[raattgift]

General Relativity (GR) is a metric theory of gravitation, with one metric to which everything couples.

In GR gravitational waves (GW) have lightlike worldlines. Consequently, a source emitting both electromagnetic and gravitational radiation will have its GWs and EMWs (or more generally its optical image and the direction in which things indicating its gravitational influence point) line up. This has been well-tested observationally, for example by watching the deflection of light from distant objects (like quasars) around Jupiter (whose mass, orbit, and distance from us are all very well characterized).

However, one can write down a bimetric theory of gravitation with different couplings. It's possible to write down a bimetric theory in which gravitational waves move more slowly or more quickly than electromagnetic waves.

It was fairly popular some years to take this kind of approach to solve some cosmological problems relating to the homogeneity within the horizon [1]. These were often cast as "variable speed of light", for aesthetic reasons fixing the speed of the gravitational interaction. However, it is perfectly reasonable to call the same models "variable speed of gravitational radiation" fixing the speed of light, as one has many freedoms with respect to coordinate conditions in General Relativity.

The problem is that these "variable speed of gravitational radiation" theories do not match observations of the galaxy-filled parts of the universe that we can see, and also does not match what we see in the Cosmic Microwave Background. (Some bimetric models fail to match the results of laboratory-scale physics experiments too.) Viable bimetric theories thus have the second metric decay in the very very early universe, such that in the galaxy-filled epoch the speeds of light and gravitational radiation are identical, and physics becomes (outside of the very early universe) indistinguishable from their "standard" single-metric General Relativity based generally covariant formulations. Such decaying-bimetric theories usually are designed to do away with cosmic inflation, but it becomes difficult to distinguish between cosmic inflation and viable bimetric-decay models because the observables eventually have to become identical, and the time at which they can differ gets pushed back further as we develop observatories which can resolve objects at ever higher redshifts, or as we can get better data on the anisotropies of the CMB.

> we're slowly narrowing in on C in measurements

We should determine c empirically, but we have already done so to exquisite precision.

However, we can also fix c to some exact value (e.g. the CODATA value, or 1) and be mindful of the side effects of doing so. This is, by far, the most common approach; you will be hard-pressed to find any formulation of a physical law which introduces uncertainty into the value of c, although it's certainly doable.

The fixed CODATA value is extremely good. The relative uncertainty in the speed of light is principally driven by the uncertainties in interferometry, which at the time of the 1983 redefinition of the metre was less than 0.1 part per billion (and is now less than a part per trillion, and so for all practical purposes is unimportant at scales of the observable universe).

Finally, one should note that in a general curved spacetime, while the constant factor "c" arises everywhere, it can only be taken as a speed when comparing two objects that co-occupy exactly the same infinitesimal point in spacetime. Comparing the speeds of distant objects is something that one should avoid in General Relativity. However, everywhere in every spacetime, in vacuum conditions one should find the same "c" as the upper limit of relative speeds of objects just as they enter, co-occupy, and exit the same point.

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[1] https://www.wikiwand.com/en/Horizon_problem

[Filligree]

> We should determine c empirically, but we have already done so to exquisite precision.

As interesting as that is, what I meant we're narrowing down is the speed of gravity -- that is, that it's almost certainly equal to C. Not that we're narrowing down C itself, it's way more convenient to define that as 1.

Since you're here, though... the horizon problem. I can kind of understand the logic that makes it a problem, but...

If you have the same initial conditions everywhere, and the same laws of physics, wouldn't you expect everything we can see of the universe to look similar even if there hasn't been any communication?

That seems like an obvious implication of having deterministic physics, and sure, "same initial conditions" is a big assumption -- but I never see this hypothesis offered.

[raattgift]

> more convenient to define that as 1

Yes indeed. :-)

> ... I never see this hypothesis offered.

The tl;dr is that that hypothesis is a (subjectively) boring and (less subjectively) non-Copernican answer.

I'll expand on this.

Let's just accept arguendo that the laws of physics are everywhere the same and in particular that gravitation matched General Relativity from well before photon decoupling to the end of recombination.

You can indeed then take the position that the initial conditions of a hot big bang were extremely finely tuned, with overdensities at photon decoupling (and thus reflected in our CMB) being exactly encoded in the initial values.

However, the trend is to explore mechanisms that allow for much higher (Boltzmann) entropy in the early universe with overdensities evolving from fluctuations.

ETA, I'll steal a line from slide 6 at [1] : "don't explain low entropy by positing even lower entropy".

This trend has been productive in the sense that it has produced several progressive research programmes amenable to empirical tests.

For example, cosmic inflation (eta: in part by allowing the particle horizon and the Hubble horizon to be very different) allows for a much wider set of initial conditions (some with much higher entropy than is implied by maximally finely tuned initial conditions) that could produce our CMB and ultimately galaxies. Cosmic inflation in the broadest sense has produced a number successful predictions, so work is certain to continue.

Pragmatism, aesthetics, philosophy aside, I have trouble imagining in detail what observatories we would have constructed had late 1990s cosmologists simply pursued a programme of discovering the details of values surfaces at progressively earlier times, with the goal of simply discovering the initial conditions eventually, all without doing much theoretical speculating about what as yet unexplored early values surfaces might contain. What would have succeeded BOOMERANG? Instead, that sort of speculating raises all sorts of interesting questions about the behaviour of matter at extremely high densities and temperatures, how those behaviours might be encoded in a CMB that was not strictly fixed at the hot big bang, and what alternatives to the hot big bang (e.g. a big bounce cf. slides 7- @ [1]) could lead to our galaxy-filled sky.

So you practically hit the nail on the head in recognizing that assumptions about initial conditions is crucial to whether one sees the horizon problem as a problem in the first place. If you don't care about complaints about the apparent non-genericity of the initial conditions, there's no problem; likewise, if you are pretty sure that initial conditions can be highly generic yet lead to the universe we see, then there's also no problem. This is fertile ground for philosophers (and historians) of science. [2]

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[1] http://www.preposterousuniverse.com/blog/2017/01/25/what-hap...

[2] A quick search leads to Casey McCoy (University of Edinburgh)'s http://jamesowenweatherall.com/wp-content/uploads/2014/10/Wh... section 3 (edit: notably from the bottom half of p 15 where he has some pleasantly difficult questions in parentheses) and the second half of p 18.

[Filligree]

I'm with the first one, then. I consider the Kolomogorov complexity of the laws of physics to be important, and initial conditions to be part of said laws, but I don't think it adds much complexity to posit that the initial conditions are very, very simple (and thus low-entropy).

[raattgift]

Taking that position also gives you the answer to why we won't see black holes fissioning into outspiralling pairs of neutron stars, or shards of glass spontaneously reassembling into a wineglass that leaps off the floor into someone's hand. The only cost is extremely exquisitely finely placed stress-energy-momentum in the early universe into the infinite past.

If you have no problem with an infinite history of ever lower entropy, then luckily observations to date do not contradict this sort of cosmology, and under time-reversal the "movie" showing the universe crunching into an ever more orderly state forever isn't very shocking other than we arrange our lives with the movie playing the other way, sweeping up broken shards of glass rather than catching rising stemware. Maybe conscious life somewhere else in our Hubble volume arranges their lives in that way, though, unbreaking and unmaking their artefacts and thinking our way of doing things is strange.

Some other questions for you: how far does this cosmology's future grow? Do we get infinite entropy in the infinite future? If we don't have an infinite future, why do we have an infinite ever-more-orderly past? If we have an infinite future, do you suppress fluctuations? Do you wholly suppress Poincaré recurrence? How would you distinguish us here as us here now from a recurrence? Are we a recurrence? etc.

These questions do not seem as amenable to testing with current technology as questions arising from other cosmologies.

[Filligree]

Well, let's see. I don't have much time to spend on this, and I'm well aware that I'm an amateur anyway, but...

One point, at least. I see no reason why a theory being difficult to test should count even slightly against the probability of its being true. Sure, it's inconvenient -- and there's something to be said for focusing on ideas that can be tested -- but since when was nature ever constrained to convenience?

As to your other questions...

I'm not sure where the 'infinite past' comes from. There's a single slice of space-time that needs to be constrained; everything else can flow from there, in both directions. If it's a low-entropy constraint then causality would naturally flow away from that slice. The two sides should be mirrors of each other, though.

As to the far future... that's harder.

- My intuition is that, when you're trying to guess at which of many possible universes you actually live in, the amount of "runtime" the theoretical universe spends on computing you (that is, some form of life, at least) should matter. But that's problematic, since it seems to predict higher complexity of the physical laws in exchange for a universe that's denser with life, and it's not what we see.

- More realistically, perhaps, the big rip might be a thing. Recurrence could fail to happen because the far future doesn't allow for structures complex enough to have thoughts, not even by random chance.

I prefer to punt on the question, though. I don't know enough to have an informed opinion.

[pcnix]

I must admit, I really enjoyed reading your comments on this thread. Good work, and thanks for the effort!

[Koshkin]

What are the side effects of fixing the value of c?

[raattgift]

Let's flip the question and ask what happens if we let the value of c depend on the location in spacetime.

Ellis provides a "short checklist of issues that should be satisfactorily handled by" theories that have a value of c that depends on location in spacetime. https://arxiv.org/abs/astro-ph/0703751

The tl;dr version is that relativistic physical theories will need revising because in the face of a location-dependent c, and in particular the problem arises because we necessarily break lengths and durations when c is not everywhere-and-everywhen identical.

More narrowly observables like spectral lines are sensitive to ratios involving the quantity hc which are taken to be constant. Since cosmological redshifting of spectral lines fits in a substantial web of observables related to distance (e.g. angular diameter, surface brightness) we can practically rule out variation in hc to a very high redshift (z > 10).

You might instead be asking what happens if c is the same everywhere but we've settled on a very slightly wrong value of c. The answer is that this will almost be absorbed into our system of units, in particular in various ratios involving hc. If we change the value of c to the slightly improved value, we also probably slightly change things like the fine structure constant and the electron-to-proton mass ratio, which are dimensionless quantities with ratios involving hc. These dimensionless quantities are good checks on the many ways in which we might measure c empirically.