## Self-Force Calculations with Matched Expansions and Quasinormal Mode Sums

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Casals, Marc

Dolan, Sam R.

Ottewill, Adrian C.

Wardell, Barry

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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 uses two expansions for the Green function,
valid in `quasilocal' and `distant past' regimes, which are matched within the
normal neighbourhood. We perform our calculation in a static region of the
spherically symmetric Nariai spacetime (dS_2 x S^2), on which scalar
perturbations are governed by a radial equation with a P\"oschl-Teller
potential. We combine (i) a very high order quasilocal expansion, and (ii) an
expansion in quasinormal modes, to determine the Green function globally. We
show it is singular everywhere on the null wavefront (even outside the normal
neighbourhood), and apply asymptotic methods to determine its singular
structure. We find the Green function undergoes a transition every time the
null wavefront passes through a caustic: the singular part follows a repeating
four-fold sequence $\delta(\sigma)$, $1/\pi \sigma$, $-\delta(\sigma)$, $-1/\pi
\sigma$ etc., where $\sigma$ is Synge's world function.
The matched expansion method provides new insight into the non-local
properties of the self-force; we find the contribution from the segment of the
worldline lying outside the normal neighbourhood is significant. We compute the
scalar self-force acting on a static particle, and validate against an
alternative method. Finally, we discuss wave propagation on black hole
spacetimes (where any expansion in quasinormal modes will be augmented by a
branch cut integral) and predict that the Green function in Schwarzschild
spacetime will inherit the four-fold singular structure found here.

Comment: Added references, corrected minor typos. 43 pages, 20 figures

Comment: Added references, corrected minor typos. 43 pages, 20 figures

##### Keywords

General Relativity and Quantum Cosmology, Astrophysics - Cosmology and Nongalactic Astrophysics