Extreme Gravitational Interactions in the Problem of Three Black Holes in General Relativity

TL;DR

This work extends the classical three-body problem into General Relativity by identifying Extreme Gravitational Interactions (EGIs) in a Newtonian Burrau configuration and seeding GR simulations with those pre-EGI states. Using the SFINGE code, a $3+1$ BSSN evolution with a moving-puncture, conformal formulation and spectral methods, the authors compare three-black-hole dynamics to a two-body inspiral, extracting gravitational waves via $\\Psi_4$ and performing Fourier and wavelet analyses. They demonstrate that 3-BH interactions produce irregular, multi-scale GW signals with broader frequency content and signatures suggestive of a gravitational turbulent cascade, distinguishing them from 2-BH mergers. The results offer guidance for future GW observations, motivate PN comparisons, and highlight the value of wavelet-based diagnostics in non-stationary, nonlinear multi-body spacetime dynamics.

Abstract

We study the three-body problem going from Newtonian mechanics to general relativity. In the classical case, we model the interactions in a typical chaotic configuration, identifying Extreme Gravitational Interactions (EGIs), namely transients in which the system manifests complex, highly-energetic dynamics. We then concentrate on the main part of the work, by selecting these EGIs as initial data for the general relativistic case, and performing a campaign of numerical relativity simulations. To provide a comprehensive menu of cases, we investigate different global configurations. By comparing with the more ``quiet'' two-body inspiral, we observe strong nonlinear emission of gravitational waves. The multi-body signals have been inspected by employing both Fourier and wavelet analyses, showing net differences among the global configurations. The wavelet analysis reveals the reminiscence of the EGIs in the three black holes problem. Such a survey of simulations might be a guide for future observations.

Figures (8)

• Figure 1: Overview of the Burrau three-body problem in the classical case. Left: a) Trajectories starting from the Burrau configuration (the initial position of the bodies is emphasized by the colored balls), up to $t = 70$. (b) Time history of the potential energy. (c) Second time-derivative of the momentum of inertia for the system. Spikes in (b) and (c) correspond to typical EGIs. Right: time evolution of the trajectories of the three bodies starting from the Burrau configuration. One can easily distinguish the times at which the objects are very close to each other and interact strongly, i.e. the EGIs.
• Figure 2: Overview of the kinetic energy time history for each of the classic cases, namely, Run a--d in Table \ref{['TABLE1']} (each label corresponds to a configuration). In the insets, we show the position of the bodies, with mass-proportional bullets size, and the direction of the velocity vectors at the pre-EGI time $t_0$. This time has been highlighted with an arrow.
• Figure 3: Shaded contours of the extrinsic curvature $K$ for the two black holes (RUN I, top row) and the three-body 'Classical Burrau' interaction (RUN II, bottom row), at three different times of the evolution, that is before, during, and after merger.
• Figure 4: 3D representation of the three-black hole interaction, at time $t/M = 26$ for RUN II ("Classical Burrau" in Table \ref{['TABLE2']}). The (red and blue) isosurfaces represent constant values of the extrinsic curvature $K$, outside from the final black hole (central black-shaded region). The 2D transparent plane represents a color contour of the Newman-Penrose scalar $\Psi_4$, with maxima in light-green color.
• Figure 5: Reconstruction of the gravitational wave signal for (a) the binary black hole system and (b)--(f) the three-black holes systems. Figures (a)--(f) refer respectively to RUNs I--VI listed in Table \ref{['TABLE2']}. The solid (dash-dotted) lines represent the real (imaginary) part of the spherical projection of $\Psi_4$; the vertical dotted black lines represent the merging time(s) of the black holes ($t^*$ is related to the 2-BHs merger, while $t^{*}_{1}$ and $t^{*}_{2}$ are related to the 3-BHs mergers), defined as in brugmann2008calibrationBuonanno2007inspiralBaker2006binary. All the simulations are performed up to $t/M = 145$. The three-black hole cases reveal a more structured and nonlinear behavior.