Space-time in motion: An evolving relativistic binary black hole metric for GIZMO

¹ Como Lake centre for AstroPhysics, DiSAT, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy
² INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy
³ INAF – Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Gobetti 93/3, I-40129 Bologna, Italy
⁴ Dipartimento di Fisica “A. Pontremoli”, Università degli Studi di Milano, Via Giovanni Celoria 16, 20134 Milano, Italy
⁵ Dipartimento di Fisica “G. Occhialini”, Università degli Studi di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy
⁶ INAF – Osservatorio Astronomico di Brera, Via Brera 20, I-20121 Milano, Italy

^⋆ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 13 November 2025
Accepted: 26 February 2026

Abstract

The last evolutionary stages of massive black hole binaries prior to coalescence are dominated by the emission of gravitational waves, which will be probed by the future Laser Interferometer Space Antenna. If gas is present around the two black holes, the associated electromagnetic emission can provide additional information about the binary properties and location before the merger event. For this reason, a proper characterisation of the electromagnetic emission during these phases is of fundamental importance, and requires a detailed description of the gas dynamics close to the event horizon of the two black holes; this is only achievable via numerical simulations. Within this context, we present the implementation of the superposed Kerr-Schild dynamic metric in the relativistic scheme in the meshless code GIZMO. Our code can now simulate black hole binaries approaching a merger with high computational efficiency and accuracy, taking relativistic effects on the gas into account. To validate our implementation, we performed two tests. First, we explored the case of a relativistic Bondi flow around a binary, finding very good agreement with numerical relativity simulations. Then, we explored the case of an inviscid relativistic circumbinary disc, comparing our results with a similar simulation run assuming Newtonian gravity. In this second case, we find moderate differences in the mass accretion rate and in the inflow dynamics, which suggest that the presence of a non-Keplerian potential and apsidal precession in the orbiting gas trajectories produce stronger shocks and boost angular momentum transport in the disc. Our work highlights the importance of accounting for relativistic corrections in accretion disc simulations around black hole binaries approaching a merger, even at scales much larger than those currently probed by numerical relativity simulations.