Philippe Spindel

A Primer for black hole quantum physics

Abstract 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 Hawkings famous equation for the loss of mass of the black hole due to evaporation d M d t − −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.

The fluctuating seven-sphere in eleven-dimensional supergravity

Abstract The full bosonic spectrum of eleven-dimensional supergravity compactified on the seven-sphere is obtained. The spectrum agrees with OSp(4|8) supersymmetry despite the appearance of 294 massless scalars which are not members of the N=8 massless supermultiplet. Negative (mass)2 arises only in the scalar sector with 112 nodes whose common (mass)2 is equal to the threshold value for background stability.

Non-oscillatory behaviour in vacuum Kaluza-Klein cosmologies

Abstract It is argued that the generalized Kasner solution, with monotonic power-law behaviour of the spatial distances, becomes a general solution of the Einstein vacuum field equations near the cosmological singularity in spacetime dimensions ⩾11.

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 spacetimes, 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.

Cosmological models in eleven-dimensional supergravity

Abstract Spatially homogeneous, anisotropic, cosmological models are investigated in the case of D = 11 supergravity. It is shown how the field equations reduce with this assumption to ordinary differential equations with respect to time. These equations are solved when the homogeneity group is abelian and admits non-trivial solutions with various splittings of the 11 dimensions into n + (11 − n). Other exact solutions with different homogeneity groups are exhibited, one of them being such that the physical dimensions expand much faster than the extra ones.

Supersymmetry breaking by torsion and the Ricci-flat squashed seven-spheres

Abstract In eleven-dimensional supergravity, unbroken supersymmetries may survive spontaneous compactification to V 4 × V 7 only for Freund-Rubin solutions to the equations of motion. A torsion which renders V 7 Ricci-flat and breaks supersymmetry can then be induced spontaneously. Two squashed seven-spheres with torsion are obtained in this way: the flattening torsion components are constant and given by the octonion multiplication table.

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

Abstract Using the Chern–Simons 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.

Star products on extended massive non-rotating BTZ black holes

AdS 3 space-time admits a foliation by two-dimensional twisted conjugacy classes, stable under the identification subgroup yielding the non-rotating massive BTZ black hole. Each leaf constitutes a classical solution of the space-time Dirac-Born-Infeld action, describing an open D-string in AdS 3 or a D-string winding around the black hole. We first describe two nonequivalent maximal extensions of the non-rotating massive BTZ space-time and observe that in one of them, each D-string worldsheet admits an action of a two-parameter subgroup () of SL(2,). We then construct non-formal, -invariant, star products that deform the classical algebra of functions on the D-string worldsheets and on their embedding space-times. We end by giving the first elements towards the definition of a Connes spectral triple on non-commutative AdS space-times.

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.

Symmetries in Eleven-dimensional Supergravity Compactified on a Parallelized Seven Sphere

Abstract We analyse, in eleven-dimensional supergravity compactified on S 7 , the spontaneous symmetry breaking induced by a spontaneous parallelization of the sphere. The eight supersymmetries are broken at a common scale and the SO(8) gauge group is reduced to Spin (7). Such a large residual symmetry has a simple geometrical significance revealed through use of octonions; this is explained in elementary terms.