11 research outputs found
Kerr-Schild type initial data for black holes with angular momenta
Generalizing previous work we propose how to superpose spinning black holes
in a Kerr-Schild initial slice. This superposition satisfies several physically
meaningful limits, including the close and the far ones. Further we consider
the close limit of two black holes with opposite angular momenta and explicitly
solve the constraint equations in this case. Evolving the resulting initial
data with a linear code, we compute the radiated energy as a function of the
masses and the angular momenta of the black holes.Comment: 13 pages, 3 figures. Revised version. To appear in Classical and
Quantum Gravit
Improved outer boundary conditions for Einstein's field equations
In a recent article, we constructed a hierarchy B_L of outer boundary
conditions for Einstein's field equations with the property that, for a
spherical outer boundary, it is perfectly absorbing for linearized
gravitational radiation up to a given angular momentum number L. In this
article, we generalize B_2 so that it can be applied to fairly general
foliations of spacetime by space-like hypersurfaces and general outer boundary
shapes and further, we improve B_2 in two steps: (i) we give a local boundary
condition C_2 which is perfectly absorbing including first order contributions
in 2M/R of curvature corrections for quadrupolar waves (where M is the mass of
the spacetime and R is a typical radius of the outer boundary) and which
significantly reduces spurious reflections due to backscatter, and (ii) we give
a non-local boundary condition D_2 which is exact when first order corrections
in 2M/R for both curvature and backscatter are considered, for quadrupolar
radiation.Comment: accepted Class. Quant. Grav. numerical relativity special issue; 17
pages and 1 figur
Conformal diagrams for the gravitational collapse of a spherical dust cloud
We present an algorithm for the construction of conformal coordinates in the
interior of a spherically symmetric, collapsing matter cloud in general
relativity. This algorithm is based on the numerical integration of the radial
null geodesics and a local analysis of their behavior close to the singularity.
As an application, we consider a collapsing spherical dust cloud, generate the
corresponding conformal diagram and analyze the structure of the resulting
singularity. A new bound on the initial data which guarantees that the
singularity is visible from future null infinity is also obtained.Comment: added a new subsection with a phase space analysis, 23 pages, 8
figure
Exploiting gauge and constraint freedom in hyperbolic formulations of Einstein's equations
We present new many-parameter families of strongly and symmetric hyperbolic
formulations of Einstein's equations that include quite general algebraic and
live gauge conditions for the lapse. The first system that we present has 30
variables and incorporates an algebraic relationship between the lapse and the
determinant of the three metric that generalizes the densitized lapse
prescription. The second system has 34 variables and uses a family of live
gauges that generalizes the Bona-Masso slicing conditions. These systems have
free parameters even after imposing hyperbolicity and are expected to be useful
in 3D numerical evolutions. We discuss under what conditions there are no
superluminal characteristic speeds
Towards absorbing outer boundaries in General Relativity
We construct exact solutions to the Bianchi equations on a flat spacetime
background. When the constraints are satisfied, these solutions represent in-
and outgoing linearized gravitational radiation. We then consider the Bianchi
equations on a subset of flat spacetime of the form [0,T] x B_R, where B_R is a
ball of radius R, and analyze different kinds of boundary conditions on
\partial B_R. Our main results are: i) We give an explicit analytic example
showing that boundary conditions obtained from freezing the incoming
characteristic fields to their initial values are not compatible with the
constraints. ii) With the help of the exact solutions constructed, we determine
the amount of artificial reflection of gravitational radiation from
constraint-preserving boundary conditions which freeze the Weyl scalar Psi_0 to
its initial value. For monochromatic radiation with wave number k and arbitrary
angular momentum number l >= 2, the amount of reflection decays as 1/(kR)^4 for
large kR. iii) For each L >= 2, we construct new local constraint-preserving
boundary conditions which perfectly absorb linearized radiation with l <= L.
(iv) We generalize our analysis to a weakly curved background of mass M, and
compute first order corrections in M/R to the reflection coefficients for
quadrupolar odd-parity radiation. For our new boundary condition with L=2, the
reflection coefficient is smaller than the one for the freezing Psi_0 boundary
condition by a factor of M/R for kR > 1.04. Implications of these results for
numerical simulations of binary black holes on finite domains are discussed.Comment: minor revisions, 30 pages, 6 figure
A minimization problem for the lapse and the initial-boundary value problem for Einstein's field equations
Cauchy horizon stability in a collapsing spherical dust cloud: I. Geometric optics approximation and spherically symmetric test fields
Hyperbolicity of the BSSN system of Einstein evolution equations
We discuss an equivalence between the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation of the Einstein evolution equations, a subfamily of the Kidder-Scheel-Teukolsky formulation, and other strongly or symmetric hyperbolic first order systems with fixed shift and densitized lapse. This allows us to show under which conditions the BSSN system is, in a sense to be discussed, hyperbolic. This desirable property may account in part for the empirically observed better behavior of the BSSN formulation in numerical evolutions involving black holes