Energy–momentum relation
In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating any object's rest (intrinsic) mass, total energy, and momentum:

( 1 )
holds for a system, such as a particle or macroscopic body, having intrinsic rest mass m_{0}, total energy E, and a momentum of magnitude p, where the constant c is the speed of light, assuming the special relativity case of flat spacetime.^{[1]}^{[2]}^{[3]}
The Dirac sea model, which was used to predict the existence of antimatter, is closely related to the energymomentum equation.
Contents
Connection to E = mc^{2}Edit
The energy–momentum relation is consistent with the familiar mass–energy relation in both its interpretations: E = mc^{2} relates total energy E to the (total) relativistic mass m (alternatively denoted m_{rel} or m_{tot} ), while E_{0} = m_{0}c^{2} relates rest energy E_{0} to (invariant) rest mass m_{0}.
Unlike either of those equations, the energy–momentum equation (1) relates the total energy to the rest mass m_{0}. All three equations hold true simultaneously.
Special casesEdit
 If the body is a massless particle (m_{0} = 0), then (1) reduces to E = pc. For photons, this is the relation, discovered in 19th century classical electromagnetism, between radiant momentum (causing radiation pressure) and radiant energy.
 If the body's speed v is much less than c, then (1) reduces to E = 1/2m_{0}v^{2} + m_{0}c^{2}; that is, the body's total energy is simply its classical kinetic energy (1/2m_{0}v^{2}) plus its rest energy.
 If the body is at rest (v = 0), i.e. in its centerofmomentum frame (p = 0), we have E = E_{0} and m = m_{0}; thus the energy–momentum relation and both forms of the mass–energy relation (mentioned above) all become the same.
A more general form of relation (1) holds for general relativity.
The invariant mass (or rest mass) is an invariant for all frames of reference (hence the name), not just in inertial frames in flat spacetime, but also accelerated frames traveling through curved spacetime (see below). However the total energy of the particle E and its relativistic momentum p are framedependent; relative motion between two frames causes the observers in those frames to measure different values of the particle's energy and momentum; one frame measures E and p, while the other frame measures E′ and p′, where E′ ≠ E and p′ ≠ p, unless there is no relative motion between observers, in which case each observer measures the same energy and momenta. Although we still have, in flat spacetime:
The quantities E, p, E′, p′ are all related by a Lorentz transformation. The relation allows one to sidestep Lorentz transformations when determining only the magnitudes of the energy and momenta by equating the relations in the different frames. Again in flat spacetime, this translates to;
Since m_{0} does not change from frame to frame, the energy–momentum relation is used in relativistic mechanics and particle physics calculations, as energy and momentum are given in a particle's rest frame (that is, E′ and p′ as an observer moving with the particle would conclude to be) and measured in the lab frame (i.e. E and p as determined by particle physicists in a lab, and not moving with the particles).
In relativistic quantum mechanics, it is the basis for constructing relativistic wave equations, since if the relativistic wave equation describing the particle is consistent with this equation – it is consistent with relativistic mechanics, and is Lorentz invariant. In relativistic quantum field theory, it is applicable to all particles and fields.^{[4]}
Origins and derivation of the equationEdit
The Energy–momentum relation was first established by Paul Dirac in 1928 under the form , where V is the amount of potential energy. ^{[5]}
The equation can be derived in a number of ways, two of the simplest include:
 From the relativistic dynamics of a massive particle,
 By evaluating the norm of the fourmomentum of the system. This method applies to both massive and masseless particles, and can be extended to multiparticle systems with relatively little effort (see § Manyparticle systems below).
Heuristic approach for massive particlesEdit
For a massive object moving at threevelocity u = (u_{x}, u_{y}, u_{z}) with magnitude u = u in the lab frame:^{[1]}
is the total energy of the moving object in the lab frame,
is the three dimensional relativistic momentum of the object in the lab frame with magnitude p = p. The relativistic energy E and momentum p include the Lorentz factor defined by:
Some authors use relativistic mass defined by:
although rest mass m_{0} has a more fundamental significance, and will be used primarily over relativistic mass m in this article.
Squaring the 3momentum gives:
then solving for u^{2} and substituting into the Lorentz factor obtains its alternative form in terms of 3momentum and mass, rather than 3velocity:
Inserting this form of the Lorentz factor into the energy equation:
followed by more rearrangement yields (1). The elimination of the Lorentz factor also eliminates implicit velocity dependence of the particle in (1), as well as any inferences to the "relativistic mass" of a massive particle. This approach is not general as massless particles are not considered. Naively setting m_{0} = 0 would mean that E = 0 and p = 0 and no energy–momentum relation could be derived, which is not correct.
Norm of the fourmomentumEdit
Special relativityEdit
In Minkowski space, energy (divided by c) and momentum are two components of a Minkowski fourvector, namely the fourmomentum;^{[6]}
(these are the contravariant components).
The Minkowski inner product ⟨ , ⟩ of this vector with itself gives the square of the norm of this vector, it is proportional to the square of the rest mass m of the body:
a Lorentz invariant quantity, and therefore independent of the frame of reference. Using the Minkowski metric η with metric signature (− + + +), the inner product is
and
so
General relativityEdit
In general relativity, the 4momentum is a fourvector defined in a local coordinate frame, although by definition the inner product is similar to that of special relativity,
in which the Minkowski metric η is replaced by the metric tensor field g:
solved from the Einstein field equations. Then:^{[7]}
Performing the summations over indices followed by collecting "timelike", "spacetimelike", and "spacelike" terms gives:
where the factor of 2 arises because the metric is a symmetric tensor, and the convention of Latin indices i, j taking spacelike values 1, 2, 3 is used. As each component of the metric has space and time dependence in general; this is significantly more complicated than the formula quoted at the beginning, see metric tensor (general relativity) for more information.
Units of energy, mass and momentumEdit
In natural units where c = 1, the energy–momentum equation reduces to
In particle physics, energy is typically given in units of electron volts (eV), momentum in units of eV·c^{−1}, and mass in units of eV·c^{−2}. In electromagnetism, and because of relativistic invariance, it is useful to have the electric field E and the magnetic field B in the same unit (Gauss), using the cgs (Gaussian) system of units, where energy is given in units of erg, mass in grams (g), and momentum in g·cm·s^{−1}.
Energy may also in theory be expressed in units of grams, though in practice it requires a large amount of energy to be equivalent to masses in this range. For example, the first atomic bomb liberated about 1 gram of heat, and the largest thermonuclear bombs have generated a kilogram or more of heat. Energies of thermonuclear bombs are usually given in tens of kilotons and megatons referring to the energy liberated by exploding that amount of trinitrotoluene (TNT).
Special casesEdit
Centreofmomentum frame (one particle)Edit
For a body in its rest frame, the momentum is zero, so the equation simplifies to
where m_{0} is the rest mass of the body.
Massless particlesEdit
If the object is massless, as is the case for a photon, then the equation reduces to
This is a useful simplification. It can be rewritten in other ways using the de Broglie relations:
if the wavelength λ or wavenumber k are given.
Correspondence principleEdit
Rewriting the relation for massive particles as:
and expanding into power series by the binomial theorem (or a Taylor series):
in the limit that u ≪ c, we have γ(u) ≈ 1 so the momentum has the classical form p ≈ m_{0}u, then to first order in (p/m_{0}c)^{2}
_{} (i.e. retain the term (p/m_{0}c)^{2n}
_{} for n = 1 and neglect all terms for n ≥ 2) we have
or
where the second term is the classical kinetic energy, and the first is the rest mass of the particle. This approximation is not valid for massless particles, since the expansion required the division of momentum by mass. Incidentally, there are no massless particles in classical mechanics.
Manyparticle systemsEdit
Addition of four momentaEdit
In the case of many particles with relativistic momenta p_{n} and energy E_{n}, where n = 1, 2, ... (up to the total number of particles) simply labels the particles, as measured in a particular frame, the fourmomenta in this frame can be added;
and then take the norm; to obtain the relation for a many particle system:
where M_{0} is the invariant mass of the whole system, and is not equal to the sum of the rest masses of the particles unless all particles are at rest (see mass in special relativity for more detail). Substituting and rearranging gives the generalization of (1);

( 2 )
The energies and momenta in the equation are all framedependent, while M_{0} is frameindependent.
Centerofmomentum frameEdit
In the centerofmomentum frame (COM frame), by definition we have:
with the implication from (2) that the invariant mass is also the centre of momentum (COM) mass–energy, aside from the c^{2} factor:
and this is true for all frames since M_{0} is frameindependent. The energies E_{COM n} are those in the COM frame, not the lab frame.
Rest masses and the invariant massEdit
Either the energies or momenta of the particles, as measured in some frame, can be eliminated using the energy momentum relation for each particle:
allowing M_{0} to be expressed in terms of the energies and rest masses, or momenta and rest masses. In a particular frame, the squares of sums can be rewritten as sums of squares (and products):
so substituting the sums, we can introduce their rest masses m_{n} in (2):
The energies can be eliminated by:
similarly the momenta can be eliminated by:
where θ_{nk} is the angle between the momentum vectors p_{n} and p_{k}.
Rearranging:
Since the invariant mass of the system and the rest masses of each particle are frameindependent, the right hand side is also an invariant (even though the energies and momenta are all measured in a particular frame).
Matter wavesEdit
Using the de Broglie relations for energy and momentum for matter waves,
where ω is the angular frequency and k is the wavevector with magnitude k = k, equal to the wave number, the energy–momentum relation can be expressed in terms of wave quantities:
and tidying up by dividing by (ħc)^{2} throughout:

( 3 )
This can also be derived from the magnitude of the fourwavevector
in a similar way to the fourmomentum above.
Since the reduced Planck constant ħ and the speed of light c both appear and clutter this equation, this is where natural units are especially helpful. Normalizing them so that ħ = c = 1, we have:
Tachyon and exotic matterEdit
The velocity of a bradyon with the relativistic energy–momentum relation
can never exceed c. On the contrary, it is always greater than c for a tachyon whose energy–momentum equation is^{[8]}
By contrast, the hypothetical exotic matter has a negative mass^{[9]} and the energy–momentum equation is
See alsoEdit
ReferencesEdit
 ^ ^{a} ^{b} Kleppner, Daniel; Robert J. Kolenkow (2010) [1973]. An Introduction to Mechanics. Cambridge University Press. pp. 499–500. ISBN 9780521198219.
 ^ J.R. Forshaw; A.G. Smith (2009). Dynamics and Relativity. Wiley. pp. 149, 249. ISBN 9780470014608.
 ^ D. McMahon (2006). Relativity. DeMystified. Mc Graw Hill (USA). p. 20. ISBN 0071455450.
 ^ D. McMahon (2008). Quantum Field Theory. DeMystified. Mc Graw Hill (USA). pp. 11, 88. ISBN 9780071543828.
 ^ Eisberg, R., Resnick, R. (1985) Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles. 2nd Edition, John Wiley & Sons. New York. p.132.ISBN 047187373X
 ^ J.R. Forshaw; A.G. Smith (2009). Dynamics and Relativity. Wiley. pp. 258–259. ISBN 9780470014608.
 ^ J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. pp. 201, 649, 1188. ISBN 0716703440.
 ^ G. Feinberg (1967). "Possibility of fasterthanlight particles". Physical Review. 159: 1089–1105. Bibcode:1967PhRv..159.1089F. doi:10.1103/PhysRev.159.1089.
 ^ Z.Y.Wang (2016). "Modern Theory for Electromagnetic Metamaterials". Plasmonics. 11 (2): 503–508. doi:10.1007/s1146801500717.
 A. Halpern (1988). 3000 Solved Problems in Physics, Schaum Series. McGrawHill. pp. 704–705. ISBN 9780070257344.
 G. Woan (2010). The Cambridge Handbook of Physics Formulas. Cambridge University Press. p. 65. ISBN 9780521575072.
 C.B. Parker (1994). McGrawHill Encyclopaedia of Physics (2nd ed.). McGrawHill. pp. 1192, 1193. ISBN 0070514003.
 R.G. Lerner; G.L. Trigg (1991). Encyclopaedia of Physics (2nd ed.). VHC Publishers. p. 1052. ISBN 0895737523.