Session X14: Approximation Methods in General Relativity

Towards a perturbative treatment of gravitational wave memory

Despite the weakness of gravitational radiation, the analysis of gravitational wave memory is usually taken to require the full nonlinear apparatus of general relativity. However, one form of gravitational wave memory has to do with fields such as the electromagnetic field and neutrinos which can get to null infinity. We show how to derive the memory effects of these fields using only first order perturbation theory. We expect that this method, when extended to second order perturbation theory, can also be used to account for the memory effect due to the loss in energy by gravitational radiation.   

Tidal heating and torquing of a Kerr black hole

Astrophysical black holes in binaries are immersed in a sea of gravitational perturbations caused by their companions. These vacuum perturbations will result in the spacetime geometry deviating from the vacuum Kerr solution and in fluxes of mass and angular momentum across the black hole horizon. These effects can alter the frequency evolution of gravitational waves emitted, a detailed modeling of which may be important in detection and crucial in parameter estimation. This talk describes a perturbative analytic calculation of these fluxes, assuming the tidal deformations are small and due to a slowly-varying external universe. This work extends previous results to next-to-leading order in the ratio of the unperturbed black hole mass to the radius of curvature of the external universe.   

Branching of the quasinormal mode spectrum in nearly extremal Kerr spacetimes

The characteristic, decaying oscillation modes of perturbed black holes, their quasinormal modes, play a role in a variety of theoretical and astrophysical situations. For example, they are of interest in determining the ringdown waveform of black hole mergers, and in the context of models of quantum gravity. In this talk I will discuss newly discovered features of the quasinormal mode spectrum of black holes with nearly extreme spins. At lower spins and for fixed angular indices, the modes form a single set of frequencies indexed by an overtone number. At very high spins, and for a certain range of fixed angular indices, this single set breaks into two branches. All the modes of one branch approach zero decay in the extreme spin limit, while the modes in the other branch retain finite decay rates. I will present a physical picture for this splitting behavior and discuss some of its implications for the study of perturbed black holes.   

Self-forced gravitational waveforms for intermediate mass ratio inspirals: estimating second order effects

We present the orbit-integrated self force effects on the gravitational waveform for an IMRI source. We consider the quasi-circular motion of a particle in the spacetime of a Schwarzschild black hole and study the dependence of the dephasing of the corresponding gravitational waveforms due to ignoring the conservative piece of the self force or the second order dissipative piece of the self force. First order self forces are modeled by the fully relativistic Barack--Sago self force. Second order effects are approximated by their post Newtonian expressions. This hybrid approach allows us to gain insight into the quantitative aspects of second order self-force effects, although the post Newtonian approximation of the second order effect does not allow us to quantitatively determine the observable quantities of interest. However, when fully relativistic second order effects become known, out method will allow us to refine our analysis by including them. We calculate the cumulative dephasing of the waveforms and their overlap integral, and discuss the importance of the conservative piece of the self force vis-\`a-vis the second order dissipative effect in detection and parameter estimation. We then study the effects for the parameter space of the problem.   

Self-Force on Accelerated Charges in Generic Spacetimes

Following the work of Barack and Ori we develop the mode-sum renormalization formalism for an accelerated charge (scalar charge, electric charge, or a mass) moving in arbitrary space time. We obtain expressions for renormalization parameters (RPs) of the mode-sum expansion in a Lorenz gauge. In particular, we show that, for a charge moving along a generic trajectory, the singular field can be described entirely by the leading and sub-leading terms (the `A' and `B' terms): the remaining contributions to its mode-sum expansion vanish at the particle. We then obtain explicit expressions for the A and B parameters. As a check, we use these RPs to recover Wiseman's result that the self-force vanishes on a static scalar charge outside a Schwarzschild black hole.   

Spin-dependent post-Newtonian parameters from EMRI computation in Kerr background

Because the extreme mass-ratio inspiral (EMRI) approximation is accurate to all orders in $v/c$, it can be used to find high order post-Newtonian parameters that are not yet analytically accessible. We report here on progress in computing spin-dependent, conservative, post-Newtonian parameters from a radiation-gauge computation for a particle in circular orbit in a family of Kerr geometries. For a particle with 4-velocity $u^\alpha = U k^\alpha$, with $k^\alpha$ the helical Killing vector of the perturbed spacetime, the renormalized perturbation $\Delta U$, when written as a function of the particle's angular velocity, is invariant under gauge transformations generated by helically symmetric vectors. The EMRI computations are done in a modified radiation gauge. Extracted parameters are compared to previously known and newly computed spin-dependent post-Newtonian terms. This work is modeled on earlier computations by Blanchet, Detweiler, Le Tiec and Whiting of spin-independent terms for a particle in circular orbit in a Schwarzschild geometry.   

A comment on the calculation of periastron precession in general relativity

Periastron precession is one of the three classical tests of General Relativity, and as such its calculation appears in virtually all text books on the subject. In almost all of these texts the calculation proceeds perturbatively from the Kepler solution to the Newtonian formulation. This calculation is rather cumbersome, typically taking a few pages of text to complete. In fact, the calculation can be completed in one line if the Kepler solution is not taken as the starting point. As far as I have been able to determine, this procedure has explicitly appeared in only one text, published in 2010. In this talk I review the perturbative procedure and compare it to the alternative. This material should be of interest to anyone who teaches a course in general relativity.   

Riemannian space-time, de Donder Conditions and Gravitational Field in Flat Space-time

Let the coordinate system $x^{i}$ of flat space-time to absorb a second rank tensor field $\Phi_{ij}$ of the flat space-time deforming into a Riemannian space-time, namely, the tensor field $\Phi_{\mu\nu}$ is regarded as a metric tensor with respect to the coordinate system $x^{\mu}$. After done this, the $x^{\mu}$ is not the coordinate system of flat space-time anymore, but is the coordinate system of the new Riemannian space-time. The inverse operation also can be done. According to these notions, The concepts of the absorption operation and the desorption operation are proposed. These notions are actually compatible with Einstein's equivalence principle. By using these concepts, the relationships of the Riemannian space-time, the de Donder conditions and the gravitational field in flat space-time are analyzed and elaborated. The tensor field of gravitation can be desorbed from the Riemannian space-time to the Minkowski space-time by using the de Donder conditions. Einstein equations with de Donder conditions can be solved in flat space-time. Base on Fock's works, the equations of gravitational field in flat space-time are obtained, and the tensor expression of the energy-momentum of gravitational field is found. They all satisfy the global Lorentz covariance.