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The Characteristic Gluing Problem for the Einstein Vacuum Equations: Linear and Nonlinear Analysis

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Abstract

This is the second paper in a series of papers addressing the characteristic gluing problem for the Einstein vacuum equations. We solve the codimension-10 characteristic gluing problem for characteristic data which are close to the Minkowski data. We derive an infinite-dimensional space of gauge-dependent charges and a 10-dimensional space of gauge-invariant charges that are conserved by the linearized null constraint equations and act as obstructions to the gluing problem. The gauge-dependent charges can be matched by applying angular and transversal gauge transformations of the characteristic data. By making use of a special hierarchy of radial weights of the null constraint equations, we construct the null lapse function and the conformal geometry of the characteristic hypersurface, and we show that the aforementioned charges are in fact the only obstructions to the gluing problem. Modulo the gauge-invariant charges, the resulting solution of the null constraint equations is \(C^{m+2}\) for any specified integer \(m\ge 0\) in the tangential directions and \(C^2\) in the transversal directions to the characteristic hypersurface. We also show that higher-order (in all directions) gluing is possible along bifurcated characteristic hypersurfaces (modulo the gauge-invariant charges).

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Acknowledgements

S.A. acknowledges support through the NSERC grant 502581 and the Ontario Early Researcher Award. S.C. acknowledges support through the NSF grant DMS-1439786 of the Institute for Computational and Experimental Research in Mathematics (ICERM). I.R. acknowledges support through NSF grants DMS-2005464, DMS-1709270 and a Simons Investigator Award.

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Correspondence to Stefan Czimek.

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Communicated by Mihalis Dafermos.

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Appendices

Appendix A: Perturbations of Sphere Data

In this section, we prove Proposition 2.21; that is, we show that

$$\begin{aligned} \begin{aligned} {{\mathcal {P}}}: \, {\mathcal {X}}^+(\tilde{\underline{{\mathcal {H}}}}_{[-\delta ,\delta ],2}) \times {\mathcal {Y}}_f \times {\mathcal {Y}}_q&\rightarrow {\mathcal {X}}(S_{0,2}), \\ (\tilde{{\underline{x}}},f,q)&\mapsto x_{0,2}:= {{\mathcal {P}}}_{f,q}(\tilde{{\underline{x}}}), \end{aligned} \end{aligned}$$

is well defined and smooth in an open neighborhood of \((\tilde{{\underline{x}}},f,q)=(\underline{{\mathfrak {m}}},0,0)\), and satisfies the estimate

$$\begin{aligned} \begin{aligned} \Vert {{\mathcal {P}}}_{f,q}(\tilde{{\underline{x}}}) - \tilde{{\underline{x}}}_{0,2} \Vert _{{\mathcal {X}}(S_{0,2})} \lesssim \Vert f \Vert _{{\mathcal {Y}}_{f}} + \Vert q \Vert _{{\mathcal {Y}}_q} + \Vert \tilde{{\underline{x}}}-\underline{{\mathfrak {m}}} \Vert _{{\mathcal {X}}^+(\tilde{\underline{{\mathcal {H}}}}_{[-\delta ,\delta ],2})}, \end{aligned} \end{aligned}$$

where we denoted \(\tilde{{\underline{x}}}_{0,2}:= \tilde{{\underline{x}}}\vert _{S_{0,2}}\).

In Sect. A.1, we derive explicit expressions for the sphere data \({{\mathcal {P}}}_{f,0}(\tilde{{\underline{x}}})\). In Sect. A.2, we prove Proposition 2.21; that is, we analyze \({{\mathcal {P}}}_{f,q}(\tilde{{\underline{x}}})\).

1.1 Explicit Formulas for Transversal Sphere Perturbations

In the following, we rigorously set up transversal perturbations \({{\mathcal {P}}}_{f,0}\) and write out explicit formulas for the resulting sphere data. In Sect. A.1.1, we recapitulate the null geometry setting. In Sect. A.1.2, we define sphere perturbations and analyze metric coefficients. In Sects. A.1.3 and A.1.4, we analyze Ricci coefficients and null curvature components, respectively.

1.1.1 Null Geometry

First we recall the null geometry setup. Let \({{\tilde{S}}}\) be a spacelike 2-sphere in a spacetime \(({{\mathcal {M}}},\textbf{g})\). Let \(({\tilde{u}},{\tilde{v}},\tilde{\theta }^1,\tilde{\theta }^2)\) be a local double null coordinate system around \({{\tilde{S}}}\), that is,

$$\begin{aligned} \begin{aligned} \textbf{g}&= - 4 \tilde{\Omega }^2 d{\tilde{u}} d{\tilde{v}} + \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}_{CD} (d\tilde{\theta }^C - {\tilde{b}}^C d{\tilde{v}})(d\tilde{\theta }^D - {\tilde{b}}^D d{\tilde{v}}), \end{aligned} \end{aligned}$$
(A.1)

such that \({{\tilde{S}}} = {\tilde{S}}_{0,2}:= \{{{\tilde{u}}}=0, {{\tilde{v}}}=2\}.\) We recall the following standard notation, see, for example, Sect. 1 of [15].

  • The geodesic null vectorfields are defined by

    $$\begin{aligned} \begin{aligned} \tilde{{\,{\underline{L}}}}' := -2 \textbf{D}{\tilde{v}}, \,\, {\tilde{L}}' := -2 \textbf{D}{\tilde{u}}, \end{aligned} \end{aligned}$$
    (A.2)

    where \(\textbf{D}\) denotes the covariant derivative on \(({{\mathcal {M}}},\textbf{g})\).

  • The normalized null vectorfields are defined by

    $$\begin{aligned} \begin{aligned} \widehat{{\tilde{L}}} := \Omega {\tilde{L}}', \,\, \widehat{\tilde{{\,{\underline{L}}}}} := \Omega \tilde{{\,{\underline{L}}}}'. \end{aligned} \end{aligned}$$
  • The equivariant null vectorfields are defined by

    $$\begin{aligned} \begin{aligned} {{\tilde{L}}} := \tilde{\Omega }^2 L' , \,\, {\tilde{{\,{\underline{L}}}}} := \tilde{\Omega }^2 {\,{\underline{L}}}'. \end{aligned} \end{aligned}$$
    (A.3)
  • The Ricci coefficients are defined with respect to the above vectorfields as follows,

    $$\begin{aligned} \begin{aligned} \tilde{\chi }_{AB}&:= \textbf{g}(\textbf{D}_{{{\tilde{A}}}} \widehat{{{\tilde{L}}}},{\partial }_{{\tilde{B}}}),&\tilde{{{\underline{\chi }}}}_{AB}&:= \textbf{g}(\textbf{D}_{{\tilde{A}}} \widehat{{\tilde{{\,{\underline{L}}}}}},{\partial }_{{\tilde{B}}}),&\tilde{\zeta }_A&:= \frac{1}{2}\textbf{g}(\textbf{D}_{{\tilde{A}}} \widehat{{{\tilde{L}}}},\widehat{{\tilde{{\,{\underline{L}}}}}}), \\ {\tilde{\eta }}&:= {\tilde{\zeta }} + {\tilde{{d \hspace{-5.11108pt}/\,}}} \log {\tilde{\Omega }},&{\tilde{\omega }}&:= {{\tilde{L}}} \log {\tilde{\Omega }},&\tilde{{\underline{\omega }}}&:= \tilde{{\,{\underline{L}}}} \log {\tilde{\Omega }}, \end{aligned} \end{aligned}$$
    (A.4)

    where \(\tilde{{d \hspace{-5.11108pt}/\,}}\) denotes the exterior derivative on spheres \({\tilde{S}}_{u,v}\).

We have the following practical lemma, see, for example, [15].

Lemma A.1

(Properties of double null coordinates). The following holds.

  1. (1)

    The inverse \(\textbf{g}^{-1}\) of (A.1) is given by

    $$\begin{aligned} \begin{aligned} \textbf{g}^{-1}&= -\frac{1}{2\tilde{\Omega }^2} \left( {\partial }_{{{\tilde{u}}}} \otimes {\partial }_{{{\tilde{v}}}} + {\partial }_{{{\tilde{v}}}} \otimes {\partial }_{{{\tilde{u}}}}\right) -\frac{{\tilde{b}}^C}{2\tilde{\Omega }^2 } \left( {\partial }_{{{\tilde{u}}}} \otimes {\partial }_{{{\tilde{C}}}} + {\partial }_{{{\tilde{C}}}} \otimes {\partial }_{{{\tilde{u}}}}\right) + \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{AB} {\partial }_{{{\tilde{A}}}} \otimes {\partial }_{{{\tilde{B}}}}. \end{aligned} \end{aligned}$$
    (A.5)

    Specifically,

    $$\begin{aligned} \begin{aligned} \textbf{g}^{{{\tilde{v}}} {{\tilde{v}}}}=\textbf{g}^{{{\tilde{v}}} {{\tilde{A}}}} =0. \end{aligned} \end{aligned}$$
    (A.6)
  2. (2)

    It holds that \(\textbf{g}\left( L',{\,{\underline{L}}}'\right) =-2\tilde{\Omega }^{-2}\), and

    $$\begin{aligned} \begin{aligned} {{\tilde{L}}} = {\partial }_{{{\tilde{v}}}} + {\tilde{b}}^A {\partial }_{{\tilde{\theta }}^A}, \,\, {\tilde{{\,{\underline{L}}}}} = {\partial }_{{{\tilde{u}}}}. \end{aligned} \end{aligned}$$
    (A.7)
  3. (3)

    It holds that for \(A=1,2\),

    $$\begin{aligned} \begin{aligned} {\partial }_{{{\tilde{u}}}} {\tilde{b}}^A = 4\tilde{\Omega }^2 \tilde{\zeta }^A. \end{aligned} \end{aligned}$$
    (A.8)
  4. (4)

    It holds that

    $$\begin{aligned} \begin{aligned} \Gamma ^{{{\tilde{v}}}}_{{{\tilde{u}}} {{\tilde{u}}}}&= \Gamma ^{{{\tilde{v}}}}_{{{\tilde{u}}} {{\tilde{v}}}} = \Gamma ^{{{\tilde{v}}}}_{{{\tilde{u}}} {{\tilde{A}}}} =0,&\Gamma ^{{{\tilde{v}}}}_{{{\tilde{A}}} {{\tilde{v}}}}&= {\partial }_{{{\tilde{A}}}} \log {\tilde{\Omega }} - \tilde{\zeta }_A - \frac{1}{2{\tilde{\Omega }}} \tilde{{{\underline{\chi }}}}_{AB} {\tilde{b}}^B,&\Gamma ^{{{\tilde{v}}}}_{{{\tilde{A}}} {{\tilde{B}}}}&= \frac{1}{2{\tilde{\Omega }}} \tilde{{{\underline{\chi }}}}_{AB}, \end{aligned} \end{aligned}$$
    (A.9)

    where the Christoffel symbols are defined by \(\Gamma ^\gamma _{\mu \nu }:= \frac{1}{2}\textbf{g}^{\gamma {\alpha }} \left( {\partial }_\mu \textbf{g}_{{\alpha }\nu }+{\partial }_\nu \textbf{g}_{{\alpha }\mu }\right. \left. -{\partial }_{\alpha }\textbf{g}_{\mu \nu }\right) \).

1.1.2 Definition of u on \(\tilde{\underline{{{\mathcal {H}}}}}_2\) and Analysis of Foliation Geometry

In the following, we change \({{\tilde{u}}}\) to u on \(\tilde{\underline{{\mathcal {H}}}}_2:=\{{{\tilde{v}}}=2 \}\) and analyze how the foliation geometry of the resulting local double null coordinates \((u, v, \theta ^1,\theta ^2)\) (with \(v={{\tilde{v}}}\) on \({{\mathcal {M}}}\)) relates to the foliation geometry of the local double null coordinates \(({{\tilde{u}}}, {{\tilde{v}}}, {\tilde{\theta }}^1,{\tilde{\theta }}^2)\).

For a given scalar function \(f=f(u,\theta ^1,\theta ^2)\), define \((u,\theta ^1,\theta ^2)\) on \(\tilde{\underline{{\mathcal {H}}}}_2\) by

$$\begin{aligned} \begin{aligned} {\tilde{u}}=u+f(u,\theta ^1,\theta ^2), \,\, {\tilde{\theta }}^1=\theta ^1, \,\, {\tilde{\theta }}^2=\theta ^2. \end{aligned} \end{aligned}$$
(A.10)

For f sufficiently small, \((u,\theta ^1, \theta ^2)\) are a coordinate system on \(\tilde{\underline{{\mathcal {H}}}}_2\) and we have that

$$\begin{aligned} \begin{aligned} {\partial }_u = \left( 1+{\partial }_u f\right) {\partial }_{{{\tilde{u}}}}, \,\, {\partial }_{\theta ^A} = {\partial }_{{\tilde{\theta }}^A} + ({\partial }_{\theta ^A} f) {\partial }_{{{\tilde{u}}}}, \,\, {\partial }_{\theta ^A} f =\left( 1+{\partial }_u f\right) {\partial }_{{\tilde{\theta }}^A} f . \end{aligned} \end{aligned}$$
(A.11)

In accordance with (A.2) and (A.7), define on \(\tilde{\underline{{\mathcal {H}}}}_2\)

$$\begin{aligned} \begin{aligned} {\,{\underline{L}}}:= {\partial }_u, \,\, {\,{\underline{L}}}' := -2 \textbf{D}{\tilde{v}} = \tilde{{\,{\underline{L}}}}', \end{aligned} \end{aligned}$$
(A.12)

and define in accordance with (A.3) the null lapse \(\Omega \) on \(\tilde{\underline{{\mathcal {H}}}}_2\) through the relation

$$\begin{aligned} \begin{aligned} {\,{\underline{L}}}= {\Omega }^2 {\,{\underline{L}}}'. \end{aligned} \end{aligned}$$
(A.13)

We can relate the foliation geometry of \((u,\theta ^1,\theta ^2)\) to the geometry of \(({{\tilde{u}}}, {\tilde{\theta }}^1, {\tilde{\theta }}^2)\) as follows.

  1. (1)

    We explicitly calculate \(\Omega \) on \(\tilde{\underline{{\mathcal {H}}}}_2\) as follows. Using (A.3), (A.10), (A.11) and (A.12), it holds that on \(\{{{\tilde{v}}} =2\}\),

    $$\begin{aligned} \begin{aligned} {\,{\underline{L}}}= \left( 1+{\partial }_u f\right) {\tilde{{\,{\underline{L}}}}} = \left( 1+{\partial }_u f\right) \tilde{\Omega }^{2}\tilde{{\,{\underline{L}}}}' = \left( 1+{\partial }_u f\right) \tilde{\Omega }^{2} {{\,{\underline{L}}}}', \end{aligned} \end{aligned}$$
    (A.14)

    from which we conclude by (A.13) that on \(\tilde{\underline{{\mathcal {H}}}}_2\),

    $$\begin{aligned} \begin{aligned} \Omega ^2 = \tilde{\Omega }^2 \left( 1+{\partial }_u f\right) . \end{aligned} \end{aligned}$$
    (A.15)
  2. (2)

    By (A.11), it follows that the induced metric \({g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}\) on level sets of u on \(\tilde{\underline{{\mathcal {H}}}}_2\) is given for \(A,B=1,2\) by

    $$\begin{aligned} \begin{aligned} {g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}_{AB} := \textbf{g}\left( {\partial }_A, {\partial }_B\right) = \textbf{g}\left( {\partial }_{{{\tilde{A}}}}, {\partial }_{{{\tilde{B}}}}\right) = \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}_{AB}. \end{aligned} \end{aligned}$$
    (A.16)

    This implies further that

    $$\begin{aligned} \begin{aligned} {g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}^{AB} = \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{AB}. \end{aligned} \end{aligned}$$
    (A.17)

    We remark that in explicit notation, (A.15) and (A.16) are

    $$\begin{aligned} \begin{aligned} \Omega ^2(u,\theta ^1,\theta ^2)&= \left( 1+({\partial }_u f)(u,\theta ^1,\theta ^2)\right) \tilde{\Omega }^2(u+f(u,\theta ^1,\theta ^2),\theta ^1,\theta ^2),\\ {g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}_{AB}(u,\theta ^1,\theta ^2)&= \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}_{AB}\left( u+f(u,\theta ^1,\theta ^2),\theta ^1,\theta ^2\right) . \end{aligned} \end{aligned}$$
  3. (3)

    The vectorfield \({\,{\underline{L}}}\) and the scalar function \(\Omega \) uniquely determine the null vectorfield L on \(\{{{\tilde{v}}}=2\}\) defined by

    $$\begin{aligned} \begin{aligned} \textbf{g}\left( L,{\,{\underline{L}}}\right) = -2 \Omega ^2, \,\, \textbf{g}\left( L,{\partial }_{\theta ^1}\right) =\textbf{g}\left( L,{\partial }_{\theta ^2}\right) =0. \end{aligned} \end{aligned}$$
    (A.18)

    An explicit calculation shows that L is given by

    $$\begin{aligned} \begin{aligned} L&= \left( \tilde{\Omega }^2\vert \nabla \hspace{-7.22214pt}/\ f \vert _{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}^2 \right) \tilde{{\,{\underline{L}}}} + {\tilde{L}} + \left( 2\tilde{\Omega }^2 \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{AC}{\partial }_C f \right) {\partial }_{{\tilde{\theta }}^A}. \end{aligned} \end{aligned}$$
    (A.19)

    where \(\vert \nabla \hspace{-7.22214pt}/\ f \vert _{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}^2:= \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{AB}{\partial }_Af {\partial }_B f\). We define further \({\widehat{L}}:= \Omega ^{-1} L\).

1.1.3 Analysis of Ricci Coefficients on \(\tilde{\underline{{\mathcal {H}}}}_2\)

The Ricci coefficients with respect to \(\left( {{\widehat{L}}}, {\widehat{{\,{\underline{L}}}}}\right) \) are defined as follows,

$$\begin{aligned} \begin{aligned} \chi _{AB}&:= \textbf{g}(\textbf{D}_{A} {\widehat{L}},{\partial }_B),&{{\underline{\chi }}}_{AB}&:= \textbf{g}(\textbf{D}_A \widehat{{\,{\underline{L}}}},{\partial }_B),&\zeta _A&:= \frac{1}{2}\textbf{g}(\textbf{D}_A {\widehat{L}},\widehat{{\,{\underline{L}}}}), \\ \eta&:= \zeta + {d \hspace{-5.11108pt}/\,}\log \Omega ,&\omega&:= L \log \Omega ,&{\underline{\omega }}&:= {\,{\underline{L}}}\log \Omega . \end{aligned} \end{aligned}$$

We analyze the Ricci coefficients in the order \(\left( {\underline{\omega }}, {{\underline{\chi }}}, \omega , \zeta , \eta , \chi , {\underline{D}}{\underline{\omega }}, D\omega \right) .\)

Analysis of \({\underline{\omega }}\). On the one hand, we have by (A.12) that

$$\begin{aligned} \begin{aligned} {\underline{\omega }}:= {\,{\underline{L}}}\log \Omega = \Omega ^{-1} {\partial }_u \Omega = \frac{1}{2\Omega ^2} {\partial }_u \left( \Omega ^2\right) . \end{aligned} \end{aligned}$$

On the other hand, we have by (A.11) and (A.15) that

$$\begin{aligned} \begin{aligned} {\partial }_u \left( \Omega ^2\right)&= {\partial }_u \left( \tilde{\Omega }^2\left( 1+{\partial }_u f\right) \right) = 2\tilde{\Omega } {\partial }_{{{\tilde{u}}}} \tilde{\Omega } \left( 1+{\partial }_u f\right) ^2 + \tilde{\Omega }^2 {\partial }_u^2 f. \end{aligned} \end{aligned}$$

Combining the above two and using (A.4), it follows that

$$\begin{aligned} \begin{aligned} {\underline{\omega }}&= \frac{1}{2\Omega ^2} \left( 2\tilde{\Omega } {\partial }_{{{\tilde{u}}}} \tilde{\Omega } \left( 1+{\partial }_u f\right) ^2 + \tilde{\Omega }^2 {\partial }_u^2 f\right) = \tilde{{\underline{\omega }}} \left( 1+{\partial }_u f\right) + \frac{1}{2}\frac{\tilde{\Omega }^2}{\Omega ^2} {\partial }_u^2 f. \end{aligned} \end{aligned}$$
(A.20)

Analysis of \({{\underline{\chi }}}\). By explicit computation, we have that

$$\begin{aligned} \begin{aligned} {{\underline{\chi }}}_{AB} := \textbf{g}(\textbf{D}_A \widehat{{\,{\underline{L}}}},{\partial }_B) =\Omega ^{-1} \left( 1+{\partial }_u f\right) \tilde{\Omega } \tilde{{{\underline{\chi }}}}_{AB}, \end{aligned} \end{aligned}$$
(A.21)

where we used that

$$\begin{aligned} \begin{aligned} \textbf{g}\left( \textbf{D}_{{\partial }_{{{\tilde{u}}}}} {\partial }_{{{\tilde{u}}}}, {\partial }_{{{\tilde{B}}}}\right) = \tilde{\Omega }^4 \textbf{g}\left( \textbf{D}_{\tilde{{\,{\underline{L}}}}'} \tilde{{\,{\underline{L}}}}', {\partial }_{{{\tilde{B}}}}\right) = 0. \end{aligned} \end{aligned}$$

We can separate (A.21) into

$$\begin{aligned} \begin{aligned} \Omega {\textrm{tr}{{\underline{\chi }}}}= \left( 1+{\partial }_u f\right) \tilde{\Omega }{\textrm{tr}\tilde{{{\underline{\chi }}}}}, \,\, {\underline{{{\widehat{\chi }}}}}_{AB} = \frac{\tilde{\Omega }}{\Omega } \left( 1+{\partial }_u f\right) \tilde{{\underline{{{\widehat{\chi }}}}}}_{AB}, \end{aligned} \end{aligned}$$
(A.22)

that is, in explicit notation,

$$\begin{aligned} \begin{aligned} \left( \Omega {\textrm{tr}{{\underline{\chi }}}}\right) (u,\theta ^1,\theta ^2)&= \left( 1+{\partial }_u f(u,\theta ^1,\theta ^2)\right) \tilde{\Omega } \textrm{tr}\tilde{{{\underline{\chi }}}}(u+f(u,\theta ^1,\theta ^2),\theta ^1,\theta ^2),\\ {\underline{{{\widehat{\chi }}}}}_{AB}(u,\theta ^1,\theta ^2)&=\frac{\tilde{\Omega }}{\Omega }(u+f(u,\theta ^1,\theta ^2),\theta ^1,\theta ^2) \left( 1+{\partial }_u f(u,\theta ^1,\theta ^2)\right) \\&\quad \tilde{{\underline{{{\widehat{\chi }}}}}}_{AB}(u+f(u,\theta ^1,\theta ^2),\theta ^1,\theta ^2). \end{aligned} \end{aligned}$$

Analysis of \(\omega \). We calculate

$$\begin{aligned} \begin{aligned} \omega := L \log \Omega \end{aligned} \end{aligned}$$

as follows. First, by construction of the double coordinates \((u, {{\tilde{v}}}, \theta ^1, \theta ^2)\), see (A.18) and (A.12), \(\Omega \) is defined on \({{\mathcal {M}}}\) through

$$\begin{aligned} \begin{aligned} L'({{\tilde{v}}}) = \Omega ^{-2}. \end{aligned} \end{aligned}$$
(A.23)

In particular, this implies with the geodesic equation satisfied by \(L'\) that

$$\begin{aligned} \begin{aligned} L'\left( \Omega ^{-2}\right) = L'\left( L'({{\tilde{v}}}) \right) =\textbf{D}_{L'}\left( \textbf{D}_{L'} {{\tilde{v}}}\right) = \textbf{D}_{\underbrace{\textbf{D}_{L'}L'}_{=0}} {{\tilde{v}}} + \textbf{D}_{L'} \textbf{D}_{L'} {{\tilde{v}}}= \Omega ^{-4} \textbf{D}_L \textbf{D}_L {{\tilde{v}}}, \end{aligned} \end{aligned}$$
(A.24)

where we note that \(\textbf{D}\textbf{D}{{\tilde{v}}}\) is the covariant Hessian.

Second, we have the algebraic relation

$$\begin{aligned} \begin{aligned} L'\left( \Omega ^{-2}\right) = -\frac{2}{\Omega ^3} L' \left( \Omega \right) . \end{aligned} \end{aligned}$$
(A.25)

By (A.24) and (A.25), we get that

$$\begin{aligned} \begin{aligned} \omega = L \left( \log \Omega \right) = \Omega L' \left( \Omega \right) = -\frac{\Omega ^4}{2} L' \left( \Omega ^{-2}\right) = -\frac{1}{2} \textbf{D}_{L}\textbf{D}_L {{\tilde{v}}}. \end{aligned} \end{aligned}$$
(A.26)

Plugging (A.19) into (A.26) and using (A.3), (A.7), we get that

$$\begin{aligned} \begin{aligned} \omega&= -\frac{1}{2} \left( \tilde{\Omega }^2 \vert \nabla \hspace{-7.22214pt}/\ f \vert ^2_{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}\right) ^2 \textbf{D}_{{{\tilde{u}}} }\textbf{D}_{{{\tilde{u}}}} {{\tilde{v}}} \underbrace{-\frac{1}{2} \textbf{D}_{{{\tilde{L}}}} \textbf{D}_{{{\tilde{L}}}} {\tilde{v}}}_{= {\tilde{\omega }}} -2\tilde{\Omega }^4 \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{AB} \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{CD}({\partial }_B f)({\partial }_C f) \textbf{D}_{{{\tilde{A}}}} \textbf{D}_{{{\tilde{D}}}} {{\tilde{v}}} \\&- \left( \tilde{\Omega }^2 \vert \nabla \hspace{-7.22214pt}/\ f \vert ^2_{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}\right) \textbf{D}_{{{\tilde{u}}}} \textbf{D}_{{\partial }_{{{\tilde{v}}}} +{\tilde{b}}^C {\partial }_{{{\tilde{C}}}} } {{\tilde{v}}} - \left( \tilde{\Omega }^2 \vert \nabla \hspace{-7.22214pt}/\ f \vert ^2_{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}\right) \left( 2\tilde{\Omega }^2 \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{AB}{\partial }_B f \right) \textbf{D}_{{{\tilde{u}}}} \textbf{D}_{{{\tilde{A}}}} {{\tilde{v}}} \\&- \left( 2\tilde{\Omega }^2 \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{AB}{\partial }_B f \right) \textbf{D}_{{\partial }_{{{\tilde{v}}}} +{\tilde{b}}^C {\partial }_{{{\tilde{C}}}}}\textbf{D}_{{{\tilde{A}}}} {{\tilde{v}}}. \end{aligned} \end{aligned}$$
(A.27)

Here, the Hessian \(\textbf{D}\textbf{D}{{\tilde{v}}}\) is given in coordinates \(\mu , \nu \in \{{{\tilde{u}}}, {{\tilde{v}}}, {\tilde{\theta }}^1, {\tilde{\theta }}^2\} \) by

$$\begin{aligned} \begin{aligned} \textbf{D}_\mu \textbf{D}_\nu {{\tilde{v}}}&= {\partial }_\mu {\partial }_\nu {{\tilde{v}}} - \Gamma ^{\lambda }_{\mu \nu } {\partial }_{\lambda } {{\tilde{v}}} = - \Gamma ^{{{\tilde{v}}}}_{\mu \nu }. \end{aligned} \end{aligned}$$
(A.28)

From (A.9) and (A.28), we conclude that \(\textbf{D}_{{{\tilde{u}}}}\textbf{D}_{{{\tilde{u}}}}{{\tilde{v}}}= \textbf{D}_{{{\tilde{u}}}} \textbf{D}_{{{\tilde{v}}}}{{\tilde{v}}} = \textbf{D}_{{{\tilde{u}}}} \textbf{D}_{{{\tilde{A}}}}{{\tilde{v}}} =0\) and

$$\begin{aligned} \begin{aligned} \textbf{D}_{{{\tilde{A}}}} \textbf{D}_{{{\tilde{v}}}} {{\tilde{v}}}= - {\partial }_{{{\tilde{A}}}} \log {\tilde{\Omega }} + \tilde{\zeta }_A + \frac{1}{2{\tilde{\Omega }}} \tilde{{{\underline{\chi }}}}_{AB} {\tilde{b}}^B, \,\, \textbf{D}_{{{\tilde{A}}}} \textbf{D}_{{{\tilde{B}}}} {{\tilde{v}}}= - \frac{1}{2{\tilde{\Omega }}} \tilde{{{\underline{\chi }}}}_{AB}. \end{aligned} \end{aligned}$$
(A.29)

Plugging (A.29) into (A.27), we get that

$$\begin{aligned} \begin{aligned} \omega&= {\tilde{\omega }} +\tilde{\Omega }^3 \tilde{{{\underline{\chi }}}}^{AB}({\partial }_A f)({\partial }_B f) +\left( 2\tilde{\Omega }^2 \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{AB}{\partial }_B f \right) \left( {\partial }_{{{\tilde{A}}}} \log {\tilde{\Omega }} - \tilde{\zeta }_A \right) . \end{aligned} \end{aligned}$$
(A.30)

Analysis of \(\zeta \) and \(\eta \). Using (A.15), we have by explicit computation that

$$\begin{aligned} \begin{aligned}&\zeta _A := \frac{1}{2}\textbf{g}(\textbf{D}_A {\widehat{L}}, \widehat{{\,{\underline{L}}}}) \\ {}&\quad = \frac{\tilde{\Omega }^2}{\Omega ^2} {\partial }_A{\partial }_u f - \frac{1}{2\Omega ^2} \left( 1+{\partial }_u f\right) \textbf{g}\left( L, \textbf{D}_A {\partial }_{{{\tilde{u}}}}\right) - {\partial }_A \log \Omega . \end{aligned} \end{aligned}$$
(A.31)

By (A.11), (A.14), (A.19) and the geodesic equation for \(\tilde{{\,{\underline{L}}}}'\), we have that

$$\begin{aligned} \begin{aligned} \textbf{g}\left( L, \textbf{D}_A {\partial }_{{{\tilde{u}}}}\right)&= -2 \tilde{\Omega }^2 \left( \tilde{\zeta }_A + {\partial }_{{{\tilde{A}}}} \log \tilde{\Omega } + 2 ({\partial }_A f) {\tilde{{\underline{\omega }}}} - \tilde{\Omega }\tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{BC}({\partial }_Bf) \tilde{{{\underline{\chi }}}}_{AC}\right) . \end{aligned} \end{aligned}$$
(A.32)

Plugging (A.32) into (A.31), we get that

$$\begin{aligned} \begin{aligned} \zeta _A&= - {\partial }_A \log \Omega + \frac{\tilde{\Omega }^2}{\Omega ^2} {\partial }_A{\partial }_u f \\&+ \frac{\tilde{\Omega }^2}{\Omega ^2} \left( 1+{\partial }_u f\right) \left( \tilde{\zeta }_A + {\partial }_{{{\tilde{A}}}} \log \tilde{\Omega } + 2 ({\partial }_A f) {\tilde{{\underline{\omega }}}} - \tilde{\Omega }\tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{BC}({\partial }_Bf) \tilde{{{\underline{\chi }}}}_{AC}\right) \end{aligned} \end{aligned}$$
(A.33)

We conclude from the above that

$$\begin{aligned} \begin{aligned} \eta _A&:= \zeta _A +{\partial }_A \log \Omega \\&= \frac{\tilde{\Omega }^2}{\Omega ^2} {\partial }_A{\partial }_u f + \frac{\tilde{\Omega }^2}{\Omega ^2} \left( 1+{\partial }_u f\right) \left( \tilde{\zeta }_A + {\partial }_{{{\tilde{A}}}} \log \tilde{\Omega } + 2 ({\partial }_A f) {\tilde{{\underline{\omega }}}} - \tilde{\Omega }\tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{BC}({\partial }_Bf) \tilde{{\underline{{{\widehat{\chi }}}}}}_{AC}\right) \\&- \frac{\tilde{\Omega }^3}{2\Omega ^2} \left( 1+{\partial }_u f\right) ({\partial }_Af) \textrm{tr}\tilde{{{\underline{\chi }}}}. \end{aligned} \end{aligned}$$
(A.34)

Analysis of \(\chi \). By (A.11), (A.19) and (A.21), we have by explicit computation that

$$\begin{aligned} \begin{aligned} \chi _{AB} := \textbf{g}(\textbf{D}_{A} {\widehat{L}},{\partial }_B)&= \frac{\tilde{\Omega }^3}{\Omega } \vert \nabla \hspace{-7.22214pt}/\ f \vert _{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}^2 \tilde{{{\underline{\chi }}}}_{AB} +\frac{{\tilde{\Omega }}}{\Omega }\tilde{\chi }_{AB} + \Omega ^{-1} ({\partial }_A f) \textbf{g}\left( \textbf{D}_{{\tilde{{\,{\underline{L}}}}}} {{\tilde{L}}}, {\partial }_{{{\tilde{B}}}}\right) \\&+ \Omega ^{-1} ({\partial }_A f)({\partial }_Bf) \textbf{g}\left( \textbf{D}_{{\tilde{{\,{\underline{L}}}}}}{{\tilde{L}}}, {\tilde{{\,{\underline{L}}}}}\right) +\Omega ^{-1}({\partial }_Bf)\textbf{g}\left( \textbf{D}_{{{\tilde{A}}}} {{\tilde{L}}}, {\tilde{{\,{\underline{L}}}}}\right) \\&+ \Omega ^{-1}\textbf{g}\left( \textbf{D}_A \left( \left( 2\tilde{\Omega }^2 \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{CD}{\partial }_C f \right) {\partial }_{{{\tilde{D}}}}\right) ,{\partial }_B\right) . \end{aligned} \end{aligned}$$
(A.35)

By (A.7), (A.11) and (A.21), we have

$$\begin{aligned} \begin{aligned} \textbf{g}\left( \textbf{D}_{{\tilde{{\,{\underline{L}}}}}} {{\tilde{L}}}, {\partial }_{{{\tilde{B}}}}\right)&=2 \tilde{\Omega }^2 \tilde{\eta }_B, \,\, \textbf{g}\left( \textbf{D}_{{{\tilde{A}}}} {{\tilde{L}}}, {\tilde{{\,{\underline{L}}}}}\right) = 2\tilde{\Omega }^2 \left( \tilde{\eta }_A - 2 {\partial }_{{{\tilde{A}}}} \log \tilde{\Omega }\right) , \end{aligned} \end{aligned}$$

as well as

$$\begin{aligned} \begin{aligned} \textbf{g}\left( \textbf{D}_{{\tilde{{\,{\underline{L}}}}}}{{\tilde{L}}}, {\tilde{{\,{\underline{L}}}}}\right)&= \tilde{\Omega }^2 \textbf{g}\left( \textbf{D}_{{\tilde{{\,{\underline{L}}}}}} {{\tilde{L}}}, \tilde{{\,{\underline{L}}}}'\right) = -\tilde{\Omega }^2 \textbf{g}\left( {{\tilde{L}}}, \textbf{D}_{{\tilde{{\,{\underline{L}}}}}} \tilde{{\,{\underline{L}}}}'\right) = -\tilde{\Omega }^4 \textbf{g}({{\tilde{L}}}, \underbrace{\textbf{D}_{\tilde{{\,{\underline{L}}}}'} \tilde{{\,{\underline{L}}}}'}_{=0}) =0, \end{aligned} \end{aligned}$$

and

$$\begin{aligned}&\textbf{g}\left( \textbf{D}_A \left( 2\tilde{\Omega } \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{CD} ({\partial }_C f) {\partial }_{{{\tilde{D}}}} \right) , {\partial }_B\right) = {\partial }_A \left( 2\tilde{\Omega }^2\right) {\partial }_B f\\&\quad + \left( 2\tilde{\Omega }^2\right) \left( ({\partial }_A{\partial }_B f) + ({\partial }_Cf) \textbf{g}\left( \textbf{D}_A \left( \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{CD}{\partial }_{{{\tilde{D}}}} \right) , {\partial }_B\right) \right) , \end{aligned}$$

where on the right-hand side we can rewrite with (A.11)

$$\begin{aligned} \begin{aligned} \textbf{g}\left( \textbf{D}_A \left( \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{CD}{\partial }_{{{\tilde{D}}}} \right) , {\partial }_B\right) = -\tilde{\Gamma }^C_{AB} - ({\partial }_Af) \tilde{\Omega } \tilde{{{\underline{\chi }}}}_{BD} \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{CD}- ({\partial }_Bf) \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{CD} \tilde{\Omega } \tilde{{{\underline{\chi }}}}_{AD}, \end{aligned} \end{aligned}$$

yielding that

$$\begin{aligned} \begin{aligned}&\textbf{g}\left( \textbf{D}_A \left( 2\tilde{\Omega } \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{CD} ({\partial }_C f) {\partial }_{{{\tilde{D}}}} \right) , {\partial }_B\right) \\&= {\partial }_A \left( 2\tilde{\Omega }^2\right) {\partial }_B f + \left( 2\tilde{\Omega }^2\right) \left( ({\partial }_A{\partial }_B f) - ({\partial }_Cf) \tilde{\Gamma }^C_{AB}\right) \\&- \left( 2\tilde{\Omega }^2\right) \left( ({\partial }_Af)\tilde{\Omega }\tilde{{{\underline{\chi }}}}_{BD}\tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{CD}({\partial }_Cf) + ({\partial }_Bf)\tilde{\Omega }\tilde{{{\underline{\chi }}}}_{AD}\tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{CD}({\partial }_Cf)\right) . \end{aligned} \end{aligned}$$

Plugging the above into (A.35), we have that

$$\begin{aligned} \chi _{AB}&= \frac{\tilde{\Omega }^3}{\Omega } \vert \nabla \hspace{-7.22214pt}/\ f \vert _{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}^2 \tilde{{{\underline{\chi }}}}_{AB} +\frac{{\tilde{\Omega }}}{\Omega }\tilde{\chi }_{AB} \nonumber \\&+\frac{2\tilde{\Omega }^2}{\Omega } \left( ({\partial }_A f)\tilde{\eta }_B + ({\partial }_B f)\tilde{\eta }_A\right) +\frac{2\tilde{\Omega }^2}{\Omega } \left( {\partial }_A{\partial }_B f - \tilde{\Gamma }^C_{AB} {\partial }_C f\right) \nonumber \\&- \frac{2\tilde{\Omega }^2}{\Omega } \left( ({\partial }_Af)\tilde{\Omega }\tilde{{{\underline{\chi }}}}_{BD}\tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{CD}({\partial }_Cf) + ({\partial }_Bf)\tilde{\Omega }\tilde{{{\underline{\chi }}}}_{AD}\tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{CD}({\partial }_Cf)\right) . \end{aligned}$$
(A.36)

Analysis of \({\underline{D}}{\underline{\omega }}\). We have by explicit computation that

$$\begin{aligned} \begin{aligned} {\underline{D}}{\underline{\omega }}:= {\partial }_u \left( {\partial }_u \log \Omega \right) = \frac{{\partial }_u^3 f}{2\left( 1+{\partial }_u f\right) } - \frac{\left( {\partial }_u^2 f\right) ^2}{2\left( 1+{\partial }_u f\right) ^2}+ \tilde{{\underline{D}}} {\tilde{{\underline{\omega }}}} \left( 1+{\partial }_uf\right) ^2 + {\tilde{{\underline{\omega }}}} {\partial }_u^2 f. \end{aligned} \end{aligned}$$
(A.37)

Analysis of \(D \omega \). Using that (see also (A.23))

$$\begin{aligned} \begin{aligned} L'\left( \Omega \right) = -\frac{\Omega ^3}{2} L'\left( \Omega ^{-2}\right) = -\frac{\Omega ^3}{2} L'\left( L'({{\tilde{v}}})\right) , \end{aligned} \end{aligned}$$

we have that

$$\begin{aligned} \begin{aligned} D\omega&= L \left( L \log \Omega \right) = \Omega ^2 L' \left( \Omega L' \left( \Omega \right) \right) = \Omega ^2 L' \left( -\frac{\Omega ^4}{2}L'\left( L'({{\tilde{v}}})\right) \right) \\&= 4 \omega ^2 - \frac{\Omega ^6}{2} L'\left( L' \left( L'({{\tilde{v}}})\right) \right) . \end{aligned} \end{aligned}$$

Using that \(\textbf{D}_{L'}L' =0\), it follows further that

$$\begin{aligned} \begin{aligned} D\omega =&4 \omega ^2 - \frac{\Omega ^6}{2} \textbf{D}_{L'} \textbf{D}_{L'} \textbf{D}_{L'} {{\tilde{v}}}= 4\omega ^2 - \frac{1}{2} \textbf{D}_L \textbf{D}_L \textbf{D}_L {{\tilde{v}}}. \end{aligned} \end{aligned}$$

By (A.19), we can furthermore expand \(\textbf{D}_L \textbf{D}_L \textbf{D}_L {{\tilde{v}}}\) (explicit calculation omitted here), to conclude that \(D\omega \) can be written as a sum of products of first angular derivatives of f and (the following all with tilde) null curvature components, first derivatives of Ricci coefficients and second derivatives of metric coefficients.

Remark A.2

The only linear terms in f in the expression for \(D\omega \) are

$$\begin{aligned} \begin{aligned} 2 \left( 2\tilde{\Omega }^2 \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{AB}({\partial }_Af)\right) \textbf{D}_{{{\tilde{L}}}} \textbf{D}_{{{\tilde{L}}}} \textbf{D}_{{{\tilde{B}}}} {{\tilde{v}}} \text { and } 2\tilde{\Omega }^2 \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{EF}({\partial }_E f) \textbf{D}_{{{\tilde{F}}}} \textbf{D}_{{{\tilde{L}}}} \textbf{D}_{{{\tilde{L}}}} {{\tilde{v}}}, \end{aligned} \end{aligned}$$

and we note that at Minkowski,

$$\begin{aligned} \begin{aligned} \textbf{D}_{{{\tilde{L}}}}\textbf{D}_{{{\tilde{L}}}}\textbf{D}_{{{\tilde{L}}}}{{\tilde{v}}}= \textbf{D}_{{{\tilde{L}}}} \textbf{D}_{{{\tilde{L}}}} \textbf{D}_{{{\tilde{B}}}} {{\tilde{v}}} = \textbf{D}_{{{\tilde{F}}}} \textbf{D}_{{{\tilde{L}}}} \textbf{D}_{{{\tilde{L}}}} {{\tilde{v}}} =0. \end{aligned} \end{aligned}$$

Hence, the linearization of \(D\omega \) vanishes at Minkowski.

1.1.4 Calculation of Null Curvature Components on \(\tilde{\underline{{\mathcal {H}}}}_2\)

We recall from (2.10) the definition of the null curvature components,

$$\begin{aligned} \begin{aligned} \alpha _{AB}&:= {\textbf{R}}({\partial }_A,{\widehat{L}}, {\partial }_B, {\widehat{L}}),&\beta _A&:= \frac{1}{2}{\textbf{R}}({\partial }_A, {\widehat{L}},\widehat{{\,{\underline{L}}}},{\widehat{L}}),&{\rho }&:= \frac{1}{4} {\textbf{R}}(\widehat{{\,{\underline{L}}}}, {\widehat{L}}, \widehat{{\,{\underline{L}}}}, {\widehat{L}}), \\ \sigma \in \hspace{-8.88885pt}/\ \hspace{-2.77771pt}_{AB}&:= \frac{1}{2}{\textbf{R}}({\partial }_A,{\partial }_B,\widehat{{\,{\underline{L}}}}, {\widehat{L}}),&{\underline{\beta }}_A&:= \frac{1}{2}{\textbf{R}}({\partial }_A, \widehat{{\,{\underline{L}}}},\widehat{{\,{\underline{L}}}},{\widehat{L}}),&{\underline{\alpha }}_{AB}&:= {\textbf{R}}({\partial }_A,\widehat{{\,{\underline{L}}}}, {\partial }_B, \widehat{{\,{\underline{L}}}}). \end{aligned} \end{aligned}$$
(A.38)

Plugging (A.11), (A.14), (A.15) and (A.19), that is,

$$\begin{aligned} \begin{aligned} {\partial }_{\theta ^A}&= {\partial }_{{\tilde{\theta }}^A} + ({\partial }_{\theta ^A} f) {\partial }_{{{\tilde{u}}}},&\Omega ^2&= \tilde{\Omega }^2 \left( 1+{\partial }_u f\right) , \\ {\,{\underline{L}}}&= \left( 1+{\partial }_u f\right) {\tilde{{\,{\underline{L}}}}},&L&= \left( \tilde{\Omega }^2 \vert \nabla \hspace{-7.22214pt}/\ f\vert ^2_{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}\right) \tilde{{\,{\underline{L}}}} + {\tilde{L}} + \left( 2\tilde{\Omega }^2 \tilde{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}^{AC}{\partial }_C f \right) {\partial }_{{\tilde{\theta }}^A}, \end{aligned} \end{aligned}$$

into (A.38), it follows that the null curvature components \(\left( {\alpha }, {\beta }, {\rho }, \sigma , {\underline{\beta }}, {\underline{\alpha }}\right) \) can be expressed as sum of products of \(\left( {\tilde{{\alpha }}}, {\tilde{{\beta }}}, {\tilde{{\rho }}}, {\tilde{\sigma }}, {\tilde{{\underline{\beta }}}}, {\tilde{{\underline{\alpha }}}}\right) \) and \(f, {\partial }_A f\), \(A=1,2\), and \({\partial }_u f\).

1.2 Proof of Proposition 2.21

In this section, we prove Proposition 2.21; that is, we discuss the mapping \({{\mathcal {P}}}_{f,q}(\tilde{{\underline{x}}})\) and prove estimates.

First, recall from (2.56) that

$$\begin{aligned} \begin{aligned} {{\mathcal {P}}}_{f,q}(\tilde{{\underline{x}}}):= {{\mathcal {P}}}_{0,q}{{\mathcal {P}}}_{f,0}(\tilde{{\underline{x}}}), \end{aligned} \end{aligned}$$

and that in Sects. A.1.1-A.1.4 we discussed the explicit formulas for \({{\mathcal {P}}}_{f,0}(\tilde{{\underline{x}}})\).

Second, recall that \(q\in {\mathcal {Y}}_q= H^8(S_{0,2})\times H^8(S_{0,2})\) and that \(H^{{\tilde{m}}}(S_{0,2})\) is an algebra for integers \({\tilde{m}}\ge 2\), the following basic estimate holds, see, for example, [28]. There is a real number \(\varepsilon _0>0\) such that for all q satisfying

$$\begin{aligned} \begin{aligned} \Vert q \Vert _{{\mathcal {Y}}_q} \le \varepsilon _0, \end{aligned} \end{aligned}$$

it holds that for a tensor \(T \in H^m(S_{0,2})\) on \(S_{0,2}\) (with \(0\le m \le 6\) an integer), its pullback \(\Phi _1(q){}^{*}(T)\) under \(\Phi _1(q)\) is well defined and bounded by

$$\begin{aligned} \begin{aligned} \Vert \Phi _1(q)^{*}(T) - T \Vert _{H^m(S_{0,2})} \le&C_{\Vert T \Vert _{H^{m+1}(S_{0,2})}} \Vert q \Vert _{{\mathcal {Y}}_q}, \\ \Vert \Phi _1(q)^{*}(T) \Vert _{H^m(S_{0,2})} \le&\left( 1+ \Vert q \Vert _{{\mathcal {Y}}_q}\right) \Vert T \Vert _{H^{m}(S_{0,2})}. \end{aligned} \end{aligned}$$
(A.39)

We emphasize that the first of (A.39) as stated loses derivatives in T but the second estimate does not.

We omit the proof that the pullback under \(\Phi _1(q)\) with \(q\in {\mathcal {Y}}_q\) is a smooth mapping from tensors in \(H^m(S_{0,2})\) to tensors in \(H^m(S_{0,2})\) for integers \(0\le m \le 6\).

We are now in position to prove Proposition 2.21. The important step is to prove that \({{\mathcal {P}}}_{f,q}\) maps into \({\mathcal {X}}(S_{0,2})\). Then, the property that \({{\mathcal {P}}}_{f,q}\) is well defined and smooth near \((\tilde{{\underline{x}}},f,q)=(\underline{{\mathfrak {m}}},0,0)\) follows in a straightforward fashion. Hence, it remains to bound the sphere data

$$\begin{aligned} \begin{aligned} x_{0,2} := {{\mathcal {P}}}_{f,q}\left( \tilde{{\underline{x}}}\right) = {{\mathcal {P}}}_{0,q}{{\mathcal {P}}}_{f,0}(\tilde{{\underline{x}}}). \end{aligned} \end{aligned}$$

In the following, we prove that

$$\begin{aligned} \begin{aligned} \Vert {{\mathcal {P}}}_{f,q}(\tilde{{\underline{x}}}) - \tilde{{\underline{x}}}_{0,2} \Vert _{{\mathcal {X}}(S_{0,2})} \lesssim \Vert f \Vert _{{\mathcal {Y}}_{f}} + \Vert q \Vert _{{\mathcal {Y}}_q} + \Vert \tilde{{\underline{x}}}-\underline{{\mathfrak {m}}} \Vert _{{\mathcal {X}}^+(\tilde{\underline{{\mathcal {H}}}}_{[-\delta ,\delta ],2})}, \end{aligned} \end{aligned}$$
(A.40)

where we recall from Definition 2.5 the sphere data norm

$$\begin{aligned} \begin{aligned} \Vert x_{0,2} \Vert _{{\mathcal {X}}(S_{0,2})}&:= \Vert \Omega \Vert _{H^{6}(S_{0,2})} +\Vert {g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}\Vert _{H^{6}(S_{0,2})} + \Vert \Omega {\textrm{tr}\chi }\Vert _{H^{6}(S_{0,2})} + \Vert {{\widehat{\chi }}}\Vert _{H^{6}(S_{0,2})}\\&+ \Vert \Omega {\textrm{tr}{{\underline{\chi }}}}\Vert _{H^{4}(S_{0,2})} + \Vert {\underline{{{\widehat{\chi }}}}}\Vert _{H^{4}(S_{0,2})} + \Vert \eta \Vert _{H^{5}(S_{0,2})} \\&+ \Vert \omega \Vert _{H^{6}(S_{0,2})}+ \Vert D\omega \Vert _{H^{6}(S_{0,2})}+\Vert {\underline{\omega }}\Vert _{H^{4}(S_{0,2})} + \Vert {\underline{D}}{\underline{\omega }}\Vert _{H^{2}(S_{0,2})} \\&+ \Vert {\alpha }\Vert _{H^{6}(S_{0,2})} +\Vert {\underline{\alpha }}\Vert _{H^{2}(S_{0,2})}, \end{aligned} \end{aligned}$$

and from Definition 2.20 the sphere perturbation function norms

$$\begin{aligned} \begin{aligned} \Vert f \Vert _{{\mathcal {Y}}_f}&:= \Vert f(0) \Vert _{H^8({\mathbb {S}}^2)} + \Vert {\partial }_u f(0) \Vert _{H^6({\mathbb {S}}^2)}+\Vert {\partial }_u^2 f(0) \Vert _{H^4({\mathbb {S}}^2)}+\Vert {\partial }_u^3 f(0) \Vert _{H^2({\mathbb {S}}^2)}, \\ \Vert q \Vert _{{\mathcal {Y}}_q}&:= \Vert q_1 \Vert _{H^8({\mathbb {S}}^2)} + \Vert q_2 \Vert _{H^8({\mathbb {S}}^2)}. \end{aligned} \end{aligned}$$

Indeed, the proof is based on three ingredients. First, we work with a higher regularity \(\tilde{{\underline{x}}}\) along \(\tilde{\underline{{\mathcal {H}}}}_{[-\delta ,\delta ,2]}\) and thus higher derivatives falling on \(\tilde{{\underline{x}}}\) can still be bounded; in other words, loss of derivatives here is acceptable. Second, there are no higher derivatives that fall onto f; this is already visible from the explicit formulas of Sects. A.1.1-A.1.4. Third, the terms which need to be estimated using the first estimate of (A.39) (which loses derivatives in T) are in fact—due to ingredient (1) above—of higher regularity, and thus, this loss can be tolerated. This can again be verified by inspection of the explicit formulas of Sects. A.1.1-A.1.4.

In other words, there is a loss of derivative in the sphere perturbation mapping but it does not involve the functions f and q, and hence, our choice of function spaces (in particular, the higher regularity \(\tilde{{\underline{x}}}\)) allows to use the implicit function theorem setup around the sphere perturbation mapping nevertheless.

Let us illustrate the third ingredient by an example. Using (A.37), that is,

$$\begin{aligned} \begin{aligned} {\underline{D}}{\underline{\omega }}(u,\theta ^1,\theta ^2)&= \left( \frac{{\partial }_u^3 f}{2\left( 1+{\partial }_u f\right) } - \frac{\left( {\partial }_u^2 f\right) ^2}{2\left( 1+{\partial }_u f\right) ^2}+ \tilde{{\underline{D}}} {\tilde{{\underline{\omega }}}} \left( 1+{\partial }_uf\right) ^2 + {\tilde{{\underline{\omega }}}} {\partial }_u^2 f\right) \\&\quad \left( u+f(u,\theta ^1,\theta ^2), \theta ^1,\theta ^2\right) , \end{aligned} \end{aligned}$$

and that by definition of \({{\mathcal {P}}}_{f,q}\),

$$\begin{aligned} \begin{aligned} {\underline{D}}{\underline{\omega }}:= {\underline{D}}{\underline{\omega }}\left( {{\mathcal {P}}}_{f,0}(\tilde{{\underline{x}}})\right) \circ \Phi _1(q), \end{aligned} \end{aligned}$$

where \({\underline{D}}{\underline{\omega }}\left( {{\mathcal {P}}}_{f,0}(\tilde{{\underline{x}}})\right) \) denotes the component \({\underline{D}}{\underline{\omega }}\) of \({{\mathcal {P}}}_{f,0}(\tilde{{\underline{x}}})\), we estimate

$$\begin{aligned} \begin{aligned}&\Vert {\underline{D}}{\underline{\omega }}- \tilde{{\underline{D}}}\tilde{{\underline{\omega }}}\Vert _{H^2(S_{0,2})}\\&\quad \le \Vert \tilde{{\underline{D}}}\tilde{{\underline{\omega }}}\circ \Phi _1(q) - \tilde{{\underline{D}}}\tilde{{\underline{\omega }}}\Vert _{H^2(S_{0,2})}\\&\qquad + \left\| \left( \frac{{\partial }_u^3f}{2(1+{\partial }_uf)} - \frac{({\partial }_u^2f)^2}{2(1+{\partial }_uf)^2} + (2{\partial }_uf + ({\partial }_uf)^2)\tilde{{\underline{D}}}\tilde{{\underline{\omega }}}+\tilde{{\underline{\omega }}}{\partial }_u^2f\right) \circ \Phi _1(q)\right\| _{H^2(S_{0,2})}\\&\quad \lesssim \Vert \tilde{{\underline{x}}}-\underline{{\mathfrak {m}}} \Vert _{{\mathcal {X}}^+(\tilde{\underline{{\mathcal {H}}}}_{[-\delta ,\delta ],2})} + \Vert q \Vert _{{\mathcal {Y}}_q} + \Vert f \Vert _{{\mathcal {Y}}_f}\\&\qquad + (1+\Vert q \Vert _{{\mathcal {Y}}_q}) \left( \Vert f \Vert _{{\mathcal {Y}}_f} + \Vert \tilde{{\underline{x}}}-\underline{{\mathfrak {m}}} \Vert _{{\mathcal {X}}^+(\tilde{\underline{{\mathcal {H}}}}_{[-\delta ,\delta ],2})}\right) , \end{aligned} \end{aligned}$$

where we applied the first and second of (A.39) to the first and second line after the first equality, respectively.

This finishes the proof of Proposition 2.21.

Appendix B: Derivation of Null Transport Equations

In this section, we prove null transport equations used in this paper. In Sect. B.1, we prove the nonlinear null transport equation (2.20) for \({\underline{D}}{\underline{\omega }}\) along \({{\mathcal {H}}}\). In Sect. B.2, we derive the linearized null transport equations of Lemma 4.14 for \({\dot{{\underline{\omega }}}}, {\underline{D}}{\dot{{\underline{\omega }}}}\) and \({\dot{{\underline{\alpha }}}}\).

1.1 Derivation of Null Transport Equation for \({\underline{D}}{\underline{\omega }}\)

In this section, we prove the transport Eq. (2.20) for \({\underline{D}}{\underline{\omega }}\) along \({{\mathcal {H}}}\). We remark that in case of a geodesic foliation on \({{\mathcal {H}}}={{\mathcal {H}}}_{0,[1,2]}\), that is, \(\Omega \equiv 1\) on \({{\mathcal {H}}}\), this transport equation is readily available in [15]. We first have the following commutator identities, see Chapter 1 in [15].

Lemma B.1

(Commutator identity). Let W be an \(S_v\)-tangent tensorfield. Then,

$$\begin{aligned} {\underline{D}}D W - D {\underline{D}}W = \mathcal {L} \hspace{-5.0pt}/\ \hspace{-3.88885pt}_{4 \Omega ^2 \zeta } W. \end{aligned}$$

We are now in position to derive the null transport equation for \({\underline{D}}{\underline{\omega }}\). From the null structure equations (2.14), we have that

$$\begin{aligned} D {\underline{\omega }}= \Omega ^2\left( 2 (\eta ,{{\underline{\eta }}}) - \vert \eta \vert ^2 -{\rho }\right) . \end{aligned}$$
(B.1)

Applying the \({\underline{D}}\)-derivative to (B.1) and using (2.11), (2.14), (2.19) and Lemma B.1 with \(W={\underline{\omega }}\), we have that

$$\begin{aligned} \begin{aligned} D{\underline{D}}{\underline{\omega }}&= -4\Omega ^2 \zeta ({\underline{\omega }}) + {\underline{D}}D{\underline{\omega }}\\&= -4\Omega ^2 \zeta ({\underline{\omega }})+ {\underline{D}}\left( \Omega ^2 \left( 2(\eta ,{{\underline{\eta }}})-\vert \eta \vert ^2 - {\rho }\right) \right) \\&=-4\Omega ^2 \zeta ({\underline{\omega }}) +2 \Omega ^2 {\underline{\omega }}\left( 2(\eta ,{{\underline{\eta }}})- \vert \eta \vert ^2 - {\rho }\right) \\&\quad +\Omega ^2\left( 4\Omega {{\underline{\chi }}}(\eta ,{{\underline{\eta }}}) + 2 \left( -\Omega \left( {{\underline{\chi }}}\cdot \eta + {\underline{\beta }}\right) + 2 {d \hspace{-5.11108pt}/\,}{\underline{\omega }}, {{\underline{\eta }}}\right) +2 \left( \eta , \Omega \left( {{\underline{\chi }}}\cdot \eta + {\underline{\beta }}\right) \right) \right) \\&\quad + \Omega ^2 \left( -2\Omega {{\underline{\chi }}}(\eta , \eta ) - 2\left( -\Omega \left( {{\underline{\chi }}}\cdot \eta + {\underline{\beta }}\right) + 2{d \hspace{-5.11108pt}/\,}{\underline{\omega }}, \eta \right) \right) \\&\quad + \Omega ^2 \left( \frac{3}{2}\Omega {\textrm{tr}{{\underline{\chi }}}}\rho +\Omega \left( {{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ {\underline{\beta }}+(2\eta -\zeta ,{\underline{\beta }})+\frac{1}{2}({{\widehat{\chi }}},{\underline{\alpha }})\right) \right) \\&= -12 \Omega ^2 (\eta -{d \hspace{-5.11108pt}/\,}\log \Omega ,{d \hspace{-5.11108pt}/\,}{\underline{\omega }})+2\Omega ^2{\underline{\omega }}\left( (\eta ,-3\eta +4{d \hspace{-5.11108pt}/\,}\log \Omega )-\rho \right) \\&\quad +4\Omega ^3{{\underline{\chi }}}(\eta ,{d \hspace{-5.11108pt}/\,}\log \Omega ) +\Omega ^3 \left( {\underline{\beta }},7\eta -3{d \hspace{-5.11108pt}/\,}\log \Omega \right) \\ {}&\quad + \frac{3}{2} \Omega ^3{\textrm{tr}{{\underline{\chi }}}}{\rho }+ \Omega ^3 {{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ {\underline{\beta }}+ \frac{\Omega ^3}{2} ({{\widehat{\chi }}}, {\underline{\alpha }}). \end{aligned} \end{aligned}$$

where we used (2.8) and (2.9). This finishes the proof of (2.20).

1.2 Derivation of Transport Equations for \({\dot{{\underline{\omega }}}}\), \({\dot{{\underline{\alpha }}}}\) and \({\underline{D}}{\dot{{\underline{\omega }}}}\)

In this section, we prove Lemma 4.14. To simplify notation, we use that in Minkowski on \({{\mathcal {H}}}={{\mathcal {H}}}_{0,[1,2]}\) it holds that \(r=v\). First we recall from (4.5) that

$$\begin{aligned} \begin{aligned} {{\mathcal {Q}}}_1&:=\frac{v}{2} \left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) + \frac{{\dot{\phi }}}{v}, \\ {{\mathcal {Q}}}_2&:= v^2 \dot{(\Omega \textrm{tr}{{\underline{\chi }}})}-\frac{2}{v}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( v^2{\dot{\eta }}+\frac{v^3}{2}{d \hspace{-5.11108pt}/\,}\left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) \right) -v^2 \left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) +2v^3 {\dot{K}}, \\ {{\mathcal {Q}}}_3&:= \frac{{\dot{{\underline{{{\widehat{\chi }}}}}}}}{v} -\frac{1}{2}\left( {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}+1\right) {\dot{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}_c}}+ {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\left( {\dot{\eta }}+ \frac{v}{2}{d \hspace{-5.11108pt}/\,}\left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) \right) - v {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}\left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) . \end{aligned} \end{aligned}$$

and that by Lemmas 4.8 and 4.11,

$$\begin{aligned} \begin{aligned} D{{\mathcal {Q}}}_1&={\dot{\mathfrak {c}}}_1-D\left( \frac{1}{2v} {\dot{\mathfrak {c}}}_2\right) , \\ D{{\mathcal {Q}}}_2&= v^2 {\dot{\mathfrak {c}}}_5 -2v {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{\mathfrak {c}}}_4 -2v({\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}+1){\dot{\mathfrak {c}}}_1+ D\left( ({\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}+1){\dot{\mathfrak {c}}}_2\right) \\ {}&\quad -\frac{1}{v}({\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}+2) {\dot{\mathfrak {c}}}_2+ \frac{1}{v}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{\mathfrak {c}}}_3, \\ D{{\mathcal {Q}}}_3&= \frac{1}{v}{\dot{\mathfrak {c}}}_6 -\frac{1}{2v^2}\left( {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}+1\right) {{\dot{\mathfrak {c}}}_3}+ {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{\dot{\mathfrak {c}}}_4-{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}{\dot{\mathfrak {c}}}_1 +D\left( \frac{1}{2v}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}{\dot{\mathfrak {c}}}_2\right) \\ {}&\quad + \frac{1}{2v^2}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}{\dot{\mathfrak {c}}}_2. \end{aligned} \end{aligned}$$

Transport equation for \({\dot{{\underline{\omega }}}}\). We have by using (2.16) that

$$\begin{aligned} \begin{aligned}&D\left( {\dot{{\underline{\omega }}}}+\frac{1}{4v^2}{{\mathcal {Q}}}_2 +\frac{1}{3v} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( {\dot{\eta }}+\frac{v}{2}{d \hspace{-5.11108pt}/\,}\left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) \right) \right) \\&= {\dot{\mathfrak {c}}}_7 + {\dot{K}}+ \frac{1}{2v} \dot{(\Omega \textrm{tr}{{\underline{\chi }}})}-\frac{1}{2v} \dot{(\Omega \textrm{tr}\chi )}+ \frac{2}{v^2} {\dot{\Omega }}+ \frac{1}{4v^2}D{{\mathcal {Q}}}_2 \\&\quad -\frac{1}{2v^3} \left( v^2\dot{(\Omega \textrm{tr}{{\underline{\chi }}})}-\frac{2}{v}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( v^2{\dot{\eta }}+\frac{v^3}{2}{d \hspace{-5.11108pt}/\,}\left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) \right) -v^2 \left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) +2v^3 {\dot{K}}\right) \\&\quad -\frac{1}{v^2} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( {\dot{\eta }}+\frac{v}{2}{d \hspace{-5.11108pt}/\,}\left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) \right) \\&\quad +\frac{1}{3v^3} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{{{\widehat{\chi }}}}}+v^2{\dot{\mathfrak {c}}}_4+v^2 {d \hspace{-5.11108pt}/\,}{\dot{\mathfrak {c}}}_1+\frac{1}{2}{d \hspace{-5.11108pt}/\,}{\dot{\mathfrak {c}}}_2-D\left( \frac{v}{2}{d \hspace{-5.11108pt}/\,}{\dot{\mathfrak {c}}}_2\right) \right) \\&= {\dot{\mathfrak {c}}}_7+ \frac{1}{4v^2}D{{\mathcal {Q}}}_2+\frac{1}{3v^3} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{{{\widehat{\chi }}}}}+v^2{\dot{\mathfrak {c}}}_4+v^2 {d \hspace{-5.11108pt}/\,}{\dot{\mathfrak {c}}}_1+\frac{1}{2}{d \hspace{-5.11108pt}/\,}{\dot{\mathfrak {c}}}_2-D\left( \frac{v}{2}{d \hspace{-5.11108pt}/\,}{\dot{\mathfrak {c}}}_2\right) \right) \\&= {\dot{\mathfrak {c}}}_7+ \frac{1}{4v^2}D{{\mathcal {Q}}}_2+\frac{1}{3v^3} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{{{\widehat{\chi }}}}}+v^2{\dot{\mathfrak {c}}}_4+v^2 {d \hspace{-5.11108pt}/\,}{\dot{\mathfrak {c}}}_1 - {d \hspace{-5.11108pt}/\,}{\dot{\mathfrak {c}}}_2\right) -D\left( \frac{1}{6v^2}{\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}{\dot{\mathfrak {c}}}_2\right) \\&= {\dot{\mathfrak {c}}}_7+ \frac{1}{4}{\dot{\mathfrak {c}}}_5 + \frac{1}{4v^3} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{\mathfrak {c}}}_3 - \frac{1}{6v} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{\mathfrak {c}}}_4-\frac{1}{6v}({\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}+3) {\dot{\mathfrak {c}}}_1 -\frac{1}{12v^3} {\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}{\dot{\mathfrak {c}}}_2 + \frac{1}{3v^3} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{{{\widehat{\chi }}}}}. \end{aligned} \end{aligned}$$

Transport equation for \({\dot{{\underline{\alpha }}}}\). Using the above definition of \({{\mathcal {Q}}}_1\), \({{\mathcal {Q}}}_2\) and \({{\mathcal {Q}}}_3\), we have by the system (4.2),

$$\begin{aligned}&D\left( \frac{{\dot{{\underline{\alpha }}}}}{v} +\frac{2}{v}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {Q}}}_3 - \frac{1}{2v^2} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}{{\mathcal {Q}}}_2 - \frac{2}{v} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}\left( {\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}+2\right) {{\mathcal {Q}}}_1 \right) \\&= \frac{1}{v}{\dot{\mathfrak {c}}}_8 +\frac{2}{v}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\left( \frac{1}{v^2}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{{\underline{{{\widehat{\chi }}}}}}}-\frac{1}{2}{d \hspace{-5.11108pt}/\,}\dot{(\Omega \textrm{tr}{{\underline{\chi }}})}-\frac{1}{v}{\dot{\eta }}\right) -\frac{2}{v^2}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {Q}}}_3 +\frac{2}{v}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( D{{\mathcal {Q}}}_3\right) \\&\quad +\frac{1}{v^3}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}{{\mathcal {Q}}}_2 -\frac{1}{2v^2}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}\left( D{{\mathcal {Q}}}_2\right) +\frac{2}{v^2}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}\left( {\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}+2\right) {{\mathcal {Q}}}_1 -\frac{2}{v}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}\left( {\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}+2\right) \left( D{{\mathcal {Q}}}_1\right) \\&= \frac{1}{v}{\dot{\mathfrak {c}}}_8 +\frac{2}{v}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\left( \frac{1}{v^2}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{{\underline{{{\widehat{\chi }}}}}}}-\frac{1}{2}{d \hspace{-5.11108pt}/\,}\dot{(\Omega \textrm{tr}{{\underline{\chi }}})}-\frac{1}{v}{\dot{\eta }}\right) \\&\quad +\frac{2}{v}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( D{{\mathcal {Q}}}_3\right) -\frac{1}{2v^2}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}\left( D{{\mathcal {Q}}}_2\right) -\frac{2}{v}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}\left( {\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}+2\right) \left( D{{\mathcal {Q}}}_1\right) \\&\quad -\frac{2}{v^2}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( \frac{{\dot{{\underline{{{\widehat{\chi }}}}}}}}{v} -\frac{1}{2}\left( {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}+1\right) {\dot{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}_c}}+ {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\left( {\dot{\eta }}+ \frac{v}{2}{d \hspace{-5.11108pt}/\,}\left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) \right) \right. \\ {}&\left. - v {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}\left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) \right) \\&\quad +\frac{1}{v^3}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}\left( v^2 \dot{(\Omega \textrm{tr}{{\underline{\chi }}})}-\frac{2}{v}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( v^2{\dot{\eta }}+\frac{v^3}{2}{d \hspace{-5.11108pt}/\,}\left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) \right) \right. \\ {}&\left. -v^2 \left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) +2v^3 {\dot{K}}\right) \\&\quad +\frac{2}{v^2}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}\left( {\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}+2\right) \left( \frac{v}{2} \left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) + \frac{{\dot{\phi }}}{v}\right) , \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&-\frac{2}{3}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\left( {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*+1 + {d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\right) D\left( \frac{1}{v^3}\left( v^2{\dot{\eta }}+\frac{v^3}{2}{d \hspace{-5.11108pt}/\,}\left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) \right) \right) \\&\quad =\frac{2}{v^2} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\left( {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*+1 + {d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\right) \left( {\dot{\eta }}+\frac{v}{2}{d \hspace{-5.11108pt}/\,}\left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) \right) \\&\qquad -\frac{2}{3v^3} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\left( {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*+1 + {d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\right) \left( {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{{{\widehat{\chi }}}}}+ v^2{\dot{\mathfrak {c}}}_4+v^2 {d \hspace{-5.11108pt}/\,}{\dot{\mathfrak {c}}}_1+\frac{1}{2}{d \hspace{-5.11108pt}/\,}{\dot{\mathfrak {c}}}_2-D\left( \frac{v}{2}{d \hspace{-5.11108pt}/\,}{\dot{\mathfrak {c}}}_2\right) \right) . \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\left( {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*+ 1 + {d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\right) {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}D\left( \frac{1}{v}{\dot{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}_c}}\right) \\&\quad = -\frac{1}{v^2}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\left( {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*+ 1 + {d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\right) {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}_c}}+ \frac{1}{v} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\left( {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*+ 1 + {d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\right) {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( \frac{2}{v^2}{\dot{{{\widehat{\chi }}}}}+\frac{1}{v^2}{{\dot{\mathfrak {c}}}_3}\right) . \end{aligned} \end{aligned}$$

Summing up the above and using that, see Sect. D.3,

$$\begin{aligned} \begin{aligned} 2{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}+ {d \hspace{-5.11108pt}/\,}({\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}+2)= 2 \left( {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*+ \frac{1}{2}{d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}+ 1\right) {d \hspace{-5.11108pt}/\,}=0 \end{aligned} \end{aligned}$$

yield the transport equation for \({\dot{{\underline{\alpha }}}}\) in Lemma 4.14.

Transport equation for \({\underline{D}}{\dot{{\underline{\omega }}}}\). We have that

$$\begin{aligned} \begin{aligned}&D\left( {\underline{D}}{\dot{{\underline{\omega }}}}-\frac{1}{6v^3} \left( {\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}-3\right) {{\mathcal {Q}}}_2+ \frac{1}{2v^2} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {Q}}}_3 +\frac{1}{v^2} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}{{\mathcal {Q}}}_1\right) \\&\quad ={\dot{\mathfrak {c}}}_9 +\frac{3}{v} \left( {\dot{K}}+\frac{1}{2v}\dot{(\Omega \textrm{tr}{{\underline{\chi }}})}-\frac{1}{2v}\dot{(\Omega \textrm{tr}\chi )}+ \frac{2}{v^2}{\dot{\Omega }}\right) \\&\qquad + \frac{1}{v^2} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( \frac{1}{v^2}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{{\underline{{{\widehat{\chi }}}}}}}-\frac{1}{v}{\dot{\eta }}-\frac{1}{2}{d \hspace{-5.11108pt}/\,}\dot{(\Omega \textrm{tr}{{\underline{\chi }}})}\right) \\&\qquad -\frac{1}{6v^3}({\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}-3)D{{\mathcal {Q}}}_2 +\frac{1}{2v^2}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}D{{\mathcal {Q}}}_3 +\frac{1}{v^2}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}D{{\mathcal {Q}}}_1\\&\qquad +\frac{1}{2v^4}({\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}-3)\left( v^2\dot{(\Omega \textrm{tr}{{\underline{\chi }}})}-\frac{2}{v}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( v^2{\dot{\eta }}+\frac{v^3}{2}{d \hspace{-5.11108pt}/\,}\left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) \right) \right. \\ {}&\quad \left. -v^2 \left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) +2v^3 {\dot{K}}\right) \\&\qquad -\frac{1}{v^3}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( \frac{{\dot{{\underline{{{\widehat{\chi }}}}}}}}{v} -\frac{1}{2}\left( {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}+1\right) {\dot{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}_c}}\right. \\ {}&\quad \left. + {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\left( {\dot{\eta }}+ \frac{v}{2}{d \hspace{-5.11108pt}/\,}\left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) \right) - v {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}\left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) \right) \\&\qquad -\frac{2}{v^3} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}\left( \frac{v}{2}\left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) +\frac{{\dot{\phi }}}{v}\right) , \end{aligned} \end{aligned}$$

as well as

$$\begin{aligned} \begin{aligned}&-D\left( \frac{1}{4v^2} \left( \left( {\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}-2\right) {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}+ {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\right) \left( {\dot{\eta }}+ \frac{v}{2}{d \hspace{-5.11108pt}/\,}\left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) \right) \right) \\&\quad = \frac{1}{v^3} \left( \left( {\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}-2\right) {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}+ {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\right) \left( {\dot{\eta }}+ \frac{v}{2}{d \hspace{-5.11108pt}/\,}\left( \dot{(\Omega \textrm{tr}\chi )}-\frac{4}{v}{\dot{\Omega }}\right) \right) \\&\qquad -\frac{1}{4v^4} \left( \left( {\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}-2\right) {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}+ {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\right) \left( {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{{{\widehat{\chi }}}}}+v^2{\dot{\mathfrak {c}}}_4+v^2 {d \hspace{-5.11108pt}/\,}{\dot{\mathfrak {c}}}_1+\frac{1}{2}{d \hspace{-5.11108pt}/\,}{\dot{\mathfrak {c}}}_2 -D\left( \frac{v}{2}{d \hspace{-5.11108pt}/\,}{\dot{\mathfrak {c}}}_2\right) \right) , \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&D\left( \frac{1}{8v^2} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}_c}}\right) = -\frac{1}{4v^3}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}_c}}\\ {}&\quad + \frac{1}{8v^2}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( \frac{2}{v^2}{\dot{{{\widehat{\chi }}}}}+\frac{1}{v^2}{\dot{\mathfrak {c}}}_3\right) . \end{aligned} \end{aligned}$$

Summing up the above and using Lemma 4.11 yield the transport equation for \({\underline{D}}{\dot{{\underline{\omega }}}}\) and thus finish the proof of Lemma 4.14.

Appendix C: Linearization at Schwarzschild

In this section, we first derive the linearizations of \({{\mathcal {P}}}_f\) and \({{\mathcal {P}}}_{j}\) at Schwarzschild of mass \(M\ge 0\), see Sect. C.1. The linearizations are used in (5.13) in Sect. 5.3 for proving the perturbation estimate (3.6) for \(({\textbf{E}},{\textbf{P}},{\textbf{L}},{\textbf{G}})\). Then in Sects. C.2 and C.3, we calculate the linearizations of the constraint functions and the null transport equations for the linearizations of \(({\textbf{E}},{\textbf{P}},{\textbf{L}},{\textbf{G}})\) at Schwarzschild of mass \(M\ge 0\), respectively. The latter are used for (5.24) and (5.29) in Sect. 5.4 to prove the transport estimate (3.7) for \(({\textbf{E}},{\textbf{P}},{\textbf{L}},{\textbf{G}})\).

1.1 Linearizations \(\dot{{{\mathcal {P}}}}^M_f\) and \(\dot{{{\mathcal {P}}}}^M_{q}\) at Schwarzschild of Mass \(M\ge 0\)

In this section, we define the linearization \(\dot{{{\mathcal {P}}}}^M_f\) and \(\dot{{{\mathcal {P}}}}^M_{q}\). As visible in the proofs of Lemmas C.1 and C.2, their linearization is closely connected to the more general linearized pure gauge solutions of [27].

First, we have the following lemma for \(\dot{{{\mathcal {P}}}}^M_f\).

Lemma C.1

(Linearization \(\dot{{{\mathcal {P}}}}^M_f\) of \({{\mathcal {P}}}_{f,q}\)). Let \(\dot{{{\mathcal {P}}}}_{f}^M\) denote the linearization of \({{\mathcal {P}}}_{f,q}\) in f at \(f=0\), \(q=0\), and Schwarzschild of mass \(M\ge 0\). For a given linearized perturbation function \(\dot{f}\),

$$\begin{aligned} \begin{aligned} \dot{f} := \left( {\dot{f}}(0,\theta ^1,\theta ^2), {\partial }_u {\dot{f}}(0,\theta ^1,\theta ^2), \,\, {\partial }_u^2 {\dot{f}}(0,\theta ^1,\theta ^2), \,\, {\partial }_u^3 {\dot{f}}(0,\theta ^1,\theta ^2) \right) , \end{aligned} \end{aligned}$$

the non-trivial components of \(\dot{{{\mathcal {P}}}}^M_{f}\left( {\dot{f}}\right) \) are given by

$$\begin{aligned} \begin{aligned} {\dot{\Omega }}&= \frac{1}{2\Omega _M} {\partial }_{u} \left( {\dot{f}} \Omega _M^2\right) ,&{\dot{\phi }}&=-\Omega _M^2 {\dot{f}},&{\dot{\eta }}&= \frac{r_M}{\Omega _M^2} {d \hspace{-5.11108pt}/\,}\left( {\partial }_u \left( \frac{\Omega _M^2}{r_M}{\dot{f}}\right) \right) ,\\ {\dot{{{\widehat{\chi }}}}}&= - 2\Omega _M {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}{\dot{f}},&\dot{(\Omega \textrm{tr}{{\underline{\chi }}})}&= -2{\partial }_u \left( \frac{{\dot{f}}\Omega _M^2}{r_M}\right) ,&\dot{(\Omega \textrm{tr}\chi )}&= \frac{2\Omega _M^2}{r_M^2} \left( {\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}{\dot{f}} - {\dot{f}}\left( 1-2\Omega _M^2\right) \right) , \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} {\dot{{\underline{\omega }}}}={\partial }_u\left( \frac{1}{2\Omega ^2_M} {\partial }_{u} \left( {\dot{f}} \Omega _M^2\right) \right) , \,\, {\underline{D}}{\dot{{\underline{\omega }}}}= {\partial }_u^2 \left( \frac{1}{2\Omega ^2_M} {\partial }_{u} \left( {\dot{f}} \Omega _M^2\right) \right) , \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \dot{\beta }=-\frac{6M\Omega _M}{r_M^3} {d \hspace{-5.11108pt}/\,}{\dot{f}}, \,\, {\dot{\rho }}&= -\frac{6M\Omega _M^2}{r_M^4} {\dot{f}}. \end{aligned} \end{aligned}$$

where we tacitly evaluated at \(u=0\).

Proof of Lemma C.1

The direct way to prove Lemma C.1 is to linearize \({{\mathcal {P}}}_{f,0}\) by hand, using the explicit formulas of Appendix A. In the following, we argue that \(\dot{{{\mathcal {P}}}}^M_f\) is readily calculated in [27].

Indeed, in [27] the following mapping is studied. Let \(({\tilde{u}},{\tilde{v}},\tilde{\theta }^1,\tilde{\theta }^2)\) be Eddington–Finkelstein coordinates on the exterior region of a Schwarzschild spacetime of small mass \(M>0\), see (2.24). Consider the sphere

$$\begin{aligned} \begin{aligned} {{\tilde{S}}} = {\tilde{S}}_{0,2}:= \{ {\tilde{u}}=0, {\tilde{v}}=2 \}. \end{aligned} \end{aligned}$$

Given a smooth and sufficiently small scalar function \(f=f(u,\theta ^1,\theta ^2)\), define new local coordinates \((u,v,\theta ^1,\theta ^2)\)

$$\begin{aligned} \begin{aligned} {\tilde{u}}= u+ f(u,\theta ^1,\theta ^2),\,\, {\tilde{v}}=v, \,\, \tilde{\theta }^1=\theta ^1, \,\, \tilde{\theta }^2=\theta ^2. \end{aligned} \end{aligned}$$

The coordinate system \((u,v,\theta ^1,\theta ^2)\) is not double null. However, it is shown in (173) in [27] that \((u,v,\theta ^1,\theta ^2)\) is double null to first order in f.

Hence, to first order in f, the sphere data calculated with respect to \((u,v,\theta ^1,\theta ^2)\) on the sphere

$$\begin{aligned} \begin{aligned} S= S_{0,2} := \{u=0, v=2\}, \end{aligned} \end{aligned}$$

agree with the sphere data \(x_{0,2}:={{\mathcal {P}}}_{f,0}({\underline{{\mathfrak {m}}}}^M)\) constructed by our mapping \({{\mathcal {P}}}_{f,q}\). Consequently, their linearizations in f (evaluated at \(f=0\), \(q=0\), and Schwarzschild of mass \(M>0\)) agree. This linearization is calculated in Lemma 6.1.1 in [27] and agrees with our expressions in Lemma 2.22. We note that the expression for \({\dot{{\underline{\omega }}}}\) follows from (2.48).

We remark that in [27] the linearization is calculated at Schwarzschild of mass \(M>0\), but a straightforward inspection shows that the calculation goes through for \(M\ge 0\). This finishes the proof of Lemma C.1. \(\square \)

Second, we have the following lemma for \(\dot{{{\mathcal {P}}}}_{q}\). It is a corollary of Lemma 6.1.3 in [27], where we note that our notation connects to [27] as follows,

$$\begin{aligned} \begin{aligned} \widehat{\dot{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}} = r_M^2 {\dot{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}_c}}, \,\, \frac{\dot{\sqrt{\det {g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}}}{\sqrt{\det {g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}} = \frac{2{\dot{\phi }}}{r_M}. \end{aligned} \end{aligned}$$

Lemma C.2

(Linearization \(\dot{{{\mathcal {P}}}}_{q}\) of \({{\mathcal {P}}}_{f,q}\)) . Let \(\dot{{{\mathcal {P}}}}_{q}^M\) denote the linearization of \({{\mathcal {P}}}_{f,q}\) in q at \(f=0\), \(q=0\), and Schwarzschild sphere data \({\mathfrak {m}}^M\). The non-trivial components of \(\dot{{{\mathcal {P}}}}^M_{q} ({\dot{q}})\) are given by

$$\begin{aligned} {\dot{\phi }}= \frac{r_M}{2} {\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}{\dot{q}}_1, \,\, {\dot{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}_c}}= 2 {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_1^*({\dot{q}}_1,{\dot{q}}_2). \end{aligned}$$

1.2 Linearized Constraint Functions at Schwarzschild of Mass \(M\ge 0\)

In this section, we linearize the constraint functions \(({{\mathcal {C}}}_i(x))_{1\le i \le 10}\) at Schwarzschild of mass \(M\ge 0\), that is, at \(x={\mathfrak {m}}^M\). The linearization procedure is adapted from [27]: We expand the sphere data

$$\begin{aligned} \begin{aligned} x&= \left( \Omega , {g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}, \Omega {\textrm{tr}\chi }, {{\widehat{\chi }}}, \Omega {\textrm{tr}{{\underline{\chi }}}}, {\underline{{{\widehat{\chi }}}}}, \eta , \omega , D\omega , {\underline{\omega }}, {\underline{D}}{\underline{\omega }}, {\alpha }, {\underline{\alpha }}\right) \\&= \left( \Omega _M, r_M^2 {\overset{\circ }{\gamma }}, \frac{2\Omega _M}{r}, 0, -\frac{2\Omega _M}{r}, 0, 0, \frac{M}{r_M^2}, -\frac{2M\Omega _M^2}{r_M^3}, -\frac{M}{r_M^2}, -\frac{2M\Omega _M^2}{r_M^3}, 0,0\right) \\&+ \varepsilon \cdot \left( {\dot{\Omega }}, \dot{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}}, \dot{(\Omega \textrm{tr}\chi )}, {\dot{{{\widehat{\chi }}}}}, \dot{(\Omega \textrm{tr}{{\underline{\chi }}})}, {\dot{{\underline{{{\widehat{\chi }}}}}}}, {\dot{\eta }}, {\dot{\omega }}, D{\dot{\omega }}, {\dot{{\underline{\omega }}}}, {\underline{D}}{\dot{{\underline{\omega }}}}, {\dot{\alpha }}, {\dot{{\underline{\alpha }}}}\right) + {\mathcal {O}}(\varepsilon ^2), \end{aligned} \end{aligned}$$

and differentiate in \(\varepsilon \) at \(\varepsilon =0\). Here, the Schwarzschild quantities \(r_M=r_M(u,v)\) and \(\Omega _M=\Omega _M(u,v)\) are defined in (2.25) and (2.26), respectively.

The proof of the next lemma follows by explicit calculation, see also Sect. 5 in [27].

Lemma C.3

(Linearization of constraint functions at Schwarzschild). Let \(M\ge 0\) be a real number, and let \((\dot{{{\mathcal {C}}}}_i^M)_{1\le i \le 10}\) denote the linearization of the constraint functions \(({{{\mathcal {C}}}}_i)_{1\le i \le 10}\) at Schwarzschild of mass M. Then, it holds that

$$\begin{aligned} \begin{aligned} \dot{{{\mathcal {C}}}}^M_1&= D^2 {\dot{\phi }}- 2 \Omega _M^2 {\dot{\omega }}- \frac{M}{r_M} \dot{(\Omega \textrm{tr}\chi )}- \frac{2M\Omega _M^2}{r_M^3} {\dot{\phi }}, \\ \dot{{{\mathcal {C}}}}^M_2&= r_M^2 \left( 2D\left( \frac{{\dot{\phi }}}{r_M}\right) -\dot{(\Omega \textrm{tr}\chi )}\right) ,\\ \dot{{{\mathcal {C}}}}^M_3&= r_M^2 D{\dot{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}_c}}- 2 \Omega _M {\dot{{{\widehat{\chi }}}}},\\ \dot{{{\mathcal {C}}}}^M_4&= \frac{1}{r_M^2} D\left( r_M^2 {\dot{\eta }}\right) - \Omega _M \left( \frac{1}{r_M^2} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{{{\widehat{\chi }}}}}- \frac{1}{2\Omega _M} {d \hspace{-5.11108pt}/\,}\dot{(\Omega \textrm{tr}\chi )}+ \frac{4}{r_M} {d \hspace{-5.11108pt}/\,}{\dot{\Omega }}\right) , \\ \dot{{{\mathcal {C}}}}^M_5&= \frac{1}{r_M^2} D\left( r_M^2 \dot{(\Omega \textrm{tr}{{\underline{\chi }}})}\right) - \frac{2\Omega _M^2}{r_M} \dot{(\Omega \textrm{tr}\chi )}+ \frac{2\Omega _M^2}{r_M^2} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( {\dot{\eta }}-\frac{2}{\Omega _M}{d \hspace{-5.11108pt}/\,}{\dot{\Omega }}\right) + \frac{4\Omega _0}{r_M^2} {\dot{\Omega }}+2 \Omega _M^2 {\dot{K}}, \end{aligned} \end{aligned}$$

and moreover

$$\begin{aligned} \begin{aligned} \dot{{{\mathcal {C}}}}^M_6&= \Omega _M r_M D\left( \frac{{\dot{{\underline{{{\widehat{\chi }}}}}}}}{r_M}\right) + {\dot{{\underline{{{\widehat{\chi }}}}}}}\frac{\Omega _M M}{r_M^2} - 2 \Omega _M^2 {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\left( {\dot{\eta }}-\frac{2}{\Omega _M}{d \hspace{-5.11108pt}/\,}{\dot{\Omega }}\right) -\frac{\Omega _M^3}{r_M} {\dot{{{\widehat{\chi }}}}},\\ \dot{{{\mathcal {C}}}}^M_7&= D{\dot{{\underline{\omega }}}}- \frac{2\Omega _M}{r_M^2} {\dot{\Omega }}- \Omega _M^2 {\dot{K}}+ \frac{\Omega _M^2}{2r_M} \left( \dot{(\Omega \textrm{tr}\chi )}-\dot{(\Omega \textrm{tr}{{\underline{\chi }}})}\right) , \\ \dot{{{\mathcal {C}}}}^M_8&= D{\dot{{\underline{\alpha }}}}- \frac{\Omega _M^2}{r_M} {\dot{{\underline{\alpha }}}}+ \frac{2M}{r_M^2} {\dot{{\underline{\alpha }}}}- 2 {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\left( \frac{\Omega _M}{r_M^2} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{{\underline{{{\widehat{\chi }}}}}}}- \frac{1}{2}{d \hspace{-5.11108pt}/\,}\dot{(\Omega \textrm{tr}{{\underline{\chi }}})}- \frac{\Omega _M^2}{r_M} {\dot{\eta }}\right) - \frac{6M\Omega _M}{r_M^3} {\dot{{\underline{{{\widehat{\chi }}}}}}}, \\ \dot{{{\mathcal {C}}}}^M_9&= D \left( {\underline{D}}{\dot{{\underline{\omega }}}}\right) - \frac{2M}{r_M^3} \left( 2\Omega _M^2 {\dot{{\underline{\omega }}}}-\frac{3}{2}\Omega _M^2\dot{(\Omega \textrm{tr}{{\underline{\chi }}})}- \left( -\frac{6}{r_M}+\frac{16M}{r_M^2}\right) \Omega _M {\dot{\Omega }}\right) \\&-\Omega _M^2 \left( \frac{3}{r_M}-\frac{8M}{r_M^2}\right) \left( {\dot{K}}+\frac{1}{2r_M} \left( \dot{(\Omega \textrm{tr}{{\underline{\chi }}})}-\dot{(\Omega \textrm{tr}\chi )}\right) +\frac{2\Omega _M{\dot{\Omega }}}{r_M^2}\right) \\&- \frac{\Omega _M^3}{r_M^2} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( \frac{1}{r_M^2} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{{\underline{{{\widehat{\chi }}}}}}}- \frac{1}{2\Omega _M} {d \hspace{-5.11108pt}/\,}\dot{(\Omega \textrm{tr}{{\underline{\chi }}})}- \frac{\Omega _M}{r_M} {\dot{\eta }}\right) .\\ {\dot{{{\mathcal {C}}}}}^M_{10}&= \Omega _M {\dot{\alpha }}+ D{\dot{{{\widehat{\chi }}}}}- \frac{M}{r_M^2} {\dot{{{\widehat{\chi }}}}}. \end{aligned} \end{aligned}$$

Remark C.4

In addition to the above, we have by (2.6) and (2.8) that

$$\begin{aligned} \begin{aligned} \dot{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}} = 2r_M {\dot{\phi }}{\overset{\circ }{\gamma }}+ r_M^2 {\dot{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}_c}}, \,\, {\dot{\omega }}= D\left( \frac{{\dot{\Omega }}}{\Omega _M}\right) , \,\, {\dot{{\underline{\omega }}}}= {\underline{D}}\left( \frac{{\dot{\Omega }}}{\Omega _M}\right) , \,\, {\dot{{{\underline{\eta }}}}}&= - {\dot{\eta }}+ \frac{2}{\Omega _M} {d \hspace{-5.11108pt}/\,}{\dot{\Omega }}. \end{aligned} \end{aligned}$$
(C.1)

Moreover, by (242) in [27] the linearization of the Gauss curvature \({\dot{K}}\) is given by

$$\begin{aligned} {\dot{K}}&= \frac{1}{2r_M^2} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{{g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}_c}}- \frac{1}{r_M^3} ({\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}+2) {\dot{\phi }}. \end{aligned}$$
(C.2)

Moreover, using that for Schwarzschild sphere data, \(\phi = r_M\) and \(r=r_M\), we have that \({\dot{r}}^{[\ge 1]}=0\) and

$$\begin{aligned} \begin{aligned} {\dot{r}}^{(0)} = {\dot{\phi }}^{(0)}. \end{aligned} \end{aligned}$$
(C.3)

1.3 Linearized Transport Equations for \(({\textbf{E}},{\textbf{P}},{\textbf{L}},{\textbf{G}})\) at Schwarzschild

In this section, we linearize the charges \((\dot{{\textbf{E}}},\dot{{\textbf{P}}},\dot{{\textbf{L}}},\dot{{\textbf{G}}})\) at Schwarzschild of mass \(M\ge 0\) and analyze their transport equations along \({{\mathcal {H}}}\).

First, from Definition 2.10 and (2.15) and (2.16), we recall that

$$\begin{aligned} \begin{aligned} -\frac{8\pi }{\sqrt{4\pi }} {\textbf{E}}&:= \left( r^3 \left( {\rho }+r{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ {\beta }\right) \right) ^{(0)} \\&=- \left( r^3 \left( K+\frac{1}{4} {\textrm{tr}\chi }{\textrm{tr}{{\underline{\chi }}}}-\frac{1}{2}({{\widehat{\chi }}},{\underline{{{\widehat{\chi }}}}})\right) \right) ^{(0)}\\&- \left( r^4 \left( {{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ {{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ {{\widehat{\chi }}}- \frac{1}{2}\triangle \hspace{-6.66656pt}/\ {\textrm{tr}\chi }+ {{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ \left( {{\widehat{\chi }}}\cdot \zeta - \frac{1}{2}{\textrm{tr}\chi }\zeta \right) \right) \right) ^{(0)}, \end{aligned} \end{aligned}$$
(C.4)

and

$$\begin{aligned} \begin{aligned} - \frac{8\pi }{\sqrt{\frac{4\pi }{3}}} {\textbf{P}}^m&:= \left( r^3 \left( {\rho }+r{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ {\beta }\right) \right) ^{(1m)}\\&= - \left( r^3 \left( K+\frac{1}{4} {\textrm{tr}\chi }{\textrm{tr}{{\underline{\chi }}}}-\frac{1}{2}({{\widehat{\chi }}},{\underline{{{\widehat{\chi }}}}})\right) \right) ^{(1m)}\\&- \left( r^4\left( {{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ {{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ {{\widehat{\chi }}}- \frac{1}{2}\triangle \hspace{-6.66656pt}/\ {\textrm{tr}\chi }+ {{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ \left( {{\widehat{\chi }}}\cdot \zeta - \frac{1}{2}{\textrm{tr}\chi }\zeta \right) \right) \right) ^{(1m)},\\ \frac{16\pi }{\sqrt{\frac{8\pi }{3}}}{\textbf{L}}^m&:= \left( r^3 \left( {d \hspace{-5.11108pt}/\,}{\textrm{tr}\chi }+ {\textrm{tr}\chi }\left( \eta -{d \hspace{-5.11108pt}/\,}\log \Omega \right) \right) \right) ^{(1m)}_H, \\ \frac{16\pi }{\sqrt{\frac{8\pi }{3}}} {\textbf{G}}^m&:= \left( r^3 \left( {d \hspace{-5.11108pt}/\,}{\textrm{tr}\chi }+ {\textrm{tr}\chi }\left( \eta -{d \hspace{-5.11108pt}/\,}\log \Omega \right) \right) \right) ^{(1m)}_E. \end{aligned} \end{aligned}$$
(C.5)

Linearizing these expressions at Schwarzschild of mass \(M\ge 0\), see (2.28), and using (2.49) and (2.50), we get the explicit expressions

$$\begin{aligned} \begin{aligned} -\frac{8\pi }{\sqrt{4\pi }}\dot{{\textbf{E}}}&= -\frac{6M{\dot{\phi }}^{(0)}}{ r_M} +2 {\dot{\phi }}^{(0)}-2 r_M \Omega _M {\dot{\Omega }}^{(0)} + \frac{r_M^2}{2} \dot{(\Omega \textrm{tr}\chi )}^{(0)}-\frac{r_M^2}{2} \dot{(\Omega \textrm{tr}{{\underline{\chi }}})}^{(0)}, \\ - \frac{8\pi }{\sqrt{\frac{4\pi }{3}}} \dot{{\textbf{P}}}^m&= 2 r_M\left( 2-\Omega _M\right) {\dot{\Omega }}^{(1m)} + \frac{r_M^2}{2}\left( 1-\frac{2}{\Omega _M}\right) \dot{(\Omega \textrm{tr}\chi )}^{(1m)} \\&- \frac{r_M^2}{2}\dot{(\Omega \textrm{tr}{{\underline{\chi }}})}^{(1m)} + r_M\Omega _M {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{\eta }}^{(1m)} ,\\ \frac{16\pi }{\sqrt{\frac{8\pi }{3}}}\dot{{\textbf{L}}}^m&= 2 r_M^2 \Omega _M {\dot{\eta }}^{(1m)}_H, \\ \frac{16\pi }{\sqrt{\frac{8\pi }{3}}}\dot{{\textbf{G}}}^m&= \frac{r_M^3}{\Omega _M} {d \hspace{-5.11108pt}/\,}\dot{(\Omega \textrm{tr}\chi )}^{(1m)}_E +2 r_M^2 \Omega _M {\dot{\eta }}^{(1m)}_E - 4 r_M^2 {d \hspace{-5.11108pt}/\,}{\dot{\Omega }}^{(1m)}_E, \end{aligned} \end{aligned}$$

where we used that for scalar functions f, \(({\overset{\circ }{\triangle \hspace{-6.66656pt}/\ }}f)^{[1]} = -2 f^{[1]}\), see Appendix D.

By applying the homogeneous linearized null constraint equations at Schwarzschild, see Lemma C.3, together with (C.2) and (C.3), it is straightforward to derive transport equations for these linearized charges at Schwarzschild. The resulting equations are summarized in the following lemma.

Lemma C.5

(Linearized transport equations for charges at Schwarzschild). The following hold for \(M\ge 0\) and \(m=-1,0,1\),

$$\begin{aligned} \begin{aligned} -\frac{8\pi }{\sqrt{4\pi }} D\left( \dot{{\textbf{E}}}\right)&= M \dot{(\Omega \textrm{tr}\chi )}^{(0)}, \\ - \frac{8\pi }{\sqrt{\frac{4\pi }{3}}} D\left( \dot{{\textbf{P}}}^m\right)&= -2 \left( 2(1-\Omega _M) -\frac{6M}{r_M}\right) {\dot{\Omega }}^{(1m)}\\&- \left( M\left( \frac{1}{\Omega _M}-3\right) +r_M (1-\Omega _M)\right) \dot{(\Omega \textrm{tr}\chi )}^{(1m)} \\&- \left( \frac{M}{r_M}\left( 2-3\Omega _M\right) + (\Omega _M -1)\right) ({\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{\eta }})^{(1m)}, \\ \frac{16\pi }{\sqrt{\frac{8\pi }{3}}}D\left( \dot{{\textbf{L}}}^m\right)&= 2\Omega _M M {\dot{\eta }}_H^{(1m)}, \\ \frac{16\pi }{\sqrt{\frac{8\pi }{3}}}D\left( \dot{{\textbf{G}}}^m\right)&= -\frac{M r_M}{\Omega _M} ({d \hspace{-5.11108pt}/\,}\dot{(\Omega \textrm{tr}\chi )})_E^{(1m)}+2 \Omega _M M {\dot{\eta }}_E^{(1m)} \\&- \frac{4M \Omega _M}{r_M} ({d \hspace{-5.11108pt}/\,}{\dot{\phi }})_E^{(1m)}-4M \left( {d \hspace{-5.11108pt}/\,}{\dot{\Omega }}\right) _E^{(1m)} \end{aligned} \end{aligned}$$

By definition of \(\Omega _M\) in (2.28) it holds that for M small,

$$\begin{aligned} \begin{aligned} \vert \Omega _M-1 \vert \lesssim M, \end{aligned} \end{aligned}$$
(C.6)

so that we can write the equations of Lemma C.5 for \(M\ge 0\) small as

$$\begin{aligned} \begin{aligned} D\left( \dot{{\textbf{E}}}\right)&= {{\mathcal {O}}}(M) \dot{(\Omega \textrm{tr}\chi )}^{(0)}, \\ D\left( \dot{{\textbf{P}}}^m\right)&= {{\mathcal {O}}}(M){\dot{\Omega }}^{(1m)}+ {{\mathcal {O}}}(M) \dot{(\Omega \textrm{tr}\chi )}^{(1m)} + {{\mathcal {O}}}(M) ({\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{\dot{\eta }})^{(1m)}, \\ D\left( \dot{{\textbf{L}}}^m\right)&= {{\mathcal {O}}}(M) {\dot{\eta }}_H^{(1m)}, \\ D\left( \dot{{\textbf{G}}}^m\right)&= {{\mathcal {O}}}(M) ({d \hspace{-5.11108pt}/\,}\dot{(\Omega \textrm{tr}\chi )})_E^{(1m)}+{{\mathcal {O}}}(M) {\dot{\eta }}_E^{(1m)} \\&+ {{\mathcal {O}}}(M)({d \hspace{-5.11108pt}/\,}{\dot{\phi }})_E^{(1m)}+{{\mathcal {O}}}(M) \left( {d \hspace{-5.11108pt}/\,}{\dot{\Omega }}\right) _E^{(1m)}, \end{aligned} \end{aligned}$$
(C.7)

where \({{\mathcal {O}}}(M)\) denotes terms that are bounded by M.

Appendix D: Hodge Systems and Fourier Theory on 2-Spheres

In this Sect. D.1, we recall the theory of 2-dimensional Hodge systems, see also [16]. In Sect. D.2, we recapitulate the definition and properties of tensor spherical harmonics, following the notation of [26]. In Sect. D.3, we use tensor spherical harmonics to analyze differential operators which appear in this paper.

1.1 Hodge Systems on Riemannian 2-Spheres

Definition D.1

(Hodge operators). Let \((S, {g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}})\) be a Riemannian 2-sphere. Define

  1. (1)

    for a 1-form \(X_A\),

    $$\begin{aligned} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_1(X) := ({{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ X, \mathop {\mathrm {\textrm{curl}}}\limits \hspace{-9.44443pt}/\ X). \end{aligned}$$
  2. (2)

    for a 2-tensor \(W_{AB}\),

    $$\begin{aligned} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2(W)_C := ({{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ W)_C. \end{aligned}$$
  3. (3)

    for a pair of functions \((f_1,f_2)\),

    $$\begin{aligned} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_1^{*}(f_1,f_2):= -{d \hspace{-5.11108pt}/\,}f_1 + {}^*{d \hspace{-5.11108pt}/\,}f_2. \end{aligned}$$
  4. (4)

    for a 1-form \(X_A\),

    $$\begin{aligned} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^{*}(X)_{AB} := -\frac{1}{2}\left( \nabla \hspace{-7.22214pt}/\ _A X_B + \nabla \hspace{-7.22214pt}/\ _B X_A - ({{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ X) {g \hspace{-4.44443pt}/\ \hspace{-2.77771pt}}_{AB} \right) . \end{aligned}$$

Throughout the paper, we abuse notation by denoting \({{\mathcal {D}}}_2\) as \({{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ \). In the following, we use on the round sphere \((S_{u,v},(v-u)^2{\overset{\circ }{\gamma }})\) the notation

$$\begin{aligned} {\overset{\circ }{{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}}}_1 := (v-u)^2 {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_1, \,\, {\overset{\circ }{{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}}}_2 := (v-u)^2 {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2. \end{aligned}$$

The following lemma is a paraphrase of the material in [16].

Lemma D.2

The following holds.

  1. (1)

    The kernels of \({{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_1\) and \({{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2\) are trivial.

  2. (2)

    The kernel of \({{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_1^*\) consists of pairs of constant functions \((f_1,f_2)=(c_1,c_2)\).

  3. (3)

    The kernel of \({{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\) consists of the set of conformal Killing vectorfields (a 6-dimensional space on the round sphere).

  4. (4)

    The \(L^2\)-range of \({{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_1\) consists of all pairs of functions \((f_1,f_2)\) on S with vanishing mean.

  5. (5)

    The \(L^2\)-range of \({{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2\) consists of all \(L^2\)-integrable 1-forms on S which are orthogonal to the conformal Killing vectorfields.

  6. (6)

    The operators \({{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_1^*\) and \({{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\) are conformally invariant.

1.2 Tensor Spherical Harmonics

Tensor spherical harmonics are defined on the standard round unit sphere as follows.

Definition D.3

(Tensor spherical harmonics). Introduce the following spherical harmonics functions, vectorfields and tracefree symmetric 2-tensors.

  1. (1)

    For integers \(l\ge 0\), \(-l \le m \le l\), let \(Y^{(lm)}\) be the standard (real-valued) spherical harmonics on the round unit sphere \(S_1\).

  2. (2)

    For \(l\ge 1\), \(-l \le m \le l\), define the vectorfields

    $$\begin{aligned} \begin{aligned} E^{(lm)} := \frac{1}{\sqrt{l(l+1)}} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_1^*(Y^{(lm)},0), \,\, H^{(lm)} :=\frac{1}{\sqrt{l(l+1)}} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_1^*(0, Y^{(lm)}). \end{aligned} \end{aligned}$$

    The vectorfields \(E^{(lm)}\) and \(H^{(lm)}\) are called electric and magnetic, respectively.

  3. (3)

    For \(l\ge 2\), \(-l \le m \le l\), define the tracefree symmetric 2-tensors

    $$\begin{aligned} \begin{aligned} \psi ^{(lm)}:= \frac{1}{\sqrt{\frac{1}{2}l(l+1)-1}} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\left( E^{(lm)}\right) , \,\, \phi ^{(lm)} := \frac{1}{\sqrt{\frac{1}{2}l(l+1)-1}} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\left( H^{(lm)}\right) . \end{aligned} \end{aligned}$$

    The tensors \(\psi ^{(lm)}\) and \(\phi ^{(lm)}\) are called electric and magnetic, respectively.

The following lemma is a summary of properties of spherical harmonics, see, for example, [26] for more details and proofs.

Lemma D.4

The following holds.

  1. (1)

    On the round unit sphere \(S_1\), \(L^2\)-integrable functions f, vectorfields X and tracefree symmetric 2-tensors V can be decomposed as follows,

    $$\begin{aligned} f&= \sum \limits _{l\ge 0} \sum \limits _{-l\le m \le l} f^{lm} Y^{(lm)}, \\ X&= \sum \limits _{l\ge 1} \sum \limits _{-l\le m \le l} X^{lm}_E E^{(lm)} + X^{lm}_H H^{(lm)},\\ V&= \sum \limits _{l\ge 2} \sum \limits _{-l\le m \le l} V^{lm}_\psi \psi ^{(lm)} +V^{lm}_\phi \phi ^{(lm)}, \end{aligned}$$

    where

    $$\begin{aligned} \begin{aligned} f^{(lm)}&:= \int \limits _{S_1} f Y^{(lm)} d\mu _{{\overset{\circ }{\gamma }}},{} & {} \\ X_E^{(lm)}&:= \int \limits _{S_1} X \cdot E^{(lm)} d\mu _{{\overset{\circ }{\gamma }}},&X_H^{(lm)}&:= \int \limits _{S_1} X \cdot H^{(lm)} d\mu _{{\overset{\circ }{\gamma }}}, \\ V_\psi ^{(lm)}&:= \int \limits _{S_1} V \cdot \psi ^{(lm)} d\mu _{{\overset{\circ }{\gamma }}},&V_\phi ^{(lm)}&:= \int \limits _{S_1} V \cdot \phi ^{(lm)} d\mu _{{\overset{\circ }{\gamma }}}, \end{aligned} \end{aligned}$$

    where \(d\mu _{\overset{\circ }{\gamma }}\) denotes the volume element of the standard round unit metric on \(S_1\) and \(\cdot \) denotes the product with respect to \({\overset{\circ }{\gamma }}\).

  2. (2)

    It holds that for \(l\ge 1\),

    $$\begin{aligned} \begin{aligned} ({d \hspace{-5.11108pt}/\,}f)_E^{(lm)}&= - \sqrt{l(l+1)} f^{(lm)},&({d \hspace{-5.11108pt}/\,}f)_H^{(lm)}&= 0,\\ ({{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_1^*(0,f))_E^{(lm)}&= 0,&({{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_1^*(0,f))_H^{(lm)}&= \sqrt{l(l+1)} f^{(lm)},\\ ({\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}X)^{(lm)}&= \sqrt{l(l+1)} X_E^{(lm)},{} & {} \end{aligned}\end{aligned}$$
    (D.1)

    and for \(l\ge 2\),

    $$\begin{aligned} \begin{aligned} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*(X)_\psi ^{(lm)}&= \sqrt{\frac{1}{2}l(l+1)-1} \, X_E^{(lm)},&{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*(X)_\phi ^{(lm)}&= \sqrt{\frac{1}{2}l(l+1)-1} \, X_H^{(lm)}, \\ ({\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}V)_E^{(lm)}&= \sqrt{\frac{1}{2}l(l+1)-1} \, V_\psi ^{(lm)},&({\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}V)_H^{(lm)}&= \sqrt{\frac{1}{2}l(l+1)-1} \, V_\phi ^{(lm)}. \end{aligned}\end{aligned}$$
    (D.2)
  3. (3)

    The operator \({{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_1\) is a bijection between vectorfields and pairs of functions (fg) with vanishing means,

    $$\begin{aligned} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_1: X^{[l\ge 1]} \rightarrow (Y^{[l\ge 1]}, Y^{[l\ge 1]}). \end{aligned}$$

    Moreover, the following restrictions are bijections:

    $$\begin{aligned} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_1:&E^{[l\ge 1]} \rightarrow (Y^{[l\ge 1]}, 0), \\ {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_1:&H^{[l\ge 1]} \rightarrow (0, Y^{[l\ge 1]}). \end{aligned}$$

    The spherical harmonics vectorfields of mode \(l=1\) form the space of conformal Killing vectorfields of the unit round sphere.

  4. (4)

    The operator \({{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2\) is a bijection between tracefree symmetric 2-tensors and vectorfields of modes \(l\ge 2\),

    $$\begin{aligned} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2: V^{[l\ge 2]} \rightarrow X^{[l\ge 2]}. \end{aligned}$$

    Moreover, the following mappings are bijections:

    $$\begin{aligned} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2: \psi ^{[l\ge 2]} \rightarrow E^{[l\ge 2]}, \,\, {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2: \phi ^{[l\ge 2]} \rightarrow H^{[l\ge 2]}. \end{aligned}$$
  5. (5)

    Let \(k\ge 0\) be an integer. There exists a constant \(C_k>0\), depending only on k, such that for scalar functions f, vectorfields X and symmetric tracefree 2-tensors V on \(S_1\), we have the following equivalence of norms,

    $$\begin{aligned} \begin{aligned} \sum \limits _{0 \le k' \le k} \Vert \nabla \hspace{-7.22214pt}/\ ^{k'} f \Vert ^2_{L^2(S_1)} \sim&\sum \limits _{l\ge 0} \sum \limits _{-l\le m \le l} (l+1)^{2k} \left( f^{(lm)}\right) ^2, \\ \sum \limits _{0 \le k' \le k} \Vert \nabla \hspace{-7.22214pt}/\ ^{k'} X \Vert ^2_{L^2(S_1)} \sim&\sum \limits _{l\ge 1} \sum \limits _{-l\le m \le l} (l+1)^{2k} \left( \left( X_E^{(lm)}\right) ^2+ \left( X_H^{(lm)}\right) ^2\right) , \\ \sum \limits _{0 \le k' \le k} \Vert \nabla \hspace{-7.22214pt}/\ ^{k'} V \Vert ^2_{L^2(S_1)} \sim&\sum \limits _{l\ge 2} \sum \limits _{-l\le m \le l} (l+1)^{2k} \left( \left( V_\psi ^{(lm)}\right) ^2+ \left( V_\phi ^{(lm)}\right) ^2\right) . \end{aligned} \end{aligned}$$

Notation

Given a scalar function

$$\begin{aligned} \begin{aligned} f= \sum \limits _{l\ge 0} \sum \limits _{-l\le m \le l} f^{lm} Y^{(lm)}, \end{aligned} \end{aligned}$$
(D.3)

we denote, for integers \(l' \ge 0\),

$$\begin{aligned} f^{[l']}&= \sum \limits _{l=l'} \sum \limits _{-l\le m \le l} f^{lm} Y^{(lm)}, \,\, f^{[\ge l']} = \sum \limits _{l\ge l'} \sum \limits _{-l\le m \le l} f^{lm} Y^{(lm)}, \,\,f^{[\le l']} \\&\quad = \sum \limits _{0\le l\le l'} \sum \limits _{-l\le m \le l} f^{lm} Y^{(lm)}, \end{aligned}$$

similarly for vectorfields X and symmetric tracefree 2-tensors V. Moreover, denote the electric part and the magnetic part of a vectorfield X by \(X_E\) and \(X_H\), respectively, and similarly for symmetric tracefree 2-tensors V by \(V_\psi \) and \(V_\phi \), respectively.

1.3 Spectral Analysis of Differential Operators

In this section, we discuss the differential operators that appeared in Sect. 4.

Analysis of \({{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}+1\). Let V be a tracefree symmetric 2-tensor,

$$\begin{aligned} V= \sum \limits _{l\ge 2}\sum \limits _{-l\le m \le l} V_\psi ^{(lm)} \psi ^{lm} + V_\phi ^{(lm)} \phi ^{lm}. \end{aligned}$$

Then,

$$\begin{aligned} ({{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}+1)V = \sum \limits _{l\ge 2}\sum \limits _{-l\le m \le l} \underbrace{\left( \frac{1}{2}l(l+1) -1 +1\right) }_{>0 \text { for } l\ge 2}\left( V_\psi ^{(lm)} \psi ^{lm} + V_\phi ^{(lm)} \phi ^{lm}\right) . \end{aligned}$$

Hence, the operator has no kernel and we have the following elliptic estimate. Let W be a given tracefree symmetric 2-tensor. Then, there exists a unique solution V to

$$\begin{aligned} \begin{aligned} \left( {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}+1\right) V = W, \end{aligned} \end{aligned}$$

and for integers \(k\ge 0\) we have the estimate,

$$\begin{aligned} \begin{aligned} \Vert V \Vert _{H^{k+2}(S_1)} \lesssim \Vert W \Vert _{H^{k}(S_1)}. \end{aligned} \end{aligned}$$
(D.4)

Analysis of \(({\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*+1+ {d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }})\). Let X be a vectorfield,

$$\begin{aligned} X= \sum \limits _{l\ge 1}\sum \limits _{-l\le m \le l} X_E^{(lm)} E^{(lm)} + X_H^{(lm)} H^{(lm)}. \end{aligned}$$

Then, it holds that

$$\begin{aligned} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*X&= \sum \limits _{l\ge 1}\left( \frac{1}{2}l(l+1)-1\right) \left( X_E^{(lm)} E^{(lm)} + X_H^{(lm)} H^{(lm)}\right) , \end{aligned}$$

and

$$\begin{aligned} {d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}X&= \sum \limits _{l\ge 1}\sum \limits _{-l\le m \le l}(-l(l+1)) X_E^{(lm)}E^{(lm)}. \end{aligned}$$

Therefore,

$$\begin{aligned}&\left( {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*+ 1 + {d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\right) X \\&=\sum \limits _{l\ge 1}\sum \limits _{-l\le m \le l} X_E^{(lm)} E^{(lm)} \left( \left( \frac{1}{2}l(l+1)-1\right) + 1 - l(l+1) \right) \\&\quad +\sum \limits _{l\ge 1}\sum \limits _{-l\le m \le l} X_H^{(lm)} H^{lm} \left( \left( \frac{1}{2}l(l+1)-1\right) +1\right) \\&=\sum \limits _{l\ge 1}\sum \limits _{-l\le m \le l} X_E^{(lm)} E^{lm} \underbrace{\left( -\frac{1}{2}l(l+1) \right) }_{<0 \text { for } l\ge 1} +\sum \limits _{l\ge 1} X_H^{(lm)} H^{lm} \underbrace{\left( \frac{1}{2}l(l+1) \right) }_{>0 \text { for } l\ge 1}. \end{aligned}$$

Hence, the operator has no kernel and we have the following elliptic estimate. Let Y be a given vectorfield. Then, there exists a unique solution X to

$$\begin{aligned} \begin{aligned} ({\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*+1+ {d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }})X = Y, \end{aligned} \end{aligned}$$

and for integers \(k\ge 0\) we have the estimate,

$$\begin{aligned} \begin{aligned} \Vert X \Vert _{H^{k+2}(S_1)} \lesssim \Vert Y \Vert _{H^{k}(S_1)}. \end{aligned} \end{aligned}$$
(D.5)

By (D.2) and (D.5), it follows in particular that the operator

$$\begin{aligned} \begin{aligned} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*({\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*+1+ {d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}) {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\end{aligned} \end{aligned}$$

has no kernel and admits the following estimate. For any given symmetric tracefree 2-tensor W, there exists a unique solution V to

$$\begin{aligned} \begin{aligned} {{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*({\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*+1+ {d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}) {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}V =W \end{aligned} \end{aligned}$$

satisfying

$$\begin{aligned} \begin{aligned} \Vert V \Vert _{H^{k+4}(S_1)} \lesssim \Vert W \Vert _{H^{k}(S_1)}. \end{aligned} \end{aligned}$$
(D.6)

Analysis of \(({\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*+1+\frac{1}{2}{d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}){d \hspace{-5.11108pt}/\,}\). Let f be a scalar function,

$$\begin{aligned} f= \sum \limits _{l\ge 0} \sum \limits _{-l\le m \le l} f^{(lm)} Y^{lm}. \end{aligned}$$

Then,

$$\begin{aligned} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*{d \hspace{-5.11108pt}/\,}f = \sum \limits _{l\ge 0} \sum \limits _{-l\le m \le l} \left( \frac{1}{2}l(l+1)-1\right) \left( -\sqrt{l(l+1)}\right) E^{(lm)}, \end{aligned}$$

and

$$\begin{aligned} {d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{d \hspace{-5.11108pt}/\,}f = \sum \limits _{l\ge 0} \sum \limits _{-l\le m \le l} \left( -\sqrt{l(l+1)}\right) \left( -l(l+1)\right) f^{(lm)} E^{(lm)}. \end{aligned}$$

Therefore,

$$\begin{aligned}&\left( {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*+ 1 + \frac{1}{2}{d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\right) {d \hspace{-5.11108pt}/\,}f \\&\quad = \sum \limits _{l\ge 0} \sum \limits _{-l\le m \le l} \underbrace{\left( \left( \frac{1}{2}l(l+1)-1\right) +1 +\frac{1}{2}\left( - l(l+1)\right) \right) }_{=0} \left( -\sqrt{l(l+1)}\right) E^{lm}. \end{aligned}$$

This shows that

$$\begin{aligned} \left( {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*+ 1 + \frac{1}{2}{d \hspace{-5.11108pt}/\,}{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\right) {d \hspace{-5.11108pt}/\,}f = 0. \end{aligned}$$
(D.7)

Analysis of the operator \( \left( 2-{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\right) \). Let X be a vectorfield,

$$\begin{aligned} X= \sum \limits _{l\ge 1}\sum \limits _{-l\le m \le l} X_E^{(lm)} E^{(lm)} + X_H^{(lm)} H^{(lm)}. \end{aligned}$$

Then, it holds that

$$\begin{aligned} (2-{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*) X&= \sum \limits _{l\ge 1}\left( 2- \left( \frac{1}{2}l(l+1)-1\right) \right) \left( X_E^{(lm)} E^{(lm)} + X_H^{(lm)} H^{(lm)}\right) \\&= \sum \limits _{l\ge 1}\frac{1}{2}\left( 6- l(l+1)\right) \left( X_E^{(lm)} E^{(lm)} + X_H^{(lm)} H^{(lm)}\right) \end{aligned}$$

We conclude that the kernel of the operator \( \left( 2-{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\right) \) is given by the set of vectorfields

$$\begin{aligned} \begin{aligned} \{ X: X = X^{[2]} \}. \end{aligned} \end{aligned}$$
(D.8)

Further, let Y be a vectorfield such that \(Y=Y^{[\ge 3]}\). Then, there exists a unique vectorfield X such that \(X=X^{[\ge 3]}\) and

$$\begin{aligned} \begin{aligned} (2-{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*) X=Y, \end{aligned} \end{aligned}$$

with the estimate

$$\begin{aligned} \begin{aligned} \Vert X \Vert _{H^{k+2}(S_1)} \lesssim \Vert Y \Vert _{H^{k}(S_1)}. \end{aligned} \end{aligned}$$

In particular, it follows moreover that for any given function f with \(f=f^{[\ge 3]}\), there is a unique solution V with \(V=V^{[\ge 3]}\) of

$$\begin{aligned} \begin{aligned} {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}\left( 2-{\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}{{\mathcal {D}}}\hspace{-5.55542pt}/\ \hspace{-2.77771pt}_2^*\right) {\overset{\circ }{{{\,\mathrm{\textrm{div}}\,}}\hspace{-9.44443pt}/\ }}V = f. \end{aligned} \end{aligned}$$

with the estimate

$$\begin{aligned} \begin{aligned} \Vert X \Vert _{H^{k+4}(S_1)} \lesssim \Vert f \Vert _{H^{k}(S_1)}. \end{aligned} \end{aligned}$$
(D.9)

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Aretakis, S., Czimek, S. & Rodnianski, I. The Characteristic Gluing Problem for the Einstein Vacuum Equations: Linear and Nonlinear Analysis. Ann. Henri Poincaré 25, 3081–3205 (2024). https://doi.org/10.1007/s00023-023-01394-y

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