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 $s\neq0$ the fields $\Psi^{(s)}$ and $\Psi^{(-s)}$ are independent, so the conserved current method yields no single-field energy norm. We overcome this with the Teukolsky--Starobinsky identities, which relate $\Psi^{(-2)}$ to a fourth-order operator on $\Psi^{(2)}$ 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 $\omega$-independent two-field hierarchy. We find that the pseudospectrum tightens by $1.1$ to $3.5$ decades at each step of $L^2\to H^1\to H^2$, yet stays large in every norm. The $s=+2$ and $s=-2$ operators are isospectral but not isopseudospectral: at twelve of fourteen probes the $\psi_0$ resolvent exceeds the $\psi_4$ 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 $\omega M=0.47-0.17\,i$, 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.