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

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.

Figure 1

Contour plots of the field ${\phi }_{\mathrm{ext}}$ in the vicinity of a rotating BH, as given by Eq. (16). Top: The infinite charged plane is at an angle $\gamma =0$, and the BH has dimensionless spin $a/M=0$ (left) and $a/M=0.99$ (right). The value of ${\phi }_{\mathrm{ext}}/(2\pi \sigma )$ 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 ${\phi }_{10}$ (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. (73). 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 ${\phi }_{10}$ (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. (73). 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. 4.

Figure 4

Absolute value of the real part of the scalar multipoles $|\mathrm{Re}({\phi }_{l0})|$ 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 ${\Psi }_4$, $|\mathrm{Re}({\psi }_{lm})|$, extracted at $r=56M$ for $l=m$ (the imaginary parts are identical, modulo a phase shift). Left: $M\sigma =0$. Right: $M\sigma =2\times {10}^{-7}$.

Figure 6

Numerical results for a BH binary inspiralling in a scalar -field gradient, with the orbital angular momentum perpendicular to the gradient. Left: Dependence of the various components of the scalar radiation $\mathrm{Re}({\phi }_{11})/(M\sigma )$ on the extraction radius (top to bottom: $112M$ to $56M$ in equidistant steps). The dashed line corresponds instead to ${10}^{-3}\mathrm{Im}({\phi }_{11})/(M\sigma )$ at the largest extraction radius. Right: Time derivative of the scalar field at the largest and smallest extraction radii, rescaled by radius and shifted in time. Notice how the two waveforms show a clean and typical merger pattern, and that they overlap showing that the field scales to good approximation as $1/\tilde{r}$.