Gravitational lensing beyond geometric optics: II. Metric independence (2019)
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 solution-dependent 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 Kerr-Schild or related type. The most general interesting transformations may be viewed as compositions of two Kerr-Schild transformations which are transverse to the rays, an extended Kerr-Schild transformation along the rays, and a conformal transformation. The two transverse Kerr-Schild 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 solution-generating technique. Various examples are discussed. In one, the scattering of high-frequency 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 spherically-symmetric deformations of the metric. Similarly, plane-fronted waves are shown to be unaffected by plane-fronted deformations of the geometry (in the same direction).
Persistent gravitational wave observables: General framework (2019)
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 previously-identified memory effects of various kinds (displacement, velocity, rotation), but more as well.
Gravitational lensing beyond geometric optics: I. Formalism and observables (2019)
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 Newman-Penrose 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 high-frequency scalar field, how well does that characterize a high-frequency 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 self-force problem in arbitrary dimensions (2018)
This paper derives a unified theory of motion for charged extended bodies in arbitrary spacetimes and in any number of dimensions. Self-interaction is allowed with an object's own electromagnetic field. There are three major classes of results: I) The derivation of a fully non-perturbative description of motion, including all extended-body, self-force, and self-torque effects, II) the specialization of these results using point-particle limits, and III) physical consequences of the point-particle equations of motion.
Precise definitions were found for well-behaved 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 point-particle contexts). All moments of the stress-energy tensor are finitely renormalized by the self-field, but reduce to Dixon's definitions in the test-body limit. All renormalizations are precisely defined, and although the self-field extends to large distances, its contributions to a charge's inertia do not depend on the distant past (unlike naive definitions which include the stress-energy tensor of the self-field 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 self-force and self-torque are not unique. While this lack of uniqueness is always present, it is more easily ignored in even numbers of dimensions and in the point-particle 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 Detweiler-Whiting axioms (which are logically not axioms). The "axioms" obtained here are valid in any number of dimensions, while the Detweiler-Whiting ones make sense only in even numbers of dimensions. One difference is that the requirement that the effective field satisfy the source-free field equation is dropped and replaced by something weaker.
As an application, self-forces 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 self-field 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!
Self-forces in arbitrary dimensions (2017) [arXiv]
This is essentially a summary of certain results in the more extended paper "Foundations of the self-force problem in arbitrary dimensions."
Metric-independence of vacuum and force-free electromagnetic fields (2017)
This paper shows that solutions to Maxwell's equations are invariant under large classes of transformations to the background metric. While it is well-known 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 Kerr-Schild transformations. They preserve not only electromagnetic fields away from sources, but also those which exist inside of force-free 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 non-unique. And this non-uniqueness is not the only subtlety of the metric-reconstruction problem: The different types of metric-transformation 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 more-specific applications are discussed. One result is that certain force-free solutions which had previously been identified in the Kerr spacetime were shown here to be flat-spacetime solutions in disguise. There is a sense in which the black hole in the middle of the system has no effect, which explains the previously-puzzling absence of tails.
Generating exact solutions to Einstein's equation using linearized approximations (2016)
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 "Kerr-Schild" 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 first-order (as opposed to exact) gauge transformation.
Various examples are worked out: TT-gauge gravitational wave solutions (which are not exact) are transformed to exact plane wave solutions. A second-order approximation to the Reissner-Nordstrøm 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).
Self-forces on static bodies in arbitrary dimensions (2016)
This paper non-perturbatively derived the static self-force and self-torque 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, non-rigorous 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 stress-energy tensor. Moreover, an explicit formula for the renormalized stress-energy tensor was obtained, which was new even in 4D. It was shown that self-field 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 self-force 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 stress-energy 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.
Point-particle limits were discussed as well, and it was shown that self-force effects are necessarily less significant in higher numbers of dimensions; they compete with gravitational forces of increasingly-large multipole order.
- Optics in a nonlinear gravitational plane wave (2015) [arXiv] [DOI]
- Motion in classical field theories and the foundations of the self-force problem (2015) [arXiv] [DOI]
- The not-so-nonlinear nonlinearity of Einstein's equation (2014) [arXiv] [DOI]
- Gravitational self-torque and spin precession in compact binaries (2014) [arXiv] [DOI]
- Tails of plane wave spacetimes: Wave-wave 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 stress-energy tensors, self-force, and broken symmetry (2010) [arXiv] [DOI]
- A Rigorous Derivation of Electromagnetic Self-force (2009) [arXiv] [DOI]
- Electromagnetic self-forces and generalized Killing fields (2009) [arXiv] [DOI]
- Self-forces from generalized Killing fields (2008) [arXiv] [DOI]
- Approximate spacetime symmetries and conservation laws (2008) [arXiv] [DOI]
- Extended-body effects in cosmological spacetimes (2007) [arXiv] [DOI]
- Multipole structure of current vectors in curved spacetime (2007) [arXiv] [DOI]
- Self-forces on extended bodies in electrodynamics (2006) [arXiv] [DOI]
- Mass loss by a scalar charge in an expanding universe (2002) [arXiv] [DOI]