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

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.

Commutators of a Spin-Weighted Wave Operator and rYr (or rVr)

Proposition 11

Let ${\mathbf {L}}_s$ be a spin-weighted wave operator

$$\begin{aligned} {\mathbf {L}}_s={}&\varSigma \Box _g+\tfrac{2is\cos \theta }{\sin ^2 \theta }\partial _{\phi }-s^2\cot ^2 \theta -s^2. \end{aligned}$$

(302)

For any scalar $\psi$ with spin weight s, we have the following commutators

$$\begin{aligned}{}[{\mathbf {L}}_s, -rVr]\psi ={}&-\tfrac{2(r^2-3Mr+2a^2)}{r^3}r^2V(rV(r\psi )) +\tfrac{4}{r}(a^2\partial _t +a\partial _{\phi })(rV(r\psi ))\nonumber \\&-\tfrac{2(r^2-Mr+3a^2)}{r^2}rV(r\psi ) -2(a^2\partial _t +a\partial _{\phi })\psi +\tfrac{2Mr-4a^2}{r}\psi , \end{aligned}$$

(303)

$$\begin{aligned} \psi ={}&-\tfrac{2(r^2-3Mr+2a^2)}{r^3}r^2Y(rY(r\psi )) +\tfrac{4}{r}(a^2\partial _t +a\partial _{\phi })(rY(r\psi ))\nonumber \\&+\tfrac{2(r^2-Mr+3a^2)}{r^2}rY(r\psi ) +2(a^2\partial _t +a\partial _{\phi })\psi +\tfrac{2Mr-4a^2}{r}\psi . \end{aligned}$$

(304)

Proof

Expand ${\mathbf {L}}_s \psi$ into the form of

$$\begin{aligned} {\mathbf {L}}_s\psi ={}&\left( \tfrac{1}{\sin {\theta }} \partial _{\theta }(\sin \theta \partial _{\theta })+\tfrac{\partial _{\phi \phi }^2}{\sin ^2\theta } +2a\partial _{t\phi }^2+a^2 \sin ^2 \theta \partial _{tt}^2 +\tfrac{2is\cos \theta }{\sin ^2 \theta }\partial _{\phi }-\tfrac{s^2}{\sin ^2 \theta }\right) \psi \nonumber \\&-rY\left( \tfrac{\varDelta }{r^2} V(r\psi )\right) +\tfrac{a^2\varDelta }{r^2(r^2+a^2)}(V+Y)(r\psi ) +\tfrac{2ar}{r^2+a^2}\partial _{\phi }\psi -2ias\cos \theta \partial _t \psi \nonumber \\&-\left[ \tfrac{2Mr^3+a^2r^2-4a^2Mr+a^4}{(r^2+a^2)^2}+r\sqrt{r^2+a^2}\partial _r\left( \tfrac{a^2 \varDelta }{r^2 (r^2+a^2)^{3/2}}\right) \right] \psi . \end{aligned}$$

(305)

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

$$\begin{aligned} &[rY\left( \tfrac{\varDelta }{r^2} Vr\right) , -rVr]\psi \nonumber \\ ={}&r^3 [Y,V]\left( \tfrac{\varDelta }{r^2}V(r\psi )\right) -2r^2 V\left( \tfrac{\varDelta }{r^2}V(r\psi )\right) \nonumber \\&- 2r\tfrac{\varDelta }{r^2}V(r\psi ) -rY\left( r^2 \partial _r\left( \tfrac{\varDelta }{r^2}\right) V(r\psi )\right) , \end{aligned}$$

(306)

$$\begin{aligned} & [\tfrac{a^2\varDelta }{r^2(r^2+a^2)}(V+Y)r,-rVr]\psi \nonumber \\ ={}&-\tfrac{a^2\varDelta }{r^2+a^2}[Y,V](r\psi ) +\tfrac{2a^2r\varDelta }{r^2(r^2+a^2)} V(r\psi )\nonumber \\&+r\partial _r\left( \tfrac{a^2\varDelta }{r(r^2+a^2)}\right) Y(r\psi ) +r\partial _r\left( \tfrac{a^2\varDelta }{r^3(r^2+a^2)}\right) r^2V(r\psi ), \end{aligned}$$

(307)

$$\begin{aligned} & [\tfrac{2ar}{r^2+a^2}\partial _{\phi },-rVr]\psi ={}-\tfrac{2ar^2(r^2-a^2)}{(r^2+a^2)^2}\partial _{\phi }\psi . \end{aligned}$$

(308)

The commutator of the last line with $-rVr$ equals

$$\begin{aligned} &-r^2\partial _r\left( \tfrac{2Mr^3+a^2r^2-4a^2Mr+a^4}{(r^2+a^2)^2}+r\sqrt{r^2+a^2}\partial _r\left( \tfrac{a^2 \varDelta }{r^2 (r^2+a^2)^{3/2}}\right) \right) \psi \nonumber \\ ={}&-r^2 \partial _r\left( \tfrac{2(Mr-a^2)}{r^2}\right) \psi \nonumber \\ ={}&\tfrac{2Mr-4a^2}{r}\psi . \end{aligned}$$

(309)

It remains to calculate the commutator [Y, V] which is present in both (306) and (307). For a general field $\psi$,

$$\begin{aligned}{}[Y,V]\psi ={}&-2\partial _r\left( \tfrac{r^2+a^2}{\varDelta }\right) \partial _t \psi -2\partial _r\left( \tfrac{a}{\varDelta }\right) \partial _{\phi }\psi \nonumber \\ ={}&\tfrac{4M(r^2-a^2)}{\varDelta ^2}\partial _t\psi +\tfrac{4a(r-M)}{\varDelta ^2}\partial _{\phi }\psi . \end{aligned}$$

(310)

Collecting the above discussions and calculations, we arrive at

$$\begin{aligned} &[{\mathbf {L}}_s, -rVr]\psi \nonumber \\ ={}&-\left( \tfrac{2\varDelta }{r}-\tfrac{2(Mr-a^2)}{r}\right) V(rV(r\psi )) +\tfrac{4}{r}(a^2\partial _t +a\partial _{\phi })(rV(r\psi ))\nonumber \\&-\tfrac{2(r^2-Mr+3a^2)}{r^2}rV(r\psi ) -2a^2\partial _t \psi -\tfrac{2a^3(3r^2+a^2)+2ar^2(r^2-a^2)}{(r^2+a^2)^2}\partial _{\phi }\psi \nonumber \\&+\tfrac{2Mr-4a^2}{r}\psi . \end{aligned}$$

(311)

The relation (303) follows by calculating the coefficient of each single term in the above equation. $\quad \square$