1,520 research outputs found
Constraint Propagation of -adjusted Formulation - Another Recipe for Robust ADM Evolution System
With a purpose of constructing a robust evolution system against numerical
instability for integrating the Einstein equations, we propose a new
formulation by adjusting the ADM evolution equations with constraints. We apply
an adjusting method proposed by Fiske (2004) which uses the norm of the
constraints, C2. One of the advantages of this method is that the effective
signature of adjusted terms (Lagrange multipliers) for constraint-damping
evolution is pre-determined. We demonstrate this fact by showing the
eigenvalues of constraint propagation equations. We also perform numerical
tests of this adjusted evolution system using polarized Gowdy-wave propagation,
which show robust evolutions against the violation of the constraints than that
of the standard ADM formulation.Comment: 11 pages, 5 figures. To be published in Phys. Rev.
Constraints and Reality Conditions in the Ashtekar Formulation of General Relativity
We show how to treat the constraints and reality conditions in the
-ADM (Ashtekar) formulation of general relativity, for the case of a
vacuum spacetime with a cosmological constant. We clarify the difference
between the reality conditions on the metric and on the triad. Assuming the
triad reality condition, we find a new variable, allowing us to solve the gauge
constraint equations and the reality conditions simultaneously.Comment: LaTeX file, 12 pages, no figures; to appear in Classical and Quantum
Gravit
Adjusted ADM systems and their expected stability properties: constraint propagation analysis in Schwarzschild spacetime
In order to find a way to have a better formulation for numerical evolution
of the Einstein equations, we study the propagation equations of the
constraints based on the Arnowitt-Deser-Misner formulation. By adjusting
constraint terms in the evolution equations, we try to construct an
"asymptotically constrained system" which is expected to be robust against
violation of the constraints, and to enable a long-term stable and accurate
numerical simulation. We first provide useful expressions for analyzing
constraint propagation in a general spacetime, then apply it to Schwarzschild
spacetime. We search when and where the negative real or non-zero imaginary
eigenvalues of the homogenized constraint propagation matrix appear, and how
they depend on the choice of coordinate system and adjustments. Our analysis
includes the proposal of Detweiler (1987), which is still the best one
according to our conjecture but has a growing mode of error near the horizon.
Some examples are snapshots of a maximally sliced Schwarzschild black hole. The
predictions here may help the community to make further improvements.Comment: 23 pages, RevTeX4, many figures. Revised version. Added subtitle,
reduced figures, rephrased introduction, and a native checked. :-
Constraint propagation in the family of ADM systems
The current important issue in numerical relativity is to determine which
formulation of the Einstein equations provides us with stable and accurate
simulations. Based on our previous work on "asymptotically constrained"
systems, we here present constraint propagation equations and their eigenvalues
for the Arnowitt-Deser-Misner (ADM) evolution equations with additional
constraint terms (adjusted terms) on the right hand side. We conjecture that
the system is robust against violation of constraints if the amplification
factors (eigenvalues of Fourier-component of the constraint propagation
equations) are negative or pure-imaginary. We show such a system can be
obtained by choosing multipliers of adjusted terms. Our discussion covers
Detweiler's proposal (1987) and Frittelli's analysis (1997), and we also
mention the so-called conformal-traceless ADM systems.Comment: 11 pages, RevTeX, 2 eps figure
Generation of scalar-tensor gravity effects in equilibrium state boson stars
Boson stars in zero-, one-, and two-node equilibrium states are modeled
numerically within the framework of Scalar-Tensor Gravity. The complex scalar
field is taken to be both massive and self-interacting. Configurations are
formed in the case of a linear gravitational scalar coupling (the Brans-Dicke
case) and a quadratic coupling which has been used previously in a cosmological
context. The coupling parameters and asymptotic value for the gravitational
scalar field are chosen so that the known observational constraints on
Scalar-Tensor Gravity are satisfied. It is found that the constraints are so
restrictive that the field equations of General Relativity and Scalar-Tensor
gravity yield virtually identical solutions. We then use catastrophe theory to
determine the dynamically stable configurations. It is found that the maximum
mass allowed for a stable state in Scalar-Tensor gravity in the present
cosmological era is essentially unchanged from that of General Relativity. We
also construct boson star configurations appropriate to earlier cosmological
eras and find that the maximum mass for stable states is smaller than that
predicted by General Relativity, and the more so for earlier eras. However, our
results also show that if the cosmological era is early enough then only states
with positive binding energy can be constructed.Comment: 20 pages, RevTeX, 11 figures, to appear in Class. Quantum Grav.,
comments added, refs update
Finding Principal Null Direction for Numerical Relativists
We present a new method for finding principal null directions (PNDs). Because
our method assumes as input the intrinsic metric and extrinsic curvature of a
spacelike hypersurface, it should be particularly useful to numerical
relativists. We illustrate our method by finding the PNDs of the
Kastor-Traschen spacetimes, which contain arbitrarily many black holes in
a de Sitter back-ground.Comment: 10 pages, LaTeX style, WU-AP/38/93. Figures are available (hard
copies) upon requests [[email protected] (H.Shinkai)
Formation of naked singularities in five-dimensional space-time
We numerically investigate the gravitational collapse of collisionless
particles in spheroidal configurations both in four and five-dimensional (5D)
space-time. We repeat the simulation performed by Shapiro and Teukolsky (1991)
that announced an appearance of a naked singularity, and also find that the
similar results in 5D version. That is, in a collapse of a highly prolate
spindle, the Kretschmann invariant blows up outside the matter and no apparent
horizon forms. We also find that the collapses in 5D proceed rapidly than in
4D, and the critical prolateness for appearance of apparent horizon in 5D is
loosened compared to 4D cases. We also show how collapses differ with spatial
symmetries comparing 5D evolutions in single-axisymmetry, SO(3), and those in
double-axisymmetry, U(1)U(1).Comment: 5 pages, 5 figures, To be published in Phys. Rev.
Hyperbolic formulations and numerical relativity II: Asymptotically constrained systems of the Einstein equations
We study asymptotically constrained systems for numerical integration of the
Einstein equations, which are intended to be robust against perturbative errors
for the free evolution of the initial data. First, we examine the previously
proposed "-system", which introduces artificial flows to constraint
surfaces based on the symmetric hyperbolic formulation. We show that this
system works as expected for the wave propagation problem in the Maxwell system
and in general relativity using Ashtekar's connection formulation. Second, we
propose a new mechanism to control the stability, which we call the ``adjusted
system". This is simply obtained by adding constraint terms in the dynamical
equations and adjusting its multipliers. We explain why a particular choice of
multiplier reduces the numerical errors from non-positive or pure-imaginary
eigenvalues of the adjusted constraint propagation equations. This ``adjusted
system" is also tested in the Maxwell system and in the Ashtekar's system. This
mechanism affects more than the system's symmetric hyperbolicity.Comment: 16 pages, RevTeX, 9 eps figures, added Appendix B and minor changes,
to appear in Class. Quant. Gra
Algebraic stability analysis of constraint propagation
The divergence of the constraint quantities is a major problem in
computational gravity today. Apparently, there are two sources for constraint
violations. The use of boundary conditions which are not compatible with the
constraint equations inadvertently leads to 'constraint violating modes'
propagating into the computational domain from the boundary. The other source
for constraint violation is intrinsic. It is already present in the initial
value problem, i.e. even when no boundary conditions have to be specified. Its
origin is due to the instability of the constraint surface in the phase space
of initial conditions for the time evolution equations. In this paper, we
present a technique to study in detail how this instability depends on gauge
parameters. We demonstrate this for the influence of the choice of the time
foliation in context of the Weyl system. This system is the essential
hyperbolic part in various formulations of the Einstein equations.Comment: 25 pages, 5 figures; v2: small additions, new reference, publication
number, classification and keywords added, address fixed; v3: update to match
journal versio
Fate of Kaluza-Klein Bubble
We numerically study classical time evolutions of Kaluza-Klein bubble
space-time which has negative energy after a decay of vacuum. As the zero
energy Witten's bubble space-time, where the bubble expands infinitely, the
subsequent evolutions of Brill and Horowitz's momentarily static initial data
show that the bubble will expand in terms of the area. At first glance, this
result may support Corley and Jacobson's conjecture that the bubble will expand
forever as well as the Witten's bubble. The irregular signatures, however, can
be seen in the behavior of the lapse function in the maximal slicing gauge and
the divergence of the Kretchman invariant. Since there is no appearance of the
apparent horizon, we suspect an appearance of a naked singularity as the final
fate of this space-time.Comment: 13 pages including 10 figures, RevTeX, epsf.sty. CGPG-99/12-8,
RESCEU-6/00 and DAMTP-2000-30. To appear in Phys. Rev.
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