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Ph. Spindel

Publications and source records attributed to Ph. Spindel.

At least 19 recordsLinked to original sources
arXiv2013

Solutions of massive gravity theories in constant scalar invariant geometries

We solve massive gravity field equations in the framework of locally homogenous and vanishing scalar invariant (VSI) Lorentzian spacetimes, which in three dimensions are the building blocks of constant scalar invariant (CSI) spacetimes. At first, we provide an exhaustive list of all Lorentzian three-dimensional homogeneous spaces and then we determine the Petrov type of the relevant curvature tensors. Among these geometries we determine for which values of their structure constants they are solutions of the field equations of massive gravity theories with cosmological constant. The homogeneous solutions founded are of all various Petrov types : I_C, I_R, II, III, D_t, D_s, N, O; the VSI geometries which we found are of Petrov type III. The Petrov types II and III are new explicit CSI spacetimes solutions of these types. We also examine the conditions under which the obtained anti-de Sitter solutions are free of tachyonic massive graviton modes.

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arXiv2010

G3-homogeneous gravitational instantons

We provide an exhaustive classification of self-dual four-dimensional gravitational instantons foliated with three-dimensional homogeneous spaces, i.e. homogeneous self-dual metrics on four-dimensional Euclidean spaces admitting a Bianchi simply transitive isometry group. The classification pattern is based on the algebra homomorphisms relating the Bianchi group and the duality group SO(3). New and general solutions are found for Bianchi III.

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arXiv2009

Gravitational instantons, self-duality and geometric flows

We discuss four-dimensional "spatially homogeneous" gravitational instantons. These are self-dual solutions of Euclidean vacuum Einstein's equations with potentially non-vanishing cosmological constant. They are endowed with a product structure R \times M_3 leading to a natural foliation into three-dimensional subspaces evolving in Euclidean time. For a large class of three-dimensional subspaces, the dynamics coincides with the geometric flow on the three-dimensional homogeneous slice, driven by the Ricci tensor plus an so(3) gauge connection. The metric on the three-dimensional space is related to the vielbein of the three-dimensional subspace, while the gauge field is inherited from the anti-self-dual component of the four-dimensional Levi--Civita connection.

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arXiv2008

A Special Class of Rank 10 and 11 Coxeter Groups

In the course of investigating regular subalgebras of E(10) related to cosmological solutions of 11-dimensional supergravity supporting an electric 4-form field, a class of rank 10 Coxeter subgroups of the Weyl group of E(10) was uncovered (hep-th/0606123). These Coxeter groups all share the property that their Coxeter graphs have incidence index 3, i.e. that each node is incident to three and only three single lines. Furthermore, the Coxeter exponents are either 2 or 3, but never infinity. We here go beyond subgroups of the Weyl group of E(10) and classify all rank 10 Coxeter groups with these properties. We find 21 distinct Coxeter groups of which 7 were already described in hep-th/0606123. Moreover, we extend the classification to the rank 11 case and we find 252 inequivalent rank 11 Coxeter groups with incidence index 4, of which at least 28 can be regularly embedded into E(11).

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arXiv2005

BTZ black holes, WZW models and noncommutative geometry

This note is based on a talk given by one of the authors (S. D.) at the "Rencontres Mathématiques de Glanon", held in Glanon in July 2004. We will first introduce the BTZ black hole, solution of Einstein's gravity in 2+1 dimensions, and emphasize some remarkable properties of its geometry. We will essentially pay attention to the non-rotating black hole, whose structure is significantly different to the generic case. We will then turn the some aspects of string theory, namely the emergence of non-commutative geometry and the embedding of the BTZ black hole as an exact string background using the Wess-Zumino-Witten (WZW) model. We will show the existence of winding symmetric WZW D1-branes in this space-time from the geometrical properties of the non-rotating black hole. Finally, we will introduce strict deformations of these spaces, yielding an example of non-commutative lorentzian non-compact space, with non-trivial causal structure.

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arXiv2003

Global geometry of the 2+1 rotating black hole

The generic rotating BTZ black hole, obtained by identifications in AdS3 space through a discrete subgroup of its isometry group, is investigated within a Lie theoretical context. This space is found to admit a foliation by two-dimensional leaves, orbits of a two-parameter subgroup of SL(2,R) and invariant under the BTZ identification subgroup. A global expression for the metric is derived, allowing a better understanding of the causal structure of the black hole.

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arXiv2002

About maximally localized states in quantum mechanics

We analyze the emergence of a minimal length for a large class of generalized commutation relations, preserving commutation of the position operators and translation invariance as well as rotation invariance (in dimension higher than one). We show that the construction of the maximally localized states based on squeezed states generally fails. Rather, one must resort to a constrained variational principle.

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arXiv2002

Regular Poisson structures on massive non-rotating BTZ black holes

We revisit the non-rotating massive BTZ black hole within a pseudo-Riemannian symmetric space context. Using classical symmetric space techniques we find that every such space intrinsically carries a regular Poisson structure whose symplectic leaves are para-hermitian symmetric surfaces. We also obtain a global expression of the metric yielding a dynamical description of the black hole from its initial to its final singularity.

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arXiv2001

Uniqueness of the asymptotic AdS3 geometry

We explicitly show that in (2+1) dimensions the general solution of the Einstein equations with negative cosmological constant on a neigbourhood of timelike spatial infinity can be obtained from BTZ metrics by coordinate transformations corresponding geometrically to deformations of their spatial infinity surface. Thus, whatever the topology and geometry of the bulk, the metric on the timelike extremities is BTZ.

gr-qc↗
arXiv2000

Holonomies, anomalies and the Fefferman-Graham ambiguity in AdS3 gravity

Using the Chern-Simon formulation of (2+1) gravity, we derive, for the general asymptotic metrics given by the Fefferman-Graham-Lee theorems, the emergence of the Liouville mode associated to the boundary degrees of freedom of (2+1) dimensional anti de Sitter geometries. Holonomies are described through multi-valued gauge and Liouville fields and are found to algebraically couple the fields defined on the disconnected components of spatial infinity. In the case of flat boundary metrics, explicit expressions are obtained for the fields and holonomies. We also show the link between the variation under diffeomorphisms of the Einstein theory of gravitation and the Weyl anomaly of the conformal theory at infinity.

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arXiv2000

Quantum charged fields in Rindler space

We study, using Rindler coordinates, the quantization of a charged scalar field interacting with a constant, external, electric field. First we establish the expression of the Schwinger vacuum decay rate, using the operator formalism. Then we rederive it in the framework of the Feynman path integral method. Our analysis reinforces the conjecture which identifies the zero winding sector of the Minkowski propagator with the Rindler propagator. Moreover we compute the expression of the Unruh's modes that allow to make connection between Minkowskian and Rindlerian quantization scheme by purely algebraic relations. We use these modes to study the physics of a charged two level detector moving in an electric field whose transitions are due to the exchange of charged quanta. In the limit where the Schwinger pair production mechanism of the exchanged quanta becomes negligible we recover the Boltzman equilibrium ratio for the population of the levels of the detector. Finally we explicitly show how the detector can be taken as the large mass and charge limit of an interacting fields system.

gr-qc↗
arXiv2000

The Fefferman-Graham Ambiguity and AdS Black Holes

Asymptotically anti-de Sitter space-times in pure gravity with negative cosmological constant are described, in all space-time dimensions greater than two, by classical degrees of freedom on the conformal boundary at space-like infinity. Their effective boundary action has a conformal anomaly for even dimensions and is conformally invariant for odd ones. These degrees of freedom are encoded in traceless tensor fields in the Fefferman-Graham asymptotic metric for any choice of conformally flat boundary and generate all Schwarzschild and Kerr black holes in anti-de Sitter space-time. We argue that these fields describe components of an energy-momentum tensor of a boundary theory and show explicitly how this is realized in 2+1 dimensions. There, the Fefferman-Graham fields reduce to the generators of the Virasoro algebra and give the mass and the angular momentum of the BTZ black holes. Their local expression is the Liouville field in a general curved background.

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arXiv1998

(2+1) dimensional stars

We investigate, in the framework of (2+1) dimensional gravity, stationary, rotationally symmetric gravitational sources of the perfect fluid type, embedded in a space of arbitrary cosmological constant. We show that the matching conditions between the interior and exterior geometries imply restrictions on the physical parameters of the solutions. In particular, imposing finite sources and absence of closed timelike curves privileges negative values of the cosmological constant, yielding exterior vacuum geometries of rotating black hole type. In the special case of static sources, we prove the complete integrability of the field equations and show that the sources' masses are bounded from above and, for vanishing cosmological constant, generally equal to one. We also discuss and illustrate the stationary configurations by explicitly solving the field equations for constant mass--energy densities. If the pressure vanishes, we recover as interior geometries Godel like metrics defined on causally well behaved domains, but with unphysical values of the mass to angular momentum ratio. The introduction of pressure in the sources cures the latter problem and leads to physically more relevant models.

gr-qc↗
arXiv1998

Mimimal Length Uncertainty Principle and the Transplanckian Problem of Black Hole Physics

The minimal length uncertainty principle of Kempf, Mangano and Mann (KMM), as derived from a mutilated quantum commutator between coordinate and momentum, is applied to describe the modes and wave packets of Hawking particles evaporated from a black hole. The transplanckian problem is successfully confronted in that the Hawking particle no longer hugs the horizon at arbitrarily close distances. Rather the mode of Schwarzschild frequency $ω$ deviates from the conventional trajectory when the coordinate $r$ is given by $| r - 2M|\simeq β_H ω/ 2 π$ in units of the non local distance legislated into the uncertainty relation. Wave packets straddle the horizon and spread out to fill the whole non local region. The charge carried by the packet (in the sense of the amount of "stuff" carried by the Klein--Gordon field) is not conserved in the non--local region and rapidly decreases to zero as time decreases. Read in the forward temporal direction, the non--local region thus is the seat of production of the Hawking particle and its partner. The KMM model was inspired by string theory for which the mutilated commutator has been proposed to describe an effective theory of high momentum scattering of zero mass modes. It is here interpreted in terms of dissipation which gives rise to the Hawking particle into a reservoir of other modes (of as yet unknown origin). On this basis it is conjectured that the Bekenstein--Hawking entropy finds its origin in the fluctuations of fields extending over the non local region.

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arXiv1998

Godel metric as a squashed anti-de Sitter geometry

We show that the non flat factor of the Godel metric belongs to a one parameter family of 2+1 dimensional geometries that also includes the anti-de Sitter metric. The elements of this family allow a generalization a la Kaluza-Klein of the usual 3+1 dimensional Godel metric. Their lightcones can be viewed as deformations of the anti-de Sitter ones, involving tilting and squashing. This provides a simple geometric picture of the causal structure of these space-times, anti-de Sitter geometry appearing as the boundary between causally safe and causally pathological spaces. Furthermore, we construct a global algebraic isometric embedding of these metrics in 4+3 or 3+4 dimensional flat spaces, thereby illustrating in another way the occurrence of the closed timelike curves.

gr-qc↗
arXiv1997

Chiral supersymmetric pp-wave solutions of IIA supergravity

We describe solutions of type IIA (N=2, D=10) supergravity built under the assumption of the existence of at least one residual chiral supersymmetry. Their geometry is of pp-wave type. Explicit parametrization of the metric and matter field components, in terms of Killing spinors and arbitrary functions, is provided.

hep-th↗
arXiv1997

Interacting Charged Particles in an Electric Field and the Unruh Effect

We compute the transition amplitudes between charged particles of mass $M$ and $m$ accelerated by a constant electric field and interacting by the exchange of quanta of a third field. We work in second quantization in order to take into account both recoil effects induced by transitions and the vacuum instability of the charged fields. In spite of both effects, when the exchanged particle is neutral, the equilibrium ratio of the populations is simply $\exp(π(M^2 - m^2)/eE)$. Thus, in the limit $(M-m)/M \to 0$, one recovers Unruh's result characterized by the temperature $a/2π$ where $a$ is the acceleration. When the exchanged particle is charged, its vacuum instability prevents a simple description of the equilibrium state. However, in the limit wherein the charge of the exchanged particle tends to zero, the equilibrium distribution is once more Boltzmanian, but characterized not only by a temperature but also by the electric potential felt by the exchanged particle. This work therefore confirms that thermodynamics in the presence of horizons does not rely on a semi-classical treatment. The relationship with thermodymanics of charged black holes is stressed.

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