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Register a new userGeneral Relativitys energy and positivity -- a brief history
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I give a brief review of the search for a proper definition of energy in General Relativity (GR), a far from trivial quest, which was only completed after four and a half decades. The equally (or perhaps more) difficult task of establishing its positivity -- it was to take another fifteen plus years -- will then be summarized. Extension to cosmological GR is included. Mention is made of some recent offshoots.
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Gurzadyan-Xue Dark Energy was derived in 1986 (twenty years before the paper of Gurzadyan-Xue). The paper by the present author, titled The Planck Length as a Cosmological Constant, published in Astrophysics Space Science, Vol. 127, p.133-137, 1986 contains the formula claimed to have been derived by Gurzadyan-Xue (in 2003).
During the First World War, the status of energy conservation in general relativity was one of the most hotly debated questions surrounding Einsteins new theory of gravitation. His approach to this aspect of general relativity differed sharply from another set forth by Hilbert, even though the latter conjectured in 1916 that both theories were probably equivalent. Rather than pursue this question himself, Hilbert chose to charge Emmy Noether with the task of probing the mathematical foundations of these two theories. Indirect references to her results came out two years later when Klein began to examine this question again with Noethers assistance. Over several months, Klein and Einstein pursued these matters in a lengthy correspondence, which culminated with several publications, including Noethers now famous paper Invariante Variationsprobleme. The present account focuses on the earlier discussions from 1916 involving Einstein, Hilbert, and Noether. In these years, a Swiss student named R.J. Humm was studying relativity in Gottingen, during which time he transcribed part of Noethers lost manuscript on Hilberts invariant energy vector. By making use of this 9-page manuscript, it is possible to reconstruct the arguments Noether set forth in response to Hilberts conjecture. Her results turn out to be closely related to the findings Klein published two years later, thereby highlighting, once again, how her work significantly deepened contemporary understanding of the mathematical underpinnings of general relativity.
I review the meaning of General Relativity (GR), viewed as a dynamical field, rather than as geometry, as effected by the 1958-61 anti-geometrical work of ADM. This very brief non-technical summary, is intended for historians.
A classic problem in general relativity, long studied by both physicists and philosophers of physics, concerns whether the geodesic principle may be derived from other principles of the theory, or must be posited independently. In a recent paper [Geroch & Weatherall, The Motion of Small Bodies in Space-Time, Comm. Math. Phys. (forthcoming)], Bob Geroch and I have introduced a new approach to this problem, based on a notion we call tracking. In the present paper, I situate the main results of that paper with respect to two other, related approaches, and then make some preliminary remarks on the interpretational significance of the new approach. My main suggestion is that tracking provides the resources for eliminating point particles---a problematic notion in general relativity---from the geodesic principle altogether.
We adopt in this work the idea that the building blocks of the visible Universe belong to a class of the irreducible representations of the Poincare group of transformations (the things) endowed with classificatory quantum numbers (the properties). After a discussion of this fundamentality, the question of the nature of both dark components of the Universe which are deemed necessary, but have not been observed, is analyzed within this context. We broadly discuss the ontology of dark matter/dark energy in relation to the irreducible representations of the Poincare group + quantum numbers, pointing out some cases in which the candidates can be associated to them, and others for which a reclassification of both the dark and visible (ordinary) components would be needed.
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