arXiv subjects

Adam Pound

Publications and source records attributed to Adam Pound.

At least 73 records · Page 4Linked to original sources

arXiv2015

Second-order perturbation theory: problems on large scales

In general-relativistic perturbation theory, a point mass accelerates away from geodesic motion due to its gravitational self-force. Because the self-force is small, one can often approximate the motion as geodesic. However, it is well known that self-force effects accumulate over time, making the geodesic approximation fail on long timescales. It is less well known that this failure at large times translates to a failure at large distances as well. At second perturbative order, two large-distance pathologies arise: spurious secular growth and infrared-divergent retarded integrals. Both stand in the way of practical computations of second-order self-force effects. Utilizing a simple flat-space scalar toy model, I develop methods to overcome these obstacles. The secular growth is tamed with a multiscale expansion that captures the system's slow evolution. The divergent integrals are eliminated by matching to the correct retarded solution at large distances. I also show how to extract conservative self-force effects by taking local-in-time "snapshots" of the global solution. These methods are readily adaptable to the physically relevant case of a point mass orbiting a black hole.

gr-qc↗

arXiv2015

Gauge and motion in perturbation theory

Through second order in perturbative general relativity, a small compact object in an external vacuum spacetime obeys a generalized equivalence principle: although it is accelerated with respect to the external background geometry, it is in free fall with respect to a certain \emph{effective} vacuum geometry. However, this single principle takes very different mathematical forms, with very different behaviors, depending on how one treats perturbed motion. Furthermore, any description of perturbed motion can be altered by a gauge transformation. In this paper, I clarify the relationship between two treatments of perturbed motion and the gauge freedom in each. I first show explicitly how one common treatment, called the Gralla-Wald approximation, can be derived from a second, called the self-consistent approximation. I next present a general treatment of smooth gauge transformations in both approximations, in which I emphasise that the approximations' governing equations can be formulated in an invariant manner. All of these analyses are carried through second perturbative order, but the methods are general enough to go to any order. Furthermore, the tools I develop, and many of the results, should have broad applicability to any description of perturbed motion, including osculating-geodesic and two-timescale descriptions.

gr-qc↗

arXiv2015

Motion of small objects in curved spacetimes: An introduction to gravitational self-force

In recent years, asymptotic approximation schemes have been developed to describe the motion of a small compact object through a vacuum background to any order in perturbation theory. The schemes are based on rigorous methods of matched asymptotic expansions, which account for the object's finite size, require no "regularization" of divergent quantities, and are valid for strong fields and relativistic speeds. Up to couplings of the object's multipole moments to the external background curvature, these schemes have established that at least through second order in perturbation theory, the object's motion satisfies a generalized equivalence principle: it moves on a geodesic of a certain smooth metric satisfying the vacuum Einstein equation. I describe the foundations of this result, particularly focusing on the fundamental notion of how a small object's motion is represented in perturbation theory. The three common representations of perturbed motion are (i) the "Gralla-Wald" description in terms of small deviations from a reference geodesic, (ii) the "self-consistent" description in terms of a worldline that obeys a self-accelerated equation of motion, and (iii) the "osculating geodesics" description, which utilizes both (i) and (ii). Because of the coordinate freedom in general relativity, any coordinate description of motion in perturbation theory is intimately related to the theory's gauge freedom. In this review I describe asymptotic solutions of the Einstein equations adapted to each of the three representations of motion, and I discuss the gauge freedom associated with each. I conclude with a discussion of how gauge freedom must be refined in the context of long-term dynamics.

gr-qc↗

arXiv2015

Linear-in-mass-ratio contribution to spin precession and tidal invariants in Schwarzschild spacetime at very high post-Newtonian order

Using black hole perturbation theory and arbitrary-precision computer algebra, we obtain the post-Newtonian (pN) expansions of the linear-in-mass-ratio corrections to the spin-precession angle and tidal invariants for a particle in circular orbit around a Schwarzschild black hole. We extract coefficients up to 20pN order from numerical results that are calculated with an accuracy greater than 1 part in $10^{500}$. These results can be used to calibrate parameters in effective-one-body models of compact binaries, specifically the spin-orbit part of the effective Hamiltonian and the dynamically significant tidal part of the main radial potential of the effective metric. Our calculations are performed in a radiation gauge, which is known to be singular away from the particle. To overcome this irregularity, we define suitable Detweiler-Whiting singular and regular fields in this gauge, and we devise a rigorous mode-sum regularization method to compute the invariants constructed from the regular field.

gr-qc↗

arXiv2014

A conservative effect of the second-order gravitational self-force on quasicircular orbits in Schwarzschild spacetime

A compact object moving on a quasicircular orbit about a Schwarzschild black hole gradually spirals inward due to the dissipative action of its gravitational self-force. But in addition to driving the inspiral, the self-force has a conservative piece. Within a second-order self-force formalism, I derive a second-order generalization of Detweiler's redshift variable, which provides a gauge-invariant measure of conservative effects on quasicircular orbits. I sketch a frequency-domain numerical scheme for calculating this quantity. Once this scheme has been implemented, its results may be used to determine high-order terms in post-Newtonian theory and parameters in effective-one-body theory.

gr-qc↗

arXiv2014

Research Update on Extreme-Mass-Ratio Inspirals

The inspirals of stellar-mass mass compact objects into massive black holes in the centres of galaxies are one of the most important sources of gravitational radiation for space-based detectors like LISA or eLISA. These extreme-mass-ratio inspirals (EMRIs) will enable an ambitious research program with implications for astrophysics, cosmology, and fundamental physics. This article is a summary of the talks delivered at the plenary session on EMRIs at the 10th International LISA Symposium. It contains research updates on the following topics: astrophysics of EMRIs; EMRI science potential; and EMRI modeling.

astro-ph.CO↗

arXiv2014

A practical, covariant puncture for second-order self-force calculations

Accurately modeling an extreme-mass-ratio inspiral requires knowledge of the second-order gravitational self-force on the inspiraling small object. Recently, numerical puncture schemes have been formulated to calculate this force, and their essential analytical ingredients have been derived from first principles. However, the \emph{puncture}, a local representation of the small object's self-field, in each of these schemes has been presented only in a local coordinate system centered on the small object, while a numerical implementation will require the puncture in coordinates covering the entire numerical domain. In this paper we provide an explicit covariant self-field as a local expansion in terms of Synge's world function. The self-field is written in the Lorenz gauge, in an arbitrary vacuum background, and in forms suitable for both self-consistent and Gralla-Wald-type representations of the object's trajectory. We illustrate the local expansion's utility by sketching the procedure of constructing from it a numerically practical puncture in any chosen coordinate system.

gr-qc↗

arXiv2013

Gravitational self-force from radiation-gauge metric perturbations

Calculations of the gravitational self-force (GSF) in curved spacetime require as input the metric perturbation in a sufficiently regular gauge. A basic challenge in the program to compute the GSF for orbits around a Kerr black hole is that the standard procedure for reconstructing the perturbation is formulated in a class of radiation gauges, in which the particle singularity is non-isotropic and extends away from the particle's location. Here we present two practical schemes for calculating the GSF using a radiation-gauge reconstructed metric as input. The schemes are based on a detailed analysis of the local structure of the particle singularity in the radiation gauges. We identify 3 types of radiation gauges: two containing a radial string-like singularity emanating from the particle, either in one direction ("half-string" gauges) or both directions ("full-string" gauges); and a third type containing no strings but with a jump discontinuity across a surface intersecting the particle. Based on a flat-space example, we argue that the standard mode-by-mode reconstruction procedure yields the "regular half" of a half-string solution, or (equivalently) either of the regular halves of a no-string solution. For the half-string case, we formulate the GSF in a locally deformed radiation gauge that removes the string singularity near the particle. We derive a mode-sum formula for the GSF in this gauge, analogous to the standard Lorenz-gauge formula but with modified regularization parameters. For the no-string case, we formulate the GSF directly, without a local deformation, and we derive a mode-sum formula that requires no correction to the parameters but involves a certain averaging procedure. We explain the consistency of our results with Gralla's invariance theorem, and discuss the correspondence between our method and a related approach by Friedman et al.

gr-qc↗

arXiv2013

Second-order gravitational self-force -- a quick summary

In order to extract physical parameters from the waveform of an extreme-mass-ratio binary, one requires a second-order--accurate description of the motion of the smaller of the two objects in the binary. Using a method of matched asymptotic expansions, I derive the second-order equation of motion of a small, nearly spherical and non-rotating compact object in an arbitrary vacuum spacetime. I find that the motion is geodesic in a certain locally defined effective metric satisfying the vacuum Einstein equation through second order, and I outline a method of numerically calculating this effective metric.

gr-qc↗

arXiv2012

Nonlinear gravitational self-force. I. Field outside a small body

A small extended body moving through an external spacetime $g_{αβ}$ creates a metric perturbation $h_{αβ}$, which forces the body away from geodesic motion in $g_{αβ}$. The foundations of this effect, called the gravitational self-force, are now well established, but concrete results have mostly been limited to linear order. Accurately modeling the dynamics of compact binaries requires proceeding to nonlinear orders. To that end, I show how to obtain the metric perturbation outside the body at all orders in a class of generalized wave gauges. In a small buffer region surrounding the body, the form of the perturbation can be found analytically as an expansion for small distances $r$ from a representative worldline. Given only a specification of the body's multipole moments, the field obtained in the buffer region suffices to find the metric everywhere outside the body via a numerical puncture scheme. Following this procedure at first and second order, I calculate the field in the buffer region around an arbitrarily structured compact body at sufficiently high order in $r$ to numerically implement a second-order puncture scheme, including effects of the body's spin. I also define $n$th-order (local) generalizations of the Detweiler-Whiting singular and regular fields and show that in a certain sense, the body can be viewed as a skeleton of multipole moments.

gr-qc↗

arXiv2012

Second-order gravitational self-force

Using a rigorous method of matched asymptotic expansions, I derive the equation of motion of a small, compact body in an external vacuum spacetime through second order in the body's mass (neglecting effects of internal structure). The motion is found to be geodesic in a certain locally defined regular geometry satisfying Einstein's equation at second order. I outline a method of numerically obtaining both the metric of that regular geometry and the complete second-order metric perturbation produced by the body.

gr-qc↗

arXiv2012

Magneto-optical conductivity in graphene including electron-phonon coupling

We show how coupling to an Einstein phonon $ω_E$ affects the absorption peaks seen in the optical conductivity of graphene under a magnetic field $B$. The energies and widths of the various lines are shifted, and additional peaks arise in the spectrum. Some of these peaks are Holstein sidebands, resulting from the transfer of spectral weight in each Landau level (LL) into phonon-assisted peaks in the spectral function. Other additional absorption peaks result from transitions involving split LLs, which occur when a LL falls sufficiently close to a peak in the self-energy. We establish the selection rules for the additional transitions and characterize the additional absorption peaks. For finite chemical potential, spectral weight is asymmetrically distributed about the Dirac point; we discuss how this causes an asymmetry in the transitions due to left- and right-handed circularly polarized light and therefore oscillatory behavior in the imaginary part of the off-diagonal Hall conductivity. We also find that the semiclassical cyclotron resonance region is renormalized by an effective-mass factor but is not directly affected by the additional transitions. Last, we discuss how the additional transitions can manifest in broadened, rather than split, absorption peaks due to large scattering rates seen in experiment.

cond-mat.mes-hall↗

arXiv2011

The motion of point particles in curved spacetime

This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime. The field's action on the particle is difficult to calculate because of its singular nature: the field diverges at the position of the particle. But it is possible to isolate the field's singular part and show that it exerts no force on the particle. What remains after subtraction is a smooth field that is fully responsible for the self-force. The mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime are developed here from scratch. The review begins with a discussion of the basic theory of bitensors. It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle's word line. It continues with a thorough discussion of Green's functions in curved spacetime. The review presents a detailed derivation of each of the three equations of motion. Because the notion of a point mass is problematic in general relativity, the review concludes with an alternative derivation of the equations of motion that applies to a small body of arbitrary internal structure.

gr-qc↗

arXiv2011

Effects of electron-phonon coupling on Landau levels in graphene

We calculate the density of states (DOS) in graphene for electrons coupled to a phonon in an external magnetic field. We find that coupling to an Einstein mode of frequency $ω_E$ not only shifts and broadens the Landau levels (LLs), but radically alters the DOS by introducing a new set of peaks at energies $E_n\pmω_E$, where $E_n$ is the energy of the $n$th LL. If one of these new peaks lies sufficiently close to a LL, it causes the LL to split in two; if the system contains an energy gap, a LL may be split in three. The new peaks occur outside the interval $(-ω_E,ω_E)$, leaving the LLs in that interval largely unaffected. If the chemical potential is greater than the phonon frequency, the zeroth LL lies outside the interval and can be split, eliminating its association with a single Dirac point. We find that coupling to an extended phonon distribution such as a Lorentzian or Debye spectrum does not qualitatively alter these results.

cond-mat.mes-hall↗

arXiv2011

Phonon structures in the electronic density of states of graphene in magnetic field

Unlike in ordinary metals, in graphene, phonon structure can be seen in the quasiparticle electronic density of states, because the latter varies on the scale of the phonon energy. In a magnetic field, quantization into Landau levels creates even more significant variations. We calculate the density of states incorporating electron-phonon coupling in this case and find that the coupling has pronounced new effects: shifting and broadening of Landau levels, creation of new peaks, and splitting of any Landau levels falling near one of the new peaks. Comparing our calculations with a recent experiment, we find evidence for a phonon with energy similar to but somewhat greater than the optical $E_{2g}$ mode and a coupling corresponding to a mass enhancement parameter $λ\simeq 0.07$.

cond-mat.mes-hall↗

arXiv2011

Optical properties of the pseudogap state in underdoped cuprates

Recent optical measurements of deeply underdoped cuprates have revealed that a coherent Drude response persists well below the end of the superconducting dome. In addition, no large increase in optical effective mass has been observed, even at dopings as low as 1%. We show that this behavior is consistent with the resonating valence bond spin-liquid model proposed by Yang, Rice, and Zhang. In this model, the overall reduction in optical conductivity in the approach to the Mott insulating state is caused not by an increase in effective mass, but by a Gutzwiller factor, which describes decreased coherence due to correlations, and by a shrinking of the Fermi surface, which decreases the number of available charge carriers. We also show that in this model, the pseudogap does not modify the low-temperature, low-frequency behavior, though the magnitude of the conductivity is greatly reduced by the Gutzwiller factor. Similarly, the profile of the temperature dependence of the microwave conductivity is largely unchanged in shape, but the Gutzwiller factor is essential in understanding the observed difference in magnitude between ortho-I and -II YBa$_2$Cu$_3$O$_y$.

cond-mat.supr-con↗

arXiv2010

Motion of small bodies in general relativity: foundations and implementations of the self-force

Extreme mass-ratio inspirals, in which solar-mass compact bodies spiral into supermassive black holes, are an important potential source for gravitational wave detectors. Because of the extreme mass-ratio, one can model these systems using perturbation theory. However, in order to relate the motion of the small body to the emitted waveform, one requires a model that is accurate on extremely long timescales. Additionally, in order to avoid intractable divergences, one requires a model that treats the small body as asymptotically small rather than exactly pointlike. Both of these difficulties can be resolved by using techniques of singular perturbation theory. I begin this dissertation with an analysis of singular perturbation theory on manifolds, including the common techniques of matched asymptotic expansions and two-timescale expansions. I then formulate a systematic asymptotic expansion in which the metric perturbation due to the body is expanded while a representative worldline is held fixed, and I contrast it with a regular expansion in which both the metric and the worldline must be expanded. This results in an approximation that is potentially uniformly accurate on long timescales. The equation of motion for the body's fixed worldline is determined by performing a local-in-space expansion in the neighbourhood of the body. Using this local expansion as boundary data, I construct a global solution to the perturbative Einstein equation. To concretely characterize orbits, I next devise a relativistic generalization of the Newtonian method of osculating orbits. Making use of this method and two-timescale expansions, I examine the utility of adiabatic approximations that can forgo an explicit calculation of the force.

gr-qc↗

arXiv2010

Singular perturbation techniques in the gravitational self-force problem

Much of the progress in the gravitational self-force problem has involved the use of singular perturbation techniques. Yet the formalism underlying these techniques is not widely known. I remedy this situation by explicating the foundations and geometrical structure of singular perturbation theory in general relativity. Within that context, I sketch precise formulations of the methods used in the self-force problem: dual expansions (including matched asymptotic expansions), for which I identify precise matching conditions, one of which is a weak condition arising only when multiple coordinate systems are used; multiscale expansions, for which I provide a covariant formulation; and a self-consistent expansion with a fixed worldline, for which I provide a precise statement of the exact problem and its approximation. I then present a detailed analysis of matched asymptotic expansions as they have been utilized in calculating the self-force. Typically, the method has relied on a weak matching condition, which I show cannot determine a unique equation of motion. I formulate a refined condition that is sufficient to determine such an equation. However, I conclude that the method yields significantly weaker results than do alternative methods.

gr-qc↗