119 research outputs found
Geometrical optics for scalar, electromagnetic and gravitational waves in curved spacetime
The geometrical-optics expansion reduces the problem of solving wave
equations to one of solving transport equations along rays. Here we consider
scalar, electromagnetic and gravitational waves propagating on a curved
spacetime in general relativity. We show that each is governed by a wave
equation with the same principal part. It follows that: each wave propagates at
the speed of light along rays (null generators of hypersurfaces of constant
phase); the square of the wave amplitude varies in inverse proportion to the
cross section of the beam; and the polarization is parallel-propagated along
the ray (the Skrotskii/Rytov effect). We show that the optical scalars for a
beam, and various Newman-Penrose scalars describing a parallel-propagated null
tetrad, can be found by solving transport equations in a second-order
formulation. Unlike the Sachs equations, this formulation makes it
straightforward to find such scalars beyond the first conjugate point of a
congruence, where neighbouring rays cross, and the scalars diverge. We discuss
differential precession across the beam which leads to a modified phase in the
geometrical-optics expansion.Comment: 17 pages, 1 figure. Proceedings for IV Amazonian Symposium on
Physics, Belem, Brazil at UFPA on 18-22 Sep 201
Self-force via Green functions and worldline integration
A compact object moving in curved spacetime interacts with its own
gravitational field. This leads to both dissipative and conservative
corrections to the motion, which can be interpreted as a self-force acting on
the object. The original formalism describing this self-force relied heavily on
the Green function of the linear differential operator that governs
gravitational perturbations. However, because the global calculation of Green
functions in non-trivial black hole spacetimes has been an open problem until
recently, alternative methods were established to calculate self-force effects
using sophisticated regularization techniques that avoid the computation of the
global Green function. We present a method for calculating the self-force that
employs the global Green function and is therefore closely modeled after the
original self-force expressions. Our quantitative method involves two stages:
(i) numerical approximation of the retarded Green function in the background
spacetime; (ii) evaluation of convolution integrals along the worldline of the
object. This novel approach can be used along arbitrary worldlines, including
those currently inaccessible to more established computational techniques.
Furthermore, it yields geometrical insight into the contributions to
self-interaction from curved geometry (back-scattering) and trapping of null
geodesics. We demonstrate the method on the motion of a scalar charge in
Schwarzschild spacetime. This toy model retains the physical history-dependence
of the self-force but avoids gauge issues and allows us to focus on basic
principles. We compute the self-field and self-force for many worldlines
including accelerated circular orbits, eccentric orbits at the separatrix, and
radial infall. This method, closely modeled after the original formalism,
provides a promising complementary approach to the self-force problem.Comment: 18 pages, 9 figure
Conversion of electromagnetic and gravitational waves by a charged black hole
In a strong electromagnetic field, gravitational waves are converted into
electromagnetic waves of the same frequency, and vice versa. Here we calculate
the scattering and conversion cross sections for a planar wave impinging upon a
Reissner-Nordstr\"om black hole in vacuum, using the partial-wave expansion and
numerical methods. We show that, at long wavelengths, the conversion cross
section matches that computed by Feynman-diagram techniques. At short
wavelengths, the essential features are captured by a geometric-optics
approximation. We demonstrate that the converted flux can exceed the scattered
flux at large scattering angles, for highly-charged black holes. In the
short-wavelength regime, the conversion effect may be understood in terms of a
phase that accumulates along a ray. We compute the scattering angle for which
the converted and scattered fluxes are equal, as a function of charge-to-mass
ratio. We show that this scattering angle approaches degrees in the
extremal limit.Comment: 20 pages, 7 figures. Added a proof that the angle for half-conversion
is 90 degrees in the extremal case Q=
Gravitational self-torque and spin precession in compact binaries
We calculate the effect of self-interaction on the "geodetic" spin precession
of a compact body in a strong-field orbit around a black hole. Specifically, we
consider the spin precession angle per radian of orbital revolution for
a particle carrying mass and spin in a circular orbit
around a Schwarzschild black hole of mass . We compute
through in perturbation theory, i.e, including the correction
(obtained numerically) due to the torque exerted by the
conservative piece of the gravitational self-field. Comparison with a
post-Newtonian (PN) expression for , derived here through 3PN
order, shows good agreement but also reveals strong-field features which are
not captured by the latter approximation. Our results can inform
semi-analytical models of the strong-field dynamics in astrophysical binaries,
important for ongoing and future gravitational-wave searches.Comment: 5 pages, 1 table, 1 figure. Minor changes to match published versio
Metric perturbations of Kerr spacetime in Lorenz gauge: Circular equatorial orbits
We construct the metric perturbation in Lorenz gauge for a compact body on a
circular equatorial orbit of a rotating black hole (Kerr) spacetime, using a
newly-developed method of separation of variables. The metric perturbation is
formed from a linear sum of differential operators acting on Teukolsky mode
functions, and certain auxiliary scalars, which are solutions to ordinary
differential equations in the frequency domain. For radiative modes, the
solution is uniquely determined by the Weyl scalars, the trace,
and gauge scalars whose amplitudes are determined by imposing
continuity conditions on the metric perturbation at the orbital radius. The
static (zero-frequency) part of the metric perturbation, which is handled
separately, also includes mass and angular momentum completion pieces. The
metric perturbation is validated against the independent results of a 2+1D time
domain code, and we demonstrate agreement at the expected level in all
components, and the absence of gauge discontinuities. In principle, the new
method can be used to determine the Lorenz-gauge metric perturbation at a
sufficiently high precision to enable accurate second-order self-force
calculations on Kerr spacetime in future. We conclude with a discussion of
extensions of the method to eccentric and non-equatorial orbits.Comment: 88 pages, 14 figure
Gravitational Self-Force Correction to the Innermost Stable Circular Equatorial Orbit of a Kerr Black Hole
For a self-gravitating particle of mass \mu in orbit around a Kerr black hole
of mass M >> \mu, we compute the O(\mu/M) shift in the frequency of the
innermost stable circular equatorial orbit (ISCEO) due to the conservative
piece of the gravitational self-force acting on the particle. Our treatment is
based on a Hamiltonian formulation of the dynamics in terms of geodesic motion
in a certain locally-defined effective smooth spacetime. We recover the same
result using the so-called first law of binary black-hole mechanics. We give
numerical results for the ISCEO frequency shift as a function of the black
hole's spin amplitude, and compare with predictions based on the post-Newtonian
approximation and the effective one-body model. Our results provide an accurate
strong-field benchmark for spin effects in the general relativistic two-body
problem.Comment: 5 pages, 1 table, 1 figure, matches version published in PRL. Raw
data of H_int/mu are available at
http://link.aps.org/supplemental/10.1103/PhysRevLett.113.16110
- …