I think people make an obvious oversight when it comes to the term "speed of light". In Landau, Classical Theory of Fields, they formulate Einstein's second postulate as:
"There is a finite limit speed c with which interaction can be transmitted."
According to the First Postulate (Principle of Relativity), this limit must have the same value in every inertial reference frame.
Then, they go on to prove that the space-time interval:
<br />
ds^{2} \equiv c^{2} \, dt^{2} - dx^{2} - dy^{2} - dz^{2}<br />
is an invariant and derive everything else from there.
In particular, they build electromagnetism "from the ground up". It turns out from Maxwell's equations then, that electromagnetic fields can exist independent from any charges and currents, but have to be time-dependent. These fields propagate as waves and their speed of propagation in vacuum (free space) is equal to the same c as above. That is why this is called speed of light in vacuum.
However, as we all know, the speed of propagation of electromagnetic waves need not always be c, as in some materials, where it is also frequency dependent (dispersion).
For the case of non-inertial reference frames (or the case where a gravitational field is present), they simply define the space-time interval as a general quadratic form:
<br />
ds^{2} = g_{\mu \nu} \, dx^{\mu} \, dx^{\nu}<br />
with g_{\mu \nu} = g_{\nu \mu} - the metric tensor, containing all the information about the space-time and the particular coordinate system. Because at any point we may diagonalize this quadratic matrix, it must resemble the Minkowski metric tensor g^{(0)}_{\mu \nu} = \mathrm{diag}(1, -1, -1, -1). In particular, for real space-time, we must have:
<br />
g < 0<br />
The metric tensor can be used to deduce real time intervals, distances and synchronize clocks.
For example, if we are at a particular point in space (x^{i} = \mathrm{const.} \Rightarrow dx^{i} = 0) and consider two infinitesimally closed events, than, by analogy with SR we define the proper time interval as ds = c \, d\tau
<br />
ds^{2} = g_{0 0} \, (dx^{0})^{2} = (c \, d\tau)^{2}<br />
<br />
g_{00} \ge 0 \Rightarrow d\tau = \frac{\sqrt{g_{0 0}}}{c} \, dx^{0}<br />
The case g_{0 0} < 0 does not necessarily mean that that space-time is impossible, but simply that the particular coordinate system we are using is unsuitable.
Now comes the important point: Distances are defined through the same "radar procedure" using something that moves along null geodesics. Namely, let us consider two points A with space coordinates x^{i} and B, which is infinitely close and with space coordinates x^{i} + dx^{i}. We shine a light ray from B at x^{0} + (dx^{0})_{2}, it propagates to A, reflects at x^{0} and arrives back at B at x^{0} + (dx^{0})_{2}. Since the wave is traveling along a null geodesic, we may find (dx^{0})_{1/2} by equating ds = 0 and solving the quadratic equation:
<br />
g_{0 0} (dx^{0})^{2} + 2 \, g_{0 i} \, dx^{i} \, dx^{0} + g_{i j} \, dx^{i} \, dx^{j} = 0<br />
where a summation from 1 to 3 over a repeated Latin superscript and subscript is implied. The solution of this equation is:
<br />
(dx^{0})_{1/2} = \frac{-g_{0 i} \, dx^{i} \mp \sqrt{(g_{0 i} \, g_{0 j} - g_{0 0} \, g_{i j}) \, dx^{i} \, dx^{j}}}{g_{0 0}}<br />
According to what has been said above for proper time intervals, the round trip time, according to B is:
<br />
d\tau = \frac{\sqrt{g_{0 0}}}{c} \, \left((dx^{0})_{2} - (dx^{0})_{1}\right)<br />
and this, by definition, corresponds to a distance:
<br />
dl = \frac{c \, d\tau}{2}<br />
which gives the following:
<br />
dl^{2} = \gamma_{i j} \, dx^{i} \, dx^{j}, \gamma_{i j} = \frac{g_{0 i} \, g_{0 j}}{g_{0 0}} - g_{i j}<br />
for the spatial distance and \gamma_{i j} = \gamma_{j i} is the spatial metric tensor. For coordinate systems attainable by physical bodies, the quadratic form dl^{2} \ge 0 must be positive definite.
The moment of reflection of the signal at point A, according to B, corresponds to the time coordinate x^{0} + \Delta x^{0}, where:
<br />
\Delta x^{0} = \frac{(dx^{0})_{1} + (dx^{0})_{2}}{2} = g_{; i} \, dx^{i}, \; g_{; i} = -\frac{g_{0 i}}{g_{0 0}}<br />
which is the synchronization offset. In this way, we can synchronize clocks along any open line in space, bu not, in general, over closed loops, since:
<br />
-\oint{\frac{g_{0 i}}{g_{0 0}} \, dx^{i}} \neq 0<br />
in general. As a conclusion, signals that travel along null geodesics have speed c by definition.
But, if you write the equations of electromagnetism in curved spacetime, you will see that they predict propagation of electromagnetic waves at different speeds than c.
