Four-momentum

Or 4-momentum. Not to be confused with energy-momentum tensor.

Four-momentum $P$ or $p^{μ}$ is an extension of the classical concept of momentum to special relativity, adapted to accommodate the principles of Einstein's theory. It's a 4-vector that includes both energy and three-dimensional momentum:

The first component is the energy term, expressed as $E / c$ , where $E$ is energy and $c$ is the speed of light.
The next three components are the standard momentum components in the x, y, and z directions.

So $P = (E / c, p_{x}, p_{y}, p_{z})$ .

The norm of this four-momentum vector is a special quantity. It's calculated using the Minkowski metric and is related to the invariant mass of the particle. This norm is constant for all observers, regardless of their relative motion, and it's equal to $m_{0} c$ , where $m_{0}$ is the invariant mass (this is the definition, indeed, of $m_{0}$ ). Indeed, it is

P = m_{0} U,

with $U$ the four-velocity.
In the reference frame of the particle, the spatial components of the four-momentum are zero. So we have

∥ P ∥^{2} = (E / c)^{2} - p_{x}^{2} - p_{y}^{2} - p_{z}^{2} = (E / c)^{2} = m_{0}^{2} c^{2},

and then

E = m_{0} c^{2} .

And in an arbitrary inertial frame we have:

E^{2} = (p_{x}^{2} + p_{y}^{2} + p_{z}^{2}) c^{2} + m_{0}^{2} c^{4} .

Relation to four-current

The four-current is a kind of four-momentum density. See four-current#Relation to four-momentum.