# Can someone explain Einstein's light clock?

The explanations for time dilation that I have seen all use the thought experiment of a photon bouncing between two parallel mirrors, which are themselves moving in a direction perpendicular to the photon’s motion. From the Pythagorean theorem, along with the formula velocity * time = distance, and assuming a constant speed of light, one can easily derive the Lorentz transformation and the phenomenon of time dilation.

What I don’t understand is this: it seems to me that in order for an observer, who is presumed to be “at rest”, to perceive such an effect, that photon would have to be in two places at the same time: bouncing back and forth forever between the mirrors, and also in the eye of the observer (or external measuring apparatus). This seems quite impossible. Either that, or there would have to be another photon – a stream of photons – emanating out from the bouncing photon itself back to the eye of the observer or his measuring device. This seems just as impossible. It would destroy the simple geometry of the intended thought experiment. In other words, either one thing would have to be in two places at the same time, or there would have to be two things, in two different places, that are “really” one thing.

It seems to me that the explanatory power of such a thought experiment depends on a “slippage”, or conflation, of two points of view: one is the point of view of an outsider – the all-knowing theorist who creates this diagram (or those who read it and understand it) and computes the Lorentz gamma factor. The other is an observer who is presumed to be located inside the thought experiment and is subject to its posited parameters. The theorist is outside, not subject to any constraints, and knows everything that is going to happen. The observer who is said to witness the time dilation is inside, subject to the effects that are a result of the geometry and the postulate of the constant speed of light.

In other words, it seems to me that the thought experiment illegitimately claims that an actual situated observer would be able to perceive something which an external, omniscient theorist can only conceive.

• You've already gotten a good answer, but in case you're still not happy, note that time dilation can be derived in about 20 different ways, most of which don't involve a light clock. The light clock is just the simplest way, which we show to the laypeople. – knzhou May 30 '18 at 7:38
• Can you please direct me to a source (preferably a web site) which does not rely on the light clock? – sumwunyumaynotno May 30 '18 at 15:44
• @sumwunyumaynotno Here's a classic paper that only requires a bit of calculus and linear algebra (multiplying matrices): arxiv.org/pdf/physics/0302045 – Mark H May 31 '18 at 0:32

## 2 Answers

The at-rest observer doesn't need to observe the actual photon that is bouncing between the mirrors, only some signal of when and where the photon contacted the mirror. For example, the mirrors could be mounted on pressure-sensitive switches that are so sensitive that a single reflecting photon is enough to activate it. This switch activates another light that is visible to the at-rest observer. The photon is then free to keep bouncing. Since this is a thought experiment, we can ignore the energy lost from the photon in doing work to press the switch--or make it negligibly small.

You are right about the need to be careful about what each observer in a relativity thought experiment is allowed to observe. There should never be a God's-eye-view of the situation that gets to say what's "really" going on. Relativity is all about correlating the observations of two observers in different reference frames without privileging either one as more correct.

• I believe this would still have the same problem that I have described in my answer to tfb, below. The observer would have no way of knowing that any light signal he receives has anything to do with some bouncing photon moving away from him. The saw-tooth pattern of right-triangles, from which the Lorentz contraction can be derived, would be gone. The "God's-eye-view" is presupposed by the graphic representations in all the explications that I have seen of how the Lorentz gamma factor is derived. – sumwunyumaynotno May 30 '18 at 15:42
• @sumwunyumaynotno Why would the light signal not be related to the clock photon? Since this is an idealized thought experiment, we can make the inside of the train car dark so that only the clock photon activates the light switches. – Mark H May 30 '18 at 18:58
• Because it is not the photon. You would still have the problem of unifying two different entities. The conventional explanation makes sense only if you can assume that the at-rest observer - who is never depicted , as far as I have seen, in such diagrams - could actually see the ongoing path of the photon as it oscillates. And that is what I am questioning. The time dilation and the Lorentz transformation are derived via simple calculation via the Pythagorean theorem from the right triangle whose hypotenuse is traced out by the bouncing photon. – sumwunyumaynotno May 30 '18 at 19:33
• That triangle - sequence of triangles, actually - a sawtooth pattern - can only be conceived by the God's-eye-view theorist, not perceived by a presumably at-rest observer. That observer would only receive a sequence of individual "blips". How could you use the Pythagorean theorem and the postulate of the constant velocity of light to derive the Lorentz factor from that? You would have to provide a diagram showing how it would happen. – sumwunyumaynotno May 30 '18 at 19:45
• @sumwunyumaynotno Is inferring the path of the clock photon by drawing straight lines between the blips not satisfactory? – Mark H May 30 '18 at 20:10

Special Relativity and General Relativity are classical, not quantum theories: they don't deal in photons or any of the associated observational questions.

Instead consider a light clock where short, bright, pulses of light are bouncing between the mirrors, which are slightly semi-silvered so that a detector, behind the mirrors, can measure the arrival time of the pulses, while allowing a single pulse to travel many times between the mirrors.

Practically, such a thing can be made: people can make bright EM pulses of the order of femtoseconds ($10^{-15}\,\mathrm{s}$) long, which are therefore of the order of $10^{-6}\,\mathrm{m}$ long.

• I was taught that all of relativity theory is considered "non classical", because it overturns some fundamental notions of classical Newtonian physics - the conception that there exists an absolute, universal time, the same for all observers. As I mentioned, every explanation of the phenomenon of time dilation explicitly refers to the light clock, the bouncing photon, and the moving pair of mirrors. – sumwunyumaynotno May 30 '18 at 15:18
• In your modified version, you would still have to show how an observer (or apparatus) could receive a photon which would designate the pulse between the mirrors. The nice right-triangle and the Pythagorean geometry would be gone, and there would still be the problem for the observer of associating the the light signals he receives with some bouncing photon, which he knows nothing about, since he doesn't see it. – sumwunyumaynotno May 30 '18 at 15:24
• The geometry is exactly the same. The whole point of the half-silvered mirror is that some of the light leaks out so you can observe it. Unless you somehow believe that we have to observe the light at all the intermediate points along the path rather than when it hits the mirrors, which would be pretty silly. – tfb May 30 '18 at 17:24
• The geometry is not the same. You're back to relying on two entities again: the bouncing photon, and the photon that reaches the observer's eye or measuring device. I would need to see a diagram depicting the various parts of this thought experiment, including the at-rest observer, from which we could use simple geometry to deduce the consequence of the Lorentz factor and the observer's perception of time dilation in the moving frame. – sumwunyumaynotno May 30 '18 at 19:51
• @sumwunyumaynotno I give up. – tfb May 30 '18 at 22:48