FRW vs. Minkowski: Variable Speed of Light?

[nicksauce]

We can write the Minkowski metric as
$$ds^2 = -c^2dt^2 + d\mathbf{x}^2$$
or if we wanted different units for the metric
$$ds^2 = -dt^2 + \frac{d\mathbf{x}^2}{c^2}$$
If we make c a function of time we have
$$ds^2 = -dt^2 + \frac{d\mathbf{x}^2}{c(t)^2}$$

Which looks exactly like the FRW metric where c(t) = 1/a(t). So two questions: is my logic here correct? and if so, is it possible to tell the difference, in a purely gravitational way, between an FRW universe and a Minkowski universe with variable speed of light?

 

[Chronos]

looks about right to me, nick. just missing einstein's constant. Integrating these functions requires a constant to be mathematically complete.

 

[Chalnoth]

We can write the Minkowski metric as
$$ds^2 = -c^2dt^2 + d\mathbf{x}^2$$
or if we wanted different units for the metric
$$ds^2 = -dt^2 + \frac{d\mathbf{x}^2}{c^2}$$
If we make c a function of time we have
$$ds^2 = -dt^2 + \frac{d\mathbf{x}^2}{c(t)^2}$$

Which looks exactly like the FRW metric where c(t) = 1/a(t). So two questions: is my logic here correct? and if so, is it possible to tell the difference, in a purely gravitational way, between an FRW universe and a Minkowski universe with variable speed of light?

Sure, because the speed of light is intimately coupled with the strength of the electromagnetic force, so all we have to do is observe to ensure that the properties of electromagnetism were the same in the early universe as they are today. The answer we find is that they agree to within about a percent, though there are some tentative hints that there are some very small deviations at very high redshift (e.g. a fraction of a percent at z=6 or so). This deviation is too small for a varying speed of light to explain the expansion.

 

[nicksauce]

Is there an intuitive way to explain why a variable speed of light would lead to an expanding universe?