As John Wheeler said "Physics is simple when it is local."
That is the case here.
Mach's Principle in this classical case is astrology in my opinion.
Frankly I am not able to follow most of Jim's informal language when he does not include some concrete mathematical examples.
The general situation about the speed of light is simple.
ds = 0 for light (null geodesic)
the speed of light in a unaccelerated non-rotating local inertial frame (LIF) is c = 3x10^10 cm and it is invariant for all locally coincident LIF --> LIF'.
The local metric for any weightless observer clamped to any LIF (even in strong curvature fields) is
ds^2 = (cdt)^2 - dL^2
dL^2 = dx^2 + dy^2 + dz^2
I don't think what Jim says about distant observers is correct.
Next consider the LOCALLY MEASURED speed of light in a translationally accelerating and/or rotating locally coincident (with the LIFs) non-inertial frame LNIF.
The mapping from any LIF to any LOCALLY COINCIDENT LNIF is the TETRAD MAPPING with 16 components and the SPIN-CONNECTION mapping with 24 components.
The LOCAL METRIC for the observer CLAMPED to the LNIF is guv(LNIF) where
ds^2(LIF) = ds^2(LNIF) = guvdx^udx^v
= g00(cdt')^2 + A.dxcdt' + dL'^2
A.dx = g0idx'^i
i,j = 1,2,3
dL'^2 = -gijdx'^idx'^j
Formally
g00 = 1 + phi/c^2
this is general no, weak field post-Newtonian approximation yet.
Therefore, there is no sufficient reason in this general case to treat phi and A as if they were a Lorentz 4-vector and as if they were the gauge potentials of a U(1) gravity field.
If you want to do that you have to start from Einstein's LOCAL FIELD EQUATIONS
Guv + 8piLp^2/hcTuv = 0
The speed of light LOCALLY MEASURED IN THE LNIF IS NOT INVARIANT UNDER THE TETRAD MAP AND IT'S NOT INVARIANT UNDER THE GENERAL COORDINATE TRANSFORMATIONS BETWEEN LOCALLY COINCIDENT LNIF --> LNIF'.
This is easily seen from
ds^2 = 0
0 = g00(cdt')^2 + A.dxcdt' - dL'^2
The LNIF observer's number for the locally measured speed of light is
c' = dL'/dt'
g00(cdt')^2 = - A.dxcdt' + dL'^2
g00c^2 = - cA.dx/dt + dL'^2/dt'^2
(1 + phi/c^2)c^2 = - cA.dx/dt + c'^2
(1 + phi/c^2)c^2 + cA.dx/dt = c'^2
(1 + phi/c^2)c^2 + cc'A.n = c'^2
n is the normal unit vector of the light ray in space
this is a quadratic equation for c' with TWO ROOTS when there is gravimagnetism.
The LOCALLY MEASURED speed of light at the horizon g00 = 0
is a degenerate single root
c' = A.n
this is NOT LOCALLY INVARIANT under general coordinate transformations.
On Jan 10, 2012, at 3:56 PM, Paul Zielinski wrote:
OK Jim I'll do some reading on this notion of a "locally measured invariant" and reply later. I see you've discussed this in
detail in some of your published papers.
On 1/10/2012 3:43 PM,
Paul,
The locally measured invariance of c (and phi that is its square) in both inertial and accelerated frames does not have the consequence that covariant derivatives and such like vanish. The derivatives look at non-local things. As an example, look at the speed of light in the vicinity of the Sun (or some star). A distant observer measures the speed in general to vary in both time and space, notwithstanding that locally measured invariance as I have stated it obtains point by point throughout the field. When you compute the derivatives of c, you are calculating the sort of variation seen by the distant observer, and they do not vanish.
Jim