Soft graviton exchange and the information paradox

Abstract

We show that there is a remarkable phase in quantum gravity where gravitational scattering amplitudes mediated by virtual gravitons can be calculated explicitly in effective field theory, when the impact parameter <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>b</a:mi></a:math> satisfies <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:msub><c:mrow><c:mi>L</c:mi></c:mrow><c:mrow><c:mi>Pl</c:mi></c:mrow></c:msub><c:mo>≪</c:mo><c:mi>b</c:mi><c:mo>≲</c:mo><c:msub><c:mrow><c:mi>R</c:mi></c:mrow><c:mrow><c:mi>S</c:mi></c:mrow></c:msub></c:mrow></c:math>, with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>R</e:mi><e:mi>S</e:mi></e:msub></e:math> being the Schwarzschild radius. This phase captures collisions with energies satisfying <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:msqrt><g:mrow><g:mi>s</g:mi></g:mrow></g:msqrt><g:mo>≫</g:mo><g:mi>γ</g:mi><g:msub><g:mrow><g:mi>M</g:mi></g:mrow><g:mrow><g:mi>Pl</g:mi></g:mrow></g:msub></g:mrow></g:math> (with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mrow><i:mi>γ</i:mi><i:mo>∼</i:mo><i:msub><i:mrow><i:mi>M</i:mi></i:mrow><i:mrow><i:mi>Pl</i:mi></i:mrow></i:msub><i:mo>/</i:mo><i:msub><i:mrow><i:mi>M</i:mi></i:mrow><i:mrow><i:mi>BH</i:mi></i:mrow></i:msub></i:mrow></i:math>) near the horizon. We call this the black hole eikonal phase, in contrast to its flat space analog where collisions are trans-Planckian. Hawking’s geometric optics approximation neglects gravitational interactions near the horizon, and results in thermal occupation numbers in the Bogoliubov coefficients. We show that these interactions are mediated by graviton exchange in <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mn>2</k:mn><k:mo stretchy="false">→</k:mo><k:mn>2</k:mn></k:math> scattering near the horizon, and explicitly calculate the <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>S</n:mi></n:math>-matrix nonperturbatively in <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:msub><p:mi>M</p:mi><p:mi>Pl</p:mi></p:msub><p:mo>/</p:mo><p:msub><p:mi>M</p:mi><p:mi>BH</p:mi></p:msub></p:math>. This involves a resummation of infinitely many ladder diagrams near the horizon, all mediated by virtual soft gravitons. The <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mi>S</r:mi></r:math>-matrix turns out to be a pure phase upon this resummation and is agnostic of Planckian physics and any specific ultraviolet completion. In contrast to the flat-space eikonal limit, the black hole eikonal phase captures collisions of extremely low energy near the horizon. Published by the American Physical Society 2024

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