New parametrization for spherically symmetric black holes in metric theories of gravity

We propose a new parametric framework to describe in generic metric theories of gravity the spacetime of spherically symmetric and slowly rotating black holes. In contrast to similar approaches proposed so far, we do not use a Taylor expansion in powers of $M/r$, where $M$ and $r$ are the mass of the black hole and a generic radial coordinate, respectively. Rather, we use a continued-fraction expansion in terms of a compactified radial coordinate. This choice leads to superior convergence properties and allows us to approximate a number of known metric theories with a much smaller set of coefficients. The measure of these coefficients via observations of near-horizon processes can be used to effectively constrain and compare arbitrary metric theories of gravity. Although our attention is here focussed on spherically symmetric black holes, we also discuss how our approach could be extended to rotating black holes.

Figure 1

Left panel: Difference between the exact values of the dilaton black hole orbit impact parameter for a circular orbit $b_{\text{ph}}$ and the values obtained using the continued-fraction expansions (19). The results are shown as a function of the dimensionless strength of the dilaton parameter $b/\mu$. Different lines refer to different levels of approximation, i.e., $a_2=0$ (blue line), $a_3=0$ (red line), and $a_4=0$ (magenta line). Note that even when $a_2=0$, the differences are $\lesssim 10^{-4}$ for $b\lesssim \frac{1}{2}\mu$. Right panel: The same as in the left panel but for the ISCO frequency.

Figure 2

Left panel: Relative difference in the photon circular orbit impact parameter $b_{\text{ph}}$ between general relativity and the alternative Einstein-aether theory (cf., Fig. 4 of Ref. [17]). The differences are reported within the mathematically allowed ranges for the aether parameters $c_{+}$ and $c_{-}$. The contours correspond to the following values (from left to right): 0.01, 0.025, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5. Right panel: the same as in the left panel but for the ISCO frequency (cf., Fig. 2 of Ref. [17]). The impact parameter and ISCO frequencies were calculated using continued-fraction expansions with $a_3=0$.

Figure 3

Left panel: Evolution of the scattered scalar field $|\mathrm{\Psi }|$ for the $\ell =0$ perturbations at $r=2r_0$ as computed using the exact dilaton black hole metric with $b/\mu =1$ (blue solid line) or the corresponding parametrized form with $a_3=0=b_3$ (red dashed line). Right panel: relative difference in the evolution of $|\mathrm{\Psi }|$ shown in the left panel.

Figure 4

Difference between the exact values of the ISCO frequencies in the equatorial plane for the dilaton black hole in the slow-rotation approximation regime and the values obtained using the continued-fraction expansions. Different curves, shown as functions of the rotation parameter, refer to the different levels of approximation: $a_2=0={\omega }_2$ (blue line), $a_3=0={\omega }_3$ (red line), and $a_4=0={\omega }_4$ (magenta line). In all cases we have taken a reference value of $b=\mu /2$.

Figure 5

Difference between the exact values of the dilaton black-hole orbit impact parameter for a circular orbit $b_{\text{ph}}$ and the values obtained using the generalized Johannsen-Psaltis metric with the coefficients calculated by comparing the asymptotic expansions for the metric functions. The results are shown as a function of the dimensionless strength of the dilaton parameter $b/\mu$. Different lines refer to different levels of approximation, i.e., $0={\epsilon }_4^t={\epsilon }_5^t={\epsilon }_6^t=\cdots$ (blue line), $0={\epsilon }_5^t={\epsilon }_6^t={\epsilon }_7^t=\cdots$ (red line), and $0={\epsilon }_6^t={\epsilon }_7^t={\epsilon }_8^t=\cdots$ (magenta line), and so on. The dashed lines of the same color correspond to our continued-fraction approximation having the same number of parameters; hence the first three lines should be compared with those of Fig. 1 with the same color.