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

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.