arXiv subjects

Clifford M. Will

Publications and source records attributed to Clifford M. Will.

At least 19 recordsLinked to original sources

arXiv2025

Equations of motion for compact binary systems in general relativity: Do they depend on the bodies' internal structure at the third post-Newtonian order?

We present and discuss the possibility, derived from work carried out 20 years ago, that the equations of motion for compact binary neutron stars at the third post-Newtonian (3PN) order in general relativity might actually depend on the internal structure of the bodies. These effects involve integrals over the density and internal gravitational potentials of the bodies that are independent of the mass and radius of the bodies, but dependent on their equations of state. These effects could alter the coefficients in the 3PN equations derived using ``point mass'' methods by as much as 100 percent. They were found in independent calculations done at Washington University using the Direct Integration of the Relaxed Einstein Equations (DIRE) approach, and at the Institut d'Astrophysique de Paris using the Multipolar post-Minkowskian (MPPM) approach. Neither calculation was completed because of the enormous complexity of the algebraic computations and the limitations of software of the day (Maple, Mathematica), and because of an assumption that the effects would somehow cancel or be removable by some transformation. This assumption was rooted in the Strong Equivalence Principle (SEP), which would suppress such effects up to the stage where tidal interactions become important, effectively 5PN order for compact bodies. SEP was well supported at lower PN orders and in special examples. We argue that this assumption needs to be verified by calculations. If the results show that these terms exactly cancel or can be absorbed into renormalized masses or shifted positions of each body, this would provide support for the Strong Equivalence Principle. But if they do not cancel and are not incorporated into gravitational waveforms, they could impact efforts using next-generation gravitational-wave interferometers to extract information about the equation of state for neutron star matter.

gr-qc↗

arXiv2025

Compact binary systems in Einstein-Æther gravity. II. Radiation reaction to 2.5 post-Newtonian order

We obtain the equations of motion for compact binary systems (black holes or neutron stars) in an alternative theory of gravity known as Einstein-AEther theory, which supplements the standard spacetime metric with a timelike four-vector (the AEther field) that is constrained to have unit norm. The equations make use of solutions obtained in Paper I for the gravitational and AEther field potentials within the near zone of the system, evaluated to 2.5 post-Newtonian (PN) order ($O(v/c)^5$ beyond Newtonian gravity), sufficient to obtain the effects of gravitational radiation reaction to the same order as the quadrupole approximation of general relativity. Those potentials were derived by applying the post-Minkowskian method to the field equations of the theory. Using a modified geodesic equation that is a consequence of the effects of the interaction between the AEther field and the internal strong-gravity fields of the compact bodies, we obtain explicit equations of motion in terms of the positions and velocities of the bodies, focussing on the radiation-reaction terms that contribute at 1.5PN and 2.5PN orders. We obtain the rate of energy loss by the system, including the effects of dipole gravitational radiation (conventionally denoted $-1$PN order) and the analogue of quadrupole radiation (denoted $0$PN order). We find significant disagreements with published results, based on calculating the energy flux in the far zone using a ``Noether current'' construction.

gr-qc↗

arXiv2025

Higher-order effects in the dynamics of hierarchical triple systems. III. Astrophysical implications of second-order and dotriacontapole terms

We study the long-term evolution of selected hierarchical triple systems in Newtonian gravity. We employ analytic equations derived in Paper II for the evolution of orbit-averaged orbital elements for both inner and outer orbits, which include two classes of contributions. One class consists of linear-order contributions, including quadrupole, octupole, hexadecapole and dotriacontapole orders, the latter scaling as $ε^6$, where $ε= a/A$, the ratio of the semimajor axes of the inner and outer orbits. The second class consists of contributions at {\em second} order in the fundamental perturbation parameter; they contribute at orders $ε^{9/2}$, $ε^{5}$, $ε^{11/2}$ and $ε^{6}$. For well studied triples such as star-planet systems perturbed by a low-mass third body (``hot Jupiters''), second-order and dotriacontapole (SOD) effects induce only small corrections. For stellar-mass binaries orbiting supermassive black holes, SOD corrections can suppress orbital flips that are generated by purely first-order effects. Planets orbiting binary star systems are susceptible to significant variations in the planetary semimajor axis, an effect that does not occur at first perturbative order. SOD effects in triple black hole systems can induce migrations of the eccentricity to significantly larger values than predicted by first-order perturbations, with implications for the gravitational-wave induced inspiral of the inner binary. We also show that in most cases, evolutions using our SOD equations are in better agreement with those from direct integration of the N-body equations of motion than those from first-order perturbations through hexadecapole order.

astro-ph.EP↗

arXiv2024

Higher-order effects in the dynamics of hierarchical triple systems. II. Second-order and dotriacontapole-order effects

We analyze the long-term evolution of hierarchical triple systems in Newtonian gravity to second order in the quadrupolar perturbation parameter, and to sixth order in $ε= a/A$, the ratio of the semimajor axes of the inner and outer orbits. We apply the ``two-timescale'' method from applied mathematics to the Lagrange Planetary Equations for the inner and outer orbits, in which each osculating orbit element is split into an orbit averaged part that evolves on the long perturbative timescale, and an ``average-free'' part that is oscillatory in the orbital timescales. Averages over the two orbital timescales are performed using the well-known secular approximation. We also incorporate perturbative corrections to the relation between time and the orbital phases. We place no restrictions on the masses, on the relative orbit inclinations or on the eccentricities, beyond the requirement that the quadrupolar parameter and $ε$ both be small. The result is a complete set of long-timescale evolution equations for the averaged elements of the inner and outer orbits. At first order in perturbation theory, we obtain the dotriacontapole contributions explicitly at order $ε^6$. At second order in perturbation theory, i.e. quadratic in the quadrupole perturbation amplitude, we find contributions that scale as $ε^{9/2}$ (found in earlier work), $ε^{5}$, $ε^{11/2}$, and $ε^{6}$. At first perturbative order and dotriacontapole order, the two averaged semimajor axes are constant in time (and we prove that this holds to arbitrary multipole orders); but at second perturbative order, beginning at $O( ε^{5})$, they are no longer constant. Nevertheless we verify that the total averaged energy of the system is conserved, and we argue that this behavior is not incompatible with classical theorems on secular evolution of the semimajor axes.

astro-ph.EP↗

arXiv2023

Compact binary systems in Einstein-Aether gravity: Direct integration of the relaxed field equations to 2.5 post-Newtonian order

The Einstein-Aether theory is an alternative theory of gravity in which the spacetime metric is supplemented by a long-range timelike vector field (the "aether" field). Here, for the first time, we apply the full formalism of post-Minkowskian theory and of the Direct Integration of the Relaxed Einstein Equations (DIRE), to this theory of gravity, with the goal of deriving equations of motion and gravitational waveforms for orbiting compact bodies to high orders in a post-Newtonian expansion. Because the aether field is constrained to have unit norm, a naive application of post-Minkowskian theory leads to contributions to the effective energy momentum tensor that are {\em linear} in the perturbative fields. We show that a suitable redefinition of fields using an array of "superpotentials" can eliminate such linear terms to any desired post-Newtonian order, resulting in flat spacetime wave equations for all fields, with sources consisting of matter terms and terms quadratic and higher in the fields. As an initial application of this new method, and as a foundation for obtaining the equations of motion for compact binaries, we obtain explicit solutions of the relaxed equations sufficient to obtain the metric in the near zone through 2.5 post-Newtonian order, or $O[(v/c)^5]$ beyond the Newtonian approximation.

gr-qc↗

arXiv2023

Constraining a companion of the galactic center black hole, Sgr A*

We use 23 years of astrometric and radial velocity data on the orbit of the star S0-2 to constrain a hypothetical intermediate-mass black hole orbiting the massive black hole Sgr A* at the Galactic center. The data place upper limits on variations of the orientation of the stellar orbit (inclination, nodal angle, and pericenter) at levels between 0.02 and 0.07 degrees per year. We use a combination of analytic estimates and full numerical integrations of the orbit of S0-2 in the presence of a black-hole binary. For a companion IMBH whose semi-major axis $a_c$ is larger than that of S0-2 (1020 a.u.), we find that in the region between 1000 and 4000 a.u., a companion black hole with mass $m_c$ between $10^3$ and $10^5 M_\odot$ is excluded, with a boundary behaving as $a_c \sim m_c^{1/3}$. For a companion with $a_c < 1020$ a.u., we find that a black hole with mass between $10^3$ and $10^5 \, M_\odot$ is again excluded, with a boundary behaving as $a_c \sim m_c^{-1/2}$. These bounds arise from quadrupolar perturbations of the orbit of S0-2. However, significantly stronger bounds on the mass of an inner companion arise from the fact that the location of S0-2 is measured relative to the bright emission of Sgr A*. As a consequence, that separation is perturbed by the ``wobble'' of Sgr A* about the center of mass between it and the companion, leading to ``apparent'' perturbations of S0-2's orbit that also include a dipole component. The result is a set of bounds as small as $400 \, M_\odot$ at 200 a.u.; the numerical simulations suggest a bound from these effects varying as $a_c \sim m_c^{-1}$. We compare and contrast our results with those from a recent analysis by the GRAVITY collaboration.

astro-ph.GA↗

arXiv2023

A Stability Timescale for Non-Hierarchical Three-Body Systems

The gravitational three-body problem is a fundamental problem in physics and has significant applications to astronomy. Three-body configurations are often considered stable as long the system is hierarchical; that is, the two orbital distances are well-separated. However, instability, which is often associated with significant energy exchange between orbits, takes time to develop. Assuming two massive objects in a circular orbit and a test particle in an eccentric orbit, we develop an analytical formula estimating the time it takes for the test particle's orbital energy to change by an order of itself. We show its consistency with results from N-body simulations. For eccentric orbits in particular, the instability is primarily driven not by close encounters of the test particle with one of the other bodies, but by the fundamental susceptibility of eccentric orbits to exchange energy at their periapsis. Motivated by recent suggestions that the galactic center may host an intermediate-mass black hole (IMBH) as a companion to the massive black hole Sgr A*, we use our timescale to explore the parameter space that could harbor an IMBH for the lifetime of the S-cluster of stars surrounding Sgr A*. Furthermore, we show that the orbit of an S-star can be stable for long timescales in the presence of other orbital crossing stars, thus suggesting that the S-cluster may be stable for the lifetimes of its member stars.

astro-ph.GA↗

arXiv2022

Modified geodesic equations of motion for compact bodies in alternative theories gravity

We derive exact, modified geodesic equations for a system of non-spinning, self-gravitating interacting bodies in a class of alternative theories of gravity to general relativity. We use a prescription proposed by Eardley for incorporating the effects of self-gravity within gravitationally bound bodies, in which their masses may depend on invariant quantities constructed from the auxiliary scalar, vector or tensor fields introduced by such theories, evaluated in the vicinity of each body. The forms of the equations are independent of the field equations of the chosen theory. In the case where the masses are strictly constant, the equations reduce to the conventional geodesic equations of general relativity. These equations may be useful tools for deriving equations of motion for compact bodies to high post-Newtonian orders in alternative theories of gravity.

gr-qc↗

arXiv2021

Residual eccentricity of inspiralling orbits at the gravitational-wave detection threshold: Accurate estimates using post-Newtonian theory

We use equations of motion containing gravitational radiation-reaction terms through 4.5 post-Newtonian order to calculate the late-time eccentricities of inspiraling binary systems of non-spinning compact bodies as they cross the detection threshold of ground-based gravitational-wave interferometers. The initial eccentricities can be as large as 0.999. We find that the final eccentricities are systematically smaller than those predicted by the leading quadrupole approximation, by as much as 30 percent for a 300 solar mass binary crossing the LIGO/Virgo detection threshold at 10 Hz, or eight percent smaller for a 60 solar mass binary. We find an analytic formula for the late-time eccentricity that accurately accounts for the higher-order post-Newtonian effects, generalizing a formula derived by Peters and Mathews in the 1960s. We also find that the final eccentricities are independent of the ratio of the masses of the two compact bodies to better than two percent.

gr-qc↗

arXiv2020

Higher-order effects in the dynamics of hierarchical triple systems. Quadrupole-squared terms

We analyze the secular evolution of hierarchical triple systems to second-order in the quadrupolar perturbation induced on the inner binary by the distant third body. The Newtonian three-body equations of motion, expanded in powers of the ratio of semimajor axes $a/A$, become a pair of effective one-body Keplerian equations of motion, perturbed by a sequence of multipolar perturbations, denoted quadrupole, $O[(a/A)^3]$, octupole, $O[(a/A)^4]$, and so on. In the Lagrange planetary equations for the evolution of the instantaneous orbital elements, second-order effects arise from obtaining the first-order solution for each element, consisting of a constant (or slowly varying) piece and an oscillatory perturbative piece, and reinserting it back into the equations to obtain a second-order solution. After an average over the two orbital timescales to obtain long-term evolutions, these second-order quadrupole ($Q^2$) terms would be expected to produce effects of order $(a/A)^6$. However we find that the orbital average actually enhances the second-order terms by a factor of the ratio of the outer to the inner orbital periods, $ \sim (A/a)^{3/2}$. For systems with a low-mass third body, the $Q^2$ effects are small, but for systems with a comparable-mass or very massive third body, such as a Sun-Jupiter system orbiting a solar-mass star, or a $100 \, M_\odot$ binary system orbiting a $10^6 \, M_\odot$ massive black hole, the $Q^2$ effects can completely suppress flips of the inner orbit from prograde to retrograde and back that occur in the first-order solutions. These results are in complete agreement with those of Luo, Katz and Dong, derived using a "Corrected Double-Averaging" method.

astro-ph.EP↗

arXiv2019

A hidden friend for the galactic center black hole, Sgr A*

The hierarchical nature of galaxy formation suggests that a supermassive black hole binary could exist in our galactic center. We propose a new approach to constraining the possible orbital configuration of such a binary companion to the galactic center black hole Sgr A* through the measurement of stellar orbits. Focusing on the star S0-2, we show that requiring its orbital stability in the presence of a companion to Sgr A* yields stringent constraints on the possible configurations of such a companion. Furthermore, we show that precise measurements of {\it time variations} in the orbital parameters of S0-2 could yield stronger constraints. Using existing data on S0-2 we derive upper limits on the binary black hole separation as a function of the companion mass. For the case of a circular orbit, we can rule out a 10^5 M_sun companion with a semimajor axis greater than 170 astronomical units or 0.8 mpc. This is already more stringent than bounds obtained from studies of the proper motion of Sgr A*. Including other stars orbiting the galactic center should yield stronger constraints that could help uncover the presence of a companion to Sgr A*. We show that a companion can also affect the accretion process, resulting in a variability which may be consistent with the measured infrared flaring timescales and amplitudes. Finally, if such a companion exists, it will emit gravitational wave radiation, potentially detectable with LISA.

astro-ph.GA↗

arXiv2019

Pericenter advance in general relativity: Comparison of approaches at high post-Newtonian orders

The advance of the pericenter of the orbit of a test body around a massive body in general relativity can be calculated in a number of ways. In one method, one studies the geodesic equation in the exact Schwarzschild geometry and finds the angle between pericenters as an integral of a certain radial function between turning points of the orbit. In another method, one describes the orbit using osculating orbit elements, and analyzes the "Lagrange planetary equations" that give the evolution of the elements under the perturbing effects of post-Newtonian (PN) corrections to the motion. After separating the perturbations into periodic and secular effects, one obtains an equation for the secular rate of change of the pericenter angle. While the different methods agree on the leading post-Newtonian contribution to the advance, they do not agree on the higher-order PN corrections. We show that this disagreement is illusory. When the orbital variables in each case are expressed in terms of the invariant energy and angular momentum of the orbit and when account is taken of a subtle difference in the meaning of "pericenter advance" between the two methods, we show to the third post-Newtonian order that the different methods actually agree perfectly.

gr-qc↗

arXiv2019

Compact binary inspiral: Nature is perfectly happy with a circle

It is standard lore that gravitational radiation reaction circularizes the orbits of inspiralling binary systems. But in recent papers, Loutrel et al. have argued that at late times in such inspirals, one measure of eccentricity actually increases, and that this could have observable consequences. We show that this variable, the magnitude of the Runge-Lenz vector ($e_{\rm RL}$), is not an appropriate measure of orbital eccentricity, when the eccentricity is smaller than the leading non-Keplerian perturbation of the orbit. Following Loutrel et al., we use Newtonian equations of motion plus the leading gravitational radiation-reaction terms, the osculating-orbits approach for characterizing binary orbits, and a two-timescale analysis for separating secular from periodic variations of the orbit elements. We find that $e_{\rm RL}$ does grow at late times, but that the actual orbital variables $r$ and $dr/dt$ show no such growth in oscillations. This is in complete agreement with Loutrel et al. We reconcile this apparent contradiction by pointing out that it is essential to take into account the direction of the Runge-Lenz vector, not just its magnitude. At late times in an inspiral, that direction, which defines the pericenter angle, advances at the same rate as the orbital phase. The correct picture is then of a physically circular orbit whose osculating counterpart is indeed eccentric but that resides permanently at the orbit's latus rectum at $-90^{\rm o}$, therefore exhibiting no oscillations. Including first post-Newtonian effects in the equations of motion, we show that $e_{\rm RL}$ grows even more dramatically. But the phase of the Runge-Lenz vector again rotates with the orbit at late times, but now the osculating orbit resides at "perpetual apocenter", so again the physical orbit circularizes.

gr-qc↗

arXiv2018

Solar system vs. gravitational-wave bounds on the graviton mass

TThe detection of gravitational waves from merging binary black holes has led to a bound on the mass of a hypothetical massive carrier of the gravitational interaction predicted by some modified gravity theories (a massive graviton, for short), corresponding to a bound on the Compton wavelength $λ_g > 1.6 \times 10^{13}$ km. This bound is six times more stringent than a 1988 bound inferred from solar-system dynamics. Using 30 years of improvements in solar system data, chiefly from missions involving orbiters and probes of planets from Mercury to Saturn, we revisit this bound. We show that data on the perihelion advance of Mars obtained from the Mars Reconnaissance Orbiter leads to a credible lower bound on $λ_g$ between $1.2$ and $2.2 \times 10^{14}$ km, surpassing the gravitational-wave bound by an order of magnitude. We discuss ways in which each of these competing bounds may improve in the future.

gr-qc↗

arXiv2018

A new general relativistic contribution to Mercury's perihelion advance

We point out the existence of a new general relativistic contribution to the perihelion advance of Mercury that, while smaller than the contributions arising from the solar quadrupole moment and angular momentum, is 100 times larger than the second-post-Newtonian contribution. It arises in part from relativistic "cross-terms" in the post-Newtonian equations of motion between Mercury's interaction with the Sun and with the other planets, and in part from an interaction between Mercury's motion and the gravitomagnetic field of the moving planets. At a few parts in $10^6$ of the leading general relativistic precession of 42.98 arcseconds per century, these effects are likely to be detectable by the BepiColombo mission to place and track two orbiters around Mercury, scheduled for launch around 2018.

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arXiv2018

Testing general relativity with compact-body orbits: A modified Einstein-Infeld-Hoffmann framework

We describe a general framework for analyzing orbits of systems containing compact objects (neutron stars or black holes) in a class of Lagrangian-based alternative theories of gravity that also admit a global preferred reference frame. The framework is based on a modified Einstein-Infeld-Hoffmann (EIH) formalism developed by Eardley and by Will, generalized to include the possibility of Lorentz-violating, preferred-frame effects. It uses a post-Newtonian $N$-body Lagrangian with arbitrary parameters that depend on the theory of gravity and on "sensitivities" that encode the effects of the bodies' internal structure on their motion. We determine the modified EIH parameters for the Einstein-Æther and Khronometric vector-tensor theories of gravity. We find the effects of motion relative to a preferred universal frame on the orbital parameters of binary systems containing neutron stars, such as a class of ultra-circular pulsar-white dwarf binaries, the amplitudes of the effects depend upon "strong-field" preferred-frame parameters $\hat{\,\,α}_1$ and ${\hat{\,\,α}}_2$, which we relate to the fundamental modified EIH parameters. We also determine the amplitude of the "Nordtvedt effect" in a triple system containing the pulsar J0337+1715 in terms of the modified EIH parameters.

gr-qc↗

arXiv2018

Dark matter spikes in the vicinity of Kerr black holes

The growth of a massive black hole will steepen the cold dark matter density at the center of a galaxy into a dense spike, enhancing the prospects for indirect detection. We study the impact of black hole spin on the density profile using the exact Kerr geometry of the black whole in a fully relativistic adiabatic growth framework. We find that, despite the transfer of angular momentum from the hole to the halo, rotation increases significantly the dark matter density close to the black hole. The gravitational effects are still dominated by the black hole within its influence radius, but the larger dark matter annihilation fluxes might be relevant for indirect detection estimates.

astro-ph.CO↗

arXiv2017

Orbital flips in hierarchical triple systems: relativistic effects and third-body effects to hexadecapole order

We analyze the secular evolution of hierarchical triple systems in the post-Newtonian approximation to general relativity. We expand the Newtonian three-body equations of motion in powers of the ratio $a/A$, where $a$ and $A$ are the semimajor axis of the inner binary's orbit and of the orbit of the third body relative to the center of mass of the inner binary, respectively. The leading order "quadrupole" terms, of order $(a/A)^3$ relative to the $1/a^2$ acceleration within the inner binary, are responsible for the well-known Kozai-Lidov oscillations of orbital inclination and eccentricity. The octupole terms, of order $(a/A)^4$ have been shown to allow the inner orbit to "flip" from prograde relative to the outer orbit to retrograde and back, and to permit excursions to very large eccentricities. We carry the expansion of the equations of motion to hexadecapole order, corresponding to contributions of order $(a/A)^5$. We also include the leading orbital effects of post-Newtonian theory, namely the pericenter precessions of the inner and outer orbits. Using the Lagrange planetary equations for the orbit elements of both binaries, we average over orbital timescales, obtain the equations for the secular evolution of the elements through hexadecapole order, and employ them to analyze cases of astrophysical interest. We find that, for the most part, the orbital flips found at octupole order are robust against both relativistic and hexadecapole perturbations. We show that, for equal-mass inner binaries, where the octupole terms vanish, the hexadecapole contributions can alone generate orbital flips and excursions to very large eccentricities.

astro-ph.EP↗