Two expression for relativistic acceleration

In summary, Rindler's result is a special case of the more general result that uses the velocity transformation.
  • #1
Pi-Bond
302
0
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:

[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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  • #3
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?
 
  • #4
Pi-Bond said:
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:

[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.

Is the particle (instantaneously) at rest in in the primed frame in Rindler's treatment?
 
  • #5
Rindler says: "Let S' be the the instantaneous rest frame of P at some time t..."
 
  • #6
What does this say about [itex]u_x[/itex] and [itex]v[/itex]?
 
  • #7
"..u=v and u'=0 at [itex]t_{0}[/itex] , but u and u' vary while v is constant"

He is using different notation, with [itex]u=u_{x}[/itex] and so on. He formulates his proof by finding [itex]\frac{dt}{dt'}[/itex] 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.
 
  • #8
Pi-Bond said:
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.

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
Pi-Bond said:
[itex]\large a'_{x} = a_{x} \frac{(1- \frac{v^{2}}{c^{2}})^{3/2}}{(1-\frac{u_{x}v}{c^{2}})^{3}} [/itex]
 
  • #9
I see. Thanks for the help again!
 

Related to Two expression for relativistic acceleration

1. What is the equation for relativistic acceleration?

The equation for relativistic acceleration is a = γ^3 * a', where γ is the Lorentz factor and a' is the classical acceleration.

2. What is the difference between relativistic acceleration and classical acceleration?

Relativistic acceleration takes into account the effects of special relativity, such as time dilation and length contraction, while classical acceleration does not. This means that the equation for relativistic acceleration includes the Lorentz factor, which is not present in the classical equation.

3. How does relativistic acceleration affect an object's motion?

Relativistic acceleration can cause an object to appear to accelerate differently depending on the observer's frame of reference. This is due to the effects of time dilation and length contraction, which can change the perceived velocity and distance of the object.

4. What is the other expression for relativistic acceleration?

The other expression for relativistic acceleration is a = (γ^2 - 1) * c^2 / r, where c is the speed of light and r is the distance from the center of the acceleration.

5. How is relativistic acceleration related to Einstein's theory of special relativity?

Einstein's theory of special relativity states that the laws of physics are the same for all observers in uniform motion. Relativistic acceleration takes into account this principle and shows how the effects of special relativity can impact an object's acceleration and motion.

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