1,094 research outputs found
Constructing Reference Metrics on Multicube Representations of Arbitrary Manifolds
Reference metrics are used to define the differential structure on multicube
representations of manifolds, i.e., they provide a simple and practical way to
define what it means globally for tensor fields and their derivatives to be
continuous. This paper introduces a general procedure for constructing
reference metrics automatically on multicube representations of manifolds with
arbitrary topologies. The method is tested here by constructing reference
metrics for compact, orientable two-dimensional manifolds with genera between
zero and five. These metrics are shown to satisfy the Gauss-Bonnet identity
numerically to the level of truncation error (which converges toward zero as
the numerical resolution is increased). These reference metrics can be made
smoother and more uniform by evolving them with Ricci flow. This smoothing
procedure is tested on the two-dimensional reference metrics constructed here.
These smoothing evolutions (using volume-normalized Ricci flow with DeTurck
gauge fixing) are all shown to produce reference metrics with constant scalar
curvatures (at the level of numerical truncation error).Comment: 37 pages, 16 figures; additional introductory material added in
version accepted for publicatio
Fuchsian methods and spacetime singularities
Fuchsian methods and their applications to the study of the structure of
spacetime singularities are surveyed. The existence question for spacetimes
with compact Cauchy horizons is discussed. After some basic facts concerning
Fuchsian equations have been recalled, various ways in which these equations
have been applied in general relativity are described. Possible future
applications are indicated
Hydrodynamic Flows on Curved Surfaces: Spectral Numerical Methods for Radial Manifold Shapes
We formulate hydrodynamic equations and spectrally accurate numerical methods
for investigating the role of geometry in flows within two-dimensional fluid
interfaces. To achieve numerical approximations having high precision and level
of symmetry for radial manifold shapes, we develop spectral Galerkin methods
based on hyperinterpolation with Lebedev quadratures for -projection to
spherical harmonics. We demonstrate our methods by investigating hydrodynamic
responses as the surface geometry is varied. Relative to the case of a sphere,
we find significant changes can occur in the observed hydrodynamic flow
responses as exhibited by quantitative and topological transitions in the
structure of the flow. We present numerical results based on the
Rayleigh-Dissipation principle to gain further insights into these flow
responses. We investigate the roles played by the geometry especially
concerning the positive and negative Gaussian curvature of the interface. We
provide general approaches for taking geometric effects into account for
investigations of hydrodynamic phenomena within curved fluid interfaces.Comment: 14 figure
Fast integral equation methods for the Laplace-Beltrami equation on the sphere
Integral equation methods for solving the Laplace-Beltrami equation on the
unit sphere in the presence of multiple "islands" are presented. The surface of
the sphere is first mapped to a multiply-connected region in the complex plane
via a stereographic projection. After discretizing the integral equation, the
resulting dense linear system is solved iteratively using the fast multipole
method for the 2D Coulomb potential in order to calculate the matrix-vector
products. This numerical scheme requires only O(N) operations, where is the
number of nodes in the discretization of the boundary. The performance of the
method is demonstrated on several examples
Post-Newtonian Freely Specifiable Initial Data for Binary Black Holes in Numerical Relativity
Construction of astrophysically realistic initial data remains a central
problem when modelling the merger and eventual coalescence of binary black
holes in numerical relativity. The objective of this paper is to provide
astrophysically realistic freely specifiable initial data for binary black hole
systems in numerical relativity, which are in agreement with post-Newtonian
results. Following the approach taken by Blanchet, we propose a particular
solution to the time-asymmetric constraint equations, which represent a system
of two moving black holes, in the form of the standard conformal decomposition
of the spatial metric and the extrinsic curvature. The solution for the spatial
metric is given in symmetric tracefree form, as well as in Dirac coordinates.
We show that the solution differs from the usual post-Newtonian metric up to
the 2PN order by a coordinate transformation. In addition, the solutions,
defined at every point of space, differ at second post-Newtonian order from the
exact, conformally flat, Bowen-York solution of the constraints.Comment: 41 pages, no figures, accepted for publication in Phys. Rev. D,
significant revision in presentation (including added references and
corrected typos
The Simplicial Ricci Tensor
The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of
gravitation. The 3-dimensional Ric of a spacelike surface vanishes at the
moment of time symmetry for vacuum spacetimes. The 4-dimensional Ric is the
Einstein tensor for such spacetimes. More recently the Ric was used by Hamilton
to define a non-linear, diffusive Ricci flow (RF) that was fundamental to
Perelman's proof of the Poincare conjecture. Analytic applications of RF can be
found in many fields including general relativity and mathematics. Numerically
it has been applied broadly to communication networks, medical physics,
computer design and more. In this paper, we use Regge calculus (RC) to provide
the first geometric discretization of the Ric. This result is fundamental for
higher-dimensional generalizations of discrete RF. We construct this tensor on
both the simplicial lattice and its dual and prove their equivalence. We show
that the Ric is an edge-based weighted average of deficit divided by an
edge-based weighted average of dual area -- an expression similar to the
vertex-based weighted average of the scalar curvature reported recently. We use
this Ric in a third and independent geometric derivation of the RC Einstein
tensor in arbitrary dimension.Comment: 19 pages, 2 figure
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