Length Contraction & Time Dilation: Proving It

[UuserForMe]

I have been able to prove to myself that, based on Einstein's two postulates and the the Pythagorean theorem, that time dilates. From here how do I prove that length contracts? (All of this observing a frame that is moving relative to the proper frame at uniform velocity.)

 

[etotheipi]

If you want a heuristic argument, why don't you try to analyse a particle traveling in a straight line between two points, in both the rest frame of the particle, and another inertial frame (preferably where the two points are at rest)?

 

[PeroK]

I have been able to prove to myself that, based on Einstein's two postulates and the the Pythagorean theorem, that time dilates. From here how do I prove that length contracts? (All of this observing a frame that is moving relative to the proper frame at uniform velocity.)

That's actually a bit trickier. What about considering an object of length $L$ moving relative to you. It measures its rest length by firing a light beam from one end to the other and back again and taking the length to be $L = \Delta t/2c$.

Then, analyse that in your frame (where the object is moving with speed $v$ in the direction of its length) and calculate the length of the object in your frame.

 

[Dale]

From here how do I prove that length contracts?

Try an L-shaped light clock with one arm perpendicular to the relative velocity and the other arm parallel. Consider the length of a “tick” to be the time between when a light pulse is emitted at the intersection between the arms, travels down each arm to the mirror, and returns to the intersection. The length of the perpendicular arm is fixed at L and the length of the parallel arm is adjusted such that the return pulses reach the intersection at the same time. Calculate the length of the parallel arm as a function of v.

 

[Sagittarius A-Star]

From here how do I prove that length contracts?

The following video visualizes (from time 13:33 on) length contraction and relativity of simultaneity. It is self-explaining, so you can ignore the German explantion in the audio track.

 

[robphy]

Here's a video showing, on a spacetime diagram,
how length contraction is needed so that there is no time-difference between "ticks" for the transverse and longitudinal light clocks (i.e., the arms of a Michelson-Morley apparatus).



and here a GeoGebra visualization of the situation
https://www.geogebra.org/m/XFXzXGTq

But why stop with two clocks?
Here's a circular light-clock (similar to what is shown in the German-language video posted earlier)