The answer to this very good question seems to be favored by a large amount of users.
Yet it seems to imply that the constancy of the speed of light and its finiteness stems from the underlying space-time symmetries.
First it says:
... let me note that speed itself is a coordinate system dependent concept. If you had a bunch of identical rulers and clocks, you could even make a giant grid of rulers and put clocks at every intersection, to try to build up a "physical" version of a coordinate system with spatial differences being directly read off of rulers, and time differences being read from clocks. Even in this idealized situation we cannot yet measure the speed of light. Why? Because we still need to specify one more piece: how remote clocks are synchronized. It turns out the Einstein convention is to synchronize them using the speed of light as a constant. So in this sense, it is a choice ... a choice of coordinate system. There are many coordinate systems in which the speed of light is not constant, or even depends on the direction.
How is agreeing on the speed of light a choice? If we didn't accept a priori its constance, what sense would it make agreeing on a value? You would end up with very different conclusions about the world depending on how fast was light speed locally for you.
Further it says:
...It is because of the symmetry of spacetime that we can make an infinite number of inertial coordinate systems that all agree on the speed of light. It is the structure of spacetime, its symmetry, that makes special relativity...
and ends with:
...The modern statement of special relativity is usually something like: the laws of physics have Poincare symmetry (Lorentz symmetry + translations + rotations)... Not everything is relative in SR, and speed being a coordinate system dependent quantity can have any value you want with appropriate choice of coordinate system. If we design our coordinate system to describe space isotropically and homogenously and describe time uniformly to get our nice inertial reference frames, the causal structure of spacetime requires the speed of light to be isotropic and finite and the same constant in all of the inertial coordinate systems.
From what I understand, this would seem to imply that you have freedom in choosing your coordinate systems and this would yield different results for the measured speed.
While it is equivalent to derive the constancy of light speed from the symmetries of space-time, as opposed to the inverse process, I don't think it yields the same result in the construction of knowledge. The former order would imply that light speed can vary, depending on the local space-time properties, and we are choosing a convention in which the value is the same. This would mean that the comparison of light speed with other, say sound speed, would be region dependent, since the latter depends at least on the medium properties if not also on space-time.
However, the second one, where light constancy assumption yields the symmetries, would imply that the local space-time properties however altered, will always hold the light speed, which seems to me more natural and consistent with reality.
My question is then: Are the answer's assertions correct? If so, where am I misunderstanding?
Answer
Special relativity is the spacetime geometry described by the Minkowksi metric:
$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 $$
where $c$ is a constant. The Minkowksi metric is the solution to the equations of general relativity when no mass or energy is around to curve spacetime$^1$. All the symmetries you alluded to are encapsulated in the Minkowski metric - indeed all of special relativity is encapsulated in this metric.
Experiment confirms that the Minkowski metric correctly predicts observations i.e. no deviation from it has ever been observed. So our working hypothesis is that the Minkowski metric is the correct description of flat spacetime.
From the metric it's easy to show that $c$ is a velocity and indeed is the maximum possible velocity anything can have. Maxwell's equations also tell us that $c$ is the velocity with which electromagnetic waves propagate. Therefore we conclude that the constant $c$ is the speed of light and therefore that the speed of light is constant.
Any deviation from the predictions of relativity could be evidence that the speed of light isn't constant, and there is no shortage of scientists looking for them. So far no such deviations have been found.
Note that relativity doesn't tell us what the value of $c$ is, only that it is a constant.
To address the specific point in the cited answer:
There are many coordinate systems in which the speed of light is not constant, or even depends on the direction.
For example we can rewrite the Minkowski metric using accelerated coordinates and the result is the Rindler metric:
$$ c^2d\tau^2 = \left(1 + \frac{a}{c^2}x \right)^2 c^2 dt^2 - dx^2 $$
It is very important to emphasise that in this metric $c$ is still a constant, but now it is the local speed of light i.e. the speed any observer will measure if they do a measurement at their location. However if you use the metric to calculate the speed of light as a function of the distance $x$ from the observer you get:
$$ \frac{dx}{dt} = c\,\left(1 + \frac{a}{c^2}x \right) $$
and this speed can be greater or less than $c$ depending on the value of $x$. See my answer to Acceleration and its effect on the speed of light for more on this.
The speed $dx/dt$ is the coordinate velocity of light, and it can differ from $c$ because we have complete freedom to choose our coordinates. However it remains the case that any local measurement of the speed of light by any observer will always return the same value of $c$.
Incidentally, this principle remains true even in curved spacetimes so it is true in general relativity as well as special relativity.
$^1$ strictly speaking the Minkowski metric is only one of several vacuum solutions - the Schwarzschild and Kerr black holes are also vacuum solutions. However it is the only one with zero ADM mass.
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