# Is the non-existence of a rest frame for photon in vacuum a consequence of the second postulate?

So there are multiple reasons given as to why does a photon (or any massless particle) have no rest frame (inertial, of course). I perfectly understand all the possible explanations one can give - it gives nonsensical results in terms of length contraction and time dilation, the standard argument where the energy momentum relation applied in such a frame gives $E^2=m^2+p^2=0$ implying 'no photon', etc.

But I am rather surprised to find very little mention of what I think should be the most obvious answer-The postulate of relativity. In all inertial frames, light (hence photons) must travel at $c$; so it is impossible to have an inertial frame where it is moving at anything other than $c$, let alone 'at rest'. So, no inertial rest frame for photons. This is the way I've understood it so far.

Is there a flaw in my reasoning? i.e. does the existence of a rest frame NOT violate the second postulate, but is wrong because of the other reasons mentioned above? (Hence nobody ever mentions it?) Or is it just too trivial to mention when there are more sophisticated arguments?

• It seems the most simple and logic way to see it. – Alchimista Dec 19 '17 at 10:38
• – Qmechanic Dec 19 '17 at 11:15

Given that the second postulate is what distinguishes Galiliean relativity from Einsteinian relativity, then the answer is yes.(*) An observer cannot move with the invariant speed because sie would necessarily have to see things in hir frame that moved with that speed be at rest, yet at the same time by the notion of invariance of the speed, in motion - a contradiction, which proves that no such reference frame can exist.

(*) Actually there are those that will claim that you can derive SR from only the first postulate, plus symmetry of space and time. This is, depending on how you interpret it and more accurately how you interpret math, something that may or may not be correct. You can get from the first postulate and the symmetry of space that the necessary transformation group on the space-time that gives the transformations relating reference frames can be one of three possible groups: the Euclidean group, the Poincare group, or the Galilean group. The Poincare group gives SR, it corresponds to taking the second postulate as well. The Galilean group gives Galilean relativity (the space-time background of Newtonian mechanics - note not "Newtonian mechanics" itself, that's a dynamical theory set therein; you can also set quantum mechanics in either foundation, indeed "undergrad QM" is just QM in Galilean background.) and approximates SR at low speeds. The Euclidean group was not the one that who/whatever decided the laws of nature used for our Universe used. (If you want, there are some very nice sci-fi novels by the Australian author Greg Egan called "Orthogonal" which explore the possibilities of a universe built using this case. It's very, very weird I'll tell you, but astonishingly, it manages to work out and could perhaps even support life. I've read it a bit; I'd recommend it a lot if you are into that kind of stuff.) Which one of these is actually the case is not determined from the first postulate alone.

The reason I say that "interpret math" is important is because technically when you derive this in the most "natural" way (again, interpretation interpretation) you get that the frame transformation group has a free parameter $K$, and which of the sets above you can get depends on the domain you allow for that parameter (which must be consistent enough for the logic to work out). If you allow your invariant speed $K$ to take values in the mathematical set $\bar{\mathbb{R}} \cup i\mathbb{R}$, that is, either imaginary or extended real values meaning you admit $\infty$ as an actual number, these three become unified into a single mathematical entity, and in the cases where the speed is not imaginary, _including $K = \infty$, the speed $K$ will have the property you mention. If $K$ is imaginary, every speed (since actually moving at imaginary speed doesn't make sense here because our spatial dimensions are strictly real-valued coordinates - I have no idea what happens if you try to extend them to be complex, but that would not be our universe or anything like it although it's a natural speculative possibility) including infinite speed will have a rest frame. So you could say that the most general solution to the first postulate in its full extent is a Poincare-like group with a free parameter $K$ which can range in this set. But we could apply constraints to $K$ then from considerations of "what we call mathematically meaningful" such that, starting from this formalism, those other groups would be weeded out.

However, you could also argue that the restrictions on choice of domain are essentially equivalent to assuming some form of the second postulate (you could say in a way Newtonian mechanics even assumes its own "second postulate" which is $K = \infty$. A weaker postulate that both the Newtonian and SR one is some statement to the effect that $K$ is extended real only. The fantastic postulate is $K$ is imaginary; I wonder how you'd formulate that in "physical" terms - I haven't read enough of Greg Egan, he probably knows :) Technically the SR postulate is stronger than "$K$ is real", it's actually $K = c$ where $c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}$ is taken from the natural speed in Maxwell's equations.). Thus I am a bit leery of saying "SR is derivable from the first postulate alone".

• Thanks a lot! What fascinated me the most is the 2nd part of your answer, the part about deriving SR from the 1st postulate entirely. Never heard of that one. I'll try to read up on it! – GRrocks Dec 19 '17 at 16:33
• @GRrocks : Thanks :) Another thing to point out is that it does not follow from purely geometric constraints that the invariant speed $K$ must be a limiting speed (indeed in the Euclidean case it's not a speed anything can travel at at all as it's imaginary, and there is no speed limit.). That $K$ is a limiting speed follows from the additional imposition of the requirement of unidirectional causality, that effect must always temporally precede cause or, that an "arrow of time" exists. This also rules out the Euclidean case as well - but it's important to point out (cont'd) – The_Sympathizer Dec 20 '17 at 2:09
• (cont'd) that this is an additional assumption, while the derivation can rest purely on geometry alone. Even without causality, $K$ will be a limit to speed achieved by acceleration , but there may also be particles that always move with speed higher than $K$ (and cannot decelerate to below it), these are called "tachyons". But we have not seen that, and furthermore observe strict causality directly, so this assumption holds empirically. – The_Sympathizer Dec 20 '17 at 2:10

There is no flaw. You are correct in your reasoning.

The first nine paragraphs of the Fourth Revised English Edition of Course Of Theoretical Physics Volume 2: The Classical Theory Of Fields by Landau and Lifshitz constitute probably the best introduction to relativity that I've personally read:

For the description of processes taking place in nature, one must have a system of reference. By a system of reference, we understand a system of coordinates serving to indicate the position of a particle in space, as well as clocks fixed in this system serving to indicate the time.

There exist systems of reference in which a freely moving body, i.e. a moving body which is not acted upon by external forces, proceeds with constant velocity. Such reference systems are said to be inertial.

If two reference systems move uniformly relative to each other, and if one of them is an inertial system, then clearly the other is also inertial (in this system too every free motion will be linear and uniform). In this way one can obtain arbitrarily many inertial systems of reference, moving uniformly relative to one another.

Experiments show that the so-called principle of relativity is valid. According to this principle all the laws of nature are identical in all inertial systems of reference. In other words, the equations expressing the laws of nature are invariant with respect to transformations of coordinates and time from one inertial system to another. This means that the equation describing any law of nature, when written in terms of coordinates and time in different inertial reference systems, has one and the same form.

The interaction of material particles is described in ordinary mechanics by means of a potential energy of interaction, which appears as a function of the coordinates of the interacting particles. It is easy to see that this manner of describing interactions contains the assumption of instantaneous propagation of interactions. For the forces exerted on each of the particles by the other particles at a particular instant of time depend, according to this description, only on the positions of the particles at this one instant. A change in the position of any of the interacting particles influences the other particles immediately.

However, experiment shows that instantaneous interactions do not exist in nature. Thus a mechanics based on the assumption of instantaneous propagation of interactions contains within itself a certain inaccuracy. In actuality, if any change takes place in one of the interacting bodies, it will influence the other bodies only after the lapse of a certain interval of time. It is only after this time interval that processes caused by the initial change begin to take place in the second body. Dividing the distance between the two bodies by this time interval, we obtain the velocity of propagation of the interaction.

We note that this velocity should, strictly speaking, be called the maximum velocity of propagation of interaction. It determines only that interval of time after which a change occurring in one body begins to manifest itself in another. It is clear that the existence of a maximum velocity of propagation of interaction implies, at the same time, that motions of bodies with greater velocity than this are in general impossible in nature. For if such a motion could occur, then by means of it one could realize an interaction with a velocity exceeding the maximum possible velocity of propagation of interactions.

Interactions propagating from one particle to another are frequently called "signals", sent out from the first particle and "informing" the second particle of changes which the first has experienced. The velocity of propagation of interaction is then referred to as the signal velocity.

From the principle of relativity, it follows in particular that the velocity of propagation of interaction is the same in all inertial systems of reference. Thus the velocity of propagation of interactions is a universal constant. This constant velocity (as we shall show later) is also the velocity of light in empty space. The velocity of light is usually designated by the letter $c$, and its numerical value is $$c = 2.998 \times 10^{10} \mathrm{cm/sec} \tag{1.1}$$

• Thanks a lot for the resource recommendation! Appreciate it. – GRrocks Dec 19 '17 at 16:34