CRITICS about Uncle Al's writing
(Schwartz Alan M. 2004.
Affine versus metric gravitation parity.
12 pages. 31.1.2005 online Uncle Al's homepage.
12 Nov 2004, 04.80.Cc, 11.30.Er
http://www.mazepath.com/uncleal/qz.pdf ):
"A teleparallel gravitation stress-energy pseudotensor
antisymmetric to parity transformation constructs volume
integral for total gravitation four-momentum and total
angular momentum. It obtains by comparing vectors at
different points of spacetime- a coframe field - unlike GR.
When the coframe field changes the pseudotensor changes
(not gauge - invariant; not covariant under general
coordinate transformations) [8].
This defines an integral energy-momentum as a redistribution
of energy between material and gravitation (coframe) fields
obeying an exact conservation law.
The Lagrangian for GR can arise from the coframe field only
and be antisymmetric to parity transformation. Extremal
parity test masses may violate the EP." (Schwartz)
EP = Equivalence Principle
"The Weak EP assumes a flat gravitation field is a local
approximation around a given world-point.
Stronger EP statements include the Weak EP [1]." (Schwartz)
"Affine / teleparallel theories embody spacetime torsion.
They ignore the EP and can violate it [3]." (Schwartz)
Above text are from (Uncle Al = Alan M. Schwartz)
Schwartz Alan M. 2004.
Affine versus metric gravitation parity.
12 pages. 31.1.2005 online Uncle Al's homepage.
12 Nov 2004, 04.80.Cc, 11.30.Er
http://www.mazepath.com/uncleal/qz.pdf
Critics:
The weak point of the writing (and hence the whole writing
may be QUESTIONABLE) may be the following:
"It obtains by comparing vectors at different points of
spacetime- a coframe field - unlike GR."
I think that "comparing vectors at different points
of spacetime" is a mathematical problem which
is not solved in GR .
Parallel transport is one tool in trying to solve this
in GR but it depends on the path in question.
The problem in trying to generalize equation
covariat_div(T) + sum_mu nabla_mu(T) = 0
,where T is called the stress-energy tensor into integral
form is best explained by Michael Weiss and John Baez
("Is Energy Conserved in General Relativity" in Physics
FAQ dated 7.11.2000): We would need an extension of
Gauss's theorem.
Now the flux through a face is not a scalar, but a vector
(the flux of energy-momentum through the face). The argument
just sketched involves adding these vectors, which are
defined at different points in spacetime.
Such "remote vector comparison" runs into trouble precisely
for curved spacetimes.
The mathematician Levi-Civita invented the standard solution
to this problem, and dubbed it "parallel transport". It is
easy to picture parallel transport: just move the vector
along a path, keeping its direction "as constant as possible".
The parallel transport of a vector depends on the
transportation path.
But parallel transportation over an "infinitesimal distance"
suffers no such ambiguity. (It's not hard to see the connection
with curvature).
To compute a divergence, we need to compare quantities
(here vectors) on opposite faces.
Using parallel transport for this leads to the covariant
divergence. This is well-defined, because we are dealing
with an infinitesimal hypervolume.
But to add up fluxes all over a finite-sized hypervolume
(as in the contemplated extension of Gauss's theorem)
runs smack into dependence on transportation path.
So the flux integral is not well-defined, and we have no
analogue for Gauss's theorem (Weiss, Baez).
Pseudotensors may indicate that total energy is not properly
defined at least in GR ?
I investigated also related problem when I defined
+, -, * and / operations for my directed geodesic lines
on a sphere surface (special case) in my surface algebras
but I think that question remained somehow open to further
investigations (definitions succeeded when directed
geodesic lines started from the same point
but how to define +, -, * and / operations for
directed geodesic lines which starts from different
points remained somehow open except the case of - operation
in first case where one directed geodesic line
was moved to start from the same starting point as others) ?
Steve Carlip wrote: Local conservation of energy holds only
in a stationary gravitational field in GR. In particular,
an expanding Universe is not stationary, and there is
no local energy conservation law.
Also in GR, no local gravitational potential energy density
can be defined.
The principle of equivalence implies that there can be no
covariant gravitational stress-energy tensor due one can
always choose coordinates in which the geodesics are
arbitrarily close to straight line in a small region,
which implies that the gravitational energy in that region
is arbitrarily close to zero; but a tensor that vanishes in
any coordinate system vanishes in every coordinate system.
The best one can hope for is a "quasilocal gravitational energy",
energy defined in a finite region.
(Steve Carlip: "Re: energy in a comoving volume,
where is the energy going ?", sci.astro.research, 10.12.2001
and "Gravitational Potential in GR", sci.physics.relativity,
23.2.1998).
As my opinion use of this "quasilocal gravitational energy",
may indicate the possible failure of tensor formalism in
this case ?
If I remember right (matters from years ago) that
the Weak Equivalence Principle (local property) is correct
but the Strong Equivalence Principle (global property) is wrong ?
Best Regards,
Hannu Poropudas
Vesaisentie 9E
90900 Kiiminki
Finland
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