Scalar fields and parametrized spherically symmetric black holes: Can one hear the shape of space-time?

In this work we study whether parametrized spherically symmetric black hole solutions in metric theories of gravity can appear to be isospectral when studying perturbations. From a theory agnostic point of view, the test scalar field wave equation is the ideal starting point to approach the quasinormal mode spectrum in alternative black hole solutions. We use a parametrization for the metric proposed by Rezzolla and Zhidenko, as well as the higher order WKB method in the determination of the quasinormal mode spectra. We look for possible degeneracies in a tractable subset of the parameter space with respect to the Schwarzschild quasinormal modes. Considering the frequencies and damping times of the expected observationally most relevant quasinormal modes, we find such degeneracies. We explicitly demonstrate that the leading Schwarzschild quasinormal modes can be approximated by alternative black hole solutions when their mass is treated as free parameter. In practice, we conclude that the mass has to be known with extremely high precision in order to restrict the leading terms in the metric expansion to currently known limits coming from the PPN expansion. Possible limitations of using the quasinormal mode ringdown to investigate black hole space-times are discussed.

Figure 1

Relative errors for $n=0$ and different $l$ for $[a_0,b_0]$. In the following we show the logarithm ${\log }_{10}(x)$ of the relative errors from left to right for: the real part ${\omega }_{\mathrm{r}}$, the imaginary part ${\omega }_{\mathrm{i}}$, and the combination ${\omega }_{\mathrm{c}}$ Eq. (16). From top to bottom we show $l=2$ and $l=3$.

Figure 2

Relative errors for $n=0$ and different $l$ for $[a_1,b_1]$. In the following we show the logarithm ${\log }_{10}(x)$ of the relative errors from left to right for: the real part ${\omega }_{\mathrm{r}}$, the imaginary part ${\omega }_{\mathrm{i}}$, and the combination ${\omega }_{\mathrm{c}}$ Eq. (16). From top to bottom we show $l=2$ and $l=3$.

Figure 3

Here we show the combined relative errors ${\omega }_{\mathrm{c}}$ for $n=0$ and different $M$. The top panels are the $l=2$ (left) and $l=3$ (right) case for $[a_0,b_0]$, while the same $l$ cases are shown in the bottom panels for $[a_1,b_1]$. Each layer is obtained for different mass $M$, while setting the nonvarying parameters to the general relativity value of 0.

Figure 4

Relative errors for $n=1$ and different $l$ for $[a_0,b_0]$. In the following we show the logarithm ${\log }_{10}(x)$ of the relative errors from left to right for: the real part ${\omega }_{\mathrm{r}}$, the imaginary part ${\omega }_{\mathrm{i}}$, and the combination ${\omega }_{\mathrm{c}}$ Eq. (16). From top to bottom we show $l=2$ and $l=3$.

Figure 5

Relative errors for $n=1$ and different $l$ for $[a_1,b_1]$. In the following we show the logarithm ${\log }_{10}(x)$ of the relative errors from left to right for: the real part ${\omega }_{\mathrm{r}}$, the imaginary part ${\omega }_{\mathrm{i}}$, and the combination ${\omega }_{\mathrm{c}}$ Eq. (16). From top to bottom we show $l=2$ and $l=3$.

Figure 6

Here we show the combined relative errors ${\omega }_{\mathrm{c}}$ for $n=1$ and different $M$. The top panels are the $l=2$ (left) and $l=3$ (right) case for $[a_0,b_0]$, while the same $l$ cases are shown in the bottom panels for $[a_1,b_1]$. Each layer is obtained for different mass $M$, while setting the nonvarying parameters to the general relativity value of 0.

Figure 7

Here we show various potential barriers $V(r)$ as function of the tortoise coordinate $r^{*}$ for different parameters $[M,a_0,b_0]$ (top panels) and $[M,a_1,b_1]$ (bottom panels). From left to right we consider input from $n=0$ for the $l=2$ and $l=3$ cases. In each panel we show the standard Schwarzschild potentials (Sch) for different $M=[0.99,1.00,1.01]$ (solid lines). The dashed lines are the best matches of $[a_0,b_0]$ and $[a_1,b_1]$ for $M=[0.99,1.01]$ in the RZ metric (dashed lines). The parameters were chosen by taking the smallest combined error $\delta {\omega }_c$ with respect to the $M=1$ Schwarzschild case. To demonstrate that these modified masses change the Schwarzschild potential noticeably, we also show the Schwarzschild potentials for $M=[0.99,1.01]$ (red and blue solid) for comparison. Note that the RZ results (dashed lines) overlap with the canonic Schwarzschild potential (black solid) around the maximum, while the other Schwarzschild cases (red and blue solid) are clearly distinguishable.