Four-acceleration

In the theory of relativity, four-acceleration is a four-vector (vector in four-dimensional spacetime) that is analogous to classical acceleration (a three-dimensional vector, see three-acceleration in special relativity). Four-acceleration has applications in areas such as the annihilation of antiprotons, resonance of strange particles and radiation of an accelerated charge.

In inertial coordinates in special relativity, four-acceleration ${\displaystyle \mathbf {A} }$ is defined as the rate of change in four-velocity ${\displaystyle \mathbf {U} }$ with respect to the particle's proper time along its worldline. We can say:

where

and

and ${\displaystyle \gamma _{u}}$ is the Lorentz factor for the speed ${\displaystyle u}$ (with ${\displaystyle |\mathbf {u} |=u}$). A dot above a variable indicates a derivative with respect to the coordinate time in a given reference frame, not the proper time ${\displaystyle \tau }$ (in other terms, ${\displaystyle {\dot {\gamma }}_{u}={d\gamma _{u} \over dt}}$).

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