A typical Lorentz Contraction proof relies on the axiom that "the speed of light is constant" and goes as follows. Given:
- Frame $F_1$ moves at speed $v$ relative to frame $F_0$. In frame $F_1$ sit 2 parallel mirrors.
- Distance between mirrors is measured as $l_0$ in $F_1$ (at rest relative to mirrors).
- Distance between mirrors is measured as $l$ in $F_0$ (while mirrors move past in $F_1$ at speed $v$).
- Time for light to make "roundtrip" between mirrors measured as $t_0$ in $F_1$ (at rest relative to mirrors).
- Time for light to make "roundtrip" between mirrors measured as $t$ in $F_0$ (while mirrors move past in $F_1$ at speed $v$).
- Already proved $t = \frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}=\gamma t_0$ (time dilation).
A "roundtrip" of light passing between mirrors takes two trips; measured from $F_0$, those trips take times $t_1$ and $t_2$. During those trips, the ship travels $vt_1$ and $vt_2$, meaning light travels $l+vt_1$ and $l-vt_2$ when light moves in the same and opposite directions as $F_1$, respectively, all measured in $F_0$. The constancy of the speed of light gives:
- Trip 1 (light moves same direction as $F_1$ relative to $F_0$): $c = \frac{l + vt_1}{t_1}$ $\Rightarrow$ $t_1 = \frac{l}{c-v}$
- Trip 2 (light moves opposite direction as $F_1$ relative to $F_0$): $c = \frac{l - vt_2}{t_2}$ $\Rightarrow$ $t_2 = \frac{l}{c+v}$
- So, $\color{red}{t = t_1 + t_2 = \frac{l}{c-v} + \frac{l}{c+v} =\frac{2lc}{c^2-v^2}= \frac{2l/c}{1-\frac{v^2}{c^2}} = \frac{2\gamma^2}{c} l}$.
Measured in $F_1$, the "roundtrip" distance is simply $2l_0$, and so $c=\frac{2l_0}{t_0} \Rightarrow t_0 = \frac{2l_0}{c}$.
Combining this with time dilation yields $t=\gamma t_0 = \gamma\frac{2l_0}{c} = \frac{2\gamma}{c}l_0$.
Putting it all together yields $$\frac{2\gamma^2}{c}l =t = \frac{2\gamma}{c}l_0 \Rightarrow l = \frac{l_0}{\gamma} \tag*{$\Box$}$$
Question:
Can I shorten this proof to just use "one trip" between the mirrors instead of a "round trip"? I have tried, and cannot! I $\color{red}{\text{have highlighted in red}}$ the part of the proof where the round trip yields some nice cancellation.
What am I missing?
There are proofs that rely on axioms other than "the speed of light is constant", but I'm looking for a proof that just relies on that.
The typical proof for time dilation $t=\frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}$ involves light bouncing between two mirrors that are PERPENDICULAR to the motion of the reference frames. I went through this proof, and it absolutely does NOT break down when only one trip between the mirrors is considered. The proof in this question involves mirrors separated by a distance PARALLEL to the motion of the reference frames.
If "roundtrip" is unclear, here are two animations, each depicting two "roundtrips":
First image made by me. Second image from Help Me Gain an Intuitive Understanding of Lorentz Contraction , which goes through this same proof based on the speed of light being constant.
A proof I’ve seen for time dilation is as follows, and only seems to require a single trip of a light beam:
Suppose a pair of mirrors separated by distance $L$ is moving past at speed $v$, such that the displacement between the mirrors is perpendicular to the motion of the mirrors. In the reference frame of the mirrors, light bouncing between the mirrors travels distance $L$ in $t_0$ seconds at speed $c=\frac{L}{t_0}$. In the reference frame relative to which the mirrors are moving at speed $v$, however, light bouncing between the mirrors takes time $t$ to do so and travels $\sqrt{(vt)^2 + L^2}$. So, $c = \frac{\sqrt{(vt)^2 + L^2}}{t}$ as well because the speed of light is constant to all observers. Solving for $t$ and substituting $t_0=\frac{L}{c}$ yields $t=\frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}$.
Image used is from what about doing the laser beam in a moving reference frame but with a ball
