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Physical Review D

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Second-order perturbation theory: The problem of infinite mode coupling

Jeremy Miller¹, Barry Wardell^2,3, and Adam Pound¹

¹Mathematical Sciences and STAG Research Centre, University of Southampton, Southampton SO17 1BJ, United Kingdom
²School of Mathematical Sciences and Complex and Adaptive Systems Laboratory, University College Dublin, Belfield, Dublin 4, Ireland
³Department of Astronomy, Cornell University, Ithaca, New York 14853, USA

Phys. Rev. D 94, 104018 – Published 7 November, 2016

DOI: https://doi.org/10.1103/PhysRevD.94.104018

Abstract

Second-order self-force computations, which will be essential in modeling extreme-mass-ratio inspirals, involve two major new difficulties that were not present at first order. One is the problem of large scales, discussed in Pound [Phys. Rev. D 92, 104047 (2015)]. Here we discuss the second difficulty, which occurs instead on small scales: if we expand the field equations in spherical harmonics, then because the first-order field contains a singularity, we require an arbitrarily large number of first-order modes to accurately compute even a single second-order mode. This is a generic feature of nonlinear field equations containing singularities, allowing us to study it in the simple context of a scalar toy model in flat space. Using that model, we illustrate the problem and demonstrate a robust strategy for overcoming it.

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References (41)

E. Poisson, A. Pound, and I. Vega, Living Rev. Relativ. 14, 7 (2011).
A. Pound, Fundam. Theor. Phys. 179, 399 (2015).
A. I. Harte, Fundam. Theor. Phys. 179, 327 (2015).
L. Barack, Classical Quantum Gravity 26, 213001 (2009).
P. Amaro-Seoane, J. R. Gair, A. Pound, S. A. Hughes, and C. F. Sopuerta, J. Phys. Conf. Ser. 610, 012002 (2015).
L. Blanchet, S. L. Detweiler, A. Le Tiec, and B. F. Whiting, Phys. Rev. D 81, 084033 (2010).
S. Akcay, L. Barack, T. Damour, and N. Sago, Phys. Rev. D 86, 104041 (2012).
L. Blanchet, G. Faye, and B. F. Whiting, Phys. Rev. D 90, 044017 (2014).
A. Le Tiec, Int. J. Mod. Phys. D 23, 1430022 (2014).
S. Bernuzzi, A. Nagar, T. Dietrich, and T. Damour, Phys. Rev. Lett. 114, 161103 (2015).
T. Damour, P. Jaranowski, and G. Schäfer, Phys. Rev. D 93, 084014 (2016).
B. Wardell, Fundam. Theor. Phys. 179, 487 (2015).
M. Colleoni, L. Barack, A. G. Shah, and M. van de Meent, Phys. Rev. D 92, 084044 (2015).
D. Bini, T. Damour, and A. Geralico, Phys. Rev. D 93, 064023 (2016).
T. Osburn, N. Warburton, and C. R. Evans, Phys. Rev. D 93, 064024 (2016).
S. Hopper, C. Kavanagh, and A. C. Ottewill, Phys. Rev. D 93, 044010 (2016).
C. Kavanagh, A. C. Ottewill, and B. Wardell, Phys. Rev. D 93, 124038 (2016).
M. van de Meent, Phys. Rev. D 94, 044034 (2016).
S. Akcay, D. Dempsey, and S. Dolan, arXiv:1608.04811.
T. Hinderer and E. E. Flanagan, Phys. Rev. D 78, 064028 (2008).
E. Rosenthal, Phys. Rev. D 73, 044034 (2006).
E. Rosenthal, Phys. Rev. D 74, 084018 (2006).
S. Detweiler, Phys. Rev. D 85, 044048 (2012).
A. Pound, Phys. Rev. Lett. 109, 051101 (2012).
S. E. Gralla, Phys. Rev. D 85, 124011 (2012).
A. Pound, Phys. Rev. D 86, 084019 (2012).
A. Pound and J. Miller, Phys. Rev. D 89, 104020 (2014).
A. Pound, Phys. Rev. D 92, 044021 (2015).
A. Pound, Phys. Rev. D 92, 104047 (2015).
A. Pound (to be published).
N. Warburton and B. Wardell, Phys. Rev. D 89, 044046 (2014).
B. Wardell and N. Warburton, Phys. Rev. D 92, 084019 (2015).
L. Barack and C. O. Lousto, Phys. Rev. D 72, 104026 (2005).
L. Barack and N. Sago, Phys. Rev. D 75, 064021 (2007).
S. L. Detweiler and B. F. Whiting, Phys. Rev. D 67, 024025 (2003).
A. Pound, GR21, Columbia University, 2016; slides are available at http://www.gr21.org/files/Adam_Pound_-_Pound_(1).pdf.gz.
A. Heffernan, A. Ottewill, and B. Wardell, Phys. Rev. D 86, 104023 (2012).
R. Haas and E. Poisson, Phys. Rev. D 74, 044009 (2006).
S. Orszag, Mon. Weather Rev. 102, 56 (1974).
G. Lohöfer, J. Approx. Theory 95, 178 (1998).
L. Barack, A. Ori, and N. Sago, Phys. Rev. D 78, 084021 (2008).