104 research outputs found
Analytic Continuation for Asymptotically AdS 3D Gravity
We have previously proposed that asymptotically AdS 3D wormholes and black
holes can be analytically continued to the Euclidean signature. The analytic
continuation procedure was described for non-rotating spacetimes, for which a
plane t=0 of time symmetry exists. The resulting Euclidean manifolds turned out
to be handlebodies whose boundary is the Schottky double of the geometry of the
t=0 plane. In the present paper we generalize this analytic continuation map to
the case of rotating wormholes. The Euclidean manifolds we obtain are quotients
of the hyperbolic space by a certain quasi-Fuchsian group. The group is the
Fenchel-Nielsen deformation of the group of the non-rotating spacetime. The
angular velocity of an asymptotic region is shown to be related to the
Fenchel-Nielsen twist. This solves the problem of classification of rotating
black holes and wormholes in 2+1 dimensions: the spacetimes are parametrized by
the moduli of the boundary of the corresponding Euclidean spaces. We also
comment on the thermodynamics of the wormhole spacetimes.Comment: 28 pages, 14 figure
Geometric Finiteness, Holography and Quasinormal Modes for the Warped AdS_3 Black Hole
We show that there exists a precise kinematical notion of holography for the
Euclidean warped black hole. This follows from the fact that the
Euclidean warped black hole spacetime is a geometrically finite
hyperbolic manifold. For such manifolds a theorem of Sullivan provides a
one-to-one correspondence between the hyperbolic structure in the bulk and the
conformal structure of its boundary. Using this theorem we obtain the
holographic quasinormal modes for the warped black hole.Comment: Latex file, 7 pages. Journal versio
Black Hole Thermodynamics and Riemann Surfaces
We use the analytic continuation procedure proposed in our earlier works to
study the thermodynamics of black holes in 2+1 dimensions. A general black hole
in 2+1 dimensions has g handles hidden behind h horizons. The result of the
analytic continuation is a hyperbolic 3-manifold having the topology of a
handlebody. The boundary of this handlebody is a compact Riemann surface of
genus G=2g+h-1. Conformal moduli of this surface encode in a simple way the
physical characteristics of the black hole. The moduli space of black holes of
a given type (g,h) is then the Schottky space at genus G. The (logarithm of
the) thermodynamic partition function of the hole is the Kaehler potential for
the Weil-Peterson metric on the Schottky space. Bekenstein bound on the black
hole entropy leads us to conjecture a new strong bound on this Kaehler
potential.Comment: 17+1 pages, 9 figure
Geometrical Finiteness, Holography, and the BTZ Black Hole
We show how a theorem of Sullivan provides a precise mathematical statement
of a 3d holographic principle, that is, the hyperbolic structure of a certain
class of 3d manifolds is completely determined in terms of the corresponding
Teichmuller space of the boundary. We explore the consequences of this theorem
in the context of the Euclidean BTZ black hole in three dimensions.Comment: 6 pages, Latex, Version to appear in Physical Review Letter
Surface homeomorphisms with zero dimensional singular set
We prove that if f is an orientation-preserving homeomorphism of a closed
orientable surface M whose singular set is totally disconnected, then f is
topologically conjugate to a conformal transformation.Comment: 22 page
Systoles on Compact Riemann Surfaces with Symbolic Dynamics
In this chapter, systolic inequalities are established, precise values are
computed, and their behavior is also examined with the variation of the Fenchel–
Nielsen coordinates on a compact Riemann surface of genus 2
Fuchsian convex bodies: basics of Brunn--Minkowski theory
The hyperbolic space \H^d can be defined as a pseudo-sphere in the
Minkowski space-time. In this paper, a Fuchsian group is a group of
linear isometries of the Minkowski space such that \H^d/\Gamma is a compact
manifold. We introduce Fuchsian convex bodies, which are closed convex sets in
Minkowski space, globally invariant for the action of a Fuchsian group. A
volume can be associated to each Fuchsian convex body, and, if the group is
fixed, Minkowski addition behaves well. Then Fuchsian convex bodies can be
studied in the same manner as convex bodies of Euclidean space in the classical
Brunn--Minkowski theory. For example, support functions can be defined, as
functions on a compact hyperbolic manifold instead of the sphere.
The main result is the convexity of the associated volume (it is log concave
in the classical setting). This implies analogs of Alexandrov--Fenchel and
Brunn--Minkowski inequalities. Here the inequalities are reversed
Grafting and Poisson structure in (2+1)-gravity with vanishing cosmological constant
We relate the geometrical construction of (2+1)-spacetimes via grafting to
phase space and Poisson structure in the Chern-Simons formulation of
(2+1)-dimensional gravity with vanishing cosmological constant on manifolds of
topology , where is an orientable two-surface of genus
. We show how grafting along simple closed geodesics \lambda is
implemented in the Chern-Simons formalism and derive explicit expressions for
its action on the holonomies of general closed curves on S_g. We prove that
this action is generated via the Poisson bracket by a gauge invariant
observable associated to the holonomy of . We deduce a symmetry
relation between the Poisson brackets of observables associated to the Lorentz
and translational components of the holonomies of general closed curves on S_g
and discuss its physical interpretation. Finally, we relate the action of
grafting on the phase space to the action of Dehn twists and show that grafting
can be viewed as a Dehn twist with a formal parameter satisfying
.Comment: 43 pages, 10 .eps figures; minor modifications: 2 figures added,
explanations added, typos correcte
Geometrical (2+1)-gravity and the Chern-Simons formulation: Grafting, Dehn twists, Wilson loop observables and the cosmological constant
We relate the geometrical and the Chern-Simons description of
(2+1)-dimensional gravity for spacetimes of topology , where
is an oriented two-surface of genus , for Lorentzian signature and general
cosmological constant and the Euclidean case with negative cosmological
constant. We show how the variables parametrising the phase space in the
Chern-Simons formalism are obtained from the geometrical description and how
the geometrical construction of (2+1)-spacetimes via grafting along closed,
simple geodesics gives rise to transformations on the phase space. We
demonstrate that these transformations are generated via the Poisson bracket by
one of the two canonical Wilson loop observables associated to the geodesic,
while the other acts as the Hamiltonian for infinitesimal Dehn twists. For
spacetimes with Lorentzian signature, we discuss the role of the cosmological
constant as a deformation parameter in the geometrical and the Chern-Simons
formulation of the theory. In particular, we show that the Lie algebras of the
Chern-Simons gauge groups can be identified with the (2+1)-dimensional Lorentz
algebra over a commutative ring, characterised by a formal parameter
whose square is minus the cosmological constant. In this
framework, the Wilson loop observables that generate grafting and Dehn twists
are obtained as the real and the -component of a Wilson loop
observable with values in the ring, and the grafting transformations can be
viewed as infinitesimal Dehn twists with the parameter .Comment: 50 pages, 6 eps figure
Dirichlet fundamental domains and complex-projective varieties
We prove that for every finitely-presented group G there exists a
2-dimensional irreducible complex-projective variety W with the fundamental
group G, so that all singularities of W are normal crossings and Whitney
umbrellas.Comment: 1 figur
- …