Imagine a boxcar with a lamp on the left end and a mirror on the right end, so that a light signal can be sent down and back. The boxcar is moving to the right with velocity v. They find the length of the boxcar with respect to the car itself and then with respect to an observer on the ground using time dilation. They use the round trip time, which is fine and makes sense. However, what if you don't use the round trip time? What if you just measure the distance and time it takes light to travel to the end of the boxcar and thats it? For an observer on the car, it would be $t'=d'/c$. For an observer on the ground, it would be the distance he measures the boxcar to be plus how much the boxcar has progressed in time t. So for him it would be $t=(d+vt)/c$. Now, if you use the fact that Lorentz contraction says that d' and d should be related by $d'=\gamma d$, and plug d into the time it takes the light to reach the end according to the observer and relate $t$ to $t'$, you do not get the time dilation relationship. You actually just get $t=t'\sqrt{\frac{c+v}{c-v}}$. This is a problem because $t$ and $t'$ should be related by gamma, not this other factor. I thought the time it takes for light to travel to the end in the frame of the car should be related by gamma to the time it takes to reach the end in the frame of the ground. Can you please explain why this is not so?


Say your friend Harry stands on the left end of the boxcar.

Harry says "When the light hit the right side of the boxcar, I was just scratching my nose. At that moment, exactly $t'$ minutes had elapsed".

You (standing on the platform) are going to say that it took $t_1$ minutes till Harry scratched his nose and $t_2$ minutes till the light hit the right end of the boxcar. But $t_1$ won't equal $t_2$. That's because unlike Harry, you don't believe that the nose-scratch was simultaneous with the light hitting the right end of the boxcar.

The factor $\gamma$ isn't supposed to measure $t'/t_2$. It's supposed to measure $t'/t_1$. This, I think, is what's misleading you.

If you stare long enough at the space-time diagram, things like this always become clear:

enter image description here

The vertical axis is your worldline; the blue lines are the worldlines of the left and right sides of the boxcar; the red lines are lines of simultaneity for the denizens of the boxcar. (These red lines are supposed to be equally spaced; sorry if my artistry is imperfect on this point.) The gold lines are the path of the light ray.

The distances along the axes are measured taking the speed of light to be $1$.

The green point is where Harry scratches his nose. If you drew a horizontal line from here over the the vertical axis, it would hit at point $t_1$. The yellow point is where the light ray hits the far end of the boxcar. I've drawn a horizontal line from here to the vertical axis which hits at point $t_2$. You can see that $d'/d=t'/t_1$, as it should be. This is the ratio you're calling $\gamma$.

But $t'/t_2$ can't be equal to $\gamma$ (as you've noted), which is okay because it's not supposed to be.

(PS--I hope the labels are clear. I've marked distances rather than points, so that the $t_2$ on the left, for example, is the distance from the origin to the black dot above it, so that black dot is at the point $t_2$.)


This issue is how you derived the gamma factor and the misconception of the factor itself, as you've already stated. You can derive the gamma factor geometrically via tensor analysis of an invariant rotation. I'll spare you the bloody work, and let you know what the result comes to be. (It's mostly trigonometry)

For the relation $ \tau = \gamma t $ where $ \tau $ is the proper time as in the frame of an observer stationary with respect to the moving object ( in your case the train ), and $ t $ is the time in the frame of the moving object. Gamma here is defined as $$ \gamma = \frac{1} {(1-(\frac{v}{c})^{2})^{1/2}} $$ Where $ v $ is the velocity of the car.

Your title mentions time dilation, so I've given it to you in terms of time. General relativity will also show that space is affected in the same way ( as in the length of the boxcar that you've mentioned in your example ). The length measured by your outside observer who uses the candles to figure out the length of the car will be $ x^{'} = \gamma x $ where x prime and x are interchangeable with $ \tau $ and $ t $ respectively.

For the geometric derivation I direct you here: http://www.unitytheory.info/lorentz_gamma_factor_basics.html .

In that , they introduce $ \beta = \frac{v}{c} $.

  • $\begingroup$ Also Introduction to Tensor Calculus, Relativity and Cosmology by D.F. Lawden is a very good and cheap book to add to the collection from link. $\endgroup$ Jan 11 '14 at 22:42
  • $\begingroup$ I understand the derivation of gamma. My question is, why am I not getting the gamma relationship (time dilation formula) when I find the time it takes the light beam to travel to the end of the car from the left side in the frame of the "train" versus the frame of the observer on the ground? $\endgroup$
    – arch1234
    Jan 11 '14 at 22:47
  • $\begingroup$ The error stems from the term ((c+v)/(c-v))^1/2. Compare how you're getting that to the method that the link finds beta. $\endgroup$ Jan 11 '14 at 22:55
  • $\begingroup$ The error does not stem from the term $((c+v)/(c-v))^{1/2}$. Instead the error leads to the term $((c+v)/(c-v))^{1/2}$. The error stems from computing the wrong ratio in the first place, as explained in my answer. $\endgroup$
    – WillO
    Jan 13 '14 at 2:14

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