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Variations on this question have been asked a few times (e.g. here, here, and a few YouTube videos here and here). The claim seems to be that because we can only measure the round-trip speed of light, this leaves open the door for the one-way speed of light to an-isotropic. In the extreme case, advocates seem to even suggest that the speed of light can be infinite in one direction so long as it travels at only half the speed of light in the other.

I've read parts of Anderson et al. (here), and what I think I understand is that this is essentially a gauge choice for the synchronization of clocks in an inertial reference frame. Since the physics of special (and maybe general) relativity only care about spacetime lengths, the absolute value of a time at any given location in an inertial reference frame is unimportant. Per Anderson et al.:

An illustration from electrostatics may help; it is more closely related to our topic than may at first appear (Section 2.3.3). Voltmeters are useful, despite the dependence of the value of the voltage at any point on an arbitrary electromagnetic gauge choice. The conventional content of the voltage concept is well understood and is evidenced by the manufacturers’ provision of a second (earth) probe on each voltmeter. We would not counsel the removal of “Danger High Voltage” signs, even though the implied convention allows the re-classification of an electrical feeder wire as being at low voltage. On the other hand, if a company were to advertise a product with just one external probe which purported to “test for the absolute zero of voltage in seawater”, the matter becomes more serious than one of taste and judgement. This kind of claim is corrected in the following. Our fundamental aim is to clarify, if by analysis of such counterexamples, those testable facts which are independent of convention (see for example the discussion in Section 1.3.2).

So... I think that for special relativity, the specifics of how one synchronizes clocks across an inertial reference frame don't actually have any bearing on the predictions of the theory.

This seems to be a result of conceptualizing synchronization by having a "master" clock at the origin and sending out some signalling to "remote" clocks in the reference frame. We essentially can't necessarily all agree on when t = 0 (or rather, agreeing on when when t = 0 is not important for predictions of the theory). Since round trip times are limited by the speed of light, there is no way to confirm that a signal actually took a given amount of time to get from one point to another. You need to have an event that literally happened everywhere all at once that all observers can agree on. But the finite speed of light prevents that.

However, we do have another potential candidate: the Big Bang. If our current view of cosmology is right, the Big Bang did happen everywhere all at once. That is, I think that all observers in an inertial reference frame will agree on how long ago the Big Bang occurred. Can this be used to synchronize clocks across a frame?

The thought experiment is something like this: imagine we sent a light signal to another galaxy a billion light years away, and the content of the signal was something like, "We sent this signal 13.8 billion years after the beginning of the universe. When did you receive it?" If the speed of light is isotropic, I would expect our distant aliens to receive the light signal when they would measure the Big Bang to have been 14.8 billion years in the past. They would send a signal back telling us they observed the universe to be 14.8 billion years old, and we would receive the message when the universe was 15.8 billion years old. If the speed of light is not isotropic, then there would be a discrepancy as to the age our distant alien friends would assess the universe when they receive the message (although we would always assess the universe to be 15.8 billion years old when we receive the reply).

In the extreme example of the speed of light being infinite in one of the directions, our alien partners would receive the message when they assess the universe to be only 13.8 billion years old. I guess if some really weird gauges are correct, it might be the case our alien partners assess the age of the universe to be younger than 13.8 billion years when they receive the message (that would certainly have some weird implications, but I think that the gauge invariance technically allows for this possibility).

I realize there are many practical and technical problems with this thought experiment (not the least of which identifying some alien partners in a galaxy a billion light years away who could help us out). The question is more of, in principle, does this scheme work? Or would any an-isotropy in the speed of light cause distant observers to assess a different age for the universe?

UPDATE FOR CLARIFICATION

Based on some of the comments, I wanted to try to clarify my question. I think it is in two parts. Part 1 is if all inertial observers agree on the age of the universe? This question was inspired by Lewis et al. (https://arxiv.org/pdf/2012.12037.pdf) who seem to conclude that an-isotropy in the one-way speed of light may not be evident in observational astronomy; specifically that:

The conclusion is that the presence of an anisotropic speed of light leads to anisotropic time dilation effects, and hence observers in the Milne universe would be presented with an isotropic view of the distant cosmos.

Essentially, even though the sky presents as isotropic, this cannot be used as evidence that the one-way speed of light is isotropic because the very an-isotropy in the one-way speed of light would give rise to anisotropic time dilation effects and make the universe appear isotropic. The question is essentially, has any conducted a similar analysis on any effects that an-isotropic one-way speed of light would have on estimate of the age of the universe for observers located in physically different locations in the universe.

The second part of the question in contingent on the first part of the question. If, in fact, all inertial observers do agree on the age of the universe, can that be used to measure the one-way speed of light in principle? That is, can the beginning of the universe be used as a standard clock to establish some notion of an absolute time (i.e. time after beginning of the universe) that would get around the issues of synchronizing clocks (e.g. synchronization by light signal or the problem of slowly transported clocks discussed by Anderson et al.). There is, perhaps, some philosophical issue I'm missing that truly makes the one-way speed of light an undefined quantity that even in the presences of a definable standard clock makes the one-way speed of light undefined. So that's the nature of the second part of the question.

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    $\begingroup$ I don't think this question is a duplicate of the linked ones (though it is related). The OP has put some thought into this and so I'm voting to reopen (although I would suggest that an edit to make the question a bit more concise might be helpful) $\endgroup$
    – Eric Smith
    Commented Nov 20, 2023 at 2:26
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    $\begingroup$ I agree with @EricSmith . This is a thoughtful question and it is different from the linked (and far less thoughtful) questions. It should certainly be open. $\endgroup$
    – WillO
    Commented Nov 20, 2023 at 3:13
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    $\begingroup$ The impossibility to measure the one-way speed of light is a direct consequence of the way spacetime works with light. It is the reason why Special Theory of Relativity is successful. The language is very confusing if you argue it the way Veritasium did; it is however extremely obvious if you consider Minkowski diagrams. Once you understand what physical concept is being conveyed here, it would no longer make sense to keep asking questions about the one-way speed of light. $\endgroup$ Commented Nov 20, 2023 at 3:38
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    $\begingroup$ @naturallyinconsistent : The reasoning you're referring to concerns the speed of light traveling from one event to another within a given coordinate patch. It does not clearly apply to the OP's question, which involves light traveling great distances and also (more or less) assumes a cosmological model with a preferred global time coordinate, which is entirely missing from the usual argument. (So Minkowski diagrams don't apply.) I share your frustration with posters who ignore the usual well-established arguments, but I don't think this poster is the right target for that frustration. $\endgroup$
    – WillO
    Commented Nov 20, 2023 at 4:25
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    $\begingroup$ @WillO Yes, I am aware of that; indeed, I casted the 3rd reopen vote because I saw that it is much better than usual questions of this sort. But I disagree that, just because of a cosmological model, it circumvents the objection. The OP thinks that the original problem's resolution is merely a gauge choice on the synchronisation of clocks. I am saying that it is a fundamental property of spacetime. In more detail, this means that any acceptable cosmological model would have to satisfy this property too. $\endgroup$ Commented Nov 20, 2023 at 5:29

3 Answers 3

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… the Big Bang did happen everywhere all at once. <…> Can this be used to synchronize clocks across a frame?

Yes. Here is a paper that provides definition for such cosmological time function:

The main idea is seen from the abstract:

Let $(M,g)$ be a time oriented Lorentzian manifold and $d$ the Lorentzian distance on $M$. The function $\tau(q):=\sup_{p< q} d(p,q)$ is the cosmological time function of $M$, where as usual $p< q$ means that $p$ is in the causal past of $q$. This function is called regular iff $\tau(q) < \infty$ for all $q$ and also $\tau \to 0$ along every past inextendible causal curve. If the cosmological time function $\tau$ of a space time $(M,g)$ is regular it has several pleasant consequences: (1) It forces $(M,g)$ to be globally hyperbolic, (2) every point of $(M,g)$ can be connected to the initial singularity by a rest curve (i.e., a timelike geodesic ray that maximizes the distance to the singularity), (3) the function $\tau$ is a time function in the usual sense, in particular (4) $\tau$ is continuous, in fact locally Lipschitz and the second derivatives of $\tau$ exist almost everywhere.

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If the speed of light is not isotropic, then there would be a discrepancy as to the age our distant alien friends would assess the universe (14.8 billion years old) when they receive the message (although we would always assess the universe to be 15.8 billion years old when we receive the reply)

This turns out to be incorrect. A one way speed of light is also accompanied by a position-dependent time dilation. If the aliens lived in the "fast light" direction from us, then their part of the universe is older than ours. Our signal would instantaneously (or at least rapidly depending on the variation) get to the aliens, who live in a 14.8 billion year old part of the universe, then their return signal would take the remainder of the 2 billion years to reach us.

That is, I think that all observers in an inertial reference frame will agree on how long ago the Big Bang occurred. Can this be used to synchronize clocks across a frame?

You cannot have inertial reference frames over cosmological scales. However, you can use the age of the universe as a synchronization convention. This is the standard synchronization convention used in the cosmological literature. As with other synchronization conventions, this is just a matter of convention and alternative conventions are not ruled out by experiment.

Part 1 is if all inertial observers agree on the age of the universe?

Yes, with some small changes. All observers at a given event agree on the age of the universe, regardless of whether they are inertial or not. The age of the universe is calculated according to the proper time of a co-moving clock. This is is invariant, and is not based on the motion of the observer.

Observers at different events may or may not agree on the age of the universe. Furthermore, in different coordinate systems the age of the universe may or may not be spatially constant.

If, in fact, all inertial observers do agree on the age of the universe, can that be used to measure the one-way speed of light in principle?

No. The one way speed of light depends on the coordinate system chosen, but the choice of coordinate system cannot affect the outcome of any measurement.

For further details on this point see my answers here:

Can One-Way Speed of Light be Instantaneous?

What is wrong with these ways of measuring one-way speed of light?

Are the non-standard one-way speed of light conventions just transformations of coordinates?

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  • $\begingroup$ This makes sense (and embarrassingly is what Lewis et al. wrote). Quick follow-up question on the different patches of the universe being different ages, so aliens would always see the signal at an cosmological age of 14.8 billion years regardless of what we assign the one-way speed of light. The question is: does this reasoning regarding position-dependent time dilation work for 1<=abs(kappa) (or using Reichenbach's notation, 1<=epsilon)? Or can we exclude those gauges as being completely nonphysical, at least from the point of view of using cosmological age. $\endgroup$
    – user218912
    Commented Nov 21, 2023 at 0:47
  • $\begingroup$ @user218912 those are iffy. Anderson defined his synchronization criteria such that he would accept such conventions. Others prefer to require that the simultaneity convention foliate spacetime into spacelike surfaces, which would reject those conventions. I tend to prefer the foliation approach, but opinions vary. $\endgroup$
    – Dale
    Commented Nov 21, 2023 at 1:08
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RETRACTION: My first answer agreed that an anisotropic one way speed of light should be noticeable when making cosmic scale observations. After giving it some more thought I am retracting that position: I initially assumed that if light was travelling at 2c its travel time from emission to reception would be approximately half the age of the universe. That seems reasonable, but unfortunately, it was a wrong assumption because light travels differently in an expanding universe.

Take a look at the diagram below (I borrowed the background from the excellent website on Cosmology by Ned Wright)

enter image description here

This diagram represents the spacetime of galaxies being carried outward by the Hubble flow. Due to the galaxies co-moving with the expanding space in this model, the outer galaxies do not experience time dilation relative to the observer at the centre so proper time advances equally everywhere and a line of simultaneity according to the central observer is simply a horizontal line. The proper time of the receding galaxies is the same as the coordinate time of the central observer. The red curves are the light paths of isotropic light where the speed of light is equal in all directions. The light blue curve is the path of light that is travelling at 1.5c from left to right in an anisotropic universe. The green curve is the path of light that is travelling at 0.75c on its return journey from right to left in the same anisotropic universe. The horizontal yellowish line at the bottom represents the time of 'first light' after the recombination event when the plasma of ions produced by the big bang condensed into normal matter that allowed light to travel normally. This is essentially what we observe as the CMB. It can be seen from the diagram that light emitted simultaneously at first light arrives simultaneously no matter which direction it comes from, in an anisotropic universe. The observed redshift is proportional to the amount of expansion of the universe during the light travel time and since the coordinate travel time of the light is the same in either direction, the observed redshift of the CMB will the same from either direction. Whichever direction you look in, you see just as far back in time and there is no way to deduce the one way speed of light here.

enter image description here

This second diagram is a coordinate transformation of the first model, to the Minkowski space of Special Relativity. The galaxies are treated as if they are projectiles in flat space, moving outward from the central observer. In this model the outer galaxies do experience time dilation relative to the central observer. The lines of simultaneity are now hyperbolas because proper time is no longer equal everywhere. The orange hyperbola is the time of first light in this diagram. Calculations of velocity from redshift now have to take into account, time dilation due to relative motion in this model, but nevertheless distance/velocity/redshift measurements are the same as in the first model. Observations in either direction both see right back to the time of emission of the CMB and both directions see the same redshift for the CMB in either direction. The time of emission in terms of proper time (age of the emitting object) is the same in either direction. The 'surface' of the CMB is effectively moving away from us at close to the speed of light. As a result of this, there is extreme time dilation and while light emitted at event A was emitted at a later coordinate time than light emitted at event B, the proper time of emission is the same. The two models are almost indistinguishable, but we cannot safely conclude that the galaxies may be simply moving as projectiles in flat space. For one, the flat space model cannot account for the accelerating expansion (which is not modelled in the above simplified diagrams). Projectiles necessarily move with constant velocity in flat space.

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