On Mar 28, 2005, at 11:47 PM, wrote:
J: I keep showing you that you completely misunderstand what is going on
here. It is precisely the linear elastic cross terms IB that IS the
Newtonian limit because I0^0B0^0 limits to V(Newton)/c^2, where
V(Newton) is the gravity potential energy per unit test mass in the
Newtonian "inertial frame" in which the UNIVERSAL gravity acceleration is
g = - GradV(Newton)
Voila - the ANSWER to your question!
Z: So your position is that you do recover the Newtonian separation of
inertial and gravitational effects in the weak field slow
frame-acceleration limit, but this is lost when you go to the
strong-field Einsteinian regime?
J: When I say "inertial" I mean what happens when you GCT
Iu^a(Minkowski)abIv^b with any GCT matrix Xu'^u.
Z: OK, that's basically what I mean; however, I am limiting my claims to
the *local* interaction of the effects of the inertial and gravitational
fields.
J: All my equations are LOCAL. No integrals anywhere - YET.
Z: The decomposition I am talking about is *strictly local*, i.e., point
by point on the spacetime manifold.
J:cWhen I say "gravity" I mean the Bu^a piece of the tetrad.
Z: Fine. I assume this B transformation also contains information about
spacetime curvature?
J: When B =/= 0 then the GCT Xu'^u operates on
guv(LNIF) = (Iu^a + Bu^a)(Minkowski)ab(Iv^b + Bv^b)
Z: So your (Bu^a)( )(Bu^b) acts just like the regular tetrad
transformation (Iu^a)( )(Iu^b)?
Or in addition to producing metric gradients, does this "B"
transformation also produce
Riemann curvature?
J: Yes & Yes.
Bu^a is a first rank GCT tensor in the base space and it is also a first
rank O(1,3) tensor in the tangent space. Therefore, its vanishing at a
LOCAL EVENT P is OBJECTIVELY REAL COORDINATE-INDEPENDENT (even if the
manifold points change as in p -> q in an ACTIVE DIFFEOMORPHISM, i.e. p
~ q where ~ is an equivalence relation of a gauge freedom).
That is
gu'v'(LNIF') = Xu'^u(Iu^a + Bu^a)(Minkowski)abXv'^v(Iv^b + Bv^b)
The Newtonian physics limit is when B^2 << IB, only the 2 terms IB + BI
contribute, hence the famous factor of 2 and in fact in slow speed weak
curvature limit
Z: OK, so I got this wrong. In the weak field B^2 << IB and is thus
negligible -- but not in the strong field.
J: I0^0B0^0 = V(Newton)/c^2
Now this is BEAUTIFUL!
Z: No, it looks good, but I am trying to reconcile it with the usual
treatment. I can see how the inertial and gravitational fields get
entangled in the strong field regime over a *finite* region of
spacetime, but I still say they must be separable at each point on the
manifold. Otherwise, you entangle the effects of manifold geometry with
its coordinate expressions, which doesn't make sense -- any more than it
would make sense to think you can change the objective geometry of the
manifold by changing an observer's worldline on the manifold.
J: We are talking about different ideas of "inertial" and
"gravitational". I mean by "inertial" I (trivial tetrad) ONLY the action
of certain dangerous subsets of GCT, e.g. Xu^u' like in the Galilean
limit where
x -> x' = x - (1/2)gt^2
t' = t
etc.
By "gravitational" I mean the objectively real local tensor field B =/=
0 that is ALSO the compensating tetrad field from locally gauging GLOBAL
T4 down to LOCAL Diff(4) with the Xu^u connecting COINCIDENT LNIFs at
same P. Einstein's geometrodynamic field guv is BI-LINEAR in (I + B) and
the entangled "inertial-gravity" terms are only the WEAK FIELD ELASTIC
terms IB, the STRONG FIELD PLASTIC terms B^2 are PURE GRAVITY. Of course
some geometrodynamic observables may have terms like I^nB^n', n, n' =
0,1,2, ...
Z: The objective intrinsic geometry of the manifold, and the choice of
coordinates with respect to that geometry,
*should* be unconditionally separable at each point on the manifold.
J: This is false. Or, at least, you have to PROVE it with mathematics.
You first have to define precisely what you mean mathematically by
"inertial" and "gravitational" and I do not see where you have done
that? Until you do that I do not even understand what you are saying and
have no way to compare it to what I am saying.
Z: That is why I think the LC connection is pointwise decomposable into
geometric and coordinate contributions, which in GR would correspond
respectively to the gravitational and inertial contributions to the LC
connection.
J: This is meaningless to me until I see adequate equations.
Z: In addition to making proper Newtonian correspondence impossible.
J: Utter hogwash!
Z: OK, I see your point about B^2 << IB.So you do recover the
inertial-gravitational decomposition in the Newtonian limit.
J: Yes, trivially, because I0^0B0^0 ~ V(Newton)/c^2 in the Galilean
inertial frame F.
where in F
g = - GradVx(Newton) along x-axis
A "dangerous" Xu'^u transformation like
x -> x' - (1/2)at^2 to a non-inertial frame F'
So that in F'
g' = g - a = real gravity acceleration in the inertial frame - inertial
force acceleration of accelerated frame
*But this is not a fundamental requirement from any deep principle. It
is an artifact of a limiting case. You have over-generalized this
fragment for the whole. Your error is Bohm's "fragmentary thinking"
mistaking the part for the whole when there is no scale-invariant
conceptual "self-similarity" in the meta-theoretical structure!
a is the Xu'^u inertial acceleration subtracted from the REAL gravity
acceleration from the B-field potential V(Newton)/c^2
Therefore in general ALL INERTIAL EFFECTS come from the Xu'^u
transformations of GCT (even in the weak field IB >> B^2 and slow-speed
v/c << 1 limit of Galilean relativity of Newton's gravity force
contained INSIDE of Einstein's GR!
Remember B is a pure dimensionless real number over the entire real line
I think?
[B] ~ [Potential Energy per unit mass]/c^2
Now that is EXACTLY what I mean by "inertial" and "gravitational" and I
do not know what you mean in order to compare.
Z: Again: How do you recover Newtonian linear composition of inertial
and gravitational effects in the weak-field domain, for arbitrary
acceleration of the observer's frame of reference?
J: I showed you exactly.
Z: Of course I meant in the *conventional* formulation of GR.
J: I showed you exactly above.
That's what Einstein did! I have shown you above exactly how Einstein's
1916 limit translates simply in the tetrad formulation. It works
beautifully!
Z: The algebra looks OK, but I'm having difficulty relating your I and B
transformations to the subtleties of how the LC connection decomposes in
the conventional formalism.
J: Of course, because what you are attempting is logically impossible -
unless you pin down your definitions, which you have not to my satisfaction.
Z: The metric components g_uv -- both diagonal and off-diagonal -- are
obviously themselves locally tensorial under GCTs,
J: Already you are in error. The tensor is ALL of the components. Under
GCT all the initial components MIX together to form a single new
component. It's like a hologram sort of.
Tu' = Xu'^uTu
Is a local linear superposition at fixed P.
Higher rank tensors are multi-linear superpositions.
Remember also Penrose's "spinor" formulation of GR. There must be a
close connection of the tetrads to the spinors.
Of course the tetrad Bu^a is a bi-tensor in the two spaces. The
base-space tensor index u is two spinor indices U,U' and similarly a -> A,A'
For example
Bu^a = Bu^A^A'(^a,AA')
(^a,AA') = |A>a|A'>a
|A>a is a spinor "wave function" for system "a"
So that the tetrad Bu^a is partly an EPR correlated pair spinor state!
A = 0,1 qubit basis
One tetrad is 2 qubits in tangent space.
Similarly in base space.
This is
IT FROM QUBIT?
"What I cannot create, I do not understand." (Feynman)
Z: while the *derivatives* of the g_uv cause the 3-index Christoffel
symbols built from them to change non-tensorially. So the LC
decomposition I've proposed can still work, because you do not have to
split the g_uv themselves in order to get a coordinate-independent
decomposition.
J: Show me with math. I do not understand your words.
Z: The idea is that if you simply remove the "non-tensorial" inertial
changes in the metric *derivatives* g_uv, w, and only use the residual
"tensorial" components, then you are left with a 3-index tensor quantity
G^u_vw which together with the Riemann tensor R^u_vwl locally represents
the actual gravitational field; while from the g_uv and the
"non-tensorial" parts of the g_uv, w, you can build a 3-index non-tensor
I^u_vw that locally represents the pure inertial field.
J: What you just wrote is meaningless to me unless you can show the math
explicitly.
.
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