Why speed of light in vacuum remains constant over space and time?

Question

Imagine a Pulse of light traveling through space at $c$, coming towards an observer on Earth, while at the same time, Space-time fabric (metric) is continuously changing (expanding), then why is the speed of light constant throughout space-time, since the separation of the very two-points in space, light is traveling in between, is not constant?

My guess to that was $c=\lambda\nu$ but, how does Frequency of light and its Wavelength changes in just the right way so there product gives the speed of light and not a speed lesser than that of light(because space is Expanding)?

A few articles also argued about its effect on Sommerfield's constant, but I've read that String theory allows Sommerfield's constant to change over time.

I am not a GR head (yet) so, this post is bound to have a lot of things wrong (or maybe, all of them) so kindly keep your explanations as descriptive as possible. It'll be really helpful if you could provide some intuitions or examples for the same.

Answer 1

This became too long for the comments. Before I continue, maybe you should take a look at this answer too. I don't claim the following is a good answer but maybe it gives you ideas...

So I think the point is that the speed of light will vary if you are in a reference frame, that is experiencing acceleration/gravity. If you are in an inertial reference frame, the speed of light is $c$. This is in one of the links that I mentioned in the comments, but let's just illustrate it through an example. Suppose we consider an observer in the background of a Schwarzschild black hole with a Schwarzschild radius $r_S$ and a distance from the singularity of $r$. The metric is

\begin{equation} d \tau^2 = - \left(1 - \frac{r_S}{r} \right) c^2 d t^2 + \left(1 - \frac{r_S}{r} \right)^{-1} dr^2 + r^2 d\theta^2 + r^2 \sin (\theta)^2 d\phi^2. \end{equation}

Now if we are a particle of light following a null geodesic, we have that $d \tau=0$. The instantaneous radial velocity is \begin{equation} c'=\frac{dr}{dt} = \left(1 - \frac{r_S}{r} \right) c. \end{equation}

So you see that far from the singularity, when $r\gg r_s$, we have that $c' \rightarrow c$. Whereas, in the vicinity of the black hole horizon, $c'$ can become arbitrarily small.

Now I think that to answer your question about the variability of space-time, you might have to repeat the same calculation for the FLRW metric, for example. You will get some variation, that I'm not sure how you can measure, but if you were to measure the speed of light locally, you'd still get $c$. I hope someone else can give a better answer to this.

Answer 2

These are different ideas. The local speed of light is constant. That is the speed of light as measured locally by an observer. This is unrelated to the increasing distance of the point from which the light was originally emitted.

To deal with large regions of space, involving expansion, we have to use maps involving scaling distortions, much as we do when we map the surface of the Earth. Usually we use coordinates in which objects (galaxies) remain the same size, and distances increase over time. In such a map the coordinate speed of light does not remain constant. An equivalent way of mapping shows galaxies getting smaller over time. In such a map the radial coordinate speed of light can be constant

For this to work, the rate of time on has to be increasing, so that the observed laws of physics are the locally always the same. Since the rate of time increases, the observed frequency of light decreases. So basically, the answer to the question is that the local laws of physics remain always the same, and this means that the wavelength and frequency of light must change in just such a way that the local speed of light remains constant.