Two expression for relativistic acceleration

[Pi-Bond]

Suppose a particle in frame S moves with acceleration a_{x} and velocity u_{x} 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:

a_{x}=\frac{du_{x}}{dt} and a'_{x}=\frac{du'_{x}}{dt'}

Using the chain rule,

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'}

Using the velocity transformation,

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

Similarly from the Lorentz transformations:

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

Thus,

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

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:

\large a'_{x} = \gamma^{3} a_{x}

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.

 

[bcrowell]

Do you know about four-vectors? A nice, simple way to get the result and be sure it was right would be to use the Lorentz transformation on the acceleration four-vector: http://en.wikipedia.org/wiki/Four-acceleration

 

[Pi-Bond]

I am aware of four-vectors but not of the acceleration four vector (I have only done a first year basic course on relativity, and the books have not mentioned it till yet either). Both results are right though, aren't they?

 

[George Jones]

Suppose a particle in frame S moves with acceleration a_{x} and velocity u_{x} 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:

a_{x}=\frac{du_{x}}{dt} and a'_{x}=\frac{du'_{x}}{dt'}

Using the chain rule,

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'}

Using the velocity transformation,

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

Similarly from the Lorentz transformations:

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

Thus,

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

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:

\large a'_{x} = \gamma^{3} a_{x}

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.

Is the particle (instantaneously) at rest in in the primed frame in Rindler's treatment?

 

[Pi-Bond]

Rindler says: "Let S' be the the instantaneous rest frame of P at some time t..."

 

[George Jones]

What does this say about u_x and v?

 

[Pi-Bond]

"..u=v and u'=0 at t_{0} , but u and u' vary while v is constant"

He is using different notation, with u=u_{x} and so on. He formulates his proof by finding \frac{dt}{dt'} and then diffrenciating the velocity transformation with respect to dt'. I don't understand his proof very well, but I have seen the gamma cubed formula at other places.

 

[George Jones]

Suppose a particle in frame S moves with acceleration a_{x} and velocity u_{x} 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.

Rindler's result is a special case of your more general result.

If S' moves with speed v with respect to S, and if the particle is at rest in S', then the particle moves with speed v in S, i.e., Rindler looks at the special case u_x = v. Use this in

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

 

[Pi-Bond]

I see. Thanks for the help again!