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Topic: Coordinates and intrinsic objects in General Relativity
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Jack Sarfatti

Posts: 1,942
Registered: 12/13/04
Coordinates and intrinsic objects in General Relativity
Posted: Sep 18, 2007 7:10 PM
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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.









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