Numerical simulations of single and binary black holes in scalar-tensor theories: circumventing the no-hair theorem

TL;DR

The paper investigates whether scalar-tensor theories allow scalar radiation from black holes in the presence of a background scalar-field gradient, potentially circumventing the no-hair theorems. It develops a unified Einstein-frame formulation, derives analytical solutions for single and binary BHs in a gradient, and implements a 3+1 numerical relativity framework (BSSN) to evolve the system with appropriate boundary conditions. The key findings show that isolated BHs relax to static scalar configurations in gradients, accelerated BHs emit scalar radiation, and quasi-circular BH binaries radiate dipole scalar radiation at twice the orbital frequency, with the gravitational waveform largely unaffected for small gradients. The work highlights a mechanism by which no-hair constraints can be bypassed in non-isolated or gradient-rich environments and discusses the implications for astrophysical systems and potential observational bounds on scalar-field gradients, while noting limitations due to gradient strength and the need for longer, spinning-BH simulations.

Abstract

Scalar-tensor theories are a compelling alternative to general relativity and one of the most accepted extensions of Einstein's theory. Black holes in these theories have no hair, but could grow "wigs" supported by time-dependent boundary conditions or spatial gradients. Time-dependent or spatially varying fields lead in general to nontrivial black hole dynamics, with potentially interesting experimental consequences. We carry out a numerical investigation of the dynamics of single and binary black holes in the presence of scalar fields. In particular we study gravitational and scalar radiation from black-hole binaries in a constant scalar-field gradient, and we compare our numerical findings to analytical models. In the single black hole case we find that, after a short transient, the scalar field relaxes to static configurations, in agreement with perturbative calculations. Furthermore we predict analytically (and verify numerically) that accelerated black holes in a scalar-field gradient emit scalar radiation. For a quasicircular black-hole binary, our analytical and numerical calculations show that the dominant component of the scalar radiation is emitted at twice the binary's orbital frequency.

Figures (6)

• Figure 1: Contour plots of the field $\varphi_{\rm ext}$ in the vicinity of a rotating BH, as given by Eq. (\ref{['rotating_solution']}). Top: The infinite charged plane is at an angle $\gamma=0$, and the BH has dimensionless spin $a=0$ (left) and $a/M=0.99$ (right). The value of $\varphi_{\rm ext}/(2\pi\sigma)$ is shown along selected contour lines; the two panels only differ because of the different size of the horizon. Bottom: A BH with $a/M=0.99$ is immersed in a field gradient at angles $\pi/4$ (left) and $\pi/2$ (right). All contour plots refer to the plane $y=0$. Selected contour lines correspond to the same values as the top panels.
• Figure 2: Real part of the scalar dipole mode ${\varphi}_{10}$ (the imaginary part vanishes) for $M\sigma=10^{-5}$ and extraction radii (from top to bottom) $\tilde{r}/M=50$, $40$, $30$, $20$, $15$, $10$ and $5$, compared to the predictions of Eq. (\ref{['an_prediction']}). Solid lines refer to the numerical evolution; dashed lines refer to the analytical solution evaluated at time-dependent areal radii $r$, which are computed dynamically during the evolution. The inset shows the percentage discrepancy between the numerical and analytical prediction as a function of extraction radius.
• Figure 3: Real part of the scalar dipole mode ${\varphi}_{10}$ (rescaled by $M\sigma$) at the largest extraction radius $\tilde{r}/M=50$ for $M\sigma=10^{-4}$ and $M\sigma=10^{-5}$, compared to the predictions of Eq. (\ref{['an_prediction']}). Solid lines refer to the numerical evolution; dashed lines refer to the analytical solution evaluated at time-dependent areal radii $r$, which are computed dynamically during the evolution. The evolution does not settle to the analytical solution for $M\sigma=10^{-4}$: there is an exponentially growing mode. This also shows up as an exponential growth of the subleading multipoles, as can be seen in Fig. \ref{['fig:phi_sigma']}.
• Figure 4: Absolute value of the real part of the scalar multipoles $\left|{\rm Re}({\varphi}_{l0})\right|$ evaluated at the largest extraction radius $\tilde{r}=50M$ for different values of $l$ and two values of the scalar-field gradient, $M\sigma=10^{-5}$ and $M\sigma=10^{-4}$ (left and right panel, respectively).
• Figure 5: Numerical results for a BH binary inspiralling in a scalar field gradient, with the orbital angular momentum perpendicular to the gradient. We show the spin-weighted spheroidal harmonic components of the Weyl scalar $\Psi_4$, $|{\rm Re}(\psi_{lm})|$, extracted at $r=56~M$ for $l=m$ (the imaginary parts are identical, modulo a phase shift). Left: $M \sigma=0$, right: $M \sigma=2\times 10^{-7}$.