Comment on “Some exact quasinormal frequencies of a massless scalar field in Schwarzschild spacetime”

A new branch of quasinormal modes for a massless scalar field propagating on the Schwarzschild spacetime was recently announced [D. Batic, M. Nowakowski, and K. Redway, Phys. Rev. D 98, 024017 (2018)]. We review the quasinormal modes characterization and arguments and identify the flaws in their proof. Then, we preset explicit counterexamples to such arguments. Finally, we study the modes via alternative methods and do not find the new branch. We conclude against their interpretation.

Figure 1

Top Panel: for $s_0^{(-)}=(-1-\sqrt{13}i)/2$ (top plot), $\{H_k^{(-)}\}$ grows as $k^{\zeta }{e}^{\kappa \sqrt{k}}$ for $k\gg 1$, while $\{H_{k,-}^{(-)}\}$ does not satisfy $H_{k,-}^{(-)}=0$ for $k<0$. For $s=s^{\mathrm{QNM}}$ (bottom plot), both sequences become linearly dependent. Bottom panel: For $s_0^{(-)}=(-1-\sqrt{13}i)/2$ (blue circle), $y_k-1>0$ contradicting the choice made in Eq. (44) of Ref. [1]. For $s=s^{\mathrm{QNM}}$ (red triangle), one obtains $y_k-1<0$ as expected.

Figure 2

Top panel: QNMs as the spectra of the operator $\mathbf{M}$ (solid red circles). The extra panel displays the numerical error to each QNM. The values $s_n^{(-)}$ in Eq. (17) (empty blue circles) are not found as eigenvalues of $\mathbf{M}$. Bottom: Time evolution (red) according to Ref. [13]. The filtered field (blue) shows the second QNM $s_1^{\mathrm{QNM}}$. No mode with $s_0^{\mathrm{new}}=(-1-\sqrt{13}i)/2$ is detected in the evolution.