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

Aspects of (2+1) dimensional gravity: AdS3 asymptotic dynamics in the framework of Fefferman-Graham-Lee theorems

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.Comment: 6 pages, Latex, presented at the Journees Relativistes 99, Weimar, September 12-17, 1999; revised version with minor modifications and 1 reference adde

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

Einstein-Podolsky-Rosen correlations between two uniformly accelerated oscillators

We consider the quantum correlations, i.e. the entanglement, between two systems uniformly accelerated with identical acceleration a in opposite Rindler quadrants which have reached thermal equilibrium with the Unruh heat bath. To this end we study an exactly soluble model consisting of two oscillators coupled to a massless scalar field in 1+1 dimensions. We find that for some values of the parameters the oscillators get entangled shortly after the moment of closest approach. Because of boost invariance there are an infinite set of pairs of positions where the oscillators are entangled. The maximal entanglement between the oscillators is found to be approximately 1.4 entanglement bits.Comment: 11 page

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