Black hole scattering near the transition to plunge: Self-force and resummation of post-Minkowskian theory
Oliver Long, Christopher Whittall, Leor Barack
Abstract
Geodesic scattering of a test particle off a Schwarzschild black hole can be parametrized by the speed-at-infinity <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>v</a:mi> </a:math> and the impact parameter <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>b</c:mi> </c:math> , with a “separatrix,” <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>b</e:mi> <e:mo>=</e:mo> <e:msub> <e:mi>b</e:mi> <e:mi>c</e:mi> </e:msub> <e:mo stretchy="false">(</e:mo> <e:mi>v</e:mi> <e:mo stretchy="false">)</e:mo> </e:math> , marking the threshold between scattering and plunge. Near the separatrix, the scattering angle diverges as <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mo>∼</i:mo> <i:mi>log</i:mi> <i:mo stretchy="false">(</i:mo> <i:mi>b</i:mi> <i:mo>−</i:mo> <i:msub> <i:mi>b</i:mi> <i:mi>c</i:mi> </i:msub> <i:mo stretchy="false">)</i:mo> </i:math> . The self-force correction to the scattering angle (at fixed <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:mi>v</m:mi> </m:mrow> </m:math> , <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mrow> <o:mi>b</o:mi> </o:mrow> </o:math> ) diverges even faster, like <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mo>∼</q:mo> <q:msub> <q:mi>A</q:mi> <q:mn>1</q:mn> </q:msub> <q:mo stretchy="false">(</q:mo> <q:mi>v</q:mi> <q:mo stretchy="false">)</q:mo> <q:msub> <q:mi>b</q:mi> <q:mi>c</q:mi> </q:msub> <q:mo>/</q:mo> <q:mo stretchy="false">(</q:mo> <q:mi>b</q:mi> <q:mo>−</q:mo> <q:msub> <q:mi>b</q:mi> <q:mi>c</q:mi> </q:msub> <q:mo stretchy="false">)</q:mo> </q:math> . Here we numerically calculate the divergence coefficient <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:msub> <w:mi>A</w:mi> <w:mn>1</w:mn> </w:msub> <w:mo stretchy="false">(</w:mo> <w:mi>v</w:mi> <w:mo stretchy="false">)</w:mo> </w:math> in a scalar-charge toy model. We then use our knowledge of <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:msub> <ab:mi>A</ab:mi> <ab:mn>1</ab:mn> </ab:msub> <ab:mo stretchy="false">(</ab:mo> <ab:mi>v</ab:mi> <ab:mo stretchy="false">)</ab:mo> </ab:math> to inform a resummation of the post-Minkowskian expansion for the scattering angle, and demonstrate that the resummed series agrees remarkably well with numerical self-force results even in the strong-field regime. We propose that a similar resummation technique, applied to a mass particle subject to a gravitational self-force, can significantly enhance the utility and regime of validity of post-Minkowskian calculations for black-hole scattering. Published by the American Physical Society 2024