940 research outputs found
Dynamical Triangulations, a Gateway to Quantum Gravity ?
We show how it is possible to formulate Euclidean two-dimensional quantum
gravity as the scaling limit of an ordinary statistical system by means of
dynamical triangulations, which can be viewed as a discretization in the space
of equivalence classes of metrics. Scaling relations exist and the critical
exponents have simple geometric interpretations. Hartle-Hawkings wave
functionals as well as reparametrization invariant correlation functions which
depend on the geodesic distance can be calculated. The discretized approach
makes sense even in higher dimensional space-time. Although analytic solutions
are still missing in the higher dimensional case, numerical studies reveal an
interesting structure and allow the identification of a fixed point where we
can hope to define a genuine non-perturbative theory of four-dimensional
quantum gravity.Comment: Review, 44 pages, tar compressed uuencoded ps-file (after removing
header, type csh filename.uu
Counting a black hole in Lorentzian product triangulations
We take a step toward a nonperturbative gravitational path integral for
black-hole geometries by deriving an expression for the expansion rate of null
geodesic congruences in the approach of causal dynamical triangulations. We
propose to use the integrated expansion rate in building a quantum horizon
finder in the sum over spacetime geometries. It takes the form of a counting
formula for various types of discrete building blocks which differ in how they
focus and defocus light rays. In the course of the derivation, we introduce the
concept of a Lorentzian dynamical triangulation of product type, whose
applicability goes beyond that of describing black-hole configurations.Comment: 42 pages, 11 figure
Scattered data fitting on surfaces using projected Powell-Sabin splines
We present C1 methods for either interpolating data or for fitting scattered data associated with a smooth function on a two-dimensional smooth manifold Ω embedded into R3. The methods are based on a local bivariate Powell-Sabin interpolation scheme, and make use of local projections on the tangent planes. The data fitting method is a two-stage method. We illustrate the performance of the algorithms with some numerical examples, which, in particular, confirm the O(h3) order of convergence as the data becomes dens
Interpolation and scattered data fitting on manifolds using projected Powell–Sabin splines
We present methods for either interpolating data or for fitting scattered data on a two-dimensional smooth manifold. The methods are based on a local bivariate Powell-Sabin interpolation scheme, and make use of a family of charts {(Uξ , ξ)}ξ∈ satisfying certain conditions of smooth dependence on ξ. If is a C2-manifold embedded into R3, then projections into tangent planes can be employed. The data fitting method is a two-stage method. We prove that the resulting function on the manifold is continuously differentiable, and establish error bounds for both methods for the case when the data are generated by a smooth function
The geometry of dynamical triangulations
We discuss the geometry of dynamical triangulations associated with
3-dimensional and 4-dimensional simplicial quantum gravity. We provide
analytical expressions for the canonical partition function in both cases, and
study its large volume behavior. In the space of the coupling constants of the
theory, we characterize the infinite volume line and the associated critical
points. The results of this analysis are found to be in excellent agreement
with the MonteCarlo simulations of simplicial quantum gravity. In particular,
we provide an analytical proof that simply-connected dynamically triangulated
4-manifolds undergo a higher order phase transition at a value of the inverse
gravitational coupling given by 1.387, and that the nature of this transition
can be concealed by a bystable behavior. A similar analysis in the
3-dimensional case characterizes a value of the critical coupling (3.845) at
which hysteresis effects are present.Comment: 166 pages, Revtex (latex) fil
Equivariant smoothing of piecewise linear manifolds
We prove that every piecewise linear manifold of dimension up to four on
which a finite group acts by piecewise linear homeomorphisms admits a
compatible smooth structure with respect to which the group acts smoothly. This
solves a challenge posed by Thurston in dimension three and confirms a
conjecture by Kwasik and Lee in dimension four in a stronger form.Comment: revised version, accepted by Math. Proc. Cambridge Philos. So
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