Speed of light in accelerating local frames of reference

On Sep 7, 2011, at 12:43 AM, JACK SARFATTI wrote:

On Sep 6, 2011, at 1:07 PM, Paul Zielinski wrote:

In 1905 SR, it is the coordinate speed that is an absolute invariant under the Einstein-Lorentz transformations and this is at the core of Einstein's 1905 version of relativity.

I replied:

(ct)^2 - L^2 = (ct')^2 - L'^2  S and S' both GLOBAL inertial frames

L^2 = x^2 + y^2 + z^2

PZ: In modern GR, the coordinate speed is not invariant under general transformations; so it is deprecated as "coordinate speed" in favor of invariant measures, in the spirit of general covariance.

I replied:

(cdt)^2 - dx^2 - dy^2 - dz^2 = guv(LNIF)dx^udx^v = gu'v'dx^u'dx^v'

first = is the tetrad map

2nd = is a GCT

You really can't define the speed of light very well when there is gravimagnetism g0i = Ai =/= 0

Define dL^2 = gijdx^idx^j

i,j = 1,2,3

for a light ray ds^2 = 0

dT = g00^1/2dt

c^2dT^2 - dL^2 + cAidtdx^i = 0

(dL/dT)^2 = c^2 + cAidx^i/dT

c(LNIF) = [c(LIF)^2 + cAi(dt/dT)dx^i/dT]^1/2

v^I = dx^i/dT

dt/dT = 1/g00^1/2

c(LNIF) = c(LIF)[1 + Aiv^i'/c(LIF)g00^1/2]^1/2

Only when Ai = 0   

c(LNIF) = c(LIF)  and in this case speed of light in a non-dispersive vacuum is an absolute invariant.


PZ: Of course in Minkowski's version of SR, we already have the Lorentz-invariant interval s which gives us the time actually read by moving clocks. And of course this goes over to the generally covariant infinitesimal interval ds^2 = g_uv dx^udx^v in GR. I think this illustrates very clearly how GR supersedes 1905 SR, and how the definition of "light speed" is not the same in both  theories. I think the most one can say is that  in LIFs, the locally observed empirical predictions of the two theories agree, since  the laws of SR "work" in such local frames.


On 9/4/2011 10:05 PM, JACK SARFATTI wrote:

On Sep 4, 2011, at 9:24 PM, Paul Zielinski wrote:

Jim and Jack,

I think it's high time we got this straightened out. I can;t believe this is still in dispute.

Here's Baez:

"The problem here comes from the fact that speed is a coordinate-dependent quantity, and is therefore somewhat ambiguous.  To determine speed (distance moved/time taken) you must first choose some standards of distance and time, and different choices can give different answers."

"This is already true in special relativity: if you measure the speed of light in an accelerating reference frame, the answer will, in general, differ from c."