Abstract
This second part of the series treats spin \(\pm 2\) components (or extreme components), that satisfy the Teukolsky master equation, of the linearized gravity in the exterior of a slowly rotating Kerr black hole. For each of these two components, after performing a first-order differential operator once and twice, the resulting equations together with the Teukolsky master equation itself constitute a linear spin-weighted wave system. An energy and Morawetz estimate for spin \(\pm 2\) components is proved by treating this system. This is a first step in a joint work (Andersson et al. in Stability for linearized gravity on the Kerr spacetime, arXiv:1903.03859, 2019) in addressing the linear stability of slowly rotating Kerr metrics.
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Notes
\(\kappa \) is one of the spin coefficients used in [12, Chapter 1.8].
This application of the first-order differential operators to the spin \(\pm 2\) components is closely related to Chandrasekhar transformation [13].
We should distinguish among these different notations that a tilde means that there is no extra \(r^{-\delta }\) power in the coefficients of \(\partial _r\)- and \(\partial _{t^*}\)-derivatives term and a subscript \(\text {deg}\) means there is the trapping degeneracy in the trapped region, and vice versa.
In fact, the N–P components should be viewed as sections of a complex line bundle. Therefore, “smooth” means that these components and their derivatives to any order with respect to \((\partial _{t^*},\partial _r, {\nabla \!\!\!/}_1, {\nabla \!\!\!/}_2, {\nabla \!\!\!/}_3)\) are continous.
The dependence of \({\hat{\epsilon }}_1\) in \(C_1\) is not needed for spin \(-2\) component.
Note that in Schwarzschild case, \(\varLambda _{m\ell }=A+s+s^2\), with A being the separation constant in [44].
A solution to (27a) or (27b) is integrable if for every integer \(n\ge 0\), every multi-index \(0\le |i|\le n\) and any \(r'>r_+\), we have
$$\begin{aligned} \sum _{0\le |i|\le n}\int _{{\mathcal {D}}(-\infty ,\infty )\cap \{r=r'\}}(|\partial ^i \psi |^2+|\partial ^i F|^2)<\infty . \end{aligned}$$(168).
The authors in [17] missed one term \(-4aMrm\omega \) in the Equation (33), but what is used thereafter is the Schrödinger equation (34) in Section 9 which is correct. Therefore, the validity of the proof will not be influenced by the missing term.
\(r_{\infty }^*\) is chosen based on \(\tau \) from the property of finite speed of propagation.
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The author is grateful to Lars Andersson, Pieter Blue and Claudio Paganini for many helpful discussions and comments.
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Commutators of a Spin-Weighted Wave Operator and rYr (or rVr)
Commutators of a Spin-Weighted Wave Operator and rYr (or rVr)
Proposition 11
Let \({\mathbf {L}}_s\) be a spin-weighted wave operator
For any scalar \(\psi \) with spin weight s, we have the following commutators
Proof
Expand \({\mathbf {L}}_s \psi \) into the form of
We prove the commutator relation (303) below, and the commutator (304) is manifest from (303) by letting \(t\rightarrow -t\) and \(\phi \rightarrow -\phi \) (hence \(\partial _t\rightarrow -\partial _t\), \(\partial _{\phi } \rightarrow -\partial _{\phi }\) and \(V \rightarrow -Y\)). We calculate the commutators between each term and \(-rVr\). The first line of (305) commutes with \(-rVr\), and hence their commutators vanish. The last term on the second line commutes with \(-rVr\), and for the other terms on the second line, we have
The commutator of the last line with \(-rVr\) equals
It remains to calculate the commutator [Y, V] which is present in both (306) and (307). For a general field \(\psi \),
Collecting the above discussions and calculations, we arrive at
The relation (303) follows by calculating the coefficient of each single term in the above equation. \(\quad \square \)
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Ma, S. Uniform Energy Bound and Morawetz Estimate for Extreme Components of Spin Fields in the Exterior of a Slowly Rotating Kerr Black Hole II: Linearized Gravity. Commun. Math. Phys. 377, 2489–2551 (2020). https://doi.org/10.1007/s00220-020-03777-2
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DOI: https://doi.org/10.1007/s00220-020-03777-2