Spacetime
Extending momentum to four dimensions

In classical mechanics, the state of motion of a particle is characterized by its mass and its velocity. Linear momentum, the product of a particle's mass and velocity, is a vector quantity, possessing the same direction as the velocity: p = m v. It is a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum cannot change.

In relativistic mechanics, the momentum vector is extended to four dimensions. Added to the momentum vector is a time component that allows the spacetime momentum vector to transform like the spacetime position vector (x, t) {\displaystyle (x,t)}. In exploring the properties of the spacetime momentum, we start, in Fig. 3-8a, by examining what a particle looks like at rest. In the rest frame, the spatial component of the momentum is zero, i.e. p = 0, but the time component equals mc.

We can obtain the transformed components of this vector in the moving frame by using the Lorentz transformations, or we can read it directly from the figure because we know that (m c) ′ = γ m c {\displaystyle (mc)^{\prime }=\gamma mc} and p ′ = − β γ m c {\displaystyle p^{\prime }=-\beta \gamma mc}, since the red axes are rescaled by gamma. Fig. 3-8b illustrates the situation as it appears in the moving frame. It is apparent that the space and time components of the four-momentum go to infinity as the velocity of the moving frame approaches c.

We will use this information shortly to obtain an expression for the four-momentum.
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