63 research outputs found
A complete gauge-invariant formalism for arbitrary second-order perturbations of a Schwarzschild black hole
Using recently developed efficient symbolic manipulations tools, we present a
general gauge-invariant formalism to study arbitrary radiative
second-order perturbations of a Schwarzschild black hole. In particular, we
construct the second order Zerilli and Regge-Wheeler equations under the
presence of any two first-order modes, reconstruct the perturbed metric in
terms of the master scalars, and compute the radiated energy at null infinity.
The results of this paper enable systematic studies of generic second order
perturbations of the Schwarzschild spacetime. In particular, studies of
mode-mode coupling and non-linear effects in gravitational radiation, the
second-order stability of the Schwarzschild spacetime, or the geometry of the
black hole horizon.Comment: 14 page
Late-time Kerr tails: generic and non-generic initial data sets, "up" modes, and superposition
Three interrelated questions concerning Kerr spacetime late-time scalar-field
tails are considered numerically, specifically the evolutions of generic and
non-generic initial data sets, the excitation of "up" modes, and the resolution
of an apparent paradox related to the superposition principle. We propose to
generalize the Barack-Ori formula for the decay rate of any tail multipole
given a generic initial data set, to the contribution of any initial multipole
mode. Our proposal leads to a much simpler expression for the late-time power
law index. Specifically, we propose that the late-time decay rate of the
spherical harmonic multipole moment because of an initial
multipole is independent of the azimuthal number , and is
given by , where for and
for . We also show explicitly that the angular symmetry group of
a multipole does not determine its late-time decay rate.Comment: 12 pages, 13 figures, 4 tables. Substantially revised manuscrip
Mode coupling of Schwarzschild perturbations: Ringdown frequencies
Within linearized perturbation theory, black holes decay to their final
stationary state through the well-known spectrum of quasinormal modes. Here we
numerically study whether nonlinearities change this picture. For that purpose
we study the ringdown frequencies of gauge-invariant second-order gravitational
perturbations induced by self-coupling of linearized perturbations of
Schwarzschild black holes. We do so through high-accuracy simulations in the
time domain of first and second-order Regge-Wheeler-Zerilli type equations, for
a variety of initial data sets. We consider first-order even-parity
perturbations and odd-parity ones, and all
the multipoles that they generate through self-coupling. For all of them and
all the initial data sets considered we find that ---in contrast to previous
predictions in the literature--- the numerical decay frequencies of
second-order perturbations are the same ones of linearized theory, and we
explain the observed behavior. This would indicate, in particular, that when
modeling or searching for ringdown gravitational waves, appropriately including
the standard quasinormal modes already takes into account nonlinear effects
How much energy do closed timelike curves in 2+1 spacetimes need?
By noticing that, in open 2+1 gravity, polarized surfaces cannot converge in
the presence of timelike total energy momentum (except for a rotation of 2 pi),
we give a simple argument which shows that, quite generally, closed timelike
curves cannot exist in the presence of such energy condition.Comment: 3 pages, with no figures. Accepted in PRD as Rapid Communicatio
Exotic spacetimes, superconducting strings with linear momentum, and (not quite) all that
We derive the general exact vacuum metrics associated with a stationary (non
static), non rotating, cylindrically symmetric source. An analysis of the
geometry described by these vacuum metrics shows that they contain a subfamily
of metrics that, although admitting a consistent time orientation, display
"exotic" properties, such as "trapping" of geodesics and closed causal curves
through every point. The possibility that such spacetimes could be generated by
a superconducting string, endowed with a neutral current and momentum, has
recently been considered by Thatcher and Morgan. Our results, however, differ
from those found by Thatcher and Morgan, and the discrepancy is explained. We
also analyze the general possibility of constructing physical sources for the
exotic metrics, and find that, under certain restrictions, they must always
violate the dominant energy condition (DEC). We illustrate our results by
explicitly analyzing the case of concentric shells, where we find that in all
cases the external vacuum metric is non exotic if the matter in the shells
satisfies the DEC.Comment: 13 pages with no figures. Accepted in PR
Hamiltonian Relaxation
Due to the complexity of the required numerical codes, many of the new
formulations for the evolution of the gravitational fields in numerical
relativity are not tested on binary evolutions. We introduce in this paper a
new testing ground for numerical methods based on the simulation of binary
neutron stars. This numerical setup is used to develop a new technique, the
Hamiltonian relaxation (HR), that is benchmarked against the currently most
stable simulations based on the BSSN method. We show that, while the length of
the HR run is somewhat shorter than the equivalent BSSN simulation, the HR
technique improves the overall quality of the simulation, not only regarding
the satisfaction of the Hamiltonian constraint, but also the behavior of the
total angular momentum of the binary. The latest quantity agrees well with
post-Newtonian estimations for point-mass binaries in circular orbits.Comment: More detailed description of the numerical implementation added and
some typos corrected. Version accepted for publication in Class. and Quantum
Gravit
A numerical study of the quasinormal mode excitation of Kerr black holes
We present numerical results from three-dimensional evolutions of scalar perturbations of Kerr black holes. Our simulations make use of a high-order accurate multi-block code which naturally allows for fixed adaptivity and smooth inner (excision) and outer boundaries. We focus on the quasinormal ringing phase, presenting a systematic method for extraction of the quasinormal mode frequencies and amplitudes and comparing our results against perturbation theory. The amplitude of each mode depends exponentially on the starting time of the quasinormal regime, which is not defined unambiguously. We show that this time-shift problem can be circumvented by looking at appropriately chosen relative mode amplitudes. From our simulations we extract the quasinormal frequencies and the relative and absolute amplitudes of corotating and counterrotating modes (including overtones in the corotating case). We study the dependence of these amplitudes on the shape of the initial perturbation, the angular dependence of the mode and the black hole spin, comparing against results from perturbation theory in the so-called asymptotic approximation. We also compare the quasinormal frequencies from our numerical simulations with predictions from perturbation theory, finding excellent agreement. Finally we study under what conditions the relative amplitude between given pairs of modes gets maximally excited and present a quantitative analysis of rotational mode-mode coupling. The main conclusions and techniques of our analysis are quite general and, as such, should be of interest in the study of ringdown gravitational waves produced by astrophysical gravitational wave sources
Late-time Kerr tails revisited
The decay rate of late time tails in the Kerr spacetime have been the cause
of numerous conflicting results, both analytical and numerical. In particular,
there is much disagreement on whether the decay rate of an initially pure
multipole moment is according to , where
is the least multipole moment whose excitation is not disallowed,
or whether the decay rate is according to , where . We do
careful 2+1D numerical simulations, and explain the various results. In
particular, we show that pure multipole outgoing initial data in either
Boyer--Lindquist on ingoing Kerr coordinates on the corresponding slices lead
to the same late time tail behavior. We also show that similar initial data
specified in terms of the Poisson spherical coordinates lead to the simpler
late time tail. We generalize the rule to
subdominant modes, and also study the behavior of non--axisymmetric initial
data. We discuss some of the causes for possible errors in 2+1D simulations,
demonstrate that our simulations are free of those errors, and argue that some
conflicting past results may be attributed to them.Comment: 16 pages, 26 figures, 2 tables; Many changes compared with previous
versio
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