Gauge Theory of Gravity 1-22-11

On Jan 22, 2011, at 11:15 AM, Paul Zielinski wrote:

On Sat, Jan 22, 2011 at 1:11 AM, JACK SARFATTI <sarfatti@pacbell.net> wrote:

On Jan 21, 2011, at 11:18 PM, Paul Zielinski wrote:

On Fri, Jan 21, 2011 at 7:23 PM, JACK SARFATTI <sarfatti@pacbell.net> wrote:

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?

all that shows is that your intuition detached from the mathematical machinery leads you to wrong hunches.

It's a question.  A perfectly reasonable one in my view.

What is the point of general covariance if physical quantities are not generally covariant?

Fair question.

General covariance is simply the local gauge invariance of the translation group T4(x).

Mathematically this is just a fancy recipe for generating GCTs. It's not at all clear to me that such local
invariance has any more meaning than that in gauge gravity.

The local gauge principle is an organizing meta-principle that unifies and works. Also it give physical meaning to GCTs when combined with the equivalence principle as the computation of invariants by locally coincident Alice and Bob each independently in arbitrary motion measuring the same observables. SR is restricted to inertial motions and constant acceleration hyperbolic motion (Rindler horizons & maybe extended to special conformal boosts).

Locally gauging SR with T4 ---> T4(x) gives 1916 GR.

However, the INDUCED spin 1 vector tetrad gravity fields e^I are fundamental with guv spin 2 fields as secondary. Nick's problem why no spin 1 & spin 0 in addition to spin 2 still needs a good answer of course.

In terms of reference frames, doesn't this simply mean that the observer's velocity is allowed to vary from point to point in spacetime?

No. It means that and a lot more. The coincident observers also can have, acceleration, jerk, snap, crackle, pop, i.e. D^nx^u(Alice, Bob ...)/ds^n =/= 0 for all n.

Physically it corresponds to locally coincident frame transformations between Alice and Bob each of which is on any world line that need not be geodesic, but can be. 

I think you should say here that it is the invariance of tensor quantities under such transformations.

It's COVARIANCE not INVARIANCE. Invariants can be constructed by contractions of COVARIANTS.

e.g. in non-Abelian gauge fields SU2 & SU3, unlike the U1 Maxwell electrodynamics the curvature 2-form F^a is not invariant, but is covariant

i.e.

F^a = dA^a + f^abcA^b/\A^c

[A^b,A^c] = fa^b^cA^a

F^a ---> F^a' = G^a'aF^a

This is COVARIANCE not INVARIANCE (U1 is a degenerate case exception).

G^a'a is a matrix irrep of G (Lie gauge group of relevant frame transformations).

when the gauge connection Cartan 1-form transforms inhomogeneously (not a G-tensor)

A^a --> A^a' = G^a'aA^a + G^bc'G^a'b,c'

In gravity G is a universal space-time symmetry group for all actions of all physical fields including their couplings. This is the EEP in most fundamental form.

But if "physical" quantities (e.g., LC^) are not invariant under such transformations, what is the point of general covariance?

As far as I can see calling GCTs "gauge transformations" based on a superficial analogy with internal parameter gauge theory
doesn't change anything.

The intrinsic induced pure gravity fields are the four tetrads e^I that form a Lorentz group 4-vector hence spin 1.

Well this is tricky. It is the tetrad *transformations* e^u_a that represent the Einstein field. Such transformation take
you from a coordinate LNIF basis to an LIF orthonormal non-coordinate tetrad basis. Thus the e^u_a pick up both
the intrinsic geometry *and* the coordinate representation of the LNIF

Of course the e^u_a and the e^a_u can also be treated as the components of the LNIF coordinate basis vectors in the tetrad
basis, and vice versa, but that is another matter.

Each e^I is generally INVARIANT i.e. scalar under GCTs T4(x).

Right. Local Lorentz frames and LLTs are represented by orthonormal tetrad basis vectors in this model, while we
are free to apply arbitrary GCTs in the local frames. I think it is this subtlety of the tetrad model that has led Chen
and Zhu astray as to their attempted decomposition of the LC connection into physical and "spurious" parts in the
context of plain vanilla GTR (coordinate frame model).

What they appear to have done is extract the first order variation of the metric g_uv from the part that encodes the
Riemann curvature, attributing such first order variation in its entirety to the choice of coordinates. If so then the
entire paper is misconceived IMO.

The LC connection is not gauge invariant nor even gauge covariant - that's an effect of the equivalence principle that Newton's "gravity force" is a chimera - 100% inertial force from the acceleration of the detector in curved spacetime.

Of course and no one is saying that it is. We are talking about the "physical part" LC^ that Chen and Zhu claim
to have extracted from LC, after removiing what they call the "spurious" part LC_ that according to them simply reflects
the choice of coordinates.

But that choice is also physical though not intrinsic. Its physical because its a state of motion of a detector - ultimately at the operational level where the hard rubber hits the ground of experience.

My point is that if their "physical part" LC^ is not a covariant quantity, then its intrinsic value likewise depends on
the choice of coordinates. This makes no sense to me. Not only that, but they claim to be able to derive a tensor
vacuum stress-energy density from such a quantity. Since the whole problem with the Einstein and various other
stress-energy pseudotensors is precisely that they are not covariant quantities, what exactly *is* the point of their
paper?

It depends what you mean by "physical". Arbitrary concomitant g-forces are observables even though they are are not tensor covariants or part of the intrinsic curvature geometry, which is 100% geodesic deviations.

It's not clear to me whether Chen and Zhu are saying this must be the case in gauge gravity, or in the GTR,
or both. Their reasoning strikes me as obscure.

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 (principle bundle etc) needed to keep the extended action of the source matter field (associated bundle etx) invariant can never be a tensor relative to G(x). That's in the very definition of local gauging'.

You're talking here about a connection. Of course, everyone knows that. If the connection itself is a tensor, then you
don't get a covariant derivative. A connection has to be non-covariant. In order to correct for curved coordinate artifacts
in partial derivatives, it has do depend non-tensorially on the coordinates.

But Chen and Xhu said they were going to remove the coordinate dependent part LC_ from the LC connection to get
their "physical part" LC^. If so, then why is the resulting LC^ not a covariant quantity?

And if it isn't, how does it help with the construction of a vacuum stress-energy tensor?

Clear as mud.

 

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/\

Jack, no one is saying that connections are tensors. Please. 

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.

I guess you mean the tetrad *transformations*, starting from an LNIF coordinate basis.

The induced A clearly depends on the initial coordinates and on the geometry in the general
case.

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.

OK fine but beside the point. No one is arguing that a connection is a tensor. As far as I know
no one ever has.

Exactly what is Chen and Zhu's so-called "geometric part" LC_ ? Do you know? 

 

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.