Light speed in a non Inertial frame of reference

No, it doesn't. The Principle of Relativity is is always stated for inertial frames only. Different inertial frames are related by Galilei and Lorentz transforms in Newtonian and special relativistic mechanics respectively.

 

It's also hard to define an object's speed in a non-inertial frame. Ignore relativity for a moment: a car is moving at 30 mph in the street's rest frame. You're accelerating at 1 m/s^2. What's the car's speed in your frame? You'll see there is no single number - it depends on what you are measuring and when/how you are measuring it.

 

I thought that it held locally since any region of spacetime is locally flat.

 

The question is about Minkowski spacetime, which is globally flat. And the coordinate speed of light can be anything you want it to be, depending on what coordinate system you use.

It's a bit strange to say that something like this holds "locally" in Minkowski spacetime, because any "local inertial frame" (=comoving inertial frame) on Minkowski spacetime (which is globally flat) is actually a global inertial frame. So the claim that the speed of light is invariant "locally" in SR doesn't really mean anything, unless you explain what you mean of course. (It sounds like it could mean that the comoving inertial frame is an inertial frame, which is sort of implied by the name)

If we use the "proper reference frame" of an accelerating observer (i.e. the coordinate system constructed using the standard synchronization procedure), the coordinate speed of light emitted by the observer will depend on a lot of different things, but if he emits the light at the origin of his coordinates, it will at least start out with speed c. I guess that's one thing we could mean by "holds locally" (but I'd rather not use phrases like that).

 

Yes, locally could mean this. If any "photon" is (sufficiently close to being) coincident with any observer, accelerated or non-accleerated, in both special and general relativity, then, by using (sufficiently small) standard clocks and rulers, the observer measures the speed of the photon to be c. If the photon and the observer are not coincident, then, as Vanadium 50 has pointed out, the concept of speed speed is subtle (even more so in relativity), and, as pointed out by atyy and Fredrik, coordinate speed can take on any value.

Fredrik: sometimes (but not always) when a physicist says "local," the physicist means "in the tangent space." If an observer and a photon are coincident at event [itex]p[/itex] and [itex]\left\{\mathbf{e}_0, \mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3 \right\}[/itex] is an orthonormal frame Fermi-transported by the observer, then (if alignment is right), the tangent vector at [itex]p[/itex] of the photon's worldline is [itex]\mathbf{e}_0 + c \mathbf{e}_1[/itex]. Even for Minkowski spacetime, this is naturally interpreted as a statement about objects in the tangent space at [itex]p[/itex].

Unfortunately, "local" can mean quite different things for mathematicians and physicists.

 

By "locally" I just meant "nearby" - in the sense that the surface of a sphere is flat if one looks at a small enough piece of it. Maybe I used the term inappropriately.

I don't see why "non-inertial frame of reference" implies flat spacetime. Am I not right now in a non-inertial frame of reference (I'm not freefalling as I type) in a spacetime that is curved?

 

Yes. I gave my answer for the "global" speed of light in special relativity, which is restricted to flat spacetime.

If we talk about the "local" speed of light in a "proper reference frame", then it is always c, even in curved spacetime.

A proper reference frame (George Jones's "Fermi-transported orthonormal frame") is one that in which the metric is Minkowskian at the origin.

If the worldline is accelerated, then a proper reference frame is not locally inertial, because the Christoffel symbols, which contain first derivatives of the metric and represent inertial forces due to an accelerated frame of reference or "fake gravity", are not all zero.

If the worldline is free-falling, then a proper reference frame is locally inertial, because the Christoffel symbols are all zero. Even in a local inertial frame, the first derivatives of the Christoffel symbols, which are second derivatives of the metric, cannot all be zero, if spacetime curvature or "true gravity" is present.

Strictly speaking "locally" in the above means "at a point" (George Jones's "tangent space"). It is not strictly true for any finite piece of spacetime (unless spacetime is flat, and an inertial frame is used), but practically speaking, a finite piece of spacetime that is small enough will be close enough to a "point" given limited sensitivity of experimental equipment.

There are other contexts where other mathematical statements (eg. about the extremal length of spacetime geodesics) are strictly true though locally means a finite piece of spacetime that is more than a point.

 

When they say the speed of light is "locally" c in GR, this also implies that you are using a locally inertial reference frame, does it not? Even in an arbitrarily small region, couldn't you pick a locally non-inertial frame where the speed of light would be other than c?

 

The idea of 'speed' is really not useful, or even defied in a non-inertial frame. In a general non-inertial frame, the coordiantes x^μ are pretty much on the same footing, and there is no special 'time' coordinate. This already makes speed meaningless, but even if one could identify a natural time coordinate, the speed defined with respect to this would be different at different points due to the fact that the coordinates can twist and turn from point to point.

But a non-inertial observer can always construct a local inertial frame, which he can Fermi-Walker transport with himself. In this local inertial frame, the speed of light is always c.