Time gap between explosions in moving frame of reference

[Vijay Bhatnagar]

Came across the following interesting problem :

Two explosions take place at the same place in a rest frame with a time separation of 10 s in that frame. A) Find the time between explosions, as measured in a frame moving with a speed 0.9 c with respect to the rest frame according to classical physics.

B) Find the time between explosions, as measured in the frame moving with a speed 0.9 c with respect to the rest frame according to the special theory of relativity.

Here I wish to discuss the solution for A).


Time gap between explosions as measured in the moving frame can be found applying Galilean Transformation i.e. t' = t. Thus, the time gap as measured from moving frame = 10 sec.

The above explanation is most obvious. However, came across the following counter view point :

Classical (or Galilean) transformation is based upon every day observations/experiences where in the velocities encountered are negligible as compared to the velocity of light (v << c). As such an event taking place in a stationary reference frame S appears to take place at the same instant in another moving reference frame S' with velocity v<<c. Hence, t' = t provided v<<c (this assumption is implied even if not stated).

Further, the Lorentz Transformations t' = (t - vx/c^2)/ \/(1 - v^2/c^2) reduces to Galilean transformation t' = t for v << c. Problem is, in the given problem v is not very very less than c. Hence, strictly speaking t' = t does not apply. To an observer in frame S' the events will not appear to occur at the same instant as in frame S even from the classical point of view. The classical treatment of the problem should be similar to that of the Doppler effect of sound where the source of sound is stationary and the listener is moving away with a constant velocity.

Any comments/views?

 

[bernhard.rothenstein]

simultaneity?

Came across the following interesting problem :

Two explosions take place at the same place in a rest frame with a time separation of 10 s in that frame. A) Find the time between explosions, as measured in a frame moving with a speed 0.9 c with respect to the rest frame according to classical physics.

B) Find the time between explosions, as measured in the frame moving with a speed 0.9 c with respect to the rest frame according to the special theory of relativity.

Here I wish to discuss the solution for A).


Time gap between explosions as measured in the moving frame can be found applying Galilean Transformation i.e. t' = t. Thus, the time gap as measured from moving frame = 10 sec.

The above explanation is most obvious. However, came across the following counter view point :

Classical (or Galilean) transformation is based upon every day observations/experiences where in the velocities encountered are negligible as compared to the velocity of light (v << c). I think that the essential difference between Galileo's transformations and the Lorentz-Einstein's one consist in the fact that in the first case the synchronization of the clocks is performed by a signal that propagates with infinite speed whereas in the second case the synchronization is performed by a light signal that propagates with finite and invariant speed. So we expect that the LE transformation holds at each relative velocity becoming the Galileo one for c infinityAs such an event taking place in a stationary reference frame S appears to take place at the same instant in another moving reference frame S' with velocity v<<c. Hence, t' = t provided v<<c (this assumption is implied even if not stated).It is not advisable to make such an assumption!


Further, the Lorentz Transformations t' = (t - vx/c^2)/ \/(1 - v^2/c^2) reduces to Galilean transformation t' = t for v << c. The transformation you mention becomes t=t' only for c equal infiniteProblem is, in the given problem v is not very very less than c. Hence, strictly speaking t' = t does not apply. To an observer in frame S' the events will not appear to occur at the same instant as in frame S even from the classical point of view. The classical treatment of the problem should be similar to that of the Doppler effect of sound where the source of sound is stationary and the listener is moving away with a constant velocity.
Have a look at the problem of the relativity of simultaneity in a textbook devoted to special relativity.
Any comments/views?

I will send you a link to a reference

 

[neutrino]

Here I wish to discuss the solution for A).


Time gap between explosions as measured in the moving frame can be found applying Galilean Transformation i.e. t' = t. Thus, the time gap as measured from moving frame = 10 sec.

The above explanation is most obvious. However, came across the following counter view point :

Classical (or Galilean) transformation is based upon every day observations/experiences where in the velocities encountered are negligible as compared to the velocity of light (v << c). As such an event taking place in a stationary reference frame S appears to take place at the same instant in another moving reference frame S' with velocity v<<c. Hence, t' = t provided v<<c (this assumption is implied even if not stated).

Further, the Lorentz Transformations t' = (t - vx/c^2)/ \/(1 - v^2/c^2) reduces to Galilean transformation t' = t for v << c. Problem is, in the given problem v is not very very less than c. Hence, strictly speaking t' = t does not apply. To an observer in frame S' the events will not appear to occur at the same instant as in frame S even from the classical point of view. The classical treatment of the problem should be similar to that of the Doppler effect of sound where the source of sound is stationary and the listener is moving away with a constant velocity.

Any comments/views?

It's not another answer for/contradiction of A. The answer to A is 10 secs. The rest of the answer you quote is an explanation for B, or more like the summary of the reason behind discrepancy that arises between Galilean and Einsteinian Relativities.

In the case of Galilean transofrmations the 'c' is infinitely large. That is, the invariant velocity in Galilean Relativity is "infinity." So t = t'. But in SR, 'c' is finite. Hence t = t' becomes an approximation for low-speeds.

 

[bernhard.rothenstein]

It's not another answer for/contradiction of A. The answer to A is 10 secs. The rest of the answer you quote is an explanation for B, or more like the summary of the reason behind discrepancy that arises between Galilean and Einsteinian Relativities.

In the case of Galilean transofrmations the 'c' is infinitely large. That is, the invariant velocity in Galilean Relativity is "infinity." So t = t'. But in SR, 'c' is finite. Hence t = t' becomes an approximation for low-speeds.

How low?

 

[neutrino]

How low?

When you find that the results of your experiments show no difference between both methods within experimental accuracy.

 

[bernhard.rothenstein]

simultaneity

When you find that the results of your experiments show no difference between both methods within experimental accuracy.

But what happens when experimental accuracy increases? Do we change each time our point of view?

 

[Doc Al]

But what happens when experimental accuracy increases? Do we change each time our point of view?

Experimental results don't care about your point of view. What happens is that you realize that special relativity applies at all speeds, but that you just don't care when speeds are low enough.

 

[neutrino]

But what happens when experimental accuracy increases?

You come to the conclusion that SR is the more accurate of the two.

 

[bernhard.rothenstein]

simultaneity

Experimental results don't care about your point of view. What happens is that you realize that special relativity applies at all speeds, but that you just don't care when speeds are low enough.

I would say: When I know the space-time coordinates of an event in one of the involved reference frames then I should state precisely which is the theory I use in order to establish its space-time coordinates in the other reference frame. I would preffere special relativity! For the learner I would invite him to represent 1/sqrt(1-bb) as a function of b=V/c asking him which value of b is in the limits of the sensitivity of his measuring devices.
I think that your in your statement reffers to you as well (smile).

 

[bernhard.rothenstein]

simultaneity

You come to the conclusion that SR is the more accurate of the two.

I am convinced!