Monday, September 17, 2007

On Sep 16, 2007, at 4:56 PM, Paul Zielinski wrote:

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
No? Then how do you recover Einstein's geometrodynamic field? Tidal effects?

That's what R^a^b(T4) = DS^a^b(T4) = dS^a^b(T4) + S^ac(T4)/\S^cb(T4) is

S^a^b(T4) = w^a^bc(T4)(I^c + B^c)

The Riemann curvature tensor Ryv^w^l of 1916 GR is from

R^a^b = R^a^buvdx^u/\dx^v

Ruv^w^l = ea^web^lR^a^buv

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/\


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


[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


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


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


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?

For the time being I'll take your word for all this. :-)

Ch 2 Rovelli's "Quantum Gravity" has all of this implicitly once you make the Ansatz

e^a = I^a(flat) + B^a(curved)

My particular retro-causal dark energy world hologram model further posits

B^a = N^-1/3A^a

N = (area of dark energy future de Sitter horizon in our FUTURE retrocausal causal light cone)/4(Planck area)

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.

Linearization is really just an approximation method for solving the Einstein field equations, right? For weak fields?

Yes. In that approximation you lose background independence.

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.

Here I would say it's the *geometry* that is objective, not the pointer readings.


This is another example of Einstein's pre-1920s empiricism.

Nothing wrong with that in moderation. It's necessary but not sufficient. String theory and loop quantum gravity have gone to much the other way. Give them a Planck length and they take the Hubble radius! ;-)

I would suggest:

Objective = Frame Invariant

Exactly. That's what I mean.

e^a is a local GCT frame invariant i.e.

e^a = e^audx^u is a u-index GCT scalar.

e^a is also a Lorentz group 4-vector - hence spin 1.

ds^2 = e^aea

considered as a quantum object obviously has spin 0, spin 1 & spin 2 virtual quanta because 2 spin 1's add as

1 + 1 = 0, 1, 2

i.e. in terms of dimensions of fundamental matrix representations

9 = 1 + 3 + 5

i.e. the tensor product of an entangled spin 1 pair has dimension 3x3 = 9 which splits into irreducible representations of dimension 1, 3 & 5 for spin 0, 1, 2 virtual quanta of the geometrodynamic field zero point quantum vacuum fluctuations that have effective "rest mass" off-mass-shell. Only the far field waves are massless with 2 transverse polarizations. Macro-quantum waves are Glauber coherent states of real quanta. There will be a longitudinal spin 0 wave, a vector wave, as well as the traditional tensor wave LISA & LIGO are set up for - if these macro-quantum geometrodynamic radiation states can form and the spin 0 and spin 1 real quanta remain massless?

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.
Wait a minute -- physical coordinates are not arbitrary maps, unless you invoke a physical relativity principle.

Of course I do. This is physical relativity not string theory. ;-)

Mathematical coordinates on the other hand are *always* arbitrary maps regardless of the physics.

Pragmatically (empirically), since physical coordinates are realized by actual small detectors in motion both zero g-force free geodesic and non-zero g-force off-geodesic - the latter from external non-gravity forces (e.g. rockets firing in space), the actual domain of pragmatic physical frames is much smaller than the theoretically allowed ones. But that is contingent. "Small" means in relation to local radii of curvature of fabric of dynamical background-independent fabric of space, i.e. geometrodynamic field B^a in e^a = I^a + B^a. Also "g-force" only refers to the center of mass motion of an extended body. The curvature tidal distortions of Weyl stretch-squeeze and Ricci compression-expansion are only in the relative coordinates of the extended object.

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.

There can be any number of instances of any given abstract group structure. For example, there are active and passive instances of the diffeomorphism group Diff(R^4), as pointed out for example by Rovelli.

So far I have no need of that distinction. For example, the covariant derivatives on the matter source fields Psi are of the form, without internal symmetries for simplicity

Du = e^auPa + S^a^buP[a,b] acting on matter source fields Psi.

where e^au(x) and S^a^bu(x) are the "phases" conjugate to the Lie algebra {Pa, P[a,b]} of Poincare group P10 = T4xO(1,3) using a matrix representation of {Pa, P[a,b]} that matches the same for the source field Psi. If Psi is a 2-spinor then {Pa, P[a,b]} is a 2x2 matrix rep with Pauli matrices, if Psi is a 4 spinor then {Pa, P[a,b]} is a 4x4 matrix rep with Dirac gamma matrices. Rovelli has this all in detail in Ch 2.

There is only one Poincare group with an intrinsic Lie algebra.

In the algebraic abstract, yes. More concretely, however, you can instantiate this algebraic structure with a set of coordinate transformations, or alternatively with a set of geometric transformations. The operations are defined differently and have different effects, even while they are both models for the same abstract group structure.

I don't understand this distinction. Show with examples. How does it make a physical difference?

Trivially, any given group S_N of permutations of N objects can be instantiated over any set of N particular objects.

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.

OK. But if you are talking about Lie algebras a la Noether's theorem, then I suspect you are dealing with physical symmetry operations, as opposed to coordinate transformations defined as permutations of 4-tuple labels attached to spacetime points.

Again I say I always always mean physical symmetry operations with redundant formal relabeling factored out.

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

Again give a concrete example of what you mean here.

I consider for example the T4 group is from an INFINITESIMAL displacement

1) x^u(P) to x^u(P') = x^u(P) + a^u

Now suppose we have some local matter field Psi(x) with a global action integral S over ALL of spacetime. For now considered flat - no gravity, no torsion. When a^u is a constant, is S[Psi] invariant under 1)? If so, this is a global symmetry and the total energy-momentum of the field Pu on spacelike slices can be defined and is conserved on any foliation of spacelike slicing - Noether's theorem in action.

If, we locally gauge T4 then globally rigid homogeneous "constant" a^u -> locally variable inhomogeneous a^u(P) where P is an Einstein "local coincidence" with relabeling factored out as Rovelli explains in Ch 2 on "Einstein Hole" problem of 1917. This is "relational" "background-independent" in Lee Smolin's sense.

The global theory with globally homogeneous constant a^u has tetrads I^a. The local theory with inhomogeneous INFINITESIMAL variable a^u(P) has tetrads I^a(P) + B^a(P) where B^a(P) is the intrinsic curvature compensating gauge field. The new global action S'[Psi, Bu] is invariant under this local T4(x) group of GCTs in the sense of local physical frame transformations between locally coincident observers with redundant relabeling (i.e. "gauge orbits") factored out like the Faddeev-Popov method of quantizing internal Yang Mills fields using Feynman path integrals where a similar problem of "gauge redundancy" needs to be factored out.

Note x^u(P) is a local coordinate chart with origin at P and x^u'(P) is another overlapping local coordinate chart.

Since I am only doing local coordinate-free objective physics obviously the latter I would think.

But maybe it's not so obvious?

Too vague. I don't know what you mean without an example.

If you align your vector "objects" with a set of coordinates (coordinate basis), using directional derivatives or some other method, then where does that leave you?

Too vague. I don't know what you mean without an example.

What is the difference between a tensor quantity (in this case a vector) whose intrinsic value depends on the
choice of coordinates, and a non-tensor? Is there any difference? If so, exactly what is that difference?

Too vague. I don't know what you mean without an example.

Yet if you don't use a coordinate basis you lose the relationship with reference frames, since your tetrads are
then reduced to arbitrary basis vectors in a tangent space that are completely insensitive to the choice of coordinates, up to arbitrary tensor transformations of the components.

Too vague. I don't know what you mean without an example.

e^a = e^audx^u

e^au'dx^u' = e^audx^u is invariant under local GCT a^u(P)

i.e. e^au(P) is a first rank GCT covariant tensor and is a first rank contravariant Lorentz group tensor.

dx^u is a first rank GCT contravariant tensor.

e^a(P) is a GCT invariant local scalar field that is also a Lorentz group 4-vector.

In contrast

I^au is not a GCT tensor, it transforms as a GCT 1st rank tensor + inhomogeneous term Xu and B^au is similar, but with -Xu hence their sum e^ua is a GCT tensor.

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.

It is true that the abstract structure can be the same regardless, and so formally the math looks the same.

Therefore it's only idle chatter of a meta-theoretical kind. I see no physics here.

However, if you don't make this distinction then how do you explain the dependence of the tetrads on the coordinates?

Simple. I don't. Red Herring. It's e^a(P) where P is an objective invariant Einstein "local coincidence" explained by Rovelli Ch 2. e^a(P) is a local GCT invariant under x^u(P) -> x^u'(P) note FIXED P.

What is the difference between a vector whose orientation depends on the coordinates, and a non-vector?
Either way the bottom line is that the components do not transform according to tensor rules under coordinate

You lost me. I see no physics here. Too vague.

You are trying to recover the tetrads from a set of Goldstone phases that are properties of a BEC. Wouldn't it help to understand the relationship between the two sets of quantities if this distinction is made clear?

I don't see how. Show this with a concrete example. The Goldstone phases are also local GCT scalar 0-forms Theta^b(P) and Phi^a(P) each a Lorentz group 4-vector i.e. GCT (local T4) scalars & Lorentz group 1st rank tensors.

A formal analogy is not sufficient to establish a valid correspondence relationship IMHO.

You lost me. I see no physics here. Too vague.

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).

Yes, of course you are technically correct here, but I'm using "Lorentz group" loosely to emphasize the connection with the Lorentz transformations of the 1905 theory.

The 3 Lorentz boosts + 3 space rotations form the Lorentz group O(1,3).

Clearly, only coordinate transformations of the Lorentz type are admissible to represent the effects of inertial frame transitions, *even in a generally covariant formalism* where n_uv -> g_uv (in a
completely flat spacetime).

Not so, you can also have homogeneous displacements, i.e. a^u(P) same for all P. Since local geodesic observers feel no g-forces they locally use only special relativity (equivalence principle) hence yes, locally coincident geodesic observers relate to each other via Lorentz group.

The 6-parameter Lorentz group O(1,3) is a subgroup.
Right. Yes of course you can add the translations and get the larger Poincare group.
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.
Yes, but this is all beside the point, which is that in order to represent frame changes you only use a subgroup of the full covariance group Diff(R^4) of the fully covariant version of SR (in flat spacetime).

Huh? No comprende. Give a concrete example.

When you go to GR in flat spacetime,

Makes no sense. "GR" is for "curved spacetime". "SR" is for "flat spacetime". So that's a typo, you mean "curved" there not "flat".

you expand the class of reference frames to include accelerating frames, but the coordinates associated with such frames must still always be instantaneously co-moving Lorentz, since frame acceleration per se has no effect on clocks other than that determined in each instant by relative velocity.

What does this mean physically? Operationally? Why are these words important? To solve what kind of problem? Relative velocity has an effect on clocks? I suppose you mean that a uniformly moving clock runs slow to me, but not to the guy in the rest frame of the clock. It's not obvious that a g-force on a off-geodesic non-inertial clock from a non-gravity force will not change its operation relative to a geodesic inertial clock with zero g-force. Indeed, since the g-force is like keeping a clock off-geodesic fixed in a gravity curvature field, such a non-inertial clock will show a relative red shift compared to a locally coincident inertial clock on a (timelike) geodesic.

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.


dso^2 = I^aIa 1905 SR

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

OK. In the 1915 formulation, this simply means that ds^2 between any two spacetime points will always have exactly the same value regardless of the choice of spacetime coordinates.

Yes, also for momentarily coincident observers Alice and Bob in arbitrary relative motion including acceleration, jerks, etc ...

It's just that under Lorentz coordinate transformations,
coordinate intervals happen to agree with distance intervals determined by the metric.

It's exactly the same in ordinary 3D space under the so-called orthogonal transformations.


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

*I made a hasty error here. Scratch that last sentence.

If I^au is not a GCT tensor then ds0^2 is not a GCT invariant.

That is, ds0^2 -> (I^a + X^a)(Ia + Xa) =/= ds0^2

1905 SR only requires ds0^2 invariant under the globally rigid 10-parameter Poincare group connecting globally inertial detectors without any intrinsic Minkowski 4-acceleration.

i.e. (d^2X^a/ds0^2)(d^2Xa/ds0^2) = 0

use Minkowski nab to raise and lower Lorentz indices.

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.

The Minkowski metric n_uv is by definition only transformable under physical *Lorentz* transformations.

Yes if you want ds0^2 invariant. That's what I call nab not n_uv.

Mathematically you can substitute g_uv for n_uv in the 1907 theory, and then you have a general covariant version of the theory.

You can ask what happens when you use accelerating detectors in globally flat spacetime.

Then, in general

guv(P) = nabI^au(P)I^bv(P)


ds0^2 = guv(P)dx^udx^v is no longer a local invariant for this larger group.

You must use the compensating B^a field so that

ds^2 = nab(I^au + B^au)(I^bu + B^bu)dx^udx^v = e^aea

is the invariant

But in Minkowski spacetime

S^a^b = w^a^bce^c

R^a^b = DS^a^b = dS^a^b + S^ac/\S^cb = 0 everywhere-when i.e. globally vanishing curvature field 2-form


T^a = De^a = 0 everywhere-when i.e. globally vanishing torsion field 2-form

However, inertial frame transitions are then still described only by Lorentz transformations, even in the covariant formulation of SR: arbitrary coordinate transformations need not in that case correspond to any possible frame transition (e.g., Lorentz -> Galilean), even while they may be mathematically admissible in the generally covariant formulation of the theory.

Yes, OK.

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

Right, if here you are talking about non-linear *physical* coordinate transformations applied to fully covariant g_uv.

I don't understand your sentence. I mean Alice and Bob flitting through space with rocket engines on and off willy nilly. Note they cannot use their zero point energy induced geodesic warp drive here to actually do the gedankenexperiments envisioned here-now.

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.

Well, for example, Galilean coordinate transformations are excluded from the class of physical coordinate transformations, even while they are *mathematically* admissible in a generally covariant formulation of SR.

Why excluded? I don't think so. They are simply an admissible limiting case physically realizable.

Now if the Galilean transformations weren't mathematically admissible, then *by definition* the theory wouldn't be general covariant, would it?

Too vague.

Most simply put in operational pragmatic terms - non-inertial off-geodesic observers who feel g-forces always need guv(x).

Yes, but there are all kinds of coordinate transformations of the components of g_uv(x) that can have nothing to do with reference frames and their transformations.

Yes, and you want to erase them in computing physical numbers. Factor them out e.g. Ch 2 Rovelli.

Again, Galilean transformations are definitely verboten, even though they must
be *mathematically* admissible in any general covariant theory.


Wrong, I think. Galilean transformations simply mean that the locally coincident clocks are moving with v/c << 1 relative to each other. They could be accelerating relative to each other i.e. Galilean GR is a perfectly good limiting case in fact most experimental tests of GR are in fact in this limit!

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.

What you are saying is OK, but it doesn't address the point I'm trying to make here.

No doubt.

Yes its a slippery rascal but I consider it to be very important in understanding the meaning of Einstein's theories of relativity.

I don't know why. Your only point here that I got was about Galilean transformations, which I think is wrong. You can transform to accelerating frames in Galilean relativity for small enough accelerations and short enough times.


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


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.

No comments: