Suppose a particle in frame S moves with acceleration [itex]a_{x}[/itex] and velocity [itex]u_{x}[/itex] at a given instance in the x-direction. I wanted to find the acceleration in a frame S' moving with velocity v in the positive x-direction with respect to frame S. To do this I used the following approach:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]a_{x}=\frac{du_{x}}{dt}[/itex] and [itex]a'_{x}=\frac{du'_{x}}{dt'}[/itex]

Using the chain rule,

[itex]a'_{x} = \frac{du'_{x}}{dt'} = \frac{du'_{x}}{du_{x}} \frac{du_{x}}{dt'} = \frac{du'_{x}}{du_{x}} \frac{du_{x}}{dt} \frac{dt}{dt'} = a_{x} \frac{du'_{x}}{du_{x}} \frac{dt}{dt'} [/itex]

Using the velocity transformation,

[itex]\large \frac{du'_{x}}{du_{x}} = \frac{1- \frac{v^{2}}{c^{2}} }{ 1 - \frac{u_{x} v} {c^{2}} } [/itex]

Similarly from the Lorentz transformations:

[itex]\large \frac{dt}{dt'} = \frac{\sqrt{1-\frac{v^{2}}{c^{2}}} } {1 - \frac{u_{x} v} {c^{2}}}[/itex]

Thus,

[itex]\large a'_{x} = a_{x} \frac{(1- \frac{v^{2}}{c^{2}})^{3/2}}{(1-\frac{u_{x}v}{c^{2}})^{3}} [/itex]

Now I know this formula is correct, as it listed in Resnick's and French's introductory books on Special relativity. However in W. Rindler's book on the subject, the author shows the relativistic acceleration as:

[itex]\large a'_{x} = \gamma^{3} a_{x}[/itex]

How come there are two formulas for this quantity, one of which does not even refer to the speed of the particle? I have posted my working so that maybe someone can understand and help discriminate between these formulas.

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# Two expression for relativistic acceleration

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