We show that, in the application of Riemannian geometry to gravity, there exists a superpotential Vij of the Riemann-Christoffel tensor which is the tensor generalization of Poisson's classical potential. Leaving open the question of a zero on nonzero rest mass k of the graviton we show that, in the latter case, k2 Vij is an energy momentum density, or “Maxwell-like tensor,” of the gravity field itself, adding to the “material tensor” in the right-hand sides of both the (generalized) Poisson equation and the Einstein gravity equation, but that, nevertheless, Einstein's requirement of geodesic motion of a point particle is rigorously preserved.
Two interesting possibilities are thus opened: a tentative explanation of the cosmological “missing mass” and quantization of the Riemannian gravity field along a standard procedure.
O. Costa de Beauregard, "Massless or massive graviton?" 3 Foundations of Physics Letters 81-85 (1990).
3 comments:
This paper has two citations, in papers by the author that themselves have zero citations. All three papers came at the end of his career, they were among the very last papers he ever wrote.
The main ideas seem to be, add a very small mass to the graviton. He thinks this could add missing mass to the universe, and also allow a local definition of energy.
I suspect an actual expert in quantum gravity (familiar with the past several decades' work on field theory approaches) would be able to offer a swift evaluation of whether those ideas are promising or even sensible...
@Mitchell
Your points of skepticism are sound. But, the concept is nonetheless interesting and deserved a mention.
A related issue in my mind is whether existing classical GR models in astronomy where you are using Newtonian gravity as an approximation of weak field GR are adequately reflecting the fact that photons have mass-energy, even if they don't have rest mass. GR acknowledges this and it comes into play in cosmology work with GR and other GR strong field applications, but it isn't clear to me that the Newtonian practice of treating photons and gravitons as having zero gravitational mass-energy pull of their own is valid in galaxy scale or larger astronomy applications.
One reason that massive graviton theories may be replicating dark matter is that while gravitons should have zero rest mass, they may have mass-energy sufficient to make ignoring it a problem.
I've tried to get a better handle on this question from reading a GR textbook (MTW) and reading other works on the subject, but I admit to still being a bit fuzzy on the basis for the supposed equivalence of the self-interactions of the gravitational field via the LHS of Einstein's equations of GR and the widely stated proposition that Einstein's equations of GR are equivalent to those of a spin-2 graviton with no rest mass that interacts with every particle with a coupling strength proportional to the mass-energy of that particle. The statement can't be literally true, because the physics of quantum particles are distinguishable in principle from the physics of classically formulated GR (e.g. re the localization of gravitational energy, stochastic v. deterministic character, etc.). But without understanding the basis for the proposition better than I do, it is hard to understand how we know that they would be equivalent in some appropriate limit.
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