Why is the photon clock equivalent to all clocks?

I can understand why, if the speed of light is invariant, a photon clock would tick slower. I find this explanation very useful in terms of introducing the idea of time dilation (also because it allows for the Lorentz formula to be derived intuitively, only using Pythagora's Theorem).

But this approach has one important missing concept. A student might say; Okay I get why the photon clock would tick slower, but why is it an intrinsic property of time itself? Why is this not some effect of the mechanics of this specific clock? How are a pendulum clock, an atomic clock, circadian rhythms, a chemical clock, etc... all equivalent to the photon clock? Why the slowdown of the ticking of the photon clock is a probe on the very nature of all clocks and time itself and not just a probe on the nature of this particular clock (more so if we consider that the explanation relies on the specific mechanism of this clock to work)?

For example some students might reason; a pendulum clock would slow down on the lunar surface, since the gravity is lower and therefore the pendulum would have a larger period, but we don't immediately jump to the conclusion that time itself has slowed down on the Moon with respect to Earth (in fact, ironically, in general relativity it is the other way around), just that the technical features of this particular clock make this happen because we have altered its functionality by altering the physical enviroment where it operates. The same could be said of a spring clock submerged in water for example. But if we don't think that the Moon gravity slows time with respect to Earth's just because the pendulum clock ticks slower, or that water slows time just because the spring clock ticks slower, then why should we think that moving at a certain relative speed slows the flow of time just because the photon clock ticks slower?

Invoke the principle of relativity.
An inertial observer carries both a light clock and a mechanical wristwatch, which agree when all are at rest. If they don't agree when the inertial observer is moving [with nonzero constant velocity] carrying these clocks, then that observer can distinguish being at rest from traveling with nonzero constant velocity.

UPDATE:
Q: What makes the photon clock special among all other clocks?
A: Simplicity.
It's easier to formulate, analyze, and interpret than other clocks.
If the principle of relativity holds, it must turn out that one can eventually analyze any clock and get the same result as the light-clock---it probably takes a lot more analysis and interpretation [of the device, the surroundings, and the interactions].

• Okay, I think I somehow get it. Since the idea of relativity (since Galileo) is that there is no way to know your absolute velocity, since there is no absolute space to refer that velocity, any experiment that allows you to know if you are moving at a certain speed in absolute terms can't happen. Thus if you have many different clocks, including a photon clock, and you see a discrepancy between them you could argue that you have reasons to believe you are moving, which is impossible in abolute terms, thus all clocks must behave the same way as the photon clock. Is this right? – Swike Aug 3 '20 at 13:09
• That's essentially correct. An important feature of the principle of relativity is that the motions being compared are inertial [that is, nonaccelerated]. One cannot distinguish inertial-at-rest from inertial-with-nonzero-velocity. – robphy Aug 3 '20 at 13:25
• Your answer is not convincing. Assume that I have a light clock, as well as a mechanical one, both with the same rate in my own rest frame of reference. How would you justify that the dilated time rates remain the same (dilated by the same factor, that is) as measured from the viewpoint of a moving observer? If the moving observer has similar photon and mechanical clocks to those held by me, they would also dilate from the viewpoint of me dissimilarly. This cannot distinguish the rest from the traveling frame. – Mohammad Javanshiry Aug 3 '20 at 13:53
• @MohammadJavanshiry As each of his clocks tick off one second, he makes a mark, say, on a graph. This record is invariant, All other external observers, regardless of their states of motion, will agree that he says that all of his clocks ticked identically. So, whatever time-dilation factor is assigned by an external observer to his light-clock must also be assigned to his wristwatch. – robphy Aug 3 '20 at 15:12
• @NuclearWang robphy didn't say that all observers would agree that his clocks ticked identically. He said that all observers would agree that he says that all of his clocks ticked identically. – Dan Henderson Aug 5 '20 at 22:32

Based on my current understanding of the topic the light clock is not a proof of time dilation but simply a clue, pointing at it. You are absolutely right in stating that the light clock is not a proof. In fact there isn't any proof of Lorentz transformation at all. Lorentz transformations are not proven, Lorentz transformations are postulated. This transformations are our best guess of how time and space works in absence of gravity and acceleration. Sure we can see that our experiments, mental or physical, agree with them, but this is not a proof, is reasoning by induction at best.

This kind of mental experiments, such as the light clock, help us to guess the correct form of the transformation, but there is no way to prove them. This is a recurrent theme in physic.

• Although Einstein may have initially postulated them, it is possible to prove that the Lorentz transformations are the unique linear transformations under some key assumptions (e.g. light has speed $c$ in every reference frame). Inability to prove things is recurrent in many parts of physics, but not in this case. So please fix your wrong answer that claims there isn't any proof at all. – user21820 Aug 4 '20 at 6:44
• But you indeed have to postulate that the transformation are linear to prove them from the principles of relativity – Noumeno Aug 4 '20 at 12:04
• I didn't say anything about linearity; I highlighted that uniqueness is what you can prove. – user21820 Aug 4 '20 at 18:39

Why is the photon clock equivalent to all clocks?

Stipulate that, in some inertial reference frame, there is a photon clock and some mechanical clock that are co-located and at rest in this frame.

Further stipulate that, in this frame, the clocks run at the same rate, i.e., both clocks 'tick' simultaneously.

Now, because the two clocks are co-located, all inertial observers in relative motion to the clocks agree that the ticks are simultaneous. Whatever time dilation is observed by the relatively moving inertial observers affects both clocks identically.

A mechanical clock is made of cogs and springs which transfer forces to the clock hands along complicated pathways. But if you look really close, all of those cogs and springs are made from protons and electrons, held together by electromagnetic forces and they push on each other by exchanging photons, just like in the simplified photon clock model.

• That's really not what photon clocks are – ɪdɪət strəʊlə Aug 3 '20 at 22:44

In my answer here, you can replace the horizontal photon clock with an atomic clock, a pendulum clock, or the clock of your choice.

The "moving" photon clock is in a frame of reference that has (in principle) a set of synchronized clocks distributed along the axis of relative motion to the "stationary" frame. So whatever is deduced about the photon clock must apply to the other clocks in that frame as they are synchronized. The fact that some other observer is measuring his clock in no way affects his clock nor its synchronization with respect to all the other clocks.

You are asking "Why the slowdown of the ticking of the photon clock is a probe on the very nature of all clocks and time itself and not just a probe on the nature of this particular clock (more so if we consider that the explanation relies on the specific mechanism of this clock to work)?"

Now it is because the photon clock is the fundamental clock, it is like the "elementary particle" of clocks. I will use here the theoretical explanation of the photon clock where there is always vacuum inbetween the mirrors, that is, the photon bounces in vacuum always.

This is because the photon clock only (disregarding the mass of the mirrors) uses one fundamental underlying element of the universe we live in, that is light (EM wave) and the speed of light. All other clocks are more complex to explain, yes, even the atomic clock.

The speed of light is c always, when measured locally in vacuum. Furthermore, all inertial observers will agree on this one speed, independent of their relative motion.

You are saying, that in the case of other clocks "Why is this not some effect of the mechanics of this specific clock?" and "just that the technical features of this particular clock make this happen because we have altered its functionality by altering the physical enviroment where it operates.", now this is the crucial point in your question.

You cannot alter the functionality of the photon clock because of its simplicity, its fundamentality, its elementarity, and because it uses only one single ingredient, the speed of a single photon bouncing between two mirrors (where we disregard the mass of the mirrors).

• Thank you for this explanation, but I think it is wrong. The photon clock can alter it's functionality in a non-fundamental way if for example we use it inside the Earth's atmopshere (where the reflactive index of air will slow down the photon) in contrast with what would happen in the vacuum of space. I think the correct answer might be the one given by robphy, even if it wasn't explained in too much depth. – Swike Aug 7 '20 at 16:30
• @Swike correct, that is why I said it was about the vacuum speed of light, but I will edit. The photon clock is (as I thought) always a theoretical clock with vacuum inbetween the mirrors. But I will edit. – Árpád Szendrei Aug 7 '20 at 16:34

Your starting position is not accurate. Relativity depends on one axiom.

• The general principle of relativity: Local laws of physics are the same irrespective of the reference matter which a particular observer uses to quantify them.

It follows immediately that all clocks measure exactly the same unit of time (to the accuracy of the clock mechanism). They run at precisely one second per second. Time dilation does not affect the rate at which a clock measures time. It is an apparent effect, due to the way in which an observer views a moving clock. You can compare this to the observed length of a rod held at an angle to the line of sight of the observer. The observer does not see the true length.

Unfortunately, early on in relativity, Einstein thought that the observed properties (length and time) should be thought of as the physical properties. Later, he changed this, and introduced the general principle which makes clear that the physical properties of an object are measured by an observer moving with the object. Even more unfortunately, over 100 years later, low grade and popular accounts have not caught up.