Begin forwarded message:
From: JACK SARFATTI <
Date: November 28, 2010 6:15:18 PM PST
To: Paul Zielinski <
Subject: Re: The physical meaning of Einstein's General Coordinate Transformations (GCT)
The best way to see this is with global Penrose conformal diagrams.
http://en.wikipedia.org/wiki/File:Penrose.PNG
This is for globally flat Minkowski spacetime of 1905 SR
R^I^J = 0 identically everywhere-when.
This is an unstable solution of Ruv = 0.
You can make GCTs on this and get all kinds of screwy looking guv metrics, all kinds of Levi-Civita connection patterns - but all of them will have vanishing curl. Therefore, they are gauge transformations only describing the flight maneuvers of LNIF spacecraft.
No GCT will ever morph the above global geodesic pattern into any of these for example:
http://en.wikipedia.org/wiki/File:PENROSE2.PNG
On Nov 28, 2010, at 6:06 PM, JACK SARFATTI wrote:
GCTs are not singular, they are single-valued not multi-valued around topological obstructions, consequently they do not change the real global pattern of the geodesics. GCTs are redundant gauge transformations that describe all possible mappings of the motions of possible fields of accelerating small LNIF detectors. Classical counter-factuality is assumed. That is, if you put a detector there, they would measure a definite result. The invariant computed from that definite result is assumed to really exist there even if no actual measurement has been made.
On Nov 28, 2010, at 5:38 PM, JACK SARFATTI wrote to Z:
It is a mistaken model. It's wrong because you have made a false premise.
The geodesic equation?
No, of course not. Your false premise is that a change in the Levi-Civita connection induced by a GCT changes the objective pattern of geodesics in any way. It never does.
Your starting point is mathematically and conceptually wrong.
You seem to be saying that the geodesic equation is wrong. That is my premise here.
No, I never said that. To say that is stupid. I am not that stupid. What I said was that:
The first-order metric gradients do not change the objective pattern of the geodesics as you assume.
Can you explain this with specific reference to the textbook standard geodesic equation
d^2x^u/d lambda^2 + LC^u_rs (dx^r/d lambda) (dx^s/d lambda) = 0
and show how the second derivatives of the metric appearing in the Riemann curvature tensor enter
into the problem of solving for the geodesics in the presence of a real gravitational field? I just don't see it.
Very simple.
Start from
Guv = -kTuv + boundary/initial conditions
get a valid solution like
gtt = -1/grr = 1 - rs/r
which describes only static LNIFs. It's a representation, a shadow on the wall in contrast to the "light" that is the local invariant
ds^2 = guvdx^udx^v
Compute Ruvwl in this representation ~ rs/r^3
The source parameter here is
rs = 2GM/c^2 - that comes from a Tuv.
Now make any GCT you like. M is not changed. The GCT will change the representations of guv, Levi-Civita^uvw, Ruvwl but it will not change the global pattern of the geodesics of the solution that only depends on the source parameter M.
In other words the GCTs are non-physical gauge transformations - that do represent different patterns of possible fleets of LNIF detectors.
The only way to change the geodesic pattern is by changing M which will change the curl of the Levi-Civita connection in any representation.