Comment by raattgift
4 days ago
Physicists (and in particular the subset doing physical cosmology) don't usually explore parametrizations of c because they're not clearly physical (and sometimes even clearly unphysical), or alternatively don't help solve astrophysical or cosmological problems.
Relativists sometimes like to explore things that make using the Tangherlini transformations rather than the Lorentz transformations look positively benign. (To be clear, the Tangherlini synchronization system is clearly unphysical, requiring infinite speeds. His thesis also proposed using a distinguished global frame, which is not really philosophically different from how the standard cosmological frame is used, 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...)
my understanding is vsl physics can eliminate singularities if you have c approach zero in the presence of mass in a flat spacetime. AIUI this is equivalent to einsteinian curved spacetime except as you approach zero and deviate from linear in your c(m) formula.
you get pseudo black holes but depending on the extremeness of the deviation from linear, the difference to black holes might not be observable with current tech.
Not sure what you mean; you can't have any mass in a flat spacetime and obey the Einstein Field Equations for a Lorentzian spacetime (because T_{\mu\nu} doesn't vanish everywhere).
There are a variety of types of variable speed of light. If we foliate to 3+1 the usual picture is that c is constant on all spatial slices. Some VSL theories have the same c at all points on a given slice, but introduce a time variation of c. Other VSL theories introduce spatial variation as well (or instead). These families of theories all have significantly different equations of motion or actions from one another (cf. <https://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_actio...>). There's no obvious reason why c couldn't relate in a more complicated way to the stress-energy tensor than the Einstein gravitational constant does, but there's also no obvious reason to think such an alternative theory should produce free-fall trajectories similar to those from GR.
In any case, I think you have to choose your function on c, obtain the field equations, decide which energy conditions and constraint equations you want to impose, set appropriate boundary conditions, choose a curve along which to foliate, and run with enough different initial-value surfaces (each of which must satisfy the constraints initially), that eventually an intuition develops. A Will-like parameterized post-Newtonian formalism approach would also be a good idea (<https://en.wikipedia.org/wiki/Parameterized_post-Newtonian_f...>).
Unfortunately I'm unable to guess your choice of "c(m) formula".