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In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime. The contravariant four-momentum of a particle with three-momentum p = (px, py, pz) and energy E is


\mathbf{P} = \begin{pmatrix}
P^0 \\ P^1 \\ P^2 \\ P^3 
\end{pmatrix} = 
\begin{pmatrix}
E/c \\ p_\text{x} \\ p_\text{y} \\ p_\text{z} 
\end{pmatrix}

The four-momentum is useful in relativistic calculations because it is a Lorentz vector. This means that it is easy to keep track of how it transforms under Lorentz transformations.

The above definition applies under the coordinate convention that x = ct. Some authors use the convention x = t, which yields a modified definition with P = E/c. It is also possible to define covariant four-momentum Pμ where the sign of the energy is reversed.


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