32 research outputs found
A fully (3+1)-D Regge calculus model of the Kasner cosmology
We describe the first discrete-time 4-dimensional numerical application of
Regge calculus. The spacetime is represented as a complex of 4-dimensional
simplices, and the geometry interior to each 4-simplex is flat Minkowski
spacetime. This simplicial spacetime is constructed so as to be foliated with a
one parameter family of spacelike hypersurfaces built of tetrahedra. We
implement a novel two-surface initial-data prescription for Regge calculus, and
provide the first fully 4-dimensional application of an implicit decoupled
evolution scheme (the ``Sorkin evolution scheme''). We benchmark this code on
the Kasner cosmology --- a cosmology which embodies generic features of the
collapse of many cosmological models. We (1) reproduce the continuum solution
with a fractional error in the 3-volume of 10^{-5} after 10000 evolution steps,
(2) demonstrate stable evolution, (3) preserve the standard deviation of
spatial homogeneity to less than 10^{-10} and (4) explicitly display the
existence of diffeomorphism freedom in Regge calculus. We also present the
second-order convergence properties of the solution to the continuum.Comment: 22 pages, 5 eps figures, LaTeX. Updated and expanded versio
Apparent horizons in simplicial Brill wave initial data
We construct initial data for a particular class of Brill wave metrics using
Regge calculus, and compare the results to a corresponding continuum solution,
finding excellent agreement. We then search for trapped surfaces in both sets
of initial data, and provide an independent verification of the existence of an
apparent horizon once a critical gravitational wave amplitude is passed. Our
estimate of this critical value, using both the Regge and continuum solutions,
supports other recent findings.Comment: 7 pages, 6 EPS figures, LaTeX 2e. Submitted to Class. Quant. Gra
Constraints in Quantum Geometrodynamics
We compare different treatments of the constraints in canonical quantum
gravity. The standard approach on the superspace of 3--geometries treats the
constraints as the sole carriers of the dynamic content of the theory, thus
rendering the traditional dynamical equations obsolete. Quantization of the
constraints in both the Dirac and ADM square root Hamiltonian approaches leads
to the well known problems of time evolution. These problems of time are of
both an interpretational and technical nature. In contrast, the geometrodynamic
quantization procedure on the superspace of the true dynamical variables
separates the issues of quantization from the enforcement of the constraints.
The resulting theory takes into account states that are off-shell with respect
to the constraints, and thus avoids the problems of time. We develop, for the
first time, the geometrodynamic quantization formalism in a general setting and
show that it retains all essential features previously illustrated in the
context of homogeneous cosmologies.Comment: 36 pages, no figures, submitted to IJMPA, Rewording, Fixed Typo
Coupling Non-Gravitational Fields with Simplicial Spacetimes
The inclusion of source terms in discrete gravity is a long-standing problem.
Providing a consistent coupling of source to the lattice in Regge Calculus (RC)
yields a robust unstructured spacetime mesh applicable to both numerical
relativity and quantum gravity. RC provides a particularly insightful approach
to this problem with its purely geometric representation of spacetime. The
simplicial building blocks of RC enable us to represent all matter and fields
in a coordinate-free manner. We provide an interpretation of RC as a discrete
exterior calculus framework into which non-gravitational fields naturally
couple with the simplicial lattice. Using this approach we obtain a consistent
mapping of the continuum action for non-gravitational fields to the Regge
lattice. In this paper we apply this framework to scalar, vector and tensor
fields. In particular we reconstruct the lattice action for (1) the scalar
field, (2) Maxwell field tensor and (3) Dirac particles. The straightforward
application of our discretization techniques to these three fields demonstrates
a universal implementation of coupling source to the lattice in Regge calculus.Comment: 10 pages, no figures, Latex, fixed typos and minor corrections
The constraints as evolution equations for numerical relativity
The Einstein equations have proven surprisingly difficult to solve
numerically. A standard diagnostic of the problems which plague the field is
the failure of computational schemes to satisfy the constraints, which are
known to be mathematically conserved by the evolution equations. We describe a
new approach to rewriting the constraints as first-order evolution equations,
thereby guaranteeing that they are satisfied to a chosen accuracy by any
discretization scheme. This introduces a set of four subsidiary constraints
which are far simpler than the standard constraint equations, and which should
be more easily conserved in computational applications. We explore the manner
in which the momentum constraints are already incorporated in several existing
formulations of the Einstein equations, and demonstrate the ease with which our
new constraint-conserving approach can be incorporated into these schemes.Comment: 10 pages, updated to match published versio
(Broken) Gauge Symmetries and Constraints in Regge Calculus
We will examine the issue of diffeomorphism symmetry in simplicial models of
(quantum) gravity, in particular for Regge calculus. We find that for a
solution with curvature there do not exist exact gauge symmetries on the
discrete level. Furthermore we derive a canonical formulation that exactly
matches the dynamics and hence symmetries of the covariant picture. In this
canonical formulation broken symmetries lead to the replacements of constraints
by so--called pseudo constraints. These considerations should be taken into
account in attempts to connect spin foam models, based on the Regge action,
with canonical loop quantum gravity, which aims at implementing proper
constraints. We will argue that the long standing problem of finding a
consistent constraint algebra for discretized gravity theories is equivalent to
the problem of finding an action with exact diffeomorphism symmetries. Finally
we will analyze different limits in which the pseudo constraints might turn
into proper constraints. This could be helpful to infer alternative
discretization schemes in which the symmetries are not broken.Comment: 32 pages, 15 figure
Is the Regge Calculus a consistent approximation to General Relativity?
We will ask the question of whether or not the Regge calculus (and two
related simplicial formulations) is a consistent approximation to General
Relativity. Our criteria will be based on the behaviour of residual errors in
the discrete equations when evaluated on solutions of the Einstein equations.
We will show that for generic simplicial lattices the residual errors can not
be used to distinguish metrics which are solutions of Einstein's equations from
those that are not. We will conclude that either the Regge calculus is an
inconsistent approximation to General Relativity or that it is incorrect to use
residual errors in the discrete equations as a criteria to judge the discrete
equations.Comment: 27 pages, plain TeX, very belated update to match journal articl
Geodesic Deviation in Regge Calculus
Geodesic deviation is the most basic manifestation of the influence of
gravitational fields on matter. We investigate geodesic deviation within the
framework of Regge calculus, and compare the results with the continuous
formulation of general relativity on two different levels. We show that the
continuum and simplicial descriptions coincide when the cumulative effect of
the Regge contributions over an infinitesimal element of area is considered.
This comparison provides a quantitative relation between the curvature of the
continuous description and the deficit angles of Regge calculus. The results
presented might also be of help in developing generic ways of including matter
terms in the Regge equations.Comment: 9 pages. Latex 2e with 5 EPS figures. Submitted to CQ
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