Spacetime of rotating black holes surrounded by massive scalar charges

TL;DR

This work extends spectral-method techniques to build the spacetime of rotating black holes surrounded by massive, nonminimally coupled scalar fields in beyond-GR theories (axi-dilaton, dynamical Chern–Simons, and scalar Gauss–Bonnet gravity). By expanding around Kerr with a small coupling parameter $\zeta$ and a leading-order Klein–Gordon equation for the massive scalars, the authors obtain accurate scalar-field configurations and corresponding metric deformations up to $a\leq 0.8$, resolving scalar masses with Compton wavelengths down to a few times the BH mass. They introduce a robust numerical framework that handles the exponential radial decay via an auxiliary field $\varphi$, enforces proper asymptotics, and computes horizon properties such as the horizon angular velocity $\Omega_H^{(1)}$ and surface gravity $\kappa^{(1)}$, which in turn inform potential quasinormal-mode and ringdown observations. The results show that increasing the scalar mass mainly reduces the magnitude of deformations without substantially changing their multipolar structure, offering a practical path to testing massive scalar degrees of freedom with current and future electromagnetic and gravitational-wave data. These spacetimes provide a foundation for incorporating massive scalar charges into waveform models and for exploring BH spectroscopy as a probe of fundamental fields.

Abstract

Massive scalar charges are ubiquitous in extensions to General Relativity and the Standard Model in particle physics. We describe spectral methods which can accurately construct the spacetime of rotating black holes with dimensionless spin up to $a \leq 0.8$ surrounded by massive scalar fields nonminimally coupled to spacetime curvature. We consider axi dilaton, dynamical Chern Simons, and scalar Gauss Bonnet couplings, and obtain leading order solutions for both the scalar field and the associated metric modifications. Our method accurately resolves massive scalar fields with Compton wavelengths as short as 5 times the black hole mass, achieving residual errors $\lesssim 10^{-5}$, and yields the corresponding leading order spacetime modifications with residual errors $\lesssim 10^{-3}$. Using the constructed spacetimes, we computes the leading-order shifts in the surface gravity and the angular velocity of the event horizon, important information for computing the quasinormal modes. These results pave the way to incorporate massive scalar charges into electromagnetic observations and gravitational-wave detections of black holes, potentially enabling new probes of fundamental scalar degrees of freedom.

Figures (13)

• Figure 1: The backward-modulus difference [see Eq. \ref{['eq:BWD_scalar_field']} in the main text for definition] of the massive scalar field of mass $\mu = 0.01$ (dark-blue circles), 0.1 (dark-red squares) and 0.2 (dark-green diamonds) around a rotating black hole of dimensionless spin $a=0.1$ in dynamical Chern-Simons (dCS, left panel) and scalar Gauss-Bonnet (sGB, right panel) gravity as a function of the spectral order $N$.
• Figure 2: The error [see Eq. (\ref{['eq:Err_scalar']}) in the main text for definition] of the massive scalar field of mass $\mu = 0.01$ (circles in dark blue), 0.1 (squares in dark red) and 0.2 (diamonds in dark green) around a rotating black hole of dimensionless spin $a=0.1$ in dCS (left panel) and sGB (right panel) gravity as a function of the spectral order $N$. As there is no a universally unambiguous definition of the residual of an equation, we have normalized the residual at $N = 1$ to be unity for different $\mu$. We observe that, for all $\mu$ and both gravity theories, the error first exponentially decrease at a rate as $N$ increases to a spectral order, and then continues to decrease exponentially, but with a smaller rate. For $\mu=0.1$ and 0.2, the error could even reach a minimum for $N \leq 30$. These patterns are related to the radial variation of the scalar field introduced by the exponential factor $e^{-\mu r}$.
• Figure 3: The least error [see the text around Eq. \ref{['eq:opt_spec_order_phi']} for definition] of the massive scalar field of mass $\mu = 0.01$ (circles in dark blue), 0.1 (squares in dark red) and 0.2 (diamonds in dark green) around a rotating black hole in dCS (left panel) and sGB (right panel) gravity as a function of the dimensionless spin $a$ We observe that, relatively, the least error show less variations over $a$, but increase significantly as $\mu$ increases. This change can be explained from the rapid changes introduced to the scalar field profile by the exponential factor $e^{-\mu r}$ as $\mu$ increases.
• Figure 4: Meridional cross-sections of the massive scalar field around a rotating black hole with dimensionless spin $a = 0.8$ in dCS (left panels) and sGB (right panels) gravity. The top, middle, and bottom panels correspond to scalar-field masses $\mu = 0.01$, $\mu = 0.1$, and $\mu = 0.2$, respectively. All solutions are computed at their respective optimal spectral resolutions. In dCS gravity, the scalar field exhibits a dipolar structure and is antisymmetric about the equatorial plane ($\chi = 0$, or $\theta = \pi/2$), which acts as a nodal plane where the field vanishes at all radii. In contrast, the scalar field in sGB gravity displays a quadrupolar structure and is symmetric about the equator. In both theories, the radial profile decays exponentially with increasing radius due to the factor $e^{-\mu r}$. The overall multipolar structure of the scalar field is not significantly altered by the presence of a finite mass, consistent with previous massless results Lam:2025fzi.
• Figure 5: The backward-modulus difference [see Eq. \ref{['eq:BMD_H']} in the main text for definition] of the metric modifications due to the presence of a scalar field of mass $\mu = 0.01$ (dark-blue circles), 0.1 (dark-red squares) and 0.2 (dark-green diamonds) around a rotating black hole of dimensionless spin $a=0.1$ in dCS (left panel) and sGB (right panel) gravity as a function of the spectral order $N$.