PeterDonis said:
This is the "transverse" light clock, in which the light bounces back and forth in a path that is transverse to the direction of relative motion of the two frames.
If you also add a second light pulse and a second mirror, aligned parallel with the direction of relative motion, with the length in the clock's rest frame the same in both directions (so the two pulses, after being reflected off the mirrors, arrive back at the origin at the same event), you can then analyze this clock in the frame in which it is moving and recover length contraction as well as time dilation. (This configuration is similar to the one used in the Michelson-Morley experiment, except that we've left out measurement of interference fringes.)
This second clock is the "longitudinal light clock".
The usual story is that to make longitudinal light clock
agree with the time-dilation observed in the "transverse light clock",
the longitudinal light clock must be length contracted.
(Note: No contraction in the transverse direction using the "nails on a meterstick" argument
https://www.physicsforums.com/threads/length-contraction-question.121833/#post-997499 )
You can find this in, e.g., Griffith's Introduction to Electrodynamics 4e, Ch 12.
As
@PeterDonis notes, this is essentially the Michelson-Morley apparatus, as described in special relativity to explain the null result, which was unexpected from a pre-SpecialRelativistic viewpoint.
(
https://en.wikipedia.org/wiki/Michelson–Morley_experiment ).
The apparatus was a clever way to measure the expected time-difference in the round trips since high-precision clocks weren't available then.
You can see the geometry involved in the MM-apparatus (the transverse and longitudinal light clocks)
in a [seemingly uncommon] spacetime diagram of the MM-apparatus.
pre-SR (without length-contraction): TX and TY are distinct reception events
SR (with length-contraction): TX and TY are coincident reception events
In addition to this txy-spacetime diagram,
you may wish to view
the xy-plane (for the transverse light clock) and
the tx-plane (for the longitudinal light clock)
using
https://www.geogebra.org/m/XFXzXGTqFor an older animation,
based on an old paper of mine
(VPT) "Visualizing proper-time in Special Relativity"
https://arxiv.org/abs/physics/0505134
[which would inspire the "Relativity on Rotated Graph Paper" (RRGP) approach mentioned by
@pervect]
look at
In the above VPT-paper, the MM-apparatus is generalized into a "circular light clock"
whose ticks provide a visualization of proper-time...
which I use to physically argue what special relativity numerically predicts
from applying Einstein's postulates with light-clocks,
which can further be analyzed with textbook formulas... and various ways of interpreting the result.
I feel this argument and geometric construction is a "physics-first" approach,
which can be followed up by the typical textbook algebraic approach.
Time dilation (with length-contraction and relativity-of-simultaneity)
[has sound!] :
and the Clock-Effect/Twin-Paradox
[has sound!] :
for more details, go to my site:
http://visualrelativity.com/LIGHTCONE/LightClock/By the way,
in addition to my
https://www.physicsforums.com/insights/relativity-rotated-graph-paper/ that
@pervect mentioned,
there is one on the Bondi k-calculus
https://www.physicsforums.com/insights/relativity-using-bondi-k-calculus/From these light-clock approaches, I try to regard
"length contraction" as a [supplementary] effect (and not a [primary] cause).
What I regard as more primary is the "causal diamond" between ticks,
because that is what is Lorentz-invariant and is more connected to "causality",
which I think is primary in relativity.
Then, from the Rotated Graph Paper approach, all of (1+1)-Minkowski geometry can be extracted.
(I would think (3+1)-Minkowski geometry should also be extractable from the full causal diamond.)