Self-force calculations with matched expansions and quasinormal mode sums
- CENTRA, Instituto Superior Tecnico, Lisbon (Portugal) and School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9 (Ireland)
Accurate modeling of gravitational wave emission by extreme-mass ratio inspirals is essential for their detection by the LISA mission. A leading perturbative approach involves the calculation of the self-force acting upon the smaller orbital body. In this work, we present the first application of the Poisson-Wiseman-Anderson method of 'matched expansions' to compute the self-force acting on a point particle moving in a curved spacetime. The method employs two expansions for the Green function, which are, respectively, valid in the 'quasilocal' and 'distant past' regimes, and which may be matched together within the normal neighborhood. We perform our calculation in a static region of the spherically symmetric Nariai spacetime (dS{sub 2}xS{sup 2}), in which scalar-field perturbations are governed by a radial equation with a Poeschl-Teller potential (frequently used as an approximation to the Schwarzschild radial potential) whose solutions are known in closed form. The key new ingredients in our study are (i) very high order quasilocal expansions and (ii) expansion of the distant past Green function in quasinormal modes. In combination, these tools enable a detailed study of the properties of the scalar-field Green function. We demonstrate that the Green function is singular whenever x and x{sup '} are connected by a null geodesic, and apply asymptotic methods to determine the structure of the Green function near the null wave front. We show that the singular part of the Green function undergoes a transition each time the null wave front passes through a caustic point, following a repeating fourfold sequence {delta}({sigma}), 1/{pi}{sigma}, -{delta}({sigma}), -1/{pi}{sigma}, etc., where {sigma} is Synge's world function. The matched-expansion method provides insight into the nonlocal properties of the self-force. We show that the self-force generated by the segment of the worldline lying outside the normal neighborhood is not negligible. We apply the matched-expansion method to compute the scalar self-force acting on a static particle on the Nariai spacetime, and validate against an alternative method, obtaining agreement to six decimal places. We conclude with a discussion of the implications for wave propagation and self-force calculations. On black hole spacetimes, any expansion of the Green function in quasinormal modes must be augmented by a branch-cut integral. Nevertheless, we expect the Green function in Schwarzschild spacetime to inherit certain key features, such as a fourfold singular structure manifesting itself through the asymptotic behavior of quasinormal modes. In this way, the Nariai spacetime provides a fertile testing ground for developing insight into the nonlocal part of the self-force on black hole spacetimes.
- OSTI ID:
- 21301002
- Journal Information:
- Physical Review. D, Particles Fields, Vol. 79, Issue 12; Other Information: DOI: 10.1103/PhysRevD.79.124043; (c) 2009 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); ISSN 0556-2821
- Country of Publication:
- United States
- Language:
- English
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