Can Length Contraction be Proven Experimentally?

[RobikShrestha]

Is length contraction an assumption which is verified experimentally or can it be proved? If it can be proved, how to prove it graphically? By proof, I do not mean demonstrate. I just want to prove length1=length2 * (gamma).

 

[Dale]

If it can be proved, how to prove it graphically? By proof, I do not mean demonstrate. I just want to prove length1=length2 * (gamma).

See the derivation posted here:

https://www.physicsforums.com/showpost.php?p=3162964&postcount=11

 

[ghwellsjr]

Length contraction is just as easy to prove and demonstrate as time dilation is (which isn't easy, but it has been done). The two go hand in hand. You cannot have time dilation and have the laws of physics be invariant in different inertial frames of reference without also having length contraction. Everyone seems to accept without question that time dilation is a real phenomenon based on the atomic clocks that have been flown and shown to have a lesser time on them when returned to earth.

So all you have to do is consider a light clock. Usually, a light clock is illustrated as having the light beam bounce back and forth at right angles to the direction of motion because this does not involve length contraction. But if you take that same light clock and rotate it 90 degrees so that the light bounces back and forth along the direction of motion the mirrors will need to move closer together by the amount determined by length contraction in order for it to tick at the same rate as before. Therefore, length contraction is just as real as time dilation.

If you want some graphical animations to help understand this, see posts 78 and 79 on this thread:

https://www.physicsforums.com/showthread.php?t=458093&page=5

 

[RobikShrestha]

What is the proof of:
xA'(t) = gamma * t * v
xB'(t) = gamma * (Lo + t * v)?

The transformations of time is easy to visualize using speeding atomic clocks, but how to prove the above two equations? Can I use speeding clocks like those atomic clocks but rotated 90 degrees? If so how do we prove?

 

[Dale]

What is the proof of:
xA'(t) = gamma * t * v
xB'(t) = gamma * (Lo + t * v)?

Those come directly from the Lorentz transform of the corresponding unprimed quantities.

 

[Dale]

If it can be proved, how to prove it graphically?

You can show it graphically simply from plotting the Lorentz transform.

Note that the x'=2 line crosses the t=0 line at x<2. That is length contraction. Note also that the x=2 line crosses the t'=0 line at x'<2. That is length contraction in the other frame, showing that the effect is the same in both frames.

 

[RobikShrestha]

I got the derivations but I could not derive: tB'(t) = gamma * (t + Lo * v/ c2).
If I am correct this is supposed to be the time equation for B-end of the rod (which is Lo away from A in A's frame) as we (A) pass through time 't'. Please provide hints or derivations for this formula. It would be really helpful.

 

[RobikShrestha]

In the graph above, are the scales for primed axes plotted from Lorentz transforms? Should we always do so or is there another way? We know x=1 unit, t=1 unit, and (prime) is moving at 'v' in our frame. Now can we 'derive the scale' of (prime) axes: 1 unit of x' in our frame and 1 unit of t' in our frame without using Lorentz transforms? Anyways thank you.

 

[Dale]

I got the derivations but I could not derive: tB'(t) = gamma * (t + Lo * v/ c2).

From the Lorentz transform we have:
$$t'=\gamma(t+xv/c^2)$$

Simply substitute $$x_B=L_0$$ to obtain

$$t_B'=\gamma(t+L_0v/c^2)$$

 

[Dale]

Now can we 'derive the scale' of (prime) axes: 1 unit of x' in our frame and 1 unit of t' in our frame without using Lorentz transforms?

You can derive the Lorentz transform from the postulates of relativity, so in principle you can derive anything that comes from the Lorentz transform directly from the postulates of relativity, but it is a rather pointless exercise, IMO.

It is better to learn how the Lorentz transform arises from the postulates and then just start there. If you want to prove length contraction, then start with the Lorentz transform.

 

[RobikShrestha]

Thanks it was really helpful!