35 research outputs found
Constraint violation in free evolution schemes: comparing BSSNOK with a conformal decomposition of Z4
We compare numerical evolutions performed with the BSSNOK formulation and a
conformal decomposition of a Z4-like formulation of General Relativity. The
important difference between the two formulations is that the Z4 formulation
has a propagating Hamiltonian constraint, whereas BSSNOK has a zero-speed
characteristic variable in the constraint subsystem. In spherical symmetry we
evolve both puncture and neutron star initial data. We demonstrate that the
propagating nature of the Z4 constraints leads to results that compare
favorably with BSSNOK evolutions, especially when matter is present in the
spacetime. From the point of view of implementation the new system is a simple
modification of BSSNOK.Comment: Published in PR
Induced scalarization in boson stars and scalar gravitational radiation
The dynamical evolution of boson stars in scalar-tensor theories of gravity
is considered in the physical (Jordan) frame. We focus on the study of
spontaneous and induced scalarization, for which we take as initial data
configurations on the well-known S-branch of a single boson star in general
relativity. We show that during the scalarization process a strong emission of
scalar radiation occurs. The new stable configurations (S-branch) of a single
boson star within a particular scalar-tensor theory are also presented.Comment: 18 pages, 11 figures. Minor changes to match the published versio
Equilibrium initial data for moving puncture simulations: The stationary 1+log slicing
We propose and explore a "stationary 1+log" slicing condition for the
construction of solutions to Einstein's constraint equations. For stationary
spacetimes, these initial data will give a stationary foliation when evolved
with "moving puncture" gauge conditions that are often used in black hole
evolutions. The resulting slicing is time-independent and agrees with the
slicing generated by being dragged along a time-like Killing vector of the
spacetime. When these initial data are evolved with moving puncture gauge
conditions, numerical errors arising from coordinate evolution are minimized.
In the construction of initial data for binary black holes it is often assumed
that there exists an approximate helical Killing vector that generates the
binary's orbit. We show that, unfortunately, 1+log slices that are stationary
with respect to such a helical Killing vector cannot be asymptotically flat,
unless the spacetime possesses an additional axial Killing vector.Comment: 20 pages, 3 figures, published versio
Gauge conditions for binary black hole puncture data based on an approximate helical Killing vector
We show that puncture data for quasicircular binary black hole orbits allow a
special gauge choice that realizes some of the necessary conditions for the
existence of an approximate helical Killing vector field. Introducing free
parameters for the lapse at the punctures we can satisfy the condition that the
Komar and ADM mass agree at spatial infinity. Several other conditions for an
approximate Killing vector are then automatically satisfied, and the 3-metric
evolves on a timescale smaller than the orbital timescale. The time derivative
of the extrinsic curvature however remains significant. Nevertheless,
quasicircular puncture data are not as far from possessing a helical Killing
vector as one might have expected.Comment: 11 pages, 6 figures, 2 table
Quasi-equilibrium binary black hole sequences for puncture data derived from helical Killing vector conditions
We construct a sequence of binary black hole puncture data derived under the
assumptions (i) that the ADM mass of each puncture as measured in the
asymptotically flat space at the puncture stays constant along the sequence,
and (ii) that the orbits along the sequence are quasi-circular in the sense
that several necessary conditions for the existence of a helical Killing vector
are satisfied. These conditions are equality of ADM and Komar mass at infinity
and equality of the ADM and a rescaled Komar mass at each puncture. In this
paper we explicitly give results for the case of an equal mass black hole
binary without spin, but our approach can also be applied in the general case.
We find that up to numerical accuracy the apparent horizon mass also remains
constant along the sequence and that the prediction for the innermost stable
circular orbit is similar to what has been found with the effective potential
method.Comment: 6 pages, 3 figures, 1 tabl
Conformal-thin-sandwich initial data for a single boosted or spinning black hole puncture
Sequences of initial-data sets representing binary black holes in
quasi-circular orbits have been used to calculate what may be interpreted as
the innermost stable circular orbit. These sequences have been computed with
two approaches. One method is based on the traditional
conformal-transverse-traceless decomposition and locates quasi-circular orbits
from the turning points in an effective potential. The second method uses a
conformal-thin-sandwich decomposition and determines quasi-circular orbits by
requiring the existence of an approximate helical Killing vector. Although the
parameters defining the innermost stable circular orbit obtained from these two
methods differ significantly, both approaches yield approximately the same
initial data, as the separation of the binary system increases. To help
understanding this agreement between data sets, we consider the case of initial
data representing a single boosted or spinning black hole puncture of the
Bowen-York type and show that the conformal-transverse-traceless and
conformal-thin-sandwich methods yield identical data, both satisfying the
conditions for the existence of an approximate Killing vector.Comment: 13 pages, 2 figure
A fast stroboscopic spectral method for rotating systems in numerical relativity
We present a numerical technique for solving evolution equations, as the wave
equation, in the description of rotating astrophysical compact objects in
comoving coordinates, which avoids the problems associated with the light
cylinder. The technique implements a fast spectral matching between two domains
in relative rotation: an inner spherical domain, comoving with the sources and
lying strictly inside the light cylinder, and an outer inertial spherical
shell. Even though the emphasis is placed on spectral techniques, the matching
is independent of the specific manner in which equations are solved inside each
domain, and can be adapted to different schemes. We illustrate the strategy
with some simple but representative examples.Comment: 16 pages, 15 figure
Free and constrained symplectic integrators for numerical general relativity
We consider symplectic time integrators in numerical General Relativity and
discuss both free and constrained evolution schemes. For free evolution of
ADM-like equations we propose the use of the Stoermer-Verlet method, a standard
symplectic integrator which here is explicit in the computationally expensive
curvature terms. For the constrained evolution we give a formulation of the
evolution equations that enforces the momentum constraints in a holonomically
constrained Hamiltonian system and turns the Hamilton constraint function from
a weak to a strong invariant of the system. This formulation permits the use of
the constraint-preserving symplectic RATTLE integrator, a constrained version
of the Stoermer-Verlet method.
The behavior of the methods is illustrated on two effectively 1+1-dimensional
versions of Einstein's equations, that allow to investigate a perturbed
Minkowski problem and the Schwarzschild space-time. We compare symplectic and
non-symplectic integrators for free evolution, showing very different numerical
behavior for nearly-conserved quantities in the perturbed Minkowski problem.
Further we compare free and constrained evolution, demonstrating in our
examples that enforcing the momentum constraints can turn an unstable free
evolution into a stable constrained evolution. This is demonstrated in the
stabilization of a perturbed Minkowski problem with Dirac gauge, and in the
suppression of the propagation of boundary instabilities into the interior of
the domain in Schwarzschild space-time.Comment: 25 pages, 7 figures; This version contains minor clarifications and
correction
Comparing initial-data sets for binary black holes
We compare the results of constructing binary black hole initial data with
three different decompositions of the constraint equations of general
relativity. For each decomposition we compute the initial data using a
superposition of two Kerr-Schild black holes to fix the freely specifiable
data. We find that these initial-data sets differ significantly, with the ADM
energy varying by as much as 5% of the total mass. We find that all
initial-data sets currently used for evolutions might contain unphysical
gravitational radiation of the order of several percent of the total mass. This
is comparable to the amount of gravitational-wave energy observed during the
evolved collision. More astrophysically realistic initial data will require
more careful choices of the freely specifiable data and boundary conditions for
both the metric and extrinsic curvature. However, we find that the choice of
extrinsic curvature affects the resulting data sets more strongly than the
choice of conformal metric.Comment: 18 pages, 12 figures, accepted for publication in Phys. Rev.
Black Hole Mergers and Unstable Circular Orbits
We describe recent numerical simulations of the merger of a class of equal
mass, non-spinning, eccentric binary black hole systems in general relativity.
We show that with appropriate fine-tuning of the initial conditions to a region
of parameter space we denote the threshold of immediate merger, the binary
enters a phase of close interaction in a near-circular orbit, stays there for
an amount of time proportional to logarithmic distance from the threshold in
parameter space, then either separates or merges to form a single Kerr black
hole. To gain a better understanding of this phenomena we study an analogous
problem in the evolution of equatorial geodesics about a central Kerr black
hole. A similar threshold of capture exists for appropriate classes of initial
conditions, and tuning to threshold the geodesics approach one of the unstable
circular geodesics of the Kerr spacetime. Remarkably, with a natural mapping of
the parameters of the geodesic to that of the equal mass system, the scaling
exponent describing the whirl phase of each system turns out to be quite
similar. Armed with this lone piece of evidence that an approximate
correspondence might exist between near-threshold evolution of geodesics and
generic binary mergers, we illustrate how this information can be used to
estimate the cross section and energy emitted in the ultra relativistic black
hole scattering problem. This could eventually be of use in providing estimates
for the related problem of parton collisions at the Large Hadron Collider in
extra dimension scenarios where black holes are produced.Comment: 16 pages, 12 figures; updated to coincide with journal versio