You do get the torsion 1-form by locally gauging only T4, because the tetrad is the torsion 1-form. The torsion 2-form is what gives the gap in the parallelogram. It is what gives the contortion tensor addition to the Levi-Civita connection. Remember torsion gaps are translational defects. Curvature is a disclination defect - i.e. change in orientation of a vector around a small parallelogram - that won't close if there is a curl in the torsion 1-form.
torsion gap 2-form = D(tetrad) = d(tetrad torsion 1-form) + (spin connection 1-form)/(tetrad torsion 1-form)
IMPOSE BY HAND ad-hoc
torsion gap 2-form = 0
this still permits a spin-connection given by
that CAN have a non-vanishing curl, i.e. a curvature 2-form =/= 0
R(curvature 2-form) = D(spin connection) = d(spin connection) + (spin connection)/(spin connection) =/= 0
I admit this is very tricky conceptually.
Locally gauging SO1,3 is redundant I suppose. The key group is T4(x).
Suppose we have a matter spinor field A in the presence of gravity the covariant derivative operator on the spinor field is symbolically
Du = (tetrad)T4(generator) + (spin-connection)SO1,3(generator)
i.e., Du = eu^IPI + wu^I^JPIJ in the LNIF where I,J are the LIF indices.
where PI, PIJ generate the Poincare group Lie algebra.
We also need the Dirac 4x4 gamma matrices in the LNIFs
(gamma)^u = e^uI(gamma)^I + w^uIJ[(gamma)^I,(gamma)^J]
Then the absolute invariant operator D slash = D/ is
D/(LNIF) = (gamma)^uDu =/= D/(LIF)
On Dec 31, 2010, at 2:10 PM, Paul Zielinski wrote:
That's not true. Locally gauging T4 naturally permits torsion.
I said you don't get torsion by locally gauging T4. Is that not correct?
Torsion is the dislocation defect - the gap in closing a parallelogram where one edge is parallel transported about the other. You have to impose zero torsion 2-form as an adhoc constraint and you still can get disclination curvature because you can have a spin connection determined from the tetrads that has a non-vanishing covariant curl with itself. It's very tricky I admit.
Then why do you need to gauge the full Poincare group to get Einstein-Cartan?
On Gravity & Accelerated Frames
Synge identified the actual gravitational field of GR with non-vanishing Riemann curvature.
Fine, nothing wrong with that. However, Einstein's critics simply quibbling because Einstein was talking about Newton's idea of "gravity force" trying to show how his new idea fitted in with Newton's in the appropriate limit of small speeds of test particles and weak curvature fields. Also some Pundits mess up by not being clear that Newton's "inertial frame" is Einstein's "static LNIF"!
On Dec 31, 2010, at 8:37 AM, Paul Zielinski wrote:
I say that locally gauging *passive* T4 in Minkowski spacetime cannot possibly give you a curved connection field. All it
can give you is GCTs and a "flat space" covariant derivative, an analog of the LC covariant derivative in flat spacetime.
Jack: Red Herring - no one makes that claim.
Look at the analogous problem in U1 electromagnetism.
Start with a spinor field. You need global U1 to get conserved electric charge q Noether's theorem.
But you need U1 --> U1(x) to get Maxwell's equations
F = dA
dF = 0
d*F = &J(q)
from the INVARIANT ACTION S PRINCIPLE
&S(spinor field, A) = 0
Similarly in GR
Global T4 gives CONSERVED total energy-momentum of the SPINOR FIELD.
T4 ---> T4(x) gives the TETRAD field similar to A in EM.
&S(spinor, TETRAD) = 0
gives Einstein's Guv + kTuv = 0
as shown in detail by Kibble 1962 I think. Note the spin connection determined by the tetrads and the ad hoc constraint of zero torsion - that restricts us to T4(x).
I think Kibble's T4 was passive. The resulting gauge connection field simply compensates for curved coordinate artifacts.
Locally gauging passive T4 just gives you GCTs, which produce coordinate artifacts in tensor field derivatives that are
corrected for by the gauge connection field. Then the gauge connection field is treated as a gravitational field per the
Einstein principle -- watch that pea!
No Z, you are missing the physics here. Kibble's T4 --> T4(x) is completely PHYSICAL active
Alice and Bob are LOCALLY COINCIDENT Alice is LIF, Bob is LNIF, they both measure the same ds^2
ds^2(Alice) = ds^2(Bob)
Tetrads map Alice's data to Bob's data & vice versa.
LC is not merely a FORMAL passive artifact - it describes the "WEIGHT" felt on any object clamped to the LNIF.
On Dec 31, 2010, at 8:37 AM, Paul Zielinski wrote:
On Thu, Dec 30, 2010 at 12:21 PM, JACK SARFATTI wrote:
Fock's vague remark on the "uniformity" of space is unintelligible to me.
He says exactly what he means in his book, and in lectures on gravity published in Rev Mod Phys
in 1957:
http://www.deepdyve.com/lp/american-physical-society-aps/three-lectures-on-relativity-theory-vGK2dKg9ZK
"Uniformity of space" is a physical symmetry of spacetime. Not to be confused with covariance. Both the uniformity of Minkowski spacetime and the non-uniformity of Riemannian spacetime have generally covariant descriptions. So this has nothing to do with covariance, contrary to what Einstein once believed. That was the position that Fock was arguing against.
OK, I think this is a Red Herring whose proper solution is well known today. Of course all theories can be made covariant under general coordinate transformations T4 ---> T4(x) as you say. That is necessary, but not sufficient. The additional condition we need is the EEP that formally is expressed as for COINCIDENT LIF & LNIF
ds^2(absolute local frame invariant) = guv(LNIF)e^u(LNIF)e^v(LNIF) = nIJ(LIF)e^I(LIF)e^J(LIF)
e^u(LNIF) = (tetrad)^uIe^I(LIF)
Synge keeps confounding the two different meanings of "gravitational field" as 1) Newton's g-force and 2) Einstein's curvature. Since EEP only deals with 1) of course its irrelevant to 2) - trivial and exactly as it should be.
Synge identified the actual gravitational field of GR with non-vanishing Riemann curvature.
Fine, nothing wrong with that. However, Einstein's critics simply quibbling because Einstein was talking about Newton's idea of "gravity force" trying to show how his new idea fitted in with Newton's in the appropriate limit of small speeds of test particles and weak curvature fields. Also some Pundits mess up by not being clear that Newton's "inertial frame" is Einstein's "static LNIF"!
e.g. an accelerating LNIF in zero curvature Minkowski spacetime is equivalent to a static LNIF in a non-zero curvature field - where it is understood that no direct measurement of curvature is made. Also the acceleration can be arbitrary it need not be constant!
The flip side of above is that a LIF in a non-zero curvature field is equivalent to a LIF in a zero curvature field - again no curvature measurement.
But notice both of the above statements compare detectors far apart from each other e.g. on on Earth other far away in space.
The real statement of the EEP is the purely LOCAL tetrad map for locally coincident LNIF & LIF
ds^2(absolute local frame invariant) = guv(LNIF)e^u(LNIF)e^v(LNIF) = nIJ(LIF)e^I(LIF)e^J(LIF)
e^u(LNIF) = (tetrad)^uIe^I(LIF)
He considered the "displacement field strength" LC^u_vw(x) to be purely a matter of the choice of reference frame.
No kidding, so do I. I have been telling you this now for years.
One facet of EEP is that a LNIF in Minkowski spacetime with constant proper acceleration ~ g00^-1/2(LC)^ztt along the z-axis is locally equivalent to a static LNIF in an appropriate curvature field. It is understood that measurements of the curvature tensor are excluded. Synge confuses a static LNIF with an "inertial frame."
He does? Where?
Newton's "inertial frame" is Einstein's "static LNIF" - explaining part of a general confusion among the Pundits. Note that curvature is measured as geodesic deviation on pairs of closely spaced force-free test particles.
I don't think Fock or Synge were confused about that.
Again a lot of Pundits have lost sight of the simple physical idea because they get lost in the pipe dreams of too much pure mathematics with a huge amount of excess off-topic formal baggage.
Before Kretschmann, Einstein believed that general covariance meant general relativity. That is what Fock was arguing against. Fock's position was that once this error is exposed, the term "general relativity" ceases to have meaning, and should no longer be used.
The principle of relativity is that of objective invariance.
Well then you actually agree with Fock. You are not saying that relativity <=> coordinate covariance.
So what's the problem?
Alice and Bob measure the same pattern of events out there. They each compute a set of invariant numbers from their independent data according to the program that is the theory. If their numbers agree in many tests then we know
1) the theory is good
2) their measurements are good.
Bob and Alice can also agree on the objective translational uniformity (or non-uniformity) of spacetime.
So you do in fact agree with Fock.
1905 SR is only good when Alice and Bob are inertial observers in flat spacetime.
But it can be extended to accommodate accelerating frames in a globally flat spacetime. This has nothing
to do with gravity -- no gravitational sources. This is simply a matter of general covariance. The spacetime
is still "uniform".
Of course we have tested 1905 in our static LNIFs on Earth's surface. We get away with it because gravity effects are weak compared to the v/c effects in our high energy machines. If we do right kinds of measurements e.g. Pound-Rebka Harvard Tower gravity redshift, and also now our GPS system, we see the small GR corrections to 1905 SR.
Fine. This is a (local) correspondence principle.
The local gauge principle applies to gravity as well as to the electromagnetic-weak-strong interactions in a general way with important differences of detail owing to the EEP (Einstein Equivalence Principle).
Which is *not* Einstein's version of EP. This is a shell game with misnomers for peas!
Gravity determines the universal inertial geodesic motions of all particles.
And since the geodesics are covariantly determined by the geodesic equation, this has nothing to do with the choice
of coordinates.
Red Herring, no one claimed that ever. Forces push massive particles off timelike geodesics. Therefore gravity is not a force because it is the dynamical background needed to define "force" in the old Newtonian way. "Interaction" is better than "force" if you want a unified field theory.
I think everyone agrees on that.
Einstein's 1916 GR is simply the local gauging of global translation subgroup T4 to T4(x).
I'm not sure why you think this is so "simple". Gauge gravity is still regarded as controversial by many.
Einstein said he thought two things were infinite, the universe and human stupidity. He added he had his doubts about the universe.
But yes you can generate a gauge connection field that is formally analogous to the torsionless connection
field of 1916 GR by locally gauging passive T4 in a Minkowski spacetime.
Alice and Bob may now each be in arbitrary motion either LIF or LNIF. However, when they test GR they must be in LOCAL COINCIDENCE because of the possibility of curvature, which means anholonomic path dependence. Therefore, there is no unique way compare their invariants when they measure the SAME local differential ds^2, or when they measure the local electromagnetic field Fuv for example.
But isn't that the point of the unique metric compatible LC connection? That it uniquely "connects" one LNIF with another
in curved spacetime? Allowing covariant differentiation of vector and tensor fields on curved Riemannian manifolds independently of the curved geometry?
We have three possibilities when they are LOCALLY COINCIDENT (neglect internal U1, SU2, SU3 - a fourth possibility).
Alice and Bob are both LIFs - SO(1,3) group
Alice is LIF and Bob is LNIF - tetrad group
Alice is LNIF and Bob is LNIF - T4(x) group also with the fancy name of "active Diff4" of Rovelli's Fig. 2.4
I think Kibble's T4 was passive. The resulting gauge connection field simply compensates for curved coordinate artifacts.
Locally gauging passive T4 just gives you GCTs, which produce coordinate artifacts in tensor field derivatives that are
corrected for by the gauge connection field. Then the gauge connection field is treated as a gravitational field per the
Einstein principle -- watch that pea! I say that locally gauging *passive* T4 in Minkowski spacetime cannot possibly give you a curved connection field. All it can give you is GCTs and a "flat space" covariant derivative, an analog of the LC covariant derivative in flat spacetime.
Shipov's and Waldyr's ideas seem to be related.Start with Einstein-Cartan i.e. localize Poincare group to get a bigger affine connection with the T4(x) GCT contortion tensor added to the Levi-Civita-Christoffel connection- better yet localize the de Sitter group with a local /(x) vacuum energy field. Even better do a conformal group extension of de Sitter with the dilaton and the hyperbolic const g special conformal boosts.
If you locally gauge the active Poincare group you get a gauge connection field with torsion. Starting with tetrad frame fields instead of coordinate frames yes you end up with the Einstein-Cartan formalism.
You are parroting back what I have told you. OK
Add teleparallel constraint that the total curvature is zero. Therefore, the 1916 Einstein curvature is expressed in terms of the T4(x) torsion gap dislocations on Kleinert's world crystal lattice. If the Einstein-Hilbert action density linear in Ricci scalar R is quadratic in the torsion 2-forms then we have a spin 1 Yang-Mills type theory of gravity that should be unitary and renormalizable if quantized in the usual way.
I think Kibble's method locally gauges passive T4 in Minkowski spacetime. To me this just seems like a fancy way of generating GCTs.
I find your perception bizarre - this "fancy way" is profoundly beautiful and unifies gravity with the other interactions - though gravity is not a force in the way electromagnetic Lorentz force is.
Of course an LC-type connection in flat spacetime will then correct for curved coordinate artifacts in tensor field derivatives
which, according to his version of EP, Einstein regarded *in a certain sense* as physically equivalent to a real gravity field. But you are still just constructing a connection field that corrects for coordinate artifacts. Which is exactly what covariant derivatives are all about, regardless of whether or not you use local gauging to arrive at the connection field.
So what? Quibble, quibble, quibble. Excess verbal baggage