A class of conformally Einstein metrics

Kapadia, Devendra; Sparling, George

doi:10.1088/0264-9381/17/22/314

Cited by 4 publications

(4 citation statements)

References 3 publications

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“…As of the time of writing, the author has been able to show by direct calculation that the prototypical space-times of Kapadia & Sparling (2000) are coherent, at least for complex homogeneous input functions, giving the first known example of a curved space-time that is such. Note that this proposed classification is inherently non-perturbative and potentially gives a coordinateindependent definition of dynamical chaos (Cvitanović et al 2004).…”

Section: The X-transformmentioning

confidence: 99%

Germ of a synthesis: space–time is spinorial, extra dimensions are time-like

Sparling

2007

Proc. R. Soc. A.

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A pressing issue for modern physics is the possibility of extra dimensions of space–time. Here, a novel approach to this question is put forward, with three facets: First, an integral transform is introduced into Einstein's general relativity that is non-local and spinorial. For Minkowskian space–time, the transform intertwines three spaces of six dimensions, which a priori are on an equal footing, linked by the octavic triality of Cartan. Two of these spaces are interpreted as null twistor spaces; the third may be regarded as giving space–time two extra time-like dimensions, for which the ordinary space–time is an axis of symmetry. Second, it is suggested that the extra dimensions perdure for a general space–time: the overall structure is controlled by a generalized Fefferman tensor. Accordingly, it is posited that the additional time-like dimensions arise naturally and constitute an aspect of space–time reality that ultimately will be amenable to experimental investigation. Conceivably, devices such as the Large Hadron Collider will uncover this reality. Third, it is argued that the structure hints at a synthesis of ideas deriving from general relativity, string theory, condensed matter physics, category theory and non-commutative geometry.

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Section: The X-transformmentioning

confidence: 99%

Germ of a synthesis: space–time is spinorial, extra dimensions are time-like

Sparling

2007

Proc. R. Soc. A.

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“…Riemannian manifolds (M, g) for which there exists a pointwise conformal deformation g = e 2u g, u ∈ C ∞ (M ), such that the new metric g is Einstein. This problem has received a considerable amount of attention by mathematicians and physicists in the last decades: just to mention some old and recent papers we cite the pioneering work of Brinkmann, [4], Yano and Nagano, [35], Gover and Nurowski, [17], Kapadia and Sparling, [21], Derdzisnki and Maschler, [15], and references therein. In particular in [17] the authors describe two necessary integrability conditions for the existence of the conformal deformation g realizing the Einstein metric.…”

Section: Introductionmentioning

confidence: 99%

Conformal Ricci solitons and related integrability conditions

Catino

Mastrolia

Monticelli

et al. 2016

Advances in Geometry

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Abstract. In this paper we introduce, in the Riemannian setting, the notion of conformal Ricci soliton, which includes as particular cases Einstein manifolds, conformal Einstein manifolds and (generic and gradient) Ricci solitons. We provide here some necessary integrability conditions for the existence of these structures that also recover, in the corresponding contexts, those already known in the literature for conformally Einstein manifolds and for gradient Ricci solitons. A crucial tool in our analysis is the construction of some appropriate and highly nontrivial (0, 3)-tensors related to the geometric structures, that in the special case of gradient Ricci solitons become the celebrated tensor D recently introduced by Cao and Chen. A significant part of our investigation, which has independent interest, is the derivation of a number of commutation rules for covariant derivatives (of functions and tensors) and of transformation laws of some geometric objects under a conformal change of the underlying metric.

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“…There is only one non-conformally flat case where the relevant calculations have been carried out in detail, that of the Kapadia [15] metric:…”

Section: Introductionmentioning

confidence: 99%

Causal geometries, null geodesics, and gravity

Holland¹,

Sparling²

2011

Preprint

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The authors study a generalized notion of null geodesic defined by the Legendrian dynamics of a regular conical subbundle of the tangent bundle on a manifold. A natural extension of the Weyl tensor is shown to exist, and to depend only on this conical subbundle. Given a suitable defining function of the conical bundle, the Raychaudhuri-Sachs equations of general relativity continue to hold, and give rise to the same phenomenon of covergence of null geodesics in regions of positive energy that underlies the theory of gravitation.

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A class of conformally Einstein metrics

Cited by 4 publications

References 3 publications

Germ of a synthesis: space–time is spinorial, extra dimensions are time-like

Germ of a synthesis: space–time is spinorial, extra dimensions are time-like

Conformal Ricci solitons and related integrability conditions

Causal geometries, null geodesics, and gravity

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