arXiv subjects

Clifford M. Will

Publications and source records attributed to Clifford M. Will.

At least 37 records · Page 2Linked to original sources

arXiv2016

Relativistic orbits around spinning supermassive black holes. Secular evolution to 4.5 post-Newtonian order

We derive the secular evolution of the orbital elements of a stellar-mass object orbiting a spinning massive black hole. We use the post-Newtonian approximation in harmonic coordinates, with test-body equations of motion for the conservative dynamics that are valid through 3PN order, including spin-orbit, quadrupole and (spin)$^2$ effects, and with radiation-reaction contributions linear in the mass of the body that are valid through 4.5PN order, including the 4PN damping effects of spin-orbit coupling. The evolution equations for the osculating orbit elements are iterated to high PN orders using a two-timescale approach and averaging over orbital timescales. We derive a criterion for terminating the orbit when its Carter constant drops below a critical value, whereupon the body plunges across the event horizon at the next closest approach. The results are valid for arbitrary eccentricities and arbitrary inclinations. We then analyze numerically the orbits of objects injected into high-eccentricity orbits via interactions within a surrounding star cluster, obtaining the number of orbits and the elapsed time between injection and plunge, and the residual orbital eccentricity at plunge as a function of inclination. We derive an analytic approximation for the time to plunge in terms of initial orbital variables. We show that, if the black hole is spinning rapidly, the flux of gravitational radiation during the final orbit before plunge may be suppressed by as much as three orders of magnitude if the orbit is retrograde on the equatorial plane compared to its prograde counterpart.

gr-qc↗

arXiv2014

Constraining Lorentz-violating, Modified Dispersion Relations with Gravitational Waves

Modified gravity theories generically predict a violation of Lorentz invariance, which may lead to a modified dispersion relation for propagating modes of gravitational waves. We construct a parametrized dispersion relation that can reproduce a range of known Lorentz-violating predictions and investigate their impact on the propagation of gravitational waves. A modified dispersion relation forces different wavelengths of the gravitational wave train to travel at slightly different velocities, leading to a modified phase evolution observed at a gravitational-wave detector. We show how such corrections map to the waveform observable and to the parametrized post-Einsteinian framework, proposed to model a range of deviations from General Relativity. Given a gravitational-wave detection, the lack of evidence for such corrections could then be used to place a constraint on Lorentz violation. The constraints we obtain are tightest for dispersion relations that scale with small power of the graviton's momentum and deteriorate for a steeper scaling.

gr-qc↗

arXiv2014

Was Einstein Right? A Centenary Assessment

This article is an overview of 100 years of testing general relativity, to be published in the book General Relativity and Gravitation: A Centennial Perspective, to commemorate the 100th anniversary of general relativity. It is effectively an abridged version of the recent update of the author's Living Review in Relativity.

gr-qc↗

arXiv2014

The 1919 measurement of the deflection of light

The measurement of the deflection of starlight during a total solar eclipse on May 29, 1919 was the first verification of general relativity by an external team of scientists, brought Einstein and his theory to the attention of the general public, and left a legacy of experimental testing that continues today. The discovery of gravitational lenses turned Einstein's deflection into an important tool for astronomy and cosmology. This article reviews the history of the 1919 measurement and other eclipse measurements, describes modern measurements of the effect using radio astronomy, and of its cousin, the Shapiro time delay, and discusses gravitational lenses.

physics.hist-ph↗

arXiv2014

Post-Circular Expansion of Eccentric Binary Inspirals: Fourier-Domain Waveforms in the Stationary Phase Approximation

We lay the foundations for the construction of analytic expressions for Fourier-domain gravitational waveforms produced by eccentric, inspiraling compact binaries in a post-circular or small-eccentricity approximation. The time-dependent, "plus" and "cross" polarizations are expanded in Bessel functions, which are then self-consistently re-expanded in a power series about zero initial eccentricity to eighth order. The stationary phase approximation is then employed to obtain explicit analytic expressions for the Fourier transform of the post-circular expanded, time-domain signal. We exemplify this framework by considering Newtonian-accurate waveforms, which in the post-circular scheme give rise to higher harmonics of the orbital phase and amplitude corrections both to the amplitude and the phase of the Fourier domain waveform. Such higher harmonics lead to an effective increase in the inspiral mass reach of a detector as a function of the binary's eccentricity e_0 at the time when the binary enters the detector sensitivity band. Using the largest initial eccentricity allowed by our approximations (e_0 < 0.4), the mass reach is found to be enhanced up to factors of approximately 5 relative to that of circular binaries for Advanced LIGO, LISA, and the proposed Einstein Telescope at a signal-to-noise ratio of ten. A post-Newtonian generalization of the post circular scheme is also discussed, which holds the promise to provide "ready-to-use" Fourier-domain waveforms for data analysis of eccentric inspirals.

gr-qc↗

arXiv2014

Post-Newtonian effects in N-body dynamics: Relativistic precession and conserved quantities in hierarchical triple systems

Conventional approaches to incorporating general relativistic effects into the dynamics of N-body systems containing central black holes, or of hierarchical triple systems with a relativistic inner binary, may not be adequate when the goal is to study the evolution of the system over a timescale related to relativistic secular effects, such as the precession of the pericenter. For such problems, it may necessary to include post-Newtonian "cross terms" in the equations of motion in order to capture relativistic effects consistently over the long timescales. Cross terms are post-Newtonian (PN) terms that explicitly couple the two-body relativistic perturbations with the Newtonian perturbations due to other bodies in the system. In this paper, we show that the total energy and the normal component of total angular momentum of a hierarchical triple system is manifestly conserved to Newtonian order over the relativistic pericenter precession timescale of the inner binary if and only if PN cross-term effects in the equations of motion are taken carefully into account.

astro-ph.GA↗

arXiv2014

On incorporating post-Newtonian effects in N-body dynamics

The increasing role of general relativity in the dynamics of stellar systems with central massive black holes and in the evolution of hierarchical triple systems inspires a close examination of how post-Newtonian effects are incorporated into N-body dynamics. The majority of approaches incorporate relativity by adding to the Newtonian N-body equations the standard two-body post-Newtonian terms for a given star around the black hole or for the close binary in a triple system. We argue that, for calculating the evolution of such systems over timescales comparable to the relativistic pericenter advance timescale, it is essential to include "cross terms" in the equations of motion. These are post-Newtonian terms that represent a coupling between the potential of the central black hole and the potential due to other stars in the system. For hierarchical triple systems, these are couplings between the potential of the inner binary and that of the distant third body. Over pericenter precession timescales, the effects of such terms can actually be "boosted" to amplitudes of Newtonian order. We write down the post-Newtonian N-body equations of motion including a central black hole in a truncated form that includes all the relevant cross terms, in a format ready to use for numerical implementation. We do the same for hierarchical triple systems, and illustrate explicitly the effects of cross terms on the orbit-averaged equations of evolution for the orbit elements of the inner binary for the special case where the third body is on a circular orbit. We also describe the inspiration for this investigation: the motion of a test body about a central body with a Newtonian quadrupole moment, including the relativistic pericenter advance, whose correct solution for the conserved total Newtonian energy requires including PN cross terms between the mass monopole and quadrupole potentials.

astro-ph.GA↗

arXiv2014

The Confrontation between General Relativity and Experiment

The status of experimental tests of general relativity and of theoretical frameworks for analyzing them are reviewed and updated. Einstein's equivalence principle (EEP) is well supported by experiments such as the Eotvos experiment, tests of local Lorentz invariance and clock experiments. Ongoing tests of EEP and of the inverse square law are searching for new interactions arising from unification or quantum gravity. Tests of general relativity at the post-Newtonian level have reached high precision, including the light deflection, the Shapiro time delay, the perihelion advance of Mercury, the Nordtvedt effect in lunar motion, and frame-dragging. Gravitational-wave damping has been detected in an amount that agrees with general relativity to better than half a percent using the Hulse-Taylor binary pulsar, and a growing family of other binary pulsar systems is yielding new tests, especially of strong-field effects. Current and future tests of relativity will center on strong gravity and gravitational waves.

gr-qc↗

arXiv2013

The Schwarzschild metric: It's the coordinates, stupid!

Every general relativity textbook emphasizes that coordinates have no physical meaning. Nevertheless, a coordinate choice must be made in order to carry out real calculations, and that choice can make the difference between a calculation that is simple and one that is a mess. We give a concrete illustration of the maxim that "coordinates matter" using the exact Schwarzschild solution for a vacuum, static, spherical spacetime. We review the standard textbook derivation, Schwarzschild's original 1916 derivation, and a derivation using the Landau-Lifshitz formulation of the Einstein field equations. The last derivation is much more complicated, has one aspect for which we have been unable to find a solution, and gives an explicit illustration of the fact that the Schwarzschild geometry can be described in infinitely many coordinate systems.

gr-qc↗

arXiv2013

Compact binary systems in scalar-tensor gravity: Equations of motion to 2.5 post-Newtonian order

We calculate the explicit equations of motion for non-spinning compact objects to 2.5 post-Newtonian order, or O(v/c)^5 beyond Newtonian gravity, in a general class of scalar-tensor theories of gravity. We use the formalism of the Direct Integration of the Relaxed Einstein Equations (DIRE), adapted to scalar-tensor theory, coupled with an approach pioneered by Eardley for incorporating the internal gravity of compact, self-gravitating bodies. For the conservative part of the motion, we obtain the two-body Lagrangian and conserved energy and momentum through second post-Newtonian order. We find the 1.5 post-Newtonian and 2.5 post-Newtonian contributions to gravitational radiation reaction, the former corresponding to the effects of dipole gravitational radiation, and verify that the resulting energy loss agrees with earlier calculations of the energy flux. For binary black holes we show that the motion through 2.5 post-Newtonian order is observationally identical to that predicted by general relativity. For mixed black-hole neutron-star binary systems, the motion is identical to that in general relativity through the first post-Newtonian order, but deviates from general relativity beginning at 1.5 post-Newtonian order, in part through the onset of dipole gravitational radiation. But through 2.5 post-Newtonian order, those deviations in the motion of a mixed system are governed by a single parameter dependent only upon the scalar-tensor coupling constant and the structure of the neutron star, and are formally the same for a general class of scalar-tensor theories as they are for pure Brans-Dicke theory.

gr-qc↗

arXiv2013

Dark matter distributions around massive black holes: A general relativistic analysis

The cold dark matter at the center of a galaxy will be redistributed by the presence of a massive black hole. The redistribution may be determined using an approach pioneered by Gondolo and Silk: begin with a model distribution function for the dark matter, and ``grow'' the black hole adiabatically, holding the adiabatic invariants of the motion constant. Unlike the approach of Gondolo and Silk, which adopted Newtonian theory together with ad hoc correction factors to mimic general relativistic effects, we carry out the calculation fully relativistically, using the exact Schwarzschild geometry of the black hole. We find that the density of dark matter generically vanishes at r=2R_S, not 4R_S as found by Gondolo and Silk, where R_S is the Schwarzschild radius, and that the spike very close to the black hole reaches significantly higher densities. We apply the relativistic adiabatic growth framework to obtain the final dark matter density for both cored and cusped initial distributions. Besides the implications of these results for indirect detection estimates, we show that the gravitational effects of such a dark matter spike are significantly smaller than the relativistic effects of the black hole, including frame dragging and quadrupolar effects, for stars orbiting close to the black hole that might be candidates for testing the black hole no-hair theorems.

astro-ph.GA↗

arXiv2012

Capture of non-relativistic particles in eccentric orbits by a Kerr black hole

We obtain approximate analytic expressions for the critical value of the total angular momentum of a non-relativistic test particle moving in the Kerr geometry, such that it will be captured by the black hole. The expressions apply to arbitrary orbital inclinations, and are accurate over the entire range of angular momentum for the Kerr black hole. The expressions can be easily implemented in N-body simulations of the evolution of star clusters around massive galactic black holes, where such captures play an important role.

astro-ph.GA↗

arXiv2012

Gravitational radiation from compact binary systems in the massive Brans-Dicke theory of gravity

We derive the equations of motion, the periastron shift, and the gravitational radiation damping for quasicircular compact binaries in a massive variant of the Brans-Dicke theory of gravity. We also study the Shapiro time delay and the Nordtvedt effect in this theory. By comparing with recent observational data, we put bounds on the two parameters of the theory: the Brans-Dicke coupling parameter ω_{BD} and the scalar mass m_s. We find that the most stringent bounds come from Cassini measurements of the Shapiro time delay in the Solar System, that yield a lower bound ω_{BD}>40000 for scalar masses m_s<2.5x10^{-20} eV, to 95% confidence. In comparison, observations of the Nordtvedt effect using Lunar Laser Ranging (LLR) experiments yield ω_{BD}>1000 for m_s<2.5x10^{-20} eV. Observations of the orbital period derivative of the quasicircular white dwarf-neutron star binary PSR J1012+5307 yield ω_{BD}>1250 for m_s<10^{-20} eV. A first estimate suggests that bounds comparable to the Shapiro time delay may come from observations of radiation damping in the eccentric white dwarf-neutron star binary PSR J1141-6545, but a quantitative prediction requires the extension of our work to eccentric orbits.

gr-qc↗

arXiv2011

Testing the black hole no-hair theorem at the galactic center: Perturbing effects of stars in the surrounding cluster

Observations of the precessing orbits of stars very near the massive black hole in the galactic center could provide measurements of the spin and quadrupole moment of the hole and thereby test the no-hair theorem of general relativity. Since the galactic center is likely to be populated by a distribution of stars and small black holes, their gravitational interactions will perturb the orbit of any given star. We estimate the effects of such perturbations using analytic orbital perturbation theory, and show that for a range of possible stellar distributions, and for an observed star sufficiently close to the black hole, the relativistic spin and quadrupole effects will be larger than the effects of stellar cluster perturbations. Our results are consistent those from recent numerical N-body simulations by Merritt et al.

gr-qc↗

arXiv2011

Finally, results from Gravity Probe-B

Nearly fifty years after its inception, the Gravity Probe B satellite mission delivers the first measurements of how a spinning gyroscope precesses in the gravitational warping of spacetime.

gr-qc↗

arXiv2011

On the unreasonable effectiveness of the post-Newtonian approximation in gravitational physics

The post-Newtonian approximation is a method for solving Einstein's field equations for physical systems in which motions are slow compared to the speed of light and where gravitational fields are weak. Yet it has proven to be remarkably effective in describing certain strong-field, fast-motion systems, including binary pulsars containing dense neutron stars and binary black hole systems inspiraling toward a final merger. The reasons for this effectiveness are largely unknown. When carried to high orders in the post-Newtonian sequence, predictions for the gravitational-wave signal from inspiraling compact binaries will play a key role in gravitational-wave detection by laser-interferometric observatories.

gr-qc↗

arXiv2010

Carter-like constants of motion in the Newtonian and relativistic two-center problems

In Newtonian gravity, a stationary axisymmetric system admits a third, Carter-like constant of motion if its mass multipole moments are related to each other in exactly the same manner as for the Kerr black-hole spacetime. The Newtonian source with this property consists of two point masses at rest a fixed distance apart. The integrability of motion about this source was first studied in the 1760s by Euler. We show that the general relativistic analogue of the Euler problem, the Bach-Weyl solution, does not admit a Carter-like constant of motion, first, by showing that it does not possess a non-trivial Killing tensor, and secondly, by showing that the existence of a Carter-like constant for the two-center problem fails at the first post-Newtonian order.

gr-qc↗

arXiv2010

Resource Letter PTG-1: Precision Tests of Gravity

This resource letter provides an introduction to some of the main current topics in experimental tests of general relativity as well as to some of the historical literature. It is intended to serve as a guide to the field for upper-division undergraduate and graduate students, both theoretical and experimental, and for workers in other fields of physics who wish learn about experimental gravity. The topics covered include alternative theories of gravity, tests of the principle of equivalence, solar-system and binary-pulsar tests, searches for new physics in gravitational arenas, and tests of gravity in new regimes, involving astrophysics and gravitational radiation.

gr-qc↗