Black hole scattering near the transition to plunge: Self-force and resummation of post-Minkowskian theory

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.