Energy–momentum relation


In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating any object's rest mass, total energy, and momentum:
holds for a system, such as a particle or macroscopic body, having intrinsic rest mass, total energy, and a momentum of magnitude, where the constant is the speed of light, assuming the special relativity case of flat spacetime.
The Dirac sea model, which was used to predict the existence of antimatter, is closely related to the energy-momentum equation.

Connection to {{math|''E'' ''mc''2}}

The energy–momentum relation is consistent with the familiar mass–energy relation in both its interpretations: relates total energy to the relativistic mass , while relates rest energy to rest mass.
Unlike either of those equations, the energy–momentum equation relates the total energy to the rest mass. All three equations hold true simultaneously.

Special cases

  1. If the body is a massless particle, then reduces to. For photons, this is the relation, discovered in 19th century classical electromagnetism, between radiant momentum and radiant energy.
  2. If the body's speed is much less than, then reduces to ; that is, the body's total energy is simply its classical kinetic energy plus its rest energy.
  3. If the body is at rest, i.e. in its center-of-momentum frame, we have and ; thus the energy–momentum relation and both forms of the mass–energy relation all become the same.
A more [|general form] of relation holds for general relativity.
The invariant mass is an invariant for all frames of reference, not just in inertial frames in flat spacetime, but also accelerated frames traveling through curved spacetime. However the total energy of the particle and its relativistic momentum 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 and, while the other frame measures and, where and, 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,,, 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 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 and measured in the lab frame.
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.

Origins and derivation of the equation

The 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:
  1. From the relativistic dynamics of a massive particle,
  2. 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.

    Heuristic approach for massive particles

For a massive object moving at three-velocity with magnitude 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. The relativistic energy and momentum include the Lorentz factor defined by:
Some authors use relativistic mass defined by:
although rest mass has a more fundamental significance, and will be used primarily over relativistic mass in this article.
Squaring the 3-momentum gives:
then solving for and substituting into the Lorentz factor 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:
followed by more rearrangement yields. The elimination of the Lorentz factor also eliminates implicit velocity dependence of the particle in, 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 would mean that and and no energy–momentum relation could be derived, which is not correct.

Norm of the four-momentum

Special relativity

In Minkowski space, energy and momentum are two components of a Minkowski four-vector, namely the four-momentum;
.
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 of the body:
a Lorentz invariant quantity, and therefore independent of the frame of reference. Using the Minkowski metric with metric signature, the inner product is
and
so

General relativity

In general relativity, the 4-momentum is a four-vector defined in a local coordinate frame, although by definition the inner product is similar to that of special relativity,
in which the Minkowski metric is replaced by the metric tensor field :
solved from the Einstein field equations. Then:
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 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 for more information.

Units of energy, mass and momentum

In natural units where, the energy–momentum equation reduces to
In particle physics, energy is typically given in units of electron volts, momentum in units of eV·−1, and mass in units of eV·−2. In electromagnetism, and because of relativistic invariance, it is useful to have the electric field and the magnetic field in the same unit, using the cgs system of units, where energy is given in units of erg, mass in grams, 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.

Special cases

Centre-of-momentum frame (one particle)

For a body in its rest frame, the momentum is zero, so the equation simplifies to
where is the rest mass of the body.

Massless particles

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 or wavenumber are given.

Correspondence principle

Rewriting the relation for massive particles as:
and expanding into power series by the binomial theorem :
in the limit that, we have so the momentum has the classical form, then to first order in 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.

Many-particle systems

Addition of four momenta

In the case of many particles with relativistic momenta and energy, where 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 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. Substituting and rearranging gives the generalization of ;
The energies and momenta in the equation are all frame-dependent, while is frame-independent.

Center-of-momentum frame

In the center-of-momentum frame, by definition we have:
with the implication from that the invariant mass is also the centre of momentum mass–energy, aside from the factor:
and this is true for all frames since is frame-independent. The energies 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 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 :
so substituting the sums, we can introduce their rest masses in :
The energies can be eliminated by:
similarly the momenta can be eliminated by:
where is the angle between the momentum vectors and.
Rearranging:
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.

Matter waves

Using the de Broglie relations for energy and momentum for matter waves,
where is the angular frequency and is the wavevector with magnitude, equal to the wave number, the energy–momentum relation can be expressed in terms of wave quantities:
and tidying up by dividing by throughout:
This can also be derived from the magnitude of the four-wavevector
in a similar way to the four-momentum above.
Since the reduced Planck constant and the speed of light both appear and clutter this equation, this is where natural units are especially helpful. Normalizing them so that, we have:

Tachyon and exotic matter

The velocity of a bradyon with the relativistic energy–momentum relation
can never exceed. On the contrary, it is always greater than for a tachyon whose energy–momentum equation is
By contrast, the hypothetical exotic matter has a negative mass and the energy–momentum equation is