MostlyHarmless said:
I had a quick question about Time Dilation and Length Contraction.
Are the two just different ways of measuring/describing the same effect? Or rather they both follow as a consequence from one another?
i.e. I can find how much a length is contracted by finding the dilated time interval and multiplying by velocity.
Or I can how much a time interval is dilated by finding how much the length is contracted and then dividing by the velocity?
Length contraction, time dilation and relativity of simultaneity are all three necessary to insure that the laws of physics look the same in any inertial coordinate system.
A way to see that length contraction and time dilation must go hand-in-hand is by considering the round-trip time for a light signal from one end of a traveling spaceship to the other and back.
If the rocket were stationary, and its length were L, then the round-trip time would be simply T = \frac{2L}{c} where c is the speed of light.
Now, consider the same rocket ship moving forward at speed v. If the rocket did not undergo length contraction or time dilation, then the round-trip time would be:
T = \frac{1}{1-\frac{v^2}{c^2}} \frac{2L}{c}
Why is that: because if you send a light signal from the rear of the rocket toward the front, then the front of the rocket is moving away from the light signal, making it take longer to reach the front. Specifically, the time to get to the front would be: T_1 = \frac{1}{1-\frac{v}{c}} \frac{L}{c}. For the return trip, from the front of the rocket to the rear, the time would be shorter, since the rear would be moving toward the light signal. The time would be: T_2 = \frac{1}{1+\frac{v}{c}} \frac{L}{c}
The total time for the round-trip would be T = T_1 + T_2 = \frac{1}{1-\frac{v^2}{c^2}} \frac{2L}{c}
In terms of \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}, the round-trip time would be:
T = \gamma^2 \frac{2L}{c}
This is without length contraction or time dilation. If we assume that (as measured in the original stationary frame) moving clocks slow by a factor of \gamma, then in terms of the moving clocks, the time would be less by a factor of \gamma, so clocks aboard the rocket would measure T' = \frac{T}{\gamma} for the round-trip time for light. So they would measure:
T' = \gamma \frac{2L}{c}
If we further assume that moving rockets are length-contracted by the same factor of \gamma, then that would cut down the measured round-trip time further. Instead of
T' = \gamma \frac{2L}{c}
the measured time would be
T' = \gamma \frac{2L'}{c}
where L' = \frac{L}{\gamma}. Therefore, the measured time for a round trip would be
T' = \frac{2L}{c}
the same as for a rocket that is stationary.
