arXiv subjects

Philippe Spindel

Publications and source records attributed to Philippe Spindel.

At least 19 recordsLinked to original sources

arXiv2025

Photon rings in a holographic toy model

Light circling around an astrophysical black hole can spend a long time skirting its unstably bound photon orbits before escaping to infinity. To a distant observer, this orbiting light would appear as a bright ring encircling the image of the black hole. Though not yet resolved by radio-interferometric observations from the ground, this ``photon ring'' will be the target of future space-based black hole observations. Motivated by this experimental prospect, studies have sought to elucidate the theoretical connections between the photon ring -- an observable, classical effect -- and the putative holographic description of black holes in quantum gravity. General relativity predicts that the detailed structure of the photon ring encodes the high-frequency (eikonal) spectrum of quasinormal modes (QNMs) emitted by a perturbed black hole as it rings down, and also that the photon ring displays an emergent conformal symmetry that acts upon this spectrum. In holography, the classical QNM frequencies are expected to map to Ruelle resonances of the dual quantum theory. In this paper, we explore these connections in a lower-dimensional toy model based on Warped AdS$_3$ black holes that shares many features with the (3+1)-dimensional Kerr background -- including a photon ring at finite radius -- while still providing analytic control of the QNM frequencies.

hep-th↗

arXiv2024

Aspects of Warped AdS$_3$ geometries

We discuss the geometry of three-dimensional warped Anti-de Sitter spaces and quotients thereof, paying special attention to their underlying group manifold nature. We perform a systematic analysis of warped Anti-de Sitter geometries, focusing on their global properties and illustrating their occurrence as special solutions of various three-dimensional gravity theories.

gr-qc↗

arXiv2022

Revisiting the name variant of the two-children problem

Initially proposed by Martin Gardner in the 1950s, the famous two-children problem is often presented as a paradox in probability theory. A relatively recent variant of this paradox states that, while in a two-children family for which at least one child is a girl, the probability that the other child is a boy is $2/3$, this probability becomes $1/2$ if the first name of the girl is disclosed (provided that two sisters may not be given the same first name). We revisit this variant of the problem and show that, if one adopts a natural model for the way first names are given to girls, then the probability that the other child is a boy may take any value in $(0,2/3)$. By exploiting the concept of Schur-concavity, we study how this probability depends on model parameters.

math.PR↗

arXiv2022

Hidden Kac-Moody Structures in the Fermionic Sector of Five-Dimensional Supergravity

We study the supersymmetric quantum dynamics of the cosmological models obtained by reducing $D=5$ supergravity to one timelike dimension. This consistent truncation has fourteen bosonic degrees of freedom, while the quantization of the homogeneous gravitino field leads to a $2^{16}$--dimensional fermionic Hilbert space. We construct a consistent quantization of the model in which the wave function of the Universe is a $2^{16}$--component spinor %\textcolor{red}{of Spin(24,8)} depending on fourteen continuous coordinates, which satisfies eight Dirac-like wave equations (supersymmetry constraints) and one Klein-Gordon-like equation (Hamiltonian constraint). The fermionic part of the quantum Hamiltonian is built from operators that generate a $2^{16}$-dimensional representation of the (infinite-dimensional) maximally compact sub-algebra $K(G_2^{++})$ of the rank-4 hyperbolic Kac--Moody algebra $G_2^{++}$. The (quartic-in-fermions) squared-mass term $\widehat \mu^2$ entering the Klein-Gordon-like equation has several remarkable properties: (i) it commutes with the generators of $K(G_2^{++})$; and (ii) it is a quadratic polynomial in the fermion number $N_F \sim \overline\Psi \Psi$, and a symplectic fermion bilinear $C_F \sim \Psi C\Psi$. Some aspects of the structure of the solutions of our model are discussed, and notably the Kac-Moody meaning of the operators describing the reflection of the wave function on the fermion-dependent potential walls ("quantum fermionic Kac-Moody billiard").

hep-th↗

arXiv2021

Comment on "Spherically symmetric perturbations of a Schwarzschild black hole in torsion bigravity"

The goal of this short note is to provide a simpler derivation of the effective potential surrounding a Schwarzschild black hole for spherically symmetric perturbations in the framework of torsion bigravity than the one presented in [V. Nikiforova, Spherically symmetric perturbations of a Schwarzschild black hole in torsion bigravity, Phys. Rev. D 104, 024032 (2021).]. We also discuss the unicity of the reduction process that leads to the potential.

gr-qc↗

arXiv2021

Dynamical torsion gravity backgrounds

We write the field equations of torsion gravity theories and the N\oe ther identity they obey directly in terms of metric and contorsion tensor components expressed with respect to natural coordinates, i.e. without using vierbien but Lagrange multipliers. Then we obtain explicit solutions of these equations, under specific ans\"atze for the contorsion field, by assuming the metric to be respectively of the Bertotti-Robinson, pp-wave, Friedmann-Lema\^itre-Robertson-Walker or static spherically symmetric type. Among these various solutions we obtain some of them have their contorsion tensor depending on arbitrary functions that didn't influence their geometry. This raises question about the predictability of the theory.

gr-qc↗

arXiv2019

Three dimensional black strings: instabilities and asymptotic charges

Three-dimensional Einstein gravity coupled to zero, one and two forms is solved in terms of a polyhomogeneous asymptotic expansion, generalising stationary black string solutions. From first order terms we obtain, in closed form, a new solution evolving from the stationary black string structure to a geometry developing a singularity in the future. This new solution itself may be extended to more general ones. Taking into account subleading terms of the asymptotic expansion, both singularities in the past and the future occur. This demonstrates the unstable character of the stationary black string : tiny perturbations generated by terms breaking the rotational invariance of the stationary black string configurations lead to cosmological-like singularities. The symmetry algebra of the conserved charges also is determined: it is a finite dimensional one. In the general case the surface charge associated to energy is not integrable. However we identify a sub-class of solutions, admitting asymptotic symplectic symmetries and as a consequence conserved charges that appear to be integrable.

hep-th↗

arXiv2017

Quantum Supersymmetric Cosmological Billiards and their Hidden Kac-Moody Structure

We study the quantum fermionic billiard defined by the dynamics of a quantized supersymmetric squashed three-sphere (Bianchi IX cosmological model within D=4 simple supergravity). The quantization of the homogeneous gravitino field leads to a 64-dimensional fermionic Hilbert space. We focus on the 15- and 20-dimensional subspaces (with fermion numbers N_F=2 and N_F=3) where there exist propagating solutions of the supersymmetry constraints that carry (in the small-wavelength limit) a chaotic spinorial dynamics generalizing the Belinskii-Khalatnikov-Lifshitz classical "oscillatory" dynamics. By exactly solving the supersymmetry constraints near each one of the three dominant potential walls underlying the latter chaotic billiard dynamics, we compute the three operators that describe the corresponding three potential-wall reflections of the spinorial state describing, in supergravity, the quantum evolution of the universe. It is remarkably found that the latter, purely dynamically-defined, reflection operators satisfy generalized Coxeter relations which define a type of spinorial extension of the Weyl group of the rank-3 hyperbolic Kac-Moody algebra AE_3.

gr-qc↗

arXiv2015

Minisuperspace quantum supersymmetric cosmology (and its hidden hyperbolic Kac-Moody structures)

This work summarises recent progress obtained by the mini-superpspace quantization of $\mathcal N=1$, $d=4$ supergravity, formulated in the framework of the Bianchi IX cosmological model. The emphasis is put on three main results : the completeness of the solution space obtained, the elements suggesting a hidden Kac-Moody structure of the theory and those leading to conjecture an avoidance of the cosmological singularity by some branches of the wave function of the Universe.

gr-qc↗

arXiv2014

Investigating the origin of time with trapped ions

Even though quantum systems in energy eigenstates do not evolve in time, they can exhibit correlations between internal degrees of freedom in such a way that one of the internal degrees of freedom behaves like a clock variable, and thereby defines an internal time, that parametrises the evolution of the other degrees of freedom. This situation is of great interest in quantum cosmology where the invariance under reparametrisation of time implies that the temporal coordinate dissapears and is replaced by the Wheeler-DeWitt constraint. Here we show that this paradox can be investigated experimentally using the exquisite control now available on moderate size quantum systems. We describe in detail how to implement such an experimental demonstration using the spin and motional degrees of freedom of a single trapped ion.

gr-qc↗

arXiv2014

Quantum Supersymmetric Bianchi IX Cosmology

We study the quantum dynamics of a supersymmetric squashed three-sphere by dimensionally reducing to one timelike dimension the action of D=4 simple supergravity for a Bianchi IX cosmological model. After imposition of the diffeomorphism constraints, the wave function of the Universe becomes a spinor of Spin(8,4) depending on the three squashing parameters, which satisfies Dirac, and Klein-Gordon-like, wave equations describing the propagation of a quantum spinning particle reflecting off spin-dependent potential walls. The algebra of the susy constraints and of the Hamiltonian one is found to close. One finds that the quantum Hamiltonian is built from operators that generate a 64-dimensional representation of the maximally compact sub-algebra of the rank-3 hyperbolic Kac-Moody algebra AE3. The (quartic-in-fermions) squared-mass term entering the Klein-Gordon-like equation has several remarkable properties: 1)it commutes with all the other (Kac-Moody-related) building blocks of the Hamiltonian; 2)it is a quadratic function of the fermion number NF; 3)it is negative in most of the Hilbert space. The latter property leads to a possible quantum avoidance of the singularity ("cosmological bounce"), and suggests imposing the boundary condition that the wavefunction of the Universe vanish when the volume of space tends to zero. The space of solutions is a mixture of "discrete-spectrum states" (explicitely given) and of continuous-spectrum states (parametrized by arbitrary functions entering some initial-value problem). The predominantly negative values of the squared-mass term lead to a "bottle effect" between small and large volume-Universes and to a possible reduction of the continuous spectrum to a discrete spectrum of quantum states looking like excited versions of the Planckian-size Universes described by the discrete states at fermionic levels NF=0 and 1.

gr-qc↗

arXiv2013

Quantum Supersymmetric Cosmology and its Hidden Kac-Moody Structure

We study the quantum dynamics of a supersymmetric squashed three-sphere by dimensionally reducing (to one timelike dimension) the action of D=4 simple supergravity for an SO(3)-homogeneous (Bianchi IX) cosmological model. The quantization of the homogeneous gravitino field leads to a 64-dimensional fermionic Hilbert space. The algebra of the supersymmetry constraints and of the Hamiltonian one is found to close. One finds that the quantum Hamiltonian is built from operators that generate a 64-dimensional representation of the (infinite-dimensional) maximally compact sub-algebra of the rank-3 hyperbolic Kac-Moody algebra AE_3. Some exponentials of these operators generate a spinorial extension of the Weyl group of AE_3 which describe (in the small wavelength limit) the chaotic quantum evolution of the universe near the cosmological singularity.

gr-qc↗

arXiv2011

Quantum Einstein-Dirac Bianchi Universes

We study the mini--superspace quantization of spatially homogeneous (Bianchi) cosmological universes sourced by a Dirac spinor field. The quantization of the homogeneous spinor leads to a finite-dimensional fermionic Hilbert space and thereby to a multi-component Wheeler-DeWitt equation whose main features are: (i) the presence of spin-dependent Morse-type potentials, and (ii) the appearance of a q-number squared-mass term, which is of order ${\cal O}(\hbar^2)$, and which is affected by ordering ambiguities. We give the exact quantum solution of the Bianchi type-II system (which contains both scattering states and bound states), and discuss the main qualitative features of the quantum dynamics of the (classically chaotic) Bianchi type-IX system. We compare the exact quantum dynamics of fermionic cosmological billiards to previous works that described the spinor field as being either classical or Grassmann-valued.

gr-qc↗

arXiv2010

The hidden horizon and black hole unitarity

We motivate through a detailed analysis of the Hawking radiation in a Schwarzschild background a scheme in accordance with quantum unitarity. In this scheme the semi-classical approximation of the unitary quantum - horizonless - black hole S-matrix leads to the conventional description of the Hawking radiation from a classical black hole endowed with an event horizon. Unitarity is borne out by the detailed exclusive S-matrix amplitudes. There, the fixing of generic out-states, in addition to the in-state, yields in asymptotic Minkowski space-time saddle-point contributions which are dominated by Planckian metric fluctuations when approaching the Schwarzschild radius. We argue that these prevent the corresponding macroscopic "exclusive backgrounds" to develop an event horizon. However, if no out-state is selected, a distinct saddle-point geometry can be defined, in which Planckian fluctuations are tamed. Such "inclusive background" presents an event horizon and constitutes a coarse-grained average over the aforementioned exclusive ones. The classical event horizon appears as a coarse-grained structure, sustaining the thermodynamic significance of the Bekenstein-Hawking entropy. This is reminiscent of the tentative fuzzball description of extremal black holes: the role of microstates is played here by a complete set of out-states. Although the computations of unitary amplitudes would require a detailed theory of quantum gravity, the proposed scheme itself, which appeals to the metric description of gravity only in the vicinity of stationary points, does not.

hep-th↗

arXiv2008

The deformation quantizations of the hyperbolic plane

We describe the space of (all) invariant deformation quantizations on the hyperbolic plane as solutions of the evolution of a second order hyperbolic differential operator. The construction is entirely explicit and relies on non-commutative harmonic analytical techniques on symplectic symmetric spaces. The present work presents a unified method producing every quantization of the hyperbolic plane, and provides, in the 2-dimensional context, an exact solution to Weinstein's WKB quantization program within geometric terms. The construction reveals the existence of a metric of Lorentz signature canonically attached (or `dual') to the geometry of the hyperbolic plane through the quantization process.

math-ph↗

arXiv2008

Spacelike Singularities and Hidden Symmetries of Gravity

We review the intimate connection between (super-)gravity close to a spacelike singularity (the "BKL-limit") and the theory of Lorentzian Kac-Moody algebras. We show that in this limit the gravitational theory can be reformulated in terms of billiard motion in a region of hyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite) sequence of reflections, which are elements of a Lorentzian Coxeter group. Such Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras, suggesting that these algebras yield symmetries of gravitational theories. Our presentation is aimed to be a self-contained and comprehensive treatment of the subject, with all the relevant mathematical background material introduced and explained in detail. We also review attempts at making the infinite-dimensional symmetries manifest, through the construction of a geodesic sigma model based on a Lorentzian Kac-Moody algebra. An explicit example is provided for the case of the hyperbolic algebra E10, which is conjectured to be an underlying symmetry of M-theory. Illustrations of this conjecture are also discussed in the context of cosmological solutions to eleven-dimensional supergravity.

hep-th↗

arXiv2008

Uncertainty Relation for the Discrete Fourier Transform

We derive an uncertainty relation for two unitary operators which obey a commutation relation of the form UV=exp[i phi] VU. Its most important application is to constrain how much a quantum state can be localised simultaneously in two mutually unbiased bases related by a Discrete Fourier Transform. It provides an uncertainty relation which smoothly interpolates between the well known cases of the Pauli operators in 2 dimensions and the continuous variables position and momentum. This work also provides an uncertainty relation for modular variables, and could find applications in signal processing. In the finite dimensional case the minimum uncertainty states, discrete analogues of coherent and squeezed states, are minimum energy solutions of Harper's equation, a discrete version of the Harmonic oscillator equation.

quant-ph↗

arXiv2007

A Primer for Black Hole Quantum Physics

The mechanisms which give rise to Hawking radiation are revealed by analyzing in detail pair production in the presence of horizons. In preparation for the black hole problem, three preparatory problems are dwelt with at length: pair production in an external electric field, thermalization of a uniformly accelerated detector and accelerated mirrors. In the light of these examples, the black hole evaporation problem is then presented. The leitmotif is the singular behavior of modes on the horizon which gives rise to a steady rate of production. Special emphasis is put on how each produced particle contributes to the mean albeit arising from a particular vacuum fluctuation. It is the mean which drives the semiclassical back reaction. This aspect is analyzed in more detail than heretofore and in particular its drawbacks are emphasized. It is the semiclassical theory which gives rise to Hawking's famous equation for the loss of mass of the black hole due to evaporation $dM/dt \simeq -1/M^2$. Black hole thermodynamics is derived from the evaporation process whereupon the reservoir character of the black hole is manifest. The relation to the thermodynamics of the eternal black hole through the Hartle--Hawking vacuum and the Killing identity are displayed. It is through the analysis of the fluctuations of the field configurations which give rise to a particular Hawking photon that the dubious character of the semiclassical theory is manifest. The present frontier of research revolves around this problem and is principally concerned with the fact that one calls upon energy scales that are greater than Planckian and the possibility of a non unitary evolution as well. These last subjects are presented in qualitative fashion only, so that this review stops at the threshold of quantum gravity.

gr-qc↗