New varying speed of light theories

and seems OK because the distribution of stress-energy can pick out useful systems of coordinates in standard relativity. Unfortunately his method frustrates and probably outright breaks comparisons between inertial reference frames related by a boost, which the standard cosmology does not, and it's hard to see an alternative method that preserves his central ideas.) But why even be stuck with 3+1d spacetime like Tangherlini? He was trying to do physics. But an unphysical metric signature with 47 plusses and no minuses is really cool! In our observed universe, FAPP, c is the same everywhere after recombination, and we get that from spectral lines. You have to play really weird games to preserve the Lyman-alpha forest's apparent isotropy while introducing spacetime (or redshift-space, here) anisotropy. Things like BAOs make the problem even harder. If we strip away all that pesky radiation and the information its structure encodes, analysing variations of c gets a lot easier. A relatively recent paper (Lewis & Barnes 2021) I enjoyed considered anisotropy in the one-way speed of light in an FLRW cosmology with zero energy density (well, really the convenient Milne model, which is also far from spatially flat). "So far, we have considered two cases, where either the speed of light is isotropic, or the extreme case where the anisotropic speed is 1/2 in one direction, and infinite in the other. The question remains whether this holds true in general case, for an arbitrary κ": https://www.cambridge.org/core/journals/publications-of-the-... (arxiv: <https://arxiv.org/abs/2012.12037>). "For more general cosmological models, where the presence of mass and energy results in curved space-time, the picture is more complicated as there is no simple mapping of the modified Lorentz transformations into the general relativistic picture. We leave this discussion for a future contribution." Sadly there doesn't seem to be a future contribution yet, at least going by published citations (<https://scholar.google.com/scholar?cites=2012575105829699847...>). (Of those, I've put the Chamberlain paper on my to-read pile; you'd appreciate how it relates to Tangherlini, "credence is given to one-way infinite light-speed inward to each particle in direct comparison against Einstein’s isotropic (c=constant) light-speed"). Of course there's also the excellent Magueigo 2003 VSL overview <https://iopscience.iop.org/article/10.1088/0034-4885/66/11/R...> copy <https://cds.cern.ch/record/618057/files/0305457.pdf> preprint <https://arxiv.org/abs/astro-ph/0305457>. And even if you can make sense of an f(c) cosmology in the early visible universe, you will get to epochs before recombination and try to make sense of the later universe's chemistry, which of course relates to big bang light nucleosynthesis, baryogenesis and electroweak ssb. How do you abolish Lorentz symmetry in those epochs? Good luck! (I mean, I think if you are doing physical cosmology you ought not to ignore gauge theory...)

that fit your observational data. Technology of the past 10 years brings more data, and lets economics move away from the representative agent of the past 100 years. Good. Let's do that, but let's not forget whose shoulders we sit on - no one mocks the orbital model of the atom, or Newtonian laws of physics, even though both models are outdated [2][3]. [0] - http://curious.astro.cornell.edu/question.php?number=465 [1] - http://arxiv.org/abs/astro-ph/0305457 [2] - http://spiff.rit.edu/classes/phys314/lectures/bohrprob/bohrprob.html [3] - http://en.wikipedia.org/wiki/Classical_mechanics#Limits_of_validity

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.

including the highly investigated parity symmetry breaking in the neutrino sector. The gaps in which one tries to hide from this are small and shrinking. This is a strong constraint, and it was felt when investigating SUGRA (especially the 11-dimensional maximal SUperGRAvity) as a physical theory. The unit-timelike vector in the paper discussed in The Fine Article can, without constraints, break local time-orientability. One could grind out consequences of this, but it's not easy to do (and the authors do not do it in this paper). One might, for example, need a mechanism to suppress the popping-up of light right-handed neutrinos [1], or at least their impact on astronomical spectra and terrestrial studies of isotopes. I'd think one would want to do that in the matter side of the EFEs rather than the curvature side, but partly it's a matter of aesthetics and ease of use, and partly it's "you introduced curvature which has the effect of changing the observables predicted by the Standard Model, including regularly confirmed observables; you should repair the damage with curvature (and the obvious thing is to just hit 'undo')". There are many other natural phenomena that serve as probes of local Lorentz invariance violation ("LIV"), which I am sure you can find if you are interested. https://en.wikipedia.org/wiki/Modern_searches_for_Lorentz_vi... is a decent jumping off point. One might note that there is a lot about neutrinos there. GR can certainly completely consistently describe spacetimes of different dimension, different (or non-)orientability locally and globally, and so forth. There are good reasons to study 2-dimensional and 3-dimensional spacetimes instead of the 4-dimensional one we inhabit. There are fun reasons to study timeless, many-timed, and/or many-spaced spacetimes[1b]. There have been reasonably well-motivated-by-physics reasons to study extra spacelike dimensions. I'm not sure what your threshold of "be[ing] sure" is, but mine is certainly met. It would be a tremendous (and cool) surprise if LIV were to be shown experimentally or implied in extreme astrophysical observations, and even more so within our (less extreme) solar system. Finally, I turn to your IIRC, which is not correct. General Relativity has a single metric to which everything couples. This is the root of the universality of free fall part of the Strong Equivalence Principle. In General Relativity we can only vary the (vacuum) speed of light by giving light a rest mass. Most Variable Speed of Light (VSL) approaches introduce new objects into the field equations, making a theoretical alternative to General Relativity. Examples include bimetric theories, which add a metric of constant curvature and typically other scalar and/or vector fields; different massless species can couple to the different metrics, and if the constant curvature is Minkowski (flat), then Special Relativity is preserved on a per-sector basis, with complicated intersectoral interactions. There are a variety of other approaches, Magueijo 2003 https://arxiv.org/abs/astro-ph/0305457 is an overview that springs to mind. - -- [1] Gordon M. McCabe's 2005 paper http://philsci-archive.pitt.edu/2218/ §2.1 (pdf page 14) expands upon this. (The author writes assiduously in this area with few, but not zero, citations https://scholar.google.com/scholar?hl=en&as_sdt=7,39&q=Gordo... and with occasional collaborations -- he is among other things one of the editors of https://ui.adsabs.harvard.edu/abs/2015rrrp.book.....S/abstra... https://en.wikipedia.org/wiki/The_Road_to_Reality ) [1b] ibid., §2.2 (pdf page 18) is the start of an extended discussion of how one might couple realistic matter (in this section a toy non-interacting (except gravitationally) field, but heading towards the full Standard Model in later sections) into non-Lorentzian universes.