Stability and quasinormal modes of black holes in tensor-vector-scalar theory: Scalar field perturbations

The imminent detection of gravitational waves will trigger precision tests of gravity through observations of quasinormal ringing of black holes. While general relativity predicts just two polarizations of gravitational waves, the so-called plus and cross polarizations, numerous alternative theories of gravity predict up to six different polarizations which will potentially be observed in current and future generations of gravitational wave detectors. Bekenstein’s Tensor-Vector-Scalar (TeVeS) theory and its generalization fall into one such class of theory that predict the full gamut of six polarizations of gravitational waves. In this paper we begin the study of quasinormal modes (QNMs) in TeVeS by studying perturbations of the scalar field in a spherically symmetric background. We show that, at least in the case where superluminal propagation of perturbations is not present, black holes are generically stable to this kind of perturbation. We also make a unique prediction that, as the limit of the various coupling parameters of the theory tend to zero, the QNM spectrum tends to $1/\sqrt{2}$ times the QNM spectrum induced by scalar perturbations of a Schwarzschild black hole in general relativity due to the intrinsic presence of the background vector field. We further show that the QNM spectrum does not vary significantly from this value for small values of the theory’s coupling parameters, however can vary by as much as a few percent for larger, but still physically relevant parameters.

Figure 1

Potential, $V$, for the scalar perturbation wave equation for a black hole (i.e. $a=2$) with $\mathcal{K}=0.1$, $k=0.01$ and ${\phi }_c=0.003$. Note that $r$ is an isotropic radial coordinate implying the horizon of the spacetime is at $r=M/2$. This is the case of ${\delta }_{-}$, although at this range plots of ${\delta }_{+}$ look similar (the effect of the sign choice in ${\delta }_{\pm }$ becomes more relevant for higher $\ell$’s). Here, the thick black, dashed blue and dotted red lines represent $\ell =0$, 1 and 2, respectively.

Figure 2

A zoomed in view of Fig. 1 for ${\delta }_{-}$ (top panel) and ${\delta }_{+}$ (bottom panel). As stated in the text, as $r\to M/2$ the potential goes to zero for all $\ell$’s for the ${\delta }_{-}$ case, however diverges to negative infinity for the case of ${\delta }_{+}$. This divergent behavior is associated with the superluminal propagation of the scalar perturbations.

Figure 3

Typical temporal evolution of scalar field for $\ell =0$ with $\mathcal{K}=k=0.1$ and ${\phi }_c=0.003$. The presence of oscillations in the evolution implies nonzero ${\omega }_{\mathrm{Re}}$ which, due to the spherically symmetric nature of the background spacetime, implies the mode is stable. Because of the strong damping of the $\ell =0$ mode, only a few oscillations can be seen in the evolution.

Figure 4

Temporal evolution of scalar field for $\ell =1$ (top panel), $\ell =2$ (middle) and $\ell =3$ (bottom) with $\mathcal{K}=k=0.1$ and ${\phi }_c=0.003$. The numerical time evolution is shown in black which includes the region of quasinormal ringing as well as the late-time tail. Overlaid in red is the damped sinusoidal oscillations of the fundamental mode calculated using the WKB method (see Sec. 5b).

Figure 5

The real (top) and the imaginary (bottom) part of the fundamental $\ell =2$ QNM frequencies as a function of the scalar field coupling parameter $k$ for several values of the vector field coupling parameter $\mathcal{K}\mathrm{\text{:}}=K+K_{+}-K_4$. The cosmological value of the scalar field is ${\phi }_c=0.003$ and ${\omega }_{\mathrm{Re}}$ and ${\omega }_{\mathrm{Im}}$ are shown in units of $M$.

Figure 6

The real (top) and the imaginary (bottom) part of the fundamental $\ell =2$ QNM frequencies as a function of the cosmological value of the scalar field ${\phi }_c$ for several values of the parameters $\mathcal{K}$ and $k$. Here, ${\omega }_{\mathrm{Re}}$ and ${\omega }_{\mathrm{Im}}$ are shown in units of $M$.