In the general theory of relativity, the speed of light in vacuum is $c$. In the special theory of relativity, a postulate is made that the speed of light is the same (called $c$) in all inertial frames.
Consider the following statement:
"The speed of light is $c$ in all inertial frames but it can vary in accelerating frames."
My question is:
Does this statement violate the principle of equivalence or the special theory of relativity or any fundamental law of physics?
Can the speed of light in vacuum depend on the acceleration of the frame of reference?
Not the acceleration, but rather the difference in the (gravitational) potentials.
@MarkMoralesII gave a concise answer. In an accelerating frame, the velocity of the light is the same ($c_0$) $-$ in accord with special relativity $-$ as measured in the vicinity of the observer. However, this speed, for the photons that travel above the observer, is measured greater; and the speed of the photons that move far below the observer is measured smaller. I denote by above the locations with less negative, and by below the places with more negative gravitational potential.
Does this statement violate the principle of equivalence or the special theory of relativity or any fundamental law of physics?
The so-called fundamental laws of physics are, at least, held valid locally in non-inertial frames, unless you want to apply them non-locally.
The notion of speed is meaningless unless you make it clear how you're measuring it.
An "inertial frame" is (in Einstein's paper) a system of clocks and metersticks. His postulate means that if you set up all of this infrastructure according to a certain procedure, and then measure certain quantities and divide them, you'll get $c$.
If you set things up differently and measure different quantities, you may get other values. That's fine; there's no law that the speed of light has to be $c$ with respect to any system of coordinates you can dream up. Any theory based on that kind of rule would be incoherent, because I could define coordinates $t'{=}t, x'{=}2x$ and show that the speed of light is $2c$ and therefore $c=0$.
The postulate of the constancy of the speed of light has physical content because you can show that there are many different relatively moving inertial frames with respect to which the same beam of light has the same speed $c$, which doesn't happen in Newtonian physics.
Einstein's two-postulate presentation made sense given his audience at the time, but I think it's unnecessarily complicated, because inertial frames are such complicated objects. There's a nicer development of special relativity popularized by Hermann Bondi whose fundamental non-Newtonian postulate is the symmetry of Doppler shifts.
