Positive small-vacuum-region gravitational-energy expressions

Abstract

We construct an infinite number of new holonomic quasi-local gravitational energy-momentum density pseudotensors with good limits asymptotically and in small regions, both materially and in vacuum. For small vacuum regions they are all a positive multiple of the Bel-Robinson tensor and consequently have positive energy.PACS numbers: 04.20.Cv, 04.20.Fy The localization of energy for gravitating systems has been an important fundamental problem since before Einstein finalized his field equations (see [1]). In recent decades there has been some significant progress on this outstanding problem. New perspectives and ideas on gravitational energy continue to be considered as noteworthy (e.g., [2,3,4,5]). Recall that it is a well-known inevitable consequence of the equivalence principle that there is no covariant (reference frame independent) description for the gravitational energy-momentum density (for a discussion see Ch. 20 in [6]). The coordinate dependent pseudotensor density description used in earlier times has in recent decades largely been replaced by the quasi-local perspective (i.e., energy-momentum is to be associated with a closed 2-surface; for a comprehensive review see [7]). However it has been noted that the Hamiltonian approach to quasi-local energy-momentum includes all the possible pseudotensors simply by taking their associated superpotentials as the Hamiltonian boundary term. This not only shows the pseudotensors to be a special type of quasi-local expression but, moreover, reveals their ambiguities to be just the same as those of the quasi-local Hamiltonian boundary term. The Hamiltonian approach, however, gives these ambiguities a clear physical and geometric meaning. Specifically, the freedom in the choice of expression is associated (via the boundary term in the Hamiltonian variation) with the freedom to choose the type of boundary conditions, and the choice of quasi-local reference (which determines the zero energy or ground state) is effectively just the choice of coordinates on the boundary for the holonomic pseudotensor [8,9,10,11,12].Various criteria have been proposed for quasi-local quantities (see, e.g., [7,13]), including having the correct asymptotic limits at infinity and positivity. Positivity has been regarded as a strong condition. It is difficult to prove the positivity of the quasi-local energy determined by some expression for a general region. Just consid- * Electronic address: s0242010@webmail.tku.edu.tw † Electronic address: nester@phy.ncu.edu.tw ering a small vacuum region already gives a significant requirement, namely that the energy-momentum be proportional to the Bel-Robinson tensor [7,14]. This will guarantee positive energy within a small vacuum region.We have had some hope that positivity within a small vacuum region would be a quite strong restriction, perhaps even selecting a unique expression. We found that it is indeed a serious constraint; in particular the requirement of a positive energy density eliminates all of the classical pseudotensors, nevertheless ...

Quasi-Local Energy-Momentum and Angular Momentum in General Relativity

“…It has been shown that a quadratic expression of these coordinates with the Bel-Robinson tensor as their coefficient appears naturally in the local conservation law for the matter energy-momentum tensor [327]; the Bel-Robinson tensor can be recovered as some ‘double gradient’ of a special combination of the Einstein and the Landau-Lifshitz pseudotensors [170]; Møller’s tetrad expression, as well as certain combinations of several other classical pseudotensors, yield the Bel-Robinson tensor [473, 470, 471]. In the presence of some non-dynamical (background) metric a 11-parameter family of combinations of the classical pseudotensors exists, which, in vacuum, yields the Bel-Robinson tensor [472, 474]. (For this kind of investigation see also [465, 468, 466, 467, 469]).…”

Normal frames for general connections

“…[11][12][13]). For Einstein's general relativity it was possible to infer a suitable orthonormal normal frame (see [14][15][16]) without needing all the machinery developed here.…”

“…In accordance with (10,12,13), at a preselected point we can choose the frame and holonomic basis so that…”

Quasi-local energy from a Minkowski reference

“…3 But none of them gives positive energy for small vacuum regions [15]. An 11-parameter set of new pseudotensor superpotentials with this desirable property was constructed by So [16].…”