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.Comment: 47 pages, typos and references corrected, introduction enhanced, evolution of the horizons of the special time dependent solutions provide

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.Comment: Proceedings Fourteenth Marcel Grossmann Meeting - Rome, July 12-18, 201

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.Comment: Extended Version; 13 pages; In press in Phys. Rev. Let

On bound states of Dirac particles in gravitational fields

We investigate the quantum motion of a neutral Dirac particle bouncing on a mirror in curved spacetime. We consider different geometries: Rindler, Kasner-Taub and Schwarzschild, and show how to solve the Dirac equation by using geometrical methods. We discuss, in a first-quantized framework, the implementation of appropriate boundary conditions. This leads us to consider a Robin boundary condition that gives the quantization of the energy, the existence of bound states and of critical heights at which the Dirac particle bounces, extending the well-known results established from the Schrodinger equation. We also allow for a nonminimal coupling to a weak magnetic field. The problem is solved in an analytical way on the Rindler spacetime. In the other cases, we compute the energy spectrum up to the first relativistic corrections, exhibiting the contributions brought by both the geometry and the spin. These calculations are done in two different ways. On the one hand, using a relativistic expansion and, on the other hand, with Foldy-Wouthuysen transformations. Contrary to what is sometimes claimed in the literature, both methods are in agreement, as expected. Finally, we make contact with the GRANIT experiment. Relativistic effects and effects that go beyond the equivalence principle escape the sensitivity of such an experiment. However, we show that the influence of a weak magnetic field could lead to observable phenomena.Comment: ReVTeX, 24 pages, 2 figure

Noncommutative Locally Anti-de Sitter Black Holes

We give a review of our joint work on strict deformation of BHTZ 2+1 black holes \cite{BRS02,BDHRS03}. However some results presented here are not published elsewhere, and an effort is made for enlightening the instrinsical aspect of the constructions. This shows in particular that the three dimensional case treated here could be generalized to an anti-de Sitter space of arbitrary dimension provided one disposes of a universal deformation formula for the actions of a parabolic subgroup of its isometry group.Comment: 10 pages, based on a talk given by P.B., to appear in the proceedings of the workshop `Noncommutative Geometry and Physics 2004' (Feb. 2004, Keio University, Japan) (World Scientific

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 (\ca \cn) of \SL. We then construct non-formal, \ca \cn-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.Comment: 25 pages, 1 figur

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.Comment: An old review article on black hole evaporation and black hole thermodynamics, put on the archive following popular demand, 178 pages, 21 figures (This text differs in slightly from the published version

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.Comment: 5 page