Regularization of the the second-order gravitational perturbations produced by a compact object Eran Rosenthal University of Guelph - Canada Amos Ori Technion – Israel Institute of Technology

Presentation of the problem Problem: calculation of at the limit such that. Consider a Schwarzschild black-hole with a mass moving in a vacuum background spacetime with radius of curvature Background metric Full metric

Practical motivation Accurate calculation of the accumulating phase for long gravitational wave trains emitted from an extreme mass-ratio binary system (Lior Burko, Eric Poisson). These calculations can be used to construct accurate waveforms for LISA. This requires accurate calculation of the orbit (of the smaller mass object) in the background spacetime induced by the more massive object.

Geodesic in the background spacetime Perturbative approach to the calculation of the orbit First order self-force corrections Second order self-force corrections requires

Gravitational perturbations at the limit Solution in the external zone world-line Produced by a Schwarzschild black-hole with a mass moving in a vacuum background Produced by a Schwarzschild black-hole with a mass moving in a vacuum background spacetime Background metric (Lorenz gauge) The world line is a geodesic with respect to the background

At the leading order is a geodesic with respect to the background spacetime. Higher order corrections will be discussed later. (General gauge) Second-order gravitational perturbations

Calculation of - main difficulties Consider the linear differential equations for obtained from Einstein equations. Schematically written as Naive construction of the 2 nd order retarded solution diverges at every point in spacetime. Non-integrable source terms ! (Lorenz gauge) Distance from the world-line (Lorenz gauge)

Require Consider in Fermi coordinates. In the vicinity of the world-line: Regularization of the singularity in

Finding a causal Will be discussed now t

Schematically written: Regularization of the singularity in New 1 st order gauge

Particular solution in (1 st order) Fermi gauge Retarded solution Consider the following 1 st order gauge: Fermi gauge Geodesic: No corrections of order The world-line Corrections to a geodesic world-line which come from the first order self- force induce a 2 nd order corrections to the gravitational perturbations.

General solution in (1 st order) Fermi gauge 1. Boundary conditions at infinity: No incoming waves (requires additional regularization at infinity!) Requirements on which fix 3. Divergent boundary conditions as determined from matched asymptotic expansions 2. Causality Retarded

Kirchhoff representation (assuming boundary conditions at infinity) Only divergent boundary conditions as are required The semi-homogeneous part satisfies Required boundary conditions as Fermi coordinates

Expansions in the buffer zone (Thorne and Hartle 1985) if

2 nd order solution in (1 st order ) Fermi gauge is formally given by: Retarded solution Results in 2 nd order Lorenz gauge Schematically:

Considering the equation away from the world-line Considering the equation away from the world-line Introducing Introducing Choosing (1 st order) Fermi gauge Choosing (1 st order) Fermi gauge Determining from boundary conditions as Determining from boundary conditions as Summary of the method