Justin Vines

Canonical Hamiltonian for an extended test body in curved spacetime: To quadratic order in spin

We derive a Hamiltonian for an extended spinning test-body in a curved background spacetime, to quadratic order in the spin, in terms of three-dimensional position, momentum, and spin variables having canonical Poisson brackets. This requires a careful analysis of how changes of the spin supplementary condition are related to shifts of the body’s representative worldline and transformations of the body’s multipole moments, and we employ bitensor calculus for a precise framing of this analysis. We apply the result to the case of the Kerr spacetime and thereby compute an explicit canonical Hamiltonian for the test-body limit of the spinning two-body problem in general relativity, valid for generic orbits and spin orientations, to quadratic order in the test spin. This fully relativistic Hamiltonian is then expanded in post-Newtonian orders and in powers of the Kerr spin parameter, allowing comparisons with the test-mass limits of available post-Newtonian results. Both the fully relativistic Hamiltonian and the results of its expansion can inform the construction of waveform models, especially eective-one-body models, for the analysis of gravitational waves from compact binaries.

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 Einsteins equation. As examples, we derive the exact Kerr and gravitational plane wave metrics from standard harmonic-gauge approximations.

Hyperbolic scattering of spinning particles by a Kerr black hole

We investigate the scattering of a spinning test particle by a Kerr black hole within the Mathisson-Papapetrou-Dixon model to linear order in spin. The particles spin and orbital angular momentum are taken to be aligned with the black holes spin. Both the particles mass and spin length are assumed to be small in comparison with the characteristic length scale of the background curvature, in order to avoid backreaction effects. We analytically compute the modifications due to the particles spin to the scattering angle, the periastron shift, and the condition for capture by the black hole, extending previous results valid for the nonrotating Schwarzschild background. Finally, we discuss how to generalize the present analysis beyond the linear approximation in spin, including spin-squared corrections in the case of a black-hole-like quadrupolar structure for the extended test body.

Is motion under the conservative self-force in black hole spacetimes an integrable Hamiltonian system?

A point-like object moving in a background black hole spacetime experiences a gravitational self-force which can be expressed as a local function of the objects instantaneous position and velocity, to linear order in the mass ratio. We consider the worldline dynamics defined by the conservative part of the local self-force, turning off the dissipative part, and we ask: Is that dynamical system a Hamiltonian system, and if so, is it integrable? In the Schwarzschild spacetime, we show that the system is Hamiltonian and integrable, to linear order in the mass ratio, for generic (but not necessarily all) stable bound orbits. There exist an energy and an angular momentum, being perturbed versions of their counterparts for geodesic motion, which are conserved under the forced motion. We also discuss difficulties associated with establishing analogous results in the Kerr spacetime. This result may be useful for future computational schemes, based on a local Hamiltonian description, for calculating the conservative self-force and its observable effects. It is also relevant to the assumption of the existence of a Hamiltonian for the conservative dynamics for generic orbits in the effective-one-body formalism, to linear order in the mass ratio, but to all orders in the post-Newtonian expansion.

Prescriptions for measuring and transporting local angular momenta in general relativity

For observers in curved spacetimes, elements of the dual space of the set of linearized Poincare transformations from an observer’s tangent space to itself can be naturally interpreted as local linear and angular momenta. We present an operational procedure by which observers can measure such quantities using only information about the spacetime curvature at their location. When applied by observers near spacelike or null infinity in stationary, vacuum, asymptotically flat spacetimes, there is a sense in which the procedure yields the well-defined linear and angular momenta of the spacetime. We also describe a general method by which observers can transport local linear and angular momenta from one point to another, which improves previous prescriptions. This transport is not path independent in general, but becomes path independent for the measured momenta in the same limiting regime. The transport prescription is defined in terms of differential equations, but it can also be interpreted as parallel transport in a particular direct-sum vector bundle. Using the curvature of the connection on this bundle, we compute and discuss the holonomy of the transport law. We anticipate that these measurement and transport definitions may ultimately prove useful for clarifying the physical interpretation of the Bondi-Metzner-Sachs charges of asymptotically flat spacetimes.

Properties of an affine transport equation and its holonomy

An affine transport equation was used recently to study properties of angular momentum and gravitational-wave memory effects in general relativity. In this paper, we investigate local properties of this transport equation in greater detail. Associated with this transport equation is a map between the tangent spaces at two points on a curve. This map consists of a homogeneous (linear) part given by the parallel transport map along the curve plus an inhomogeneous part, which is related to the development of a curve in a manifold into an affine tangent space. For closed curves, the affine transport equation defines a “generalized holonomy” that takes the form of an affine map on the tangent space. We explore the local properties of this generalized holonomy by using covariant bitensor methods to compute the generalized holonomy around geodesic polygon loops. We focus on triangles and “parallelogramoids” with sides formed from geodesic segments. For small loops, we recover the well-known result for the leading-order linear holonomy (

Scattering of two spinning black holes in post-Minkowskian gravity, to all orders in spin, and effective-one-body mappings

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Erratum: Post-1-Newtonian equations of motion for systems of arbitrarily structured bodies [Phys. Rev. D71, 044010 (2005)]

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