[tagrun]

What is not true? You can't have reference frame independent (which is a term that also includes orientations) Maxwell equations and anisotropic speed of light at the same time.

If Maxwell equations are correct (which was already well-tested by then), speed is already the same for forward and backward propagating electromagnetic waves (=light), and there is no other spatial anisotropy either.

Differing one-way speed of light is an amusing "loophole" in the experiments measuring the speed of light (which requires one particular magical angular distribution of c to slip through a Michelson-Morley interferometer) but never existed in the theory that directly predicted it to begin with, so if you insist on it, one needs to ask how would that even work with the rest of physics? c doesn't have a spatial/direction preference in electrodynamics or quantum electrodynamics, vacuum permeability and permittivity (\mu_0 and \epsilon_0) don't have any observed spatial dependence. (Such a thing happens in condensed matter systems, effective mass, vacuum permittivity, g-factor, etc etc are in general anistroptic due to the medium, and is easily detectable, and their spatial derivatives do show up and need to be taken into account to match the observations as in the case of the kinetic term -\hbar^2(d/dx)(1/2m(x))(d/dx). Coulomb force doesn't get stronger or weaker when you rotate the table you perform your experiments on, current carrying wires don't produce stronger magnetic fields as you change their orientation (at least not within any observed precision). Similar goes for any field theory in the standard model.

I should add that in terms of experimental precision, quantum electrodynamics is the most accurate theory that we have, and can put very strong limits on possible anisotropic deviations if any.

[blueplanet200]

You should watch the video. There is no experiment that has been done that shows the speed of light does not have a preference because every measurement sneaks in the assumption it's symmetric.

This is a convention. It's called the Einstein synchronization convention. https://en.wikipedia.org/wiki/Einstein_synchronisation

See also: https://en.wikipedia.org/wiki/One-way_speed_of_light . From the article: "Experiments that attempt to directly probe the one-way speed of light independent of synchronization have been proposed, but none have succeeded in doing so.[3] Those experiments directly establish that synchronization with slow clock-transport is equivalent to Einstein synchronization, which is an important feature of special relativity. However, those experiments cannot directly establish the isotropy of the one-way speed of light since it has been shown that slow clock-transport, the laws of motion, and the way inertial reference frames are defined already involve the assumption of isotropic one-way speeds and thus, are equally conventional.[4] In general, it was shown that these experiments are consistent with anisotropic one-way light speed as long as the two-way light speed is isotropic.[1][5] "

I get what you're saying and I'm well aware that Maxwell's equations are rotation invariant. I'm saying it's more subtle and complicated than you think. For instance, time dilation will have an asymmetry under these assumptions.

[tagrun]

You can call in convention as many times as you want, but unfortunately, it just is not a convention. It is called theory of electrodynamics which is a well established, experimentally verified branch of physics.

What exactly is more subtle and complicated in the context of Maxwell equations? If speed of light has the anisotropy that you are describing, Maxwell equations must be incorrect. In what electromagnetic experiment has such anistropy of magnetic or electric constants have been ever observed?

You're basically saying "you haven't measured the one-way speed of light directly, so you haven't ruled it out the possibility of my exotic theory", but it is actually been ruled out by Maxwell equations a long time ago. Unless you have some experimental proof that Maxwell equations need to be modified to accommodate that elusive version of your aether, you can't claim the existence of such an anisotropy.

Physics is well connected in that you can't change one part of it (in your case, c in the context of special relatively) just because you found something that wasn't experimentally ruled out, and hope the rest of the physics (basically all massless field theories and relevant experimental results in this case) won't break.

[raattgift]

2/2 [sorry that this will be out of order]

For a (physical) relativist, the speed of light is really simple. c = 1, everywhere and everywhen. <https://en.wikipedia.org/wiki/Geometrized_unit_system> This is because we have excellent evidence for the utility of <https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold#App...>, and the further astrophysically-driven demands of global hyperbolicity or at least reasonably strong causality conditions, no isometric embeddings, geodesic incompleteness, asymptotic flatness around sources, junctions in sufficiently flat space, and energy conditions. Those further demands are the basis for continuing to rely on Special Relativity in laboratory settings.

For a theoretical relativist, well, the best metric signature is probably +,+,+,...,+ (89,0). (cf. Egan's (4,0) "Riemannian General Relativity", <https://www.gregegan.net/ORTHOGONAL/06/GRExtra.html>)

- --

[1] quoting your wikipedia link, "... inertial frames and coordinates are defined from the outset so that space and time coordinates as well as slow clock-transport are described isotropically". Well, yes. Establish points first then assign coordinate labels is the relativist's procedure, surely?

On this point, Earth laboratories are in general not in inertial frames, thanks to gravitation. No laboratory is in general in an inertial frame, thanks to the metric expansion of space. We can in principle extract a preferred foliation (e.g. the scale factor a, or some function on lunisolar tides) and use that as the basis for time coordinates instead. In effect this is what we do for high-redshift objects and many lunar laser ranging experiments <https://ssd.jpl.nasa.gov/ftp/eph/planets/ioms/>[a] <https://arxiv.org/abs/1606.08376> §3,§4. <https://link.springer.com/article/10.1007/s10569-010-9303-5> discusses aspects of how to choose a preferred foliation (in the context of gauge freedom) in the solar system, and in the context of grinding out a results-prediction for some future LLR experiment. The goal is to be able to show that the locations of the three instruments were accurately predicted, and Lorentz-invariance is thoroughly baked in (the calculations are so exceptionally sensitive to the introduction of tiny breaking parameters in the style of SME <https://arxiv.org/abs/0801.0287> that it has led to the discovery and/or better understanding of several of the features listed as parameters at [[a] LLR_Model_2020_DR.pdf §4]).

[2] <https://archive.org/details/principleofrelat00eins/page/n189...>. Einstein 1916 is adapted into the arguably handier <https://en.wikisource.org/wiki/The_Foundation_of_the_General...> §9. c.f. there pages marked in square brackets [776-777]

[3] Baez: <https://math.ucr.edu/home/baez/einstein/node2.html> 2nd and 3rd paragraph "Preliminaries".

[raattgift]

1/2

tl;dr: coincident-events first, then labels (coordinates). [Einstein 1916, p.117 [2] although I remembered to look there only after writing all of the below]. One-way speed of light arguments are in danger of being coordinates-first, and thus insufficiently general for physics.[3]

The key word in your comment is

> directly

But why do we care? We have an abundance of indirect evidence, premised on direct tests of coordinate-independent features of our best most-fundamental theory. The two important features of (general) relativity are pointwise local Lorentz covariance -- where c is the only free parameter of the Lorentz group -- and the minimal coupling. Special relativity's Minkowski space is in this view a special static time-orientable spacetime in which we have global Poincaré invariance (c again is the only free parameter of the Poincaré group; the Lorentz group is a subgroup of the Poincaré group -- the latter includes all the spacetime translations, and in the Minkowski case the space-translation and time-translation symmetries all commute). When we go blithely parallel-transporting null vectors, this is what matters.

We can certainly write down an f(c) theory. Dicke did this in in his superb 1957 "Gravitation without a Principle of Equivalence" >https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.29....? and so have several others (See Ellis or Magueijo for a review <https://link.springer.com/article/10.1007/s10714-007-0396-4> corresponding with <https://arxiv.org/abs/astro-ph/0703751> resp. <https://iopscience.iop.org/article/10.1088/0034-4885/66/11/R...> open access, but corresponds with <https://arxiv.org/abs/astro-ph/0305457>).

It is far from silly to write down a theory where c varies in spacetime. It is the foundation of several alternative-to-cosmic-inflation decaying-bimetric theories of the very early universe, where c eventually stabilizes to its value in our local spacetime having been a different (typically much much much -- ~30 orders of magnitude -- higher) value during the formation of primordial matter density variations. The faster speed of light allows for distant reaches of the early universe to reach the same temperature with uniformity up to the small fluctuations in the cosmic microwave background.

Of course we run into the same point you've been working in this thread: it's hard to discover the exact function on c in the early universe. We have to rely on indirect evidence, and strong gravitational lensing is useful there. SVOM <https://svom.cnes.fr/en/SVOM/GP_mission.htm> is looking for Lorentz-invariance-violation (LIV)-induced modifications to the photon dispersion relation in vacuum, and is a particularly good platform for test of a Taylor-series expansion like E^2=p^2 c^2 ( 1 +- \sum_{n=1}^{\infty} a_n ), since GRBs at least somewhat escape the problem that the lowest order terms dominate at small energies, and they are distributed across the sky and at different redshifts. We are also now better equipped to study light echos (oh for a galactic supernova!) and detailed strong galactic lensing studies. Spoiler: the constraints on a spacetime-translational variation of c grow tighter with every observation. However, to fully rule out a sharp phase-change in c, we will need practical ~ 10^-15 Hz gravitational-wave astronomy. LIGO is most sensitive around 10^2 Hz; eLISA would be around 10^-2 Hz. (I am fairly sure the authors of most modern variable-speed-of-light early cosmologies knew as they were writing that they probably could hide in that hard-to-explore space through a few generations of gravitational wave observatories. One might say the same about a wide variety of recent cosmic inflation theories, too.)

In a general dynamical spacetime the notion of a two-way path is tricky. Even in Minkowski space, for a two-way signal, the return detection arrives at a different, later, point in spacetime than the outbound signal, even if the spacelike coordinates are always (0,0,0) [this is somewhat reminiscent of the twin paradox]. Outbound-and-return are two future-directed null geodesics. Your argument in this setting is equivalent to saying that we are somehow in trouble because the "outbound" and "return" null geodesics may have, without rescaling, different affinely-parametrized lengths.

In SR what we care about is that the signal is Lorentz invariant at each spacetime point where it could be sampled, even as sender and receiver/reflector are moving ultrarelativistically or are ultraboosted. Given Lorentz-invariance we can determine the three relevant points on the manifold. (Poincaré invariance means we can do this same test at any time or place in the flat space universe). Your complaint is that this is not a direct one-way measurement. OK, it's not. So what? We can in principle directly test Lorentz-invariance at any point (e.g. we can have a sparse gas with a well-understood (as in at the Standard Model of Particle Physics level) low extinction coefficient). If we have no flat space violation of Lorentz-invariance, we must have symmetry of light travel time for constant light-like separation.

In a dynamical general spacetime (Lorentz invariance -> local Lorentz invariance (LLI)), we can readily move the intended recipient of a one-way light pulse outside the reach of the light pulse itself; nature already does this for us in at least a couple of ways (metric expansion and astrophysical black holes). We can also have different delays on each arm of a two-way measurement, e.g. through Shapiro delay, around a spinning mass, or in the presence of a gravitational wave. However, at each point in the (vacuum part of the) curved spacetime [a] light obeys the massless wave equation and [b] local Lorentz invariance demands that the fraction of the wave at X propagates to a neighbouring point X' at c, and that X' must be drawn only from certain available neighbouring points.

So, really, it's not so much "what is the one-way velocity of light?" but rather "how much spacetime does a pulse of light traverse between two spacetime points?". Or in other words, we are looking for an affine parametrization on a curve of zero interval and that extremizes the length between two points on the manifold m and is constructed by parallel-propagating a tangent vector on m and in its own direction.

Consequently, I think the issue at the core of your points about measuring one-way speed of light is how to best label, with coordinates, two particular coincidence-points in a Lorentzian spacetime, rather than choosing two labels[1] (via your favourite synchronization scheme, for example) which are then used as the basis for a parametrization of a null curve. Then the question you ask is: "how can we know that the spacetime is Lorentzian?" or alternatively, "how do we know there is not some vicious additional gravitational field and vacuum polarization which makes the spacetime only seem Lorentzian?" for which experimentalists have generated a century worth of answers. The answer of how to best label two points in a Lorentzian manifold is \mu : it really depends on what and how you want to calculate. The physics here are that the two points (in your null-curve-parametrizing / one-way-light-travel-time experiment) are timelike-separated.