45 research outputs found
Osculating orbits in Schwarzschild spacetime, with an application to extreme mass-ratio inspirals
We present a method to integrate the equations of motion that govern bound,
accelerated orbits in Schwarzschild spacetime. At each instant the true
worldline is assumed to lie tangent to a reference geodesic, called an
osculating orbit, such that the worldline evolves smoothly from one such
geodesic to the next. Because a geodesic is uniquely identified by a set of
constant orbital elements, the transition between osculating orbits corresponds
to an evolution of the elements. In this paper we derive the evolution
equations for a convenient set of orbital elements, assuming that the force
acts only within the orbital plane; this is the only restriction that we impose
on the formalism, and we do not assume that the force must be small. As an
application of our method, we analyze the relative motion of two massive
bodies, assuming that one body is much smaller than the other. Using the hybrid
Schwarzschild/post-Newtonian equations of motion formulated by Kidder, Will,
and Wiseman, we treat the unperturbed motion as geodesic in a Schwarzschild
spacetime whose mass parameter is equal to the system's total mass. The force
then consists of terms that depend on the system's reduced mass. We highlight
the importance of conservative terms in this force, which cause significant
long-term changes in the time-dependence and phase of the relative orbit. From
our results we infer some general limitations of the radiative approximation to
the gravitational self-force, which uses only the dissipative terms in the
force.Comment: 18 pages, 6 figures, final version to be published in Physical Review
Circular orbits of corotating binary black holes: comparison between analytical and numerical results
We compare recent numerical results, obtained within a ``helical Killing
vector'' (HKV) approach, on circular orbits of corotating binary black holes to
the analytical predictions made by the effective one body (EOB) method (which
has been recently extended to the case of spinning bodies). On the scale of the
differences between the results obtained by different numerical methods, we
find good agreement between numerical data and analytical predictions for
several invariant functions describing the dynamical properties of circular
orbits. This agreement is robust against the post-Newtonian accuracy used for
the analytical estimates, as well as under choices of resummation method for
the EOB ``effective potential'', and gets better as one uses a higher
post-Newtonian accuracy. These findings open the way to a significant
``merging'' of analytical and numerical methods, i.e. to matching an EOB-based
analytical description of the (early and late) inspiral, up to the beginning of
the plunge, to a numerical description of the plunge and merger. We illustrate
also the ``flexibility'' of the EOB approach, i.e. the possibility of
determining some ``best fit'' values for the analytical parameters by
comparison with numerical data.Comment: Minor revisions, accepted for publication in Phys. Rev. D, 19 pages,
6 figure
A generalized Damour-Navier-Stokes equation applied to trapping horizons
An identity is derived from Einstein equation for any hypersurface H which
can be foliated by spacelike two-dimensional surfaces. In the case where the
hypersurface is null, this identity coincides with the two-dimensional
Navier-Stokes-like equation obtained by Damour in the membrane approach to a
black hole event horizon. In the case where H is spacelike or null and the
2-surfaces are marginally trapped, this identity applies to Hayward's trapping
horizons and to the related dynamical horizons recently introduced by Ashtekar
and Krishnan. The identity involves a normal fundamental form (normal
connection 1-form) of the 2-surface, which can be viewed as a generalization to
non-null hypersurfaces of the Hajicek 1-form used by Damour. This 1-form is
also used to define the angular momentum of the horizon. The generalized
Damour-Navier-Stokes equation leads then to a simple evolution equation for the
angular momentum.Comment: Added subsection IV.D; corrected an error in Appendix A; added some
references; accepted for publication in Phys. Rev. D (16 pages, 4 EPS
figures
Relativistic theory of tidal Love numbers
In Newtonian gravitational theory, a tidal Love number relates the mass
multipole moment created by tidal forces on a spherical body to the applied
tidal field. The Love number is dimensionless, and it encodes information about
the body's internal structure. We present a relativistic theory of Love
numbers, which applies to compact bodies with strong internal gravities; the
theory extends and completes a recent work by Flanagan and Hinderer, which
revealed that the tidal Love number of a neutron star can be measured by
Earth-based gravitational-wave detectors. We consider a spherical body deformed
by an external tidal field, and provide precise and meaningful definitions for
electric-type and magnetic-type Love numbers; and these are computed for
polytropic equations of state. The theory applies to black holes as well, and
we find that the relativistic Love numbers of a nonrotating black hole are all
zero.Comment: 25 pages, 8 figures, many tables; final version to be published in
Physical Review
Area evolution, bulk viscosity and entropy principles for dynamical horizons
We derive from Einstein equation an evolution law for the area of a trapping
or dynamical horizon. The solutions to this differential equation show a causal
behavior. Moreover, in a viscous fluid analogy, the equation can be interpreted
as an energy balance law, yielding to a positive bulk viscosity. These two
features contrast with the event horizon case, where the non-causal evolution
of the area and the negative bulk viscosity require teleological boundary
conditions. This reflects the local character of trapping horizons as opposed
to event horizons. Interpreting the area as the entropy, we propose to use an
area/entropy evolution principle to select a unique dynamical horizon and time
slicing in the Cauchy evolution of an initial marginally trapped surface.Comment: Some references added, 5 pages, Phys. Rev. D, in pres
Inner boundary conditions for black hole Initial Data derived from Isolated Horizons
We present a set of boundary conditions for solving the elliptic equations in
the Initial Data Problem for space-times containing a black hole, together with
a number of constraints to be satisfied by the otherwise freely specifiable
standard parameters of the Conformal Thin Sandwich formulation. These
conditions altogether are sufficient for the construction of a horizon that is
instantaneously in equilibrium in the sense of the Isolated Horizons formalism.
We then investigate the application of these conditions to the Initial Data
Problem of binary black holes and discuss the relation of our analysis with
other proposals that exist in the literature.Comment: 13 pages. Major general revision. Section V comparing with previous
approaches restructured; discussion on the lapse boundary condition extended.
Appendix with some technical details added. Version accepted for publication
in Phys.Rev.
Spherical Self-Similar Solutions in Einstein-Multi-Scalar Gravity
We consider a general non-linear sigma model coupled to Einstein gravity and
show that in spherical symmetry and for a simple realization of
self-similarity, the spacetime can be completely determined. We also examine
some more specific matter models and discuss their relation to critical
collapse.Comment: 11 pages, 1 figur
Binary black holes in circular orbits. II. Numerical methods and first results
We present the first results from a new method for computing spacetimes
representing corotating binary black holes in circular orbits. The method is
based on the assumption of exact equilibrium. It uses the standard 3+1
decomposition of Einstein equations and conformal flatness approximation for
the 3-metric. Contrary to previous numerical approaches to this problem, we do
not solve only the constraint equations but rather a set of five equations for
the lapse function, the conformal factor and the shift vector. The orbital
velocity is unambiguously determined by imposing that, at infinity, the metric
behaves like the Schwarzschild one, a requirement which is equivalent to the
virial theorem. The numerical scheme has been implemented using multi-domain
spectral methods and passed numerous tests. A sequence of corotating black
holes of equal mass is calculated. Defining the sequence by requiring that the
ADM mass decrease is equal to the angular momentum decrease multiplied by the
orbital angular velocity, it is found that the area of the apparent horizons is
constant along the sequence. We also find a turning point in the ADM mass and
angular momentum curves, which may be interpreted as an innermost stable
circular orbit (ISCO). The values of the global quantities at the ISCO,
especially the orbital velocity, are in much better agreement with those from
third post-Newtonian calculations than with those resulting from previous
numerical approaches.Comment: 27 pages, 20 PostScript figures, improved presentation of the
regularization procedure for the shift vector, new section devoted to the
check of the momentum constraint, references added + minor corrections,
accepted for publication in Phys. Rev.
Nonrotating black hole in a post-Newtonian tidal environment
We examine the motion and tidal dynamics of a nonrotating black hole placed
within a post-Newtonian external spacetime. The tidal perturbation created by
the external environment is treated as a small perturbation. At a large
distance from the black hole, the gravitational field of the external
distribution of matter is assumed to be sufficiently weak to be adequately
described by the (first) post-Newtonian approximation to general relativity.
There, the black hole is treated as a monopole contribution to the total
gravitational field. There exists an overlap in the domains of validity of each
description, and the black-hole and post-Newtonian metrics are matched in the
overlap. The matching procedure produces the equations of motion for the black
hole and the gravito-electric and gravito-magnetic tidal fields acting on the
black hole. We first calculate the equations of motion and tidal fields by
making no assumptions regarding the nature of the post-Newtonian environment;
this could contain a continuous distribution of matter or any number of
condensed bodies. We next specialize our discussion to a situation in which the
black hole is a member of a post-Newtonian two-body system. As an application
of our results, we examine the geometry of the deformed event horizon and
calculate the tidal heating of the black hole, the rate at which it acquires
mass as a result of its tidal interaction with the companion body.Comment: 27 pages, 1 figur
A spin-4 analog of 3D massive gravity
A 6th-order, but ghost-free, gauge-invariant action is found for a 4th-rank
symmetric tensor potential in a three-dimensional (3D) Minkowski spacetime. It
propagates two massive modes of spin 4 that are interchanged by parity, and is
thus a spin-4 analog of linearized "new massive gravity". Also found are
ghost-free spin-4 analogs of linearized "topologically massive gravity" and
"new topologically massive gravity", of 5th- and 8th-order respectively.Comment: 16 pages, v2 : version published in Class. Quant. Gra