39 research outputs found
The York map as a Shanmugadhasan canonical transformation in tetrad gravity and the role of non-inertial frames in the geometrical view of the gravitational field
A new parametrization of the 3-metric allows to find explicitly a York map in
canonical ADM tetrad gravity, the two pairs of physical tidal degrees of
freedom and 14 gauge variables. These gauge quantities (generalized inertial
effects) are all configurational except the trace of
the extrinsic curvature of the instantaneous 3-spaces (clock
synchronization convention) of a non-inertial frame. The Dirac hamiltonian is
the sum of the weak ADM energy (whose density is coordinate-dependent due to the inertial
potentials) and of the first-class constraints. Then: i) The explicit form of
the Hamilton equations for the two tidal degrees of freedom in an arbitrary
gauge: a deterministic evolution can be defined only in a completely fixed
gauge, i.e. in a non-inertial frame with its pattern of inertial forces. ii) A
general solution of the super-momentum constraints, which shows the existence
of a generalized Gribov ambiguity associated to the 3-diffeomorphism gauge
group. It influences: a) the explicit form of the weak ADM energy and of the
super-momentum constraint; b) the determination of the shift functions and then
of the lapse one. iii) The dependence of the Hamilton equations for the two
pairs of dynamical gravitational degrees of freedom (the generalized tidal
effects) and for the matter, written in a completely fixed 3-orthogonal
Schwinger time gauge, upon the gauge variable ,
determining the convention of clock synchronization. Therefore it should be
possible (for instance in the weak field limit but with relativistic motion) to
try to check whether in Einstein's theory the {\it dark matter} is a gauge
relativistic inertial effect induced by .Comment: 90 page
Conditional probabilities in Ponzano-Regge minisuperspace
We examine the Hartle-Hawking no-boundary initial state for the Ponzano-Regge
formulation of gravity in three dimensions. We consider the behavior of
conditional probabilities and expectation values for geometrical quantities in
this initial state for a simple minisuperspace model consisting of a
two-parameter set of anisotropic geometries on a 2-sphere boundary. We find
dependence on the cutoff used in the construction of Ponzano-Regge amplitudes
for expectation values of edge lengths. However, these expectation values are
cutoff independent when computed in certain, but not all, conditional
probability distributions. Conditions that yield cutoff independent expectation
values are those that constrain the boundary geometry to a finite range of edge
lengths. We argue that such conditions have a correspondence to fixing a range
of local time, as classically associated with the area of a surface for
spatially closed cosmologies. Thus these results may hint at how classical
spacetime emerges from quantum amplitudes.Comment: 26 pages including 10 figures, some reorganization in the
presentation of results, expanded discussion of results in the context of 2+1
gravity in the Witten variables, 3 new reference
Scalar Field Dark Matter: behavior around black holes
We present the numerical evolution of a massive test scalar fields around a
Schwarzschild space-time. We proceed by using hyperboloidal slices that
approach future null infinity, which is the boundary of scalar fields, and also
demand the slices to penetrate the event horizon of the black hole. This
approach allows the scalar field to be accreted by the black hole and to escape
toward future null infinity. We track the evolution of the energy density of
the scalar field, which determines the rate at which the scalar field is being
diluted. We find polynomial decay of the energy density of the scalar field,
and use it to estimate the rate of dilution of the field in time. Our findings
imply that the energy density of the scalar field decreases even five orders of
magnitude in time scales smaller than a year. This implies that if a
supermassive black hole is the Schwarzschild solution, then scalar field dark
matter would be diluted extremely fastComment: 15 pages, 21 eps figures. Appendix added, accepted for publication in
JCA
Shape Dynamics in 2+1 Dimensions
Shape Dynamics is a formulation of General Relativity where refoliation
invariance is traded for local spatial conformal invariance. In this paper we
explicitly construct Shape Dynamics for a torus universe in 2+1 dimensions
through a linking gauge theory that ensures dynamical equivalence with General
Relativity. The Hamiltonian we obtain is formally a reduced phase space
Hamiltonian. The construction of the Shape Dynamics Hamiltonian on higher genus
surfaces is not explicitly possible, but we give an explicit expansion of the
Shape Dynamics Hamiltonian for large CMC volume. The fact that all local
constraints are linear in momenta allows us to quantize these explicitly, and
the quantization problem for Shape Dynamics turns out to be equivalent to
reduced phase space quantization. We consider the large CMC-volume asymptotics
of conformal transformations of the wave function. We then use the similarity
of Shape Dynamics on the 2-torus with the explicitly constructible strong
gravity (BKL) Shape Dynamics Hamiltonian in higher dimensions to suggest a
quantization strategy for Shape Dynamics.Comment: 15 pages, LaTeX, no figure
Notes on a paper of Mess
These notes are a companion to the article "Lorentz spacetimes of constant
curvature" by Geoffrey Mess, which was first written in 1990 but never
published. Mess' paper will appear together with these notes in a forthcoming
issue of Geometriae Dedicata.Comment: 26 page
Dirac's Observables for the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge
We define the {\it rest-frame instant form} of tetrad gravity restricted to
Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of
gauge transformations generated by the 14 first class constraints of the
theory, we define and solve the multitemporal equations associated with the
rotation and space diffeomorphism constraints, finding how the cotriads and
their momenta depend on the corresponding gauge variables. This allows to find
quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal
gauges and to find the Dirac observables for superspace in these gauges.
The construction of the explicit form of the transformation and of the
solution of the rotation and supermomentum constraints is reduced to solve a
system of elliptic linear and quasi-linear partial differential equations. We
then show that the superhamiltonian constraint becomes the Lichnerowicz
equation for the conformal factor of the 3-metric and that the last gauge
variable is the momentum conjugated to the conformal factor. The gauge
transformations generated by the superhamiltonian constraint perform the
transitions among the allowed foliations of spacetime, so that the theory is
independent from its 3+1 splittings. In the special 3-orthogonal gauge defined
by the vanishing of the conformal factor momentum we determine the final Dirac
observables for the gravitational field even if we are not able to solve the
Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted
to this completely fixed gauge.Comment: RevTeX file, 141 page
An Infinite Class of Extremal Horizons in Higher Dimensions
We present a new class of near-horizon geometries which solve Einstein's
vacuum equations, including a negative cosmological constant, in all even
dimensions greater than four. Spatial sections of the horizon are inhomogeneous
S^2-bundles over any compact Kaehler-Einstein manifold. For a given base, the
solutions are parameterised by one continuous parameter (the angular momentum)
and an integer which determines the topology of the horizon. In six dimensions
the horizon topology is either S^2 x S^2 or CP^2 # -CP^2. In higher dimensions
the S^2-bundles are always non-trivial, and for a fixed base, give an infinite
number of distinct horizon topologies. Furthermore, depending on the choice of
base we can get examples of near-horizon geometries with a single rotational
symmetry (the minimal dimension for this is eight). All of our horizon
geometries are consistent with all known topology and symmetry constraints for
the horizons of asymptotically flat or globally Anti de Sitter extremal black
holes.Comment: 42 pages, latex. v2: corrected section 6.1, two references added. v3:
modified angular momentum and corrected area comparison, version to be
published in Commun. Math. Phy