39 research outputs found
Gauge fixing in (2+1)-gravity with vanishing cosmological constant
We apply Dirac's gauge fixing procedure to (2+1)-gravity with vanishing
cosmological constant. For general gauge fixing conditions based on two point
particles, this yields explicit expressions for the Dirac bracket. We explain
how gauge fixing is related to the introduction of an observer into the theory
and show that the Dirac bracket is determined by a classical dynamical
r-matrix. Its two dynamical variables correspond to the mass and spin of a cone
that describes the residual degrees of freedom of the spacetime. We show that
different gauge fixing conditions and different choices of observers are
related by dynamical Poincar\'e transformations. This allows us to locally
classify all Dirac brackets resulting from the gauge fixing and to relate them
to a set of particularly simple solutions associated with the centre-of-mass
frame of the spacetime.Comment: Talk given at the Workshop on Noncommutative Field Theory and Gravity
Corfu, September 7 - 11, 2011; 20 pages, 6 figure
Generalised shear coordinates on the moduli spaces of three-dimensional spacetimes
We introduce coordinates on the moduli spaces of maximal globally hyperbolic
constant curvature 3d spacetimes with cusped Cauchy surfaces S. They are
derived from the parametrisation of the moduli spaces by the bundle of measured
geodesic laminations over Teichm\"uller space of S and can be viewed as
analytic continuations of the shear coordinates on Teichm\"uller space. In
terms of these coordinates the gravitational symplectic structure takes a
particularly simple form, which resembles the Weil-Petersson symplectic
structure in shear coordinates, and is closely related to the cotangent bundle
of Teichm\"uller space. We then consider the mapping class group action on the
moduli spaces and show that it preserves the gravitational symplectic
structure. This defines three distinct mapping class group actions on the
cotangent bundle of Teichm\"uller space, corresponding to different values of
the curvature.Comment: 40 pages, 6 figure
A (2+1) non-commutative Drinfel'd double spacetime with cosmological constant
We show that the Drinfel'd double associated to the standard quantum
deformation is isomorphic to the (2+1)-dimensional AdS algebra
with the initial deformation parameter related to the cosmological
constant . This gives rise to a generalisation of a
non-commutative Minkowski spacetime that arises as a consequence of the quantum
double symmetry of (2+1) gravity to non-vanishing cosmological constant. The
properties of the AdS quantum double that generalises this symmetry to the case
are sketched, and it is shown that the new non-commutative AdS
spacetime is a nonlinear -deformation of the Minkowskian one.Comment: 14 pages; some comments and references adde