On Jan 21, 2011, at 6:53 PM, Paul Zielinski wrote:
If their "physical part" LC^ represents a "true" physical quantity, why would it not be generally
covariant in GTR? Given that GTR is a generally covariant theory?
To which I replied:
"All that shows is that your intuition detached from the mathematical machinery leads you to wrong hunches. What happens in local gauging of a rigid group G to a local group G(x) is that the induced compensating connection A Cartan 1-form (principal bundle etc) needed to keep the extended action of the source matter field (associated bundle etc) invariant can never be a tensor relative to G(x). That's in the very definition of local gauging.
All you can hope for is covariance of the "field" 2-form, i.e. the 2-form A-covariant derivative of itself is a tensor under G(x).
D = d + A/
But A/A = 0 for U1(x)
a = 1
but
A/A =/= 0
for SU2(x)
a = 1,2,3
&
SU3(x)
a = 1,2,3,4,5,6,7,8
In general A/A -> fbc^aA^b/A^c
i.e. F^a = DA^a = dA^a + fbc^aA^b/A^c
[A^b,A^c] = f^abcA^c
In the special case G(x) -> U1(x) the field 2-form F = dA is actually invariant, but not so for SU2(x) & SU3(x)
If G(x) has the representation U(G(x)) then
A -> A' = UAU^-1 + dUU^-1
F --> F' = UFU^-1
Now for Einstein's GR G(x) -> T4(x)
and the induced A is NOT the spin 2 Christoffel symbol etc. but the non-trivial TETRAD set.
the internal index a is replaced by the Lorentz group index I (J,K etc).
The induced gravity spin 1 tetrad connection is A^I analog to A^a (Yang-Mills)
I = 0, 1, 2, 3
the relation to the spin 2 Christoffel symbol is very indirect and complicated.
And how do Chen and Zhu propose to derive a vacuum stress-energy *tensor* from LC^ if
LC^ is not itself covariant? How can non-covariant LC^ be a solution to the GR energy
problem?
Doesn't make sense.
On Fri, Jan 21, 2011 at 6:44 PM, JACK SARFATTI wrote:
On Jan 21, 2011, at 6:15 PM, Paul Zielinski wrote:
Yes you're right -- they start by saying that they are separating a coordinate dependent
part LC_ from LC, leaving what they call the "physical part" LC^ that represents the true
gravity field, but then on p8 they say that LC^ cannot transform covariantly under GCTs
due to the way that LLTs are represented in gauge gravity.
So I think they've been led astray by their gauge gravity template.
No, I think it means that what you want to do cannot be done.
I know you think that.
Note, that Arnowitt, Deser & Misner in 1962 have a solution, but it too is too limited in the end.
Yes I'm looking at it.
I think my approach is much simpler, much more direct -- you just remove the coordinate
correction terms from the LC connection, leaving a unique tensor residue that encodes
the intrinsic spacetime geometry. No need for perturbation expansions and so on -- exact
decomposition.
As I said your words are too vague. You just beat around the push as if saying what you want to do is the same as doing it.
Come on Jack -- the math is settled. Nothing vague about it.
Just ask Waldyr.
Physics uses math. It's not enough to have correct math if the math cannot be connected to laboratory techniques - that's the hard part.
You are doing magickal cargo cult thinking as if wishing makes it so - in my opinion.
Sure Jack.
So how do Chen and Zhu propose to build a covariant vacuum stress energy tensor from non-covariant
LC^? Aren't they just going around in circles? Isn't *that* magickal cargo cult thinking?
This brings us back to the chronic confusion between passive and active diffeomorphisms
in gauge gravity. I think this may be also the root of the confusion in this paper.
On Fri, Jan 21, 2011 at 4:35 PM, JACK SARFATTI wrote:
Z you are mistaken - you have not correctly read the text on p.8
transformations in eq. 6
On Jan 21, 2011, at 4:17 PM, JACK SARFATTI wrote:
On Jan 21, 2011, at 3:50 PM, Paul Zielinski wrote:
OK I read it.
This is exactly the same decomposition I've been talking about for years, approached from the perspective of gauge gravity. And yes it points directly to a localized tensor gravitational vacuum stress-energy density, as I have always maintained. The "physical" part of the LC connection defined in this paper is just the tensor component of LC
Where do they say that? Copy and paste the exact text please. "Tensor" with respect to what group of frame transformations? With respect to the rigid Poincare group of the background Minkowski spacetime - no problem. Remember they do perturbation theory on a non-dynamical globally flat background.
guv = nuv + huv
huv << nuv
therefore no horizons g00 -> 0 in this limit .
Z: that corrects for (and thus encodes) the spacetime geometry, as I've already explained. This part is zero everywhere in a Minkowski spacetime in *any* coordinate system. I have been calling this the "geometric" part of the LC connection. What the authors of this paper erroneously describe as the "pure geometric" part of the LC connection is the part that
corrects only for *coordinate* artifacts (what I've been calling the "curved coordinate correction term"). This part is zero in a Minkowski spacetime in *rectilinear* coordinates, but not in *curvilinear* coordinates. This is the "gauge dependent" part of the LC connection field. There is no actual need for any perturbation expansion here -- the decomposition is exact as well as unique and can be arrived at without the use of any approximations (as I've explained). So the use of perturbation methods in this paper looks
like a quirk of the authors' gauge gravity mindset. I get the impression that while they are on the right track, Chen and Zhu have not yet fully understood the fundamental meaning of the LC decomposition, blinded as they are by the arcane mysteries of gauge theory. :-)
On Wed, Jan 19, 2011 at 8:38 PM, JACK SARFATTI wrote:
Unfortunately it's a perturbation series technique - it obviously cannot describe thermal horizons. Throws the baby out with the bath water, but it's a step in the right direction.
"Because of non-linearity, we have to rely again on perturbative method, and require that the gravitational field be at most moderately strong."
On Jan 19, 2011, at 7:39 PM, JACK SARFATTI wrote:
Yes, this is relevant thanks.
On Jan 19, 2011, at 7:35 PM, Jonathan Post wrote:
If we're trying to distinguish between gravitational effects and pseudo-effects that depend on coordinatizations, is this useful?
Cross-lists for Thu, 20 Jan 11
[73] arXiv:1006.3926 (cross-list from gr-qc) [pdf, ps, other]
Title: Physical decomposition of the gauge and gravitational fields
Authors: Xiang-Song Chen, Ben-Chao Zhu
Comments: 11 pages, no figure; significant revision, with discussion on relations of various metric decompositions
Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
Physical decomposition of the non-Abelian gauge field has recently solved the two-decade-lasting problem of a meaningful gluon spin. Here we extend this approach to gravity and attack the century-lasting problem of a meaningful gravitational energy. The metric is unambiguously separated into a pure geometric term which contributes null curvature tensor, and a physical term which represents the true gravitational effect and always vanishes in a flat space-time. By this decomposition the conventional pseudo-tensors of the gravitational stress-energy are easily rescued to produce definite physical result. Our decomposition applies to any symmetric tensor, and has interesting relation to the transverse-traceless (TT) decomposition discussed by Arnowitt, Deser and Misner, and by York.