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Coordinates and intrinsic objects in General Relativity
Posted:
Sep 18, 2007 7:10 PM


Z: Even if the spacetime remains flat? If so, then nonvanishing B^au doesn't necessarily mean nonvanishing 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 gforces. 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 gforce free geodesic and nonzero gforce offgeodesic  the latter from external nongravity 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 backgroundindependent 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 "gforce" 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 stretchsqueeze and Ricci compressionexpansion 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 noninertial 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 10parameter Poincare group that IS 1905 special relativity analogous to internal symmetry YangMills theory of electroweakstrong 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 2spinor 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 nonequivalently, 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 everywherewhen is a sufficient condition for globally flat Minkowski spacetime it is not a necessary condition. That is accelerating off geodesic Minkowski noninertial 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 2forms vanish everywherewhen. 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 offgeodesic accelerating frames.
Since I^a > I^a + X^a in a nonlinear 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 noninertial frames on Minkowski nongeodesic 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 EinsteinCartan 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 simplyconnected 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 EinsteinCartan 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 4tuple 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 offgeodesic 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 pathindependent 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 LeviCivita connection is not FUNDAMENTAL, it is DERIVEDCOMPOSITE 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 LeviCivita 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 stretchsqueeze 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 everwherewhen  this is NONCAUSAL not limited to interiors of light cones. Hence it is NONPHYSICAL (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 energymomentum 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" "backgroundindependent" 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 HiggsGoldstone vacuum ODLRO field formed in the moment of INFLATION. These point nodes have a quantum foam bubble radius whose LandauGinzburg 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 largescale structure of the universe N ~ 10^122 from the retrocasual 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 HiggsGoldstone 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 stressenergy 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 stretchsqueeze 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 nonvanishing 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 FaddeevPopov 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 (nonlinear 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 BohmAharonov 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 nongeodesic detector and that is the tetrad  even in Minkowski spacetime where R^a^b = 0 and T^a = 0 globally everywherewhen even when local B^a =/= 0 induced by the nongeodesic detectors. Therefore, in that case nongravity 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 simplyconnected 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 coordinatefree 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 coordinatefree *and* coordinatedependent, according to my references.
S: Contradiction. Both X and notX. 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 4vectors.
Z: How can you say that the 4 x 4 quantities e^a_u are coordinateinvariant?
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 coordinateinvariant!
S: After setting up a straw man about e^au 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 nontensor? 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 4vector.
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 coordinateinvariant if it carries information about reference frames (even in a flat spacetime where all B^a = 0).
S: Partially correct  not under nonaffine 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 simplyconnected space  same story as for the LeviCivita 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 metatheoretical 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 nonvector? 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 0forms Theta^b(P) and Phi^a(P) each a Lorentz group 4vector 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 uindex, but
e^a = I^a + B^a
is a uindex 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 postinflation degenerate vacuum ODLRO HiggsGoldstone field.
Z: EinsteinMinkowski (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 gforces 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 6parameter 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 everywherewhen 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 nongeodesic accelerating detectors to each other and to coincident geodesic detectors with backgroundindependent
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 BohmAharonov 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 metatheoretical 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 noninertial 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 nonaccelerating 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 noninertial 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 offgeodesic 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 gforces 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 zerog force warp bubble in sense of 1916 GR? It's my conjecture that it can  I may be wrong.
Example 2 rotating noninertial frame S' in xy 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? Spinorbit coupling?
*The objective Litmus test of curved spacetime is if you have to apply a nongravity force to stand still relative to some detectable object. Also you feel gforce 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 comoving 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 gforce on a offgeodesic noninertial clock from a nongravity force will not change its operation relative to a geodesic inertial clock with zero gforce. Indeed, since the gforce is like keeping a clock offgeodesic fixed in a gravity curvature field, such a noninertial 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 socalled orthogonal transformations.
S: Yes.
Z: OK.
S: most tests of GR only use I^aBa + B^aIa with spin 1 gravivector 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 macroquantum gravity waves of scalar spin 0 and vector spin 1 in addition to the expected classical tensor spin 2. The macroquantum 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 postinflation Higgs mechanism. There should also be torsion field waves. Only in strong field, i.e. B^aBa term will you get spin 0 graviscalar & spin 2 tensor "gravitons" as well as spin 1 quanta is my counterintuitive prediction!
Z: From a mathematical POV the noncovariant 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 (wellbehaved) mathematical coordinates completely arbitrary.
S: I do not understand your use of "noncovariant"
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 10parameter Poincare group connecting globally inertial detectors without any intrinsic Minkowski 4acceleration.
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 offgeodesic 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 nonvanishing B^au doesn't necessarily mean nonvanishing 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 gforces. 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 everywherewhen i.e. globally vanishing curvature field 2form
and
T^a = De^a = 0 everywherewhen i.e. globally vanishing torsion field 2form
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 nongeodesic observers in Minkowski spacetime.
Z: Right, if here you are talking about nonlinear *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 herenow.
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 comoving Lorentz (for noninertial 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  noninertial offgeodesic observers who feel gforces 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 gforces 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, LeviCivita 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 EinsteinCartan, 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 11 Minkowski spacetime just as you can in a 2dim 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 coordinateindependent and even GCT & Lorentz localframe 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" framedependent 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 offgeodesic "accelerating" observers outside the event horizon of a nonrotating 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 comoving 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 grqc 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 coordinateinvariant 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.



