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Vanishing Frobenius coefficients for scalar perturbations at the core

Ascertain whether physically realistic initial data for massless scalar field perturbations on static, spherically symmetric spacetimes with integrable singularities can yield mode solutions near r = 0 whose Frobenius-series coefficients A1 and A2 vanish for all angular multipoles l and wavenumbers k, thereby avoiding logarithmic or inverse-power divergences and preventing non‑integrable growth of the associated energy density.

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Background

The paper analyzes linear test‑field perturbations on integrable singularity backgrounds and finds that scalar modes generically blow up near r = 0: s‑waves as ln r and non‑s‑waves as inverse powers of r, leading to a non‑integrable energy density for l > 0.

They note that the leading behavior is determined by Frobenius-series coefficients A1 and A2, which are fixed by initial conditions set at early times (e.g., horizon crossing). The authors explicitly acknowledge the possibility that specific initial data might make A1 and A2 vanish for all l and k, which would eliminate the problematic divergences.

However, they emphasize that they have not ruled out this possibility, indicating an open question about whether such initial conditions exist and are physically reasonable rather than fine‑tuned.

References

There remains the possibility that evolving a physical field from early times results in a vanishing of the $A_{1,2}$ coefficients in~eq: Spin zero, roots for all values of $l$ and $k$. We have not ruled out this possibility, but consider it quite unlikely unless initial conditions are fine tuned.

eq: Spin zero, roots:

Ψ01=r∑n=0∞anrn ,Ψ02=ln⁡(r) r∑n=0∞anrn +∑n=0∞bnrn ,Ψ0=A1Ψ01 + A2Ψ02 .\begin{split} &\Psi_{0}^{1}=r\sum_{n=0}^{\infty} a_{n}r^n\,,\\ &\Psi_{0}^{2}= \ln(r)\ r \sum_{n=0}^{\infty} a_{n}r^n\ + \sum_{n=0}^{\infty}b_{n} r^{n}\,,\\ &\Psi_{0}=A_{1}\Psi_{0}^{1}\ +\ A_{2} \Psi_{0}^{2}\,. \end{split}

— Physical and Theoretical Challenges to Integrable Singularities (2504.17863 - Arrechea et al., 24 Apr 2025) in Section 5.1, General test field perturbations (s‑waves discussion)