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:

: $mathbf\{A\}\; =frac\{dmathbf\{U\{d\; au\}=left(gamma\_udotgamma\_u\; c,gamma\_u^2mathbf\; a+gamma\_udotgamma\_umathbf\; u\; ight)$

where

: $mathbf\; a\; =\; \{dmathbf\; u\; over\; dt\}$ and $dotgamma\_u\; =\; frac\{mathbf\{a\; cdot\; u\{c^2\}\; gamma\_u^3\; =\; frac\{mathbf\{a\; cdot\; u\{c^2\}\; frac\{1\}\{left(1-frac\{u^2\}\{c^2\}\; ight)^\{3/2=\; \{udot\; u/c^2\; over\; (1\; -\; u^2/c^2)^\{3/2$

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 : $mathbf\{A\}\; =left(0,\; mathbf\; a\; ight)$

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.

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

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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