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# Four-acceleration

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

: $mathbf\left\{A\right\} =frac\left\{dmathbf\left\{U\left\{d au\right\}=left\left(gamma_udotgamma_u c,gamma_u^2mathbf a+gamma_udotgamma_umathbf u ight\right)$

where

: $mathbf a = \left\{dmathbf u over dt\right\}$ and $dotgamma_u = frac\left\{mathbf\left\{a cdot u\left\{c^2\right\} gamma_u^3 = frac\left\{mathbf\left\{a cdot u\left\{c^2\right\} frac\left\{1\right\}\left\{left\left(1-frac\left\{u^2\right\}\left\{c^2\right\} ight\right)^\left\{3/2= \left\{udot u/c^2 over \left(1 - u^2/c^2\right)^\left\{3/2$

and $gamma_u$ is the Lorentz factor for the speed $u$. It should be noted that a dot above a variable indicates a derivative with respect to the coordinate time in a given reference frame, not the proper time $au$.

In an instantaneously co-moving inertial reference frame $mathbf u = 0$, $gamma_u = 1$ and $dotgamma_u = 0$, i.e. in such a reference frame : $mathbf\left\{A\right\} =left\left(0, mathbf a ight\right)$

Therefore, the magnitude of the four-acceleration (which is an invariant scalar) is equal to the proper acceleration that a moving particle "feels" moving along a world line.The world lines having constant magnitude of four-acceleration are Minkowski-circles i.e. hyperbolas (see "hyperbolic motion")

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

: $F^mu = mA^mu$

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.

:$A^lambda := frac\left\{DU^lambda \right\}\left\{d au\right\} = frac\left\{dU^lambda \right\}\left\{d au \right\} + Gamma^lambda \left\{\right\}_\left\{mu u\right\}U^mu U^ u$

This relation holds in special relativity too when one uses curved coordinates, i.e. when the frame of reference isn't inertial.

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

ee also

* four-vector
* four-velocity
* four-momentum
* four-force

References

*

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