Eikonal quasinormal modes of black holes beyond general relativity

Much of our physical intuition about black hole quasinormal modes in general relativity comes from the eikonal/geometric optics approximation. According to the well-established eikonal model, the fundamental quasinormal mode represents wave packets orbiting in the vicinity of the black hole’s geodesic photon ring, slowly peeling off towards the event horizon and infinity. Besides its strength as a “visualisation” tool, the eikonal approximation also provides a simple quantitative method for calculating the mode frequency, in close agreement with rigorous numerical results. In this paper we move away from Einstein’s theory and its garden-variety black holes and go on to consider spherically symmetric black holes in modified theories of gravity through the lens of the eikonal approximation. The quasinormal modes of such black holes are typically described by a set of coupled wave equations for the various field degrees of freedom. Considering a general, theory-agnostic system of two equations for two perturbed fields, we derive eikonal formulas for the complex fundamental quasinormal mode frequency. In addition we show that the eikonal modes can be related to the extremum of an effective potential and its associated “photon ring.” As an application of our results, we consider a specific example of a modified theory of gravity with known black hole quasinormal modes and find that these are well approximated by the eikonal formulas.

Figure 1

The effective potentials for axial tensor-scalar QNMs of Schwarzschild black holes in dCS gravity. We show the $M^4\beta =0.5$, $\ell =2$ potentials $V_{\mathrm{eff}}(r,{\omega }_{\pm })$ and $V_{\mathrm{eff}}(r,{\omega }_{\pm }(r))$ defined in Sec. 3c. The left (right) panel corresponds to the $-$ ($+$) mode solution. The corresponding $r_m$ solutions (cf. Table 1) mark the common extrema of the two potentials. In the right panel, these extrema are both maxima, while in the left panel (see inset), it is a maximum for $V_{\mathrm{eff}}(r,{\omega }_{-}(r))$ and a minimum for $V_{\mathrm{eff}}(r,{\omega }_{-})$.