# Is the FRW metric physically distinguishable from a metric with a speed of light that changes over time?

There are many questions on this site that ask whether the expansion of space could instead be interpreted as a speed of light that changes over time, e.g.:

But they're all rather vague and so difficult to answer precisely. I have a more precise version of this question.

Q1. The Minkowski metric is

$$ds^2 = -c_0^2 dt^2 + (d{\bf x})^2, \tag{1}$$

where $$c_0$$ is the speed of light and $$(d{\bf x})^2$$ represents the ordinary Euclidean metric on $$\mathbb{R}^3$$. It seems to me that the simplest generalization of this metric that has what you might reasonably call a "variable speed of light" is just the metric

$$ds^2 = -c(t)^2 dt^2 + (d{\bf x})^2, \qquad c:\mathbb{R} \to \mathbb{R}^+, \tag{2}$$ which picks out a preferred slicing of spacetime.

This metric is conformally equivalent to the FRW metric

$$ds^2 = -c_0^2 dt^2 + a(t)^2 d{\bf x}^2 \tag{3}$$

if we let $$a(t) = c_0/c(t)$$, with the two metrics (2) and (3) related by the conformal factor $$\Omega(x) = a(t)$$.

Are these two metrics actually isometric (or "diffeomorphic" in the standard physicist terminology) as well as just conformal? If not, what experiment could you perform to distinguish them?

Q2. (Softer and more subjective.) The Minkowski metric (1) can be almost completely equivalently written as

$$-d\tau^2 = -dt^2 + d{\bf x}^2/c_0^2, \tag{4}$$ which just reformulates (1) to focus more on timelike rather than spacelike displacements. Starting from this formulation, the natural generalization to a time-variable speed of light is $$-d\tau^2 = -dt^2 + d{\bf x}^2/c(t)^2. \tag{5}$$ In other words, if we divide the metric by the constant $$c_0^2$$ (which is a completely trivial operation) before we promote it to a varying function, then we get a metric exactly proportional (not just conformal) to the FRW metric (3). To me, this gives extremely hand-wavy evidence that the metrics (2) and (3) may be physically equivalent, since (1) and (4) seem like physically equivalent starting points. If the metrics (2) and (3) are not physically equivalent, then is there any reason to think that either one is more natural to consider as "Minkowski space with a time-varying speed of light"? Which one, and why?

For both questions, I assume that nothing qualitatively new happens if we generalize the Euclidean metric $$d{\bf x}^3$$ to a more general time-independent Riemannian metric, but feel free to comment if that's wrong.

By the way, I'm not sure I agree with the answers to some of the linked questions. They correctly note that only dimensionless ratios are physically meaningful, so the absolute (dimensionful) scale of the speed of light is not physically meaningful in special relativity. Two universes described by special relativity on Minkowski space with different (finite) speeds of light would be physically equivalent as long as all of the dimensionless ratios in the Standard Model were the same.

But I don't think that argument applies to an expanding universe on curved spacetime, because the dimensionless function $$c(t)/c_0$$ (where $$c_0$$ is now just some reference constant) gives you a continuum of physically measurable dimensionless numbers. I won't get into the philosophical debate about whether metric (2) above "really" describes a universe with a time-varying speed of light; I'll just posit that it could reasonably be thought about that way (without suggesting that that's the only reasonable way to think about it).

• I think you're perfectly correct, but also that the standard way of talking about it is qualitatively better. The standard way of putting it makes it clear that everything that's going on is gravitational. Phrasing it in terms of a "variable speed of light" makes one expect that there would be additional effects that could be observed in electromagnetic experiments much smaller than the Hubble scale, but we've never seen anything of the sort. – knzhou Jun 8 at 3:47
• To say this another way, when you talk about a "variable speed of light", it's not clear how this affects, e.g. the speeds of neutrinos, or gravitational waves, or anything bound with electromagnetic forces. Do gravitational waves get redshifted as much as the photons do, for instance? The usual way of phrasing it emphasizes that the expansion is a property of the metric itself and not the fields on it, which means that photons, neutrinos, and gravitons all redshift exactly the same amount, no measurable properties of atoms change, etc. – knzhou Jun 8 at 3:48
• @knzhou : I would think that when one is talking of "variable speed of light" in this context one means "variable $c$" (or "variable 'speed of causality'" if you wish), instead, because "speed of light" is typically taken in this context as synonymous with "$c$" and not "speed of photons/EM radiation" more literally. – The_Sympathizer Jun 8 at 3:49
• @knzhou I don't understand what you're getting at. Why does the phrase "variable speed of light" imply that it could be observed in EM experiments smaller than the Hubble scale? Is your point that the phrase "speed of light" seems to suggest something that only applies to EM radiation? In that case I don't really agree - as The_Sympathizer says, I think it's pretty widely understood that the phrase "speed of light" actually refers to much broader "speed of causality". – tparker Jun 8 at 23:19

Your metric (2) is just a coordinate reparametrization of Minkowski space. Writing it with a different time variable $$ds^2 = -c(T)^2 dT^2 + d{\bf x}^2$$ to avoid confusion, they're equivalent when $$t = \int_{T_0}^T c(T) dT/c_0$$ (for some arbitrary constant $$T_0$$).

Your metric (5) is equivalent to the general spatially flat FRW metric, and it's true that you can think of it as a variable-speed-of-light metric if you want. It's actually useful to think of it that way when computing light cones. But I think it's misleading to say that a FRW spacetime is physically indistinguishable from a VSoL spacetime, for the same reason it would be misleading to say that six is physically indistinguishable from half a dozen: it suggests that there are two different things which we can't tell apart, when in reality there's just one thing which we're describing in two different ways.

So you probably won't interest any cosmologists with your argument about variable speeds of light because it's only a matter of words and doesn't touch the underlying physics. On the other hand, cosmologists already define and use a number of different quantities called speed/velocity, and the speed of light in most of those senses is not constant, so not only are you correct but they already agree with you.

It makes perfect sense when you think about it this way: your equation implies that the maximum permitted speed of causality, $$c$$, is slowing down with time. That means that everything is confined to move "slower and slower" - shorter distances, longer times.

Every physical object must then shrink: once the speed approaches from above the speed at which, say, electrons are otherwise moving, the orbit must begin to undergo length contraction, while the electron speed becomes "capped", like the circumference of an Ehrenfest disk. The atom shrinks. As atoms shrink, they pull each other closer together, and so the objects made of them also shrink.

(Note that this means your rulers also shrink, and moreover your clocks also slow down, so using them, you would still measure the same "proportional" value of $$c$$.)

In a flipped way of thinking, that's the same as the space between things getting bigger (while the things themselves do not), and your equation shows indeed that this correspondence is mathematically exact. That also means: no, there isn't any way to "physically distinguish", as you put it, the two cases. However, this may provide a starting point for thinking about things differently which might, then, lead to novel theories.