26 research outputs found
Constraint preserving boundary conditions for the linearized Einstein equations
The successful construction of long time convergent finite difference schemes approximating highly gravitating systems in general relativity remains an elusive task. The presence of constraints and the introduction of artificial time-like boundaries contribute significantly to the difficulty of this problem. Whereas in the absence of boundaries the Bianchi identities ensure that the constraints vanish during evolution provided that they are satisfied initially, this is no longer true when time-like boundaries are introduced. In this work we consider the linearization around the Minkowski space-time in Cartesian coordinates of the generalized Einstein-Christoffel system and analyze different kinds of boundary conditions that are designed to ensure that the constraints vanish throughout the computational domain: the Neumann, Dirichlet, and Sommerfeld cases. In addition to the situation in which the boundary is aligned with a coordinate surface, we examine the presence of corners in the computational domain. We find that, at a corner, there are compatibility conditions which the boundary data and its derivatives must satisfy and that, in general, achieving consistency of a finite difference scheme can be troublesome. We present several numerical experiments aimed at establishing or confirming the well-posedness or ill-posedness of a problem and the consistency of the numerical boundary conditions at the corners. In the case of a smooth boundary we are able to find stable discretizations for all three cases. However, when a corner is present no stable discretization was found for the Sommerfeld case. Finally, we propose an alternative implementation of the Sommerfeld boundary conditions that would preserve the constraints, offer a good approximation for absorbing boundary conditions, and eliminate the problem of the corners
A rigorous analysis of the cavity equations for the minimum spanning tree
We analyze a new general representation for the Minimum Weight Steiner Tree
(MST) problem which translates the topological connectivity constraint into a
set of local conditions which can be analyzed by the so called cavity equations
techniques. For the limit case of the Spanning tree we prove that the fixed
point of the algorithm arising from the cavity equations leads to the global
optimum.Comment: 5 pages, 1 figur
Exact boundary conditions in numerical relativity using multiple grids: scalar field tests
Cauchy-Characteristic Matching (CCM), the combination of a central 3+1 Cauchy
code with an exterior characteristic code connected across a time-like
interface, is a promising technique for the generation and extraction of
gravitational waves. While it provides a tool for the exact specification of
boundary conditions for the Cauchy evolution, it also allows to follow
gravitational radiation all the way to infinity, where it is unambiguously
defined.
We present a new fourth order accurate finite difference CCM scheme for a
first order reduction of the wave equation around a Schwarzschild black hole in
axisymmetry. The matching at the interface between the Cauchy and the
characteristic regions is done by transfering appropriate characteristic/null
variables. Numerical experiments indicate that the algorithm is fourth order
convergent. As an application we reproduce the expected late-time tail decay
for the scalar field.Comment: 14 pages, 5 figures. Included changes suggested by referee
Extended Lifetime in Computational Evolution of Isolated Black Holes
Solving the 4-d Einstein equations as evolution in time requires solving
equations of two types: the four elliptic initial data (constraint) equations,
followed by the six second order evolution equations. Analytically the
constraint equations remain solved under the action of the evolution, and one
approach is to simply monitor them ({\it unconstrained} evolution).
The problem of the 3-d computational simulation of even a single isolated
vacuum black hole has proven to be remarkably difficult. Recently, we have
become aware of two publications that describe very long term evolution, at
least for single isolated black holes. An essential feature in each of these
results is {\it constraint subtraction}. Additionally, each of these approaches
is based on what we call "modern," hyperbolic formulations of the Einstein
equations. It is generally assumed, based on computational experience, that the
use of such modern formulations is essential for long-term black hole
stability. We report here on comparable lifetime results based on the much
simpler ("traditional") - formulation.
We have also carried out a series of {\it constrained} 3-d evolutions of
single isolated black holes. We find that constraint solution can produce
substantially stabilized long-term single hole evolutions. However, we have
found that for large domains, neither constraint-subtracted nor constrained
- evolutions carried out in Cartesian coordinates admit
arbitrarily long-lived simulations. The failure appears to arise from features
at the inner excision boundary; the behavior does generally improve with
resolution.Comment: 20 pages, 6 figure
Constraint propagation equations of the 3+1 decomposition of f(R) gravity
Theories of gravity other than general relativity (GR) can explain the
observed cosmic acceleration without a cosmological constant. One such class of
theories of gravity is f(R). Metric f(R) theories have been proven to be
equivalent to Brans-Dicke (BD) scalar-tensor gravity without a kinetic term.
Using this equivalence and a 3+1 decomposition of the theory it has been shown
that metric f(R) gravity admits a well-posed initial value problem. However, it
has not been proven that the 3+1 evolution equations of metric f(R) gravity
preserve the (hamiltonian and momentum) constraints. In this paper we show that
this is indeed the case. In addition, we show that the mathematical form of the
constraint propagation equations in BD-equilavent f(R) gravity and in f(R)
gravity in both the Jordan and Einstein frames, is exactly the same as in the
standard ADM 3+1 decomposition of GR. Finally, we point out that current
numerical relativity codes can incorporate the 3+1 evolution equations of
metric f(R) gravity by modifying the stress-energy tensor and adding an
additional scalar field evolution equation. We hope that this work will serve
as a starting point for relativists to develop fully dynamical codes for valid
f(R) models.Comment: 25 pages, matches published version in CQG, references update