Graviton-graviton scattering, Bel-Robinson and energy (pseudo)-tensors

Deser, S.; Franklin, Jerrold; Seminara, Domenico

doi:10.1088/0264-9381/16/9/301

Cited by 30 publications

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“…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.…”

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confidence: 99%

“…It turns out that when expanded in RNC the lowest non-vanishing vacuum energy-momentum expressions are of the second or-der and are quadratic in the curvature tensor: [14] of all such possible terms (taking into account the Weyl = vacuum Riemann tensor symmetries) shows that they can be written in terms of The Bel-Robinson and some related tensors. The BelRobinson tensor has many well known remarkable properties, see e.g., [14,29]. For our considerations we are interested in it only in the vacuum, where the Riemann tensor reduces to the Weyl tensor.…”

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“…Any linear combinations of the expressions (9), aU A + bU B + cU C + dU D + eU E + f U F , may be added to the Freud expression. The Freud expression (normalized to give the correct asymptotic and small region non-vacuum values) leads to a small region vacuum second order contribution of [6,14,31] …”

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“…The usual requirement is that one should have a positive multiple of the Bel-Robinson tensor [14], which means that the parameter combinations s and v should vanish, and β, the coefficient of B, should be positive. This means, in addition to one inequality, 2 restrictions on the 6 parameters controlling the contributions from (9).…”

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“…One simple choice, a = b = c = d = e = 0 and f = 2/3, yields the one case which had been identified some time ago [14], normalized here to give both the proper small region matter interior value as well as the correct asymptotic limit. This choice results in a small region vacuum energy-momentum density of…”

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Positive small-vacuum-region gravitational-energy expressions

Nester

2009

Phys. Rev. D

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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 ...

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Positive small-vacuum-region gravitational-energy expressions

Nester

2009

Phys. Rev. D

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Some remarks on the Bel‐Robinson tensor

Garecki

2001

Annalen der Physik

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In this paper we present our point of view on the correct physical interpretation of the Bel‐Robinson tensor within the framework of standard General Relativity (GR), i.e., within the framework of GR without supplementary elements like an arbitrary vector field, a distinguished tetrad field or a second metric. We show that this tensor arises as a consequence of the Bianchi identities and, in a natural manner, it is linked to the difference of the canonical gravitational energy‐momentum as calculated in normal coordinates in a small vicinity of their origin P and in the origin P.

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Teleparallelism: A New Insight into Gravity

Pereira

2014

Springer Handbook of Spacetime

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Teleparallel gravity, a gauge theory for the translation group, turns up as fully equivalent to Einstein's general relativity. In spite of this equivalence, it provides a whole new insight into gravitation. It breaks several paradigms related to the geometric approach of general relativity, and introduces new concepts in the description of the gravitational interaction. The purpose of this chapter is to explore some of these concepts, as well as discuss possible consequences for gravitation, mainly those that could be relevant for the quantization of the gravitational field.

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Graviton-graviton scattering, Bel-Robinson and energy (pseudo)-tensors

Cited by 30 publications

References 7 publications

Positive small-vacuum-region gravitational-energy expressions

Positive small-vacuum-region gravitational-energy expressions

Some remarks on the Bel‐Robinson tensor

Teleparallelism: A New Insight into Gravity

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