There are many ways to write the Einstein
equations. The most common is the ADM or 3 + 1 form
[23]
which decomposes spacetime into layers of three-dimensional
space-like hypersurfaces, threaded by a time-like normal
congruence
, where we use Greek (Latin) indices to specify spacetime
(spatial) quantities. The general spacetime metric is written
as
It is worth noting that several alternative
formulations of Einstein’s equations have been suggested,
including hyperbolic systems
[136]
which have nice mathematical properties, and conformal traceless
systems
[147, 31]
which make use of a conformal decomposition of the 3-metric and
trace-free part of the extrinsic curvature
. Introducing
with
so that the determinant of
is unity, and
, evolution equations can be written in the conformal traceless
system for
,
,
,
and the conformal connection functions, though not all of these
variables are independent. However, it is not yet entirely clear
which of these methods is best suited for generic problems. For
example, hyperbolic forms are easier to characterize
mathematically than ADM and may potentially be more stable, but
can suffer from greater inaccuracies by introducing additional
equations or higher order derivatives. Conformal treatments are
considered to be generally more stable
[31
], but can be less accurate than traditional ADM for short term
evolutions
[6
]
.
Many numerical methods have been used to solve the Einstein equations, including variants of the leapfrog scheme, the method of McCormack, the two-step Lax-Wendroff method, and the iterative Crank-Nicholson scheme, among others. For a discussion and comparison of the different methods, the reader is referred to [43], where a systematic study was carried out on spherically symmetric black hole spacetimes using traditional ADM, and to [31, 6, 13] (and references therein) which discuss the stability and accuracy of hyperbolic and conformal treatments.
For cosmological simulations, one typically
takes the shift vector to be zero, hence
. However, the shift can be used advantageously in deriving
conditions to maintain the 3-metric in a particular form, and to
simplify the resulting differential equations
[59, 60
]
. See also
[146]
describing an approximate minimum distortion gauge condition
used to help stabilize simulations of general relativistic binary
clusters and neutron stars.
Several options can be implemented for the
lapse function, including geodesic (), algebraic, and mean curvature slicing. The algebraic condition
takes the form
The mean curvature slicing equation is derived
by taking the trace of the extrinsic curvature evolution
equation (11),
A different approach to conventional (i.e., 3
+ 1 ADM) techniques in numerical cosmology has been developed by
Berger and Moncrief
[40]
. For example, they consider Gowdy cosmologies on
with the metric
Although the resulting Einstein equations can be solved in the usual spacetime discretization fashion, an interesting alternative method of solution is the symplectic operator splitting formulation [40, 121] founded on recognizing that the second order equations can be obtained from the variation of a Hamiltonian decomposed into kinetic and potential subhamiltonians,
The symplectic method approximates the evolution operator by although higher order representations are possible. If the two Hamiltonian componentsSymplectic integration methods are applicable to other spacetimes. For example, Berger et al. [39] developed a variation of this approach to explicitly take advantage of exact solutions for scattering between Kasner epochs in Mixmaster models. Their algorithm evolves Mixmaster spacetimes more accurately with larger time steps than previous methods.
A unique approach to numerical cosmology (and numerical relativity in general) is the method of Regge Calculus in which spacetime is represented as a complex of 4-dimensional, geometrically flat simplices. The principles of Einstein’s theory are applied directly to the simplicial geometry to form the curvature, action, and field equations, in contrast to the finite difference approach where the continuum field equations are differenced on a discrete mesh.
A 3-dimensional code implementing Regge
Calculus techniques was developed recently by Gentle and
Miller
[81]
and applied to the Kasner cosmological model. They also describe
a procedure to solve the constraint equations for time asymmetric
initial data on two spacelike hypersurfaces constructed from
tetrahedra, since full 4-dimensional regions or lattices are
used. The new method is analogous to York’s procedure (see
[163]
and Section
6.3) where the conformal metric, trace of the extrinsic curvature,
and momentum variables are all freely specifiable. These early
results are promising in that they have reproduced the continuum
Kasner solution, achieved second order convergence, and sustained
numerical stability. Also, Barnett et al.
[29]
discuss an implicit evolution scheme that allows local (vertex)
calculations for efficient parallelism. However, the Regge
Calculus approach remains to be developed and applied to more
general spacetimes with complex topologies, extended degrees of
freedom, and general source terms.
![]() |
http://www.livingreviews.org/lrr-2001-2 | © Max Planck Society and
the author(s)
Problems/comments to |