PS
Consider

F = T + R = (I/\ + A/\ + S/\)(I + A + S)

OK so now it really looks like Yang-Mills F with A & S appearing symmetrically!

Note that

(I/\ + A/\ + S/\)I = 0

therefore we have the 2-form field from the ten compensating gauge potentials of Poincare group

F = T + R = d(A + S) + (A + S)/\(A + S)

F is peculiar with mixed Lorentz group indices

T^a + S^a^b = dA^a + dS^a^b + A^a/\A^b + S^ac/\A^c + S^ac/\S^c^b

which splits into

T^a = dA^a + S^ac/\A^c torsion field

R^a^b = dS^a^b + A^a/\A^b + S^ac/\S^c^b curvature field


On Oct 1, 2007, at 3:11 PM, Jack Sarfatti wrote:

v2 (expanded & corrected too many "/\" in R formulae)
On Sep 29, 2007, at 6:22 PM, Jack Sarfatti wrote:

Note also, the effect of the equivalence principle makes a difference in comparing gravity fields to Yang-Mills fields.

Look at the exterior covariant derivative 1-form

D = (d + S)/\

note that d is usually written without the /\ that is tacitly understood since

d(p-form) = (p + 1)-form

dual to boundary operator

&(p + 1) dim manifold = p-dim manifold so that Stokes theorem generalizes to

Integral of d(p-form) on p + 1 dim manifold = Integral of (p-form) on the boundary &(p + 1)manifold

for example when p = 0 in 3D space i.e. fundamental definite integral formula of calculus

d(0-form) = gradient of a function

p = 1 e.g. loop integral of vector (1-form) field is interior flux (2-form) integral

d(1-form) - curl of a vector (2-form)

p = 2 Gauss's divergence theorem - end of short digression

d(2-form) = divergence of a 1-form vector field in 3D space.

can generalize to Minkowski spacetime of 1905 SR

Stoke's theorem is key. A star gate traversable wormhole time machine to past (or equivalently in this regard a weightless zero g-force geodesic glider warp drive bubble in sense of Alcubierre's toy model metric) held open by universally antigravitating positive zero point dark energy density with equal and opposite negative pressure (w = -1 with GR source factor 1 + 3w) depends on multiply-connected spacetime corresponding to topological defects in the vacuum ODLRO field whose phase modulation determines the local curved tetrad fields and the local torsion field spin connections from localizing the 10-parameter Poincare symmetry group of special relativistic quantum field theory. The latter is background-dependent the former is not.

Multiple connectivity means, in our specific problem of metric engineering practical warp and wormhole, closed 2 surfaces (portals or wormhole mouths) without boundary that are not boundaries of the 3D space wormhole tunnel.


http://www.kahl.net/astro/graphics/wormhole.gif

the 2D mouth is a circle (1 space dimension removed in picture at a fixed "time"

the 3D space Dr. Who walks through is the 2D tube in the picture - you are flatlander bug constrained to surface in the picture.

"dark energy" = "exotic matter"

Since the vacuum ODLRO field is single-valued that means that the geometrodynamic field area density flux through a nonbounding closed 2D surface is quantized and this explains the Hawking-Bekenstein formula corresponding to point gravity monopole defects in the fabric of dynamical 3D space.

Entropy/kB = Horizon Area/4 Quantum of Area Flux = N

i.e. Newton's Planck area hG/c^3 ~ 10^-66 cm^2

with the world hologram formulae

Size of wavelet Quantum Foam Bubble ~ N^1/6(Quantum of Area)^1/2

i.e. &L ~ N^1/6Lp = (Lp^2L)^1/3

L ~ N^1/2Lp (hologram)

Horizon area ~ (10^28 cm)^2 in our pocket universe on the Cosmic Landscape with &L ~ 10^-13 cm and observed dark energy density ~ hc/NLp^4 ~ 10^-29 gm/cc


Back to main point

d in a sense is e

i.e. in a flat spacetime in a geodesic GIF coordinate basis

d(0-form) is "4-gradient" d/dx^a on a 0-form function since ea^u = Kronecker delta

ea = ea^u(d/dx^u)

e^a = e^audx^u

That is we can think of d/\ as e/\

or

D/\ = (e + S)/\

e = I + A


i.e.

D/\ = I/\ + A/\ + S/\

I is when we have globally flat Minkowski spacetime and a Global Inertial Frame (GIF)

A & S are the compensating geometrodynamic field gauge potentials that first appear in a Global Non-Inertial Frame

in globally flat Minkowski spacetime where the curvature R^a^b = 0 and the torsion T^a = 0 but in 1916 GR R^a^b =/= 0 while still T^a = 0.


where

I/\ is in flat Minkowski spacetime of 1905 SR

A comes from using a GNIF and finally from localizing T4 to LIFs & LNIFs.

Note that A induces S that is not independent when the torsion T field vanishes globally.

In a GNIF

R = D/\S = 0


i.e.

R = (I/\ + A/\ + S/\)S = 0

and also

T = De = 0

i.e.

(I/\ + A/\ + S/\)(I + A) = 0

when rigid T4 is localized to T4(x) then we have possibility that

R = (I/\ + A/\ + S/\)S =/= 0

This notation makes the Yang-Mills field structure more apparent.

Consider

F = T + R = (I/\ + A/\ + S/\)(I + A + S)

OK so now it really looks like Yang-Mills F with A & S appearing symmetrically!


In 1916 GR S = S(A) is redundant as shown in Rovelli's eq. 2.89

S has and independent part when there is a torsion field i.e. full Poincare group is locally gauged.

That is 1916 GR is really a theory of the spin 1 Yang-Mills curvature tetrad field A with a redundant S as given in Rovelli (2.89). Hence GR is renormalizable in t'Hooft's sense including vacuum ODLRO Higgs field that may give Salam's strong short-range f-gravity in addition to zero mass geometrodynamic field quanta. Indeed the composite quanta are spin 0, spin 1 and spin 2 from pairs of the fundamental spin 1 geometrodynamic quanta i.e. zero point fluctuations of the uncondensed part of the post-inflation vacuum ODLRO field. Think of the geometrodynamic field random "dark energy" quanta like the zero point motions of helium 4 atoms in the T = 0 ground state that is only 10% coherent condensate even though the effective superfluid density is 100%.





typo corrected draft 2
On Sep 29, 2007, at 5:37 PM, Jack Sarfatti wrote:

The key equation is Rovelli's (2.89) for only the torsion-free curvature-only spin connection in terms of the tetrads. It has quadratic and quartic parts. The quartic part can be put into the desired form but the quadratic part cannot. Also both parts depend on gradients in the tetrad component fields. It may be that only the torsion part of the spin connection can be put into the Yang-Mills covariant derivative form. I have not yet confirmed that. However, this is really a side issue, as in general we need to treat the 6 spin connection 1-forms S^a^b and the 4 tetrad 1-forms e^a as independent Yang-Mills type compensating local gauge field potentials in which we define the exterior covariant derivative as

D = d + S/\

Suppressing indices for simplicity. This is analogous to a Yang-Mills theory where the curvature two form field is

R = DS

i.e. curvature field 2-form = exterior covariant derivative of the spin connection Yang-Mills potential with itself, i.e. in 1916 GR

R = dS + S/\S

This is completely analogous to the Yang-Mills theory where

F = DA

= dA + A/\A

DF = 0

D*F = J*

DJ* = 0

In 1916 GR

DR = 0

D*R = *J

must translate in ordinary tensor notation to

Guv = kTuv

D*J = 0

corresponds to

Tuv^;v = 0 i.e. local energy-momentum stress current densities conserved - all bets off on global integrals over spacelike surfaces.

All of the above is for zero torsion fields

T = De = 0

This is an auxiliary equation not found in the internal Yang-Mills theories. The theory is more complex of course when T =/= 0 i.e. locally gauging the full 10-parameter Poincare spacetime symmetry group. One must be careful on how to make the analogy of GR with Yang-Mills theories. The analogy is perfect in Utiyama 1956 where there is only S and no e in the sense of the compensating field A where e = I + A because T4 is not locally gauged there. GCTs are put in adhoc - not pretty.

On Sep 28, 2007, at 4:25 PM, Jack Sarfatti wrote:

In trying to make gravity tetrad GR into a formal analog of Yang-Mills I have posited

S^ac = w^acc'e^c'

e^c' are the Einstein tetrad 1-forms

S^ac are the spin-connection 1-forms (involving gradients of the tetrads in 2.88)

Rovelli has (2.88) for example. Now I had thought I had seen the equivalent of S^ac = w^acc'e^c' in Rovelli's book, but now I cannot find it.

Using it, the torsion field 2-form is

T^a = de^a + S^ac/\e^c

= de^a + w^acc'e^c'/\e^c

which is like the Yang-Mills field 2-form

F^a = dA^a + w^acc'A^a/\A^c'

It is not clear that S^ac = w^acc'e^c' is consistent with (2.88)

Note also, the effect of the equivalence principle makes a difference in comparing gravity fields to Yang-Mills fields.

Look at the exterior covariant derivative 1-form

D = (d + S)/\

note that d is usually written without the /\ that is tacitly understood since

d(p-form) = (p + 1)-form

dual to boundary operator

&(p + 1) dim manifold = p-dim manifold so that Stokes theorem generalizes to

Integral of d(p-form) on p + 1 dim manifold = Integral of (p-form) on the boundary &(p + 1)manifold

for example when p = 0 in 3D space i.e. fundamental definite integral formula of calculus

d(0-form) = gradient of a function

p = 1 e.g. loop integral of vector (1-form) field is interior flux (2-form) integral

d(1-form) - curl of a vector (2-form)

p = 2 Gauss's divergence theorem - end of short digression

d(2-form) = divergence of a 1-form vector field in 3D space.

can generalize to Minkowski spacetime of 1905 SR

Back to main point

d in a sense is e

i.e. in a flat spacetime in a geodesic GIF coordinate basis

d(0-form) is "4-gradient" d/dx^a on a 0-form function since ea^u = Kronecker delta

ea = ea^u(d/dx^u)

e^a = e^audx^u

That is we can think of d/\ as e/\

or

D/\ = (e + S)/\

e = I + A


i.e.

D/\ = I/\ + A/\ + S/\

where

I/\ is in flat Minkowski spacetime of 1905 SR

A comes from using a GNIF and finally from localizing T4 to LIFs & LNIFs.

Note that A induces S that is not independent when the torsion T field vanishes globally.

In a GNIF

R = D/\S = 0

i.e.

R = (I/\ + A/\ + S/\)/\S = 0

and also

T = De = 0

i.e.

(I/\ + A/\ + S/\)(I + A) = 0

when rigid T4 is localized to T4(x) then we have possibility that

R = (I/\ + A/\ + S/\)/\S =/= 0

This notation makes the Yang-Mills field structure more apparent.

In 1916 GR S = S(A) is redundant as shown in Rovelli's eq. 2.89

S has and independent part when there is a torsion field i.e. full Poincare group is locally gauged.

That is 1916 GR is really a theory of the spin 1 Yang-Mills curvature tetrad field A with a redundant S as given in Rovelli (2.89). Hence GR is renormalizable in t'Hooft's sense including vacuum ODLRO Higgs field that may give Salam's strong short-range f-gravity in addition to zero mass geometrodynamic field quanta. Indeed the composite quanta are spin 0, spin 1 and spin 2 from pairs of the fundamental spin 1 geometrodynamic quanta i.e. zero point fluctuations of the uncondensed part of the post-inflation vacuum ODLRO field. Think of the geometrodynamic field random "dark energy" quanta like the zero point motions of helium 4 atoms in the T = 0 ground state that is only 10% coherent condensate even though the effective superfluid density is 100%.

The key equation is Rovelli's (2.89) for only the torsion-free curvature-only spin connection in terms of the tetrads. It has quadratic and quartic parts. The quartic part can be put into the desired form, but the quadratic part cannot. Also both parts depend on gradients in the tetrad component fields. It may be that only the torsion part of the spin connection can be put into the Yang-Mills covariant derivative form. I have not yet confirmed that. However, this is really a side issue, as in general we need to treat the 6 spin connection 1-forms S^a^b and the 4 tetrad 1-forms e^a as independent Yang-Mills type compensating local gauge field potentials in which we define the exterior covariant derivative as

D = d + S/\

Suppressing indices for simplicity. This is analogous to a Yang-Mills theory where the curvature two form field is

R = DS

i.e. curvature field 2-form = exterior covariant derivative of the spin connection Yang-Mills potential with itself, i.e. in 1916 GR

R = dS + S/\S

This is completely analogous to the Yang-Mills theory where

F = DA

= dA + A/\A

DF = 0

D*F = J*

DJ* = 0

In 1916 GR

DR = 0

D*R = *J

must translate in ordinary tensor notation to

Guv = kTuv

D*J = 0

corresponds to

Tuv^;v = 0 i.e. local energy-momentum stress current densities conserved - all bets off on global integrals over spacelike surfaces.

All of the above is for zero torsion fields

T = De = 0

This is an auxiliary equation not found in the internal Yang-Mills theories. The theory is more complex of course when T =/= 0 i.e. locally gauging the full 10-parameter Poincare spacetime symmetry group. One must be careful on how to make the analogy of GR with Yang-Mills theories. The analogy is perfect in Utiyama 1956 where there is only S and no e in the sense of the compensating field A where e = I + A because T4 is not locally gauged there. GCTs are put in adhoc - not pretty.

On Sep 28, 2007, at 4:25 PM, Jack Sarfatti wrote:

In trying to make gravity tetrad GR into a formal analog of Yang-Mills I have posited

S^ac = w^acc'e^c'

e^c' are the Einstein tetrad 1-forms

S^ac are the spin-connection 1-forms (involving gradients of the tetrads in 2.88)

Rovelli has (2.88) for example. Now I had thought I had seen the equivalent of S^ac = w^acc'e^c' in Rovelli's book, but now I cannot find it.

Using it, the torsion field 2-form is

T^a = de^a + S^ac/\e^c

= de^a + w^acc'e^c'/\e^c

which is like the Yang-Mills field 2-form

F^a = dA^a + w^acc'A^a/\A^c'

It is not clear that S^ac = w^acc'e^c' is consistent with (2.88)

I forgot that there is a missing dimensionless variable factor x -corrected below where x is the ratio of two lengths, the position of the test particle to the intrinsic scale r of the geometrodynamic field that is related to the source distortion M. What is r? Here I take it as the radius of curvature at the surface r* of the source as the closest analog to the cosmological parameter in the de Sitter vacuum solution of constant space curvature. x < 1 is inside the source x > 1 is outside the source that may actually be exotic vacuum zero point energy.

The source mass density is M/r*^3 the radius of curvature at r* is r ~ (r*3/rs)^1/2 where rs = 2GM/c^2.

Therefore N below is computed using this "r".

Dvali's et-al work on extra space dimensions arrives at Abdus Salam's 1973 f-gravity that in the static Newtonian limit is simply for the gravity potential energy per unit test particle

V(r) = -(GM/r)(1 + a*e^-r/b*)

In my world hologram tetrad model, the Einstein-Cartan gravity tetrad 1-form field is conjectured to be

e^a = I^a + N^-1/3A^a

ds^2(1916GR) = e^ae^a = guvdx^udx^v

I^aIa is ds^2(1905SR) in which only global inertial frame (GIF) transformations are allowed.

As soon as one allows global non-inertial frames, A^a =/= 0 in such a GNIF. A^a = 0 in a GIF.

For a GNIF R^a^b = 0 (vanishing curvature 2-form) and T^a = 0 (vanishing torsion 2-form)

In 1916 GR localize rigid T4 to elastic T4(x) and now in general R^a^b =/= 0 but T^a = 0 still.

In the Jack Ng et-al world hologram conjecture for simple SSS vacuum model

4pir^2 = NLp^2/4

r = Schwarzschild radial coordinate for static LNIF observers when r > 2GM/c^2, M = source mass energy

Lp^2 = hG/c^3

N = number of Bekenstein c-BITS

r = N^1/2Lp/16pi

The size of the quantum gravity foam bubble is

&r ~ (Lp^2r)^1/3 ~ N^1/6Lp/16pi

Therefore r^3/&r^3 ~ N^3/2/N^1/2 ~ N

Therefore, there is a 1-1 hologram correspondance between each area hologram quantum and it's projected "volume without volume" hologram image quantum.

Conjecture

a* = N^1/3

b* = &r

Therefore

V(N,x) ~ -(GM/Lpx)(1/N^1/2)(1 + N^1/3e^-N^1/2/N^1/6) = -(GM/Lpx)(1/N^1/2)(1 + N^1/3e^-xN^1/3)

The second rank tensor field guv(x) is composite

guv(x) = nabe^au(x)e^bv(x)

therefore the quantum foam gravity zero point vacuum fluctuations in the fabric of warped 4D spacetime guv(x) are spin 0, 1 & 2.

Therefore LISA/LIGO & Ray Chiao's "transducer" in addition to seeing the two massless spin 2 stretch-squeeze Weyl conformal vacuum gravity tensor wave linearly polarized modes at 45 deg relative to each other may also see two massless spin 1 vector linear modes 90 deg to each other just like EM transverse waves, as well as a massless spin 0 scalar longitudinal wave. On the other hand if the spin 0 and spin 1 geometrodynamic quanta have rest mass they will not be seen in far field radiation. There may also be torsion field waves as well from

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

The principle of local gauge invariance for spin 1 compensating vector fields that are renormalizable with the Higgs-Goldstone hidden ODLRO symmetry (t'Hooft, 1973) is battle-tested experimentally (electroweak force & QCD) and is a direct consequence of Einstein locality for the Bohmian IT hidden variables when the effect of the BIT quantum (super) potential is small Q. The effect of the latter is, in essence, a tiny warp-drive bubble in which the, e.g. electron as a tiny T = 0 Kelvin non-radiating "black hole", can under the action of Q can literally turn around in time giving a literal meaning to the Feynman diagrams (electron positron pair creation & annihilation) as more than simply a heuristic of perturbation theory.

The Einstein-Cartan "spin 1" tetrad 1-form Lorentz group vector field decomposes as the local T4(x) minimal coupling to matter fields Psi

e^aPa = I^aPa + A^aPa

i.e. P to P + A acting on Psi

P = I^aPa

A =A^aPa

{Pa} is the Lie algebra of the universal rigid T4 symmetry group of all material field actions in 1905 Special Relativity (SR).

This is analogous in QED to the local U1(x) minimal coupling

P to P + A acting on Psi

Now what is I^a?

I^a are the 4 tetrad 1-form fields I^a = I^audx^u that are AFFINE INVARIANTS under the restricted flat geodesic Global Inertial Frame (GIF) transformations in Minkowski spacetime under the 10-parameter Poincare group. That is, only maps from one GIF to another GIF.

s^2(1905) = I^aIa AFFINE INVARIANT

As soon as one considers a transformation from a GIF to an accelerating off-(flat) geodesic Global Non-Inertial Frame (GNIF) I^a is no longer invariant. Note these are constant accelerations the same everywhere-when (conformal boosts?)

In fact I^a to I^a + X^a

Therefore, we introduce a compensating gauge potential A^a such that

A^a to A^a - X^a

to keep the sum e^a = I^a + A^a INVARIANT under this wider NON-AFFINE group of PHYSICAL FRAME transformations.

s^2("conformal"?) = (I^a + A^a)(I^a + A^a)

This is like in electromagnetism where the gauge transformation is

A to A - Gradf

This is the first BABY STEP to Einstein's 1916 General Relativity (GR).

There is a SPIN-CONNECTION 1-form

S^a^b = w^a^bce^c = w^a^bc(I^c + A^c)

The curvature field 2-form R^a^b must vanish in the above case

R^a^b = DS^a^b = dS^a^b + S^ac/\S^cb = 0 everywhere-when

The torsion field 2-form T^a must also vanish

T^a = De^a = de^a + S^ac/\e^c

Einstein's 1915 GR locally gauges the rigid 4-parameter T4 translation subgroup of P10 to T4(x), i.e. locally inhomogeneous displacements where

I^a to I^a(x)

I^a(x) to I^a(x) + X^a(x)

A^a to A^a(x) -> I^a(x) - X^a(x)

Now the HOMOGENEOUS global spacetime interval s connecting pairs of arbitrarily separated events is localized to the INHOMOGENEOUS local ds(x) connecting pairs of closely spaced events.

Where

R^a^b(x) =/= 0 is allowed but T^a(x) = 0 is enforced.

i.e. curvature but not torsion in 1916 GR.

Note that all states connected by a redundant gauge transformation (on same gauge orbit) are the same physical state.

Curvature and torsion spin 1 vector fields in geometrodynamic sqrt substratum

On Sep 19, 2007, at 1:33 PM, Paul Zielinski wrote:

"In which case I suppose your

e^a = I^a + B^a

is an *arbitrary* basis in the tangent space?"

Yes in sense of equivalence principle that the metric in the a,b,c Latin indices is Minkowski nab i.e.

eb = nabe^a

In a coordinate basis

e^a = e^audx^u

ds^2 = guvdx^udx^v = e^aea = nabe^aue^bvdx^udx^v

hence

guv = nabe^aue^bv

Z: "As I understand it this is not a "tetrad"; the "tetrads" are given by the 4 x 4 2-index quantities e^a_u."

I call e^a_u the "tetrad components" and I call e^a the four tetrad 1-forms

Also eu = eu^aea

so eu is a basis vector in u space where each eu has 4 components eu^a

a duality

Z: "Am I right?"

OK, except for some name difference. BTW Hawking and Ellis have an interesting discussion on part of this. More on that when I get a chance.

Z: "If so, how can e^a carry information about reference frames?"Z:

Why do you keep asking this? e^a is independent of u-frames, i.e. e^audx^u is a scalar invariant under

x^u(P) morphs to x^u'(P)

P = objective local coincidence physically = gauge orbit formally

Z: " Or is it only the components I^a and B^a that individually carry such coordinate-dependent information, which exactly cancels out in the
sum e^a = I^a + B^a?"

Yes. Now this is original with me and I may be wrong, but it seems it has to be that way in order that B^a transform like the EM vector potential A in a U(1) gauge transformation where A is not a U(1) tensor but is a U(1) connection with the inhomogeneous term. e^auPa is the GCT local T4 analog of the EM U(1) gauge covariant partial derivative! Then everything works! That is, compare

U(1) id/dx^u - (e/hc)Au gauge covariant derivative on the NR electron quantum wave field to

local T4 i.e. iI^auPa - B^auPa morphs to id/dx^u - Bu

where {Pa} is Lie algebra of T4 rigid group of 1905 Special Relativity in a MATRIX REPRESENTATION matching that of the source fields under the Poincare group P10. For Dirac spinors the Pa have Dirac gamma matrices in them - see Rovelli Ch 2.

In my world hologram theory Bu = N^-1/3Au

N = Bekenstein BITS to IT of a dominating world horizon like the RETROCAUSAL dark energy future deSitter horizon.

where Au is the local T4 geometrodynamic spin 1 vector field analog to the U(1) EM 4-potential

for a curvature field only.

Similarly, if you add a new dynamically independent torsion field (e.g. Gennady Shipov) the larger localized P10 gauge covariant derivative on the matter source fields is of the form

iI^auPa - B^auPa + S^a^buP[ab] morphs to id/dx^u - Bu

where P[ab] are the 3 Lorentz boosts and 3 space rotations of Lie algebra of rigid O(1,3) of 1905 special relativity and S^a^b = S^a^budx^u is the spin connection.

Just as Bu = B^auPa = curvature tetrad spin 1 vector field

Su = S^a^buP[ab] is the torsion spin 1 vector field.

Jack Sarfatti wrote:
I never said e^u_a were coordinate invariant, I said e^a were coordinate invariant
On Sep 19, 2007, at 7:54 AM, Paul Zielinski wrote:

Jack Sarfatti wrote:

Z: If the difference vector between two spacetime points P and P' is dx, then

dx = dx^u e^u

S: No, that formula makes no sense.

I only meant that you should have written

dx = dx^ueu without the ^ in e^u - otherwise obvious since eu is a basis set

dx = dx^u e_u.

Z: "The point is that the vielbeins e^u_a are clearly not coordinate-invariant, since they depend
on the coordinate basis {e_u}."

I never said otherwise

Z: Even if the spacetime remains flat? If so, then non-vanishing B^au doesn't necessarily mean
non-vanishing intrinsic curvature.

* S: That's precisely the point. I was too sloppy earlier. B^au =/= 0 both when there are accelerating detectors in flat spacetime and for all detectors in curved spacetime. What I should have said was that B^au =/= 0 in a curved spacetime. B^au = 0 in flat spacetime using only inertial geodesic detectors that do not experience g-forces. I should have written that curved spacetime is sufficient for B^a =/= 0, but is not necessary. I put the cart before the horse.

Jack Sarfatti wrote:

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

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

S: 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.

Z: Yes, accelerating frames have event horizons, and are only locally definable, even in a flat spacetime.

S: Yes.

Also "g-force" only refers to the center of mass motion of an extended body.

Z: This refers to effects on the center of mass motion of a test body.

S: The curvature tidal distortions of Weyl stretch-squeeze and Ricci compression-expansion are only in the relative coordinates of the extended object.

Z: Meaning the center of mass is not affected by tidal forces? Of course. Also true in Newton.

S: Yes.

The MAP is NOT the Territory (Alfred Korzybski)

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

Z: 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).

S: Clearly. Yes that is the way I picture it.

Z: OK.

S: 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.

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

S: "Instance"?

Z: 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.

S: So far I have no need of that distinction.

Z: But the distinction has been made by e.g. Rovelli.

S: Really? Where? Exact reference, page - text please

Z: You asked about "instances". This is simply offered as an example of *two different instances* of the same abstract group Diff(R^4).

S: I do not understand your sentence. You mean two different matrix representations of an abstract group? That I understand. Where is "instance" used in this context?

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.

Z: OK, fine.

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

Z: 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.

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

Z: We are talking about *mathematical* meaning here.

S: I don't understand what you mean - too vague without more concrete examples. I am very suspicious of "math for math's sake" in discussion of physical problems.

Z: For example, you could define a set of geometric operations that transform the world line of any given isolated
system (say, a test particle) in spacetime while leaving all other world lines invariant. These are geometric
transformations that cannot be equated to spacetime coordinate transformations.

S: Yes, but that can only happen by changing the intrinsic geometrodynamic field B^a - exactly WARP DRIVE metric engineering where all world lines are still geodesic, OR non-equivalently, putting a charge on the test particle and switching on an electromagnetic field keeping neighboring test particles neutral. The charged particle is no longer on a geodesic. Also this is not merely a coordinate transformation because one has changed a dynamical field configuration.

Z: You could build an instance of the group T4 from those kinds of transformations, and then "locally gauge" that.

S: I don't think so. Show me. Prove that with a simple example. It cannot be done in the second case of a charged test particle. Since locally gauging T4 means introducing the real geometrodynamical compensating field B^a, then you can do it in first case. Note while B^a = 0 everywhere-when is a sufficient condition for globally flat Minkowski spacetime it is not a necessary condition. That is accelerating off geodesic Minkowski non-inertial frames induce a B^a field as well to keep ds^2 invariant but

R^a^b = 0

T^a = 0

i.e. curvature and torsion 2-forms vanish everywhere-when. So I suppose you can call B^a the "inertial field" in sense of Gennady Shipov's term?

Thus

ds0^2 = I^aIa

is an affine invariant i.e. only for the linear globally rigid Poincare group transformations connecting the preferred class of global geodesic inertial frames excluding nonlinear off-geodesic accelerating frames.

Since I^a -> I^a + X^a in a non-linear transformation ds0^2 is not invariant, however

ds^2 = (I^a + B^a)(Ia + Ba)

is invariant still under this larger group of transformations including accelerating global non-inertial frames on Minkowski non-geodesic world lines.

but R^a^b(T4) = 0 & T^ab(T4) = 0

with respect to the S^a^b(T4) spin connection in Minkowski spacetime without source masses real or virtual ZPF.

Z: You are trying to relate a set of BEC Goldstone phases to the tetrads of Einstein-Cartan GR.

S: Yes, though that does not yet arise here.

Z: You have a decomposition

e^a = I^a + B^a

which in flat spacetime reduces to e^a = I^a.

S: Yes. Tacit assumption here is only transformations from one inertial frame to another, otherwise we need a B^a which is like an EM vector potential without an EM field tensor, or like a superfluid velocity flow without vorticity in a simply-connected region for the order parameter free of nodes whose Goldstone phases give B^a as an emergent field.

Z: If your tetrad field e^a incorporates information about the observer's frame of reference, then shouldn't it be made clear exactly how this is accomplished in Einstein-Cartan based on transformations of I^a?

S: Yes. For all global inertial frames I^au = Kronecker delta. This is in Rovelli Chap 2. The a index is aligned with the u index in that case. I will work out a transformation to a rotating frame later - good problem. The case of hyperbolic motion in special relativity should be worked out as well.

Z: Trivially, any given group S_N of permutations of N objects can be instantiated over any set of N particular objects.
This is a simple example of multiple "instances" of an abstract group.

S: 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.

Z: 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.

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

Z: OK, so you mean physical operations, represented by a subset of the mathematically available coordinate transformations.

S: Yes, that's what I always mean unless explicitly otherwise noted. We want to erase all artifacts of the contingent representation and only deal with intrinsic invariant structure. Of course to get a number we need a detector. For example in Schwarzschild vacuum metric in empty space

g00 = -1/grr = 1 - rs/r

rs/r < 1

4pr^2 = area of concentric surface relative to center of mass of SSS source.

This representation is only good for static LNIF off-geodesic detectors firing rockets radially toward the center of mass such that their acceleration

g = -GM/r^2 = -c^2rs/2r^2

keeps them standing still in curved spacetime relative to source center of mass.

Also they have zero orbital angular momentum relative to the source center of mass.

The fact that an accelerating object in curved spacetime can stand still relatively speaking shows that "curved spacetime" is objectively real and is qualitatively different from flat spacetime. Stephen Hawking has a good picture of this in his "The Universe in a Nutshell."

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

S: Again give a concrete example of what you mean here.

Z: All depends on what is being rotated, translated, or reflected. It could be coordinates, or it could be selected physical
or geometric objects.

S: I always mean pairs of detectors being compared or single detectors being displaced, rotated, boosted etc. In GR we can only compare locally coincident detectors. Separated detectors cannot be compared in a path-independent way, i.e. need a connection field for parallel transport even in extra dimensions for internal symmetries. The FUNDAMENTAL connection field is always a local compensating spin 1 vector field from localizing the relevant globally rigid symmetry group of the action of the source field chosen. The Levi-Civita connection is not FUNDAMENTAL, it is DERIVED-COMPOSITE as a bilinear ENTANGLED form in the fundamental tetrads e^a in the case of the geometrodynamic field = fabric of spacetime. As a quantum field the Levi-Civita connection has spin 0, spin 1 as well as spin 2 virtual quanta. Therefore, LIGO/LISA may also find longitudinal spin 0, vector spin 1 as well as tensor spin 2 gravity waves if the spin 0 and spin 1 real quanta have no rest mass. Spin 0 has only a single compression expansion polarization. Spin 1 has 2 transverse like the EM wave (e.g. mutually orthogonal linear polarizations) and spin 2 has 2 transverse stretch-squeeze at 45 deg to each other.

Z: The Lorentz group, for example, refers to a set of coordinate transformations in Minkowski spacetime. However, you
cannot apply such transformations selectively to the world line of a single object. Such selective transformations change
the geometric relationships between world lines, which a spacetime coordinate transformation alone cannot do.

S: Yes - as above.

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

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

Z: Well, this looks like a coordinate substitution. Is that what it is?

S: No. Imagine a perfect crystal lattice where X^u(Pi) is an equilibrium for an "atom".

Let the lattice spacing be L^u, i = 1 ... N

Physically distort the position of the atom at X^u(Pi) to X^u(Pi) + a^u(Pi) where

(radius of variable disclination curvature defects) >> a^u(Pi) >> L^u

Z: Don't global translations in T4 change the coordinates of all spacetime points, and of all test objects located
at those points?

S: Only HOMOGENEOUS translations where a^u is same for each P. GLOBAL here always means RIGID or HOMOGENEOUS & ISOTROPIC, i.e. same change everwhere-when - this is NON-CAUSAL not limited to interiors of light cones. Hence it is NON-PHYSICAL (classically) needing LOCAL GAUGING.

Z: Suppose you only want to translate the world line of a single test object?

S: Then you need to either put a charge on it and apply an EM field, or you need to locally warp the geometrodynamic field on the test particle by modulating the local Goldstone vacuum ODLRO phases out of which the geometrodynamic field emerges.

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 Pu of the field Psi on spacelike slices can be defined and is conserved on any foliation of spacelike slicing - Noether's theorem in action.

Z: Right.

S: If, we locally gauge T4 then globally rigid homogeneous "constant" a^u morphs to a 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.

Z: Your translation group T4 is then said to be "locally gauged"?

S: Yes.
The global theory with globally homogeneous constant a^u has tetrads I^a.

Z: OK.

S: 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.

Z: Are you saying you can generate an intrinsic curvature tetrad component B^a in e^a simply by locally gauging a set of spacetime coordinate
transformations in a flat spacetime? I don't see how that can be possible.

S: I am NOT saying that. Indeed that is NOT possible. There are physical point monopole nodes in the Higgs-Goldstone vacuum ODLRO field formed in the moment of INFLATION. These point nodes have a quantum foam bubble radius whose Landau-Ginzburg coherence length is from the world hologram axiom

&L = N^1/6Lp

I mean "coherence length" is same way as for magnetic vortex cores in a type II superconductor. There is also a second penetration depth for the B^a geometrodynamic field.

In the case of the large-scale structure of the universe N ~ 10^122 from the retro-casual dark energy future de Sitter horizon

&L ~ 10^-13 cm

i.e. the vacuum ODLRO order parameter drops to zero over the scale &L when L ~ N^1/2Lp is the scale of the detector resolution and

&L ~ (Lp^2L)^1/3 e.g. Jack Ng's papers.

Therefore, a^u(Pi) are small actual physical distortions of the Higgs-Goldstone field nodes forming Hagen Kleinert's "world crystal lattice".

(radius of disclination curvature defects) much greater than a^u(Pi) much greater than &L

These actual physical distortions are cause by the stress-energy tensor Tuv sources both real outside the vacuum and virtual inside the vacuum which is partially coherent, i.e. ODLRO.

So, I do not mean formal coordinate relabeling making real physical differences ever.

Z: On the other hand, if T4 in this case is a set of *geometric* transformations, then this makes sense, since you can liken it to a stretch-squeeze
deformation of the flat geometry. This is equivalent to making the flat metric g_uv = n_uv vary from point to point in a manner that results in
non-vanishing R^u_vwl.

S: Yes, that's what I have been saying.
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.

Z: Local GCTs are not the same as intrinsic curvature.

S: Of course not.

Z: How do you get from GCTs (non-linear coordinate transformations) to
intrinsic spacetime curvature, by "locally gauging" a group of coordinate transformations?

S: That's not what local gauging T4 means. The GCTs are REDUNDANT gauge orbits that are factored out leaving only physical relations between locally coincident detectors. Rovelli in Ch 2 explains this clearly. He even draws pictures. (Hole Paradox). It's same as in electrodynamics where

A morphs to A + GradChi

EM fields ~ curl A etc. unaffected and closed loop Bohm-Aharonov integrals for physical quantum fringe shifts unaffected.

Local gauging T4 is same as local gauging U(1). Einstein's GCTs in the redundant sense of formal relabeling is same as EM gauge transformations functionally speaking. T4 is fundamental a physical group of moving detectors around and the relationships between detectors, Physics is operational needing a theory of measurements to be physics otherwise its untestable metaphysics of no interest. String theory and loop quantum gravity have strayed too far from a theory of measurement although string phenomenologists and quantum gravity phenomenologists looking for quantum foam in cosmological scale gamma bursts etc have begun to correct that lately - a good sign.

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

Z. Have you changed the coordinate system, or the geometry?

S: There its the coordinate system not the geometry. You have not distinguished my earlier

I. X^u(P) morphs to X^u(P) + a^u(P)

from my above

II. X^u(P) morphs to X^u'(P) change of relabeling for different coordinate charts at same local coincidence P of a pair of detectors.

What's physically important in Einstein's 1916 GR based on locally gauged T4 only (no torsion) are the relative relations of locally coincident detectors in arbitrary relative motion including accelerations to arbitrary orders i.e. d^nX^u(P)/ds^n =/= 0. The redundant formal relabeling are the gauge orbits that are divided out of the physical description. In particular local T4 connects a geodesic detector to a locally coincident non-geodesic detector and that is the tetrad - even in Minkowski spacetime where R^a^b = 0 and T^a = 0 globally everywhere-when even when local B^a =/= 0 induced by the non-geodesic detectors. Therefore, in that case non-gravity forces induce a fictitious B^a field without any covariant curl leaving ds^2 a local frame invariant. It's like a static homogeneous EM vector potential A in a simply-connected space. What happens in 1905 special relativity is the use only of globally rigid P10 = T4xO(1,3) connecting pairs of geodesic detectors only which need not be locally coincident because there is no anholonomic path dependence in parallel transport relative to a connection field that is the compensating field of localized P10 beyond 1905 SR.
Since I am only doing local coordinate-free objective physics obviously the latter I would think.

Z: Well, tetrads, which relate an arbitrary basis set e^a to a coordinate basis set e^u, i.e.,

e^u = e^a_u e^a

S: Yes

Z: are both coordinate-free *and* coordinate-dependent, according to my references.

S: Contradiction. Both X and not-X. Quantum logic? I have no idea what that means.

e^a = e^audx^u

is a GCT scalar invariant under both passive relabelings and active pairs of coincident detectors in arbitrary relative motion.

Z: But maybe it's not so obvious?

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

Z: 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?

S: Too vague. I don't know what you mean without an example.
When e^au = Kronecker delta I call that "alignment".

Z: If the difference vector between two spacetime points P and P' is dx, then

dx = dx^u e^u

S: No, that formula makes no sense.

Z: defines a coordinate basis in the tangent space. That means that the vectors e^u point along the coordinate axes,
and if you change the coordinates, you also change the orientation of the vectors e^u. Then the tetrads e^a_u
relate an arbitrary basis to the coordinate basis according to

e_u = e^a_u e_a

S: This formula is OK and when e^a_u = Kronecker delta the e_a and e_u vectors are aligned. Note that here a & u are not components but label different 4-vectors.

Z: How can you say that the 4 x 4 quantities e^a_u are coordinate-invariant?

S: Red Herring. I NEVER said that. You are not reading what I wrote carefully. I said e^a are GCT invariants NOT e^au!

e^a = e^audx^u

is a local scalar invariant under X^u(P) morphs to X^u'(P) formal relabeling - also for any two locally coincident detectors at P. The latter are an equivalence class of the former as in Rovelli's Ch 2 on the Hole Paradox. The former connects two manifold points p on same gauge orbit P. The CONFUSION is not using the PHYSICAL QUOTIENT set {p}/P factoring out the local T4 gauge freedom (orbits) leaving only physically objective relations between pairs of locally coincident detectors in arbitrary relative motion.

Z: Of course the *arbitrary* basis set {e^a} is coordinate-invariant!

S: After setting up a straw man about e^a-u you here repeat what I wrote a jillion times about e^a.

Z: 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?

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

Z: See above. I've given you a standard definition of "coordinate basis", and the tetrads relate a coordinate basis to an
arbitrary basis by definition

S: Still too vague. I don't know what you mean without an example.

Z: 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.

S: That's good.

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

Z. But the quantities e^a_u are clearly not themselves invariant under local GCTs.

S: Of course they are not and I never said they were. Red Herring.

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.

Z: In any case, it should be clear that I^a cannot be coordinate-invariant if it carries information about
reference frames (even in a flat spacetime where all B^a = 0).

S: Partially correct - not under non-affine or nonlinear transformations that relate locally coincident detectors in arbitrary relative motion. Note that B^a =/= 0 in general when we have accelerating detectors, but in the case of Minkowski spacetime, R^a^b = 0 and T^a = 0 even when these pure gauge fields B^a of the accelerating detectors are not zero. In this case there are global transformations that make B^a = 0 in a simply-connected space - same story as for the Levi-Civita connection derived from I^a & B^a - bilinear in them and their gradients.
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.

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

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

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

S: Simple. I don't. Red Herring.

Z: Do you mean that you don't think the tetrads depend on the coordinates?

S: I mean e^a(P) do not depend on the choice of X^u(P) or X^u'(P) where P = physical local coincidence = gauge orbit of equivalent manifold points p, i.e.
P = {p mod local T4}.
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.

Z: 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 transformations.

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

Z: Yes, this is really a mathematical issue.

S: Then show the math for your above distinction.

Z:
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?

S: 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.

Z: My guess would be that your Goldstone phases have no relationship to reference frames.

S: Since B^a ~ M^a^a then B^a^a has a relation to reference frames in u-index, but

e^a = I^a + B^a

is a u-index frame invariant.

That is I^au + B^au together is a first rank covariant local T4 tensor under change of local coordinate charts X^u(P) to X^u'(P) for the same gauge orbit P.

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

S: Sure but so what? What formal analogy?
You lost me. I see no physics here. Too vague.

Z: I'm talking about the relationship between your BEC Goldstone phases and e^a = I^a + B^a.

S: No formal analogy there.

B^a ~ M^a^a = (dTheta)^a(Phi)^a - (Theta)^a(dPhi)^a

Therefore, B^a -> B^a + X^a induced by a GCT does indeed correspond to Goldstone phase shifts in the post-inflation degenerate vacuum ODLRO Higgs-Goldstone field.

Z: 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).

S: No, 1905 special relativity uses the 10 parameter Poincare group P10 = T4xO(1,3).

Z: 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.

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

Z: Right.

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).

S: Not so, you can also have homogeneous displacements, i.e. a^u(P) same for all P.
I meant relative displacements of detectors.

Z: That's not what I mean by "inertial frame transitions", which I intended to imply a change from
one inertial frame to another.

S: OK you mean inertial frames related by a uniform global boost in relative unaccelerated motion.
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.

Z: Exactly.

S: The 6-parameter Lorentz group O(1,3) is a subgroup.

Z: Right. Yes of course you can add the translations and get the larger Poincare group.

S: 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.

Z: 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).

S: Huh? No comprende. Give a concrete example.
Fully covariant SR means simply a fictitious B^a tetrad field with curvature R^a^b = 0 and torsion T^a = 0 everywhere-when in the Minkowski spacetime regions where the

B^a to B^a - X^a

I^a to I^a + X^a

leaving e^a = I^a + B^a GCT local invariant at P.

under GCTs connecting coincident non-geodesic accelerating detectors to each other and to coincident geodesic detectors with background-independent

ds^2 = (I^a + B^a)(Ia + Ba) INVARIANT same for all coincident detectors at same gauge orbit P "local coincidence" (Einstein -- Rovelli Ch 2).

The point is to distill out the essence, i.e. the real geometrodynamical field changes T^a =/= 0 & R^a^b =/= 0 and Bohm-Aharonov loop integrals of B^a not zero by erasing the redundant gauge transformations of relabeling X^u(P) to X^u'(P) in fixed gauge orbit P. Dividing the space of gauge orbits into its quotient space mod local T4 is the key to avoiding meta-theoretical confusion and sticking close to the physics.

The physics is ONLY the Leibnitzian relations between locally coincident field detectors looking at the same classical signals in the first approximation. When the signals are quantum, then there may be Heisenberg uncertainty clash when two detectors try to detect the same quantum signal - if the detector observables do not commute.


Z: 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".

Z: Can't agree. There are no accelerating frames as such in SR, even if the spacetime is flat. Only "frame boosts", which are different.

S: What about hyperbolic motion in SR? What about rotating frames in SR. Both these problems involve transformations from an inertial frame to a non-inertial frame which you can do even in Galilean relativity within certain limits. True 1905 SR invariant ds^2 is only for inertial observers in relative non-accelerating motion.

Example 1 hyperbolic motion (uniform translational acceleration) in Galilean relativity of Newton's particle mechanics only for simplicity

x in inertial frame S to x' in non-inertial frame S'

x'(S') = x(S) - (1/2)gt^2

v = gt << c

with absolute simultaneity t' = t in this approximation "~"

In global inertial frame S using 1905 SR in limit v/c << 1

ds^2 = (cdt)^2 - dx^2

since t = t', therefore dt = dt'

since x = x' + (1/2)gt^2

dx = dx' + gtdt

dx^2 = dx'^2 + (gt)^2dt^2 + 2gtdx'dt

Therefore

ds^2 ~ (cdt')^2[1 - (gt'/c)^2] - dx'^2 - 2gt'dx'dt'

it looks like a "particle horizon" wants to form where g00 = 0. It does form when you remove the gt/c << 1 and use 1905 SR consistently e.g. MTW chapter on hyperbolic motion for the relativistic rocket. Accelerating frames in 1905 SR do have causal particle detector horizons. Note there is also a "gravimagnetic field" ~ 2gt' in Ray Chiao's sense. There is also an Uhruh effect.

This is a 1 + 1 spacetime model with only two tetrads e^0 and e^1 with frame (S & S')-dependent components e^0u and e^1u.

R^a^b = 0 & T^a = 0 is the physics here.

Point is that off-geodesic frames (detectors) induce curvilinear guv, i.e. B^a =/= 0 even if, so to speak, it's "curl free" or "fictitious" in sense of inertial g-forces on centers of mass without any real geometrodynamical curvature field distortions in the coordinates relative to the center of mass of the test object. Note the torsion field is assumed zero here. The issue here is whether the torsion field can generate a weak approximately zero-g force warp bubble in sense of 1916 GR? It's my conjecture that it can - I may be wrong.

Example 2 rotating non-inertial frame S' in x-y plane again in Galilean limit

x' = xcoswt + ysinwt

y' = -xsinwt + ycoswt

t' = t

algebra is more complicated - do it for homework.

Can a rotating frame's center of mass move along a timelike geodesic in, e.g. curved Schwarzschild metric? Spin-orbit coupling?

*The objective Litmus test of curved spacetime is if you have to apply a non-gravity force to stand still relative to some detectable object. Also you feel g-force in that case.

Z: 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.

S: What does this mean physically? Operationally? Why are these words important?

Z: The point is that you can formulate SR in a general covariant manner without including accelerating frames,
and that the question of general covariance can be completely separated from the question of relativity.

S: Well if you include accelerating frames in SR and you can then they force a curvilinear metric for components relative to them as I show above explicitly in a simple model.

Z: OK, maybe you don't have to bother yourself with this kind of thing.

S: To solve what kind of problem? Relative velocity has an effect on clocks?

Z: True in Lorentz 1904, Einstein 1905, and Minkowski 1907. In Einstein 1905 you have *fully reciprocal* time dilation, which depends only on the speed of the clock relative to the observer. Not so in Lorentz's theory, and arguably not actually the case in Minkowski's theory.

S: I suppose you mean that a uniformly moving clock runs slow to me, but not to the guy in the rest frame of the (to me) moving identical clock.

Z: The point is that this is also true in a general covariant version of SR where n_uv -> g_uv.

S: 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.

Z: Yes. 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.

S: Precisely

dso^2 = I^aIa 1905 SR WITHOUT accelerating detectors

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

OR also in 1905 SR with accelerating detectors, but

R^a^b = 0 & T^a = 0.

Z: 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.

S: Yes, for momentarily locally coincident detectors tapping the same signals fast enough.

Z: ... ds^2 *integrated along any given path* between two spacetime points.

S: Is path dependent even in 1905 SR hence the twin time dilation effect seen in mu meson cosmic rays. The mu meson cosmic rays live longer than do their identical twins at rest in the lab.
Yes, also for momentarily coincident observers Alice and Bob in arbitrary relative motion including acceleration, jerks, etc ...

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

S: Too imprecise. For example dT = goo^1/2dt & dR = grr^1/2dr for static observers when g0i = 0. dT and dR are the proper time and radial distances directly measured by the static observers on site so to speak. Note that the longitude and latitude distances here are same as in flat space for these static observers. Hence 2piR is not a circumference nor is 4piR^2 the concentric spherical surface area, which in fact is 4pir^2. Note also that the static observer measuring dT has dr = dtheta = dphi = 0 therefore dT is a local frame invariant for all locally coincident detectors.

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

S: Yes.

Z: OK.

S: 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.

Z: OK.

S: With one possible exception that Kip Thorne's LIGO/LISA detectors may possible observe anomalous macro-quantum gravity waves of scalar spin 0 and vector spin 1 in addition to the expected classical tensor spin 2. The macro-quantum scalar and vector gravity curvature waves would be Glauber coherent states directly showing the physical reality of the tetrad substratum if the spin 0 and spin 1 geometrodynamic quanta do not acquire large rest mass by the post-inflation Higgs mechanism. There should also be torsion field waves.
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!


Z: 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.

S: 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.

Z: OK, that makes more sense.

S: I^au cannot be a GCT tensor.
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.

Z: OK.

S: 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.

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

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

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

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

Z: That's what I call GR in flat spacetime.

S: OK you should have said that at the beginning.

Then, in general

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

but

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

Z: OK.

S: 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

Z: Even if the spacetime remains flat? If so, then non-vanishing B^au doesn't necessarily mean
non-vanishing intrinsic curvature.

* S: That's precisely the point. I was too sloppy earlier. B^au =/= 0 both when there are accelerating detectors in flat spacetime and for all detectors in curved spacetime. What I should have said was that B^au =/= 0 in a curved spacetime. B^au = 0 in flat spacetime using only inertial geodesic detectors that do not experience g-forces. I should have written that curved spacetime is sufficient for B^a =/= 0, but is not necessary. I put the cart before the horse.

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

and

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

Z: 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.

S: Yes, OK.

Z: OK, good, so we agree on this.

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

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

S: 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.

Z: The point is that only certain charts are admissible to represent such frames, and this is only a small
subset of all the charts that can be obtained by application of transformations belonging to the covariance
group Diff(4).

S: Yes. OK.

Z: 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.).

S: Not sure what you mean apart from what I said.

Z: 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.

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

Z: The only Galilean transformations that are included are those for which v = 0, in which case there is no difference
between a Galilean and a Lorentz transformation. Clearly, in general (for -oo < v < oo), Galilean coordinate
transformations

t' = t

x' = x - vt

are excluded, even though they are perfectly good diffeomorphisms in Diff(4), from a purely mathematical POV.

S: No not at all. Galilean transformations are simply the limiting case of Lorentz transformations when v/c << 1. No reason to exclude them at all.

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

S: Too vague.

Z: I'm giving you a very specific example of a class of coordinate transformations in Diff(4) that are mathematically admissible
in a generally covariant theory, but are not physically admissible. It should be obvious that Galilean coordinate transformations
are not admissible as physical coordinate transformations in Einstein's theories.

S: Not only is it not obvious it's not true! Most of the actual tests of GR so far use Galilean transformations v/c << 1.
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.
That's your (and Rovelli's) solution. From my POV we are actually dealing with two overlapping sets of transformations,
one large than the other (mathematical vs. physical coordinate transformations). Passive Diff(4) is the set of *mathematical*
transformations.
Again, Galilean transformations are definitely verboten, even though they must
be *mathematically* admissible in any general covariant theory.

Right?

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!
Galilean transformations t' = t, x' = x - vt, could have any value of v in the range -oo < v < +oo. They are simply not admissible as physical coordinate transformations,
*even in general covariant GR*, despite the fact that mathematically speaking they are perfectly good diffeomorphisms over R^4.
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 are saying that Galilean transformations with v > 0 are good *physical* coordinate transformations in general covariant SR and
general covariant GR? I can't see how.
You can transform to accelerating frames in Galilean relativity for small enough accelerations and short enough times.
As I understand it you cannot use Galilean transformations to describe frame transitions in Einstein's theories when v =/= 0.
You must use Lorentz transformations. Yet mathematically speaking, Galilean transformations are perfectly good diffeomorphisms
over R^4.

Z.

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.

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.
Right.

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/\

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).

OK.

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?

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.

Fine.

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.

"Instance"?
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
transformations.

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.

Precisely

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.

Yes.

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.

OK.

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)

but

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

and

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.

Right?

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.

Z.

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.