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First Law of Binary Black Hole Scattering

Riccardo Gonzo*

Jack Lewis† and Adam Pound‡

  • Higgs Centre for Theoretical Physics, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom

  • *Contact author: rgonzo@ed.ac.uk
  • †Contact author: J.E.Lewis@soton.ac.uk
  • ‡Contact author: A.Pound@soton.ac.uk
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Phys. Rev. Lett. 135, 131401 – Published 23 September, 2025

DOI: https://doi.org/10.1103/s85p-gh7b

Abstract

In the last decade, the first law of binary black hole mechanics played an important unifying role in the gravitational two-body problem. More recently, binary black hole scattering and the application of high-energy physics methods have provided a new avenue into this classical problem. In this Letter, we connect these two themes by extending the first law to the case of scattering orbits. We present derivations based on classical S-matrix, Hamiltonian, and pseudo-Hamiltonian methods, the last of which allows us to include dissipative effects for the first time. Finally, a “boundary to bound” map links this first law to the traditional bound-orbit version. Through this map a little-known observable for scatter orbits, the elapsed proper time, is mapped to the Detweiler redshift for bound orbits, which is an invariant building block in gravitational waveform models.

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Phys. Rev. Lett. 135, 131401– Published 23 September, 2025
  • Received 1 November 2024
  • Revised 2 July 2025
  • Accepted 26 August 2025
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