Boundedness of Massless Scalar Waves on Kerr Interior Backgrounds

Abstract

We consider solutions of the massless scalar wave equation $\Box _g\psi =0$, without symmetry, on fixed subextremal Kerr backgrounds $({{\mathcal {M}}}, g)$. It follows from previous analyses in the Kerr exterior that for solutions $\psi$ arising from sufficiently regular data on a two-ended Cauchy hypersurface, the solution and its derivatives decay suitably fast along the event horizon ${{\mathcal {H}}}^+$. Using the derived decay rate, we show that $\psi$ is in fact uniformly bounded, $|\psi |\le C$, in the black hole interior up to and including the bifurcate Cauchy horizon ${{\mathcal {C}}}{{\mathcal {H}}}^+$, to which $\psi$ in fact extends continuously. In analogy to our previous paper [31], on boundedness of solutions to the massless scalar wave equation on fixed subextremal Reissner–Nordström backgrounds, the analysis depends on weighted energy estimates, commutation by angular momentum operators and an application of Sobolev embedding. In contrast to the Reissner–Nordström case, the commutation leads to additional error terms that have to be controlled.

Appendices

Appendix A: The K-current

In order to compute all scalar currents according to (16) in (u, v) coordinates, we first derive the components of the deformation tensor which is given by

$$\begin{aligned} (\pi ^X)^{\mu \nu }=\frac{1}{2}(g^{\mu \lambda }\partial _{\lambda }X^{\nu }+g^{\nu \sigma }\partial _{\sigma }X^{\mu }+g^{\mu \lambda }g^{\nu \sigma }g_{\lambda \sigma , \delta }X^{\delta }), \end{aligned}$$

(A1)

where X is an arbitrary vector field, $X=X^u \partial _u+X^v \partial _v+X^{\theta _C}\partial _{\theta _C}$, with $X^u$, $X^v$ and $X^{\theta _C}$ depending on $u,v,\theta ^{\star }, \tilde{\phi }$. From this, we obtain

$$\begin{aligned} (\pi ^X)^{v v}= & {} -\frac{1}{2\Omega ^2}\partial _u X^v, \end{aligned}$$

(A2)

$$\begin{aligned} (\pi ^X)^{u u}= & {} -\frac{1}{2\Omega ^2}\partial _v X^u, \end{aligned}$$

(A3)

$$\begin{aligned} (\pi ^X)^{u v}= & {} -\frac{1}{4\Omega ^2}\left[ \partial _uX^u+\partial _vX^v\right] -\frac{1}{2\Omega ^2} \frac{\partial _{\eta } \Omega }{\Omega }X^{\eta } , \end{aligned}$$

(A4)

$$\begin{aligned} (\pi ^X)^{v {\theta _C}}= & {} -\frac{1}{4\Omega ^2}\partial _uX^{\theta _C}-\frac{b^{\theta _C}}{4\Omega ^2}\partial _uX^v +\frac{1}{2} (\not g\,^{-1})^{{\theta _C}{\theta _D}}\partial _{\theta _D} X^v , \end{aligned}$$

(A5)

$$\begin{aligned} (\pi ^X)^{u {\theta _C}}= & {} -\frac{b^{\theta _D}}{4\Omega ^2}\partial _{{\theta _D}}X^{\theta _C}-\frac{1}{4\Omega ^2}\partial _v X^{\theta _C}-\frac{b^{\theta _C}}{4\Omega ^2}\partial _uX^u+\frac{1}{2} (\not g\,^{-1})^{{\theta _C}{\theta _D}}\partial _{\theta _D} X^u \nonumber \\&-\frac{b^{\theta _C}}{2\Omega ^2}\frac{\partial _{\eta } \Omega }{\Omega }X^{\eta } +\frac{\delta ^{\theta _C}_{\; \; \tilde{\phi }}}{4\Omega ^2}\partial _{\eta }b^{\tilde{\phi }} X^{\eta }, \end{aligned}$$

(A6)

$$\begin{aligned} (\pi ^X)^{{\theta _C} {\theta _D}}= & {} -\frac{b^{\theta _C}}{4\Omega ^2}\partial _u X^{\theta _D}+ \frac{1}{2}(\not g\,^{-1})^{{\theta _C}{\theta _A}}\partial _{{\theta _A}}X^{\theta _D}\nonumber \\&-\frac{b^{\theta _D}}{4\Omega ^2}\partial _u X^{\theta _C}+ \frac{1}{2}(\not g\,^{-1})^{{\theta _D}{\theta _A}}\partial _{{\theta _A}}X^{\theta _C}\nonumber \\&+\frac{1}{2} (\not g\,^{-1})^{{\theta _C}{\theta _A}}(\not g\,^{-1})^{{\theta _D}{\theta _B}}\partial _{\eta } \not g\,_{{\theta _A}{\theta _B}} X^{\eta }, \end{aligned}$$

(A7)

with $A,B,C,D=1,2$, $\theta _1=\theta ^{\star }$, $\theta _2=\tilde{\phi }$ ,$\zeta =u,v$ and $\eta =u,v,\theta ^{\star }, \tilde{\phi }$. From (10), we calculate the components of the energy momentum tensor in (u, v) coordinates as

$$\begin{aligned} T^{KG}_{v v}= & {} (\partial _v \psi )^2+\frac{1}{2}|b|^2\left( \frac{1}{\Omega ^2}(\partial _u\psi )( \partial _v \psi +{b^{\theta _C}}\partial _{{\theta _C}} \psi )-|\not \nabla /\,\psi |^2\right) ,\end{aligned}$$

(A8)

$$\begin{aligned} T^{KG}_{u u}= & {} (\partial _u \psi )^2,\end{aligned}$$

(A9)

$$\begin{aligned} T^{KG}_{u v}= & {} T^{KG}_{v u}=\Omega ^2|\not \nabla /\,\psi |^2-{b^{\theta _C}}(\partial _u\psi \partial _{{\theta _C}} \psi ),\end{aligned}$$

(A10)

$$\begin{aligned} T^{KG}_{v {\theta _C}}= & {} (\partial _v \psi \partial _{{\theta _C}} \psi )-\frac{b_{\theta _C}}{2}\left( \frac{1}{\Omega ^2}(\partial _u\psi )( \partial _v \psi +{b^{\theta _D}}\partial _{{\theta _D}} \psi )-|\not \nabla /\,\psi |^2\right) ,\end{aligned}$$

(A11)

$$\begin{aligned} T^{KG}_{u {\theta _C}}= & {} (\partial _u \psi \partial _{{\theta _C}} \psi ),\end{aligned}$$

(A12)

$$\begin{aligned} T^{KG}_{{\theta _C} {\theta _D}}= & {} (\partial _{{\theta _C}} \psi \partial _{{\theta _D}} \psi )+\frac{1}{2}\not g\,_{{\theta _C}{\theta _D}}\left( \frac{1}{\Omega ^2}(\partial _u\psi )( \partial _v \psi +{b^{\theta _A}} \partial _{{\theta _A}} \psi )-|\not \nabla /\,\psi |^2 \right) ,\nonumber \\ \end{aligned}$$

(A13)

with $b^{\theta _C}b_{\theta _C}=|b|^2$ and $\not g\,_{{\theta _C}{\theta _D}}b^{\theta _D}=b_{\theta _C}$.

Multiplying the components according to (16) in Eddington–Finkelstein-like coordinates, we get

$$\begin{aligned} K^X= & {} -\left[ \frac{1}{2\Omega ^2}\partial _u X^v \right] (\partial _v \psi )^2\nonumber \\&-\left[ \frac{1}{2\Omega ^2}\partial _v X^u \right] (\partial _u \psi )^2\nonumber \\&+\left[ -\frac{\partial _{\eta } X^{\eta }}{2}-\frac{{\partial _{\theta _C} X^{\theta _C}}}{2} -\frac{\partial _{\eta } \Omega }{\Omega } X^{\eta } \right. \nonumber \\&\left. -\frac{1}{4} (\not g\,^{-1})^{{\theta _C}{\theta _D}}\partial {\eta }\not g\,_{{\theta _C}{\theta _D}}X^{\eta }+\frac{b^{\tilde{\phi }}}{2}\partial _{\tilde{\phi }} X^v \right] |\not \nabla /\,\psi |^2\nonumber \\&+\left[ {\partial _{\theta _C} X^{\theta _C}}+ \frac{1}{4} (\not g\,^{-1})^{{\theta _C}{\theta _D}}\partial {\eta }\not g\,_{{\theta _C}{\theta _D}}X^{\eta } \right. \nonumber \\&\left. -\frac{b^{\tilde{\phi }}\partial _{\tilde{\phi }} X^v}{2}\right] \frac{1}{\Omega ^2}(\partial _u \psi )(\partial _v \psi +b^{\tilde{\phi }}\partial _{\tilde{\phi }}\psi )\nonumber \\&+\left[ -\frac{b^{\tilde{\phi }}}{2\Omega ^2}\partial _{{\tilde{\phi }}}X^{\theta _C}-\frac{1}{2\Omega ^2}\partial _v X^{\theta _C}-\frac{b^{\theta _C}}{2\Omega ^2}\partial _u X^u+ (\not g\,^{-1})^{{\theta _C}{\theta _D}}\partial _{\theta _D} X^u\right. \nonumber \\&\left. -\frac{b^{\theta _C}}{\Omega ^2}\frac{\partial _{\eta } \Omega }{\Omega }X^{\eta } +\frac{\delta ^{\theta _C}_{\; \; \tilde{\phi }}}{2\Omega ^2}\partial _{\eta }b^{\tilde{\phi }} X^{\eta } \right] (\partial _u \psi \partial _{{\theta _C}} \psi )\nonumber \\&+\left[ -\frac{1}{2\Omega ^2}\partial _uX^{{\theta _C}}-\frac{b^{\theta _C}}{2\Omega ^2}\partial _uX^{v}+ (\not g\,^{-1})^{{\theta _C}{\theta _D}}\partial _{\theta _D} X^v \right] (\partial _v \psi \partial _{{\theta _C}} \psi )\nonumber \\&+\left[ -\frac{b^{\theta _C}}{4\Omega ^2}\partial _uX^{\theta _D}-\frac{b^{\theta _D}}{4\Omega ^2}\partial _uX^{\theta _C}+\frac{1}{2} (\not g\,^{-1})^{{\theta _C}{\theta _A}}\partial _{\theta _A} X^{\theta _D} \right. \nonumber \\&+\frac{1}{2} (\not g\,^{-1})^{{\theta _D}{\theta _A}}\partial _{\theta _A} X^{\theta _C}\nonumber \\&+\left. \frac{1}{2}(\not g\,^{-1})^{{\theta _C}{\theta _A}}(\not g\,^{-1})^{{\theta _D}{\theta _B}}\partial {\eta }\not g\,_{{\theta _A}{\theta _B}}X^{\eta } \right] (\partial _{{\theta _C}} \psi \partial _{{\theta _D}} \psi ), \end{aligned}$$

(A14)

also recall (93).

Appendix B: The J-currents and Normal Vectors

For the J-currents, according to Eq. (13), we obtain

$$\begin{aligned} J^X_{\mu }n^{\mu }_{v=\mathrm{const}}= & {} \frac{1}{2\Omega ^2}\left[ X^u(\partial _u \psi )^2{+}[X^{\theta _C}{-}X^vb^{\theta _C}](\partial _u\psi \partial _{{\theta _C}} \psi )+ \Omega ^2 X^v|\not \nabla /\,\psi |^2\right] , \nonumber \nonumber \\ \end{aligned}$$

(B1)

$$\begin{aligned} J^X_{\mu }n^{\mu }_{u=\mathrm{const}}= & {} \frac{1}{2\Omega ^2}\left[ X^v(\partial _v\psi )^2 +\left[ X^{\theta _C}+ X^vb^{\theta _C}\right] (\partial _v\psi \partial _{{\theta _C}} \psi )\right. \nonumber \\&+{b^{\theta _C}}X^{\theta _D}(\partial _{{\theta _C}}\psi \partial _{{\theta _D}} \psi )\left. +X^u{\Omega ^2}|\not \nabla /\,\psi |^2\right] , \end{aligned}$$

(B2)

$$\begin{aligned} J^X_{\mu }n^{\mu }_{r^{\star }=\mathrm{const}}= & {} \frac{1}{\sqrt{2\Omega ^2}}\left[ X^v(\partial _v\psi )^2 + X^u(\partial _u \psi )^2+[X^{\theta _C}-X^vb^{\theta _C}](\partial _u\psi \partial _{{\theta _C}} \psi ) \right. \nonumber \\&\left. +\left[ X^{\theta _C}+ X^vb^{\theta _C}\right] (\partial _v\psi \partial _{{\theta _C}} \psi )+{b^{\theta _C}}X^{\theta _D}(\partial _{{\theta _C}}\psi \partial _{{\theta _D}} \psi )\right. \nonumber \\&\left. +\left( X^u+X^v\right) {\Omega ^2}|\not \nabla /\,\psi |^2\right] . \end{aligned}$$

(B3)

The normal vectors $n^{\mu }_{u=\mathrm{const}}$ and $n^{\mu }_{v=\mathrm{const}}$ are already stated in (76) and (75), respectively. Further, we have

$$\begin{aligned} n^{\mu }_{r^{\star }=\mathrm{const}}= & {} \frac{1}{\sqrt{2\Omega ^2}}(\partial _u+\partial _v+b^{\theta _C}\partial _{\theta _C}). \end{aligned}$$

(B4)

Moreover, note that for large v the current $J^X_{\mu }n^{\mu }_{\gamma }$ approximates $J^X_{\mu }n^{\mu }_{r^{\star }=\mathrm{const}}$ and likewise the normal vector $n^{\mu }_{\gamma }$ approximates $n^{\mu }_{r^{\star }=\mathrm{const}}$, where $\gamma$ is as defined in Sect. 4.3.1.

Appendix C: The Redshift Vector Field

We construct the redshift vector field by defining

$$\begin{aligned} N=N^u \partial _u+N^v (\partial _v+b^{\tilde{\phi }}\partial _{\tilde{\phi }}), \end{aligned}$$

(C1)

where $N^u$ and $N^v$ depend on $(u,v,\theta ^{\star })$. We further require that N is timelike so that we have the condition

$$\begin{aligned} g(N,N)<0. \end{aligned}$$

(C2)

From the above, we obtain that $N^u$ and $N^v$ have to have the same sign and should be positive, so that the vector field is future-directed. Moreover, we have already introduced the vector field $T_{{{\mathcal {H}}}^+}$ in the end of Sect. 2.2.1, see (32). Adjusting to our coordinates, the vector field

$$\begin{aligned} T_{{{\mathcal {H}}}^+}=\frac{\partial _v}{2}-\frac{\partial _u}{2}+\omega _+ \partial _{\tilde{\phi }} \end{aligned}$$

(C3)

satisfies the condition of being null at ${{\mathcal {H}}}^+$, spacelike in the interior and timelike in the exterior. The constant $\omega _+$ is defined by $\omega _B$, see Eq. (51), evaluated at $r=r_+$. We can now choose N such that

$$\begin{aligned} g(N,T_{{{\mathcal {H}}}^+})|_{{{\mathcal {H}}}^+}=-2. \end{aligned}$$

(C4)

Further, we use (A14) to calculate

$$\begin{aligned} K^N= & {} -\left[ \frac{1}{2\Omega ^2}\partial _u N^v \right] (\partial _v \psi )^2\nonumber \\&-\left[ \frac{1}{2\Omega ^2}\partial _v N^u \right] (\partial _u \psi )^2\nonumber \\&+\left[ -\frac{\partial _{u} N^{u}}{2}-\frac{\partial _{v} N^{v}}{2} -\frac{\partial _{u} \Omega }{\Omega } N^{u}-\frac{\partial _{v} \Omega }{\Omega } N^{v}\right. \nonumber \\&\left. -\frac{1}{4} (\not g\,^{-1})^{{\theta _C}{\theta _D}}\left( \partial _{u}\not g\,_{{\theta _C}{\theta _D}}N^{u}+\partial _{v}\not g\,_{{\theta _C}{\theta _D}}N^{v}\right) \right] |\not \nabla /\,\psi |^2\nonumber \\&+\left[ \frac{1}{4} (\not g\,^{-1})^{{\theta _C}{\theta _D}}\left( \partial _{u}\not g\,_{{\theta _C}{\theta _D}}N^{u}+\partial _{v}\not g\,_{{\theta _C}{\theta _D}}N^{v}\right) \right] \frac{1}{\Omega ^2}(\partial _u \psi )(\partial _v \psi +b^{\tilde{\phi }}\partial _{\tilde{\phi }}\psi )\nonumber \\&+\left[ -\frac{b^{\tilde{\phi }}}{2\Omega ^2}\left( \partial _v N^v +\partial _u N^u\right) + (\not g\,^{-1})^{{\tilde{\phi }}\theta ^{\star }} \partial _{\theta ^{\star }}N^u -\frac{b^{\tilde{\phi }}}{\Omega ^2}\left[ \frac{\partial _u \Omega }{\Omega }N^u+\frac{\partial _v \Omega }{\Omega }N^v\right] \right. \nonumber \\&\left. +\frac{1}{2\Omega ^2}\partial _u b^{\tilde{\phi }} N^u\right] (\partial _u \psi \partial _{\tilde{\phi }} \psi )\nonumber \\&+\left[ (\not g\,^{-1})^{\theta ^{\star } \theta ^{\star }} \partial _{\theta ^{\star }}N^u \right] (\partial _u \psi \partial _{\theta ^{\star }} \psi )\nonumber \\&+\left[ -\frac{1}{2\Omega ^2} \partial _v(N^v b^{\tilde{\phi }}) -\frac{b^{\tilde{\phi }}}{2\Omega ^2}\partial _u N^v+ (\not g\,^{-1})^{{\tilde{\phi }}\theta ^{\star }} \partial _{\theta ^{\star }}N^v\right] (\partial _v \psi \partial _{\tilde{\phi }} \psi )\nonumber \\&+\left[ (\not g\,^{-1})^{\theta ^{\star }\theta ^{\star }} \partial _{\theta ^{\star }}N^v\right] (\partial _v \psi \partial _{\theta ^{\star }} \psi )\nonumber \\&+\left[ -\frac{b^{\tilde{\phi }}}{2\Omega ^2}\partial _u\left( N^vb^{\tilde{\phi }}\right) +(\not g\,^{-1})^{{\tilde{\phi }} \theta ^{\star }}\partial _{\theta ^{\star }}\left( N^vb^{\tilde{\phi }}\right) \right. \nonumber \\&\left. + \frac{1}{2}(\not g\,^{-1})^{{\tilde{\phi }}{\theta _C}}(\not g\,^{-1})^{{\tilde{\phi }}{\theta _D}}\left( \partial _{u}\not g\,_{{\theta _C}{\theta _D}}N^{u}+\partial _{v}\not g\,_{{\theta _C}{\theta _D}}N^{v}\right) \right] (\partial _{\tilde{\phi }} \psi )^2\nonumber \\&+\left[ \frac{1}{2}(\not g\,^{-1})^{{\theta ^{\star }}{\theta _C}}(\not g\,^{-1})^{{\theta ^{\star }}{\theta _D}}\left( \partial _{u}\not g\,_{{\theta _C}{\theta _D}}N^{u}+\partial _{v}\not g\,_{{\theta _C}{\theta _D}}N^{v}\right) \right] (\partial _{\theta ^{\star }} \psi )^2\nonumber \\&+\left[ (\not g\,^{-1})^{{\theta ^{\star }}{\theta ^{\star }}}\partial _{\theta ^{\star }}\left( N^vb^{\tilde{\phi }}\right) + \frac{1}{2}(\not g\,^{-1})^{{\theta ^{\star }}{\theta _C}}(\not g\,^{-1})^{{\tilde{\phi }}{\theta _D}} \right. \nonumber \\&\left. \left( \partial _{u}\not g\,_{{\theta _C}{\theta _D}}N^{u}+\partial _{v}\not g\,_{{\theta _C}{\theta _D}}N^{v}\right) \right] \times (\partial _{\theta ^{\star }} \psi \partial _{\tilde{\phi }} \psi ). \end{aligned}$$

(C5)

Recall Paragraph 2.2.5 which showed that derivatives of the metric coefficients can be bounded by $\Delta$ which is zero at ${{\mathcal {H}}}^+$. The same holds for derivatives of $b^{\tilde{\phi }}$. With the choice $N^u, N^v$ positive, $\partial _u N^u$ negative and with large absolute value and $\partial _u N^v$ negative and with big enough absolute value we can prove Proposition 4.1 and Lemma 4.3. This can be seen from noticing that all dominating terms are rendered positive, remember (107) and applying the Cauchy–Schwarz inequality.

Appendix D: Commutators

The following proposition was already proven in [23] and can also be found in [1] Section 6.2. For the sake of completeness, we will briefly repeat it here.

Proposition 6.8

Let $\psi$ be a solution of the scalar wave equation

$$\begin{aligned} \Box _g \psi =f, \end{aligned}$$

(D1)

and Y be an arbitrary vector field. Then

$$\begin{aligned} \Box _g(Y\psi )= & {} Y(f)+2(\pi ^Y)^{\alpha \beta }\nabla _{\alpha }\nabla _{\beta }\psi +2\nabla ^{\alpha }(\pi ^Y)_{\alpha \mu }\nabla ^{\mu }\psi \nonumber \\&-\nabla _{\mu }(\pi ^Y)_{~\alpha }^{\alpha }\nabla ^{\mu }\psi . \end{aligned}$$

(D2)

Proof

First we state

$$\begin{aligned} Y(\Box _g \psi )= & {} {{\mathcal {L}}}_Y(g^{\alpha \beta }\nabla _{\alpha }\nabla _{\beta }\psi )=-2(\pi ^Y)^{\alpha \beta }\nabla _{\alpha }\nabla _{\beta }\psi \nonumber \\&+g^{\alpha \beta }{{\mathcal {L}}}_Y(\nabla _{\alpha }\nabla _{\beta } \psi )=Y(f), \end{aligned}$$

(D3)

$$\begin{aligned} {{\mathcal {L}}}_Y(\nabla _{\alpha }\nabla _{\beta } \psi )-\nabla _{\alpha }{{\mathcal {L}}}_Y\nabla _{\beta } \psi= & {} \left[ (\nabla _{\beta }(\pi ^Y)_{\alpha \mu })-(\nabla _{\mu }(\pi ^Y)_{\beta \alpha })\right. \nonumber \\&\left. +(\nabla _{\alpha }(\pi ^Y)_{\mu \beta })\right] \nabla ^{\mu }\psi , \end{aligned}$$

(D4)

$$\begin{aligned} {{\mathcal {L}}}_Y\nabla _{\beta } \psi= & {} \nabla _{Y}\nabla _{\beta } \psi +\nabla _{\beta }Y^{\mu }\nabla _{\mu } \psi =\nabla _{\beta }(Y\psi ). \end{aligned}$$

(D5)

Now we use Eq. (D5) in (D4) and the result of that in (D3). It then only remains to solve the equation for $\Box _g(Y\psi )$ to obtain (D2). $\square$

Appendix E: Error Terms

In order to prove pointwise boundedness, we need to commute twice with all angular operators, as explained in Sect. 2.2.6, which are unfortunately not all Killing. Therefore, we are interested in the error term

$$\begin{aligned} {{\mathcal {E}}}^V(Y^k\psi )=\Box _g(Y^k \psi ) V(Y^k\psi ), \end{aligned}$$

(E1)

resulting from commutation with the vector field Y as defined in (94), according to (17) and with $k=1,2$. To analyze this further, recall the second term of the right-hand side of (D2) and notice that the higher-order terms of the error terms are defined by the following:

$$\begin{aligned} (\pi ^Y)^{vv}= & {} (\pi ^Y)^{u u}=(\pi ^Y)^{v{\theta _C}}=0, \end{aligned}$$

(E2)

$$\begin{aligned} (\pi ^Y)^{uv}= & {} -\frac{1}{2\Omega ^2} \frac{\partial _{\theta ^{\star }} \Omega }{\Omega }Y^{\theta ^{\star }} , \end{aligned}$$

(E3)

$$\begin{aligned} (\pi ^Y)^{u {\theta _C}}= & {} -\frac{b^{\tilde{\phi }}}{4\Omega ^2} \partial _{\tilde{\phi }}Y^{\theta _C} -\frac{b^{\theta _C}}{2\Omega ^2}\frac{\partial _{\theta ^{\star }} \Omega }{\Omega }Y^{\theta ^{\star }} \nonumber \\&+\frac{1}{4\Omega ^2}(\not g\,^{-1})^{{\theta _C}{\theta _D}} \partial _{\theta ^{\star }}b_{\theta _D} Y^{\theta ^{\star }}\nonumber \\&-\frac{b^{\theta _D}}{4\Omega ^2}(\not g\,^{-1})^{{\theta _C}{\theta _A}} \partial _{\theta ^{\star }}\not g\,_{{\theta _D}{\theta _A}} Y^{\theta ^{\star }}, \end{aligned}$$

(E4)

$$\begin{aligned} (\pi ^Y)^{{\theta _C} {\theta _D}}= & {} \frac{1}{2}(\not g\,^{-1})^{{\theta _C} {\theta _A}}\partial _{{\theta _A}}Y^{\theta _D} + \frac{1}{2}(\not g\,^{-1})^{{\theta _C}{\theta _D}} \partial _{{\theta _A}}Y^{\theta _C}\nonumber \\&+\frac{1}{2} (\not g\,^{-1})^{{\theta _C}{\theta _A}} (\not g\,^{-1})^{{\theta _D}{\theta _B}}\partial _{\theta ^{\star }} \not g\,_{{\theta _A}{\theta _B}} Y^{\theta ^{\star }}, \end{aligned}$$

(E5)

where Y is as in Sect. 2.2.6. Only $(\pi ^Y)^{vv}, (\pi ^Y)^{uu}$ and $(\pi ^Y)^{v\theta _C}$ are zero and all other terms will contribute to the error terms. In particular, we obtain a $(\partial _u \partial _v \psi )$-term as can be seen from (D2) together with (E3E4). We control this by using the wave equation itself. See Appendix F, Eq. (F5) and substitute the expression, so that as a general structure we get all first- to third-order terms except the $(\partial _u \partial _v \psi )$-term, namely

$$\begin{aligned}{}[\Box _g, Y^k \psi ]= & {} \sum \limits ^4_{i=1} \sum \limits ^2_{l=0} \sum \limits ^2_{m=0} E^{lm}_i(\partial _{{\eta }_i} \partial _{\tilde{\phi }}^l\partial _{\theta ^{\star }}^m \psi ), \qquad l+m \le k, \end{aligned}$$

(E6)

where $\eta _{i}= u,v, {\tilde{\phi }},{\theta ^{\star }}$ and all coefficients $E^{lm}_i$ are bounded, as can be seen with the help of (D2) and Sect. 2.2.5.

Appendix F: The Wave Equation and Higher Derivatives

The wave on fixed background is given by

$$\begin{aligned} \Box _g \psi =\frac{1}{\sqrt{-g}} \frac{\partial }{\partial x^{\mu }}\left( g^{\mu \nu }\sqrt{-g}\frac{\partial \psi }{\partial x^{\nu }}\right) , \end{aligned}$$

(F1)

where $\sqrt{-g}=2\Omega ^2L\sin \theta$. Therefore, the wave equation in Eddington–Finkelstein-like coordinates on fixed Kerr background yields

$$\begin{aligned} \Box _g \psi= & {} \frac{1}{2\Omega ^2}\left[ -\frac{\partial _u |L\sin \theta |}{|L\sin \theta |}\partial _v \psi -\frac{\partial _v |L\sin \theta |}{|L\sin \theta |}\partial _u \psi -\frac{\partial _u |b^{\tilde{\phi }}L\sin \theta |}{|L\sin \theta |}\partial _{\tilde{\phi }} \psi \right] \nonumber \\&-\frac{1}{\Omega ^2}\partial _u\partial _v \psi -\frac{b^{\tilde{\phi }}}{\Omega ^2}\partial _u\partial _{\tilde{\phi }} \psi +\not \Delta \,\psi \nonumber \\&+\frac{2\partial _{\theta ^{\star }}\Omega }{\Omega }\left[ \frac{R^2}{L^2}\partial _{\theta ^{\star }}\psi -\left( \frac{\partial h}{\partial \theta ^{\star }}\right) \frac{R^2}{L^2}\partial _{\tilde{\phi }}\psi \right] =0, \end{aligned}$$

(F2)

where

$$\begin{aligned} \not \Delta \,\psi =\frac{1}{\sqrt{-\not g\,}} \frac{\partial }{\partial x^{{\theta _C}}}\left( \not g\,^{{\theta _C} {\theta _D}}\sqrt{-\not g\,}\frac{\partial \psi }{\partial x^{{\theta _D}}}\right) , \end{aligned}$$

(F3)

with $\sqrt{-\not g\,}=L\sin \theta$ and $C,D=1,2$ and $\theta _1=\theta ^{\star }$, $\theta _2=\tilde{\phi }$. So we get

$$\begin{aligned} \not \Delta \,\psi= & {} \frac{\partial _{\theta ^{\star }}|L\sin \theta |}{|L\sin \theta |}\left[ \frac{R^2}{L^2}\partial _{\theta ^{\star }}\psi -\left( \frac{\partial h}{\partial \theta ^{\star }}\right) \frac{R^2}{L^2}\partial _{\tilde{\phi }}\psi \right] \nonumber \\&+\partial _{\theta ^{\star }}\left( \frac{R^2}{L^2}\right) \partial _{\theta ^{\star }}\psi -\partial _{\theta ^{\star }}\left[ \left( \frac{\partial h}{\partial \theta ^{\star }}\right) \frac{R^2}{L^2}\right] \partial _{\tilde{\phi }}\psi \nonumber \\&+\frac{R^2}{L^2}\partial _{\theta ^{\star }}\partial _{\theta ^{\star }}\psi -2\left( \frac{\partial h}{\partial \theta ^{\star }}\right) \frac{R^2}{L^2}\partial _{\theta ^{\star }}\partial _{\tilde{\phi }}\psi \nonumber \\&+\left( \frac{1}{R^2\sin \theta }+\left( \frac{\partial h}{\partial \theta ^{\star }}\right) \frac{R^2}{L^2}\right) \partial _{\tilde{\phi }}\partial _{\tilde{\phi }}\psi . \end{aligned}$$

(F4)

Solving (F2) to

$$\begin{aligned} \frac{1}{\Omega ^2}\partial _u\partial _v \psi +\frac{b^{\tilde{\phi }}}{\Omega ^2}\partial _u\partial _{\tilde{\phi }} \psi= & {} \frac{1}{2\Omega ^2}\left[ -\frac{\partial _u |L\sin \theta |}{|L\sin \theta |}\partial _v \psi -\frac{\partial _v |L\sin \theta |}{|L\sin \theta |}\partial _u \psi \right. \nonumber \\&\left. -\frac{\partial _u |b^{\tilde{\phi }}L\sin \theta |}{|L\sin \theta |}\partial _{\tilde{\phi }} \psi \right] \nonumber \\&+\not \Delta \,\psi +\frac{2\partial _{\theta ^{\star }}\Omega }{\Omega } \left[ \frac{R^2}{L^2}\partial _{\theta ^{\star }}\psi -\left( \frac{\partial h}{\partial \theta ^{\star }}\right) \frac{R^2}{L^2}\partial _{\tilde{\phi }}\psi \right] ,\nonumber \\ \end{aligned}$$

(F5)

will enable us to control the mixed term $\partial _u\partial _v \psi$ by using the right-hand side of the equation, once it is shown that all coefficients are bounded. This will be needed in order to estimate error terms.