The pseudospectrum for gravitational perturbations in Kerr spacetimes
Abstract
We study the pseudospectrum of the Kerr black hole for gravitational perturbations with spin weight s=2, the case of greatest interest in gravitational-wave astronomy. In the hyperboloidal framework with two-dimensional Chebyshev collocation, we formulate the quasinormal mode problem as a non-self-adjoint eigenvalue problem. For s≠0 the fields Ψ(s) and Ψ(−s) are independent, so the conserved current method yields no single-field energy norm. We overcome this with the Teukolsky--Starobinsky identities, which relate Ψ(−2) to a fourth-order operator on Ψ(2) and yield an H2 energy norm after symmetrization. With the H1 norm known for s=0, this suggests the pattern H|s|, verified for s=0 and s=2. All results use an ω-independent two-field hierarchy. We find that the pseudospectrum tightens by $1.1$ to $3.5$ decades at each step of L2→ H1→ H2, yet stays large in every norm. The s=+2 and s=−2 operators are isospectral but not isopseudospectral: at twelve of fourteen probes the ψ0 resolvent exceeds the ψ4 one by $0.19$ to $0.82$ decades in the H2 norm, while at the two probes furthest along the QNM line the ordering reverses by $0.18$ to $0.19$ decades. The gravitational pseudospectrum exceeds the scalar one by $0.5$ to $5.2$ decades in the window minimum and by $5.9$ decades in L2 at ωM=0.47−0.17 i, while along the damping direction the s=2 resolvent is non-monotonic and the s=0 resolvent smooth. QNM spectral instability is therefore a universal feature of Kerr, and is stronger for gravitational perturbations.
Related
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- Primordial Black Hole and Gravitational Wave by Peaked Cosmic Perturbations from Axion Curvaton
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BibTeX
@misc{zhang2026pseudospectrum,
title = {{The pseudospectrum for gravitational perturbations in Kerr spacetimes}},
author = {Hongsheng Zhang and Zong-Kuan Guo and Rong-Gen Cai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06954},
eprint = {2610.06954},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06954},
}
RIS
TY - UNPB
TI - The pseudospectrum for gravitational perturbations in Kerr spacetimes
AU - Hongsheng Zhang
AU - Zong-Kuan Guo
AU - Rong-Gen Cai
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.06954
UR - https://arxiv.org/abs/2610.06954
AB - We study the pseudospectrum of the Kerr black hole for gravitational perturbations with spin weight s=2, the case of greatest interest in gravitational-wave astronomy. In the hyperboloidal framework with two-dimensional Chebyshev collocation, we formulate the quasinormal mode problem as a non-self-adjoint eigenvalue problem. For s≠0 the fields Ψ^(s) and Ψ^(−s) are independent, so the conserved current method yields no single-field energy norm. We overcome this with the Teukolsky--Starobinsky identities, which relate Ψ^(−2) to a fourth-order operator on Ψ^(2) and yield an H^2 energy norm after symmetrization. With the H^1 norm known for s=0, this suggests the pattern H^|s|, verified for s=0 and s=2. All results use an ω-independent two-field hierarchy. We find that the pseudospectrum tightens by $1.1$ to $3.5$ decades at each step of L^2→ H^1→ H^2, yet stays large in every norm. The s=+2 and s=−2 operators are isospectral but not isopseudospectral: at twelve of fourteen probes the ψ_0 resolvent exceeds the ψ_4 one by $0.19$ to $0.82$ decades in the H^2 norm, while at the two probes furthest along the QNM line the ordering reverses by $0.18$ to $0.19$ decades. The gravitational pseudospectrum exceeds the scalar one by $0.5$ to $5.2$ decades in the window minimum and by $5.9$ decades in L^2 at ωM=0.47−0.17 i, while along the damping direction the s=2 resolvent is non-monotonic and the s=0 resolvent smooth. QNM spectral instability is therefore a universal feature of Kerr, and is stronger for gravitational perturbations.
ER -