general relativity - The vacuum light speed: Is it really constant, i.e., independent of location in space-time?

I am by no means an expert in this field, however something puzzles me about the speed of light and the relativity of time and space (space-time).

Is is universally acknowledged that the speed of light (299,792,458 m/s) is the universal speed limit, and that nothing can travel faster than light? That is a measurement based on a man-made interpretation of time (hours, minutes and seconds etc. are man made...there is nothing natural dictating how long a second should be).

For instance, according to Einstein, time and space bend around the physical matter of the universe, so for example, time near or on the surface of a "super massive black hole" should be drastically slower, relative to that of earth. Lets say for example that for every second that passes on the black hole, 10 seconds pass on earth, so essentially time on the surface of the black hole is 10 times slower than the time on earth.

Given the example above, is the speed of light at the surface of the black hole still 299,792,458 m/s, or is it 299,792,458,0 m/s?

Answer

There are more ways to understand this, and the answer is the same: yes, the speed of light is constant. I will try to explain this in the way I consider simpler, but I am sure that others have their preferred explanation. Anyway, to receive a fair explanation, I suggest you to read something more detailed about special relativity, and then about general relativity. My explanation is geometric. Being mathematical, this means that you try first to understand what I mean in an imagined world of mathematics, to see that the ideas are consistent. Think at it as a sci-fi movie, in which you are only concerned with logical possibility. I suggest you that only after you are happy with the self-consistency of the model, you try to judge this image and compare it with what you know about physical world.

Spacetime is a space with four dimensions. Near each point, the spacetime is almost flat, but as we depart from the point, it becomes curved. On very short (infinitesimal) distances, the spacetime being almost flat, we can write a Pythagoras' theorem. In four dimension it is like

$$d s^2= - c^2d t^2+d x^2 + d y^2 + d z^2.$$

We are interested to make it work also in frames which are not normalized, and have different scales (denoted here by $g_{aa}$), and hence different measurement units:

$$d s^2= g_{00}d x_0^2+g_{11}d x_1^2 + g_{22}d x_2^2 + g_{33}d x_3^2.$$

Here I replaced $ct,x,y,z$ with $x_0,x_1,x_2,x_3$. But we also want to write this in coordinates whose axes are not necessarily orthogonal, so we have to add some cosines between the axies $a$ and $b$, which are written as $g_{ab}$:

$$d s^2=\sum_{a,b}g_{ab}d x_a d x_b.$$

This also works for curvilinear coordinates (we allow the metric coefficients $g_{ab}$ to vary from point to point), which are the suitable ones for curved spacetime.

The length is given by Pythagoras' theorem. Some infinitesimal distances are $d s^2>0$, and they separate points which can be in the same space. Some are $d s^2<0$, and they measure time intervals. Some are $d s^2=0$, and such directions are called light-like direction. So, if you measure the length of a curve described by a photon in spacetime, you measure it with this theorem, and you always obtain it to be $=0$. If you choose the reference frame so that $d x_2=d x_3=0$, and go back to $x,y,z,t$ notation, you obtain that

$$g_{11}d x^2+g_{00}c^2d t^2=0,$$

and the light speed is apparently

$$\frac{d x}{d t}=c\sqrt{-\frac{g_{00}}{g_{11}}},$$

which is not necessarily $=c$. Can we conclude that it is not constant? Well, not, because this formula doesn't show the speed of light in tehe units in which $c$ is expressed, but in some other units, which are scaled. To find the correct answer, we either choose the frame to be orthonormal, which gives $g_{11}=-g_{00}=1$, or we make sure to divide each infinitesimal distance with the unit "conversion factor" along that direction (they are just $\sqrt{-g_{00}}$ and $\sqrt{g_{11}}$). Hence, the speed of light is always $c$, although in rescaled coordinates may appear not to be. You cannot make it different, no matter how you try, unless you rescale it (i.e change the units).

Now, please see that this is not a proof that the speed of light is constant. It is constant by the very construction of the spacetime. It would be circular to claim that this shows the speed of light is constant. I showed you this construction to explain how it is consistent to have constant speed of light, even if time and space intervals change in different frames. Now you can compare this model with the physical data.

Now, the "true" speed of light is that present in the wave equation describing the light, in vacuum. And this is still $c$. Can it vary from point to point? It may be possible, in principle, but it is consistent with the observations that it remains constant. If it would vary, Maxwell's equations would not be covariant. This would not be a big deal, one can imagine worlds in which they are not covariant. But the theory of relativity originated from the study of their invariance.