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Relativistic Energy-momentum Equation
Relativistic equation relating total energy to invariant mass and momentum
For bodies or systems with zero momentum, it simplifies to the mass-energy equation , where total energy in this case is equal to rest energy (also written as E0).
The Dirac sea model, which was used to predict the existence of antimatter, is closely related to the energy-momentum relation.
Connection to E = mc2
The energy-momentum relation is consistent with the familiar mass-energy relation in both its interpretations: E = mc2 relates total energy E to the (total) relativistic massm (alternatively denoted mrel or mtot ), while E0 = m0c2 relates rest energyE0 to (invariant) rest mass m0.
Unlike either of those equations, the energy-momentum equation (1) relates the total energy to the rest mass m0. All three equations hold true simultaneously.
If the body's speed v is much less than c, then (1) reduces to E = 1/2m0v2 + m0c2; that is, the body's total energy is simply its classical kinetic energy (1/2m0v2) plus its rest energy.
If the body is at rest (v = 0), i.e. in its center-of-momentum frame (p = 0), we have E = E0 and m = m0; thus the energy-momentum relation and both forms of the mass-energy relation (mentioned above) all become the same.
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 frame-dependent; 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 m0 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).
The Energy-momentum relation was first established by Paul Dirac in 1928 under the form , where V is the amount of potential energy. 
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 four-momentum of the system. This method applies to both massive and massless particles, and can be extended to multi-particle systems with relatively little effort (see § Many-particle systems below).
Heuristic approach for massive particles
For a massive object moving at three-velocity u = (ux, uy, uz) with magnitude || = u in the lab frame:
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. The relativistic energy E and momentum p include the Lorentz factor defined by:
although rest mass m0 has a more fundamental significance, and will be used primarily over relativistic mass m in this article.
Squaring the 3-momentum gives:
then solving for u2 and substituting into the Lorentz factor one obtains its alternative form in terms of 3-momentum and mass, rather than 3-velocity:
Inserting this form of the Lorentz factor into the energy equation gives:
followed by more rearrangement it 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 m0 = 0 would mean that E = 0 and p = 0 and no energy-momentum relation could be derived, which is not correct.
Norm of the four-momentum
The energy and momentum of an object measured in two inertial frames in energy-momentum space - the yellow frame measures E and p while the blue frame measures E′ and p′. The green arrow is the four-momentum P of an object with length proportional to its rest mass m0. The green frame is the centre-of-momentum frame for the object with energy equal to the rest energy. The hyperbolae show the Lorentz transformation from one frame to another is a hyperbolic rotation, and ? and are the rapidities of the blue and green frames, respectively.
Performing the summations over indices followed by collecting "time-like", "spacetime-like", and "space-like" terms gives:
where the factor of 2 arises because the metric is a symmetric tensor, and the convention of Latin indices i, j taking space-like 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 momentum
In natural units where c = 1, the energy-momentum equation reduces to
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).
Centre-of-momentum frame (one particle)
For a body in its rest frame, the momentum is zero, so the equation simplifies to
where m0 is the rest mass of the body.
If the object is massless, as is the case for a photon, then the equation reduces to
in the limit that u c, we have ?(u) ? 1 so the momentum has the classical form p ? m0u, then to first order in (p/m0c)2 (i.e. retain the term (p/m0c)2n for n = 1 and neglect all terms for n >= 2) we have
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.
Addition of four momenta
In the case of many particles with relativistic momenta pn and energy En, where n = 1, 2, ... (up to the total number of particles) simply labels the particles, as measured in a particular frame, the four-momenta in this frame can be added;
and then take the norm; to obtain the relation for a many particle system:
where M0 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);
The energies and momenta in the equation are all frame-dependent, while M0 is frame-independent.
with the implication from (2) that the invariant mass is also the centre of momentum (COM) mass-energy, aside from the c2 factor:
and this is true for all frames since M0 is frame-independent. The energies ECOM n are those in the COM frame, not the lab frame.
Rest masses and the invariant mass
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 M0 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 mn in (2):
The energies can be eliminated by:
similarly the momenta can be eliminated by:
where ?nk is the angle between the momentum vectors pn and pk.
Since the invariant mass of the system and the rest masses of each particle are frame-independent, the right hand side is also an invariant (even though the energies and momenta are all measured in a particular frame).