Dimensionally dependent tensor identities by double antisymmetrization

Edgar, S. Brian; Höglund, A.

doi:10.1063/1.1425428

Cited by 54 publications

(84 citation statements)

References 26 publications

(63 reference statements)

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“…which is divergence free and identically zero in 4 dimensions giving us in this case the first Lovelock identity (for extensions see for example [14][15][16]. In four dimensions the action involving the Gauss Bonnet invariant when coupled with a scalar field is no longer a topological invariant.…”

Section: Introductionmentioning

confidence: 99%

Lovelock Galileons and black holes

Charmousis

Tsoukalas

2015

Phys. Rev. D

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We study a scalar-tensor version of Lovelock theory with a non trivial higher order galileon term involving the coupling of the Lovelock two tensor with derivatives of the scalar galileon field. For a static and spherically symmetric spacetime we extend the Boulware-Deser solution to the presence of a Galileon field. The hairy solution has a regular scalar field on the black hole event horizon and presents certain self tuning properties for the bulk cosmological constant and the Gauss-Bonnet coupling. The combined time and radial dependence of the galileon field permits its horizon regularity. Furthermore in order to investigate the effects of linear time dependence we find spherically symmetric solutions in 4 and 5 spacetime dimensions. They are shown to have singular horizons. Afar from the Schwarzschild radius and for weak higher dimensional couplings the solutions are perturbatively close to GR representing exteriors of GR like star solutions for scalar tensor theories.

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Section: Introductionmentioning

confidence: 99%

Lovelock Galileons and black holes

Charmousis

Tsoukalas

2015

Phys. Rev. D

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“…Much earlier, Lovelock [14] had pointed out that a number of apparently unrelated results were all really consequences of a class of identities which he christened dimensionally dependent identities -identities which are a trivial, but subtle, consequence of dimension alone. Recently Edgar and Höglund [15] have generalised Lovelock's results, and demonstrated that the underlying principle in all of these investigations in [6 -13], and some new ones, was the explicit exploitation of dimensionally dependent identities. Furthermore, in algebraic Rainich theory, Bergqvist and Höglund [16] have exploited these ideas further, and obtained results in five dimensions involving cubic terms in the energy momentum tensor -motivated by the familiar results in four dimensions involving quadratic terms, [17]; while Edgar and Höglund [18] have demonstrated the crucial role that dimensionally dependent identities play in the existence of the Lanczos potential for the Weyl tensor in different dimensions.…”

Section: Introductionmentioning

confidence: 99%

“…In four dimensions, obviously it cannot be sufficient simply to substitute n = 4, since we know from the spinor version that the right hand side must disappear completely in four dimensions. But if we consider the 4-dimensional fddi [14,15] (quoted in Lemma 3 at the end of this section), we find that, when contracted with L de f , we obtain the ddi,…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, in algebraic Rainich theory, Bergqvist and Höglund [16] have exploited these ideas further, and obtained results in five dimensions involving cubic terms in the energy momentum tensor -motivated by the familiar results in four dimensions involving quadratic terms, [17]; while Edgar and Höglund [18] have demonstrated the crucial role that dimensionally dependent identities play in the existence of the Lanczos potential for the Weyl tensor in different dimensions. Deser [19] has applied the adjective 'ubiquitous' to the Bel-Robinson tensor, and this description is equally appropriate to dimensionally dependent identities as can be seen by the wide range of the investigations in [6 -13], the applications given by Lovelock [14], and the more recent applications in [15,16,18]. In fact, Deser has argued elsewhere [6], that two identities (which are examples of what we call dimensionally dependent identities in four dimensions) are in a sense implicit in the familiar definitions of the Bel-Robinson tensor in four dimensions.…”

Section: Introductionmentioning

confidence: 99%

“…However, in these particular cases, signature independent versions were obtained by use of 4-dimensional tensor fddis. in [13,15] and [11] respectively. ‡ Identity (1) is quoted in [21] with a reference to a proof by Debever in [22]; this proof is a lot more complicated than the spinor proof in [2].…”

Section: Introductionmentioning

confidence: 99%

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

Edgar

Wingbrant

2003

Journal of Mathematical Physics

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Abstract.It is known that some results for spinors, and in particular for superenergy spinors, are much less transparent and require a lot more effort to establish, when considered from the tensor viewpoint. In this paper we demonstrate how the use of dimensionally dependent tensor identities enables us to derive a number of 4-dimensional identities by straightforward tensor methods in a signature independent manner. In particular, we consider the quadratic identity for the Bel-Robinson tensor T abcx T abcy = δ y x T abcd T abcd /4 and also the new conservation laws for the Chevreton tensor, both of which have been obtained by spinor means; both of these results are rederived by tensor means for 4-dimensional spaces of any signature, using dimensionally dependent identities, and also we are able to conclude that there are no direct higher dimensional analogues. In addition we demonstrate a simple way to show non-existense of such identities via counter examples; in particular we show that there is no non-trivial Bel tensor analogue of this simple Bel-Robinson tensor quadratic identity. On the other hand, as a sample of the power of generalising dimensionally dependent tensor identities from four to higher dimensions, we show that the symmetry structure, trace-free and divergence-free nature of the four dimensional Bel-Robinson tensor does have an analogue for a class of tensors in higher dimensions. It is reassuring to know that certain important but perhaps unexpected properties -disguised in the complexities of the tensor formalism -become more transparent in the spinor formalism; but the parallel and more transparent spinor investigations are restricted to 4-dimensional spacetimes with Lorentz signature, and so this assistance is not available in higher dimensions nor in four dimensional spaces with other signatures. Deser [6] has emphasised the significance of the Bel-Robinson tensor in higher dimensions, and one of the important features of Senovilla's method of construction of superenergy tensors [3] is that it is applicable to arbitrary fields in any dimension; and so, for higher dimensions, it becomes an obvious concern whether there could be unexpected properties for superenergy tensors -disguised in the even deeper complexities of tensor formalism in higher dimensions -analogous to those properties revealed by spinor formalism in four dimensions. Deeper investigations into the interaction between dimension and tensor identities have been instrumental in illustrating the uniqueness of some of the Bel-Robinson tensor's properties in four dimensions [6], explaining

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From Lovelock to Horndeski’s Generalized Scalar Tensor Theory

Charmousis

2014

Lecture Notes in Physics

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We review and discuss some recent progress in Lovelock and Horndeski theories modifying Einstein's General Relativity. Using as our guide the uniqueness properties of these modified gravity theories we then discuss how Kaluza-Klein reduction of Lovelock theory can lead to effective scalar-tensor actions including several important terms of Horndeski theory. We show how this can be put to practical use by mapping analytic black hole solutions of one theory to the other. We then elaborate on the subset of Horndeski theory that has self-tuning properties and review a generic method giving scalar-tensor black hole solutions.

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Dimensionally dependent tensor identities by double antisymmetrization

Cited by 54 publications

References 26 publications

Lovelock Galileons and black holes

Lovelock Galileons and black holes

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

From Lovelock to Horndeski’s Generalized Scalar Tensor Theory

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