search for a relativistic theory of gravity. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present, and form the core of Einstein's general theory of relativity.^{[2]}
The Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But as early as 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the so-called Schwarzschild metric. This solution laid the groundwork for the description of the final stages of gravitational collapse, and the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to electrically charged objects were taken, which eventually resulted in the Reissner–Nordström solution, now associated with electrically charged black holes.^{[3]} In 1917, Einstein applied his theory to the universe as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to match that observational presumption.^{[4]} By 1929, however, the work of Hubble and others had shown that our universe is expanding. This is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which our universe has evolved from an extremely hot and dense earlier state.^{[5]} Einstein later declared the cosmological constant the biggest blunder of his life.^{[6]}
During that period, general relativity remained something of a curiosity among physical theories. It was clearly superior to Newtonian gravity, being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein himself had shown in 1915 how his theory explained the anomalous perihelion advance of the planet Mercury without any arbitrary parameters ("fudge factors").^{[7]} Similarly, a 1919 expedition led by Eddington confirmed general relativity's prediction for the deflection of starlight by the Sun during the total solar eclipse of May 29, 1919,^{[8]} making Einstein instantly famous.^{[9]} Yet the theory entered the mainstream of theoretical physics and astrophysics only with the developments between approximately 1960 and 1975, now known as the golden age of general relativity.^{[10]} Physicists began to understand the concept of a black hole, and to identify quasars as one of these objects' astrophysical manifestations.^{[11]} Ever more precise solar system tests confirmed the theory's predictive power,^{[12]} and relativistic cosmology, too, became amenable to direct observational tests.^{[13]}
From classical mechanics to general relativityEdit
General relativity can be understood by examining its similarities with and departures from classical physics. The first step is the realization that classical mechanics and Newton's law of gravity admit a geometric description. The combination of this description with the laws of special relativity results in a heuristic derivation of general relativity.^{[14]}
Geometry of Newtonian gravityEdit
At the base of classical mechanics is the notion that a body's motion can be described as a combination of free (or inertial) motion, and deviations from this free motion. Such deviations are caused by external forces acting on a body in accordance with Newton's second law of motion, which states that the net force acting on a body is equal to that body's (inertial) mass multiplied by its acceleration.^{[15]} The preferred inertial motions are related to the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths are geodesics, straight world lines in curved spacetime.^{[16]}
Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such as electromagnetism or friction), can be used to define the geometry of space, as well as a time coordinate. However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors (see Eötvös experiment), there is a universality of free fall (also known as the weak equivalence principle, or the universal equality of inertial and passive-gravitational mass): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties.^{[17]} A simplified version of this is embodied in Einstein's elevator experiment, illustrated in the figure on the right: for an observer in a small enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is at rest in a gravitational field, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field.^{[18]}
Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential. Space, in this construction, still has the ordinary Euclidean geometry. However, spacetime as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable. From this, one can deduce that spacetime is curved. The resulting Newton–Cartan theory is a geometric formulation of Newtonian gravity using only covariant concepts, i.e. a description which is valid in any desired coordinate system.^{[19]} In this geometric description, tidal effects—the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass.^{[20]}
Relativistic generalizationEdit
As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of (special) relativistic mechanics.^{[21]} In the language of symmetry: where gravity can be neglected, physics is Lorentz invariant as in special relativity rather than Galilei invariant as in classical mechanics. (The defining symmetry of special relativity is the Poincaré group, which includes translations and rotations.) The differences between the two become significant when dealing with speeds approaching the speed of light, and with high-energy phenomena.^{[22]}
With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see image). The light-cones define a causal structure: for each event A, there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in the image), and a set of events for which such an influence is impossible (such as event C in the image). These sets are observer-independent.^{[23]} In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the space–time's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a Conformal structure^{[24]} or conformal geometry.
Special relativity is defined in the absence of gravity, so for practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall, an analogous reasoning as in the previous section applies: there are no global inertial frames. Instead there are approximate inertial frames moving alongside freely falling particles. Translated into the language of spacetime: the straight time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.^{[25]}
A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on electromagnetism, which could have a different set of preferred frames. But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through a gravitational field (cf. below). The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity.^{[26]} The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the Einstein equivalence principle, a crucial guiding principle for generalizing special-relativistic physics to include gravity.^{[27]}
The same experimental data shows that time as measured by clocks in a gravitational field—proper time, to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it is not measured by the Minkowski metric. As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with a curved generalization of Minkowski space. The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or pseudo-Riemannian metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the Levi-Civita connection, and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable locally inertial coordinates, the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish).^{[28]}
Einstein's equationsEdit
Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source is mass. In special relativity, mass turns out to be part of a more general quantity called the energy–momentum tensor, which includes both energy and momentum densities as well as stress (that is, pressure and shear).^{[29]} Using the equivalence principle, this tensor is readily generalized to curved space-time. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the field equation for gravity relates this tensor and the Ricci tensor, which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. In special relativity, conservation of energy–momentum corresponds to the statement that the energy–momentum tensor is divergence-free. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved-manifold counterparts, covariant derivatives studied in differential geometry. With this additional condition—the covariant divergence of the energy–momentum tensor, and hence of whatever is on the other side of the equation, is zero— the simplest set of equations are what are called Einstein's (field) equations:
On the left-hand side is the Einstein tensor, a specific divergence-free combination of the Ricci tensor and the metric. Where is symmetric. In particular,
is the curvature scalar. The Ricci tensor itself is related to the more general Riemann curvature tensor as
On the right-hand side, is the energy–momentum tensor. All tensors are written in abstract index notation.^{[30]} Matching the theory's prediction to observational results for planetary orbits (or, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics), the proportionality constant can be fixed as κ = 8πG/c^{4}, with G the gravitational constant and c the speed of light.^{[31]} When there is no matter present, so that the energy–momentum tensor vanishes, the results are the vacuum Einstein equations,
There are alternatives to general relativity built upon the same premises, which include additional rules and/or constraints, leading to different field equations. Examples are Brans–Dicke theory, teleparallelism, and Einstein–Cartan theory.^{[32]}
Definition and basic applicationsEdit
The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics, namely how the theory can be used for model-building.
Definition and basic propertiesEdit
General relativity is a metric theory of gravitation. At its core are Einstein's equations, which describe the relation between the geometry of a four-dimensional, pseudo-Riemannian manifold representing spacetime, and the energy–momentum contained in that spacetime.^{[33]} Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories), correspond to inertial motion within a curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow.^{[34]} The curvature is, in turn, caused by the energy–momentum of matter. Paraphrasing the relativist John Archibald Wheeler, spacetime tells matter how to move; matter tells spacetime how to curve.^{[35]}
While general relativity replaces the scalar gravitational potential of classical physics by a symmetric rank-two tensor, the latter reduces to the former in certain limiting cases. For weak gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation.^{[36]}
As it is constructed using tensors, general relativity exhibits general covariance: its laws—and further laws formulated within the general relativistic framework—take on the same form in all coordinate systems.^{[37]} Furthermore, the theory does not contain any invariant geometric background structures, i.e. it is background independent. It thus satisfies a more stringent general principle of relativity, namely that the laws of physics are the same for all observers.^{[38]} Locally, as expressed in the equivalence principle, spacetime is Minkowskian, and the laws of physics exhibit local Lorentz invariance.^{[39]}
Model-buildingEdit
The core concept of general-relativistic model-building is that of a solution of Einstein's equations. Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular, the matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present.^{[40]}
Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly.^{[41]} Nevertheless, a number of exact solutions are known, although only a few have direct physical applications.^{[42]} The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution, the Reissner–Nordström solution and the Kerr metric, each corresponding to a certain type of black hole in an otherwise empty universe,^{[43]} and the Friedmann–Lemaître–Robertson–Walker and de Sitter universes, each describing an expanding cosmos.^{[44]} Exact solutions of great theoretical interest include the Gödel universe (which opens up the intriguing possibility of time travel in curved spacetimes), the Taub-NUT solution (a model universe that is homogeneous, but anisotropic), and anti-de Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture).^{[45]}
Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions. In the field of numerical relativity, powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes.^{[46]} In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as naked singularities. Approximate solutions may also be found by perturbation theories such as linearized gravity^{[47]} and its generalization, the post-Newtonian expansion, both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity.^{[48]} An extension of this expansion is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories.^{[49]}
Consequences of Einstein's theoryEdit
General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of many years of research that followed Einstein's initial publication.
Gravitational time dilation and frequency shiftEdit
Assuming that the equivalence principle holds,^{[50]} gravity influences the passage of time. Light sent down into a gravity well is blueshifted, whereas light sent in the opposite direction (i.e., climbing out of the gravity well) is redshifted; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect is known as gravitational time dilation.^{[51]}
Gravitational redshift has been measured in the laboratory^{[52]} and using astronomical observations.^{[53]} Gravitational time dilation in the Earth's gravitational field has been measured numerous times using atomic clocks,^{[54]} while ongoing validation is provided as a side effect of the operation of the Global Positioning System (GPS).^{[55]} Tests in stronger gravitational fields are provided by the observation of binary pulsars.^{[56]} All results are in agreement with general relativity.^{[57]} However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.^{[58]}
Light deflection and gravitational time delayEdit
General relativity predicts that the path of light is bent in a gravitational field; light passing a massive body is deflected towards that body. This effect has been confirmed by observing the light of stars or distant quasars being deflected as it passes the Sun.^{[59]}
This and related predictions follow from the fact that light follows what is called a light-like or null geodesic—a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the invariance of lightspeed in special relativity.^{[60]} As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion),^{[61]} several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending the universality of free fall to light,^{[62]} the angle of deflection resulting from such calculations is only half the value given by general relativity.^{[63]}
Closely related to light deflection is the gravitational time delay (or Shapiro delay), the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction.^{[64]} In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay determine a parameter called γ, which encodes the influence of gravity on the geometry of space.^{[65]}
Gravitational wavesEdit
Predicted in 1916^{[66]}^{[67]} by Albert Einstein, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light. These are one of several analogies between weak-field gravity and electromagnetism in that, they are analogous to electromagnetic waves. On February 11, 2016, the Advanced LIGO team announced that they had directly detected gravitational waves from a pair of black holes merging.^{[68]}^{[69]}^{[70]}
The simplest type of such a wave can be visualized by its action on a ring of freely floating particles. A sine wave propagating through such a ring towards the reader distorts the ring in a characteristic, rhythmic fashion (animated image to the right).^{[71]} Since Einstein's equations are non-linear, arbitrarily strong gravitational waves do not obey linear superposition, making their description difficult. However, for weak fields, a linear approximation can be made. Such linearized gravitational waves are sufficiently accurate to describe the exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in relative distances increasing and decreasing by or less. Data analysis methods routinely make use of the fact that these linearized waves can be Fourier decomposed.^{[72]}
Some exact solutions describe gravitational waves without any approximation, e.g., a wave train traveling through empty space^{[73]} or so-called Gowdy universes, varieties of an expanding cosmos filled with gravitational waves.^{[74]} But for gravitational waves produced in astrophysically relevant situations, such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models.^{[75]}
Orbital effects and the relativity of directionEdit
General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation (precession) of planetary orbits, as well as orbital decay caused by the emission of gravitational waves and effects related to the relativity of direction.
Precession of apsidesEdit
In general relativity, the apsides of any orbit (the point of the orbiting body's closest approach to the system's center of mass) will precess—the orbit is not an ellipse, but akin to an ellipse that rotates on its focus, resulting in a rose curve-like shape (see image). Einstein first derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body as a test particle. For him, the fact that his theory gave a straightforward explanation of the anomalous perihelion shift of the planet Mercury, discovered earlier by Urbain Le Verrier in 1859, was important evidence that he had at last identified the correct form of the gravitational field equations.^{[76]}
The effect can also be derived by using either the exact Schwarzschild metric (describing spacetime around a spherical mass)^{[77]} or the much more general post-Newtonian formalism.^{[78]} It is due to the influence of gravity on the geometry of space and to the contribution of self-energy to a body's gravity (encoded in the nonlinearity of Einstein's equations).^{[79]} Relativistic precession has been observed for all planets that allow for accurate precession measurements (Mercury, Venus, and Earth),^{[80]} as well as in binary pulsar systems, where it is larger by five orders of magnitude.^{[81]}
Orbital decayEdit
According to general relativity, a binary system will emit gravitational waves, thereby losing energy. Due to this loss, the distance between the two orbiting bodies decreases, and so does their orbital period. Within the Solar System or for ordinary double stars, the effect is too small to be observable. This is not the case for a close binary pulsar, a system of two orbiting neutron stars, one of which is a pulsar: from the pulsar, observers on Earth receive a regular series of radio pulses that can serve as a highly accurate clock, which allows precise measurements of the orbital period. Because neutron stars are immensely compact, significant amounts of energy are emitted in the form of gravitational radiation.^{[83]}
The first observation of a decrease in orbital period due to the emission of gravitational waves was made by Hulse and Taylor, using the binary pulsar PSR1913+16 they had discovered in 1974. This was the first detection of gravitational waves, albeit indirect, for which they were awarded the 1993 Nobel Prize in physics.^{[84]} Since then, several other binary pulsars have been found, in particular the double pulsar PSR J0737-3039, in which both stars are pulsars.^{[85]}
Geodetic precession and frame-draggingEdit
Several relativistic effects are directly related to the relativity of direction.^{[86]} One is geodetic precession: the axis direction of a gyroscope in free fall in curved spacetime will change when compared, for instance, with the direction of light received from distant stars—even though such a gyroscope represents the way of keeping a direction as stable as possible ("parallel transport").^{[87]} For the Moon–Earth system, this effect has been measured with the help of lunar laser ranging.^{[88]} More recently, it has been measured for test masses aboard the satellite Gravity Probe B to a precision of better than 0.3%.^{[89]}^{[90]}
Near a rotating mass, there are so-called gravitomagnetic or frame-dragging effects. A distant observer will determine that objects close to the mass get "dragged around". This is most extreme for rotating black holes where, for any object entering a zone known as the ergosphere, rotation is inevitable.^{[91]} Such effects can again be tested through their influence on the orientation of gyroscopes in free fall.^{[92]} Somewhat controversial tests have been performed using the LAGEOS satellites, confirming the relativistic prediction.^{[93]} Also the Mars Global Surveyor probe around Mars has been used.^{[94]}^{[95]}
Astrophysical applicationsEdit
Gravitational lensingEdit
The deflection of light by gravity is responsible for a new class of astronomical phenomena. If a massive object is situated between the astronomer and a distant target object with appropriate mass and relative distances, the astronomer will see multiple distorted images of the target. Such effects are known as gravitational lensing.^{[96]} Depending on the configuration, scale, and mass distribution, there can be two or more images, a bright ring known as an Einstein ring, or partial rings called arcs.^{[97]} The earliest example was discovered in 1979;^{[98]} since then, more than a hundred gravitational lenses have been observed.^{[99]} Even if the multiple images are too close to each other to be resolved, the effect can still be measured, e.g., as an overall brightening of the target object; a number of such "microlensing events" have been observed.^{[100]}
Gravitational lensing has developed into a tool of observational astronomy. It is used to detect the presence and distribution of dark matter, provide a "natural telescope" for observing distant galaxies, and to obtain an independent estimate of the Hubble constant. Statistical evaluations of lensing data provide valuable insight into the structural evolution of galaxies.^{[101]}
Gravitational wave astronomyEdit
Observations of binary pulsars provide strong indirect evidence for the existence of gravitational waves (see Orbital decay, above). Detection of these waves is a major goal of current relativity-related research.^{[102]} Several land-based gravitational wave detectors are currently in operation, most notably the interferometric detectors GEO 600, LIGO (two detectors), TAMA 300 and VIRGO.^{[103]} Various pulsar timing arrays are using millisecond pulsars to detect gravitational waves in the 10^{−9} to 10^{−6} Hertz frequency range, which originate from binary supermassive blackholes.^{[104]} A European space-based detector, eLISA / NGO, is currently under development,^{[105]} with a precursor mission (LISA Pathfinder) having launched in December 2015.^{[106]}
Observations of gravitational waves promise to complement observations in the electromagnetic spectrum.^{[107]} They are expected to yield information about black holes and other dense objects such as neutron stars and white dwarfs, about certain kinds of supernova implosions, and about processes in the very early universe, including the signature of certain types of hypothetical cosmic string.^{[108]} In February 2016, the Advanced LIGO team announced that they had detected gravitational waves from a black hole merger.^{[68]}^{[69]}^{[109]}
Black holes and other compact objectsEdit
Whenever the ratio of an object's mass to its radius becomes sufficiently large, general relativity predicts the formation of a black hole, a region of space from which nothing, not even light, can escape. In the currently accepted models of stellar evolution, neutron stars of around 1.4 solar masses, and stellar black holes with a few to a few dozen solar masses, are thought to be the final state for the evolution of massive stars.^{[110]} Usually a galaxy has one supermassive black hole with a few million to a few billion solar masses in its center,^{[111]} and its presence is thought to have played an important role in the formation of the galaxy and larger cosmic structures.^{[112]}
Astronomically, the most important property of compact objects is that they provide a supremely efficient mechanism for converting gravitational energy into electromagnetic radiation.^{[113]} Accretion, the falling of dust or gaseous matter onto stellar or supermassive black holes, is thought to be responsible for some spectacularly luminous astronomical objects, notably diverse kinds of active galactic nuclei on galactic scales and stellar-size objects such as microquasars.^{[114]} In particular, accretion can lead to relativistic jets, focused beams of highly energetic particles that are being flung into space at almost light speed.^{[115]} General relativity plays a central role in modelling all these phenomena,^{[116]} and observations provide strong evidence for the existence of black holes with the properties predicted by the theory.^{[117]}
Black holes are also sought-after targets in the search for gravitational waves (cf. Gravitational waves, above). Merging black hole binaries should lead to some of the strongest gravitational wave signals reaching detectors here on Earth, and the phase directly before the merger ("chirp") could be used as a "standard candle" to deduce the distance to the merger events–and hence serve as a probe of cosmic expansion at large distances.^{[118]} The gravitational waves produced as a stellar black hole plunges into a supermassive one should provide direct information about the supermassive black hole's geometry.^{[119]}
CosmologyEdit
The current models of cosmology are based on Einstein's field equations, which include the cosmological constant Λ since it has important influence on the large-scale dynamics of the cosmos,
where is the spacetime metric.^{[120]} Isotropic and homogeneous solutions of these enhanced equations, the Friedmann–Lemaître–Robertson–Walker solutions,^{[121]} allow physicists to model a universe that has evolved over the past 14 billion years from a hot, early Big Bang phase.^{[122]} Once a small number of parameters (for example the universe's mean matter density) have been fixed by astronomical observation,^{[123]} further observational data can be used to put the models to the test.^{[124]} Predictions, all successful, include the initial abundance of chemical elements formed in a period of primordial nucleosynthesis,^{[125]} the large-scale structure of the universe,^{[126]} and the existence and properties of a "thermal echo" from the early cosmos, the cosmic background radiation.^{[127]}
Astronomical observations of the cosmological expansion rate allow the total amount of matter in the universe to be estimated, although the nature of that matter remains mysterious in part. About 90% of all matter appears to be so-called dark matter, which has mass (or, equivalently, gravitational influence), but does not interact electromagnetically and, hence, cannot be observed directly.^{[128]} There is no generally accepted description of this new kind of matter, within the framework of known particle physics^{[129]} or otherwise.^{[130]} Observational evidence from redshift surveys of distant supernovae and measurements of the cosmic background radiation also show that the evolution of our universe is significantly influenced by a cosmological constant resulting in an acceleration of cosmic expansion or, equivalently, by a form of energy with an unusual equation of state, known as dark energy, the nature of which remains unclear.^{[131]}
A so-called inflationary phase,^{[132]} an additional phase of strongly accelerated expansion at cosmic times of around 10^{-33} seconds, was hypothesized in 1980 to account for several puzzling observations that were unexplained by classical cosmological models, such as the nearly perfect homogeneity of the cosmic background radiation.^{[133]} Recent measurements of the cosmic background radiation have resulted in the first evidence for this scenario.^{[134]} However, there is a bewildering variety of possible inflationary scenarios, which cannot be restricted by current observations.^{[135]} An even larger question is the physics of the earliest universe, prior to the inflationary phase and close to where the classical models predict the big bang singularity. An authoritative answer would require a complete theory of quantum gravity, which has not yet been developed^{[136]} (cf. the section on quantum gravity, below).
Time travelEdit
Kurt Gödel showed^{[137]} that solutions to Einstein's equations exist that contain closed timelike curves (CTCs), which allow for loops in time. The solutions require extreme physical conditions unlikely ever to occur in practice, and it remains an open question whether further laws of physics will eliminate them completely. Since then other—similarly impractical—GR solutions containing CTCs have been found, such as the Tipler cylinder and traversable wormholes.
Advanced conceptsEdit
Causal structure and global geometryEdit
In general relativity, no material body can catch up with or overtake a light pulse. No influence from an event A can reach any other location X before light sent out at A to X. In consequence, an exploration of all light worldlines (null geodesics) yields key information about the spacetime's causal structure. This structure can be displayed using Penrose–Carter diagrams in which infinitely large regions of space and infinite time intervals are shrunk ("compactified") so as to fit onto a finite map, while light still travels along diagonals as in standard spacetime diagrams.^{[138]}
Aware of the importance of causal structure, Roger Penrose and others developed what is known as global geometry. In global geometry, the object of study is not one particular solution (or family of solutions) to Einstein's equations. Rather, relations that hold true for all geodesics, such as the Raychaudhuri equation, and additional non-specific assumptions about the nature of matter (usually in the form of so-called energy conditions) are used to derive general results.^{[139]}
HorizonsEdit
Using global geometry, some spacetimes can be shown to contain boundaries called horizons, which demarcate one region from the rest of spacetime. The best-known examples are black holes: if mass is compressed into a sufficiently compact region of space (as specified in the hoop conjecture, the relevant length scale is the Schwarzschild radius^{[140]}), no light from inside can escape to the outside. Since no object can overtake a light pulse, all interior matter is imprisoned as well. Passage from the exterior to the interior is still possible, showing that the boundary, the black hole's horizon, is not a physical barrier.^{[141]}
Early studies of black holes relied on explicit solutions of Einstein's equations, notably the spherically symmetric Schwarzschild solution (used to describe a static black hole) and the axisymmetric Kerr solution (used to describe a rotating, stationary black hole, and introducing interesting features such as the ergosphere). Using global geometry, later studies have revealed more general properties of black holes. In the long run, they are rather simple objects characterized by eleven parameters specifying energy, linear momentum, angular momentum, location at a specified time and electric charge. This is stated by the black hole uniqueness theorems: "black holes have no hair", that is, no distinguishing marks like the hairstyles of humans. Irrespective of the complexity of a gravitating object collapsing to form a black hole, the object that results (having emitted gravitational waves) is very simple.^{[142]}
Even more remarkably, there is a general set of laws known as black hole mechanics, which is analogous to the laws of thermodynamics. For instance, by the second law of black hole mechanics, the area of the event horizon of a general black hole will never decrease with time, analogous to the entropy of a thermodynamic system. This limits the energy that can be extracted by classical means from a rotating black hole (e.g. by the Penrose process).^{[143]} There is strong evidence that the laws of black hole mechanics are, in fact, a subset of the laws of thermodynamics, and that the black hole area is proportional to its entropy.^{[144]} This leads to a modification of the original laws of black hole mechanics: for instance, as the second law of black hole mechanics becomes part of the second law of thermodynamics, it is possible for black hole area to decrease—as long as other processes ensure that, overall, entropy increases. As thermodynamical objects with non-zero temperature, black holes should emit thermal radiation. Semi-classical calculations indicate that indeed they do, with the surface gravity playing the role of temperature in Planck's law. This radiation is known as Hawking radiation (cf. the quantum theory section, below).^{[145]}
There are other types of horizons. In an expanding universe, an observer may find that some regions of the past cannot be observed ("particle horizon"), and some regions of the future cannot be influenced (event horizon).^{[146]} Even in flat Minkowski space, when described by an accelerated observer (Rindler space), there will be horizons associated with a semi-classical radiation known as Unruh radiation.^{[147]}
SingularitiesEdit
Another general feature of general relativity is the appearance of spacetime boundaries known as singularities. Spacetime can be explored by following up on timelike and lightlike geodesics—all possible ways that light and particles in free fall can travel. But some solutions of Einstein's equations have "ragged edges"—regions known as spacetime singularities, where the paths of light and falling particles come to an abrupt end, and geometry becomes ill-defined. In the more interesting cases, these are "curvature singularities", where geometrical quantities characterizing spacetime curvature, such as the Ricci scalar, take on infinite values.^{[148]} Well-known examples of spacetimes with future singularities—where worldlines end—are the Schwarzschild solution, which describes a singularity inside an eternal static black hole,^{[149]} or the Kerr solution with its ring-shaped singularity inside an eternal rotating black hole.^{[150]} The Friedmann–Lemaître–Robertson–Walker solutions and other spacetimes describing universes have past singularities on which worldlines begin, namely Big Bang singularities, and some have future singularities (Big Crunch) as well.^{[151]}
Given that these examples are all highly symmetric—and thus simplified—it is tempting to conclude that the occurrence of singularities is an artifact of idealization.^{[152]} The famous singularity theorems, proved using the methods of global geometry, say otherwise: singularities are a generic feature of general relativity, and unavoidable once the collapse of an object with realistic matter properties has proceeded beyond a certain stage^{[153]} and also at the beginning of a wide class of expanding universes.^{[154]} However, the theorems say little about the properties of singularities, and much of current research is devoted to characterizing these entities' generic structure (hypothesized e.g. by the so-called BKL conjecture).^{[155]} The cosmic censorship hypothesis states that all realistic future singularities (no perfect symmetries, matter with realistic properties) are safely hidden away behind a horizon, and thus invisible to all distant observers. While no formal proof yet exists, numerical simulations offer supporting evidence of its validity.^{[156]}
Evolution equationsEdit
Each solution of Einstein's equation encompasses the whole history of a universe — it is not just some snapshot of how things are, but a whole, possibly matter-filled, spacetime. It describes the state of matter and geometry everywhere and at every moment in that particular universe. Due to its general covariance, Einstein's theory is not sufficient by itself to determine the time evolution of the metric tensor. It must be combined with a coordinate condition, which is analogous to gauge fixing in other field theories.^{[157]}
To understand Einstein's equations as partial differential equations, it is helpful to formulate them in a way that describes the evolution of the universe over time. This is done in so-called "3+1" formulations, where spacetime is split into three space dimensions and one time dimension. The best-known example is the ADM formalism.^{[158]} These decompositions show that the spacetime evolution equations of general relativity are well-behaved: solutions always exist, and are uniquely defined, once suitable initial conditions have been specified.^{[159]} Such formulations of Einstein's field equations are the basis of numerical relativity.^{[160]}
Global and quasi-local quantitiesEdit
The notion of evolution equations is intimately tied in with another aspect of general relativistic physics. In Einstein's theory, it turns out to be impossible to find a general definition for a seemingly simple property such as a system's total mass (or energy). The main reason is that the gravitational field—like any physical field—must be ascribed a certain energy, but that it proves to be fundamentally impossible to localize that energy.^{[161]}
Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" (ADM mass)^{[162]} or suitable symmetries (Komar mass).^{[163]} If one excludes from the system's total mass the energy being carried away to infinity by gravitational waves, the result is the so-called Bondi mass at null infinity.^{[164]} Just as in classical physics, it can be shown that these masses are positive.^{[165]} Corresponding global definitions exist for momentum and angular momentum.^{[166]} There have also been a number of attempts to define quasi-local quantities, such as the mass of an isolated system formulated using only quantities defined within a finite region of space containing that system. The hope is to obtain a quantity useful for general statements about isolated systems, such as a more precise formulation of the hoop conjecture.^{[167]}
Relationship with quantum theoryEdit
If general relativity were considered to be one of the two pillars of modern physics, then quantum theory, the basis of understanding matter from elementary particles to solid state physics, would be the other.^{[168]} However, how to reconcile quantum theory with general relativity is still an open question.
Quantum field theory in curved spacetimeEdit
Ordinary quantum field theories, which form the basis of modern elementary particle physics, are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth.^{[169]} In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime.^{[170]} Using this formalism, it can be shown that black holes emit a blackbody spectrum of particles known as Hawking radiation, leading to the possibility that they evaporate over time.^{[171]} As briefly mentioned above, this radiation plays an important role for the thermodynamics of black holes.^{[172]}
Quantum gravityEdit
The demand for consistency between a quantum description of matter and a geometric description of spacetime,^{[173]} as well as the appearance of singularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics.^{[174]} Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.^{[175]}^{[176]}
Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems.^{[177]} Some have argued that at low energies, this approach proves successful, in that it results in an acceptable effective (quantum) field theory of gravity.^{[178]} At very high energies, however, the perturbative results are badly divergent and lead to models devoid of predictive power ("perturbative non-renormalizability").^{[179]}
One attempt to overcome these limitations is string theory, a quantum theory not of point particles, but of minute one-dimensional extended objects.^{[180]} The theory promises to be a unified description of all particles and interactions, including gravity;^{[181]} the price to pay is unusual features such as six extra dimensions of space in addition to the usual three.^{[182]} In what is called the second superstring revolution, it was conjectured that both string theory and a unification of general relativity and supersymmetry known as supergravity^{[183]} form part of a hypothesized eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.^{[184]}
Another approach starts with the canonical quantization procedures of quantum theory. Using the initial-value-formulation of general relativity (cf. evolution equations above), the result is the Wheeler–deWitt equation (an analogue of the Schrödinger equation) which, regrettably, turns out to be ill-defined without a proper ultraviolet (lattice) cutoff.^{[185]} However, with the introduction of what are now known as Ashtekar variables,^{[186]} this leads to a promising model known as loop quantum gravity. Space is represented by a web-like structure called a spin network, evolving over time in discrete steps.^{[187]}
Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced,^{[188]} there are numerous other attempts to arrive at a viable theory of quantum gravity, some examples being the lattice theory of gravity based on the Feynman Path Integral approach and Regge Calculus,^{[175]}dynamical triangulations,^{[189]} causal sets,^{[190]} twistor models^{[191]} or the path-integral based models of quantum cosmology.^{[192]}
All candidate theories still have major formal and conceptual problems to overcome. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests (and thus to decide between the candidates where their predictions vary), although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available.^{[193]}
Current statusEdit
General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed many unambiguous observational and experimental tests. However, there are strong indications the theory is incomplete.^{[194]} The problem of quantum gravity and the question of the reality of spacetime singularities remain open.^{[195]} Observational data that is taken as evidence for dark energy and dark matter could indicate the need for new physics.^{[196]} Even taken as is, general relativity is rich with possibilities for further exploration. Mathematical relativists seek to understand the nature of singularities and the fundamental properties of Einstein's equations,^{[197]} and increasingly powerful computer simulations (such as those describing merging black holes) are run.^{[198]} In February 2016, it was announced that the existence of gravitational waves was directly detected by the Advanced LIGO team on September 14, 2015.^{[70]}^{[199]}^{[200]} A century after its publication, general relativity remains a highly active area of research.^{[201]}
See alsoEdit
- Alcubierre drive (warp drive)
- Center of mass (relativistic)
- Contributors to general relativity
- Derivations of the Lorentz transformations
- Ehrenfest paradox
- Einstein–Hilbert action
- Introduction to mathematics of general relativity
- Relativity priority dispute
- Ricci calculus
- Tests of general relativity
- Timeline of gravitational physics and relativity
- Two-body problem in general relativity
- Weak Gravity Conjecture
NotesEdit
- ^ O'Connor, J.J. and Robertson, E.F. (1996), General relativity. Mathematical Physics index, School of Mathematics and Statistics, University of St. Andrews, Scotland. Retrieved 2015-02-04.
- ^ Pais 1982, ch. 9 to 15, Janssen 2005; an up-to-date collection of current research, including reprints of many of the original articles, is Renn 2007; an accessible overview can be found in Renn 2005, pp. 110ff. Einstein's original papers are found in Digital Einstein, volumes 4 and 6. An early key article is Einstein 1907, cf. Pais 1982, ch. 9. The publication featuring the field equations is Einstein 1915, cf. Pais 1982, ch. 11–15
- ^ Schwarzschild 1916a, Schwarzschild 1916b and Reissner 1916 (later complemented in Nordström 1918)
- ^ Einstein 1917, cf. Pais 1982, ch. 15e
- ^ Hubble's original article is Hubble 1929; an accessible overview is given in Singh 2004, ch. 2–4
- ^ As reported in Gamow 1970. Einstein's condemnation would prove to be premature, cf. the section Cosmology, below
- ^ Pais 1982, pp. 253–254
- ^ Kennefick 2005, Kennefick 2007
- ^ Pais 1982, ch. 16
- ^ Thorne, Kip (2003). "Warping spacetime". The future of theoretical physics and cosmology: celebrating Stephen Hawking's 60th birthday. Cambridge University Press. p. 74. ISBN 0-521-82081-2. Extract of page 74
- ^ Israel 1987, ch. 7.8–7.10, Thorne 1994, ch. 3–9
- ^ Sections Orbital effects and the relativity of direction, Gravitational time dilation and frequency shift and Light deflection and gravitational time delay, and references therein
- ^ Section Cosmology and references therein; the historical development is in Overbye 1999
- ^ The following exposition re-traces that of Ehlers 1973, sec. 1
- ^ Arnold 1989, ch. 1
- ^ Ehlers 1973, pp. 5f
- ^ Will 1993, sec. 2.4, Will 2006, sec. 2
- ^ Wheeler 1990, ch. 2
- ^ Ehlers 1973, sec. 1.2, Havas 1964, Künzle 1972. The simple thought experiment in question was first described in Heckmann & Schücking 1959
- ^ Ehlers 1973, pp. 10f
- ^ Good introductions are, in order of increasing presupposed knowledge of mathematics, Giulini 2005, Mermin 2005, and Rindler 1991; for accounts of precision experiments, cf. part IV of Ehlers & Lämmerzahl 2006
- ^ An in-depth comparison between the two symmetry groups can be found in Giulini 2006a
- ^ Rindler 1991, sec. 22, Synge 1972, ch. 1 and 2
- ^ Ehlers 1973, sec. 2.3
- ^ Ehlers 1973, sec. 1.4, Schutz 1985, sec. 5.1
- ^ Ehlers 1973, pp. 17ff; a derivation can be found in Mermin 2005, ch. 12. For the experimental evidence, cf. the section Gravitational time dilation and frequency shift, below
- ^ Rindler 2001, sec. 1.13; for an elementary account, see Wheeler 1990, ch. 2; there are, however, some differences between the modern version and Einstein's original concept used in the historical derivation of general relativity, cf. Norton 1985
- ^ Ehlers 1973, sec. 1.4 for the experimental evidence, see once more section Gravitational time dilation and frequency shift. Choosing a different connection with non-zero torsion leads to a modified theory known as Einstein–Cartan theory
- ^ Ehlers 1973, p. 16, Kenyon 1990, sec. 7.2, Weinberg 1972, sec. 2.8
- ^ Ehlers 1973, pp. 19–22; for similar derivations, see sections 1 and 2 of ch. 7 in Weinberg 1972. The Einstein tensor is the only divergence-free tensor that is a function of the metric coefficients, their first and second derivatives at most, and allows the spacetime of special relativity as a solution in the absence of sources of gravity, cf. Lovelock 1972. The tensors on both side are of second rank, that is, they can each be thought of as 4×4 matrices, each of which contains ten independent terms; hence, the above represents ten coupled equations. The fact that, as a consequence of geometric relations known as Bianchi identities, the Einstein tensor satisfies a further four identities reduces these to six independent equations, e.g. Schutz 1985, sec. 8.3
- ^ Kenyon 1990, sec. 7.4
- ^ Brans & Dicke 1961, Weinberg 1972, sec. 3 in ch. 7, Goenner 2004, sec. 7.2, and Trautman 2006, respectively
- ^ Wald 1984, ch. 4, Weinberg 1972, ch. 7 or, in fact, any other textbook on general relativity
- ^ At least approximately, cf. Poisson 2004
- ^ Wheeler 1990, p. xi
- ^ Wald 1984, sec. 4.4
- ^ Wald 1984, sec. 4.1
- ^ For the (conceptual and historical) difficulties in defining a general principle of relativity and separating it from the notion of general covariance, see Giulini 2006b
- ^ section 5 in ch. 12 of Weinberg 1972
- ^ Introductory chapters of Stephani et al. 2003
- ^ A review showing Einstein's equation in the broader context of other PDEs with physical significance is Geroch 1996
- ^ For background information and a list of solutions, cf. Stephani et al. 2003; a more recent review can be found in MacCallum 2006
- ^ Chandrasekhar 1983, ch. 3,5,6
- ^ Narlikar 1993, ch. 4, sec. 3.3
- ^ Brief descriptions of these and further interesting solutions can be found in Hawking & Ellis 1973, ch. 5
- ^ Lehner 2002
- ^ For instance Wald 1984, sec. 4.4
- ^ Will 1993, sec. 4.1 and 4.2
- ^ Will 2006, sec. 3.2, Will 1993, ch. 4
- ^ Rindler 2001, pp. 24–26 vs. pp. 236–237 and Ohanian & Ruffini 1994, pp. 164–172. Einstein derived these effects using the equivalence principle as early as 1907, cf. Einstein 1907 and the description in Pais 1982, pp. 196–198
- ^ Rindler 2001, pp. 24–26; Misner, Thorne & Wheeler 1973, § 38.5
- ^ Pound–Rebka experiment, see Pound & Rebka 1959, Pound & Rebka 1960; Pound & Snider 1964; a list of further experiments is given in Ohanian & Ruffini 1994, table 4.1 on p. 186
- ^ Greenstein, Oke & Shipman 1971; the most recent and most accurate Sirius B measurements are published in Barstow, Bond et al. 2005.
- ^ Starting with the Hafele–Keating experiment, Hafele & Keating 1972a and Hafele & Keating 1972b, and culminating in the Gravity Probe A experiment; an overview of experiments can be found in Ohanian & Ruffini 1994, table 4.1 on p. 186
- ^ GPS is continually tested by comparing atomic clocks on the ground and aboard orbiting satellites; for an account of relativistic effects, see Ashby 2002 and Ashby 2003
- ^ Stairs 2003 and Kramer 2004
- ^ General overviews can be found in section 2.1. of Will 2006; Will 2003, pp. 32–36; Ohanian & Ruffini 1994, sec. 4.2
- ^ Ohanian & Ruffini 1994, pp. 164–172
- ^ Cf. Kennefick 2005 for the classic early measurements by the Eddington expeditions; for an overview of more recent measurements, see Ohanian & Ruffini 1994, ch. 4.3. For the most precise direct modern observations using quasars, cf. Shapiro et al. 2004
- ^ This is not an independent axiom; it can be derived from Einstein's equations and the Maxwell Lagrangian using a WKB approximation, cf. Ehlers 1973, sec. 5
- ^ Blanchet 2006, sec. 1.3
- ^ Rindler 2001, sec. 1.16; for the historical examples, Israel 1987, pp. 202–204; in fact, Einstein published one such derivation as Einstein 1907. Such calculations tacitly assume that the geometry of space is Euclidean, cf. Ehlers & Rindler 1997
- ^ From the standpoint of Einstein's theory, these derivations take into account the effect of gravity on time, but not its consequences for the warping of space, cf. Rindler 2001, sec. 11.11
- ^ For the Sun's gravitational field using radar signals reflected from planets such as Venus and Mercury, cf. Shapiro 1964, Weinberg 1972, ch. 8, sec. 7; for signals actively sent back by space probes (transponder measurements), cf. Bertotti, Iess & Tortora 2003; for an overview, see Ohanian & Ruffini 1994, table 4.4 on p. 200; for more recent measurements using signals received from a pulsar that is part of a binary system, the gravitational field causing the time delay being that of the other pulsar, cf. Stairs 2003, sec. 4.4
- ^ Will 1993, sec. 7.1 and 7.2
- ^ Einstein, A (June 1916). "Näherungsweise Integration der Feldgleichungen der Gravitation". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin. part 1: 688–696.
- ^ Einstein, A (1918). "Über Gravitationswellen". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin. part 1: 154–167.
- ^ ^{a} ^{b} Castelvecchi, Davide; Witze, Witze (February 11, 2016). "Einstein's gravitational waves found at last". Nature News. doi:10.1038/nature.2016.19361. Retrieved 2016-02-11.
- ^ ^{a} ^{b} B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration) (2016). "Observation of Gravitational Waves from a Binary Black Hole Merger". Physical Review Letters. 116 (6). arXiv:1602.03837. Bibcode:2016PhRvL.116f1102A. doi:10.1103/PhysRevLett.116.061102.CS1 maint: uses authors parameter (link)
- ^ ^{a} ^{b} "Gravitational waves detected 100 years after Einstein's prediction | NSF - National Science Foundation". www.nsf.gov. Retrieved 2016-02-11.
- ^ Most advanced textbooks on general relativity contain a description of these properties, e.g. Schutz 1985, ch. 9
- ^ For example Jaranowski & Królak 2005
- ^ Rindler 2001, ch. 13
- ^ Gowdy 1971, Gowdy 1974
- ^ See Lehner 2002 for a brief introduction to the methods of numerical relativity, and Seidel 1998 for the connection with gravitational wave astronomy
- ^ Schutz 2003, pp. 48–49, Pais 1982, pp. 253–254
- ^ Rindler 2001, sec. 11.9
- ^ Will 1993, pp. 177–181
- ^ In consequence, in the parameterized post-Newtonian formalism (PPN), measurements of this effect determine a linear combination of the terms β and γ, cf. Will 2006, sec. 3.5 and Will 1993, sec. 7.3
- ^ The most precise measurements are VLBI measurements of planetary positions; see Will 1993, ch. 5, Will 2006, sec. 3.5, Anderson et al. 1992; for an overview, Ohanian & Ruffini 1994, pp. 406–407
- ^ Kramer et al. 2006
- ^ A figure that includes error bars is fig. 7 in Will 2006, sec. 5.1
- ^ Stairs 2003, Schutz 2003, pp. 317–321, Bartusiak 2000, pp. 70–86
- ^ Weisberg & Taylor 2003; for the pulsar discovery, see Hulse & Taylor 1975; for the initial evidence for gravitational radiation, see Taylor 1994
- ^ Kramer 2004
- ^ Penrose 2004, §14.5, Misner, Thorne & Wheeler 1973, §11.4
- ^ Weinberg 1972, sec. 9.6, Ohanian & Ruffini 1994, sec. 7.8
- ^ Bertotti, Ciufolini & Bender 1987, Nordtvedt 2003
- ^ Kahn 2007
- ^ A mission description can be found in Everitt et al. 2001; a first post-flight evaluation is given in Everitt, Parkinson & Kahn 2007; further updates will be available on the mission website Kahn 1996–2012.
- ^ Townsend 1997, sec. 4.2.1, Ohanian & Ruffini 1994, pp. 469–471
- ^ Ohanian & Ruffini 1994, sec. 4.7, Weinberg 1972, sec. 9.7; for a more recent review, see Schäfer 2004
- ^ Ciufolini & Pavlis 2004, Ciufolini, Pavlis & Peron 2006, Iorio 2009
- ^ Iorio L. (August 2006), "COMMENTS, REPLIES AND NOTES: A note on the evidence of the gravitomagnetic field of Mars", Classical Quantum Gravity, 23 (17): 5451–5454, arXiv:gr-qc/0606092, Bibcode:2006CQGra..23.5451I, doi:10.1088/0264-9381/23/17/N01
- ^ Iorio L. (June 2010), "On the Lense–Thirring test with the Mars Global Surveyor in the gravitational field of Mars", Central European Journal of Physics, 8 (3): 509–513, arXiv:gr-qc/0701146, Bibcode:2010CEJPh...8..509I, doi:10.2478/s11534-009-0117-6
- ^ For overviews of gravitational lensing and its applications, see Ehlers, Falco & Schneider 1992 and Wambsganss 1998
- ^ For a simple derivation, see Schutz 2003, ch. 23; cf. Narayan & Bartelmann 1997, sec. 3
- ^ Walsh, Carswell & Weymann 1979
- ^ Images of all the known lenses can be found on the pages of the CASTLES project, Kochanek et al. 2007
- ^ Roulet & Mollerach 1997
- ^ Narayan & Bartelmann 1997, sec. 3.7
- ^ Barish 2005, Bartusiak 2000, Blair & McNamara 1997
- ^ Hough & Rowan 2000
- ^ Hobbs, George; Archibald, A.; Arzoumanian, Z.; Backer, D.; Bailes, M.; Bhat, N. D. R.; Burgay, M.; Burke-Spolaor, S.; et al. (2010), "The international pulsar timing array project: using pulsars as a gravitational wave detector", Classical and Quantum Gravity, 27 (8): 084013, arXiv:0911.5206, Bibcode:2010CQGra..27h4013H, doi:10.1088/0264-9381/27/8/084013
- ^ Danzmann & Rüdiger 2003
- ^ "LISA pathfinder overview". ESA. Retrieved 2012-04-23.
- ^ Thorne 1995
- ^ Cutler & Thorne 2002
- ^ "Gravitational waves detected 100 years after Einstein's prediction | NSF - National Science Foundation". www.nsf.gov. Retrieved 2016-02-11.
- ^ Miller 2002, lectures 19 and 21
- ^ Celotti, Miller & Sciama 1999, sec. 3
- ^ Springel et al. 2005 and the accompanying summary Gnedin 2005
- ^ Blandford 1987, sec. 8.2.4
- ^ For the basic mechanism, see Carroll & Ostlie 1996, sec. 17.2; for more about the different types of astronomical objects associated with this, cf. Robson 1996
- ^ For a review, see Begelman, Blandford & Rees 1984. To a distant observer, some of these jets even appear to move faster than light; this, however, can be explained as an optical illusion that does not violate the tenets of relativity, see Rees 1966
- ^ For stellar end states, cf. Oppenheimer & Snyder 1939 or, for more recent numerical work, Font 2003, sec. 4.1; for supernovae, there are still major problems to be solved, cf. Buras et al. 2003; for simulating accretion and the formation of jets, cf. Font 2003, sec. 4.2. Also, relativistic lensing effects are thought to play a role for the signals received from X-ray pulsars, cf. Kraus 1998
- ^ The evidence includes limits on compactness from the observation of accretion-driven phenomena ("Eddington luminosity"), see Celotti, Miller & Sciama 1999, observations of stellar dynamics in the center of our own Milky Way galaxy, cf. Schödel et al. 2003, and indications that at least some of the compact objects in question appear to have no solid surface, which can be deduced from the examination of X-ray bursts for which the central compact object is either a neutron star or a black hole; cf. Remillard et al. 2006 for an overview, Narayan 2006, sec. 5. Observations of the "shadow" of the Milky Way galaxy's central black hole horizon are eagerly sought for, cf. Falcke, Melia & Agol 2000
- ^ Dalal et al. 2006
- ^ Barack & Cutler 2004
- ^ Originally Einstein 1917; cf. Pais 1982, pp. 285–288
- ^ Carroll 2001, ch. 2
- ^ Bergström & Goobar 2003, ch. 9–11; use of these models is justified by the fact that, at large scales of around hundred million light-years and more, our own universe indeed appears to be isotropic and homogeneous, cf. Peebles et al. 1991
- ^ E.g. with WMAP data, see Spergel et al. 2003
- ^ These tests involve the separate observations detailed further on, see, e.g., fig. 2 in Bridle et al. 2003
- ^ Peebles 1966; for a recent account of predictions, see Coc, Vangioni‐Flam et al. 2004; an accessible account can be found in Weiss 2006; compare with the observations in Olive & Skillman 2004, Bania, Rood & Balser 2002, O'Meara et al. 2001, and Charbonnel & Primas 2005
- ^ Lahav & Suto 2004, Bertschinger 1998, Springel et al. 2005
- ^ Alpher & Herman 1948, for a pedagogical introduction, see Bergström & Goobar 2003, ch. 11; for the initial detection, see Penzias & Wilson 1965 and, for precision measurements by satellite observatories, Mather et al. 1994 (COBE) and Bennett et al. 2003 (WMAP). Future measurements could also reveal evidence about gravitational waves in the early universe; this additional information is contained in the background radiation's polarization, cf. Kamionkowski, Kosowsky & Stebbins 1997 and Seljak & Zaldarriaga 1997
- ^ Evidence for this comes from the determination of cosmological parameters and additional observations involving the dynamics of galaxies and galaxy clusters cf. Peebles 1993, ch. 18, evidence from gravitational lensing, cf. Peacock 1999, sec. 4.6, and simulations of large-scale structure formation, see Springel et al. 2005
- ^ Peacock 1999, ch. 12, Peskin 2007; in particular, observations indicate that all but a negligible portion of that matter is not in the form of the usual elementary particles ("non-baryonic matter"), cf. Peacock 1999, ch. 12
- ^ Namely, some physicists have questioned whether or not the evidence for dark matter is, in fact, evidence for deviations from the Einsteinian (and the Newtonian) description of gravity cf. the overview in Mannheim 2006, sec. 9
- ^ Carroll 2001; an accessible overview is given in Caldwell 2004. Here, too, scientists have argued that the evidence indicates not a new form of energy, but the need for modifications in our cosmological models, cf. Mannheim 2006, sec. 10; aforementioned modifications need not be modifications of general relativity, they could, for example, be modifications in the way we treat the inhomogeneities in the universe, cf. Buchert 2007
- ^ A good introduction is Linde 1990; for a more recent review, see Linde 2005
- ^ More precisely, these are the flatness problem, the horizon problem, and the monopole problem; a pedagogical introduction can be found in Narlikar 1993, sec. 6.4, see also Börner 1993, sec. 9.1
- ^ Spergel et al. 2007, sec. 5,6
- ^ More concretely, the potential function that is crucial to determining the dynamics of the inflaton is simply postulated, but not derived from an underlying physical theory
- ^ Brandenberger 2007, sec. 2
- ^ Gödel 1949
- ^ Frauendiener 2004, Wald 1984, sec. 11.1, Hawking & Ellis 1973, sec. 6.8, 6.9
- ^ Wald 1984, sec. 9.2–9.4 and Hawking & Ellis 1973, ch. 6
- ^ Thorne 1972; for more recent numerical studies, see Berger 2002, sec. 2.1
- ^ Israel 1987. A more exact mathematical description distinguishes several kinds of horizon, notably event horizons and apparent horizons cf. Hawking & Ellis 1973, pp. 312–320 or Wald 1984, sec. 12.2; there are also more intuitive definitions for isolated systems that do not require knowledge of spacetime properties at infinity, cf. Ashtekar & Krishnan 2004
- ^ For first steps, cf. Israel 1971; see Hawking & Ellis 1973, sec. 9.3 or Heusler 1996, ch. 9 and 10 for a derivation, and Heusler 1998 as well as Beig & Chruściel 2006 as overviews of more recent results
- ^ The laws of black hole mechanics were first described in Bardeen, Carter & Hawking 1973; a more pedagogical presentation can be found in Carter 1979; for a more recent review, see Wald 2001, ch. 2. A thorough, book-length introduction including an introduction to the necessary mathematics Poisson 2004. For the Penrose process, see Penrose 1969
- ^ Bekenstein 1973, Bekenstein 1974
- ^ The fact that black holes radiate, quantum mechanically, was first derived in Hawking 1975; a more thorough derivation can be found in Wald 1975. A review is given in Wald 2001, ch. 3
- ^ Narlikar 1993, sec. 4.4.4, 4.4.5
- ^ Horizons: cf. Rindler 2001, sec. 12.4. Unruh effect: Unruh 1976, cf. Wald 2001, ch. 3
- ^ Hawking & Ellis 1973, sec. 8.1, Wald 1984, sec. 9.1
- ^ Townsend 1997, ch. 2; a more extensive treatment of this solution can be found in Chandrasekhar 1983, ch. 3
- ^ Townsend 1997, ch. 4; for a more extensive treatment, cf. Chandrasekhar 1983, ch. 6
- ^ Ellis & Van Elst 1999; a closer look at the singularity itself is taken in Börner 1993, sec. 1.2
- ^ Here one should remind to the well-known fact that the important "quasi-optical" singularities of the so-called eikonal approximations of many wave-equations, namely the "caustics", are resolved into finite peaks beyond that approximation.
- ^ Namely when there are trapped null surfaces, cf. Penrose 1965
- ^ Hawking 1966
- ^ The conjecture was made in Belinskii, Khalatnikov & Lifschitz 1971; for a more recent review, see Berger 2002. An accessible exposition is given by Garfinkle 2007
- ^ The restriction to future singularities naturally excludes initial singularities such as the big bang singularity, which in principle be visible to observers at later cosmic time. The cosmic censorship conjecture was first presented in Penrose 1969; a textbook-level account is given in Wald 1984, pp. 302–305. For numerical results, see the review Berger 2002, sec. 2.1
- ^ Hawking & Ellis 1973, sec. 7.1
- ^ Arnowitt, Deser & Misner 1962; for a pedagogical introduction, see Misner, Thorne & Wheeler 1973, §21.4–§21.7
- ^ Fourès-Bruhat 1952 and Bruhat 1962; for a pedagogical introduction, see Wald 1984, ch. 10; an online review can be found in Reula 1998
- ^ Gourgoulhon 2007; for a review of the basics of numerical relativity, including the problems arising from the peculiarities of Einstein's equations, see Lehner 2001
- ^ Misner, Thorne & Wheeler 1973, §20.4
- ^ Arnowitt, Deser & Misner 1962
- ^ Komar 1959; for a pedagogical introduction, see Wald 1984, sec. 11.2; although defined in a totally different way, it can be shown to be equivalent to the ADM mass for stationary spacetimes, cf. Ashtekar & Magnon-Ashtekar 1979
- ^ For a pedagogical introduction, see Wald 1984, sec. 11.2
- ^ Wald 1984, p. 295 and refs therein; this is important for questions of stability—if there were negative mass states, then flat, empty Minkowski space, which has mass zero, could evolve into these states
- ^ Townsend 1997, ch. 5
- ^ Such quasi-local mass–energy definitions are the Hawking energy, Geroch energy, or Penrose's quasi-local energy–momentum based on twistor methods; cf. the review article Szabados 2004
- ^ An overview of quantum theory can be found in standard textbooks such as Messiah 1999; a more elementary account is given in Hey & Walters 2003
- ^ Ramond 1990, Weinberg 1995, Peskin & Schroeder 1995; a more accessible overview is Auyang 1995
- ^ Wald 1994, Birrell & Davies 1984
- ^ For Hawking radiation Hawking 1975, Wald 1975; an accessible introduction to black hole evaporation can be found in Traschen 2000
- ^ Wald 2001, ch. 3
- ^ Put simply, matter is the source of spacetime curvature, and once matter has quantum properties, we can expect spacetime to have them as well. Cf. Carlip 2001, sec. 2
- ^ Schutz 2003, p. 407
- ^ ^{a} ^{b} Hamber 2009
- ^ A timeline and overview can be found in Rovelli 2000
- ^ t'Hooft 1974
- ^ Donoghue 1995
- ^ In particular, a perturbative technique known as renormalization, an integral part of deriving predictions which take into account higher-energy contributions, cf. Weinberg 1996, ch. 17, 18, fails in this case; cf. Veltman 1975, Goroff & Sagnotti 1985; for a recent comprehensive review of the failure of perturbative renormalizability for quantum gravity see Hamber 2009
- ^ An accessible introduction at the undergraduate level can be found in Zwiebach 2004; more complete overviews can be found in Polchinski 1998a and Polchinski 1998b
- ^ At the energies reached in current experiments, these strings are indistinguishable from point-like particles, but, crucially, different modes of oscillation of one and the same type of fundamental string appear as particles with different (electric and other) charges, e.g. Ibanez 2000. The theory is successful in that one mode will always correspond to a graviton, the messenger particle of gravity, e.g. Green, Schwarz & Witten 1987, sec. 2.3, 5.3
- ^ Green, Schwarz & Witten 1987, sec. 4.2
- ^ Weinberg 2000, ch. 31
- ^ Townsend 1996, Duff 1996
- ^ Kuchař 1973, sec. 3
- ^ These variables represent geometric gravity using mathematical analogues of electric and magnetic fields; cf. Ashtekar 1986, Ashtekar 1987
- ^ For a review, see Thiemann 2006; more extensive accounts can be found in Rovelli 1998, Ashtekar & Lewandowski 2004 as well as in the lecture notes Thiemann 2003
- ^ Isham 1994, Sorkin 1997
- ^ Loll 1998
- ^ Sorkin 2005
- ^ Penrose 2004, ch. 33 and refs therein
- ^ Hawking 1987
- ^ Ashtekar 2007, Schwarz 2007
- ^ Maddox 1998, pp. 52–59, 98–122; Penrose 2004, sec. 34.1, ch. 30
- ^ section Quantum gravity, above
- ^ section Cosmology, above
- ^ Friedrich 2005
- ^ A review of the various problems and the techniques being developed to overcome them, see Lehner 2002
- ^ See Bartusiak 2000 for an account up to that year; up-to-date news can be found on the websites of major detector collaborations such as GEO 600 and LIGO
- ^ For the most recent papers on gravitational wave polarizations of inspiralling compact binaries, see Blanchet et al. 2008, and Arun et al. 2007; for a review of work on compact binaries, see Blanchet 2006 and Futamase & Itoh 2006; for a general review of experimental tests of general relativity, see Will 2006
- ^ See, e.g., the electronic review journal Living Reviews in Relativity
ReferencesEdit
- Alpher, R. A.; Herman, R. C. (1948), "Evolution of the universe", Nature, 162 (4124): 774–775, Bibcode:1948Natur.162..774A, doi:10.1038/162774b0
- Anderson, J. D.; Campbell, J. K.; Jurgens, R. F.; Lau, E. L. (1992), "Recent developments in solar-system tests of general relativity", in Sato, H.; Nakamura, T. (eds.), Proceedings of the Sixth Marcel Großmann Meeting on General Relativity, World Scientific, pp. 353–355, ISBN 981-02-0950-9
- Arnold, V. I. (1989), Mathematical Methods of Classical Mechanics, Springer, ISBN 3-540-96890-3
- Arnowitt, Richard; Deser, Stanley; Misner, Charles W. (1962), "The dynamics of general relativity", in Witten, Louis (ed.), Gravitation: An Introduction to Current Research, Wiley, pp. 227–265
- Arun, K.G.; Blanchet, L.; Iyer, B. R.; Qusailah, M. S. S. (2007), "Inspiralling compact binaries in quasi-elliptical orbits: The complete 3PN energy flux", Physical Review D, 77 (6), arXiv:0711.0302, Bibcode:2008PhRvD..77f4035A, doi:10.1103/PhysRevD.77.064035
- Ashby, Neil (2002), "Relativity and the Global Positioning System" (PDF), Physics Today, 55 (5): 41–47, Bibcode:2002PhT....55e..41A, doi:10.1063/1.1485583
- Ashby, Neil (2003), "Relativity in the Global Positioning System", Living Reviews in Relativity, 6, Bibcode:2003LRR.....6....1A, doi:10.12942/lrr-2003-1, retrieved 2007-07-06
- Ashtekar, Abhay (1986), "New variables for classical and quantum gravity", Phys. Rev. Lett., 57 (18): 2244–2247, Bibcode:1986PhRvL..57.2244A, doi:10.1103/PhysRevLett.57.2244, PMID 10033673
- Ashtekar, Abhay (1987), "New Hamiltonian formulation of general relativity", Phys. Rev., D36 (6): 1587–1602, Bibcode:1987PhRvD..36.1587A, doi:10.1103/PhysRevD.36.1587
- Ashtekar, Abhay (2007), "LOOP QUANTUM GRAVITY: FOUR RECENT ADVANCES AND A DOZEN FREQUENTLY ASKED QUESTIONS", The Eleventh Marcel Grossmann Meeting - on Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories - Proceedings of the MG11 Meeting on General Relativity, p. 126, arXiv:0705.2222, Bibcode:2008mgm..conf..126A, doi:10.1142/9789812834300_0008, ISBN 9789812834263
- Ashtekar, Abhay; Krishnan, Badri (2004), "Isolated and Dynamical Horizons and Their Applications", Living Reviews in Relativity, 7, arXiv:gr-qc/0407042, Bibcode:2004LRR.....7...10A, doi:10.12942/lrr-2004-10, retrieved 2007-08-28
- Ashtekar, Abhay; Lewandowski, Jerzy (2004), "Background Independent Quantum Gravity: A Status Report", Class. Quant. Grav., 21 (15): R53–R152, arXiv:gr-qc/0404018, Bibcode:2004CQGra..21R..53A, doi:10.1088/0264-9381/21/15/R01
- Ashtekar, Abhay; Magnon-Ashtekar, Anne (1979), "On conserved quantities in general relativity", Journal of Mathematical Physics, 20 (5): 793–800, Bibcode:1979JMP....20..793A, doi:10.1063/1.524151
- Auyang, Sunny Y. (1995), How is Quantum Field Theory Possible?, Oxford University Press, ISBN 0-19-509345-3
- Bania, T. M.; Rood, R. T.; Balser, D. S. (2002), "The cosmological density of baryons from observations of 3He+ in the Milky Way", Nature, 415 (6867): 54–57, Bibcode:2002Natur.415...54B, doi:10.1038/415054a, PMID 11780112
- Barack, Leor; Cutler, Curt (2004), "LISA Capture Sources: Approximate Waveforms, Signal-to-Noise Ratios, and Parameter Estimation Accuracy", Phys. Rev., D69 (8): 082005, arXiv:gr-qc/0310125, Bibcode:2004PhRvD..69h2005B, doi:10.1103/PhysRevD.69.082005
- Bardeen, J. M.; Carter, B.; Hawking, S. W. (1973), "The Four Laws of Black Hole Mechanics", Comm. Math. Phys., 31 (2): 161–170, Bibcode:1973CMaPh..31..161B, doi:10.1007/BF01645742
- Barish, Barry (2005), "Towards detection of gravitational waves", in Florides, P.; Nolan, B.; Ottewil, A. (eds.), General Relativity and Gravitation. Proceedings of the 17th International Conference, World Scientific, pp. 24–34, ISBN 981-256-424-1
- Barstow, M; Bond, Howard E.; Holberg, J. B.; Burleigh, M. R.; Hubeny, I.; Koester, D. (2005), "Hubble Space Telescope Spectroscopy of the Balmer lines in Sirius B", Mon. Not. Roy. Astron. Soc., 362 (4): 1134–1142, arXiv:astro-ph/0506600, Bibcode:2005MNRAS.362.1134B, doi:10.1111/j.1365-2966.2005.09359.x
- Bartusiak, Marcia (2000), Einstein's Unfinished Symphony: Listening to the Sounds of Space-Time, Berkley, ISBN 978-0-425-18620-6
- Begelman, Mitchell C.; Blandford, Roger D.; Rees, Martin J. (1984), "Theory of extragalactic radio sources", Rev. Mod. Phys., 56 (2): 255–351, Bibcode:1984RvMP...56..255B, doi:10.1103/RevModPhys.56.255
- Beig, Robert; Chruściel, Piotr T. (2006), "Stationary black holes", in Françoise, J.-P.; Naber, G.; Tsou, T.S. (eds.), Encyclopedia of Mathematical Physics, Volume 2, Elsevier, p. 2041, arXiv:gr-qc/0502041, Bibcode:2005gr.qc.....2041B, ISBN 0-12-512660-3
- Bekenstein, Jacob D. (1973), "Black Holes and Entropy", Phys. Rev., D7 (8): 2333–2346, Bibcode:1973PhRvD...7.2333B, doi:10.1103/PhysRevD.7.2333
- Bekenstein, Jacob D. (1974), "Generalized Second Law of Thermodynamics in Black-Hole Physics", Phys. Rev., D9 (12): 3292–3300, Bibcode:1974PhRvD...9.3292B, doi:10.1103/PhysRevD.9.3292
- Belinskii, V. A.; Khalatnikov, I. M.; Lifschitz, E. M. (1971), "Oscillatory approach to the singular point in relativistic cosmology", Advances in Physics, 19 (80): 525–573, Bibcode:1970AdPhy..19..525B, doi:10.1080/00018737000101171; original paper in Russian: Belinsky, V. A.; Lifshits, I. M.; Khalatnikov, E. M. (1970), "Колебательный Режим Приближения К Особой Точке В Релятивистской Космологии", Uspekhi Fizicheskikh Nauk (Успехи Физических Наук), 102(3) (11): 463–500, Bibcode:1970UsFiN.102..463B
- Bennett, C. L.; Halpern, M.; Hinshaw, G.; Jarosik, N.; Kogut, A.; Limon, M.; Meyer, S. S.; Page, L.; et al. (2003), "First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results", Astrophys. J. Suppl., 148 (1): 1–27, arXiv:astro-ph/0302207, Bibcode:2003ApJS..148....1B, doi:10.1086/377253
- Berger, Beverly K. (2002), "Numerical Approaches to Spacetime Singularities", Living Reviews in Relativity, 5, arXiv:gr-qc/0201056, Bibcode:2002LRR.....5....1B, doi:10.12942/lrr-2002-1, retrieved 2007-08-04
- Bergström, Lars; Goobar, Ariel (2003), Cosmology and Particle Astrophysics (2nd ed.), Wiley & Sons, ISBN 3-540-43128-4
- Bertotti, Bruno; Ciufolini, Ignazio; Bender, Peter L. (1987), "New test of general relativity: Measurement of de Sitter geodetic precession rate for lunar perigee", Physical Review Letters, 58 (11): 1062–1065, Bibcode:1987PhRvL..58.1062B, doi:10.1103/PhysRevLett.58.1062, PMID 10034329
- Bertotti, Bruno; Iess, L.; Tortora, P. (2003), "A test of general relativity using radio links with the Cassini spacecraft", Nature, 425 (6956): 374–376, Bibcode:2003Natur.425..374B, doi:10.1038/nature01997, PMID 14508481
- Bertschinger, Edmund (1998), "Simulations of structure formation in the universe", Annu. Rev. Astron. Astrophys., 36 (1): 599–654, Bibcode:1998ARA&A..36..599B, doi:10.1146/annurev.astro.36.1.599
- Birrell, N. D.; Davies, P. C. (1984), Quantum Fields in Curved Space, Cambridge University Press, ISBN 0-521-27858-9
- Blair, David; McNamara, Geoff (1997), Ripples on a Cosmic Sea. The Search for Gravitational Waves, Perseus, ISBN 0-7382-0137-5
- Blanchet, L.; Faye, G.; Iyer, B. R.; Sinha, S. (2008), "The third post-Newtonian gravitational wave polarisations and associated spherical harmonic modes for inspiralling compact binaries in quasi-circular orbits", Classical and Quantum Gravity, 25 (16): 165003, arXiv:0802.1249, Bibcode:2008CQGra..25p5003B, doi:10.1088/0264-9381/25/16/165003
- Blanchet, Luc (2006), "Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries", Living Reviews in Relativity, 9, Bibcode:2006LRR.....9....4B, doi:10.12942/lrr-2006-4, retrieved 2007-08-07
- Blandford, R. D. (1987), "Astrophysical Black Holes", in Hawking, Stephen W.; Israel, Werner (eds.), 300 Years of Gravitation, Cambridge University Press, pp. 277–329, ISBN 0-521-37976-8
- Börner, Gerhard (1993), The Early Universe. Facts and Fiction, Springer, ISBN 0-387-56729-1
- Brandenberger, Robert H. (2007), "Conceptual Problems of Inflationary Cosmology and a New Approach to Cosmological Structure Formation", Inflationary Cosmology, Lecture Notes in Physics, 738, p. 393, arXiv:hep-th/0701111, Bibcode:2008LNP...738..393B, doi:10.1007/978-3-540-74353-8_11, ISBN 978-3-540-74352-1
- Brans, C. H.; Dicke, R. H. (1961), "Mach's Principle and a Relativistic Theory of Gravitation", Physical Review, 124 (3): 925–935, Bibcode:1961PhRv..124..925B, doi:10.1103/PhysRev.124.925
- Bridle, Sarah L.; Lahav, Ofer; Ostriker, Jeremiah P.; Steinhardt, Paul J. (2003), "Precision Cosmology? Not Just Yet", Science, 299 (5612): 1532–1533, arXiv:astro-ph/0303180, Bibcode:2003Sci...299.1532B, doi:10.1126/science.1082158, PMID 12624255
- Bruhat, Yvonne (1962), "The Cauchy Problem", in Witten, Louis (ed.), Gravitation: An Introduction to Current Research, Wiley, p. 130, ISBN 978-1-114-29166-9
- Buchert, Thomas (2007), "Dark Energy from Structure—A Status Report", General Relativity and Gravitation, 40 (2–3): 467–527, arXiv:0707.2153, Bibcode:2008GReGr..40..467B, doi:10.1007/s10714-007-0554-8
- Buras, R.; Rampp, M.; Janka, H.-Th.; Kifonidis, K. (2003), "Improved Models of Stellar Core Collapse and Still no Explosions: What is Missing?", Phys. Rev. Lett., 90 (24): 241101, arXiv:astro-ph/0303171, Bibcode:2003PhRvL..90x1101B, doi:10.1103/PhysRevLett.90.241101, PMID 12857181
- Caldwell, Robert R. (2004), "Dark Energy", Physics World, 17 (5): 37–42
- Carlip, Steven (2001), "Quantum Gravity: a Progress Report", Rept. Prog. Phys., 64 (8): 885–942, arXiv:gr-qc/0108040, Bibcode:2001RPPh...64..885C, doi:10.1088/0034-4885/64/8/301
- Carroll, Bradley W.; Ostlie, Dale A. (1996), An Introduction to Modern Astrophysics, Addison-Wesley, ISBN 0-201-54730-9
- Carroll, Sean M. (2001), "The Cosmological Constant", Living Reviews in Relativity, 4, arXiv:astro-ph/0004075, Bibcode:2001LRR.....4....1C, doi:10.12942/lrr-2001-1, retrieved 2007-07-21
- Carter, Brandon (1979), "The general theory of the mechanical, electromagnetic and thermodynamic properties of black holes", in Hawking, S. W.; Israel, W. (eds.), General Relativity, an Einstein Centenary Survey, Cambridge University Press, pp. 294–369 and 860–863, ISBN 0-521-29928-4
- Celotti, Annalisa; Miller, John C.; Sciama, Dennis W. (1999), "Astrophysical evidence for the existence of black holes", Class. Quant. Grav., 16 (12A): A3–A21, arXiv:astro-ph/9912186, doi:10.1088/0264-9381/16/12A/301
- Chandrasekhar, Subrahmanyan (1983), The Mathematical Theory of Black Holes, Oxford University Press, ISBN 0-19-850370-9
- Charbonnel, C.; Primas, F. (2005), "The Lithium Content of the Galactic Halo Stars", Astronomy & Astrophysics, 442 (3): 961–992, arXiv:astro-ph/0505247, Bibcode:2005A&A...442..961C, doi:10.1051/0004-6361:20042491
- Ciufolini, Ignazio; Pavlis, Erricos C. (2004), "A confirmation of the general relativistic prediction of the Lense-Thirring effect", Nature, 431 (7011): 958–960, Bibcode:2004Natur.431..958C, doi:10.1038/nature03007, PMID 15496915
- Ciufolini, Ignazio; Pavlis, Erricos C.; Peron, R. (2006), "Determination of frame-dragging using Earth gravity models from CHAMP and GRACE", New Astron., 11 (8): 527–550, Bibcode:2006NewA...11..527C, doi:10.1016/j.newast.2006.02.001
- Coc, A.; Vangioni‐Flam, Elisabeth; Descouvemont, Pierre; Adahchour, Abderrahim; Angulo, Carmen (2004), "Updated Big Bang Nucleosynthesis confronted to WMAP observations and to the Abundance of Light Elements", Astrophysical Journal, 600 (2): 544–552, arXiv:astro-ph/0309480, Bibcode:2004ApJ...600..544C, doi:10.1086/380121
- Cutler, Curt; Thorne, Kip S. (2002), "An overview of gravitational wave sources", in Bishop, Nigel; Maharaj, Sunil D. (eds.), Proceedings of 16th International Conference on General Relativity and Gravitation (GR16), World Scientific, p. 4090, arXiv:gr-qc/0204090, Bibcode:2002gr.qc.....4090C, ISBN 981-238-171-6
- Dalal, Neal; Holz, Daniel E.; Hughes, Scott A.; Jain, Bhuvnesh (2006), "Short GRB and binary black hole standard sirens as a probe of dark energy", Phys.Rev., D74 (6): 063006, arXiv:astro-ph/0601275, Bibcode:2006PhRvD..74f3006D, doi:10.1103/PhysRevD.74.063006
- Danzmann, Karsten; Rüdiger, Albrecht (2003), "LISA Technology—Concepts, Status, Prospects" (PDF), Class. Quant. Grav., 20 (10): S1–S9, Bibcode:2003CQGra..20S...1D, doi:10.1088/0264-9381/20/10/301
- Dirac, Paul (1996), General Theory of Relativity, Princeton University Press, ISBN 0-691-01146-X
- Donoghue, John F. (1995), "Introduction to the Effective Field Theory Description of Gravity", in Cornet, Fernando (ed.), Effective Theories: Proceedings of the Advanced School, Almunecar, Spain, 26 June–1 July 1995, Singapore: World Scientific, p. 12024, arXiv:gr-qc/9512024, Bibcode:1995gr.qc....12024D, ISBN 981-02-2908-9
- Duff, Michael (1996), "M-Theory (the Theory Formerly Known as Strings)", Int. J. Mod. Phys., A11 (32): 5623–5641, arXiv:hep-th/9608117, Bibcode:1996IJMPA..11.5623D, doi:10.1142/S0217751X96002583
- Ehlers, Jürgen (1973), "Survey of general relativity theory", in Israel, Werner (ed.), Relativity, Astrophysics and Cosmology, D. Reidel, pp. 1–125, ISBN 90-277-0369-8
- Ehlers, Jürgen; Falco, Emilio E.; Schneider, Peter (1992), Gravitational lenses, Springer, ISBN 3-540-66506-4
- Ehlers, Jürgen; Lämmerzahl, Claus, eds. (2006), Special Relativity—Will it Survive the Next 101 Years?, Springer, ISBN 3-540-34522-1
- Ehlers, Jürgen; Rindler, Wolfgang (1997), "Local and Global Light Bending in Einstein's and other Gravitational Theories", General Relativity and Gravitation, 29 (4): 519–529, Bibcode:1997GReGr..29..519E, doi:10.1023/A:1018843001842
- Einstein, Albert (1907), "Über das Relativitätsprinzip und die aus demselben gezogene Folgerungen" (PDF), Jahrbuch der Radioaktivität und Elektronik, 4: 411, retrieved 2008-05-05
- Einstein, Albert (1915), "Die Feldgleichungen der Gravitation", Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin: 844–847, retrieved 2006-09-12
- Einstein, Albert (1916), "Die Grundlage der allgemeinen Relativitätstheorie", Annalen der Physik, 49: 769–822, Bibcode:1916AnP...354..769E, doi:10.1002/andp.19163540702, archived from the original (PDF) on 2006-08-29, retrieved 2016-02-14
- Einstein, Albert (1917), "Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie", Sitzungsberichte der Preußischen Akademie der Wissenschaften: 142
- Ellis, George F R; Van Elst, Henk (1999), Lachièze-Rey, Marc (ed.), "Theoretical and Observational Cosmology: Cosmological models (Cargèse lectures 1998)", Theoretical and observational cosmology : proceedings of the NATO Advanced Study Institute on Theoretical and Observational Cosmology, Kluwer: 1–116, arXiv:gr-qc/9812046, Bibcode:1999ASIC..541....1E, doi:10.1007/978-94-011-4455-1_1, ISBN 978-0-7923-5946-3
- Everitt, C. W. F.; Buchman, S.; DeBra, D. B.; Keiser, G. M. (2001), "Gravity Probe B: Countdown to launch", in Lämmerzahl, C.; Everitt, C. W. F.; Hehl, F. W. (eds.), Gyros, Clocks, and Interferometers: Testing Relativistic Gravity in Space (Lecture Notes in Physics 562), Springer, pp. 52–82, ISBN 3-540-41236-0
- Everitt, C. W. F.; Parkinson, Bradford; Kahn, Bob (2007), The Gravity Probe B experiment. Post Flight Analysis—Final Report (Preface and Executive Summary) (PDF), Project Report: NASA, Stanford University and Lockheed Martin, retrieved 2007-08-05
- Falcke, Heino; Melia, Fulvio; Agol, Eric (2000), "Viewing the Shadow of the Black Hole at the Galactic Center", Astrophysical Journal, 528 (1): L13–L16, arXiv:astro-ph/9912263, Bibcode:2000ApJ...528L..13F, doi:10.1086/312423, PMID 10587484
- Flanagan, Éanna É.; Hughes, Scott A. (2005), "The basics of gravitational wave theory", New J.Phys., 7: 204, arXiv:gr-qc/0501041, Bibcode:2005NJPh....7..204F, doi:10.1088/1367-2630/7/1/204
- Font, José A. (2003), "Numerical Hydrodynamics in General Relativity", Living Reviews in Relativity, 6, Bibcode:2003LRR.....6....4F, doi:10.12942/lrr-2003-4, retrieved 2007-08-19
- Fourès-Bruhat, Yvonne (1952), "Théoréme d'existence pour certains systémes d'équations aux derivées partielles non linéaires", Acta Mathematica, 88 (1): 141–225, Bibcode:1952AcM....88..141F, doi:10.1007/BF02392131
- Frauendiener, Jörg (2004), "Conformal Infinity", Living Reviews in Relativity, 7, Bibcode:2004LRR.....7....1F, doi:10.12942/lrr-2004-1, retrieved 2007-07-21
- Friedrich, Helmut (2005), "Is general relativity 'essentially understood'?", Annalen der Physik, 15 (1–2): 84–108, arXiv:gr-qc/0508016, Bibcode:2006AnP...518...84F, doi:10.1002/andp.200510173
- Futamase, T.; Itoh, Y. (2006), "The Post-Newtonian Approximation for Relativistic Compact Binaries", Living Reviews in Relativity, 10, Bibcode:2007LRR....10....2F, doi:10.12942/lrr-2007-2, retrieved 2008-02-29
- Gamow, George (1970), My World Line, Viking Press, ISBN 0-670-50376-2
- Garfinkle, David (2007), "Of singularities and breadmaking", Einstein Online, retrieved 2007-08-03
- Geroch, Robert (1996). "Partial Differential Equations of Physics". arXiv:gr-qc/9602055
|class=
ignored (help). - Giulini, Domenico (2005), Special Relativity: A First Encounter, Oxford University Press, ISBN 0-19-856746-4
- Giulini, Domenico (2006a), "Algebraic and Geometric Structures in Special Relativity", in Ehlers, Jürgen; Lämmerzahl, Claus (eds.), Special Relativity—Will it Survive the Next 101 Years?, Springer, pp. 45–111, arXiv:math-ph/0602018, Bibcode:2006math.ph...2018G, ISBN 3-540-34522-1
- Giulini, Domenico (2006b), Stamatescu, I. O. (ed.), "An assessment of current paradigms in the physics of fundamental interactions: Some remarks on the notions of general covariance and background independence", Approaches to Fundamental Physics, Lecture Notes in Physics, Springer, 721: 105, arXiv:gr-qc/0603087, Bibcode:2007LNP...721..105G, doi:10.1007/978-3-540-71117-9_6, ISBN 978-3-540-71115-5
- Gnedin, Nickolay Y. (2005), "Digitizing the Universe", Nature, 435 (7042): 572–573, Bibcode:2005Natur.435..572G, doi:10.1038/435572a, PMID 15931201
- Goenner, Hubert F. M. (2004), "On the History of Unified Field Theories", Living Reviews in Relativity, 7, Bibcode:2004LRR.....7....2G, doi:10.12942/lrr-2004-2, retrieved 2008-02-28
- Goroff, Marc H.; Sagnotti, Augusto (1985), "Quantum gravity at two loops", Phys. Lett., 160B (1–3): 81–86, Bibcode:1985PhLB..160...81G, doi:10.1016/0370-2693(85)91470-4
- Gourgoulhon, Eric (2007). "3+1 Formalism and Bases of Numerical Relativity". arXiv:gr-qc/0703035
|class=
ignored (help). - Gowdy, Robert H. (1971), "Gravitational Waves in Closed Universes", Phys. Rev. Lett., 27 (12): 826–829, Bibcode:1971PhRvL..27..826G, doi:10.1103/PhysRevLett.27.826
- Gowdy, Robert H. (1974), "Vacuum spacetimes with two-parameter spacelike isometry groups and compact invariant hypersurfaces: Topologies and boundary conditions", Annals of Physics, 83 (1): 203–241, Bibcode:1974AnPhy..83..203G, doi:10.1016/0003-4916(74)90384-4
- Green, M. B.; Schwarz, J. H.; Witten, E. (1987), Superstring theory. Volume 1: Introduction, Cambridge University Press, ISBN 0-521-35752-7
- Greenstein, J. L.; Oke, J. B.; Shipman, H. L. (1971), "Effective Temperature, Radius, and Gravitational Redshift of Sirius B", Astrophysical Journal, 169: 563, Bibcode:1971ApJ...169..563G, doi:10.1086/151174
- Hamber, Herbert W. (2009), Quantum Gravitation - The Feynman Path Integral Approach, Springer Publishing, doi:10.1007/978-3-540-85293-3, ISBN 978-3-540-85292-6
- Gödel, Kurt (1949). "An Example of a New Type of Cosmological Solution of Einstein's Field Equations of Gravitation". Rev. Mod. Phys. 21 (3): 447. Bibcode:1949RvMP...21..447G. doi:10.1103/RevModPhys.21.447.
- Hafele, J. C.; Keating, R. E. (July 14, 1972). "Around-the-World Atomic Clocks: Predicted Relativistic Time Gains". Science. 177 (4044): 166–168. Bibcode:1972Sci...177..166H. doi:10.1126/science.177.4044.166. PMID 17779917.
- Hafele, J. C.; Keating, R. E. (July 14, 1972). "Around-the-World Atomic Clocks: Observed Relativistic Time Gains". Science. 177 (4044): 168–170. Bibcode:1972Sci...177..168H. doi:10.1126/science.177.4044.168. PMID 17779918.
- Havas, P. (1964), "Four-Dimensional Formulation of Newtonian Mechanics and Their Relation to the Special and the General Theory of Relativity", Rev. Mod. Phys., 36 (4): 938–965, Bibcode:1964RvMP...36..938H, doi:10.1103/RevModPhys.36.938
- Hawking, Stephen W. (1966), "The occurrence of singularities in cosmology", Proceedings of the Royal Society, A294 (1439): 511–521, Bibcode:1966RSPSA.294..511H, doi:10.1098/rspa.1966.0221
- Hawking, S. W. (1975), "Particle Creation by Black Holes", Communications in Mathematical Physics, 43 (3): 199–220, Bibcode:1975CMaPh..43..199H, doi:10.1007/BF02345020
- Hawking, Stephen W. (1987), "Quantum cosmology", in Hawking, Stephen W.; Israel, Werner (eds.), 300 Years of Gravitation, Cambridge University Press, pp. 631–651, ISBN 0-521-37976-8
- Hawking, Stephen W.; Ellis, George F. R. (1973), The large scale structure of space-time, Cambridge University Press, ISBN 0-521-09906-4
- Heckmann, O. H. L.; Schücking, E. (1959), "Newtonsche und Einsteinsche Kosmologie", in Flügge, S. (ed.), Encyclopedia of Physics, 53, p. 489
- Heusler, Markus (1998), "Stationary Black Holes: Uniqueness and Beyond", Living Reviews in Relativity, 1, Bibcode:1998LRR.....1....6H, doi:10.12942/lrr-1998-6, retrieved 2007-08-04
- Heusler, Markus (1996), Black Hole Uniqueness Theorems, Cambridge University Press, ISBN 0-521-56735-1
- Hey, Tony; Walters, Patrick (2003), The new quantum universe, Cambridge University Press, ISBN 0-521-56457-3
- Hough, Jim; Rowan, Sheila (2000), "Gravitational Wave Detection by Interferometry (Ground and Space)", Living Reviews in Relativity, 3, retrieved 2007-07-21
- Hubble, Edwin (1929), "A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae" (PDF), Proc. Nat. Acad. Sci., 15 (3): 168–173, Bibcode:1929PNAS...15..168H, doi:10.1073/pnas.15.3.168, PMC 522427, PMID 16577160
- Hulse, Russell A.; Taylor, Joseph H. (1975), "Discovery of a pulsar in a binary system", Astrophys. J., 195: L51–L55, Bibcode:1975ApJ...195L..51H, doi:10.1086/181708
- Ibanez, L. E. (2000), "The second string (phenomenology) revolution", Class. Quant. Grav., 17 (5): 1117–1128, arXiv:hep-ph/9911499, Bibcode:2000CQGra..17.1117I, doi:10.1088/0264-9381/17/5/321
- Iorio, L. (2009), "An Assessment of the Systematic Uncertainty in Present and Future Tests of the Lense-Thirring Effect with Satellite Laser Ranging", Space Sci. Rev., 148 (1–4): 363, arXiv:0809.1373, Bibcode:2009SSRv..148..363I, doi:10.1007/s11214-008-9478-1
- Isham, Christopher J. (1994), "Prima facie questions in quantum gravity", in Ehlers, Jürgen; Friedrich, Helmut (eds.), Canonical Gravity: From Classical to Quantum, Springer, ISBN 3-540-58339-4
- Israel, Werner (1971), "Event Horizons and Gravitational Collapse", General Relativity and Gravitation, 2 (1): 53–59, Bibcode:1971GReGr...2...53I, doi:10.1007/BF02450518
- Israel, Werner (1987), "Dark stars: the evolution of an idea", in Hawking, Stephen W.; Israel, Werner (eds.), 300 Years of Gravitation, Cambridge University Press, pp. 199–276, ISBN 0-521-37976-8
- Janssen, Michel (2005), "Of pots and holes: Einstein's bumpy road to general relativity" (PDF), Annalen der Physik, 14 (S1): 58–85, Bibcode:2005AnP...517S..58J, doi:10.1002/andp.200410130
- Jaranowski, Piotr; Królak, Andrzej (2005), "Gravitational-Wave Data Analysis. Formalism and Sample Applications: The Gaussian Case", Living Reviews in Relativity, 8, Bibcode:2005LRR.....8....3J, doi:10.12942/lrr-2005-3, retrieved 2007-07-30
- Kahn, Bob (1996–2012), Gravity Probe B Website, Stanford University, retrieved 2012-04-20
- Kahn, Bob (April 14, 2007), Was Einstein right? Scientists provide first public peek at Gravity Probe B results (Stanford University Press Release) (PDF), Stanford University News Service
- Kamionkowski, Marc; Kosowsky, Arthur; Stebbins, Albert (1997), "Statistics of Cosmic Microwave Background Polarization", Phys. Rev., D55 (12): 7368–7388, arXiv:astro-ph/9611125, Bibcode:1997PhRvD..55.7368K, doi:10.1103/PhysRevD.55.7368
- Kennefick, Daniel (2005), "Astronomers Test General Relativity: Light-bending and the Solar Redshift", in Renn, Jürgen (ed.), One hundred authors for Einstein, Wiley-VCH, pp. 178–181, ISBN 3-527-40574-7
- Kennefick, Daniel (2007), "Not Only Because of Theory: Dyson, Eddington and the Competing Myths of the 1919 Eclipse Expedition", Proceedings of the 7th Conference on the History of General Relativity, Tenerife, 2005, 0709, p. 685, arXiv:0709.0685, Bibcode:2007arXiv0709.0685K
- Kenyon, I. R. (1990), General Relativity, Oxford University Press, ISBN 0-19-851996-6
- Kochanek, C.S.; Falco, E.E.; Impey, C.; Lehar, J. (2007), CASTLES Survey Website, Harvard-Smithsonian Center for Astrophysics, retrieved 2007-08-21
- Komar, Arthur (1959), "Covariant Conservation Laws in General Relativity", Phys. Rev., 113 (3): 934–936, Bibcode:1959PhRv..113..934K, doi:10.1103/PhysRev.113.934
- Kramer, Michael (2004), Karshenboim, S. G.; Peik, E. (eds.), "Astrophysics, Clocks and Fundamental Constants: Millisecond Pulsars as Tools of Fundamental Physics", Lecture Notes in Physics, Springer, 648: 33–54, arXiv:astro-ph/0405178, Bibcode:2004LNP...648...33K, doi:10.1007/978-3-540-40991-5_3, ISBN 978-3-540-21967-5
- Kramer, M.; Stairs, I. H.; Manchester, R. N.; McLaughlin, M. A.; Lyne, A. G.; Ferdman, R. D.; Burgay, M.; Lorimer, D. R.; et al. (2006), "Tests of general relativity from timing the double pulsar", Science, 314 (5796): 97–102, arXiv:astro-ph/0609417, Bibcode:2006Sci...314...97K, doi:10.1126/science.1132305, PMID 16973838
- Kraus, Ute (1998), "Light Deflection Near Neutron Stars", Relativistic Astrophysics, Vieweg, pp. 66–81, ISBN 3-528-06909-0
- Kuchař, Karel (1973), "Canonical Quantization of Gravity", in Israel, Werner (ed.), Relativity, Astrophysics and Cosmology, D. Reidel, pp. 237–288, ISBN 90-277-0369-8
- Künzle, H. P. (1972), "Galilei and Lorentz Structures on spacetime: comparison of the corresponding geometry and physics", Annales de l'Institut Henri Poincaré A, 17: 337–362
- Lahav, Ofer; Suto, Yasushi (2004), "Measuring our Universe from Galaxy Redshift Surveys", Living Reviews in Relativity, 7, arXiv:astro-ph/0310642, Bibcode:2004LRR.....7....8L, doi:10.12942/lrr-2004-8, retrieved 2007-08-19
- Landgraf, M.; Hechler, M.; Kemble, S. (2005), "Mission design for LISA Pathfinder", Class. Quant. Grav., 22 (10): S487–S492, arXiv:gr-qc/0411071, Bibcode:2005CQGra..22S.487L, doi:10.1088/0264-9381/22/10/048
- Lehner, Luis (2001), "Numerical Relativity: A review", Class. Quant. Grav., 18 (17): R25–R86, arXiv:gr-qc/0106072, Bibcode:2001CQGra..18R..25L, doi:10.1088/0264-9381/18/17/202
- Lehner, Luis (2002), "NUMERICAL RELATIVITY: STATUS AND PROSPECTS", General Relativity and Gravitation - Proceedings of the 16th International Conference, p. 210, arXiv:gr-qc/0202055, Bibcode:2002grg..conf..210L, doi:10.1142/9789812776556_0010, ISBN 9789812381712
- Linde, Andrei (1990), Particle Physics and Inflationary Cosmology, Harwood, p. 3203, arXiv:hep-th/0503203, Bibcode:2005hep.th....3203L, ISBN 3-7186-0489-2
- Linde, Andrei (2005), "Towards inflation in string theory", J. Phys. Conf. Ser., 24: 151–160, arXiv:hep-th/0503195, Bibcode:2005JPhCS..24..151L, doi:10.1088/1742-6596/24/1/018
- Loll, Renate (1998), "Discrete Approaches to Quantum Gravity in Four Dimensions", Living Reviews in Relativity, 1, arXiv:gr-qc/9805049, Bibcode:1998LRR.....1...13L, doi:10.12942/lrr-1998-13, retrieved 2008-03-09
- Lovelock, David (1972), "The Four-Dimensionality of Space and the Einstein Tensor", J. Math. Phys., 13 (6): 874–876, Bibcode:1972JMP....13..874L, doi:10.1063/1.1666069
- Ludyk, Günter (2013). Einstein in Matrix Form (1st ed.). Berlin: Springer. ISBN 9783642357978.
- MacCallum, M. (2006), "Finding and using exact solutions of the Einstein equations", in Mornas, L.; Alonso, J. D. (eds.), A Century of Relativity Physics (ERE05, the XXVIII Spanish Relativity Meeting), 841, American Institute of Physics, p. 129, arXiv:gr-qc/0601102, Bibcode:2006AIPC..841..129M, doi:10.1063/1.2218172
- Maddox, John (1998), What Remains To Be Discovered, Macmillan, ISBN 0-684-82292-X
- Mannheim, Philip D. (2006), "Alternatives to Dark Matter and Dark Energy", Prog. Part. Nucl. Phys., 56 (2): 340–445, arXiv:astro-ph/0505266, Bibcode:2006PrPNP..56..340M, doi:10.1016/j.ppnp.2005.08.001
- Mather, J. C.; Cheng, E. S.; Cottingham, D. A.; Eplee, R. E.; Fixsen, D. J.; Hewagama, T.; Isaacman, R. B.; Jensen, K. A.; et al. (1994), "Measurement of the cosmic microwave spectrum by the COBE FIRAS instrument", Astrophysical Journal, 420: 439–444, Bibcode:1994ApJ...420..439M, doi:10.1086/173574
- Mermin, N. David (2005), It's About Time. Understanding Einstein's Relativity, Princeton University Press, ISBN 0-691-12201-6
- Messiah, Albert (1999), Quantum Mechanics, Dover Publications, ISBN 0-486-40924-4
- Miller, Cole (2002), Stellar Structure and Evolution (Lecture notes for Astronomy 606), University of Maryland, retrieved 2007-07-25
- Misner, Charles W.; Thorne, Kip. S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman, ISBN 0-7167-0344-0
- Møller, Christian (1952), The Theory of Relativity (3rd ed.), Oxford University Press
- Narayan, Ramesh (2006), "Black holes in astrophysics", New Journal of Physics, 7: 199, arXiv:gr-qc/0506078, Bibcode:2005NJPh....7..199N, doi:10.1088/1367-2630/7/1/199
- Narayan, Ramesh; Bartelmann, Matthias (1997). "Lectures on Gravitational Lensing". arXiv:astro-ph/9606001
|class=
ignored (help). - Narlikar, Jayant V. (1993), Introduction to Cosmology, Cambridge University Press, ISBN 0-521-41250-1
- Nieto, Michael Martin (2006), "The quest to understand the Pioneer anomaly" (PDF), EurophysicsNews, 37 (6): 30–34, Bibcode:2006ENews..37...30N, doi:10.1051/epn:2006604
- Nordström, Gunnar (1918), "On the Energy of the Gravitational Field in Einstein's Theory", Verhandl. Koninkl. Ned. Akad. Wetenschap., 26: 1238–1245
- Nordtvedt, Kenneth (2003). "Lunar Laser Ranging—a comprehensive probe of post-Newtonian gravity". arXiv:gr-qc/0301024
|class=
ignored (help). - Norton, John D. (1985), "What was Einstein's principle of equivalence?" (PDF), Studies in History and Philosophy of Science, 16 (3): 203–246, doi:10.1016/0039-3681(85)90002-0, retrieved 2007-06-11
- Ohanian, Hans C.; Ruffini, Remo (1994), Gravitation and Spacetime, W. W. Norton & Company, ISBN 0-393-96501-5
- Olive, K. A.; Skillman, E. A. (2004), "A Realistic Determination of the Error on the Primordial Helium Abundance", Astrophysical Journal, 617 (1): 29–49, arXiv:astro-ph/0405588, Bibcode:2004ApJ...617...29O, doi:10.1086/425170
- O'Meara, John M.; Tytler, David; Kirkman, David; Suzuki, Nao; Prochaska, Jason X.; Lubin, Dan; Wolfe, Arthur M. (2001), "The Deuterium to Hydrogen Abundance Ratio Towards a Fourth QSO: HS0105+1619", Astrophysical Journal, 552 (2): 718–730, arXiv:astro-ph/0011179, Bibcode:2001ApJ...552..718O, doi:10.1086/320579
- Oppenheimer, J. Robert; Snyder, H. (1939), "On continued gravitational contraction", Physical Review, 56 (5): 455–459, Bibcode:1939PhRv...56..455O, doi:10.1103/PhysRev.56.455
- Overbye, Dennis (1999), Lonely Hearts of the Cosmos: the story of the scientific quest for the secret of the Universe, Back Bay, ISBN 0-316-64896-5
- Pais, Abraham (1982), 'Subtle is the Lord...' The Science and life of Albert Einstein, Oxford University Press, ISBN 0-19-853907-X
- Peacock, John A. (1999), Cosmological Physics, Cambridge University Press, ISBN 0-521-41072-X
- Peebles, P. J. E. (1966), "Primordial Helium abundance and primordial fireball II", Astrophysical Journal, 146: 542–552, Bibcode:1966ApJ...146..542P, doi:10.1086/148918
- Peebles, P. J. E. (1993), Principles of physical cosmology, Princeton University Press, ISBN 0-691-01933-9
- Peebles, P.J.E.; Schramm, D.N.; Turner, E.L.; Kron, R.G. (1991), "The case for the relativistic hot Big Bang cosmology", Nature, 352 (6338): 769–776, Bibcode:1991Natur.352..769P, doi:10.1038/352769a0
- Penrose, Roger (1965), "Gravitational collapse and spacetime singularities", Physical Review Letters, 14 (3): 57–59, Bibcode:1965PhRvL..14...57P, doi:10.1103/PhysRevLett.14.57
- Penrose, Roger (1969), "Gravitational collapse: the role of general relativity", Rivista del Nuovo Cimento, 1: 252–276, Bibcode:1969NCimR...1..252P
- Penrose, Roger (2004), The Road to Reality, A. A. Knopf, ISBN 0-679-45443-8
- Penzias, A. A.; Wilson, R. W. (1965), "A measurement of excess antenna temperature at 4080 Mc/s", Astrophysical Journal, 142: 419–421, Bibcode:1965ApJ...142..419P, doi:10.1086/148307
- Peskin, Michael E.; Schroeder, Daniel V. (1995), An Introduction to Quantum Field Theory, Addison-Wesley, ISBN 0-201-50397-2
- Peskin, Michael E. (2007), "Dark Matter and Particle Physics", Journal of the Physical Society of Japan, 76 (11): 111017, arXiv:0707.1536, Bibcode:2007JPSJ...76k1017P, doi:10.1143/JPSJ.76.111017
- Poisson, Eric (2004), "The Motion of Point Particles in Curved Spacetime", Living Reviews in Relativity, 7, arXiv:gr-qc/0306052, Bibcode:2004LRR.....7....6P, doi:10.12942/lrr-2004-6, retrieved 2007-06-13
- Poisson, Eric (2004), A Relativist's Toolkit. The Mathematics of Black-Hole Mechanics, Cambridge University Press, ISBN 0-521-83091-5
- Polchinski, Joseph (1998a), String Theory Vol. I: An Introduction to the Bosonic String, Cambridge University Press, ISBN 0-521-63303-6
- Polchinski, Joseph (1998b), String Theory Vol. II: Superstring Theory and Beyond, Cambridge University Press, ISBN 0-521-63304-4
- Pound, R. V.; Rebka, G. A. (1959), "Gravitational Red-Shift in Nuclear Resonance", Physical Review Letters, 3 (9): 439–441, Bibcode:1959PhRvL...3..439P, doi:10.1103/PhysRevLett.3.439
- Pound, R. V.; Rebka, G. A. (1960), "Apparent weight of photons", Phys. Rev. Lett., 4 (7): 337–341, Bibcode:1960PhRvL...4..337P, doi:10.1103/PhysRevLett.4.337
- Pound, R. V.; Snider, J. L. (1964), "Effect of Gravity on Nuclear Resonance", Phys. Rev. Lett., 13 (18): 539–540, Bibcode:1964PhRvL..13..539P, doi:10.1103/PhysRevLett.13.539
- Ramond, Pierre (1990), Field Theory: A Modern Primer, Addison-Wesley, ISBN 0-201-54611-6
- Rees, Martin (1966), "Appearance of Relativistically Expanding Radio Sources", Nature, 211 (5048): 468–470, Bibcode:1966Natur.211..468R, doi:10.1038/211468a0
- Reissner, H. (1916), "Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie", Annalen der Physik, 355 (9): 106–120, Bibcode:1916AnP...355..106R, doi:10.1002/andp.19163550905
- Remillard, Ronald A.; Lin, Dacheng; Cooper, Randall L.; Narayan, Ramesh (2006), "The Rates of Type I X-Ray Bursts from Transients Observed with RXTE: Evidence for Black Hole Event Horizons", Astrophysical Journal, 646 (1): 407–419, arXiv:astro-ph/0509758, Bibcode:2006ApJ...646..407R, doi:10.1086/504862
- Renn, Jürgen, ed. (2007), The Genesis of General Relativity (4 Volumes), Dordrecht: Springer, ISBN 1-4020-3999-9
- Renn, Jürgen, ed. (2005), Albert Einstein—Chief Engineer of the Universe: Einstein's Life and Work in Context, Berlin: Wiley-VCH, ISBN 3-527-40571-2
- Reula, Oscar A. (1998), "Hyperbolic Methods for Einstein's Equations", Living Reviews in Relativity, 1, Bibcode:1998LRR.....1....3R, doi:10.12942/lrr-1998-3, retrieved 2007-08-29
- Rindler, Wolfgang (2001), Relativity. Special, General and Cosmological, Oxford University Press, ISBN 0-19-850836-0
- Rindler, Wolfgang (1991), Introduction to Special Relativity, Clarendon Press, Oxford, ISBN 0-19-853952-5
- Robson, Ian (1996), Active galactic nuclei, John Wiley, ISBN 0-471-95853-0
- Roulet, E.; Mollerach, S. (1997), "Microlensing", Physics Reports, 279 (2): 67–118, arXiv:astro-ph/9603119, Bibcode:1997PhR...279...67R, doi:10.1016/S0370-1573(96)00020-8
- Rovelli, Carlo (2000). "Notes for a brief history of quantum gravity". arXiv:gr-qc/0006061
|class=
ignored (help). - Rovelli, Carlo (1998), "Loop Quantum Gravity", Living Reviews in Relativity, 1, arXiv:gr-qc/9710008, Bibcode:1998LRR.....1....1R, doi:10.12942/lrr-1998-1, retrieved 2008-03-13
- Schäfer, Gerhard (2004), "Gravitomagnetic Effects", General Relativity and Gravitation, 36 (10): 2223–2235, arXiv:gr-qc/0407116, Bibcode:2004GReGr..36.2223S, doi:10.1023/B:GERG.0000046180.97877.32
- Schödel, R.; Ott, T.; Genzel, R.; Eckart, A.; Mouawad, N.; Alexander, T. (2003), "Stellar Dynamics in the Central Arcsecond of Our Galaxy", Astrophysical Journal, 596 (2): 1015–1034, arXiv:astro-ph/0306214, Bibcode:2003ApJ...596.1015S, doi:10.1086/378122
- Schutz, Bernard F. (1985), A first course in general relativity, Cambridge University Press, ISBN 0-521-27703-5
- Schutz, Bernard F. (2001), "Gravitational radiation", in Murdin, Paul (ed.), Encyclopedia of Astronomy and Astrophysics, Grove's Dictionaries, ISBN 1-56159-268-4
- Schutz, Bernard F. (2003), Gravity from the ground up, Cambridge University Press, ISBN 0-521-45506-5
- Schwarz, John H. (2007), "String Theory: Progress and Problems", Progress of Theoretical Physics Supplement, 170: 214, arXiv:hep-th/0702219, Bibcode:2007PThPS.170..214S, doi:10.1143/PTPS.170.214
- Schwarzschild, Karl (1916a), "Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie", Sitzungsber. Preuss. Akad. D. Wiss.: 189–196
- Schwarzschild, Karl (1916b), "Über das Gravitationsfeld eines Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie", Sitzungsber. Preuss. Akad. D. Wiss.: 424–434
- Seidel, Edward (1998), "Numerical Relativity: Towards Simulations of 3D Black Hole Coalescence", in Narlikar, J. V.; Dadhich, N. (eds.), Gravitation and Relativity: At the turn of the millennium (Proceedings of the GR-15 Conference, held at IUCAA, Pune, India, December 16–21, 1997), IUCAA, p. 6088, arXiv:gr-qc/9806088, Bibcode:1998gr.qc.....6088S, ISBN 81-900378-3-8
- Seljak, Uros̆; Zaldarriaga, Matias (1997), "Signature of Gravity Waves in the Polarization of the Microwave Background", Phys. Rev. Lett., 78 (11): 2054–2057, arXiv:astro-ph/9609169, Bibcode:1997PhRvL..78.2054S, doi:10.1103/PhysRevLett.78.2054
- Shapiro, S. S.; Davis, J. L.; Lebach, D. E.; Gregory, J. S. (2004), "Measurement of the solar gravitational deflection of radio waves using geodetic very-long-baseline interferometry data, 1979–1999", Phys. Rev. Lett., 92 (12): 121101, Bibcode:2004PhRvL..92l1101S, doi:10.1103/PhysRevLett.92.121101, PMID 15089661
- Shapiro, Irwin I. (1964), "Fourth test of general relativity", Phys. Rev. Lett., 13 (26): 789–791, Bibcode:1964PhRvL..13..789S, doi:10.1103/PhysRevLett.13.789
- Shapiro, I. I.; Pettengill, Gordon; Ash, Michael; Stone, Melvin; Smith, William; Ingalls, Richard; Brockelman, Richard (1968), "Fourth test of general relativity: preliminary results", Phys. Rev. Lett., 20 (22): 1265–1269, Bibcode:1968PhRvL..20.1265S, doi:10.1103/PhysRevLett.20.1265
- Singh, Simon (2004), Big Bang: The Origin of the Universe, Fourth Estate, ISBN 0-00-715251-5
- Sorkin, Rafael D. (2005), "Causal Sets: Discrete Gravity", in Gomberoff, Andres; Marolf, Donald (eds.), Lectures on Quantum Gravity, Springer, p. 9009, arXiv:gr-qc/0309009, Bibcode:2003gr.qc.....9009S, ISBN 0-387-23995-2
- Sorkin, Rafael D. (1997), "Forks in the Road, on the Way to Quantum Gravity", Int. J. Theor. Phys., 36 (12): 2759–2781, arXiv:gr-qc/9706002, Bibcode:1997IJTP...36.2759S, doi:10.1007/BF02435709
- Spergel, D. N.; Verde, L.; Peiris, H. V.; Komatsu, E.; Nolta, M. R.; Bennett, C. L.; Halpern, M.; Hinshaw, G.; et al. (2003), "First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters", Astrophys. J. Suppl., 148 (1): 175–194, arXiv:astro-ph/0302209, Bibcode:2003ApJS..148..175S, doi:10.1086/377226
- Spergel, D. N.; Bean, R.; Doré, O.; Nolta, M. R.; Bennett, C. L.; Dunkley, J.; Hinshaw, G.; Jarosik, N.; et al. (2007), "Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology", Astrophysical Journal Supplement, 170 (2): 377–408, arXiv:astro-ph/0603449, Bibcode:2007ApJS..170..377S, doi:10.1086/513700
- Springel, Volker; White, Simon D. M.; Jenkins, Adrian; Frenk, Carlos S.; Yoshida, Naoki; Gao, Liang; Navarro, Julio; Thacker, Robert; et al. (2005), "Simulations of the formation, evolution and clustering of galaxies and quasars", Nature, 435 (7042): 629–636, arXiv:astro-ph/0504097, Bibcode:2005Natur.435..629S, doi:10.1038/nature03597, PMID 15931216
- Stairs, Ingrid H. (2003), "Testing General Relativity with Pulsar Timing", Living Reviews in Relativity, 6, arXiv:astro-ph/0307536, Bibcode:2003LRR.....6....5S, doi:10.12942/lrr-2003-5, retrieved 2007-07-21
- Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; Herlt, E. (2003), Exact Solutions of Einstein's Field Equations (2 ed.), Cambridge University Press, ISBN 0-521-46136-7
- Synge, J. L. (1972), Relativity: The Special Theory, North-Holland Publishing Company, ISBN 0-7204-0064-3
- Szabados, László B. (2004), "Quasi-Local Energy-Momentum and Angular Momentum in GR", Living Reviews in Relativity, 7, Bibcode:2004LRR.....7....4S, doi:10.12942/lrr-2004-4, retrieved 2007-08-23
- Taylor, Joseph H. (1994), "Binary pulsars and relativistic gravity", Rev. Mod. Phys., 66 (3): 711–719, Bibcode:1994RvMP...66..711T, doi:10.1103/RevModPhys.66.711
- Thiemann, Thomas (2006), "Approaches to Fundamental Physics: Loop Quantum Gravity: An Inside View", Lecture Notes in Physics, 721: 185–263, arXiv:hep-th/0608210, Bibcode:2007LNP...721..185T, doi:10.1007/978-3-540-71117-9_10, ISBN 978-3-540-71115-5
- Thiemann, Thomas (2003), "Lectures on Loop Quantum Gravity", Lecture Notes in Physics, 631: 41–135, arXiv:gr-qc/0210094, doi:10.1007/978-3-540-45230-0_3, ISBN 978-3-540-40810-9
- ’t Hooft, Gerard; Veltman, Martinus (1974), "One Loop Divergencies in the Theory of Gravitation", Ann. Inst. Poincare, 20: 69
- Thorne, Kip S. (1972), "Nonspherical Gravitational Collapse—A Short Review", in Klauder, J. (ed.), Magic without Magic, W. H. Freeman, pp. 231–258
- Thorne, Kip S. (1994), Black Holes and Time Warps: Einstein's Outrageous Legacy, W W Norton & Company, ISBN 0-393-31276-3
- Thorne, Kip S. (1995), "Gravitational radiation", Particle and Nuclear Astrophysics and Cosmology in the Next Millenium: 160, arXiv:gr-qc/9506086, Bibcode:1995pnac.conf..160T, ISBN 0-521-36853-7
- Townsend, Paul K. (1997). "Black Holes (Lecture notes)". arXiv:gr-qc/9707012
|class=
ignored (help). - Townsend, Paul K. (1996). "Four Lectures on M-Theory". arXiv:hep-th/9612121
|class=
ignored (help). - Traschen, Jenny (2000), Bytsenko, A.; Williams, F. (eds.), "An Introduction to Black Hole Evaporation", Mathematical Methods of Physics (Proceedings of the 1999 Londrina Winter School), World Scientific: 180, arXiv:gr-qc/0010055, Bibcode:2000mmp..conf..180T
- Trautman, Andrzej (2006), "Einstein–Cartan theory", in Françoise, J.-P.; Naber, G. L.; Tsou, S. T. (eds.), Encyclopedia of Mathematical Physics, Vol. 2, Elsevier, pp. 189–195, arXiv:gr-qc/0606062, Bibcode:2006gr.qc.....6062T
- Unruh, W. G. (1976), "Notes on Black Hole Evaporation", Phys. Rev. D, 14 (4): 870–892, Bibcode:1976PhRvD..14..870U, doi:10.1103/PhysRevD.14.870
- Valtonen, M. J.; Lehto, H. J.; Nilsson, K.; Heidt, J.; Takalo, L. O.; Sillanpää, A.; Villforth, C.; Kidger, M.; et al. (2008), "A massive binary black-hole system in OJ 287 and a test of general relativity", Nature, 452 (7189): 851–853, arXiv:0809.1280, Bibcode:2008Natur.452..851V, doi:10.1038/nature06896, PMID 18421348
- Veltman, Martinus (1975), "Quantum Theory of Gravitation", in Balian, Roger; Zinn-Justin, Jean (eds.), Methods in Field Theory - Les Houches Summer School in Theoretical Physics., 77, North Holland
- Wald, Robert M. (1975), "On Particle Creation by Black Holes", Commun. Math. Phys., 45 (3): 9–34, Bibcode:1975CMaPh..45....9W, doi:10.1007/BF01609863
- Wald, Robert M. (1984), General Relativity, University of Chicago Press, ISBN 0-226-87033-2
- Wald, Robert M. (1994), Quantum field theory in curved spacetime and black hole thermodynamics, University of Chicago Press, ISBN 0-226-87027-8
- Wald, Robert M. (2001), "The Thermodynamics of Black Holes", Living Reviews in Relativity, 4, arXiv:gr-qc/9912119, Bibcode:2001LRR.....4....6W, doi:10.12942/lrr-2001-6, retrieved 2007-08-08
- Walsh, D.; Carswell, R. F.; Weymann, R. J. (1979), "0957 + 561 A, B: twin quasistellar objects or gravitational lens?", Nature, 279 (5712): 381–4, Bibcode:1979Natur.279..381W, doi:10.1038/279381a0, PMID 16068158
- Wambsganss, Joachim (1998), "Gravitational Lensing in Astronomy", Living Reviews in Relativity, 1, arXiv:astro-ph/9812021, Bibcode:1998LRR.....1...12W, doi:10.12942/lrr-1998-12, retrieved 2007-07-20
- Weinberg, Steven (1972), Gravitation and Cosmology, John Wiley, ISBN 0-471-92567-5
- Weinberg, Steven (1995), The Quantum Theory of Fields I: Foundations, Cambridge University Press, ISBN 0-521-55001-7
- Weinberg, Steven (1996), The Quantum Theory of Fields II: Modern Applications, Cambridge University Press, ISBN 0-521-55002-5
- Weinberg, Steven (2000), The Quantum Theory of Fields III: Supersymmetry, Cambridge University Press, ISBN 0-521-66000-9
- Weisberg, Joel M.; Taylor, Joseph H. (2003), "The Relativistic Binary Pulsar B1913+16"", in Bailes, M.; Nice, D. J.; Thorsett, S. E. (eds.), Proceedings of "Radio Pulsars," Chania, Crete, August, 2002, ASP Conference Series
- Weiss, Achim (2006), "Elements of the past: Big Bang Nucleosynthesis and observation", Einstein Online, Max Planck Institute for Gravitational Physics, retrieved 2007-02-24
- Wheeler, John A. (1990), A Journey Into Gravity and Spacetime, Scientific American Library, San Francisco: W. H. Freeman, ISBN 0-7167-6034-7
- Will, Clifford M. (1993), Theory and experiment in gravitational physics, Cambridge University Press, ISBN 0-521-43973-6
- Will, Clifford M. (2006), "The Confrontation between General Relativity and Experiment", Living Reviews in Relativity, 9, arXiv:gr-qc/0510072, Bibcode:2006LRR.....9....3W, doi:10.12942/lrr-2006-3, retrieved 2007-06-12
- Zwiebach, Barton (2004), A First Course in String Theory, Cambridge University Press, ISBN 0-521-83143-1
Further readingEdit
- Popular books
- Geroch, R (1981), General Relativity from A to B, Chicago: University of Chicago Press, ISBN 0-226-28864-1
- Lieber, Lillian (2008), The Einstein Theory of Relativity: A Trip to the Fourth Dimension, Philadelphia: Paul Dry Books, Inc., ISBN 978-1-58988-044-3
- Wald, Robert M. (1992), Space, Time, and Gravity: the Theory of the Big Bang and Black Holes, Chicago: University of Chicago Press, ISBN 0-226-87029-4
- Wheeler, John; Ford, Kenneth (1998), Geons, Black Holes, & Quantum Foam: a life in physics, New York: W. W. Norton, ISBN 0-393-31991-1
- Beginning undergraduate textbooks
- Callahan, James J. (2000), The Geometry of Spacetime: an Introduction to Special and General Relativity, New York: Springer, ISBN 0-387-98641-3
- Taylor, Edwin F.; Wheeler, John Archibald (2000), Exploring Black Holes: Introduction to General Relativity, Addison Wesley, ISBN 0-201-38423-XCS1 maint: multiple names: authors list (link)
- Advanced undergraduate textbooks
- B. F. Schutz (2009), A First Course in General Relativity (Second Edition), Cambridge University Press, ISBN 978-0-521-88705-2
- Cheng, Ta-Pei (2005), Relativity, Gravitation and Cosmology: a Basic Introduction, Oxford and New York: Oxford University Press, ISBN 0-19-852957-0
- Gron, O.; Hervik, S. (2007), Einstein's General theory of Relativity, Springer, ISBN 978-0-387-69199-2
- Hartle, James B. (2003), Gravity: an Introduction to Einstein's General Relativity, San Francisco: Addison-Wesley, ISBN 0-8053-8662-9
- Hughston, L. P. & Tod, K. P. (1991), Introduction to General Relativity, Cambridge: Cambridge University Press, ISBN 0-521-33943-XCS1 maint: multiple names: authors list (link)
- d'Inverno, Ray (1992), Introducing Einstein's Relativity, Oxford: Oxford University Press, ISBN 0-19-859686-3
- Ludyk, Günter (2013). Einstein in Matrix Form (1st ed.). Berlin: Springer. ISBN 9783642357978.
- Graduate-level textbooks
- Carroll, Sean M. (2004), Spacetime and Geometry: An Introduction to General Relativity, San Francisco: Addison-Wesley, ISBN 0-8053-8732-3
- Grøn, Øyvind; Hervik, Sigbjørn (2007), Einstein's General Theory of Relativity, New York: Springer, ISBN 978-0-387-69199-2
- Landau, Lev D.; Lifshitz, Evgeny F. (1980), The Classical Theory of Fields (4th ed.), London: Butterworth-Heinemann, ISBN 0-7506-2768-9
- Misner, Charles W.; Thorne, Kip. S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman, ISBN 0-7167-0344-0
- Stephani, Hans (1990), General Relativity: An Introduction to the Theory of the Gravitational Field, Cambridge: Cambridge University Press, ISBN 0-521-37941-5
- Wald, Robert M. (1984), General Relativity, University of Chicago Press, ISBN 0-226-87033-2
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- Brown, Kevin. "Reflections on relativity". Mathpages.com. Retrieved May 29, 2005.
- Carroll, Sean M. "Lecture Notes on General Relativity". Retrieved January 5, 2014.
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{{DEFAULTSORT:Special Relativity}} Category:Concepts in physics Category:Albert Einstein
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. It describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries gravitational force. Thus string theory is a theory of quantum gravity.
TABLE OF CONTENT
§ 1. Introduction
§ 2. Huygens
§ 3. Fresnel
§ 4. Faraday
§ 5. Maxwell
§ 6. Michelson
§ 7. Kirchhoff
§ 8. Poynting
§ 9. Lorentz
§ 10. Lenard
§ 11. Planck
§ 12. Einstein's Energy Quanta
§ 13. Einstein Electrodynamics
§ 14. Einstein Electron Inertial Mass
§ 15. Minkowski
§ 16. Einstein's Electromagnetic Ether
§ 17. General Relativity
§ 18. Relativity: Special and General Theory
§ 19. Einstein's Ether
§ 20. Quantum Mechanics
§ 21. Heisenberg
§ 22. Quantum Electrodynamics
§ 23. String Theory
§ 24. Particle Physics
§ 25. Gravitation
§ 26. Astronomy
§ 27. Maxwell's Equations
§ 28. Electromagnetic Wave Equations of Light
§ 29. Transmission and Reflection Equations
§ 30. Polarization
§ 31. Aperture Diffraction Effect of Light
__________________________________________________________________________________________
The Wave Theory of Light and its Affect on Modern Physics and Astronomy
Ben T. Ito
September 27, 2015
This paper will analyze the wave theory of light and its consequence upon modern physics and astronomy. Huygens describes the propagation of light using light waves formed by the motion of an Ethereal matter. Huygens' ether particles possess a hardness that produces a springiness used to form light waves that propagate through the optical ether. The wave theory of light was established when Fresnel (1819) described diffraction using interfering light-waves produced by the vibration of an elastic fluid (ether) yet diffraction forms in vacuum, void of Fresnel's optical ether, composed of matter; consequently, Maxwell introduces an electromagnetic theory of light, based on Faraday's induction effect, since induction forms in vacuum, but induction is not luminous. Hertz's spark gap experiment is used to structurally unite light with induction but Hertz's spark gap emits electrons yet induction is also not an ionization effect. In Einstein's (1905) electrodynamics (special relativity), Einstein alters the dimensions of Maxwell's equations to justify light propagating in vacuum but manipulating the coordinate system, of Maxwell's equations, does not change the fact that Maxwell's equations are derived using Faraday's induction effect still induction is not luminous, nor is induction an ionization effect; therefore, Maxwell's equations cannot be used to justify light propagating in vacuum or justify Maxwell's theory. In addition, the velocity of light is used to justify Maxwell's theory but Roemer's ten minute time delay is caused by numerous factors, such as, Roemer's assumption the Earth and Jupiter have circular orbits that rotate on the same plane, and Jupiter being stationary during the propagation of the Earth from L to K (fig 27), during Io's completion of a cycle of rotation, around Jupiter. Roemer's experiment is an extremely crude and inaccurate attempt at measuring the velocity of light and has absolutely no scientific merit since Roemer's measurement of the velocity of light is based on a pendulum clock. Fizeau (1849) and Foucault (1850) velocity of light experiments attempt to measure the velocity of light use rotating devices that continue to emit light, after the signal is produced, since an individual signal cannot form an intensity, after propagating a distance of 8 km. In modern physics, a pulse beam is used to measure the velocity of light since a single pulse of light produced by a Kerr shutter (nanoseconds) cannot form a measurable intensity, after propagating 50 km which proves the velocity of light has not been measured and cannot be used to justify Maxwell's theory. Furthermore, the measurement of the velocity of light does not change the fact Maxwell's theory is based on Faraday's induction effect nonetheless induction is not luminous. In addition, Maxwell's equations, describe a disturbance within a three dimensional volume, that represent the formation of a spherical wave that produces a longitudinal wave which conflicts with Maxwell's transverse waves; consequently, the electromagnetic transverse wave equations of light cannot be derived using Maxwell's equations (equ 85 - 111). Quantum mechanics, quantum electrodynamics, string theory and particle physics rely on the gauge transformation of Maxwell's equations but representing Maxwell's equations with a potential does not change the fact that Maxwell's equations are derived using Faraday's induction effect tho induction is not luminous, nor can the potential of a massless electromagnetic induction field represent the structure of an electron, proton, nuclei or subatomic particle that has a mass. Einstein attempts to structurally unify Maxwell's electromagnetic field with matter, using an energy equation Eo = mc2 but the inertia (m) of Einstein's energy equation is massless since Eo represents the energy of a massless electromagnetic photon. In particle physics, subatomic particles are described using tracks formed in liquid hydrogen, within a bubble chamber. An accelerated electron beam is incident to an external metallic target that collision produces the alleged subatomic particles, that have a mass, which propagate through the steel enclosure, of the bubble chamber that contains a high pressure liquid hydrogen, to form tracks in the liquid hydrogen but subatomic particles, that have a mass, cannot propagate through the steel enclosure of the bubble chamber without forming a hole, in the steel enclosure, and causing an explosion of the liquid hydrogen. In gravitational physics, Weber (1970) detected celestial gravity waves that have the frequency of sound (1662 Hz) yet the vacuum of celestial space does not transmit sound. Wheeler describes electromagnetic gravity waves, and Thorne, Ohanian and Gertsenshtein describe electromagnetic gravity waves that propagate at the velocity of light. Experimentally, the European pulsar timing array (EPTA), detected a gravity wave with the frequency of 10-8 Hz which forms a wavelength of λ = 1016 m which is more than a light year in length!! Furthermore, electromagnetic shielding proves gravity is not an electromagnetic phenomenon. In 2016, the LIGO collaboration announced the detection of a gravity wave that has a frequency of 35 Hz which is also the frequency of sound. Particle and gravitational physics are the result of the deception and manipulation that originate from the wave theory of light.
......................................................................................................................................................................................................
§ 1. Introduction
The ancient Greeks believed the eye sent out feelers that emanated from the eye and felt the object being observed. The ancient Greeks had numerous different theories, regarding light, which was a remarkable achievement, for the openness that the Greeks allowed their scholars. Most, if not all, of the ancient Greek writings were translated from Arabic since the Roman conquerors attempted to destroy all of the ancient Greek writings. In the 9th century AD, Middle East, Iraqi scholars studying the translations of the ancient Greek writings, regarding light, resulted in the advent of the light ray theory. In Kindi's (b. 801 AD) paper "De Aspectibus", Kindi introduced the theory of vision, where light rays, interacting with the eye, formed vision. In Haytham's (b. 965) paper "Opticae Thesaurus", Haytham enhanced Kindi's light ray theory, by dissecting the eye and analyzing the anatomy of the eye, resulting in the invention of the two lens celestial magnifier that Syrian scholar Shatir (b. 1304) used to form the theory that planets revolved around the sun, described in "The Final Quest Concerning the Rectification of Principles" (1342). Copernicus (1474) plagiarized Shatir's diagrams and calculations to describe planets revolving around the sun. Galileo (b. 1564) used the design of Shatir's two lens celestial magnifier in the construction of the astronomic telescope. In 1610, Galileo discovered the rings of Saturn, and supported Shatir's theory which, at the time, was highly controversial; consequently, Galileo was punished with a life sentence, of home incarceration, for his outspoken criticism of the Ptolemy model. Leibniz (b. 1646) studied the area problem of a planetary orbital ellipse, and discovered the mathematical derivative. Newton's (b. 1643) equations of motion are based on Leibniz's derivative.
Huygens (1690) describes a propagation mechanism of light using light waves formed by the motion of an Ethereal matter where the particles of the ether possess a hardness that produces a springiness used to form light waves which propagate through the optical ether, composed of matter (solid, liquid or gas). The wave theory of light was established when Fresnel (1819) described diffraction using interfering light-waves created by the vibration of an elastic fluid. Maxwell (1864) depicts polarization using transverse light waves, formed by the motion, of an elastic medium yet the propagation, diffraction and polarization effects of light form in vacuum, that is void of an optical ether, composed of matter. Michelson (1881) tests for the existence of Fresnel's optical ether, composed of matter, but the result was negative. Lorentz (1899) reverses the negative result of Michelson's experiment to justify the existence of Fresnel's optical ether but light propagating in vacuum is definitive and irreversible experimental proof Fresnel's optical ether does not physically exist; consequently, in Einstein's (1905) electrodynamics (special relativity), Einstein alters the dimensions of Maxwell's equations to justify light propagating in vacuum but altering the coordinate system of Maxwell's equations does not change the fact that Maxwell's equations are derived using Faraday's induction effect but induction is not luminous or that Fresnel diffraction mechanism is based on an optical ether, composed of matter, that does not exist in vacuum; consequently, Maxwell's equations cannot be used to justify light propagating in vacuum or Maxwell's thoery. In Einstein's paper "Relativity: Special and General Theory" (1917), Einstein uses the reversal of the negative result of Michelson-Morley experiment, based on Lorentz's transformation, to justify the existence of Fresnel's optical ether, composed of matter.
"More artificial theories have been tried out, assuming that the real truth lies somewhere between these two limiting cases: that the ether is only partially carried by the moving bodies. But they all failed! Every attempt to explain the electromagnetic phenomena in moving CS with the help of the motion of the ether, motion through the ether, or both these motions, proved unsuccessful. Thus arose one of the most dramatic situations in the history of science. All assumptions concerning ether led nowhere! The experimental verdict was always negative." (Weaver, p. 145).
Induction is described by Oersted (1820), Ampere (1823), Lentz (1830), Faraday (1831), Henry (1831), (Maxwell (1864) and Hertz (1887). Lenz (1830) may have discovered circuital induction effect ascribed to Faraday in "On the Laws which Govern the Action of a Magnet upon a Spiral" (1830) that was initially rejected until after Faraday published his paper "Experimental Researches in Electricity on induction" (1831). Maxwell's (1864) electromagnetic theory of light, based on Faraday's induction effect, was introduced since induction forms in vacuum but induction is not luminous; consequently, Faraday (1835) uses an induction circuit that includes a spark gap to structurally unite light with induction but the spark gap emits electrons yet induction is not an ionization effect. Poynting (1884) derived an electromagnetic energy equation of light but Poynting's current wire is not emitting light; therefore, Poynting's energy equation cannot be used to represent the energy of light. Hertz's (1887) attempts to structurally unite light with induction, using a spark gap experiment, that emits light and the radio induction effect which is similar to Faraday's (1835) spark gap experiment but Hertz's spark gap emits electrons yet induction is also not an ionization effect which contradicts Maxwell's theory. Furthermore, in 1902, Lenard proves light is composed of particles that energy is dependent on only the frequency which conflicts with Fresnel's light waves' energy that is dependent on the wave amplitude which is used to form the intensity of the diffraction pattern, and the continuity of Maxwell's electromagnetic field since dispersing light particles cannot maintain the continuity of Maxwell's electromagnetic field, during propagation. Planck (1901) seeks to structurally unify light with induction and quantize Maxwell's electromagnetic field, using the blackbody radiation effect, that emits light and the radio induction effect, in the derivation of the energy element (hv) that represents the energies of both the blackbody light and radio induction effect emissions but the blackbody emits electrons yet induction is not an ionization effect which contradicts Planck's unification and quantization of Maxwell's electromagnetic field. In Einstein's (1905) electrodynamics (special relativity), Einstein alters the dimensions of Maxwell's equations to justify Maxwell's theory but manipulating the coordinate system of Maxwell's equations does not change the fact that Maxwell's equations are derived using Faraday's induction effect but induction is not luminous. In addition, the velocity of light is used to justify Maxwell's theory but Roemer's ten minute time delay is caused by numerous factors, such as, Roemer's assumption the Earth and Jupiter have circular orbits that rotate on the same plane, and Jupiter being stationary during the propagation of the Earth from L to K (fig 22), during Io's completion of a cycle of rotation, around Jupiter. Roemer's experiment is an extremely crude and inaccurate attempt at measuring the velocity of light and has absolutely no scientific merit since the time measurement devise used by Roemer is a 300 year old pendulum clock that frequently stopped. Fizeau (1849) and Foucault (1850) velocity of light experiments attempt to measure the velocity of light use rotating devices that continue to emit light, after the signal is produced, since an individual signal cannot form an intensity, after propagating a distance of 8 km. In modern physics, a pulse beam is used to measure the velocity of light since a single pulse of light produced by a Kerr shutter (nanoseconds) cannot form a measurable intensity, after propagating 50 km which proves the velocity of light has not been measured and cannot be used to justify Maxwell's theory.
Young (1804) used Bradley's stellar aberration to justify the existence of the optical ether, composed of matter, but aberration does not alter the fact that light propagates through a glass vacuum tube that is void of Fresnel's optical ether, composed of matter. Young and all ether experiments are unnecessary since light propagating in vacuum is definitive and irreversible experimental proof Fresnel's optical ether, composed of matter, does not physically exist. Furthermore, Einstein uses the aberration effect of light, based on the Doppler effect (Einstein2, § 7), to explain the stellar red and blue shifts to justify Maxwell's electromagnetic theory of light but the aberration of light does not alter the fact that Maxwell's electromagnetic field originates from Faraday's induction effect still induction is not luminous. The red and blue shifts are caused by the earth's daily and yearly motions since at different times and positions every star forms both the red and blue shifts or the stars would visually disappear. The blue shift is cause when the observer on the earth is propagating toward a particular star and the red shift is caused when the observed on the earth is moving away from the star, based on the earth's daily and yearly rotational motions. Einstein (1910) describes an electromagnetic ether that forms light waves in vacuum but Maxwell's electromagnetic field originates from Faraday's induction effect yet induction is not luminous. Furthermore, Einstein (1917) uses the increase of an electron's inertia after absorbing an electromagnetic photon to justify Maxwell's electromagnetic theory of light but Maxwell's electromagnetic field originates from Faraday's induction effect. Also, Einstein's inertia term Eo/c2 (equ 62) represents the increase in the inertia of an electron after absorbing an electromagnetic photon but Einstein's energy term Eo represents the energy of an electromagnetic photon; consequently, the inertia (m), that is represented with Eo/c2 (equ 62), is massless. Einstein is attempting to structurally unify Maxwell's electromagnetic field with Fresnel's optical ether, composed of matter, using an energy equation Eo = mc2 but the inertia (m) of Einstein's energy equation is massless since Eo represents the energy of a massless electromagnetic photon which proves the wave theory of light is physically invalid. The analysis of the wave theory of light begins with Huygens' principle.
__________________________________________________________________________________________
§ 2. Huygens
In Huygens' paper, "Treatise on Light" (1690), Huygens' describes light, using spherical waves, based on a sound wave analogy.
"We know that by means of the air, which is an invisible and impalpable body, Sound spreads around the spot where it has been produced, by a movement which is passed on successively from one part of the air to another; and that the spreading of this movement, taking place equally rapidly on all sides, ought to form spherical surfaces ever enlarging and which strike our ears. Now there is no doubt at all that light also comes from the luminous body to our eyes by some movement impressed on the matter which is between the two; since, as we have already seen, it cannot be by the transport of a body which passes from one to the other. If, in addition, light takes time for its passage—which we are now going to examine—it will follow that this movement, impressed on the intervening matter, is successive; and consequently it spreads, as Sound does, by spherical surfaces and waves" (Huygens, p. 5).
"It is true that we are here supposing a strange velocity that would be a hundred thousand times greater than that of Sound. For Sound, according to what I have observed, travels about 180 Toises in the time of one Second, or in about one beat of the pulse. But this supposition ought not to seem to be an impossibility; since it is not a question of the transport of a body with so great a speed, but of a successive movement which is passed on from some bodies to others. I have then made no difficulty, in meditating on these things, in supposing that the emanation of light is accomplished with time, seeing that in this way all its phenomena can be explained, and that in following the contrary opinion everything is incomprehensible. For it has always seemed tome that even Mr. Des Cartes, whose aim has been to treat all the subjects of Physics intelligibly, and who assuredly has succeeded in this better than any one before him, has said nothing that is not full of difficulties, or even inconceivable, in dealing with Light and its properties." (Huygens, p. 7).
"the velocity of Light is more than six hundred thousand times greater than that of Sound. This, however, is quite another thing from being instantaneous, since there is all the difference between a finite thing and an infinite. Now the successive movement of Light being confirmed in this way, it follows, as I have said, that it spreads by spherical waves, like the movement of Sound." (Huygens, p. 10).
Huygens' spherical waves are formed by the motion of an optical ether, composed of matter, yet light propagates in vacuum that is void of matter which contradicts the existence of Huygens' spherical waves. A wave is a mechanical entity that is formed by the motion of a medium, composed of matter since a force that vibrates the medium, composed of matter, produces a wave. Air is the medium that forms sound waves which are produced by the collective motion of air molecules but sound cannot propagate in a vacuum since vacuum is void of air molecules required in forming sound waves yet Huygens is using a sound wave analogy to represent the propagation of light. One of the most important physical characteristic of sound is not applicable to light since light propagates in vacuum that is void of a medium, composed of matter, which is experimental proof Huygens' sound wave analogy cannot be applied to light.
__________________________________________________________________________________________
Huygens is describing the formation of light waves produced by the motion of an Ethereal matter.
"Now if one examines what this matter may be in which the movement coming from the luminous body is propagated, which I call Ethereal matter" (Huygens, p. 11).
"But the extreme velocity of Light, and other properties which it has, cannot admit of such a propagation of motion, and I am about to show here the way in which I conceive it must occur. For this, it is needful to explain the property which hard bodies must possess to transmit movement from one to another." (Huygens, p. 13).
"But it is still certain that this progression of motion is not instantaneous, but successive, and therefore must take time. For if the movement, or the disposition to movement, if you will have it so, did not pass successively through all these spheres, they would all acquire the movement at the same time, and hence would all advance together; which does not happen. For the last one leaves the whole row and acquires the speed of the one which was pushed. Moreover there are experiments which demonstrate that all the bodies which we reckon of the hardest kind, such as quenched steel, glass, and agate, act as springs and bend somehow, not only when extended as rods but also when they are in the form of spheres or of other shapes." (Huygens, p. 13).
"Now in applying this kind of movement to that which produces Light there is nothing to hinder us from estimating the particles of the ether to be of a substance as nearly approaching to perfect hardness and possessing a springiness as prompt as we choose. It is not necessary to examine here the causes of this hardness, or of that springiness, the consideration of which would lead us too far from our subject. I will say, however, in passing that we may conceive that the particles of the ether" (Huygens, p. 14).
"But though we shall ignore the true cause of springiness we still see that there are many bodies which possess this property; and thus there is nothing strange in supposing that it exists also in little invisible bodies like the particles of the Ether. Also if one wishes to seek for any other way in which the movement of Light is successively communicated, one will find none which agrees better, with uniform progression, as seems to be necessary, than the property of springiness; because if this movement should grow slower in proportion as it is shared over a greater quantity of matter, in moving away from the source of the light, it could not conserve this great velocity over great distances. But by supposing springiness in the ethereal matter, its particles will have the property of equally rapid restitution whether they are pushed strongly or feebly; and thus the propagation of Light will always go on with an equal velocity." (Huygens, p. 15).
According to Huygens, light forms waves by the motion of an ether, composed of matter. The ether particles possess a springiness that produces a successive motion of the ether particles. Huygens principle is based on a sound wave analogy but Huygens is describing an ether that is similar to a solid since the ether particles are continuously touching (fig 1 & 2). The touching ether particles transmit the wave energy through the ether which conflicts with Huygens' sound wave analogy where sound is formed by interacting gas molecules that are randomly propagating in a volume where the gas molecules have spaces between gas molecules and are not constantly touching. The exchange of the kinetic energies of the gas molecules form sound waves which conflicts with Huygens' touching ether particles (fig 1 & 2) that are forming waves by the successive motion of ether particles that are continuously touching. The collective motion of the gas molecules' kinetic energies form sound waves which does not correspond with Huygens' propagation mechanism of light.
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Huygens states optical ether, composed of matter, penetrates glass and exists in vacuum.
"This may be proved by shutting up a sounding body in a glass vessel from which the air is withdrawn by the machine which Mr. Boyle has given us, and with which he has performed so many beautiful experiments. But in doing this of which I speak, care must be taken to place the sounding body on cotton or on feathers, in such a way that it cannot communicate its tremors either to the glass vessel which encloses it, or to the machine; a precaution which has hitherto been neglected. For then after having exhausted all the air one hears no Sound from the metal, though it is struck. One sees here not only that our air, which does not penetrate through glass, is the matter by which Sound spreads; but also that it is not the same air but another kind of matter in which Light spreads; since if the air is removed from the vessel the Light does not cease to traverse it as before. And this last point is demonstrated even more clearly by the celebrated experiment of Torricelli, in which the tube of glass from which the quicksilver has withdrawn itself, remaining void of air, transmits Light just the same as when air is in it. For this proves that a matter different from air exists in this tube, and that this matter must have penetrated the glass or the quicksilver, either one or the other, though they are both impenetrable to the air. And when, in the same experiment, one makes the vacuum after putting a little water above the quicksilver, one concludes equally that the said matter passes through glass or water, or through both." (Huygens, p. 11 & 12).
Huygens' wave theory of light is based on a sound wave analogy; consequently, the formation of a light wave requires a medium, composed of matter. The motion of Huygens' optical ether, composed of matter, forms light waves that are used to represent the propagation of light yet vacuum is void of matter. Huygens states the optical ether, composed of matter, propagates through glass and exists in vacuum yet the optical ether, composed of matter, propagating through glass would produce a hole, in the glass, or shatter the glass, which would eliminate the vacuum. Vacuum is void of matter (solid, liquid or gas); consequently, light propagating through a glass vacuum tube proves the propagation of light does not involve Huygens' optical ether that motion forms propagating light waves. Huygens' wave theory of light is based on mechanical waves that are formed by the motion of the ether, composed of matter. A force that vibrates Huygens' light medium produces light waves but light propagates in vacuum that is void of matter which contradicts Huygens' wave theory of light. Light propagating in vacuum is definitive and irreversible experimental proof that Huygens' optical ether, composed of matter, does not physically exist; consequently, Huygens states that the ether, composed of matter, propagates through glass and exists in vacuum. Maxwell (1864) also states the optical ether, composed of matter, exists within Geissler's glass vacuum tube and forms light waves in vacuum (Maxwell, Intro).
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Huygens describes optical spherical waves originate from a candle flame (fig 3).
"I have then shown in what manner one may conceive Light to spread successively, by spherical waves, and how it is possible that this spreading is accomplished with as great a velocity as that which experiments and celestial observations demand. Whence it may be further remarked that although the particles are supposed to be in continual movement (for there are many reasons for this) the successive propagation of the waves cannot be hindered by this; because the propagation consists nowise in the transport of those particles but merely in a small agitation which they cannot help communicating to those surrounding, notwithstanding any movement which may act on them causing them to be changing positions amongst themselves.
But we must consider still more particularly the origin of these waves, and the manner in which they spread. And, first, it follows from what has been said on the production of Light, that each little region of a luminous body, such as the Sun, a candle, or a burning coal, generates its own waves of which that region is the centre. Thus in the flame of a candle, having distinguished the points A, B, C, concentric circles described about each of these points represent the waves which come from them. And one must imagine the same about every point of the surface and of the part within the flame." (Huygens, p. 17).
Huygens' candle flame produces propagating optical spherical waves, from points A, B and C, by the motion of an optical ether, composed of matter, that does not physically exist (vacuum). Huygens continues to ignore the fact that light propagates through in vacuum that is void of matter by describing, in detail, the "agitation" of the ether particles that form Huygens' light waves.
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Huygens represents the propagation of light with expanding partial waves that are used to construct the wave DCF (fig 4).
"There is the further consideration in the emanation of these waves, that each particle of matter in which a wave spreads, ought not to communicate its motion only to the next particle which is in the straight line drawn from the luminous point, but that it also imparts some of it necessarily to all the others which touch it and which oppose themselves to its movement. So it arises that around each particle there is made a wave of which that particle is the centre. Thus if DCF is a wave emanating from the luminous point A, which is its centre, the particle B, one of those comprised within the sphere DCF, will have made its particular or partial wave KCL, which will touch the wave DCF at C at the same moment that the principal wave emanating from the point A has arrived at DCF; and it is clear that it will be only the region C of the wave KCL which will touch the wave DCF, to wit, that which is in the straight line drawn through AB. Similarly the other particles of the sphere DCF, such as bb, dd, etc., will each make its own wave. But each of these waves can be infinitely feeble only as compared with the wave DCF, to the composition of which all the others contribute by the part of their surface which is most distant from the centre A." (Huygens, p. 19).
Huygens' expanding partial waves KCL originate from points b, b, b, along the wave HI. The far points C, C, C, of the expanding partial waves KCL, are used to construct the wave DCF which represents Huygens' propagation mechanism of light but Huygens' wave HI is arbitrary creating energy (partial waves), from points b, b, b, along the wave HI, away from the light source, which violates energy conservation. In addition, only the far points C, C, C, of the partial waves KCL, are used to construct the wave DCF. For every partial wave used to construct the wave DCF, the partial waves' structures between K and C, and between C and L, are destroyed, after the wave DCF is constructed. An enormous amount of energy (partial waves) is created then destroyed in Huygens' propagation mechanism of light.
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Huygens describes a light beam using partial waves but dispersing partial waves would form outside the light beam and would not allow for the formation of a light beam, using Huygens' propagation mechanism (fig 4).
"To come to the properties of Light. We remark first that each portion of a wave ought to spread in such a way that its extremities lie always between the same straight lines drawn from the luminous point. Thus the portion BG of the wave, having the luminous point A as its centre, will spread into the arc CE bounded by the straight lines ABC, AGE. For although the particular waves produced by the particles comprised within the space CAE spread also outside this space, they yet do not concur at the same instant to compose a wave which terminates the movement, as they do precisely at the circumference CE, which is their common tangent.
And hence one sees the reason why light, at least if its rays are not reflected or broken, spreads only by straight lines, so that it illuminates no object except when the path from its source to that object is open along such lines." (Huygens, p. 21).
According to Huygens propagation mechanism of light (fig 4), it would not be physically possible to form a light beam since a light ray has distinct borders that bounds the light intensity but in Huygens's propagation mechanism, Huygens' partial waves are expanding. The expanding partial waves are used to form the wave DCF which represents the propagation of light and the formation of a light beam; therefore, the expansion of the partial waves is an essential component of Huygens' propagation mechanism of light but at the edges of the wave DCF, next to the boundary lines ABC and AGE, the partial waves have a substantial amplitude and cannot be eliminated; consequently, the unwanted partial waves' structures would produce an intensity outside the boundaries of a light beam which would not allow the formation of a light beam. Huygens arbitrarily eliminates the partial waves structures, at the boundary lines ABC and AGE, in the formation of a light beam, which violates energy conservation.
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Huygens describes the transmission and reflection effects of light (fig 5 & 6) using spherical waves generated by the transmission and reflection surface.
"If one considers further the other pieces H of the wave AC, it appears that they will not only have reached the surface AB by straight lines HK parallel to CB, but that in addition they will have generated in the transparent air, from the centres K, K, K, particular spherical waves, represented here by circumferences the semi-diameters of which are equal to KM, that is to say to the continuations of HK as far as the line BG parallel to AC." (Huygens, p. 24).
Huygens' spherical waves originate from points K, K, K, along the transmission and reflection surface AB. The generation of spherical waves, by the transmission and reflection surface, represents the arbitrary creation of energy (spherical waves), away from the physical light source. A candle flame, light bulb or burning piece of coal represent a physical light source; the representation of the transmission and reflection surface that is generating spherical waves implies that the transmission and reflection surface is a physical source which violates energy conservation. Furthermore, the spherical waves, that originate from the transmission and reflection surface, form inconsistent amplitudes when the points, of the spherical waves, are used to construct the transmission and reflection waves since the spherical waves that are used to construct the transmission and reflection waves have varying circumferences because every spherical wave is propagating a different distance to reach the transmission and reflection wave; therefore, when the spherical waves are used to construct the transmission and reflection waves (fig 5 & 6) inconsistent amplitudes would form along the transmission and reflection waves which conflicts with Huygens' propagation mechanism where the partial wave KCL that are used to form the wave DCF, all have identical circumferences. Huygens' transmission and reflection mechanism conflicts with Huygens propagation mechanism of light, using expanding partial waves (fig 4). The distances from the transmission and reflection surface where the spherical originates and the points along the transmission and reflection wave represent different distances; henceforth, the points, of the spherical waves, would arrive at the transmission and reflection waves at different times which would not allow for the formation of the transmission and reflection waves, using Huygens expanding (propagating) spherical waves.
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§ 3. Fresnel
In Fresnel's paper, "Memorie su la Diffraction de la Lumiere" (1819), Fresnel describes diffraction using interfering light waves formed by the vibration of the elastic fluid.
"21. If we call λ the length of a light-wave, that is to say, the distance between two points in the ether where vibrations of the same kind are occurring at the same time" (Fresnel, § 21).
"Admitting that light consists in vibrations of the ether similar to sound-waves, we can easily account for the inflection of rays of light at sensible distances from the diffraction body." (Fresnel, § 33).
"To understand how a single luminous particle may perform a large series of oscillations all of which are nearly equal, we have only to imagine that its density is much greater than that of the fluid in which it vibrates---and, indeed, this is only what has already been inferred from the uniformity of the motions of the planets through this same fluid which fills planetary space." (Fresnel, § 33).
"APPLICATIONS OF HUYGENS'S PRINCIPLE TO THE PHENOMENA OF DIFFRACTION
43. Having determined the resultant of any number of trains of light-waves. I shall now show how by the aid of these interference formulae and by the principle of Huygens alone it is possible to explain, and even to compute, all the phenomena of diffraction. This principle, which I consider as a rigorous deduction from the basal hypothesis, may be expressed thus: The vibrations at each point in the wave-front may be considered as the sum of the elementary motions which at any one instant are sent to that point from all parts of this same wave in any one of its pervious* positions, each of these parts acting independently the one of the other. It follows from the principle of the superposition of small motions that the vibrations produced at any point in an elastic fluid" (Fresnel, § 43).
Fresnel's diffraction mechanism is based on light waves formed by the motion of an optical ether, composed of matter, based on a sound wave analogy where sound waves are formed by the motion of air molecules. Fresnel describes the ether with a elastic fluid and states that this fluid exists in celestial space but the vacuum of celestial space is void of a fluid, composed of matter. In addition, the motion of Fresnel's elastic fluid (ether) forms interfering light waves that produce the diffraction effect of light, based on Huygens' principle yet diffraction forms in vacuum that is void of an elastic fluid (ether), composed of matter, which is experimental proof Fresnel's diffraction mechanism of light is physically invalid. In addition, Huygens's principle represents the propagation of light using expanding partial waves formed by the wave HI but Huygens' partial wave propagation mechanism of light is not an interference effect yet Fresnel's diffraction mechanism is based on Huygens' principle. Huygens' principle and Fresnel's diffraction mechanism are incompatible since according to Fresnel, Huygens' partial waves KCL (fig 3) would interfere and form an interference effect yet light does not always produce an interference effect.
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Fresnel describes diffraction using interfering light waves that resultant amplitudes are used to form the intensity and dark fringes of the diffraction pattern (fig 7).
"In order to compute the total effect, I refer these partial resultants to the wave emitted by the point M on the straight line CP, and to another wave displaced a quarter of a wave-length with reference to the preceding. This is the process already employed (p. 101) in the general solution of the interference problem. We shall consider only a section of the wave made by the plane perpendicular to the edge of the screen, and shall indicate by dz an element, nn', of the primary wave, and by z its distance from the point M. These, as I have shown, suffice to determine the position and the relative intensities of the bright and dark bands." (Fresnel, § 53).
Fresnel's light wave AMI forms expanding secondary waves, at the diffraction object. The expanding secondary light waves propagate to the diffraction screen and interfere, forming the diffraction pattern. The formation of expanding secondary light waves, from points along the wave AMI, represents the arbitrary generation of energy, away from the light source, which violates energy conservation. In addition, Fresnel fails to explain how the wave AMI produces the secondary waves at the diffraction object. The Sun, a light bulb or a candle flame are physical light source that light emission is represented with spherical light waves. The formation of Fresnel's expanding secondary wave is representing the wave AMI as a physical light source that is generating energy, away from the physical light source. A wave that originates from a single point and expands represents light that is originating from a physical light source; consequently, Fresnel's diffraction mechanism that is based on the light wave AMI forming secondary waves is physically invalid since Fresnel's light wave AMI is creating its own self-energy in the form of expanding secondary light waves which violates energy conservation.
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Fresnel derives a diffraction intensity equation by summating the interfering light waves' amplitudes, at the diffraction screen, using a line integral (equ 1).
"Hence the intensity of the vibration at P resulting from all these small disturbances is
{ [ ʃ dz cos (π z2 (a + b) / abλ) ]2 + [ ʃ dz sin (π z2 (a + b) / abλ)]2 }1/2 "..................................1
(Fresnel, § 53). Fresnel is using a line integral to summate the interfering light waves' amplitudes, at the diffraction screen, but a line integral represents a length. Fresnel is violating the definition of a line integral in the derivation of the diffraction intensity equations of light (equ 1) which proves Fresnel's derivation is mathematically invalid. Furthermore, Fresnel's integration (equ 1) is used to summate the light waves' amplitudes, at a point P on the diffraction screen, to represent the intensity and dark fringes of the diffraction pattern yet the formation of the intensity fringes, of the diffraction pattern, at point P, by the light waves' amplitudes represents a light energy that is dependent on the wave amplitude which conflicts with Lenard's photoelectric effect that proves light is composed of particles that energy is dependent on only the frequency (Lenard, Intro).
The formation of the small circular aperture's diffraction pattern (fig 8) is represented using wave interference but the destructive interference of Fresnel's light waves' amplitudes, to form the dark fringes of the diffraction pattern, would result in a measurable reduction in the total light intensity of the diffraction pattern since the destroyed light waves' amplitudes (intensities) do not contribute to the total light intensity of the diffraction pattern yet, experimentally, more than 10%, of the diffraction pattern is composed of dark areas which would result in at least a 10% reduction of the total light intensity of the diffraction pattern yet the total light intensity, that enters a small circular aperture (dt = 1s), equals the total light intensity that forms the diffraction pattern (dt = 1s) which is experimental proof the aperture diffraction effect of light is not formed by Fresnel's wave interference mechanism. In addition, Fresnel's light waves' amplitudes, that produce the intensity (energy) of the diffraction pattern, represent a light energy that is dependent on the wave amplitude which conflicts with Lenard's photoelectric effect that proves light is composed of particles that energy is dependent on only the frequency (Lenard, Intro).
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§ 4. Faraday
In Faraday's paper, "Experimental Researches In Electricity" (First Series, 1831) , Faraday describes induction formed by two wire helix and battery.
6. About twenty-six feet of copper wire one twentieth of an inch in diameter were wound round a cylinder of wood as a helix, the different spires of which were prevented from touching by a thin interposed twine. This helix was covered with calico, and then a second wire applied in the same manner. In this way twelve helices were superposed, each containing an average length of wire of twenty-seven feet, and all in the same direction. The first, third, fifth, seventh, ninth, and eleventh of these helices were connected at their extremities end to end, so as to form one helix; the others were connected in a similar manner; and thus two principal helices were produced, closely interposed, having the same direction, not touching anywhere, and each containing one hundred and fifty-five feet in length of wire.
7. One of these helices was connected with a galvanometer, the other with a voltaic battery of ten pairs of plates four inches square, with double coppers and well charged; yet not the slightest sensible reflection of the galvanometer-needle could be observed.
8. A similar compound helix, consisting of six lengths of copper and six of soft iron wire, was constructed. The resulting iron helix contained two hundred and fourteen feet of wire, the resulting copper helix two hundred and eight feet; but whether the current from the trough was passed through the copper or the iron helix, no effect upon the other could be perceived at the galvanometer.
9. In these and many similar experiments no difference in action of any kind appeared between iron and other metals.
10. Two hundred and three feet of copper wire in one length were coiled round a large block of wood; other two hundred and three feet of similar wire were interposed as a spiral between the turns of the first coil, and metallic contact everywhere prevented by twine. One of these helices was connected with a galvanometer, and the other with a battery of one hundred pairs of plates four inches square, with double coppers, and well charged. When the contact was made, there was a sudden and very slight effect at the galvanometer, and there was also a similar slight effect when the contact with the battery was broken. But whilst the voltaic current was continuing to pass through the one helix, no galvanometrical appearances nor any effect like induction upon the other helix could be perceived, although the active power of the battery was proved to be great, by its heating the whole of its own helix, and by the brilliancy of the discharge when made through charcoal.
11. Repetition of the experiments with a battery of one hundred and twenty pairs of plates produced no other effects; but it was ascertained, both at this and the former time, that the slight deflection of the needle occurring at the moment of completing the connexion, was always in one direction, and that the equally slight deflection produced when the contact was broken, was in the other direction; and also, that these effects occurred when the first helices were used (6. 8.).
12. The results which I had by this time obtained with magnets led me to believe that the battery current through one wire, did, in reality, induce a similar current through the other wire, but that it continued for an instant only, and partook more of the nature of the electrical wave passed through from the shock of a common Leyden jar than of the current from a voltaic battery, and therefore might magnetise a steel needle, although it scarcely affected the galvanometer.
There is evidence and precedence that Lenz (1830) discovered the circuital induction effect using the wire helix but Lenz's paper was initially rejected until after Faraday published his First Series in "Experimental Researches In Electricity" (1831) that describe the circuital induction effect using the wire helix.
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Faraday's electric current is formed within the conductor of the helix wire.
"§ 1. Induction of Electric Currents.
6. About twenty-six feet of copper wire one twentieth of an inch in diameter were wound round a cylinder of wood as a helix, the different spires of which were prevented from touching by a thin interposed twine. This helix was covered with calico, and then a second wire applied in the same manner. In this way twelve helices were superposed, each containing an average length of wire of twenty-seven feet, and all in the same direction. The first, third, fifth, seventh, ninth, and eleventh of these helices were connected at their extremities end to end, so as to form one helix; the others were connected in a similar manner; and thus two principal helices were produced, closely interposed, having the same direction, not touching anywhere, and each containing one hundred and fifty-five feet in length of wire." (Faraday, sec 6).
"1048. The following investigations relate to a very remarkable inductive action of electric currents, or of the different parts of the same current (74.), and indicate an immediate connexion between such inductive action and the direct transmission of electricity through conducting bodies, or even that exhibited in the form of a spark." Faraday, sec 1048).
"1101. As an electric current acts by induction with equal energy at the moment of its commencement as at the moment of its cessation (10. 26.), but in a contrary direction, the reference of the effects under examination to an inductive action, would lead to the conclusion that corresponding effects of an opposite nature must occur in a long wire, a helix, or an electro-magnet, every time that contact is made with the electromotor. These effects will tend to establish a resistance for the first moment in the long conductor, producing a result equivalent to the reverse of a shock or a spark. Now it is very difficult to devise means fit for the recognition of such negative results; but as it is probable that some positive effect is produced at the time, if we knew what to expect, I think the few facts bearing upon this subject with which I am acquainted are worth recording." (Faraday, sec 1101).
"§ 19. Nature of the electric current.
1617. The word current is so expressive in common language, that when applied in the consideration of electrical phenomena we can hardly divest it sufficiently of its meaning, or prevent our minds from being prejudiced by it (283. 511.). I shall use it in its common electrical sense, namely, to express generally a certain condition and relation of electrical forces supposed to be in progression.
1618. A current is produced both by excitement and discharge; and whatsoever the variation of the two general causes may be, the effect remains the same. Thus excitement may occur in many ways, as by friction, chemical action, influence of heat, change of condition, induction, &c.; and discharge has the forms of conduction, electrolyzation, disruptive discharge, and convection; yet the current connected with these actions, when it occurs, appears in all cases to be the same. This constancy in the character of the current, notwithstanding the particular and great variations which may be made in the mode of its occurrence, is exceedingly striking and important; and its inv