arXiv subjects

Abraham I. Harte

Publications and source records attributed to Abraham I. Harte.

At least 19 recordsLinked to original sources

arXiv2026

Octupole moments and the non-universality of free-fall in general relativity

Extended bodies in general relativity do not necessarily fall along geodesics, but can be accelerated. These accelerations depend on an object's angular momentum, as well as on its quadrupole, octupole, and higher-order moments. However, these multipole moments can evolve differently from one body to another. Different bodies can thus fall differently, even with identical initial data. This paper examines how octupole moments contribute to the non-universality of free-fall in general relativity. We begin by showing that in arbitrary vacuum spacetimes, only the trace-free component of an octupole moment can affect an object's motion. It follows that at least 16 out of 40 octupole components decouple from the laws of motion. Then, we obtain two decompositions for trace-free octupole moments, one in terms of a timelike frame vector and the other in terms of a null tetrad. These decompositions are applied to motion both in generic Newtonian spacetimes and in fully-relativistic vacuum spacetimes that are of Petrov type D. In Newtonian spacetimes, the mass moments are shown to have their ordinary Newtonian effects, while the momentum moments determine a body's hidden momentum---the misalignment between its momentum and its velocity. In Petrov type D spacetimes (such as Kerr), we show that some torques that are impossible with quadrupole moments are possible with octupole moments. Octupole moments can thus have qualitatively-different effects from quadrupole moments.

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arXiv2026

Conservative and dissipative sectors in a nonlinear scalar model for the gravitational self-force problem

When considering how self-interaction affects an object's motion, it can be convenient to decompose the self-force into conservative and dissipative pieces. As a toy model for understanding such decompositions of the gravitational self-force, we consider objects that do not affect the spacetime, but are instead coupled to a nonlinear scalar field. There is then a standard splitting of the first-order scalar self-force into conservative and dissipative components. Multiple criteria can be used to obtain this splitting, all of which imply the same result. However, the implications of these criteria generically differ at higher orders. Demanding that any reasonable conservative sector be Hamiltonian, we identify multiple possible definitions of the conservative second-order self-force. Motivations for these possibilities and their properties are discussed and relevant Hamiltonians are obtained. We assume the existence of a three-point function with certain properties that is a generalization of the Detweiler-Whiting two-point function. These results apply to the two-body problem but are restricted to unbound scattering trajectories, due to infrared divergences that arise for bound orbits.

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arXiv2025

Nonlinearly self-interacting extended bodies move as test bodies in effective external fields

In electromagnetism, linearized general relativity, and other contexts, previous work has shown that the laws of motion which govern compact, self-interacting bodies can be obtained by applying "Detweiler-Whiting prescriptions" to the laws of motion which govern test bodies. These prescriptions replace any field which appears in a test-body law of motion with a certain effective field which is a quasilocal functional of the physical variables -- a functional that can be interpreted as a regularization procedure in a point-particle limit. We generalize these results, presenting a formalism which allows Detweiler-Whiting prescriptions to be be directly derived for extended bodies, even in nonlinear field theories. If a generating functional with particular properties can be constructed, we find effective linear and angular momenta which evolve via Mathisson-Papapetrou-Dixon equations involving appropriate effective fields. These equations implicitly incorporate all self-force, self-torque, and extended-body effects. Although our main focus is on bodies coupled to nonlinear scalar fields, we also remark on the gravitational case.

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arXiv2025

Gravitational wave memory and its effects on particles and fields

Gravitational wave memory is said to arise when a gravitational wave burst produces changes in a physical system that persist even after that wave has passed. This paper analyzes gravitational wave bursts in plane wave spacetimes, deriving memory effects for timelike and null geodesics, massless scalar fields, and massless spinning particles whose motion is described by the spin Hall equations. We find that all such effects are characterized by four "memory tensors," three of which are independent. We also show that memory effects for null geodesics can have strong longitudinal components, even in vacuum general relativity. When considering massless particles with spin, we solve the spin Hall equations analytically by showing that there exists a conservation law associated with each conformal Killing vector. For the scattering of fields by gravitational waves, we show that given any solution to the massless scalar field equation in flat spacetime, a weak-field solution in a plane wave spacetime can be generated just by applying an appropriate differential operator -- an operator that is constructed from the aforementioned memory tensors. Memory effects for scalar fields are illustrated for both incoming plane waves and higher-order Gaussian beams. We also present a numerical comparison between the spin Hall equations and the full evolution of localized wave packets with angular momentum. Although we work in plane wave spacetimes, which are physically idealized, similar results are also expected to apply for sufficiently small systems affected by distantly generated gravitational waves. Using the Penrose limit, our results may also apply to ultrarelativistic systems at arbitrary locations in arbitrary (even nonradiating) spacetimes.

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arXiv2023

Local symmetries as constraints on the motion of freely-falling extended bodies

Different extended objects can fall in different ways, depending on their internal structures. Some motions are nevertheless impossible, regardless of internal structure. This paper derives universal constraints on extended-body motion, both in Newtonian gravity and in general relativity. In both theories, we identify a weak notion of "local symmetry" which precludes certain force and torque combinations. Local symmetries imply that certain components of a body's quadrupole moment cannot affect its motion. They also imply that some forces arise only in combination with appropriate torques. Many of these symmetries are shown to be determined by the algebraic structure of the tidal tensor. In general relativity, we thus relate qualitative features of extended-body motion to the Petrov type of the spacetime. Doing so shows that local symmetries are in fact ubiquitous. In general relativity, there are at least two in all algebraically-special spacetimes. Some of these are generated by Killing vectors and some by conformal Killing-Yano tensors. However, many local symmetries do not fall into either of these classes.

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arXiv2022

Spin Hall effects and the localization of massless spinning particles

The spin Hall effects of light represent a diverse class of polarization-dependent physical phenomena involving the dynamics of electromagnetic wave packets. In a medium with an inhomogeneous refractive index, wave packets can be effectively described by massless spinning particles following polarization-dependent trajectories. Similarly, in curved spacetime the gravitational spin Hall effect of light is represented by polarization-dependent deviations from null geodesics. In this paper, we analyze the equations of motion describing the gravitational spin Hall effect of light. We show that these equations are a special case of the Mathisson-Papapetrou equations for spinning objects in general relativity. This allows us to use several known results for the Mathisson-Papapetrou equations, and apply them to the study of electromagnetic wave packets. We derive conservation laws, we discuss the limits of validity of the spin Hall equations, and we study how the energy centroids of wave packets, effectively described as massless spinning particles, depend on the external choice of a timelike vector field, representing a family of observers. In flat spacetime, the relativistic Hall effect and the Wigner(-Souriau) translations are recovered, while our equations also provide a generalization of these effects in arbitrary spacetimes. We construct a large class of wave packets that can be described by the spin Hall equations, but also find its limits by giving examples of wave packets which are more general and are not described by the spin Hall equations. Lastly, we examine the assumption that electromagnetic wave packets are massless. While this is approximately true in many contexts, it is not exact. We show that failing to carefully account for the limitations of the massless approximation results in the appearance of unphysical centroids which are nowhere near the wave packet itself.

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arXiv2022

Persistent gravitational wave observables: general framework

The gravitational wave memory effect is characterized by the permanent relative displacement of a pair of initially comoving test particles that is caused by the passage of a burst of gravitational waves. Recent research on this effect has clarified the physical origin and the interpretation of this gravitational phenomenon in terms of conserved charges at null infinity and "soft theorems." In this paper, we describe a more general class of effects than the gravitational wave memory that are not necessarily associated with these charges and soft theorems, but that are, in principle, measurable. We shall refer to these effects as persistent gravitational wave observables. These observables vanish in non-radiative regions of a spacetime, and their effects "persist" after a region of spacetime which is radiating. We give three examples of such persistent observables, as well as general techniques to calculate them. These examples, for simplicity, restrict the class of non-radiative regions to those which are exactly flat. Our first example is a generalization of geodesic deviation that allows for arbitrary acceleration. The second example is a holonomy observable, which is defined in terms of a closed loop. It contains the usual "displacement" gravitational wave memory; three previously identified, though less well known memory effects (the proper time, velocity, and rotation memories); and additional new observables. Finally, the third example we give is an explicit procedure by which an observer could measure a persistent effect using a spinning test particle. We briefly discuss the ability of gravitational wave detectors (such as LIGO and Virgo) to measure these observables.

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arXiv2020

Extended-body motion in black hole spacetimes: What is possible?

Free-fall is only approximately universal in general relativity. Different extended bodies can fall in different ways, depending on their internal dynamics. Nevertheless, certain aspects of their motion are universal. This paper examines the universal constraints on extended-body motion in vacuum type D spacetimes. Working in the quadrupole approximation, we show that in addition to the (previously-known) constraints imposed by Killing vectors, two components of the gravitational torque must vanish. Furthermore, of the ten components of a body's quadrupole moment, four are found to be irrelevant, two can affect only the force, and the remaining four can affect both forces and torques. As an application, we consider the capabilities of a hypothetical spacecraft which controls its motion by controlling its internal structure. In the Schwarzschild spacetime, such a spacecraft can control its mass, and in doing so, it can stabilize unstable orbits, escape from bound orbits, and more -- all without a rocket.

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arXiv2020

Extended-body effects and rocket-free orbital maneuvering

The trajectory of a spherical object which falls freely in a gravitational field is fixed by its initial position and velocity. However, an object which can control its shape can also control its motion: Except where forbidden by symmetries and their associated conservation laws, a shape-changing (but rocket-free) spacecraft can have complete control over its trajectory. We discuss a general formalism which allows rocket-free maneuvers to be understood without constructing detailed interior models. A spacecraft's interior is abstracted to the specification of a quadrupole moment, and in some cases, it is only a single eigenvalue of that moment which is relevant. For orbits around a spherically-symmetric mass, we show that appropriately varying the relevant eigenvalue allows the energy and eccentricity of an orbit to be increased or decreased and its apsides to be rotated arbitrarily. Strategies are identified which optimize these maneuvers. In other contexts, we show that extended-body effects can be used to stabilize orbits which would otherwise be unstable.

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arXiv2020

Persistent gravitational wave observables: Nonlinear plane wave spacetimes

In the first paper in this series, we introduced "persistent gravitational wave observables" as a framework for generalizing the gravitational wave memory effect. These observables are nonlocal in time and nonzero in the presence of gravitational radiation. We defined three specific examples of persistent observables: a generalization of geodesic deviation that allowed for arbitrary acceleration, a holonomy observable involving a closed curve, and an observable that can be measured using a spinning test particle. For linearized plane waves, we showed that our observables could be determined just from one, two, and three time integrals of the Riemann tensor along a central worldline, when the observers follow geodesics. In this paper, we compute these three persistent observables in nonlinear plane wave spacetimes, and we find that the fully nonlinear observables contain effects that differ qualitatively from the effects present in the observables at linear order. Many parts of these observables can be determined from two functions, the transverse Jacobi propagators, and their derivatives (for geodesic observers). These functions, at linear order in the spacetime curvature, reduce to the one, two, and three time integrals of the Riemann tensor mentioned above.

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arXiv2019

Gravitational lensing beyond geometric optics: II. Metric independence

Typical applications of gravitational lensing use the properties of electromagnetic or gravitational waves to infer the geometry through which those waves propagate. Nevertheless, the optical fields themselves - as opposed to their interactions with material bodies - encode very little of that geometry: It is shown here that any given configuration is compatible with a very large variety of spacetime metrics. For scalar fields in geometric optics, or observables which are not sensitive to the detailed polarization content of electromagnetic or gravitational waves, seven of the ten metric components are essentially irrelevant. With polarization, five components are irrelevant. In the former case, this result together with diffeomorphism invariance allows essentially any geometric-optics configuration associated with a particular spacetime to be embedded into any other spacetime, at least in finite regions. Going beyond the geometric-optics approximation breaks some of this degeneracy, although much remains even then. Overall, high-frequency wave propagation is shown to be insensitive to compositions of certain conformal, Kerr-Schild, and related transformations of the background metric. One application is that new solutions for scalar, electromagnetic, and gravitational waves may be generated from old ones. In one example described here, the high-frequency scattering of a plane wave by a point mass is computed by transforming a plane wave in flat spacetime.

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arXiv2019

Gravitational lensing beyond geometric optics: I. Formalism and observables

The laws of geometric optics and their corrections are derived for scalar, electromagnetic, and gravitational waves propagating in generic curved spacetimes. Local peeling-type results are obtained, where different components of high-frequency fields are shown to scale with different powers of their frequencies. Additionally, finite-frequency corrections are identified for a number of conservation laws and observables. Among these observables are a field's energy and momentum densities, as well as several candidates for its corrected "propagation directions."

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arXiv2017

Self-forces in arbitrary dimensions

Bodies coupled to electromagnetic or other long-range fields are subject to radiation reaction and other effects in which their own fields can influence their motion. Self-force phenomena such as these have been poorly understood for spacetime dimensions not equal to four, despite the relevance of differing dimensionalities for holographic duals, effectively two-dimensional condensed matter and fluid systems, and so on. We remedy this by showing that forces and torques acting on extended electromagnetic charges in all dimensions $d \geq 3$ have the same functional form as the usual test body expressions, except that the electromagnetic field appearing in those expressions is not the physical one; it is an effective surrogate. For arbitrary even $d \geq 4$, our surrogate field locally satisfies the source-free field equations, and is conceptually very similar to what arises in the Detweiler-Whiting prescription previously established when $d=4$. The odd-dimensional case is different, involving effective fields which are not necessarily source-free. Moreover, we find a 1-parameter family of natural effective fields for each odd $d$, where the free parameter--a lengthscale--is degenerate with (finite) renormalizations of a body's stress-energy tensor. While different parameter choices can result in different forces, they do so without affecting physical observables. Having established these general results, explicit point-particle self-forces are derived in odd-dimensional Minkowski spacetimes. Simple examples are discussed for $d=3$ and $d=5$, one of which illustrates that the particularly slow decay of fields in three spacetime dimensions results in particles creating their own "preferred rest frames:" Initially-static charges which are later perturbed have a strong tendency to return to rest. Our results easily extend also to the scalar and gravitational self-force problems.

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arXiv2017

Generating exact solutions to Einstein's equation using linearized approximations

We show that certain solutions to the linearized Einstein equation can---by the application of a particular type of linearized gauge transformation---be straightforwardly transformed into solutions of the exact Einstein equation. In cases with nontrivial matter content, the exact stress-energy tensor of the transformed metric has the same eigenvalues and eigenvectors as the linearized stress-energy tensor of the initial approximation. When our gauge exists, the tensorial structure of transformed metric perturbations identically eliminates all nonlinearities in Einstein's equation. As examples, we derive the exact Kerr and gravitational plane wave metrics from standard harmonic-gauge approximations.

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arXiv2017

Metric-independence of vacuum and force-free electromagnetic fields

Electromagnetic fields which solve the vacuum Maxwell equations in one spacetime are well-known to also be solutions in all spacetimes with conformally-related metrics. This provides a sense in which electromagnetism alone cannot be used to measure certain aspects of geometry. We show that there is actually much more which cannot be so measured; relatively little of a spacetime's geometry is in fact imprinted in any particular electromagnetic field. This is demonstrated by finding a much larger class of metric transformations---involving five free functions---which preserve Maxwell solutions both in vacuum, without local currents, and also for the force-free electrodynamics associated with a tenuous plasma. One consequence of this is that many of the exact force-free fields which have previously been found around Schwarzschild and Kerr black holes are also solutions in appropriately-identified flat backgrounds. As a more direct application, we use our metric transformations to write down a large class of electromagnetic waves which remain unchanged by a large class of gravitational waves propagating "in the same direction."

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arXiv2016

Self-forces on static bodies in arbitrary dimensions

We derive exact expressions for the scalar and electromagnetic self-forces and self-torques acting on arbitrary static extended bodies in arbitrary static spacetimes with any number of dimensions. Non-perturbatively, our results are identical in all dimensions. Meaningful point particle limits are quite different in different dimensions, however. These limits are defined and evaluated, resulting in simple "regularization algorithms" which can be used in concrete calculations. In these limits, self-interaction is shown to be progressively less important in higher numbers of dimensions; it generically competes in magnitude with increasingly high-order extended-body effects. Conversely, we show that self-interaction effects can be relatively large in $1+1$ and $2+1$ dimensions. Our motivations for this work are twofold: First, no previous derivation of the self-force has been provided in arbitrary dimensions, and heuristic arguments presented by different authors have resulted in conflicting conclusions. Second, the static self-force problem in arbitrary dimensions provides a valuable testbed with which to continue the development of general, non-perturbative methods in the theory of motion. Several new insights are obtained in this direction, including a significantly improved understanding of the renormalization process. We also show that there is considerable freedom to use different "effective fields" in the laws of motion---a freedom which can be exploited to optimally simplify specific problems. Different choices give rise to different inertias, gravitational forces, and electromagnetic or scalar self-forces, but there is a sense in which none of these quantities are individually accessible to experiment. Certain combinations are observable, however, and these remain invariant under all possible field redefinitions.

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arXiv2015

Motion in classical field theories and the foundations of the self-force problem

This article serves as a pedagogical introduction to the problem of motion in classical field theories. The primary focus is on self-interaction: How does an object's own field affect its motion? General laws governing the self-force and self-torque are derived using simple, non-perturbative arguments. The relevant concepts are developed gradually by considering motion in a series of increasingly complicated theories. Newtonian gravity is discussed first, then Klein-Gordon theory, electromagnetism, and finally general relativity. Linear and angular momenta as well as centers of mass are defined in each of these cases. Multipole expansions for the force and torque are then derived to all orders for arbitrarily self-interacting extended objects. These expansions are found to be structurally identical to the laws of motion satisfied by extended test bodies, except that all relevant fields are replaced by effective versions which exclude the self-fields in a particular sense. Regularization methods traditionally associated with self-interacting point particles arise as straightforward perturbative limits of these (more fundamental) results. Additionally, generic mechanisms are discussed which dynamically shift --- i.e., renormalize --- the apparent multipole moments associated with self-interacting extended bodies. Although this is primarily a synthesis of earlier work, several new results and interpretations are included as well.

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arXiv2015

Optics in a nonlinear gravitational plane wave

Gravitational waves can act like gravitational lenses, affecting the observed positions, brightnesses, and redshifts of distant objects. Exact expressions for such effects are derived here in general relativity, allowing for arbitrarily-moving sources and observers in the presence of plane-symmetric gravitational waves. At least for freely falling sources and observers, it is shown that the commonly-used predictions of linear perturbation theory can be generically overshadowed by nonlinear effects; even for very weak gravitational waves, higher-order perturbative corrections involve secularly-growing terms which cannot necessarily be neglected when considering observations of sufficiently distant sources. Even on more moderate scales where linear effects remain at least marginally dominant, nonlinear corrections are qualitatively different from their linear counterparts. There is a sense in which they can, for example, mimic the existence of a third type of gravitational wave polarization.

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