Publications

Extendedbody motion in black hole spacetimes: What is possible? (2020)
[arXiv]
[DOI]
Description
This paper considers extendedbody motion in vacuum type D spacetimes (such as Kerr or Schwarzschild). It is essentially a relativistic generalization of the Newtonian analysis in the previous paper.
The paper has two main parts. The first asks which forces and torques are possible in the quadrupole approximation. As in the Newtonian case, symmetries (more specifically, Killing fields) imply the existence of conservation laws which preclude certain force and torque combinations. This is not new. What is new is that in the quadrupole approximation, there are additional torque constraints which do not appear to be associated with any conservation laws. These are however associated with the presence of conformal KillingYano tensors. In Schwarzschild, there is one constraint with this form. It is genuinely relativistic in the sense that it admits no Newtonian limit. In Kerr backgrounds, there are instead two torque constraints which do not appear to be related to conservation laws.
The second part of this paper specializes to the motion of an extended spacecraft in the Schwarzschild spacetime, allowing that spacecraft to control its internal structure. The spinfree, torquefree case is considered for simplicity, which requires that the spacecraft control its structure in such a way that the conformal KillingYano tensors are eigenbivectors of its quadrupole moment. This leaves one quadrupole component which can be freely manipulated in order to control the motion. Without making any postNewtonian approximations, the problem is shown to be almost identical to its Newtonian counterpart; eccentricities and apsidal orientations can be manipulated at will using periodic internal deformations. Physical interpretations nevertheless differ in the Newtonian and relativistic contexts. In the Newtonian problem, mass is constant while energy is not; internal (chemical or other) energy can be traded for orbital energy, and vice versa. In the relativistic context, it is instead the mass which varies while the energy does not. This is not due to the ejection or absorption of any matter. Rather, extendedbody effects generically change an object's rest mass, and in this context, those changes have a direct effect on the orbit.
One context where the relativistic and Newtonian problems do differ is at small radii. Below ISCO, circular pointparticle orbits are of course unstable. This instability can however be cured using extendedbody effects. The mechanism is nevertheless similar to the one used to stabilize ordinarilyunstable orbits in certain (nonspherical) Newtonian backgrounds.
Genuinelyrelativistic effects could arise via the ability to control the hidden momentum, which requires a lifting of the torquefree assumption. This would allow direct control over the velocity of an object, rather than only its acceleration.

Extendedbody effects and rocketfree orbital maneuvering (2020)
[arXiv]
[DOI]
Description
Can a spacecraft change its orbit without the use of a rocket? Indeed it can. This paper explores how, in Newtonian gravity, objects can accelerate and spin themselves up by rearranging their internal mass distributions (at fixed total mass): Objects can "grab onto" or "push off of" gradients in the gravitational field. More precisely, the paper shows that the forces and torques which can be produced using extendedbody effects are constrained mainly by the symmetries of the gravitational field gradient.
The case of a deformable spacecraft in orbit around a sphericallysymmetric gravitational mass is analyzed in detail. Considering torquefree spacecraft for simplicity, certain components of the structure are constrained. However, one component of the quadrupole moment which affects the motion can still be controlled. If that component is manipulated appropriately, orbital eccentricities and apsidal angles can be controlled at will. Merely by performing periodic internal motions, spacecraft can escape from initiallybound orbits or circularize initiallyeccentric orbits. The process may be slow, but effects build up secularly over many orbits. Different notions of energy which are relevant to these processes are analyzed and discussed in detail.
It is also shown that extendedbody effects can be used to stabilize orbits which would ordinarily be unstable. This does not apply to the Kepler problem in three dimensions, as all bound, pointparticle orbits are stable in that case. It does however apply to higherdimensional Kepler problems, and also to motion around certain highlyoblate masses in three dimensions. The process may be understood as carving out a small well at the peak of the effective potential.

Persistent gravitational wave observables: Nonlinear plane wave spacetimes (2020)
[arXiv]
[DOI]
Description
This paper evaluates certain generalizations of the memory effect in fullynonlinear gravitational plane waves. To what extent can the measurement of these observables determine the nature of the waveform? In the linear case, only a limited amount of information can be gained by such measurements. Although one might expect more information to be measurable in the nonlinear case, this is unfortunately not the case. Instead, there are a complicated set of nonlinear relations between the different observables; their physical content does not change. Nevertheless, there are other senses in which nonlinear effects are qualitatively different from their linear counterparts.

Gravitational lensing beyond geometric optics: II. Metric independence (2019)
[arXiv]
[DOI]
Description
This paper shows that geometric optics and some of its corrections depend relatively little on the geometry of spacetime. There are a large class of solutiondependent symmetries in the equations for null geodesics, amplitudes, polarization states, etc. which make these structures insensitive to much of the geometry. This may be stated in terms of transformations which may be applied to a metric without affecting an associated optical quantity. In many contexts, seven of the ten metric components are shown to be irrelevant in this sense.
Besides the conformal transformations which might be expected here, other metric transformations which preserve optical observables are of KerrSchild or related type. The most general interesting transformations may be viewed as compositions of two KerrSchild transformations which are transverse to the rays, an extended KerrSchild transformation along the rays, and a conformal transformation. The two transverse KerrSchild transformations may be individually complex, although their composition is real.
One consequence of these results is that using optical observations to infer a metric can produce results which are far from unique.
Another consequence is that these results can be used as a solutiongenerating technique. Various examples are discussed. In one, the scattering of highfrequency waves by a compact mass is determined by applying a suitable transformation to a plane wave in flat spacetime. Also, fields with spherical wavefronts are shown to be unaffected by sphericallysymmetric deformations of the metric. Similarly, planefronted waves are shown to be unaffected by planefronted deformations of the geometry (in the same direction).

Persistent gravitational wave observables: General framework (2019)
[arXiv]
[DOI]
Description
This paper introduces a number of observables which might be used to characterize bursts of gravitational waves, and which persist after those waves have passed. They include previouslyidentified memory effects of various kinds (displacement, velocity, rotation), but more as well.

Gravitational lensing beyond geometric optics: I. Formalism and observables (2019)
[arXiv]
[DOI]
Description
This paper focuses on how various observables and conservation laws are affected by corrections to geometric optics in general relativity. Scalar, electromagnetic, and gravitational waves are all discussed.
One focus is on a field's direction of propagation. While this is an unambiguous concept in geometric optics, it is shown to be problematic more generally. For scalar fields, there are no difficulties at one order beyond geometric optics. Things are more complicated for electromagnetic or gravitational waves: There can be multiple propagation directions which must be taken into account simulataneously. While unique propagation directions may be defined in various ways, these generically require the introduction of preferred observers or other external structures; different choices lead to different results and none appears to be particularly natural.
Another focus here is on the frequency dependence of different tensorial components of electromagnetic fields and Weyl tensors. Different NewmanPenrose scalars are shown to depend differently on a wave's frequency. This is reminiscent (and not unrelated) to the peeling theorems, which predict that these components decay with different powers of 1/r at large distances. Unlike those theorems, the "frequency peeling" obtained here is locally valid; it does not assume anything about the asymptotic structure of the spacetime.
Lastly, relations between different types of fields are discussed. Given knowledge of a highfrequency scalar field, how well does that characterize a highfrequency electromagnetic field? Similarly, how well does knowing something about an electromagnetic wave constrain a gravitational wave? A kind of "double copy" is shown to appear emerge naturally when answering these questions.

Foundations of the selfforce problem in arbitrary dimensions (2018)
[arXiv]
[DOI]
Description
This paper derives a unified theory of motion for charged extended bodies in arbitrary spacetimes and in any number of dimensions. Selfinteraction is allowed with an object's own electromagnetic field. There are three major classes of results: I) The derivation of a fully nonperturbative description of motion, including all extendedbody, selfforce, and selftorque effects, II) the specialization of these results using pointparticle limits, and III) physical consequences of the pointparticle equations of motion.
Precise definitions were found for wellbehaved linear and angular momenta. Their laws of motion were shown to be identical in form to Dixon's, but with a particular effective field replacing the physical field (what is often called in a regular field in pointparticle contexts). All moments of the stressenergy tensor are finitely renormalized by the selffield, but reduce to Dixon's definitions in the testbody limit. All renormalizations are precisely defined, and although the selffield extends to large distances, its contributions to a charge's inertia do not depend on the distant past (unlike naive definitions which include the stressenergy tensor of the selffield in the momenta).
There are in fact a large class of reasonable momenta which might be defined, all of which obey laws of motion with somewhat different effective fields. It follows that the selfforce and selftorque are not unique. While this lack of uniqueness is always present, it is more easily ignored in even numbers of dimensions and in the pointparticle limits which are typically considered. The lack of uniqueness must be confronted directly in odd numbers of dimensions.
Replacements are obtained for what might be called the 4D DetweilerWhiting axioms (which are logically not axioms). The "axioms" obtained here are valid in any number of dimensions, while the DetweilerWhiting ones make sense only in even numbers of dimensions. One difference is that the requirement that the effective field satisfy the sourcefree field equation is dropped and replaced by something weaker.
As an application, selfforces in 2+1D Minkowski spacetime are shown to behave completely differently than in 3+1D, and also to be connected to certain realistic systems. Most strikingly, a charge which is initially at rest and is later kicked always returns to rest; a charge's selffield decays so slowly that its initial state of motion creates a kind of "rest frame" to which the charge always returns. In this sense, inertia in one lower dimension has an almost Aristotelian (as opposed to Galilean) character!

Selfforces in arbitrary dimensions (2017) [arXiv]
Description
This is essentially a summary of certain results in the more extended paper "Foundations of the selfforce problem in arbitrary dimensions."

Metricindependence of vacuum and forcefree electromagnetic fields (2017)
[arXiv]
[DOI]
Description
This paper shows that solutions to Maxwell's equations are invariant under large classes of transformations to the background metric. While it is wellknown that conformal transformations preserve electromagnetic fields, it is shown here that there are in fact four more transformations which also preserve solutions. The forms of these additional transformations depend on the principal null direction(s) of the solution which is to be preserved, and are related to KerrSchild transformations. They preserve not only electromagnetic fields away from sources, but also those which exist inside of forcefree plasmas.
One consequence is that knowledge of a single electromagnetic field cannot be used to infer the metric (even up to a conformal factor). This is an inverse problem whose solutions are far from nonunique. And this nonuniqueness is not the only subtlety of the metricreconstruction problem: The different types of metrictransformation laws which arise for fields with one or two principal null directions imply that metrics can depend very sensitively—or even discontinuously—on the electromagnetic field.
Certain morespecific applications are discussed. One result is that certain forcefree solutions which had previously been identified in the Kerr spacetime were shown here to be flatspacetime solutions in disguise. There is a sense in which the black hole in the middle of the system has no effect, which explains the previouslypuzzling absence of tails.

Generating exact solutions to Einstein's equation using linearized approximations (2016)
[arXiv]
[DOI]
Description
It's usually thought that gauge choices in general relativity are just a matter of convenience. Here, it's shown that the choice of perturbative gauge can completely change the accuracy of a calculation: If a certain "KerrSchild" gauge exists, transforming to it will turn a linearized solution—which is not an exact solution to Einstein's equation—into a solution which is exact. This may be viewed as a kind of resummation technique, although it is the only one in this context whose efficacy is derived and not just empirically observed. There is a sense in which the error in the seed solution is exactly and controllably canceled by the error in applying a firstorder (as opposed to exact) gauge transformation.
Various examples are worked out: TTgauge gravitational wave solutions (which are not exact) are transformed to exact plane wave solutions. A secondorder approximation to the ReissnerNordstrøm solution is made exact. Lastly, the Kerr metric is derived from a linearized approximation in Lorenz gauge. This last calculation introduces some useful techniques to exactly sum series which involve contributions from all of the multipole moments of Kerr.
It may be concluded that although it is common to think of black holes as being "highly nonlinear" objects, their full structure is essentially embedded in the linearized theory; strong gravity does not imply strong nonlinearity!
Unfortunately, the gauge considered here exists only in special cases. The hope is therefore that these ideas may be incorporated into a more general scheme where gauge choices are used to systematically optimize the accuracy of a perturbative calculation (even if they cannot remove all inaccuracy).

Selfforces on static bodies in arbitrary dimensions (2016)
[arXiv]
[DOI]
Description
This paper nonperturbatively derived the static selfforce and selftorque in arbitrary dimensions. More precisely, it obtained the "holding force" and "holding torque" which must be applied to a charge in order for it to remain static.
Although this appears to be a simple problem, nonrigorous arguments had previously been used by various authors to make conflicting claims. Most provocatively, it had been suggested that the 5D problem depends on a body's internal structure in a way which is qualitatively different from what occurs in 4D.
The analysis here showed that there was no such dependence on internal structure: In any number of dimensions, such effects were shown to be removable by finitely renormalizating the stressenergy tensor. Moreover, an explicit formula for the renormalized stressenergy tensor was obtained, which was new even in 4D. It was shown that selffield contributions to the momenta can be written as functional derivatives of certain propagators with respect to the metric.
Another general result was that there is a certain lack of uniqueness which is inherent in selfforce problems: It is possible to define different reasonable momenta, for example, and these are associated with, e.g., different masses. They are also associated with different electromagnetic forces, due to changes in the effective field, and different gravitational forces, due to changes in the effective stressenergy tensor. Such shifts may be interpreted as a new kind of gauge freedom. Only certain nontrivial combinations of quantities remain invariant. This is not simply a difficulty which must be recognized; it is shown that it can also be exploited to simplify problems.
Pointparticle limits were discussed as well, and it was shown that selfforce effects are necessarily less significant in higher numbers of dimensions; they compete with gravitational forces of increasinglylarge multipole order.
 Optics in a nonlinear gravitational plane wave (2015) [arXiv] [DOI]
 Motion in classical field theories and the foundations of the selfforce problem (2015) [arXiv] [DOI]
 The notsononlinear nonlinearity of Einstein's equation (2014) [arXiv] [DOI]
 Gravitational selftorque and spin precession in compact binaries (2014) [arXiv] [DOI]
 Tails of plane wave spacetimes: Wavewave scattering in general relativity (2013) [arXiv] [DOI]
 Magnetorotational instability in relativistic hypermassive neutron stars (2013) [arXiv] [DOI]
 Strong lensing, plane gravitational waves and transient flashes (2013) [arXiv] [DOI]
 Caustics and wave propagation in curved spacetimes (2012) [arXiv] [DOI]
 Mechanics of extended masses in general relativity (2012) [arXiv] [DOI]
 Bobbing and Kicks in Electromagnetism and Gravity (2010) [arXiv] [DOI]
 Effective stressenergy tensors, selfforce, and broken symmetry (2010) [arXiv] [DOI]
 A Rigorous Derivation of Electromagnetic Selfforce (2009) [arXiv] [DOI]
 Electromagnetic selfforces and generalized Killing fields (2009) [arXiv] [DOI]
 Selfforces from generalized Killing fields (2008) [arXiv] [DOI]
 Approximate spacetime symmetries and conservation laws (2008) [arXiv] [DOI]
 Extendedbody effects in cosmological spacetimes (2007) [arXiv] [DOI]
 Multipole structure of current vectors in curved spacetime (2007) [arXiv] [DOI]
 Selfforces on extended bodies in electrodynamics (2006) [arXiv] [DOI]
 Mass loss by a scalar charge in an expanding universe (2002) [arXiv] [DOI]