| Subject: Request for opinion on the Cauchy problem/paradox
Date: Fri, 19 Nov 2004 15:31:51 +0200 From: Dimi Chakalov <dimi@chakalov.net> To: Fred I Cooperstock <cooperstock@uvphys.phys.uvic.ca> Dear Professor Cooperstock, I am respectfully requesting you opinion on the following issue. Please correct me if I'm wrong, and please be assured that I'll keep your feedback strictly private and confidential. Given your hypothesis that the gravitational contribution to energy is "localized" in the non-vacuum regions, i.e., the regions of the non-vanishing energy-momentum tensor [Ref. 1, p. 1068], how are we supposed to formulate the Cauchy problem in GR [Ref. 2]? My understanding of the latter is that we cannot, even in principle, define a global/total dynamics for the solutions of Einstein's equation on the 3-D Cauchy surface, because of some -- here comes my total confusion -- "geodesic incompleteness" that leads inevitably (?) to some closed time curves (CTCs) or singularities. It seems to me that the origin of this problem is that we cannot have a "localized" gravitational contribution to energy nor "localized expression for total angular momentum" [Ref. 1, p. 1081] at ANY point, with the sole exception of the initial/starting point at which we define the Cauchy surface and the supposed "evolution" of matter fields. Perhaps this Cauchy problem is actually a paradox, since we've never encountered any CTCs or naked singularity in the past 13.7 billion years. In the case of a spherically symmetric homogeneous scalar field (the work on the inhomogeneous generalization is in progress; see [Ref. 3]), the formation of naked singularity depends on the initial distribution of matter fields and the geometry of trapped surfaces which may subsequently form as the "collapse" evolves, so we should have already registered at least one naked singularity in the past 13.7 billion years. Similar considerations hold for CTCs [Ref. 4], I believe. If true, what is the crux of this Cauchy paradox? Kindest regards, Dimi Chakalov P.S. I personally am inclined to take the position that *all* conservation laws do not hold in GR (why would anything be "conserved" in curved spacetime?), and re-examine the very idea of 'empty space' as some virtual pool of "dark" mass&energy, but I'm not sure if it is necessary to adopt such an iconoclastic position. D.C.
References [Ref. 1] F.I. Cooperstock, Energy and Angular Momentum of Systems in General Relativity, Foundations of Physics, Vol. 31, No. 7, July 2001, pp. 1067-1082. p. 1068: "However, old problems tend to return to haunt one and this researcher never found comfort in the idea that energy localization cannot be sustained in GR. There remained the hope that eventually, a true resolution of the outstanding problems would be achieved and a clear understanding would emerge. The key to that quest came with the realization that the energy-momentum conservation laws in GR have content only in the regions of the energy-momentum tensor while they lead to a pseudo-tensor that purports to describe flows of energy in vacuum where the laws themselves are devoid of content.[1-3] This convolution might be seen as a natural explanation for the nebulous character of the pseudotensor. This, plus the fact that gravitational plane waves in vacuum can be expressed in KerrSchild form where all components of the pseudotensor vanish globally, led to the hypothesis that the gravitational contribution to energy is localized in the non-vacuum regions, i.e., the regions of the non-vanishing energy-momentum tensor.[1-3] If correct, this has major ramifications. "Gravitational waves would not be carriers of energy in
vacuum. In turn, this raises the question concerning the quantization of
gravity since one would assume that the concept of a graviton demands the
presence of energy. As well, the traditional belief that the transfer of
information is inexorably linked to energy is challenged if gravitational
waves are capable of transmitting information.
p. 1081: "Also to be considered is the connection between information and the transmission of energy and the general view that these are inexorably linked. If gravity waves are both non-energetic and capable of transmitting information, then the information-energy link has to be reexamined. "Yet to be found is a localized expression for total angular
momentum, including the contribution from gravity. This is the next immediate
challenge."
[Ref. 2] Robert Geroch, Gauge, Diffeomorphisms,
Initial-Value Formulation, Etc
"Einstein's equation as it stands does not admit an initial-value
formulation in the traditional sense, precisely because the gauge freedom
prohibits this."
[Ref. 3] Rituparno Goswami and Pankaj
S. Joshi, Naked Singularity formation in scalar field collapse,
gr-qc/0410144;
Rituparno Goswami, Pankaj S. Joshi, Cenalo Vaz, and Louis Witten, A Time-Like
Naked Singularity,
gr-qc/0410041.
[Ref. 4] W.B. Bonnor, Closed timelike curves in general relativity, Int. J. Mod. Phys. D12 (2003) 1705-1708; gr-qc/0211051.
Note: I think studying Fred Cooperstock's papers is a must for every researcher in classical and quantum gravity. Just two examples: Cooperstock F.I., The Role of Energy and a New Approach to Gravitational Waves in General Relativity, Annals Phys. 282 (2000) 115-137; gr-qc/9904046, and Cooperstock F.I. and Tieu S., The Energy of a Dynamical Wave-Emitting System in General Relativity, Found. Phys. 33 (2003) 1033-1059; gr-qc/0302020. Regarding the second paper, read more here. It's such a shame that science can be contaminated with money and politics.
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