Four-acceleration

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In special relativity, four-acceleration is a four-vector and is defined as the change in four-velocity over the particle's proper time:

<math>A^\mu=\frac{dU^\mu}{d\tau}=\left(\gamma_u\dot\gamma_u,\gamma_u^2\mathbf a+\gamma_u\dot\gamma_u\mathbf u\right)</math>

where

<math>\mathbf a = {d\mathbf u \over dt}</math> and <math>\dot\gamma_u = {u\dot u/c^2 \over (1 - u^2/c^2)^{3/2}}</math>

and <math>\gamma_u</math> is the Lorentz factor for the speed <math>u</math>. It should be noted that a dot above a variable indicates a derivative with respect to the time in a given reference frame, not the proper time <math>\tau</math>.

The scalar product of a four-velocity and the corresponding four-acceleration is always 0.

Even at relativistic speeds four-acceleration is related to the four-force such that

where m is the invariant mass of a particle.

In general relativity the elements of the acceleration four-vector are related to the elements of the four-velocity through a covariant derivative with respect to proper time.

<math>A^\lambda := \frac{DU^\lambda }{d\tau} = \frac{dU^\lambda }{d\tau } + \Gamma^\lambda {}_{\mu \nu}U^\mu U^\nu </math>

When the four-force is zero one has gravitation acting along, and the four-vector version of Newton's second law above reduces to the geodesic equation.