Undergrad Thought Exp: Gen Relativity & Time Dilation

[starkreactor]

TL;DR

I am trying to understand a thought experiment I just posed, which is: if an observer is travelling near the speed of light, and sends out two photons; one in the direction of travel and one in the opposite direction, how does general relativity account for time dilation? The photon "in front" of you will be trying to escape at the same velocity as the photon "behind" you. Wouldn't that make any moving body more or less "stand still" in this frame of reference in order for C to be conserved?

I am trying to understand a thought experiment I just posed, which is: if an observer is traveling near the speed of light, and sends out two photons; one in the direction of travel and one in the opposite direction, how does general relativity account for time dilation? The photon "in front" of you will be trying to escape at the same velocity as the photon "behind" you. Wouldn't that make any moving body more or less "stand still" in this frame of reference in order for C to be conserved?

 

[Nugatory]

if an observer is traveling near the speed of light,

Traveling near the speed of light relative to what?

You can choose to describe this situation using a frame in which the observer is at rest, and the observer is indeed standing still in that frame.

(You mentioned general relativity in the subject line, but your question is about special relativity)

 

[Ibix]

In your own frame of reference, you are stationary by definition. So the two light pulses (don't talk about photons - you don't need to complicate this with quantum effects) propagate at $c$ relative to you always.

"Travelling near the speed of light" has no meaning. This is relativity, and you can only specify speed relative to something else. "Travelling near the speed of light with respect to the Earth" would be fine. But the observer would say she is at rest and the Earth is traveling near the speed of light, so is still able to regard herself as at rest and get the expected behaviour that light moves at $c$.

If your problem is that the light pulses, emitted by the observer as viewed from the Earth are not equidistant from their source because (in this frame) the source is almost keeping up with the "forward" pulse, the solution is the relativity of simultaneity. The question, precisely stated, is "why are the two pulses not at the same distance from the ship at the same time". But Einstein's train and embankment thought experiment shows that the frames have different definitions of "at the same time". That difference resolves the problem.

 

[PeroK]

Time dilation is not relevant here. First, we have an object (A, say) that emits two pulses of light in opposite directions. In A's reference frame both pulses move at speed $c$, of course.

Now, we have a frame of reference in which A is moving at nearly the speed of light. In this frame of reference, both light pulses will move at $c$ and A will move at $v = 0.99c$, or whatever. There is not the same symmetry in this frame.

The key to understanding this is relativistic velocity addition. The formula for this is: $$u'= \frac{u + v}{1 + uv/c^2}$$ This is different from Newtonian velocity addition, which is simply $u' = u + v$.

Here $u$ is the velocity of a particle in one frame and $v$ is the relative velocity of that frame to a second frame; and $u'$ is the velocity of the particle in the second frame.

A key point of this formula is that it keeps the speed of light invariant (this means the same in all reference frame - note that conserved means it stays the same over time). I.e. if we put $u = c$, then: $$u' = \frac{c + v}{1 + cv/c^2} = c \frac{1 + v/c}{1+ v/c} = c$$ And we see that indeed $u' = c$ and light has the same speed in all reference frames (independent of $v$).

If we apply it in your scenario, we see that the speed of both light pulses is $c$ in both frames.

 

[starkreactor]

Thank you for your thought out and quick responses! I can see now how time dilation need not be considered and that this is actually a solved by Einstein’s original train/platform thought experiment- I think I was momentarily confused by thinking of sound waves and wondering how light could get “compressed” in front and “stretched behind while still remaining a constant velocity. It’s very clear again :) thanks! First post- success!