four-momentum

The four-momentum is related to the four-velocity whose components are.

In the theory of relativity, this momentum vector is taken as the four-momentum.

A concrete choice could be the four-momentum of a test body acted on by gravitational and non-gravitational forces.

The four-momentum for a massive particle is given by:

To say more about the state, we will use the generic form of fermion four-momentum:

Choosing the four-momentum, it can be written.

The four-momentum of a particle (either massless or massive) transforms under this representation.

The invariant mass is the ratio of four-momentum to four-velocity:

In SR the energy and the linear momentum are two aspects of a single entity, the four-momentum.

The four-momentum of an object is straightforward, identical in form to the classical momentum, but replacing 3-vectors with 4-vectors:

The energy of the two photons, or, equivalently, their frequency, may be determined from conservation of four-momentum.

A fermion 4-component wave function, may be decomposed into states with definite four-momentum:

There is also a stability assumption which restricts the spectrum of the four-momentum to the positive light cone (and its boundary).

The four-momentum is useful in relativistic calculations because it is a Lorentz vector.

In the language of special relativity, a tachyon would be a particle with space-like four-momentum and imaginary proper time.

Similarly, the four-momentum of a body is equivalent to the energy-momentum tensor of said body.

The net four-momentum is zero, but the negative energy photon violates the requirement that real photons have positive energy:.

In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime.

Thus, conservation of four-momentum is Lorentz-invariant and implies conservation of both mass and energy.

Often, when using the Dirac equation and solving for cross sections, one finds the slash notation used on four-momentum:

This is analogous to the way that special relativity mixes space and time into spacetime, and mass, momentum and energy into four-momentum.

Any time-like four-momentum possesses a reference frame where the momentum (3-dimensional) is zero, which is a center of momentum frame.

For a massive particle, the four-momentum is given by the particle's invariant mass m multiplied by the particle's four-velocity:

The electron on the left side of the diagram, represented by the solid line, starts out with four-momentum and ends up with four-momentum .

This last equation is the standard SR formula relating rest mass to four-momentum: and eqn (8.2) is its equivalent in Schwarzschild space-time.