Planck units
In particle physics and physical cosmology, Planck units are a set of units of measurement defined exclusively in terms of five universal physical constants, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units.
Originally proposed in 1899 by German physicist Max Planck, these units are also known as natural units because the origin of their definition comes only from properties of nature and not from any human construct. Planck units are only one system of several systems of natural units, but Planck units are not based on properties of any prototype object or particle (that would be arbitrarily chosen), but rather on only the properties of free space. They are relevant in research on unified theories such as quantum gravity.
The term Planck scale refers to the magnitudes of space, time, energy and other units, below which (or beyond which) the predictions of the Standard Model, quantum field theory and general relativity are no longer reconcilable, and quantum effects of gravity are expected to dominate. This region may be characterized by energies around 1.22×10^{19} GeV (the energyequivalent of the Planck mass), time intervals around 5.39×10^{−44} s (the Planck time) and lengths around 1.62×10^{−35} m (the Planck length). At the Planck scale, current models are not expected to be a useful guide to the cosmos, and physicists have no scientific model to suggest how the physical universe behaves. The best known example is represented by the conditions in the first 10^{−43} seconds of our universe after the Big Bang, approximately 13.8 billion years ago.
The five universal constants that Planck units, by definition, normalize to 1 are:
 the speed of light in a vacuum, c,
 the gravitational constant, G,
 the reduced Planck constant, ħ,
 the Boltzmann constant, k_{B}
 the Coulomb constant, k_{e} = 1/4πε_{0}
Each of these constants can be associated with a fundamental physical theory or concept: c with special relativity, G with general relativity, ħ with quantum mechanics, k_{B} with thermodynamics, and ε_{0} with electromagnetism.
IntroductionEdit
Any system of measurement may be assigned a mutually independent set of base quantities and associated base units, from which all other quantities and units may be derived. In the International System of Units, for example, the SI base quantities include length with the associated unit of the metre. In the system of Planck units, a similar set of base quantities may be selected, and the Planck base unit of length is then known simply as the Planck length, the base unit of time is the Planck time, and so on. These units are derived from the five dimensional universal physical constants of Table 1, in such a manner that these constants are eliminated from fundamental selected equations of physical law when physical quantities are expressed in terms of Planck units. For example, Newton's law of universal gravitation,
can be expressed as:
Both equations are dimensionally consistent and equally valid in any system of units, but the second equation, with G missing, is relating only dimensionless quantities since any ratio of two likedimensioned quantities is a dimensionless quantity. If, by a shorthand convention, it is understood that all physical quantities are expressed in terms of Planck units, the ratios above may be expressed simply with the symbols of physical quantity, without being scaled explicitly by their corresponding unit:
This last equation (without G) is valid only if F, m_{1}, m_{2}, and r are the dimensionless numerical values of these quantities measured in terms of Planck units. This is why Planck units or any other use of natural units should be employed with care. Referring to G = c = 1, Paul S. Wesson wrote that, "Mathematically it is an acceptable trick which saves labour. Physically it represents a loss of information and can lead to confusion."^{[1]}
DefinitionEdit
Constant  Symbol  Dimension in SI Quantities  Value (SI units) 

Speed of light in vacuum  c  L T^{−1}  299792458 m/s^{[2]} (exact by definition) 
Gravitational constant  G  L^{3} M^{−1} T^{−2}  6.67430(15)×10^{−11} m^{3}⋅kg^{−1}⋅s^{−2}^{[3]} 
Reduced Planck constant  ħ = h/2π where h is the Planck constant 
L^{2} M T^{−1}  1.054571817...×10^{−34} J⋅s^{[4]} (defined as 6.62607015×10^{−34} J⋅s/2π exactly) 
Boltzmann constant  k_{B}  L^{2} M T^{−2} Θ^{−1}  1.380649×10^{−23} J⋅K^{−1}^{[5]} (exact by definition) 
Coulomb constant  k_{e} = 1/4πε_{0} where ε_{0} is the permittivity of free space 
L^{3} M T^{−2} Q^{−2}  8.9875517923(14)×10^{9} kg⋅m^{3}⋅s^{−4}⋅A^{−2}^{[6]} 
Key: L = length, M = mass, T = time, Q = electric charge, Θ = temperature.
A property of Planck units is that in order to obtain the value of any of the physical constants above it is enough to replace the dimensions of the constant with the corresponding Planck units. For example, the gravitational constant (G) has as dimensions L^{3} M^{−1} T^{−2}. By replacing each dimension with the value of each corresponding Planck unit one obtains the value of (1 l_{P})^{3} × (1 m_{P})^{−1} × (1 t_{P})^{−2} = (1.616255×10^{−35} m)^{3} × (2.176435×10^{−8} kg)^{−1} × (5.391247×10^{−44} s)^{−2} = 6.674...×10^{−11} m^{3} kg^{−1} s^{−2} (which is the value of G).
This is the consequence of the fact that the system is internally coherent. For example, the gravitational attractive force of two bodies of 1 Planck mass each, set apart by 1 Planck length is 1 coherent Planck unit of force. Likewise, the distance traveled by light during 1 Planck time is 1 Planck length.
To determine, in terms of SI or another existing system of units, the quantitative values of the five base Planck units, those two equations and three others must be satisfied:
Solving the five equations above for the five unknowns results in a unique set of values for the five base Planck units:
Name  Dimension  Expression  Value (SI units) 

Planck length  Length (L)  1.616255(18)×10^{−35} m^{[7]}  
Planck mass  Mass (M)  2.176435(24)×10^{−8} kg^{[8]}  
Planck time  Time (T)  5.391247(60)×10^{−44} s^{[9]}  
Planck temperature  Temperature (Θ)  1.416785(16)×10^{32} K^{[10]}  
Planck charge  Electric charge (Q)  1.875545956(41)×10^{−18} C^{[11]}^{[4]}^{[2]} 
Table 2 clearly defines Planck units in terms of the fundamental constants. Yet relative to other units of measurement such as SI, the values of the Planck units are only known approximately. This is due to uncertainty in the values of the gravitational constant G and ε_{0} in SI units.
The values of c, h, e and k_{B} in SI units are exact due to the definition of the second, metre, kilogram and kelvin in terms of these constants, and contribute no uncertainty to the values of the Planck units expressed in terms of SI units. The vacuum permittivity ε_{0} has a relative uncertainty of 1.5×10^{−10}.^{[11]} The numerical value of G has been determined experimentally to a relative uncertainty of 2.2×10^{−5}.^{[3]} G appears in the definition of every Planck unit other than for charge in Tables 2 and 3. Hence the uncertainty in the values of the Table 2 and 3 SI equivalents of the Planck units derives almost entirely from uncertainty in the value of G. (The propagation of the error in G is a function of the exponent of G in the algebraic expression for a unit. Since that exponent is ±1/2 for every base unit other than Planck charge, the relative uncertainty of each base unit is about one half that of G.)
Although the values of the single units can be known only with some uncertainty, one of the consequences of the definitions of c h and k_{B} in SI units is that one Planck mass multiplied by one Planck length is equal exactly to 1 l_{P} × 1 m_{P} = ħ/c = 6.62607015×10^{−34}/2π × 299792458 m⋅kg, while one Planck mass divided by one Planck temperature is equal exactly to 1 m_{P}/1 T_{P} = k_{B}/c^{2} = 1.380649×10^{−23}/299792458^{2} kg/K, and finally one Planck length divided by one Planck time is equal exactly to 1 l_{P}/1 t_{P} = c = 299792458 m/s. As for the gravitational constant and the Coulomb constant instead, although their value is not exact by definition in SI units and must be measured experimentally, the attractive gravitational force F that two Planck masses placed at a distance r exert on each other is equal to the attractive/repulsive electrostatic force between two Planck charges placed at the same distance, which is equal exactly to F = ħc/r^{2} = 6.62607015×10^{−34} × 299792458/2π r^{2} N.
Derived unitsEdit
In any system of measurement, units for many physical quantities can be derived from base units. Table 3 offers a sample of derived Planck units, some of which in fact are seldom used. As with the base units, their use is mostly confined to theoretical physics because most of them are too large or too small for empirical or practical use and there are large uncertainties in their values.
Derived unit of  Expression  Approximate SI equivalent 

area (L^{2})  2.6121×10^{−70} m^{2}  
volume (L^{3})  4.2217×10^{−105} m^{3}  
momentum (LMT^{−1})  6.5249 kg⋅m/s  
energy (L^{2}MT^{−2})  1.9561×10^{9} J  
force (LMT^{−2})  1.2103×10^{44} N  
density (L^{−3}M)  5.1550×10^{96} kg/m^{3}  
acceleration (LT^{−2})  5.5608×10^{51} m/s^{2}  
frequency (T^{−1})  1.8549×10^{43} Hz 
Most Planck units are many orders of magnitude too large or too small to be of practical use, so that Planck units as a system are typically only relevant to theoretical physics. In fact, 1 Planck unit is often the largest or smallest value of a physical quantity that makes sense within presentday theories of physics. For example, our understanding of the Big Bang begins with the Planck epoch, when the universe was 1 Planck time old and 1 Planck length in diameter. Understanding the universe when it was less than 1 Planck time old requires a theory of quantum gravity that would incorporate quantum effects into general relativity. Such a theory does not yet exist.
An exception to the general pattern of Planck units being "extreme" in magnitude is the Planck mass, which is about 22 micrograms: very large compared to subatomic particles, but well within the mass range of living things.
HistoryEdit
The concept of natural units was introduced in 1881, when George Johnstone Stoney, noting that electric charge is quantized, derived units of length, time, and mass, now named Stoney units in his honor, by normalizing G, c, and the electron charge, e, to 1.
In 1899 (one year before the advent of quantum theory), Max Planck introduced what became later known as the Planck constant.^{[12]}^{[13]} At the end of the paper, Planck proposed, as a consequence of his discovery, the base units later named in his honor. The Planck units are based on the quantum of action, now usually known as the Planck constant. Planck called the constant b in his paper, though h (or the closely related ħ) is now common. However, at that time it was part of Wien's radiation law, which Planck thought to be correct. Planck underlined the universality of the new unit system, writing:
... die Möglichkeit gegebenist, Einheiten für Länge, Masse, Zeit und Temperatur aufzustellen, welche, unabhängig von speciellen Körpern oder Substanzen, ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Culturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können.
... it is possible to set up units for length, mass, time and temperature, which are independent of special bodies or substances, necessarily retaining their meaning for all times and for all civilizations, including extraterrestrial and nonhuman ones, which can be called "natural units of measure".
Planck considered only the units based on the universal constants G, ħ, c, and k_{B} to arrive at natural units for length, time, mass, and temperature.^{[13]} Planck's paper also gave numerical values for the base units that were close to modern values.
The original base units proposed by Planck in 1899 differed by a factor of from the Planck units in use today.^{[12]}^{[13]} This is due to the use of the reduced Planck constant ( ) in the modern units, which did not appear in the original proposal.
Name  Dimension  Expression  Value in SI units  Value in modern Planck units 

Original Planck length  Length (L)  4.05135×10^{−35} m  
Original Planck mass  Mass (M)  5.45551×10^{−8} kg  
Original Planck time  Time (T)  1.35138×10^{−43} s  
Original Planck temperature  Temperature (Θ)  3.55135×10^{32} K 
Planck did not adopt any electromagnetic units. However, since the spirit of the system is that of setting all constants to 1, the scientific community has gradually adopted the common habit of setting the Coulomb constant to 1 as well and include the electric charge among the Planck base units.^{[14]}^{[15]}^{[16]}^{[17]}^{[18]}^{[19]}^{[20]}^{[21]} Setting the Coulomb constant to 1 yields for the charge a value identical to the charge unit used in QCD units. Depending on the focus, however, other physicists have kept a more minimalist approach and have referred to the Planck units mentioning only length, mass and time.^{[22]}
A 2006 internal proposal of the SI Working Group of fixing the Planck charge instead of the elementary charge (since "fixing q_{P} would have kept μ_{0} at its familiar value of 4π × 10^{−7} H/m and made e dependent on measurements of α") was rejected, and instead the value of the elementary charge was chosen to be fixed by definition.^{[23]} Currently to calculate the Planck charge it is necessary to use the elementary charge (whose value is currently exact by definition) and the finestructure constant (whose value needs to be measured and is susceptible to measurement errors).
SignificanceEdit
Planck units have little anthropocentric arbitrariness, but do still involve some arbitrary choices in terms of the defining constants. Unlike the metre and second, which exist as base units in the SI system for historical reasons, the Planck length and Planck time are conceptually linked at a fundamental physical level. Consequently, natural units help physicists to reframe questions. Frank Wilczek puts it succinctly:
We see that the question [posed] is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in natural (Planck) units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number [1/(13 quintillion)].^{[24]}
While it is true that the electrostatic repulsive force between two protons (alone in free space) greatly exceeds the gravitational attractive force between the same two protons, this is not about the relative strengths of the two fundamental forces. From the point of view of Planck units, this is comparing apples to oranges, because mass and electric charge are incommensurable quantities. Rather, the disparity of magnitude of force is a manifestation of the fact that the charge on the protons is approximately the unit charge but the mass of the protons is far less than the unit mass.
Year  Quantity  Interpretation  Principal scientist 

1954^{[25]}  length  gravitational limit of quantum theory  Oskar Klein 
1955^{[26]}  length  quantum limit of general relativity  John Wheeler 
1965^{[27]}  mass  upper limit on the mass of elementary particles  Moisey Markov 
1966^{[28]}  temperature  upper limit of temperature (absolute hot)  Andrei Sakharov 
1971^{[29]}  mass  lower limit on the mass of a black hole  Stephen Hawking 
1982^{[30]}  density  limiting density of matter  Moisey Markov 
Planck scaleEdit
In particle physics and physical cosmology, the Planck scale is an energy scale around 1.22 × 10^{19} GeV (the Planck energy, corresponding to the mass–energy equivalence of the Planck mass, 2.17645 × 10^{−8} kg) at which quantum effects of gravity become strong. At this scale, present descriptions and theories of subatomic particle interactions in terms of quantum field theory break down and become inadequate, due to the impact of the apparent nonrenormalizability of gravity within current theories.
Relationship to gravityEdit
At the Planck length scale, the strength of gravity is expected to become comparable with the other forces, and it is theorized that all the fundamental forces are unified at that scale, but the exact mechanism of this unification remains unknown. The Planck scale is therefore the point where the effects of quantum gravity can no longer be ignored in other fundamental interactions, and where current calculations and approaches begin to break down, and a means to take account of its impact is required.^{[31]}^{[32]}
While physicists have a fairly good understanding of the other fundamental interactions of forces on the quantum level, gravity is problematic, and cannot be integrated with quantum mechanics at very high energies using the usual framework of quantum field theory. At lesser energy levels it is usually ignored, while for energies approaching or exceeding the Planck scale, a new theory of quantum gravity is required. Other approaches to this problem include string theory and Mtheory, loop quantum gravity, noncommutative geometry, scale relativity, causal set theory and Padic quantum mechanics.^{[33]}
In cosmologyEdit
In Big Bang cosmology, the Planck epoch or Planck era is the earliest stage of the Big Bang, before the time passed was equal to the Planck time, t_{P}, or approximately 10^{−43} seconds.^{[34]} There is no currently available physical theory to describe such short times, and it is not clear in what sense the concept of time is meaningful for values smaller than the Planck time. It is generally assumed that quantum effects of gravity dominate physical interactions at this time scale. At this scale, the unified force of the Standard Model is assumed to be unified with gravitation. Immeasurably hot and dense, the state of the Planck epoch was succeeded by the grand unification epoch, where gravitation is separated from the unified force of the Standard Model, in turn followed by the inflationary epoch, which ended after about 10^{−32} seconds (or about 10^{10} t_{P}).^{[35]}
Relative to the Planck epoch, the observable universe today looks extreme when expressed in Planck units, as in this set of approximations:^{[36]}^{[37]}
Property of presentday Observable Universe 
Approximate number of Planck units 
Equivalents 

Age  8.08 × 10^{60} t_{P}  4.35 × 10^{17} s, or 13.8 × 10^{9} years 
Diameter  5.4 × 10^{61} l_{P}  8.7 × 10^{26} m or 9.2 × 10^{10} lightyears 
Mass  approx. 10^{60} m_{P}  3 × 10^{52} kg or 1.5 × 10^{22} solar masses (only counting stars) 10^{80} protons (sometimes known as the Eddington number) 
Density  1.8 × 10^{−123} ρ_{P}  9.9 × 10^{−27} kg m^{−3} 
Temperature  1.9 × 10^{−32} T_{P}  2.725 K temperature of the cosmic microwave background radiation 
Cosmological constant  5.6 × 10^{−122} t^{ −2} _{P} 
1.9 × 10^{−35} s^{−2} 
Hubble constant  1.18 × 10^{−61} t^{ −1} _{P} 
2.2 × 10^{−18} s^{−1} or 67.8 (km/s)/Mpc 
The recurrence of large numbers close or related to 10^{60} in the above table is a coincidence that intrigues some theorists. It is an example of the kind of large numbers coincidence that led theorists such as Eddington and Dirac to develop alternative physical hypotheses (e.g. a variable speed of light or Dirac varyingG hypothesis).^{[38]} After the measurement of the cosmological constant in 1998, estimated at 10^{−122} in Planck units, it was noted that this is suggestively close to the reciprocal of the age of the universe squared.^{[39]} Barrow and Shaw (2011) proposed a modified theory in which Λ is a field evolving in such a way that its value remains Λ ~ T^{−2} throughout the history of the universe.^{[40]}
Other usesEdit
The Planck length is related to Planck energy by the uncertainty principle. At this scale, the concepts of size and distance break down, as quantum indeterminacy becomes virtually absolute. Because the Schwarzschild radius of a black hole is roughly equal to the Compton wavelength at the Planck scale, a photon with sufficient energy to probe this realm would yield no information whatsoever. Any photon energetic enough to precisely measure a Plancksized object could actually create a particle of that dimension, but it would be massive enough to immediately become a black hole (a.k.a. Planck particle), thus completely distorting that region of space, and swallowing the photon. This is the most extreme example possible of the uncertainty principle, and explains why only a quantum gravity theory reconciling general relativity with quantum mechanics will allow us to understand the dynamics of spacetime at this scale. Planck scale dynamics are important for cosmology because by tracing the evolution of the cosmos back to the very beginning, at some very early stage the universe should have been so hot that processes involving energies as high as the Planck energy (corresponding to distances as short as the Planck length) may have occurred. This period is therefore called the Planck era or Planck epoch.
Analysis of the unitsEdit
Planck areaEdit
Planck chargeEdit
Planck densityEdit
The Planck density is a very large unit, about equivalent to 10^{23} solar masses squeezed into the space of a single atomic nucleus. The Planck density is thought to be the upper limit of density.
Planck energyEdit
Most Planck units are extremely small, as in the case of Planck length or Planck time, or extremely large, as in the case of Planck temperature or Planck acceleration. For comparison, the Planck energy is approximately equal to the energy stored in an automobile gas tank (57.2 L of gasoline at 34.2 MJ/L of chemical energy). The ultrahighenergy cosmic ray observed in 1991 had a measured energy of about 50 joules, equivalent to about 2.5×10^{−8} E_{P}.^{[41]} Theoretically, the highest energy photon carries about 1 E_{P} of energy (see Ultrahighenergy gamma ray), after which it becomes indistinguishable from a Planck particle carrying the same energy.
Planck forceEdit
The Planck force is the derived unit of force resulting from the definition of the base Planck units for time, length, and mass. It is equal to the natural unit of momentum divided by the natural unit of time.
The Planck force is associated^{[42]} with the equivalence of gravitational potential energy and electromagnetic energy: the gravitational attractive force of two bodies of 1 Planck mass each, set apart by 1 Planck length is 1 Planck force; equivalently, the electrostatic attractive/repulsive force of two Planck charges set apart by 1 Planck length is 1 Planck force.
The Planck force appears in the Einstein field equations, describing the properties of a gravitational field surrounding any given mass:
where is the Einstein tensor and is the energy–momentum tensor. The Planck force thus describes how much or how easily spacetime is curved by a given amount of massenergy.
Since 1993, various authors (De Sabbata & Sivaram, Massa, Kostro & Lange, Gibbons, Schiller) have argued that the Planck force is the maximum force value that can be observed in nature. This limit property is valid both for gravitational force and for any other type of force.
Planck lengthEdit
The Planck length, denoted ℓ_{P}, is a unit of length that is the distance light in a perfect vaccum travels in one unit of Planck time. It is equal to 1.616255(18)×10^{−35} m.^{[7]}
Planck massEdit
Planck momentumEdit
The Planck momentum is equal to the Planck mass multiplied by the speed of light. Unlike most of the other Planck units, Planck momentum occurs on a human scale. By comparison, running with a fivepound object (10^{8} × Planck mass) at an average running speed (10^{−8} × speed of light in a vacuum) would give the object Planck momentum. A 70 kg human moving at an average walking speed of 1.4 m/s (5.0 km/h; 3.1 mph) would have a momentum of about 15 . A baseball, which has mass 0.145 kg, travelling at 45 m/s (160 km/h; 100 mph) would have a Planck momentum.
Planck temperatureEdit
The Planck temperature of 1 (unity), equal to 1.416785(16)×10^{32} K^{[10]}, is considered a fundamental limit of temperature.^{[43]} An object with the temperature of 1.42×10^{32} kelvin (T_{P}) would emit a black body radiation with a peak wavelength of 1.616×10^{−35} m (Planck length), where each photon and each individual collision would have the energy to create a Planck particle. There are no known physical models able to describe temperatures greater than or equal to T_{P}.
Planck timeEdit
A Planck time unit is the time required for light to travel a distance of 1 Planck length in a vacuum, which is a time interval of approximately 5.39 × 10^{−44} s.^{[44]} All scientific experiments and human experiences occur over time scales that are many orders of magnitude longer than the Planck time,^{[45]} making any events happening at the Planck scale undetectable with current scientific technology. As of November 2016^{[update]}, the smallest time interval uncertainty in direct measurements was on the order of 850 zeptoseconds (8.50 × 10^{−19} seconds).^{[46]}
While there is currently no known way to measure time intervals on the scale of the Planck time, researchers in 2020 proposed a theoretical apparatus and experiment that, if ever realized, could be capable of being influenced by effects on time as short as 10^{−33} seconds, thus establishing an upper detectable limit for the quantization of a time that is roughly 20 billion times longer than the Planck time.^{[47]}^{[48]}
List of physical equationsEdit
Physical quantities that have different dimensions (such as time and length) cannot be equated even if they are numerically equal (1 second is not the same as 1 metre). In theoretical physics, however, this scruple can be set aside, by a process called nondimensionalization. Table 7 shows how the use of Planck units simplifies many fundamental equations of physics, because this gives each of the five fundamental constants, and products of them, a simple numeric value of 1. In the SI form, the units should be accounted for. In the nondimensionalized form, the units, which are now Planck units, need not be written if their use is understood.
SI form  Planck units form  

Newton's law of universal gravitation  
Einstein field equations in general relativity  
Mass–energy equivalence in special relativity  
Energy–momentum relation  
Thermal energy per particle per degree of freedom  
Boltzmann's entropy formula  
Planck–Einstein relation for energy and angular frequency  
Planck's law (surface intensity per unit solid angle per unit angular frequency) for black body at temperature T.  
Stefan–Boltzmann constant σ defined  
Bekenstein–Hawking black hole entropy^{[49]}  
Schrödinger's equation  
Hamiltonian form of Schrödinger's equation  
Covariant form of the Dirac equation  
Unruh temperature  
Coulomb's law  
Maxwell's equations 


Ideal gas law 
As the Planck base units are derived from multidimensional constants, they can be also expressed as relationships between the latter and other base units.
Planck length (l_{P})  Planck mass (m_{P})  Planck time (t_{P})  Planck temperature (T_{P})  Planck charge (q_{P})  

Planck length (l_{P})  —  
Planck mass (m_{P})  —  
Planck time (t_{P})  —  
Planck temperature (T_{P})  —  
Planck charge (q_{P})  — 
Alternative choices of normalizationEdit
As already stated above, Planck units are derived by "normalizing" the numerical values of certain fundamental constants to 1. These normalizations are neither the only ones possible nor necessarily the best. Moreover, the choice of what factors to normalize, among the factors appearing in the fundamental equations of physics, is not evident, and the values of the Planck units are sensitive to this choice.
The factor 4π is ubiquitous in theoretical physics because the surface area of a sphere of radius r is 4πr^{2}. This, along with the concept of flux, are the basis for the inversesquare law, Gauss's law, and the divergence operator applied to flux density. For example, gravitational and electrostatic fields produced by point charges have spherical symmetry (Barrow 2002: 214–15). The 4πr^{2} appearing in the denominator of Coulomb's law in rationalized form, for example, follows from the flux of an electrostatic field being distributed uniformly on the surface of a sphere. Likewise for Newton's law of universal gravitation. (If space had more than three spatial dimensions, the factor 4π would have to be changed according to the geometry of the sphere in higher dimensions.)
Hence a substantial body of physical theory developed since Planck (1899) suggests normalizing not G but either 4πG (or 8πG or 16πG) to 1. Doing so would introduce a factor of 1/4π (or 1/8π or 1/16π) into the nondimensionalized form of the law of universal gravitation, consistent with the modern rationalized formulation of Coulomb's law in terms of the vacuum permittivity. In fact, alternative normalizations frequently preserve the factor of 1/4π in the nondimensionalized form of Coulomb's law as well, so that the nondimensionalized Maxwell's equations for electromagnetism and gravitoelectromagnetism both take the same form as those for electromagnetism in SI, which do not have any factors of 4π. When this is applied to electromagnetic constants, ε_{0}, this unit system is called "rationalized" Lorentz–Heaviside units. When applied additionally to gravitation and Planck units, these are called rationalized Planck units^{[51]} and are seen in highenergy physics.
The rationalized Planck units are defined so that .
There are several possible alternative normalizations.
Gravitational constantEdit
In 1899, Newton's law of universal gravitation was still seen as exact, rather than as a convenient approximation holding for "small" velocities and masses (the approximate nature of Newton's law was shown following the development of general relativity in 1915). Hence Planck normalized to 1 the gravitational constant G in Newton's law. In theories emerging after 1899, G nearly always appears in formulae multiplied by 4π or a small integer multiple thereof. Hence, a choice to be made when designing a system of natural units is which, if any, instances of 4π appearing in the equations of physics are to be eliminated via the normalization.
 Normalizing 4πG to 1 (and therefore setting G = 1/4π):
 Gauss's law for gravity becomes Φ_{g} = −M (rather than Φ_{g} = −4πM in Planck units).
 Eliminates 4πG from the Poisson equation.
 Eliminates 4πG in the gravitoelectromagnetic (GEM) equations, which hold in weak gravitational fields or locally flat spacetime. These equations have the same form as Maxwell's equations (and the Lorentz force equation) of electromagnetism, with mass density replacing charge density, and with 1/4πG replacing ε_{0}.
 Normalizes the characteristic impedance Z_{0} of gravitational radiation in free space to 1. (Normally expressed as 4πG/c)^{[note 1]}
 Eliminates 4πG from the Bekenstein–Hawking formula (for the entropy of a black hole in terms of its mass m_{BH} and the area of its event horizon A_{BH}) which is simplified to S_{BH} = πA_{BH} = (m_{BH})^{2}.
 Setting 8πG = 1 (and therefore setting G = 1/8π). This would eliminate 8πG from the Einstein field equations, Einstein–Hilbert action, and the Friedmann equations, for gravitation. Planck units modified so that 8πG = 1 are known as reduced Planck units, because the Planck mass is divided by √8π. Also, the Bekenstein–Hawking formula for the entropy of a black hole simplifies to S_{BH} = (m_{BH})^{2}/2 = 2πA_{BH}.
 Setting 16πG = 1 (and therefore setting G = 1/16π). This would eliminate the constant c^{4}/16πG from the Einstein–Hilbert action. The form of the Einstein field equations with cosmological constant Λ becomes R_{μν} + Λg_{μν} = 1/2(Rg_{μν} + T_{μν}).
Coulomb constantEdit
Planck units normalize to 1 the Coulomb force constant k_{e} = 1/4πε_{0} (as does the cgs system of units). This sets the derived unit of impedance, Z_{P} equal to Z_{0}/4π, where Z_{0} is the characteristic impedance of free space.
 Normalizing the permittivity of free space ε_{0} to 1 (and therefore setting k_{e} = 1/4π):
 Sets the permeability of free space μ_{0} = 1 (because c = 1).
 Sets the derived unit of impedance to the characteristic impedance of free space, Z_{P′} = Z_{0} (or sets the characteristic impedance of free space Z_{0} to 1).
 Eliminates 4π from the nondimensionalized form of Maxwell's equations.
 Eliminates ε_{0} from the nondimensionalized form of Coulomb's law, but has 4πr^{2} remaining in the denominator (which is the surface area of the enclosing sphere at radius r).
 Equates the notions of flux density and field strength in free space.
 In this case the elementary charge, measured in terms of this rationalized Planck charge, is
 where is the finestructure constant. This convention is seen in highenergy physics.
Boltzmann constantEdit
Planck normalized to 1 the Boltzmann constant k_{B}.
 Normalizing 1/2k_{B} to 1 (and therefore setting k_{B} = 2):
 Removes the factor of 1/2 in the nondimensionalized equation for the thermal energy per particle per degree of freedom.
 Introduces a factor of 2 into the nondimensionalized form of Boltzmann's entropy formula.
 Does not affect the value of any of the base or derived Planck units listed in Tables 3 and 4.
Reduced Planck constantEdit
Modern Planck units normalize to 1 the reduced Planck constant. This is the only constant in the system that affects all base units altogether in the same proportional way.
 Normalizing h (instead of ħ) to 1 (and therefore setting ħ = 1/2π):
 Restores the original form of the units as proposed by Max Planck (see § History)
 Multiplies all the Planck base units by √2π (i.e. all base units will be 2.5066 times larger)
 Normalizing αħ to 1 (and therefore setting ħ = 1/α):
 Sets the Planck charge equal to the elementary charge (q_{P} = e).
 Multiplies all the other Planck base units by √α (i.e. all base units will be 11.7 times smaller)
 Lets all fundamental constants with the exception of ħ maintain their unitary value (i.e. c = 1, G = 1, ħ = 1/α ≈ 137.036, k_{B} = 1, k_{e} = 1)
Planck units and the invariant scaling of natureEdit
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Some theorists (such as Dirac and Milne) have proposed cosmologies that conjecture that physical "constants" might actually change over time (e.g. a variable speed of light or Dirac varyingG theory). Such cosmologies have not gained mainstream acceptance and yet there is still considerable scientific interest in the possibility that physical "constants" might change, although such propositions introduce difficult questions. Perhaps the first question to address is: How would such a change make a noticeable operational difference in physical measurement or, more fundamentally, our perception of reality? If some particular physical constant had changed, how would we notice it, or how would physical reality be different? Which changed constants result in a meaningful and measurable difference in physical reality? If a physical constant that is not dimensionless, such as the speed of light, did in fact change, would we be able to notice it or measure it unambiguously? – a question examined by Michael Duff in his paper "Comment on timevariation of fundamental constants".^{[52]}^{[53]}
George Gamow argued in his book Mr Tompkins in Wonderland that a sufficient change in a dimensionful physical constant, such as the speed of light in a vacuum, would result in obvious perceptible changes. But this idea is challenged:
[An] important lesson we learn from the way that pure numbers like α define the world is what it really means for worlds to be different. The pure number we call the fine structure constant and denote by α is a combination of the electron charge, e, the speed of light, c, and Planck's constant, h. At first we might be tempted to think that a world in which the speed of light was slower would be a different world. But this would be a mistake. If c, h, and e were all changed so that the values they have in metric (or any other) units were different when we looked them up in our tables of physical constants, but the value of α remained the same, this new world would be observationally indistinguishable from our world. The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature. If all masses were doubled in value [including the Planck mass m_{P} ] you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged.
— Barrow 2002^{[36]}
Referring to Duff's "Comment on timevariation of fundamental constants"^{[52]} and Duff, Okun, and Veneziano's paper "Trialogue on the number of fundamental constants",^{[54]} particularly the section entitled "The operationally indistinguishable world of Mr. Tompkins", if all physical quantities (masses and other properties of particles) were expressed in terms of Planck units, those quantities would be dimensionless numbers (mass divided by the Planck mass, length divided by the Planck length, etc.) and the only quantities that we ultimately measure in physical experiments or in our perception of reality are dimensionless numbers. When one commonly measures a length with a ruler or tapemeasure, that person is actually counting tick marks on a given standard or is measuring the length relative to that given standard, which is a dimensionless value. It is no different for physical experiments, as all physical quantities are measured relative to some other likedimensioned quantity.
We can notice a difference if some dimensionless physical quantity such as finestructure constant, α, changes or the protontoelectron mass ratio, m_{p}/m_{e}, changes (atomic structures would change) but if all dimensionless physical quantities remained unchanged (this includes all possible ratios of identically dimensioned physical quantity), we cannot tell if a dimensionful quantity, such as the speed of light, c, has changed. And, indeed, the Tompkins concept becomes meaningless in our perception of reality if a dimensional quantity such as c has changed, even drastically.
If the speed of light c, were somehow suddenly cut in half and changed to 1/2c (but with the axiom that all dimensionless physical quantities remain the same), then the Planck length would increase by a factor of 2√2 from the point of view of some unaffected observer on the outside. Measured by "mortal" observers in terms of Planck units, the new speed of light would remain as 1 new Planck length per 1 new Planck time – which is no different from the old measurement. But, since by axiom, the size of atoms (approximately the Bohr radius) are related to the Planck length by an unchanging dimensionless constant of proportionality:
Then atoms would be bigger (in one dimension) by 2√2, each of us would be taller by 2√2, and so would our metre sticks be taller (and wider and thicker) by a factor of 2√2. Our perception of distance and lengths relative to the Planck length is, by axiom, an unchanging dimensionless constant.
Our clocks would tick slower by a factor of 4√2 (from the point of view of this unaffected observer on the outside) because the Planck time has increased by 4√2 but we would not know the difference (our perception of durations of time relative to the Planck time is, by axiom, an unchanging dimensionless constant). This hypothetical unaffected observer on the outside might observe that light now propagates at half the speed that it previously did (as well as all other observed velocities) but it would still travel 299792458 of our new metres in the time elapsed by one of our new seconds (1/2c × 4√2 ÷ 2√2 continues to equal 299792458 m/s). We would not notice any difference.
This contradicts what George Gamow writes in his book Mr. Tompkins; there, Gamow suggests that if a dimensiondependent universal constant such as c changed significantly, we would easily notice the difference. The disagreement is better thought of as the ambiguity in the phrase "changing a physical constant"; what would happen depends on whether (1) all other dimensionless constants were kept the same, or whether (2) all other dimensiondependent constants are kept the same. The second choice is a somewhat confusing possibility, since most of our units of measurement are defined in relation to the outcomes of physical experiments, and the experimental results depend on the constants. Gamow does not address this subtlety; the thought experiments he conducts in his popular works assume the second choice for "changing a physical constant". And Duff or Barrow would point out that ascribing a change in measurable reality, i.e. α, to a specific dimensional component quantity, such as c, is unjustified. The very same operational difference in measurement or perceived reality could just as well be caused by a change in h or e if α is changed and no other dimensionless constants are changed. It is only the dimensionless physical constants that ultimately matter in the definition of worlds.^{[52]}^{[55]}
This unvarying aspect of the Planckrelative scale, or that of any other system of natural units, leads many theorists to conclude that a hypothetical change in dimensionful physical constants can only be manifest as a change in dimensionless physical constants. One such dimensionless physical constant is the finestructure constant. There are some experimental physicists who assert they have in fact measured a change in the fine structure constant^{[56]} and this has intensified the debate about the measurement of physical constants. According to some theorists^{[57]} there are some very special circumstances in which changes in the finestructure constant can be measured as a change in dimensionful physical constants. Others however reject the possibility of measuring a change in dimensionful physical constants under any circumstance.^{[52]} The difficulty or even the impossibility of measuring changes in dimensionful physical constants has led some theorists to debate with each other whether or not a dimensionful physical constant has any practical significance at all and that in turn leads to questions about which dimensionful physical constants are meaningful.^{[54]}
See alsoEdit
NotesEdit
 ^ General relativity predicts that gravitational radiation propagates at the same speed as electromagnetic radiation.
ReferencesEdit
CitationsEdit
 ^ Wesson, P. S. (1980). "The application of dimensional analysis to cosmology". Space Science Reviews. 27 (2): 117. Bibcode:1980SSRv...27..109W. doi:10.1007/bf00212237.
 ^ ^{a} ^{b} "2018 CODATA Value: speed of light in vacuum". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
 ^ ^{a} ^{b} "2018 CODATA Value: Newtonian constant of gravitation". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
 ^ ^{a} ^{b} "2018 CODATA Value: reduced Planck constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 28 August 2019.
 ^ "2018 CODATA Value: Boltzmann constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
 ^ Derived from k_{e} = 1/(4πε_{0}) – "2018 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
 ^ ^{a} ^{b} "2018 CODATA Value: Planck length". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
 ^ "2018 CODATA Value: Planck mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
 ^ "2018 CODATA Value: Planck time". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
 ^ ^{a} ^{b} "2018 CODATA Value: Planck temperature". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
 ^ ^{a} ^{b} "2018 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
 ^ ^{a} ^{b} Planck (1899), p. 479.
 ^ ^{a} ^{b} ^{c} Tomilin, K. A. (1999). Natural Systems of Units. To the Centenary Anniversary of the Planck System (PDF). Proceedings Of The XXII Workshop On High Energy Physics And Field Theory. pp. 287–296.
 ^ Pavšic, Matej (2001). The Landscape of Theoretical Physics: A Global View. Fundamental Theories of Physics. 119. Dordrecht: Kluwer Academic. pp. 347–352. arXiv:grqc/0610061. doi:10.1007/0306471361. ISBN 9780792370062.
 ^ Zeidler, Eberhard (2006). Quantum Field Theory I: Basics in Mathematics and Physics (PDF). Springer. p. 953. ISBN 9783540347620.
 ^ Deza, Michel Marie; Deza, Elena (2016). Encyclopedia of Distances. Springer. p. 602. ISBN 9783662528433.
 ^ Newell, D. B.; Mohr, P. J.; Taylor, B. N. (12 May 2016), The New International System of Units: The Role of the Committee on Data for Science and Technology (CODATA), National Institute of Standards and Technology, doi:10.1080/19315775.2011.11721576
 ^ Gray, Reginald Irvan (1988). Unified Physics. Dahlgren, Virginia: Naval Surface Warfare Center. p. 339. LCCN 88602336.
 ^ Makela, Jarmo; Repo, Pasi (1998). "A Quantum Mechanical Model of the ReissnerNordstrom Black Hole". Physical Review D. 57: 4899–4916. arXiv:grqc/9708029. doi:10.1103/PhysRevD.57.4899.
 ^ Suhendro, Indranu (October 2007). "A New Conformal Theory of SemiClassical Quantum General Relativity" (PDF). Progress in Physics. 4 (2007): 96–103.
 ^ ^{a} ^{b} Elert, Glenn. "Blackbody Radiation". The Physics Hypertextbook.
 ^ Wilczek, Frank (2005). "On Absolute Units, I: Choices" (PDF). Physics Today. American Institute of Physics. 58 (10): 12–13. doi:10.1063/1.2138392.
 ^ Goldfarb, Ronald B. (2017). "The Permeability of Vacuum and the Revised International System of Units". IEEE Magnetics Letters. IEEE Magnetics Society. 8 (1110003): 1–3. doi:10.1109/LMAG.2017.2777782. PMC 5907514.
 ^ Wilczek, Frank (2001). "Scaling Mount Planck I: A View from the Bottom". Physics Today. 54 (6): 12–13. Bibcode:2001PhT....54f..12W. doi:10.1063/1.1387576.
 ^ Klein, Oskar (1954). "Aktuella problem kring fysikens små och stora tal" [Current problems surrounding the physics of small and large numbers]. Kosmos. 32: 33.
 ^ Wheeler, John Archibald (15 January 1955). "Geons". Phys. Rev. 97: 511. doi:10.1103/PhysRev.97.511.
 ^ M. A., Markov (1 January 1965). "Can the Gravitational Field Prove Essential for the Theory of Elementary Particles?". Progress of Theoretical Physics Supplement. E65: 85–95. doi:10.1143/PTPS.E65.85.
 ^ Sakharov, Andrei Dmitrievich (1966). "О максимальной температуре теплового излучения" [Maximum Temperature of Thermal Radiation]. Письма в Журнал Экспериментальной и Теоретической Физики (ЖЭТФ) [Letters to Journal of Experimental and Theoretical Physics (JETP)]. 3 (11): 439–441.
 ^ Hawking, Stephen (1 April 1971). "Gravitationally Collapsed Objects of Very Low Mass". Monthly Notices of the Royal Astronomical Society. 152 (1): 75–78. doi:10.1093/mnras/152.1.75.
 ^ Moisey Aleksandrovich, Markov (1982). "Предельная плотность материи как универсальный закон природы" [The ultimate density of matter as a universal law of nature]. Письма в Журнал Экспериментальной и Теоретической Физики (ЖЭТФ) [Letters to Journal of Experimental and Theoretical Physics (JETP)]. 36 (6): 214–216.
 ^ The Planck scale – Symmetry magazine
 ^ Can experiment access Planckscale physics?, CERN Courier
 ^ Number Theory as the Ultimate Physical Theory, Igor V. Volovich, empslocal.ex.ac.uk/~mwatkins/zeta/volovich1.pdf, 10.1134/S2070046610010061
 ^ Staff. "Birth of the Universe". University of Oregon. Retrieved 24 September 2016.  discusses "Planck time" and "Planck era" at the very beginning of the Universe
 ^ Edward W. Kolb; Michael S. Turner (1994). The Early Universe. Basic Books. p. 447. ISBN 9780201626742. Retrieved 10 April 2010.
 ^ ^{a} ^{b} John D. Barrow, 2002. The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe. Pantheon Books. ISBN 0375422218.
 ^ Barrow, John D.; Tipler, Frank J. (1986). The Anthropic Cosmological Principle 1st edition 1986 (revised 1988). Oxford University Press. ISBN 9780192821478. LCCN 87028148.
 ^ P.A.M. Dirac (1938). "A New Basis for Cosmology". Proceedings of the Royal Society A. 165 (921): 199–208. Bibcode:1938RSPSA.165..199D. doi:10.1098/rspa.1938.0053.
 ^ J.D. Barrow and F.J. Tipler, The Anthropic Cosmological Principle, Oxford UP, Oxford (1986), chapter 6.9.
 ^ Barrow, John D.; Shaw, Douglas J. (2011). "The value of the cosmological constant". General Relativity and Gravitation. 43 (10): 2555–2560. arXiv:1105.3105. Bibcode:2011GReGr..43.2555B. doi:10.1007/s1071401111991.
 ^ "HiRes  The High Resolution Fly's Eye Ultra High Energy Cosmic Ray Observatory". www.cosmicray.org. Retrieved 21 December 2016.
 ^ "Gravity and the Photon". HyperPhysics. Georgia State University. Retrieved 12 September 2012.
 ^ Nova: Absolute Hot
 ^ "Planck Era" and "Planck Time"
 ^ "First Second of the Big Bang". How The Universe Works 3. 2014. Discovery Science.
 ^ MacDonald, Fiona (14 November 2016). "Scientists have measured the smallest fragment of time ever". Science Alert. Retrieved 14 November 2016.
 ^ Yirka, Bob (26 June 2020). "Theorists calculate upper limit for possible quantization of time". Phys.org. Retrieved 27 June 2020.
 ^ Wendel, Garrett; Martínez, Luis; Bojowald, Martin (19 June 2020). "Physical Implications of a Fundamental Period of Time". Phys. Rev. Lett. 124 (24): 241301. arXiv:2005.11572. doi:10.1103/PhysRevLett.124.241301.
 ^ Also see Roger Penrose (1989) The Road to Reality. Oxford Univ. Press: 71417. Knopf.
 ^ Haug, Espen Gaarder (December 2016). "The gravitational constant and the Planck units. A simplification of the quantum realm". Physics Essays. 29 (4): 558–561. doi:10.4006/0836139829.4.558.
 ^ Sorkin, Rafael (1983). "KaluzaKlein Monopole". Phys. Rev. Lett. 51 (2): 87–90. Bibcode:1983PhRvL..51...87S. doi:10.1103/PhysRevLett.51.87.
 ^ ^{a} ^{b} ^{c} ^{d} Michael Duff (2002). "Comment on timevariation of fundamental constants". arXiv:hepth/0208093.
 ^ Michael Duff (2014). How fundamental are fundamental constants?. arXiv:1412.2040. doi:10.1080/00107514.2014.980093 (inactive 22 January 2020).
 ^ ^{a} ^{b} Duff, Michael; Okun, Lev; Veneziano, Gabriele (2002). "Trialogue on the number of fundamental constants". Journal of High Energy Physics. 2002 (3): 023. arXiv:physics/0110060. Bibcode:2002JHEP...03..023D. doi:10.1088/11266708/2002/03/023.
 ^ John Baez How Many Fundamental Constants Are There?
 ^ Webb, J. K.; et al. (2001). "Further evidence for cosmological evolution of the fine structure constant". Phys. Rev. Lett. 87 (9): 884. arXiv:astroph/0012539v3. Bibcode:2001PhRvL..87i1301W. doi:10.1103/PhysRevLett.87.091301. PMID 11531558.
 ^ Davies, Paul C.; Davis, T. M.; Lineweaver, C. H. (2002). "Cosmology: Black Holes Constrain Varying Constants". Nature. 418 (6898): 602–3. Bibcode:2002Natur.418..602D. doi:10.1038/418602a. PMID 12167848.
SourcesEdit
 Barrow, John D. (2002). The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe. New York: Pantheon Books. ISBN 9780375422218. Easier.
 Barrow, John D.; Tipler, Frank J. (1986). The Anthropic Cosmological Principle. Oxford: Claredon Press. ISBN 9780198519492. Harder.
 Penrose, Roger (2005). "Section 31.1". The Road to Reality. New York: Alfred A. Knopf. ISBN 9780679454434.
 Planck, Max (1899). "Über irreversible Strahlungsvorgänge". Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (in German). 5: 440–480. pp. 478–80 contain the first appearance of the Planck base units other than the Planck charge, and of Planck's constant, which Planck denoted by b. a and f in this paper correspond to k and G in this entry.
 Tomilin, K. A. (1999). Natural Systems of Units: To the Centenary Anniversary of the Planck System (PDF). Proceedings Of The XXII Workshop On High Energy Physics And Field Theory. pp. 287–296. Archived from the original (PDF) on 17 June 2006.
External linksEdit
This article's use of external links may not follow Wikipedia's policies or guidelines. (June 2020) (Learn how and when to remove this template message) 
 Value of the fundamental constants, including the Planck base units, as reported by the National Institute of Standards and Technology (NIST).
 Sections CE of collection of resources bear on Planck units. As of 2011, those pages had been removed from the planck.org web site. Use the Wayback Machine to access pre2011 versions of the website. Good discussion of why 8πG should be normalized to 1 when doing general relativity and quantum gravity. Many links.
 "Planck Era" and "Planck Time" (up to 10^{−43} seconds after birth of Universe) (University of Oregon).
 Constants of nature: Quantum Space Theory offers a different set of Planck units and defines 31 physical constants in terms of them.
 See the Tool bag.
 The Planck scale: relativity meets quantum mechanics meets gravity from 'Einstein Light' at UNSW
 The Planck Era from U of Tennessee Astrophysics pages
 HigherDimensional Algebra and PlanckScale Physics by John C. Baez
 Six easy roads to the Planck scale
 The Planck Epoch.
 Evolution of the Universe through the Planck Epoch.
 The Planck Era from U of Tennessee Astrophysics pages
 The Planck Era from U of Oregon Cosmology pages
 The Planck Era by Sten Odenwald from Astronomy Cafe
 The Plank Epoch by professor James Schombert 39O
 The Planck Era  definition from U of Ottawa's Astronomy Knowledge Base