Is This the Proper Frame for Your Needs?

[Abhishek11235]

Homework Statement

This is continuation of my post https://www.physicsforums.com/threads/calculating-elapsed-time-in-different-frames.970167/ .Suppose we have 2 bombs which are kept on platform with distance between bombs as L. A train with speed V moves past this platform. Suppose the bombs explode simultaneously in the frame of platform. Let us look the situation in train frame. In this frame, the events are not simultaneous. The rear clock is always $LV/c^2$ ahead of front clock. Since,the events are simultaneous in platform frame,the front bomb will explode at the same when the rear bomb has exploded(i.e,if the rear clock reads 0 when it exploded then the front bomb clock should show us time $-LV/c^2$. Now,when the front clock shows 0 time in our frame,then the bomb explodes). Now,an observer in train measures the event that front bomb explodes is longer then that clock attached on front bomb(he sees the clock of front bomb running slow due to time dilation). Now which frame is proper frame? Should I take the bomb frame as proper or the train frame since the event that the front bomb explodes doesn't take at same place in train frame?

Relevant Equations

$T=T'\gamma$

The attempt is above

 

[TSny]

Now which frame is proper frame?

I'm not sure what you mean by a "proper frame"? Do you mean a frame for which the time between the two explosions is a "proper time interval"?

The explosions of the two bombs do not occur at the same place in either the platform frame or the train frame. So, the time between the two events is not a proper time interval for either frame.

In fact, there will not exist any inertial frame for which the two explosions occur at the same place. In general, if two events occur simultaneously at different locations in some particular inertial frame, then there cannot exist any inertial frame in which the two events occur at the same place. If you are in the inertial frame for which the two events occur simultaneously at different locations, how fast would an observer need to move relative to you in order to be present at both events?

 

[Abhishek11235]

I'm not sure what you mean by a "proper frame"? Do you mean a frame for which the time between the two explosions is a "proper time interval"?

how fast would an observer need to move relative to you in order to be present at both events?

Well,the question I wanted to ask is in the platform frame,the train is contracted. The platform observer sees the bomb to explode at the length of $\gamma L$. Now,how will train observer explains this phenomenon? The train observer sees that events are not simultaneous. Also,once the rear bomb explodes,how much time has elapsed when the 2nd bomb will explode? Now,the time taken is $LV/c^2$ in train frame(due to loss of simultaneity). Now,I see that the clock on bomb is running slow. So,shouldn't the time be $LV/\gamma c^2$ instead of $\gamma LV/c^2$. My textbook mentions later result. Can you explain this please?

 

[fresh_42]

@Abhishek11235
It is not very nice to speak of bombs nowadays when nearly daily someone on Earth dies because of them. Please choose less problematic metaphors in the future.

 

[TSny]

Now,the time taken is $LV/c^2$ in train frame(due to loss of simultaneity). Now,I see that the clock on bomb is running slow. So,shouldn't the time be $LV/\gamma c^2$ instead of $\gamma LV/c^2$. My textbook mentions later result. Can you explain this please?

When you say that $LV/c^2$ is the amount by which the train clocks are desynchronized according to the platform, what is the meaning of $L$ in the expression? Is it the distance between the two train clocks according to the platform or according to the train?

 

[Abhishek11235]

When you say that $LV/c^2$ is the amount by which the train clocks are desynchronized according to the platform, what is the meaning of $L$ in the expression? Is it the distance between the two train clocks according to the platform or according to the train?

L is distance according to platform

 

[TSny]

L is distance according to platform

I'm not sure that's correct. We have to be careful with the notation. Reading the statement of the problem, $L$ is the distance between the two firecrackers according to the platform. The amount of desynchronization of the train clocks according to the platform will be of the form $xV/c^2$ where you need to use the correct distance $x$. Can you use the Lorentz transformation equations to determine if $x$ is the distance between the train clocks according to the platform or according to the train?

 

[Abhishek11235]

Can you use the Lorentz transformation equations to determine if $x$ is the distance between the train clocks according to the platform or according to the train?

Sure:
Since t is 0 in platform frame,

$T'= \gamma(t-vL/c^2) \implies T'=\gamma Lv/c^2$ which is the result. But I want to solve this problem using knowledge of simultaneity,time dilation.

 

[TSny]

Sure:
Since t is 0 in platform frame,

$T'= \gamma(t-vL/c^2) \implies T'=\gamma Lv/c^2$ which is the result.

Yes. Note that $\gamma L$ is the distance, $L'$, between the two clocks on the train according to the train. So, the desynchronization is $L'V/c^2$.

But I want to solve this problem using knowledge of simultaneity,time dilation.

If you want to get the amount of desynchronization of the two train clocks according to the platform without using the Lorentz transformation, you can try the following.Imagine a light signal sent out from the rear train clock towards the front train clock. Let the time on the rear clock be 0 at the instant the signal is sent out.

(a) What will be the reading on the front train clock when the signal arrives at the front clock? Give an answer in terms of $L$, $\gamma$, $V$, and $c$. Keep in mind that $L$ is defined as the distance between the train clocks according to the platform. This question is easily answered by considering the reference frame of the train.

(b) According to the platform, how much time does it take the signal to travel between the two train clocks?

(c) Using the fact that the platform “sees” the train clocks running slow (“time dilation”), use the answer to (b) to find the reading on the train’s rear clock according to the platform at the instant the signal reaches the front clock.

(d) According to the platform, what is the difference in reading of the two train clocks at the instant the signal reaches the front clock? This gives the amount of desynchronization according to the platform.