# Can two inertial frames differ by the speed of light?

• I
I know that special relativity is based on the postulate that an object that travels at the speed of light with respect to one inertial observer must travel with the speed of light with respect to all inertial observers. This postulate implies that no two inertial frames can differ by a relative speed equal to that of light.

But the issue of whether two such inertial frames exist is experimentally unfalsifiable, for the following reason: Suppose we modify the constancy-of-speed-of-light postulate of special relativity to the following instead:

If an object travels at the speed of light with respect to one inertial observer (call him Observer 1), then the object must travel at the speed of light with respect to all inertial observers that move at a speed lesser than that of light with respect to Observer 1. Moreover, for an observer (call him Observer 2) and a massless object that both travel at the speed of light with respect to Observer 1, either the object travels at the speed of light also with respect to Observer 2, or the object is nonobservable with respect to Observer 2 (suggestive of the correct fact that a massless object only gains relativistic energy and becomes physically observable through traveling at the speed of light).

It is easy to see that the new version of the special theory of relativity that would form under this new postulate would predict identical experimental results to the original version, and yet the "new" version does not ban two inertial frames from differing by the speed of light, unlike the original version. This shows that the issue of whether two different inertial frames can differ by the speed of light is itself experimentally meaningless.

Yet, I am confused because special relativity textbooks seem to emphasize that no inertial frames can travel at the speed of light with respect to another inertial frame, even though this is something that has not been (and cannot be) experimentally verified or proven (unlike time dilation and E=mc^2, which are very well tested by experiment). What scientific purpose is there to even deal with issues concerning whether an inertial observer can actually travel at the speed of light or not if there is no way to experimentally test any claims made on these issues? Physics is an experimental science and I don't see how physicists can seriously make "scientific" claims about the natural world that cannot be tested experimentally?

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Drakkith
Staff Emeritus
Physics is an experimental science and I don't see how physicists can seriously make "scientific" claims about the natural world that cannot be tested experimentally?
Can you elaborate on why you think this is not falsefiable? You say that it's easy to see that predictions between relativity and your modified theory would be identical, but I remain skeptical since, in my opinion, you haven't shown how this is so.

Ibix
I know that special relativity is based on the postulate that an object that travels at the speed of light with respect to one inertial observer must travel with the speed of light with respect to all inertial observers.
Actually, you can derive relativity without assuming the invariance of the speed of light - see for example https://arxiv.org/abs/physics/0302045.

Furthermore, a consequence of special relativity is that objects not travelling at the speed of light cannot reach the speed of light and vice versa. You introduced an observer (necessarily a system of moving parts) travelling at the speed of light - so its components must move below, at, and above the speed of light. So you are assuming things that would falsify relativity in order to construct a theory that you claim is experimentally indistinguishable from relativity.

bhobba
PeterDonis
Mentor
2019 Award
It is easy to see that the new version of the special theory of relativity that would form under this new postulate would predict identical experimental results to the original version
No, it doesn't. It predicts that two inertial observers can be traveling at the speed of light relative to each other. Standard SR predicts that this is impossible.

It also predicts that an inertial observer can be seen by another inertial observer to be keeping up with a light ray moving in the same direction. Your Observer 1, for example, would see Observer 2 keeping up with a light ray moving in the same direction. Standard SR predicts that this is impossible.

bhobba
PeterDonis
Mentor
2019 Award
Suppose we modify the constancy-of-speed-of-light postulate of special relativity
Doing this would also require modifying the entire mathematical structure of the theory, since the mathematical description of inertial frames and the Lorentz transformation requires the standard SR constancy of speed of light to be true, and has the fact that the relative speed of any two inertial frames must be less than the speed of light built into it.

bhobba
Dale
Mentor
the "new" version does not ban two inertial frames from differing by the speed of light
I am highly skeptical that this modification is logically self consistent. Yes, physics is an experimental science, but we also require logical self-consistency as well as empirical evidence.

Do you have any professional scientific references which can support the logical self consistency of your approach? Particularly, what is the coordinate transform between two inertial frames differing by c?

It is easy to see that the new version of the special theory of relativity that would form under this new postulate would predict identical experimental results to the original version,
It is not easy for me to see. I am not convinced that it is even logically self consistent unless the set of inertial frames moving at c is empty. Can you show that it is non-empty?

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bhobba
bhobba
Mentor
As Ibix alluded to SR is not really about the speed of light thing - it is about the symmetries imposed by the Principle Of Relativity. He posted one paper doing it that way - here is another very similar one:
http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf

It predicts the existence of an invariant speed in all frames - it may be infinity. Experiments from many different sources shows it is the speed of light. Now imagine you are in an inertial frame and there is a light beam passing by. You go to a faster frame to try and catch up to it - but because its the same speed in all frames you can not catch up to it. It not possible - as Wheeler expressed it - forward is always forward.

Thanks
Bill

FactChecker
Sorry, I wasn't clear before. What I was trying to say is that there does not seem to be any experiment one can perform that can show that one inertial observer can or cannot travel at the speed of light with respect to a second observer. For one to show that an inertial observer can move at the speed of light, one would either need to accelerate one of the observers to the speed of light (with respect to the other observer) which requires an infinite amount of energy, or one would need to find a massive particle that from the very start of the universe has been traveling at the speed of light (so that no acceleration was needed). However, no such particle has been observed so far. Neither has one proven that no such particle will ever in the future be found. So there is one key prediction of the theory of special relativity that has yet to be experimentally verified and that cannot ever be disproven.

However, we may modify the premise of special relativity to not make any assertions about the possibility or impossibility of observers traveling at the speed of light, but to simply state that for any two observers that travel at non-luminous speeds relative to each other, any particle that moves with the speed of light with respect to the first observer will also move at the speed of light with respect to the second observer. This way, in order to completely verify all predictions of the theory, one does not have to worry about designing an experiment to test the special limiting case of whether or not it is possible for a massive particle to travel at the speed of light (starting from the very beginning of the universe).

I was wondering whether there is a flaw in this argument in any way, because clearly, physics textbooks seem to emphasize the scientific truth of the statement that no observer can travel at the speed of light with respect to any other observer. If what I said above was true, then this special limiting case is merely included in the theory for mathematical elegance and completeness, and does not have any physical or experimental significance.

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P.S.

What I was trying to say in my earlier post was that we can modify Einstein's "constancy-of-speed-of-light" postulate to say (for example) the following two things:

1) For two inertial observers that have a relative speed less than that of light, a particle will travel at the speed of light with respect to one observer if and only if it does so with respect to the other observer.

2) If there exist two observers having a relative speed of c such that some massless particle moves at c relative to the first observer, then there are two options for the second observer: Either the second observer also observes the particle to move at c, or the particle completely disappears out of existence with respect to the second observer (as would any other massless particle that fails to travel at the speed of light).

If what I said earlier was true, then it is obvious (at least to me) that the second condition is entirely metaphysical (at least today) until we are able to show that luminous travel by a massive particle (created at the beginning of the universe) is or isn't possible.

Ibix
which requires an infinite amount of energy
That result, of course, assumes a speed of light that's invariant in all inertial frames. And therefore that only massless objects can travel at the speed of light, since in relativity "massless" and "travels at the speed of light" are literally two ways of saying the same thing (the four momentum is null).

If you are saying that massive objects can travel at the speed of light then you are not using relativity theory. You cannot, therefore, use the results of relativity in your argument.

Dale
Mentor
What I was trying to say is that there does not seem to be any experiment one can perform that can show that one inertial observer can or cannot travel at the speed of light with respect to a second observer
That doesn’t address my objection. Please respond specifically to the specific objections above, mine and others. This is not a blog, it is a discussion forum.

we may modify the premise of special relativity to not make any assertions about the possibility or impossibility of observers traveling at the speed of light
No modification is needed. The postulates already do not make such assertions.

Let me be a little more explicit here. The two postulates are (1) the laws of physics are the same in all inertial reference frames and (2) c is invariant in all inertial reference frames. There is no assertion about the possibility or impossibility of observers travelling at c in the postulates. I.e. the postulates are already inherently completely "agnostic" on that point. From the postulates you can then derive the result, but it is derived not assumed.

So, now you want to modify (2) to state that c is invariant in all inertial reference frames travelling at less than c wrt each other (subluminal). There are two possibilities, either the set of superluminal and luminal inertial frames is empty or it is not empty. If it is empty then there is no point to the whole caveat, so we should just use the original formulation for brevity. Therefore, this statement is in fact an assertion that such luminal or superluminal inertial frames do exist. Your modification changes the formerly agnostic statement to one that asserts the existence of superluminal or luminal inertial frames. Therefore you need to demonstrate that the existence of such frames is logically consistent. I am highly skeptical and this is the point that you have not justified.

one would need to find a massive particle that from the very start of the universe has been traveling at the speed of light
Can you show that this is logically self consistent? (Hint: it isn't)

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pervect
Staff Emeritus
Sorry, I wasn't clear before. What I was trying to say is that there does not seem to be any experiment one can perform that can show that one inertial observer can or cannot travel at the speed of light with respect to a second observer.
It's logically inconsistent within the context of special relativity to assume that an observer "moving at the speed of light" exists at all.

See for instance the FAQ https://www.physicsforums.com/threads/rest-frame-of-a-photon.511170/, which I'll quote.

A rest frame of some object is a reference frame in which the object's velocity is zero. One of the key axioms of special relativity is that light moves at c in all reference frames. The rest frame of a photon would require the photon to be at rest (velocity=0) and moving at c (velocity=299792458 m/s). That of course is contradictory. In other words, the concept doesn't make sense.
To be clear, it is my position that special relativity, when properly formulated, is self consistent, and that it does make testable predictions. I believe the problem in this instance lies in the particular formulation that is being proposed as representing "special relativity".

For an example of a testable prediction made by special relativity, one might ask for instance "if one accelerates an electron via an applied voltage, does the electron's velocity increase without limit as the voltage is increased, or is there some sort of limiting speed.

This is an example of an experiment that has been done. It's not one of the most precise tests of special relativity that has ever been done, but it is simple and basic test that has been done and is easy to communicate. The results of this experiment were consistent with special relativity and inconsistent with Newtonian physics..

,

There are many other examples of experimental tests of special relativity, this is just one particular one that's easy to understand. There is also a peer-reviewed paper documenting the above expeirment.

Wiki gives the published paper as Bertozzi, William (1964), "Speed and Kinetic Energy of Relativistic Electrons", American Journal of Physics, 32 (7): 551–555,

Published papers are a much more reliable source of information than youtube videos, though the youtube vidoes are "friendlier" to a non-technical audience. I thought I'd cover both bases by giving the video link, and also a link to the peer-reviewed published paper.

bhobba and Ibix
bhobba
Mentor
It's logically inconsistent within the context of special relativity to assume that an observer "moving at the speed of light" exists at all.
Exactly. No experiment necessary - its an inconsistency - as I said it would violate what Wheeler said - forward is always forward. Physicists take care in what they decide to test for - a logical inconsistency is usually not high on the list - even though the pay-offf would be great - because you would have disproved SR - something so well tested it's almost in the category of beyond doubt.

Thanks
Bill

So, now you want to modify (2) to state that c is invariant in all inertial reference frames travelling at less than c wrt each other (subluminal). There are two possibilities, either the set of superluminal and luminal inertial frames is empty or it is not empty. If it is empty then there is no point to the whole caveat, so we should just use the original formulation for brevity. Therefore, this statement is in fact an assertion that such luminal or superluminal inertial frames do exist. Your modification changes the formerly agnostic statement to one that asserts the existence of superluminal or luminal inertial frames.
You correctly said that if we instead postulate that the speed of light is invariant in all inertial reference frames that have a relative speed less than light with respect to each other, then there are two possibilities: Either there does exist inertial frames that travel with the speed of light with respect to each other, or there does not. But we actually don't know which of the two possibilities does hold, unless you can show me an experiment that proves that one of the two possibilities is true while the other one is false.

You said that my postulate is an assertion that such luminal inertial frames exist, but it is not. Like I said above, I am assuming that there are two possibilities: Either there exists such luminal inertial frames, or there does not. I have not told you which one of the possibilities is actually true. Yes, it is true that in the case that there does not exist such luminal inertial frames, we get back the original "constancy-of-speed-of-light" postulate, and thus if we were to assume this case is true (i.e. that there exists no luminal inertial frames), we should revert back to the original "constancy-of-speed-of-light" postulate for brevity. But this does not mean that just because I don't revert back to the original postulate, I must be assuming that the other possibility (namely that luminal frames exist) is definitely true. If I don't revert back to the original postulate, all that means is that I am not assuming that luminal frames don't exist. That does not mean that I am assuming that luminal frames exist. One is not confined to only either assuming that luminal frames exist or assuming that they don't exist; there is a third option, namely that one neither assumes that luminal frames exist nor assumes that they don't exist. This last possibility describes my stance.

Dale
Mentor
You said that my postulate is an assertion that such luminal inertial frames exist, but it is not.
If it is not then it is pointless and we should just use the standard formulation for brevity.

there is a third option, namely that one neither assumes that luminal frames exist nor assumes that they don't exist. This last possibility describes my stance.
But that is already the case of the existing postulates. There is no need to change them to allow this possibility. Without changing the postulates they are already agnostic and make no assumptions either way. The agnostic postulates then lead to an inevitable logical conclusion that such frames are logically contradicted. It is a derived result from the existing postulates, not an assumption.

Because the existing postulates are already agnostic, the modification intended to make them agnostic is unnecessary.

If I don't revert back to the original postulate, all that means is that I am not assuming that luminal frames don't exist.
That is pointless because the original postulate already does that. So why not revert unless you are in fact making the assertion?

unless you can show me an experiment that proves that one of the two possibilities is true while the other one is false.
First, you need to demonstrate that both possibilities are logically self consistent. Please do not post further without some professional scientific references to support your claim that the idea of such frames is even logically self consistent.

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PeterDonis and bhobba
Ibix
The simplest solution to this is to read Pal's paper, which I linked in #3. He starts with the principle of relativity. He makes no statement about the speed of light or any invariant speed. He concludes that only two systems of physics are consistent with this one postulate - one with an infinite invariant speed (Galilean relativity) and one with a finite invariant speed (Einsteinian relativity). Experiment shows we're in a universe of the latter sort. There is no maths more complex than matrix multiplication and one use of differentiation.

At no point was any assumption made about the existence or not of frames with a relative velocity of c. No mention of an invariant speed was even made in the assumptions. Adding the one you want to make is either changing the theory (if your "luminal frames" exist) or stating a consequence of the theory's postulates (if they do not).

Dale, PeroK and FactChecker
FactChecker