Black holes and fundamental fields in Numerical Relativity: initial data construction and evolution of bound states

TL;DR

This work tackles the nonlinear evolution of black holes immersed in fundamental, massive scalar fields by solving the Einstein–Klein–Gordon equations with constraint-satisfying initial data. Using 3+1 numerical relativity and the BSSN formulation, it constructs analytic and semi-analytic initial data for Schwarzschild and Kerr spacetimes with scalar clouds, and evolves them to reveal long-lived, multipolar scalar condensates and correlated gravitational radiation. The results show mass accretion, ringdown, and the formation of scalar clouds around both non-rotating and rapidly rotating BHs, including frame-dragging effects and beating between overtones, with potential signatures for ultralight fields and beyond-standard-model physics. The study also discusses hints of nonlinear gravitational superradiance and lays out a versatile framework for constraint-preserving initial data applicable to scalar-tensor theories and multi-field scenarios, enabling broader explorations of BH physics and gravitational-wave phenomenology.

Abstract

Fundamental fields are a natural outcome in cosmology and particle physics and might therefore serve as a proxy for more complex interactions. The equivalence principle implies that all forms of matter gravitate, and one therefore expects relevant, universal imprints of new physics in strong field gravity, such as that encountered close to black holes. Fundamental fields in the vicinities of supermassive black holes give rise to extremely long-lived, or even unstable, configurations which slowly extract angular momentum from the black hole or simply evolve non-linearly over long timescales, with important implications for particle physics and gravitational-wave physics. Here, we perform a fully non-linear study of scalar-field condensates around rotating black holes. We provide novel ways to specify initial data for the Einstein-Klein-Gordon system, with potential applications in a variety of scenarios. Our numerical results confirm the existence of long-lived bar-modes which act as lighthouses for gravitational wave emission: the scalar field condenses outside the black hole geometry and acts as a constant frequency gravitational-wave source for very long timescales. This effect could turn out to be a potential signature of beyond standard model physics and also a promising source of gravitational waves for future gravitational wave detectors.

Figures (15)

• Figure 1: Evolution of constraint satisfying (left panel, for further details on the construction see Table \ref{['tab:SetupSchwarzschildmassless']} and section \ref{['ssec:InitDataSBH']}) and constraint violating (mid) initial data, corresponding to run S00_m0_e and S00_CV in Table \ref{['tab:SetupSchwarzschildmassless']}. In both cases the spherically symmetric scalar shells are located at $r_0=12M$ with a width of $w=2M$ and amplitude $M A=0.075$. Figs. \ref{['fig:SchwarzschildMasslessHamilitonian']} and \ref{['fig:violation_H_time']} show the Hamiltonian constraint along the $x$-axis at different times throughout the evolution. Constraints become violated with larger magnitude at late times for constraint-violating initial data, specially close to the horizon. At late-times this impacts on physical quantities such as the AH area, shown in Fig. \ref{['fig:violation_AH']}, where $A_{AH,t}$ denotes the AH area at time $t$. The violation of the constraints causes the AH area to decrease (red dotted line) violating the area theorem while this unphysical feature does not occur with constraint satisfying initial data (blue solid line). The inset normalizes the area variation to zero at $t=50$, in order to better gauge the changes induced by both types of initial data.
• Figure 2: We present the relation between the initial BH mass $M_{0}=M_{\rm{BH}}(0)$ normalized by the ADM mass. and the scalar field amplitude $A_{lm}$ for initial data I (left) and II (right) for various sets of parameters. Specifically, the Gaussian scalar shells with widths $w=0.5M$ or $w=2.0M$ are centered around $r_0=6M$ or $r_0=12M$. The BH mass decreases as the amplitude increases because we have fixed the ADM mass.
• Figure 3: The relation between the amplitude of a scalar field and the initial BH mass for a non-rotating (top) and rotating BH (bottom). The BH mass (in units of the ADM mass) decreases as the scalar field amplitude increases. Fig. \ref{['fig:init_SBH_ID34']} and Fig. \ref{['fig:init_KBH_ID34']} show a dipole Gaussian or pseudo-bound state scalar field coupled to a non-rotating BH and a rotating BH with $a_0/M = 0.4$ and $0.8$. The Gaussian scalar wave packet is localized at $r_0=12M$ and has width $w=2M$.
• Figure 4: Results for a massless scalar field around a non-rotating BH. Fig. \ref{['fig:SchMasslessBHprop']} depicts the change in the BH mass (top) and the (dimensionless) angular momentum $a/M_{\rm{BH}}$ of the BH (bottom) as a function of time. The field has been set up as a shell with spherically symmetric or dipole angular configuration. Fig. \ref{['fig:SchMasslessWaveforms']} illustrates the $l=m=1$ waveform (top) of the massless scalar field with an initial dipole configuration and the dominant $l=m=2$ gravitational waveform (bottom) emitted after the accretion. Exemplarily, we present results produced by model S11_m0_c in Table \ref{['tab:SetupSchwarzschildmassless']}. Both waveforms, shifted in time by the extraction radius $r_{\rm{ex}}=40M$, show a clear quasi-normal ringdown signal.
• Figure 5: Evolution of a spherically symmetric massive scalar field with $M_{0}\mu_S=0.29$ around a non-rotating BH. Fig. \ref{['fig:SchMassiveBHpropl0']} depicts the relative area of the AH (top), the relative BH mass (middle) as compared to its value at $t=0$ and the dimensionless spin parameter $a/M_{\rm{BH}}$ (bottom) as functions of time. Fig. \ref{['fig:SchMassiveWaveformsl0']} presents the $l=m=0$ waveform of the scalar field measured at $r_{\rm{ex}}=40M$. In addition to the numerical data (black solid curve) we show a fit to the late-time tail (red dashed curve) with $t^{-0.83}$ in excellent agreement with linearized analysis.