Suppose a train is travelling forwards with speed $v$ as perceived in the inertial reference frame of the railroad. Suppose moreover that there is a lightbulb at the leftmost-end of the train, and that it emits a ray of light that hits the rightmost-end of the train. If the length of the train in its rest frame is $\ell$, then in its rest frame the time $t'$ taken for the light ray to reach the rightmost-end of the train should be: $$t'=\frac{\ell}{c}$$ On the other hand, in the rest frame of the railroad, taking the Lorentz-Fitzgerald contraction of the train along its direction of motion into account, the time $t$ that the light ray appears to take to hit the opposite end should satisfy: $$ct=\frac{\ell}{\gamma_v}+vt$$ However, when combining these equations together, I get: $$t=\sqrt{\frac{c+v}{c-v}}t'$$ rather than the usual time dilation formula $t=\gamma_vt'$. Can someone explain the error in this derivation?
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2$\begingroup$ What is your question? $\endgroup$– trulaCommented Apr 24, 2022 at 14:46
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$\begingroup$ sorry, I accidentally posted before I had asked my question $\endgroup$– SurfaceIntegralCommented Apr 24, 2022 at 14:49
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$\begingroup$ Your equation c*t=... is not valid, it says the speed of light is (c-v), if you write it in the form $ct-vt=l/\gamma$ $\endgroup$– trulaCommented Apr 24, 2022 at 15:00
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1$\begingroup$ As I mention here, you can't just use the $\gamma$ separately on distance & time and combine them, you need to use the full Lorentz transformation to transform $(t, x)$ coordinates between frames. $\endgroup$– PM 2RingCommented Apr 24, 2022 at 15:03
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$\begingroup$ Even when I apply the full Lorentz transformation, I get the same answer as above: $$ct'=x'=\ell$$ so $$ct=\gamma_v(ct'+\beta_vx')=\gamma_v(1+\beta_v)ct'=c\sqrt{\frac{c+v}{c-v}}t'$$ Hence $$t=\gamma_v(1+\beta_v)t'=\sqrt{\frac{c+v}{c-v}}t'$$ $\endgroup$– SurfaceIntegralCommented Apr 24, 2022 at 19:01
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1 Answer
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The time dilation formula is a special case that applies only to the interval between two events that are in the same location in one frame and in two different locations in another. The example you considered is not such a case, and so the time dilation formula dos not apply.
answered Apr 24, 2022 at 19:44