On the meaning of Einstein's General Relativity of the Gravitational Field
On Dec 23, 2004, at 9:41 PM, firstname.lastname@example.org wrote:
J: The ORTHOGONAL LOCAL CS basis set in this particular representation is
The eu are orthonormal basis vectors at each point (r,theta,phi,t).
Z: OK, so how does this CS relate to the corresponding global Cartesian CS that represents the *same* observer FR in Minkowski spacetime when the permanent field is switched off?
J: Meaningless question apart from obvious formal limit M -> 0. You are not asking a proper physics question if it's not this trivial one.
Z: It's really a mathematical question about the relationship between the two LNIF CSs.
J: Again that's a trivial question with an easy answer.
Xu^u' Jacobian matrix of the GCT goes from LNIF to LNIF' AT SAME EVENT E.
Z: In some sense the flat-manifold LNIF CS and corresponding curved-manifold LNIF CS must converge locally around the point of interest. I am now thinking about this in terms of a local projection of the global Cartesian CS defined over the tangent space onto the curved manifold in the neighborhood of a point.
J: Learn the math Paul. These are all novice level questions. READ PENROSE.
Lim M -> 0 SSS GR vacuum solution = Minkowski space
M -> 0 above is Minkowski metric. What's your problem?
Z: But this doesn't answer my question.
J: There is no real question here.
Z: Or that you cannot answer it.
J: Of course I cannot answer something that is not even there.
"The Question is: What is The Question?" (Wheeler)
Z: Obviously when M -> 0 you are in Minkowski spacetime and you can have a global Cartesian CS. But what is the exact mathematical relationship between the local CS on the curved manifold and the global CS on the flat manifold?
Z: How can you say this question is meaningless?
J: Because it is.
The trivial answer IS THAT THEY ARE FORMALLY IDENTICAL!
That's if your question is intelligible. I am guessing at what you are incoherently stumbling about for.
Z: If the the global Cartesian CS is defined on the flat tangent plane intersecting at the point of interest on the curved manifold, what is the geometrical relationship between the two CSs?
J: "The tangent space Tp to an n-manifold M at a point p may be intuitively understood as the limiting space, when smaller and smaller neighborhoods of p in M are examined at correspondingly greater and greater magnifications. The resulting Tp is flat: an n-dimensional vector space." Penrose p. 224
Note wavelet transform ZOOM feature built into the concept.
Also there is no implication that any curvature tensor is zero because
Curvature(abcd LIF Tangent Space) = ea^ueb^vec^wed^lCurvature(uvwl LNIF Manifold)
ea^u = tetrad components
This is EEP tensor/tetrad structure built into (embedded) the differential geometry at its core. This structure cannot be fragmented/unraveled as you, and also Puthoff in a different way, try to do.
The tangent space idea extends to infinity but of course breaks down on small scale. However, we cannot quantize guv top -> down because it is an emergent bottom-up "More is different" ODLRO effective c-number macro-quantum field theory from the micro-quantum globally flat (no gravity no inertia) pre-inflationary false vacuum substratum. The coherence in the inflationary phase transition to the Big Bang is only PARTIAL leaving dark energy/matter exotic vacuum fragments at different scales as seen in the "cosmological constant", "galactic halo", Pioneer anomaly and other phenomena. The "dark energy" partial incoherence is like the "normal fluid" component in the "two-fluid model". Even in superfluid helium T = 0 ground state the zero momentum BEC condensate is only ~ 10% of total particle number.
Z: How, mathematically speaking, can they both represent the same observer FR?
J: THEY DON'T ! If you mean the LIF?
Z: I don't.
J: You don't know what you mean.
the LIF observer Alice is on a geodesic, the LNIF observer Bob is on a non-geodesic because of a non-gravity force that he feels as "weight". Alice and Bob can reach out and touch hands momentarily - figuratively - they are coincident.
Z: I meant the stationary LNIF when the permanent field is on (curved manifold), and when the field is switched off (flat tangent manifold).
That's the TETRAD relation
guv(curved LNIF) = eu^aev^bnab(LIF ~ FLAT TO FIRST ORDER ONLY)
You keep asking novice questions. None of your concerns here are publishable in a top journal - except for AJP on how to teach elementary concepts in relativity! That's fine, but you think you have come up with a deep new alternative to Riemann and Einstein that will make Wheeler, Thorne and Misner et-al look like Dummies. You have said as much many times now.
Z: Same observer FR, two different CS representations. How does this work, mathematically? In what sense are the two CSs the same in the immediate neighborhood of the tangent point?
It should be possible to smoothly stretch-deform one into the other.
J: Yeah M -> 0 TRIVIAL.
This is hardly a problem.
Z: OK. So you can stretch-deform the Riemann surface
J: Malapropism. "Riemann surface" is inappropriate here.
Z: Are we not dealing with Riemann manifolds? They are abstract any-dimensional analogs of Gaussian *surfaces*. So they are "hypersurfaces" or just "surfaces" for short.
J: Look up "Riemann surface" - it's not used in the context you just described. It's a standard term in functions w = f(z) of complex variables w, z for multi-valued functions - e.g.
z = re^itheta
w = Logz = logr + itheta
Every 2pi rotation takes you to a new sheet of a "spiral winding space" Riemann surface z-domain of f(z) with a branch point (here at z = 0) on which logz is single-valued.
Fig. 8.1 p. 135 Penrose
Z: from curved to flat, and *locally* there is no change to the CS around any given point? The homeomorphic mapping of spacetime points to and from R^4 is locally invariant under the deformation within an infinitesimal neighborhood around the point of interest?
J: It's called geodesic normal coordinates.
Z: I'm talking about the CSs that represent the stationary LNIF. These have nothing to do with normal coordinates.
J: WORK OUT AN EXAMPLE.
That CS is unique mod static rotations, e.g. in SSS
gtt = (1 - 2GM/c^2r) = - 1/grr gtheta,theta = gphi,phi = -1
You have unique local spherical polar Cartan triad of unit vectors er,etheta,ephi mod choice of "celestial sphere" lattitudes and longitudes.
I use basis convention
dx^1 = dr along er
dx^2 = rdtheta along etheta
dx^3 = rsinthetadphi along ephi
dx^0 = cdt
to keep guv DIMENSIONLESS with derivative operators like
(1/rsintheta)d/dphi = d/dx^3
(1/r)d/dtheta = d/dx^2
d/dr = d/dx^1
(1/c)d/dt = d/dx^0
ds^2 = guvdx^udx^v = local differential frame invariant interval
guv(LNIF) = eu^aev^bnab(LIF)
Z: Then it follows from this that if around any given point on the flat manifold, the curved-coordinate contributions to the g_uv, w are zero for the Cartesian CS, then they are also zero for a corresponding local Cartesian CS on the curved manifold, right?
J: Why are we going over this AGAIN! It's the Taylor series I wrote.
Z: But since as we know there is no curved-coordinate correction to the g_uv, w, and thus to (LC), in the global flat-manifold Cartesian CS, we must conclude that there is also no local curved-coordinate correction to these quantities in the corresponding curved-manifold local CS.
J: Hogwash. False reasoning.
Z: OK, why? Don't the two CSs in some sense locally converge in the infinitesimal neighborhood of a tangent point?
J: Meaningless question.
Z: In which case, any residual g_uv, w =/= 0 must be purely geometric in origin (meaning that they reflect the true variation of the g_uv along the manifold), since the curved-coordinate corrections to the g_uv, w (as incorporated into (LC)) all vanish in both CSs.
J: Drivel - angels on head of pin.
Z: Do you think there is a true variation of the g_uv along the manifold around a given point that reflects the intrinsic geometry? Does this concept have any meaning to you?
Paul, again all answers are here if you ask a sensible question:
guv(P') = guv(P) + guv,w(P'- P)^w + (1/2)guv,w,l(P' - P)^w(P' - P)^l + ...
INTRINSIC GEOMETRY OBVIOUSLY BEGINS WITH THE THIRD TIDAL TERM IN THE TAYLOR SERIES
The EINSTEIN EQUIVALENCE PRINCIPLE IS FROM THE ACTION PRINCIPLE
EXTREMUM OF POINT TEST PARTICLE DYNAMICAL ACTION IS A CRITICAL POINT in the space of worldline alternative micro-quantum histories
guv,w = 0
Like dy/dx = 0 in calculus!
The STABILITY physics is in d^2y/dx^2
Therefore, when you and Puthoff wish to eschew the equivalence principle and tensors you also renounce differential geometry and the action principle. This is TOO MUCH to renounce.
Both you and Puthoff in PV, also Haisch in ZPF/SED "inertia" violate Einstein's Golden Rule, you are attempting to make physics simpler than IS possible!
PS Also you can expand the Jacobian matrices of the GCT's Xu^u' as Taylor series.
PPS Your conjecture
Levi-Civita metric connection = GCT Tensor + Non-Tensor
Is wrongly posed.
The only possible GCT 3rd rank tensor is the non-metricity tensor guv;w = 0 with ;w the LC covariant derivative.
The more interesting conjecture is in terms of Cartan's METRIC-INDEPENDENT exterior calculus of GLOBAL topology
Levi-Civita Connection 1-form = Exact 1-form + Non-Exact 1-form
In that case
Curvature 2-form = d(Levi-Civita Connection 1-form) = d(Non-Exact 1-form)
Then your "coordinate part" from Minkowski space is in the Exact 1-form (analog to static electric field potential in EM for Gauss's law and to irrotational fluid flow and defect free lattice).
The intrinsic geometry is in the Non-Exact 1-form! (analog to magnetic field vector potential in EM for Ampere's law).
This also clearly allows for NONLOCAL GLOBAL TOPOLOGY EFFECTS such as
1. NONLOCALITY OF THE PURE GRAVITY ENERGY (e.g. Penrose "Road to Reality")
2. CURVATURE WITHOUT CURVATURE as in Vilenken's thin dark energy wall solution as pointed out by Taub
3. Pioneer Anomaly a_g = -cH pointed back to Sun as a hedgehog topological defect in the post-inflationary macro-quantum vacuum coherence Higgs-Goldstone field.