REVIEW
Quasinormal modes of Reissner-Nordstr\"om black hole from bound state spectrum
General Relativity and Quantum Cosmology · Submitted 14 Sep 2026 · 51 references
Pith's verdict: Solid incremental application of the bound state method to RN black holes, with honest validation and a clear advantage for the extremal case, but no fundamental novelty.
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Referee report
The paper
The paper claims
A single bound-state computation, analytically continued to the imaginary axis, reproduces Reissner-Nordström black hole quasinormal modes to roughly one part in 100,000 and treats the extremal charge limit without changing the algorithm.
The reading
What carries the argument
The load-bearing object is the auxiliary scaling parameter $\alpha$: after the quasinormal-mode-to-bound-state map $x\to -ix$, the substitution $-ix\to \alpha x$ makes the inverted potential depend on the real parameter $\alpha$, so that for real $\alpha$ the differential equation is a genuine bound-state problem with discrete energies $E_n(\alpha)$. A shooting method gives $E_n(\alpha)$ on 31 uniformly spaced points with $\alpha\in[0.2,0.8]$, and diagonal or subdiagonal Padé approximants continue $E_n(\alpha)$ to $\alpha=-i$, where the original outgoing-wave problem is recovered. The homotopy deformation multiplies the potential well by $H(\alpha,\lambda)=(\lambda-1)\alpha^2+\lambda$; since $H(-i,\lambda)=1$ for every $\lambda$, the quasinormal-mode problem is unchanged at the target point even though the real-$\alpha$ well is made deeper, which moves the singularities of $E_n(\alpha)$ away from the continuation path. The extremal case enters only through the form of the tortoise coordinate, the single place where the horizon structure changes the calculation.
Extended reading notes
Core claim
The central claim is that the quasinormal frequencies of odd-parity electromagnetic and gravitational perturbations of a Reissner-Nordström black hole follow from a bound-state spectrum. In tortoise coordinate $x$, the map $x\to -ix$ turns the potential barrier into a well; introducing a real scaling parameter $\alpha$ through $-ix\to \alpha x$ produces an ordinary Schrödinger eigenvalue problem with energies $E_n(\alpha)$. These energies are computed numerically on a grid of real $\alpha$, represented by Padé approximants, and continued analytically to $\alpha=-i$, where the quasinormal frequency is $\omega_n=\sqrt{-E_n(-i)}$. For the tabulated nonextremal cases ($\ell=2$, $Q$ from $0.2$ to $0.99$), the agreement with continued-fraction reference values is better than about $10^{-5}$ for the lowest overtones, with gravitational modes computed through $n=3$ and electromagnetic modes through $n=4$; the homotopy deformation $H(\alpha,\lambda)=(\lambda-1)\alpha^2+\lambda$ with $\lambda=4$ gives reliable higher overtones, reaching $n=4$ for gravitational modes, $n=5$ for electromagnetic modes, and $n=6$ for electromagnetic modes with $Q>0.8$. In the extremal case the same procedure, using $x=r-M^2/(r-M)+2M\ln(r-M)$ as the tortoise coordinate, reproduces the known frequencies to five decimal places for $n\le2$ and exhibits the isospectrality between electromagnetic modes of index $\ell$ and gravitational modes of index $\ell+1$, a degeneracy already visible in the bound-state energies.
Load-bearing premise
The load-bearing premise is that a rational-function fit to the bound-state energies at 31 real values of the scaling parameter ($0.2\le\alpha\le0.8$) continues accurately to $\alpha=-i$; the paper itself reports this fails for gravitational overtones $n\ge4$ and electromagnetic $n\ge5$ before deformation, and it gives no general guarantee that the grid, fit order, or the fixed homotopy value $\lambda=4$ is optimal.
Editorial extensions
If this is right
- Nonextremal Reissner-Nordström low-lying quasinormal modes can be obtained from bound-state energies with relative errors near or below $10^{-5}$, giving an independent numerical cross-check of continued-fraction results.
- Extremal black hole quasinormal modes can be computed with the same code path as nonextremal ones: only the tortoise coordinate must be replaced by its extremal form, with no new recurrence relations or singularity analysis required.
- The homotopy parameter $\lambda=4$ enlarges the computable range by roughly one or two overtones, extending gravitational results to $n=4$ and electromagnetic results to $n=5$, and to $n=6$ when $Q>0.8$.
- The extremal isospectrality between electromagnetic modes of index $\ell$ and gravitational modes of index $\ell+1$ is reproduced, and it is shown to be present already at the level of the underlying bound-state energies.
Reading between the lines
- One testable extension is to treat the homotopy parameter as a convergence diagnostic: plotting where the Padé approximants' singularities move as $\lambda$ varies would give a practical warning when a claimed high-overtone frequency is about to become unreliable, since the paper identifies those singularities approaching $\alpha=-i$ as the cause of failure.
- The same real-$\alpha$ continuation strategy should carry over to other spherically symmetric spacetimes with a single effective potential barrier, but the grid span, sampling density, and optimal homotopy value would need to be re-tuned; the paper does not claim a universal prescription.
- For gravitational-wave template construction, a practical consequence would be a single code covering the whole charge range $0\le Q\le M$, including the maximally charged endpoint, because the extremal calculation is not algorithmically special.
- A direct check of whether the limiting singularities of $E_n(\alpha)$ coincide with the branch points of the inverse relation $\omega^2=-E$ would separate a fundamental obstruction to high overtones from an artifact of the chosen Padé grid.
Original abstract
We apply a newly proposed bound state method to revisit the computation of gravitational and electromagnetic quasinormal modes (QNMs) for Reissner-Nordstr\"om black holes (RNBHs). It is found that the method yields QNM frequencies of high accuracy for low-lying modes with overtones $n\leq3$. The accuracy degrades for higher overtones, however, a homotopy deformation to the potential enables us to compute more high overtone modes reliably. It is known that the continued fraction method for computing QNMs of extremal RNBHs differs significantly from that used for nonextremal ones. In contrast, the bound state method demonstrates advantage of simplicity: one only need to account for the definition of tortoise coordinate in the extremal case, and the rest of the computational procedure remains exactly the same as that for the nonextremal case.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We apply a newly proposed bound state method to revisit the computation of gravitational and electromagnetic quasinormal modes (QNMs) for Reissner-Nordstr\"om black holes (RNBHs). It is found that the method yields QNM frequencies of high accuracy for low-lying modes with overtones $n\leq3$. The accuracy degrades for higher overtones, however, a homotopy deformation to the potential enables us to compute more high overtone modes reliably. It is known that the continued fraction method for computing QNMs of extremal RNBHs differs significantly from that used for nonextremal ones. In contrast, the bound state method demonstrates advantage of simplicity: one only need to account for the definition of tortoise coordinate in the extremal case, and the rest of the computational procedure remains exactly the same as that for the nonextremal case.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We apply a newly proposed bound state method to revisit the computation of gravitational and electromagnetic quasinormal modes (QNMs) for Reissner-Nordstr\"om black holes (RNBHs). | contribution 0.97 | 0.93 | 0.10 | asserted 0.98 |
| s1 | It is found that the method yields QNM frequencies of high accuracy for low-lying modes with overtones $n\leq3$. | result 1.00 | 0.93 | 0.31 | asserted 1.00 |
| s2 | The accuracy degrades for higher overtones, however, a homotopy deformation to the potential enables us to compute more high overtone modes reliably. | limitation 0.44 | 0.89 | 0.18 | asserted 1.00 |
| s3 | It is known that the continued fraction method for computing QNMs of extremal RNBHs differs significantly from that used for nonextremal ones. | background 1.00 | 0.05 | 0.04 | asserted 1.00 |
| s4 | In contrast, the bound state method demonstrates advantage of simplicity: one only need to account for the definition of tortoise coordinate in the extremal case, and the rest of the computational procedure remains exactly the same as that for the nonextremal case. | result 0.44 | 0.77 | 0.30 | asserted 1.00 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:2609.14979 by the authors.
References
Discussion
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