On Sep 16, 2007, at 2:51 PM, Jack Sarfatti wrote:

yes - also it's intuitive and really not very different from Yang-Mills theory.

----------------Internal Yang-Mills Field----- Curvature-Torsion Field

Symmetry Group G = U(1), SU(2), SU(3) ----------- P10 = T4xSO(1,3)

Compensating gauge 1-form B^i ---------------- B^a where e^a = I^a + B^a, ds^2 = e^aea = guvdx^udx^v

i,j,k = 1 U(1); i,j,k = 1,2,3 SU(2); i,j,k = 1,2,3, ... 8 SU(3); a,b,c = 0,1,2,3 T4

Yang-Mills field 2-form

F^i = DB^i = dB^i + c^ijkB^j/\B^k

c^ijk = structure constants of Lie algebra {Q^i} of G

[Q^i,Q^j] = c^i^jkQ^k

Compare to the INTRINSIC torsion field 2-form

T'^a = D'B^a = dB^a + w^abcB^b/\B^c

Where the INTRINSIC spin-connection 1-form is

S'^a^b = w^a^bcB^c

w's are the spin-connection tetrad coefficients - but not directly structure constants of the P10 Lie algebra.

Also, there seems to be really no Yang-Mills analog to Riemann curvature 2-form in this tetrad POV, which is additional structure

R'^a^b = dS'^a^b + S'^ac/\S'^cb

Flash back to I^a for Minkowski spacetime

ds^2(Minkowski) = I^aIa

I^a = I^ausx^u

in global Minkowski geodesic inertial frames GIFs

a geodesic means zero intrinsic acceleration of the center of mass of the local frame (detector).

I^au = Kronecker-delta - ALIGNMENT

In LOCAL intrinsically accelerating off-geodesic non-inertial frames LNIFs or "rocket frames" that may also rotate, I^au(x) is a 4x4 matrix of functions of local coordinate charts encoding the inertial g-forces that we eliminate in globally superluminal geodesic zero g-force warp drive with "time travel" in all significant meanings - the goal of "metric engineering".

Define the Minkowski spin connection

So^a^b = w^a^bcI^a

Do = d + So^ac/\

Obviously

To^a = Doe^a = 0

Ro^a^b = DoSo^a^b = 0

Furthermore, if we only locally gauge 4-parameter T4 as in Einstein's 1916 GR we get w^a^bc(T4).

Let the Lie algebra of T4 be {Pa} with structure constants

C^a^bc

[P^a,P^b] = C^a^bcP^c

a,b,c = 0,1,2,3 0 timelike, 1,2,3 spacelike

note the deformed Lie algebra of non-commutative spacetime is non-Abelian

What is the relation of w^a^bc(T4) to C^a^bc?

If as in Utiyama 1956 we only locally gauge the 6-parameter SO(1,3) with Lie algebra {P^[a,b]}

[P^[a,b],P^[a',b']] = C^[a,b]^[a',b'][a",b"]P^[a",b"]

we get w^a^bc(SO(1,3)

If we do Kibble 1961 locally gauge 10-parameter P10 we get

w^a^bc = w^a^bc(T4) + w^a^bc(SO(1,3) + w^a^bc(T4,SO(1,3))

3rd term of RHS from cross commutators of 4 P^a with 6 P^[a',b']

Sticking now only with Einstein's 1916 GR we have ONLY w^a^bc(T4) that I will call w^a^bc belopw

D = d + S^ac/\

S^a^b = w^a^bc(T4)e^c

T^a = De^a = 0 zero total torsion field 2-form

where

De^a = d(I^a + B^a) + w^ac'c(I^c' + B^c')/\(I^c + B^c) = 0

where

dI^a + w^ac'cI^c'/\I^c = 0

Therefore,

De^a = dB^a + w^ac'c[I^c'/\B^c + B^c'/\I^c + B^c'/\B^c] = 0

Similarly for the curvature 2-form

R^a^b = DS^a^b

In general w^a^bc are functions of the local coordinates not fixed constants - I think?

On Sep 16, 2007, at 12:51 PM, Paul Zielinski wrote:

If the tetrad formalism facilitates the description of interactions with spinning test particles and
helps to re-formulate Einstein's theory of gravitational in terms of a gauge-field model, of course
that's a strong argument in favor of the tetrad model.

Jack Sarfatti wrote:
I agree with Rovelli Ch 2 "Quantum Gravity" that the tetrad 1-forms

e^a = e^audx^u

are the most natural choice for the gravitational field with the most compact way of doing GR

ds^2 = guvdx^udx^v = (Minkowski)abe^ae^b = e^aea

Curvature 2-form is R^a^b = R^a^buvdx^u/\dx^v = DS^a^b = dS^a^b + S^ac/\S^cb

S^a^b = S^a^budx^u is spin-connection 1-form

e^a = I^a + B^a

I^a is for Minkowski spacetime

Flat base-space geometry.

Yes, but no perturbation theory on a fixed non-dynamical background implied. Of course, you can do that if you further require

B^a << I^a

but you do not NEED to do that. The general theory is obviously background-independent in Lee Smolin's sense.


B^a is analogous to v = vudx^u = (h/m)dTheta = superflow 3-velocity 1-form

B^a expresses the deviation of the actual base geometry from flat.

Yes, key word is "actual" i.e. "intrinsic", "locally objectively real" in terms of Einstein's 1917 operational "local coincidences" of small detectors in arbitrary relative motion.

But like I^a it also carries information about reference frames once you introduce a coordinate basis where the basis vectors are aligned with the physical coordinates (using directional coordinate derivatives).

Yes, those are the choices for I^au(x) & B^au(x) that are completely arbitrary maps of the territory.

The MAP is NOT the Territory (Alfred Korzybski)

with possible exception of a STRANGE LOOP of the physical conscious mind?

Clearly this must be so if e^a = I^a + B^a, and the tetrads e^a are to represent both intrinsic spacetime geometry *and* frame acceleration (in non-inertial frames).

Clearly. Yes that is the way I picture it.

Since the B^a field is a spin 1 vector field under Lorentz group, it is obviously the natural starting point to understand gravity and torsion SIMPLY as the local gauging of the global 10-parameter Poincare group that IS 1905 special relativity analogous to internal symmetry Yang-Mills theory of electro-weak-strong forces.

Well, you still haven't specified which instance of the Poincare group you are talking about here.

"Instance"? There is only one Poincare group with an intrinsic Lie algebra. Rovelli shows how to use different representations of Poincare Lie algebra {P^a, P^[a',b']} in the minimal covariant coupling of tetrad gravity fields to scalar, spinor and vector matter source fields. Explicit formulae in his Ch 2.

Is this a coordinate invariance group, or a geometric symmetry group? Both can be described as "Poincare".

Since I am only doing local coordinate-free objective physics obviously the latter I would think. I don't see how this informal language distinction changes the actual formal structure? I mean how is that a difference that makes a physical difference in the calculation of observables? Maybe, but I have no need of that distinction so far.

Einstein-Minkowski (1907 version) clearly involves a *metric* n_uv that reflects the geometry of the Minkowski
manifold, subject to the SR definition of the "line element" (which implies the restriction of covariance to the Lorentz group).

No, 1905 special relativity uses the 10 parameter Poincare group P10 = T4xO(1,3). The 6-parameter Lorentz group O(1,3) is a subgroup. All actions are invariant under RIGID P10. Curvature without torsion is when T4 is localized where the GCTs are the local T4 gauge transformations. Torsion induces curvature if you only localize O(1,3) as Utiyama did in 1956. The complete story of curvature + torsion is Kibble 1961. Now you can also localize GL(4,R) with up to 16 parameters.

The distance between spacetime points is in this case determined by the Minkowski metric, and not directly
by the coordinates as in Einstein's 1905 theory.

Precisely

dso^2 = I^aIa 1905 SR

ds^2 = I^aIa + I^aBa + B^aIa + B^aB^a 1916 GR

most tests of GR only use I^aBa + B^aIa with spin 1 gravi-vector quanta if you could do a quantum gravity test is my prediction. All the weak field curvature classical solutions are same of course.

Only in strong field, i.e. B^aBa term will you get spin 0 gravi-scalar & spin 2 tensor "gravitons" as well as spin 1 quanta is my counter-intuitive prediction!


From a mathematical POV the non-covariant Minkowski metric is an artificial construct, since once you have
a geometric model for the 1905 theory you might as well write the generally covariant expression

ds^2 = g_uv dx^u dx^v

which renders the choice of (well-behaved) mathematical coordinates completely arbitrary.

I do not understand your use of "non-covariant"

ds0^2 = I^aIa

is even GCT covariant

it is of form guv(x)dx^udx^v for local off-geodesic observers who need to use generally curvilinear I^au(x) not the constant Kronecker delta for geodesic inertial observers.

Torsion and curvature vanish even when you use curvilinear guv(x) for non-geodesic observers in Minkowski spacetime.

Of course the *physical* coordinates must in this case still be Lorentz (for inertial frames) or pointwise instantaneously co-moving Lorentz (for non-inertial frames). All other choices of physical coordinates are excluded, even while the choice of *mathematical* coordinates is still arbitrary (subject to purely technical considerations such as differentiability etc.).

Not sure what you mean apart from what I said. Most simply put in operational pragmatic terms - non-inertial off-geodesic observers who feel g-forces always need guv(x). Geodesic observers without g-forces of any kind on their centers of mass and ignorable tidal curvature and torsion distortions in their relative coordinates to the center of mass, only need Minkowski metric nab for local measurements, i.e. equivalence principle to a sufficient approximation.

In fact The Pundits have not done so - inventing cumbersome formalisms at the 2nd rank GCT metric tensor, Levi-Civita connection level, that are not needed to get to the important physics quickly IMHO.

If you want to handle spinning particles, then I agree that you need a more sophisticated formalism.

Basic matter fields have spin, therefore, 1916 tensor theory is seriously physically incomplete.


On Sep 16, 2007, at 9:06 AM, Paul Zielinski wrote:

Agreed, and this is the advantage of Einstein-Cartan, but my point here is that you don't need tetrads in order to separate the mathematical coordinate charts from the physical coordinate charts.

Maybe, but I never use mathematical coordinate charts in my conceptualization of the objective reality physics.

But there is nothing to stop you. You can use, e.g., polar coordinates in a 1-1 Minkowski spacetime just as you can in a 2-dim Euclidean space. There is no need in this context to worry about any relationship with observer frames, detector arrays, or the motion of test particles -- none of which is affected (except as to mathematical *description*) by the choice of abstract coordinates.

I do not understand your above remark. You have eliminated the physics completely. Physics is only about "observer frames, detector arrays, or the motion of test particles" they are the babies in the bathwater. Basic tetrad equations are coordinate-independent and even GCT & Lorentz local-frame independent

e.g.

Scalar action density of pure gravity field

~ (antisymmetric tensor)abcd {R^a^b/\e^c/\e^d + (Lambda)e^a/\e^b/\e^c/\e^d}


It is only when you "piggyback" frame-dependent coordinate stipulations on geometric spacetime transformations (as in Einstein's 1905 treatment) and use coordinates to describe metrical changes that you have to worry about such things.

No one does that anymore. It's beating a dead horse. Yes, if you select say static off-geodesic "accelerating" observers outside the event horizon of a non-rotating black hole for example then you use the particular metric representation

g00 = - grr^-1 = (1 - 2rs/r)

rs/r < 1

note I^aBa + B^aIa dominates when rs/r << 1

Note that the "accelerating" observers do not move in this curved spacetime - unlike common sense in flat space of Euclid with Newton's physics!


This is part of what I mean by the "dualism" in the use of spacetime coordinates in Einstein's theories, subject to reformulation by Minkowski in terms of a geometric model (4 dimensional "spacetime").

Fine, but excess baggage.

I only think of relationships between locally coincident tiny detector observers "Alice" and "Bob" in strictly operational pragmatic terms.

This is what I'm calling "physical" spacetime coordinates. You are simply implementing the Einsteinian concept of a *relativistic reference frame* using spacetime coordinates to describe the effects of observer motion on the results of measurements carried out in such frames, as in Einstein's 1905 model.
I agree with the Cornell Physics Department philosophy of the late 1950's (Feynman, Bethe, Salpeter, Morrison, Gold, Kinoshita ...) keep the math minimal.
But you cannot conflate coordinates applied to the fully covariant expression dx^2 = g_uv dx^u dx^v with physical coordinates, which latter must be pointwise Lorentz or instantaneously co-moving Lorentz for *physical* reasons. If you do you will get bogged down in contradictions -- certainly in the context of GR.

You lost me. Give an example. In any case I have not done that.

Fancy math is usually a cover for lack of a good physical idea as we see in string theory and loop quantum gravity - not always of course.

The idea of a generally covariant formalism -- independently of physical considerations -- is hardly "fancy math". It's built into the 1915 theory. Yes Einstein got very confused about these questions, but that doesn't mean we should also be confused about them.

The symbols I use, e.g.

ds^2 = e^aea

are generally covariant and do not rely on any particular choice of coordinates or observers.


It was Kretschmann in 1917 who pointed out that there is no direct logical relationship between general covariance and physical relativity, and Einstein agreed. Physical relativity is defined with respect to *reference frame invariance*, whereas general covariance is a metatheoretic property that depends on the way in which a physical theory is mathematically formulated. They are two very different animals.

Fine, but so what? How does that affect what I propose? Tetrads are immune from that disease.

So in developing a gauge field model of gravity, I think one must be very clear in specifying exactly which instance of the Poincare group we are talking about that is to be "locally gauged". Is it a coordinate invariance group, or a physical symmetry group?

I have always explicitly said "physical symmetry group," i.e. dynamical actions of source matter fields are invariant under it.

Indeed, start with SPINOR matter fields, therefore, MUST use tetrads. However, I do not see where your distinction makes a significant difference in any actual calculation of physical importance. Give an example.

If the latter, exactly which physical symmetries of spacetime are we referring to?

In every text book. 4 displacements of local detectors, and 6 spacetime rotations of local detectors in sum are P10. The complete RIGID Lie algebra is given in Schwinger's notes for example.

Almost every theory paper on the gr-qc archive goes nowhere in terms of physical insight on the real problems, e.g. dark energy, dark matter - much fancy math, but very little physics. After looking at many of those papers I ask "So what?" "Why bother?" "What's your point?" Perhaps I am mistaken? ;-)

Well, I actually agree, but I'm afraid if this is not carefully resolved, it will simply perpetuate the confusion.

OK, if you prefer not to deal with such issues, how do you explain the use of a coordinate basis in the tetrad model?

Give an example. I don't know what you mean. I have not needed any particular choice of e^au(x) in any statement of the fundamental structure of the theory if that's what you mean?

How can you have your cake and also eat it?

Just make sure Crazy Roy is not close to the Caffe Trieste counter to spit on it? :-)


PS as a purely mathematical exercise fine - i.e. conceptual art. Let's see if LHC makes a difference if it ever goes online, similarly for LISA/LIGO.

What is purely mathematical here is the issue of formal covariance. What is not purely mathematical is how the tetrads represent reference frames, when *as vectors in a tangent space* they are clearly coordinate-invariant objects.

From your POV, how do you explain this?

I see nothing needing explanation.

Jack Sarfatti wrote:

On Sep 14, 2007, at 2:58 PM, Paul Zielinski wrote:

This is how the tetrad model accounts for the dual role of spacetime coordinates in 1915 GR. It is this tacit duality in Einstein's approach that has caused much confusion in the teaching of relativity theory IMHO. Of course one of my personal hobbyhorses is that you can also do all this without tetrads,

You need the tetrads to couple gravity to spinor fields - that's important since all the basic matter fields (leptons & quarks) are spinor fields. The spin 1 vector fields (including the curvature tetrads and torsion spin connections) are induced by localizing the different symmetry groups of the action of the spinor fields.