On Nov 22, 2005, at 10:41 AM, Alex Poltorak wrote:
"Dear Prof. Kiehn,
Thank you for taking the time to study my paper “Gravity as Nonmetricity.” Without addressing all of the points in your letter, let me address the one point you chose to summarize in your email broadcasted on this list.
As I understand, you take an issue with my statement in that paper that “…the metric g is the potential and the connection is the strength of the gravitational field.” "
[Sarfatti wrote: Yes, this is quite standard text book seen in the Schwarzschild solution for r > 2GM/c^2 where
g00 = (1 + 2V(Newton)/c^2) = - 1/grr
V(Newton) = - GM/r
g = -GradV(Newton) = - GM/r^2 ~ connection]
"Your source for this objection is a quote from Wolfram encyclopedia, which identifies gravitational force with curvature.
Although, Wolfram encyclopedia is hardly an authoritative sources for this debate, your point is well taken as it has been debated for a long time. Indeed, the tidal gravitational forces, which cause geodesic deviation, are due to the Riemannian curvature. Nevertheless, the question of what to consider as potential and what to consider as the strength of the field is a murky and apparently does not have a unique answer (see a discussion of this subject in MTW.)"
[Sarfatti wrote: Indeed, this is made even more apparent when one looks at gravity as a local gauge theory for space-time symmetries in contrast to internal symmetry gauge force theories without the equivalence principle.
In the latter case e.g. Maxwell U(1) EM theory we have the very simple template
F = dA 2-form
dF = 0 3-form
d*F = *j(source) 3-form
d*j = 0 4-form
Note that the A 1-form potential is also the connection in the exterior gauge covariant derivative
D = d + A/\ on source fields for *j(source)
In contrast in 1915 GR, at first it looks like the T4 ---> Diff(4) curvature 2-form R is like the EM 1-form F, i.e. suppressing the Lorentzian indices I,J for brevity, since we have seen them now many times we all understand that e.g.
W/\W means W^IJ/\W^JK =/= 0
R = DW = dW + W/\W
DR = 0
So that W looks like U(1) A
and R looks like U(1) F
However, in U(1) EM the source equation is
D*F = *j 3-form
So if the analogy was true we would have
D*R = *J 3-form
but this is WRONG as pointed out by Gennady Shipov.
Define the Einstein curvature 4-form as
*G = D*W
Then the source 4-form equation is
*G = *J
D*J = 0 (5-form in 4-space)
Because of the EQUIVALENCE PRINCIPLE (EEP) absent completely in U(N) internal gauge theories
g(curved) = e(Flat)e
e = 1 + B tetrad field in "subspace"
There is no "subspace" in internal symmetry gauge theory where it's all on the "surface".
in 1915 GR there is zero torsion
De = 0
dB + W/\(1 + B) = 0
That I solve as
W = -*[dB/\(1 - B)]
Note that W depends on dB exactly like F depends on dA, but we have additional nonlinearity in (1 - B) factor from EEP.
"Although, I cannot readily cite you chapter and verse, as I recall, the upshot of the MTW’s discussion is that it is unclear what is the potential (metric or connection) and what is the field strength (connection of curvature) as all three geometric objects, metric g, connection Г and curvature R describe gravitational field, as far as geometrodynamics is concerned."
Sarfatti wrote: I think I have clarified this somewhat above using the Cartan "Feynman diagram" heuristic.
EEP is subtle, but not malicious.
3-manifold integral of *W ~ 4-manifold integral of d*W = 4-manifold integral of( D*W - W/\*W) = -4-manifold integralW/\*W
This also sheds light on the nonlocality of gravity energy as a geometrodynamic Bohm-Aharonov effect. In vacuum, *G = D*W = 0. But DeRham global integral of gravity flux *W (Hodge star dual of the zero-torsion SPIN-CONNECTION) over a non-bounding 3-cycle in space-time need not vanish even though the integrand *G 4-form is locally strictly zero if there is an appropriate topological defect in the vacuum order parameter generating emergent gravity in the first place! If we take a spacelike slice of this factoring out the "time", then it seems this explains the non-zero flux of gravity waves through asymptotic flat 2D cycles that do not bound.
Basically, the nonlocality of the gravity energy is from the nonlinear term W/\*W that comes from the EEP.
Therefore, any attempt to localize gravity energy violates EEP exactly as MTW says it does! This seems very clear heurstically now using the Cartan "Feynman diagrams" as it were.]
"Needless to say, various approaches to General Relativity highlight different roles of these geometric objects. For example, in a context of classical field theory, the role of field potential is played by the metric and the role of field strength is played by the connection, as seen from the Lagrangian. On the other hand, a gauge theory approach always uses connection as a potential arising in Lie derivative leaving the role of field strength to be played by the curvature. I agree, this approach is more in line with Maxwell electrodynamics, which is a gauge theory, but ultimately, this is a matter of semantics. Another example that highlights how inconsequential is the choice of field potential would be the tetrad formulation of GR wherein the tetrads play the role of potentials of gravitational field. However, this non-traditional choice of field potential (instead of the traditional metric) does not in any way undermine the equivalence of the tetrad formulation to the Einstein traditional metric formulation of GR, as emphasized by MTW.
I am ambivalent as to the choice of words. My point was merely to emphasize that both potential and field strength must be covariant geometric objects (tensors) in order to have a fully covariant theory of gravitation. Obviously, metric and curvature are true tensors. The problem is with the connection, which is not. Therefore, whether you consider connection as the potential or as the field strength, it is a poor choice, because connection is not covariant under general coordinate transformation. A much better choice would be the tensor of affine deformation (the difference between two affine connections, which is always a tensor). The tensor of nonmetricity is a special case of a tensor of affine deformation and is a true tensor. Whether you are going to call it a potential or a field strength (or, in your terminology, field intensity), I propose to describe gravity by a triplet of geometrical objects
As we say in the US, call me what you want, just don’t call me late to lunch J. In this vein, call it what you will, just don’t use non-covariant objects to describe the gravitational field. This is the moral of my paper.
I will be studying your other remarks expressed in your letter to me and your paper on the strong equivalence principle as soon as I get some free time.
From: RKiehn2352@aol.com [mailto:RKiehn2352@aol.com]
Sent: Sunday, November 20, 2005 2:27 PM
To: Alex Poltorak; email@example.com; firstname.lastname@example.org
Subject: Re: Physical torsion
Letter to apoltorak (attachment in pdf format)
Non-metricity seems to imply that the potentials are scalar functions,
and forces are related to to the Connections.
I do not buy it.
This is counter to the view that the
connection matrix of 1-forms are the potentials
and the curvature matrix of 2-forms are the forces.
(A fact well subscribed to)
First draft of a strong equivalence principle (attachment in pdf format)