Old and new results for superenergy tensors from dimensionally dependent tensor identities

Edgar, S. Brian; Wingbrant, Ola

doi:10.1063/1.1624094

Cited by 8 publications

(13 citation statements)

References 27 publications

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“…If a tensorial proof of our result can be found, then one may consider generalizations to arbitrary dimension or arbitrary signature of the metric. Various generalizations of the contracted identity (3) were given in [4]. There are many other possible generalizations.…”

Section: Discussionmentioning

confidence: 99%

Unique characterization of the Bel–Robinson tensor

Bergqvist

Lankinen

2004

Class. Quantum Grav.

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Section: Discussionmentioning

confidence: 99%

Unique characterization of the Bel–Robinson tensor

Bergqvist

Lankinen

2004

Class. Quantum Grav.

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“…The remaining components of y ab not included in y ab n b are related to curvature jumps via differentiation as well, as in (13). Again (14) can be seen to be redundant, being a linear combination of (13) and (12) using (94).…”

Section: Remarks and Alternative Forms Of The Field Equationsmentioning

confidence: 99%

“…In summary, (13) and (12) can be taken as the fundamental set of equations in all possible cases. There are several interesting and more useful ways to write down these two fundamental equations.…”

Section: Remarks and Alternative Forms Of The Field Equationsmentioning

confidence: 99%

“…This follows because the rigged connection ∇ Σ is not volume preserving in general [39,29]. Define a volume element m-form on Σ bȳ Despite its simplicity and obvious interpretation, relations (51) and (52) are just a small subset of the whole set of field equations on the shell, given by (12)(13)(14). For instance, (14) can be rewritten, on using (25), like…”

Section: Analysis Of the Shell Field Equationsmentioning

confidence: 99%

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Equations for general shells

Senovilla

2018

J. High Energ. Phys.

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The complete set of (field) equations for shells of arbitrary, even changing, causal character are derived in arbitrary dimension. New equations that seem to have never been considered in the literature emerge, even in the traditional cases of everywhere non-null, or everywhere null, shells. In the latter case there arise field equations for some degrees of freedom encoded exclusively in the distributional part of the Weyl tensor. For non-null shells the standard Israel equations are recovered but not only, the additional relations containing also relevant information. The results are applicable to a widespread literature on domain walls, branes and braneworlds, gravitational layers, impulsive gravitational waves, and the like. Moreover, they are of a geometric nature, and thus they can be used in any theory based on a Lorentzian manifold.In [29], the questions of the junction conditions and of the existence of thin shells supported on hypersurfaces of arbitrary, even changing, causal character were addressed. Such hypersurfaces or shells are called general. The importance of dealing with hypersurfaces with possibly changing signature resides in the fact that they actually appear in many physical situations. Furthermore, because they are the generic type of hypersurface in any given spacetime. Some examples of relevant general hypersurfaces are, for instance, the spherically symmetric apparent horizon in Vaidya's spacetime -if the radiation stops at some times; more generally, dynamical horizons which eventually merge with the event horizon in asymptotically flat black-hole spacetimes; any achronal boundary, that is to say, the boundary of any future set [16]; Kerr's stationary limit surface, which is timelike everywhere but tangent to the event horizon at two lines intersecting the axis of symmetry, where it is null [16,48]; the interfaces arising in phase transitions if they occur in a concentrated spatial region and then propagate causally; signature changing braneworlds; and there are many others.Thus, for instance, if one wished to analyze the possibility of placing a surface layer on the stationary limit surface of a Kerr black hole, this should be done on a general, signaturechanging, hypersurface. Similarly, descriptions of signature change in the brane scenario -if we happen to live in a brane of a higher-dimensional world-are described by general shells, see [30,31,32]. The case of achronal boundaries is of particular interest because intuitive arguments lead to the expectation that an initial value problem for the gravitational field equations on such boundaries is well posed, see the interesting discussion in [27]. In addition to the above, a unifying framework for all type of shells, independently of their causal character, is desirable, and this may be achieved by describing general shells.The purpose of this paper is triple: (i) to find the full set of equations for general shells, (ii) to do it in arbitrary spacetime dimension, and (iii) the completion of known results for constant-signature shell...

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“…The inbuilt four-dimensional simplicity for spinors has its counterpart in the much more complicated exploitation of four-dimensional identities in tensors [10]. In the above analysis it is clear that the four-dimensional identity (7) plays a crucial role in the tensor proof of part (b) of the BES result (as it also does in part (a)); in addition the four-dimensional Rainich identity is essential to simplify (5) in the proof of part (b).…”

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confidence: 99%