Appendix A: Perturbations of Sphere Data

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

$$\begin \begin }}: \, }^+(\tilde}}}_) \times }_f \times }_q&\rightarrow }(S_), \\ (\tilde}},f,q)&\mapsto x_:= }}_(\tilde}}), \end \end$$

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

$$\begin \begin \Vert }}_(\tilde}}) - \tilde}}_ \Vert _}(S_)} \lesssim \Vert f \Vert _}_} + \Vert q \Vert _}_q} + \Vert \tilde}}-\underline}} \Vert _}^+(\tilde}}}_)}, \end \end$$

where we denoted \(\tilde}}_:= \tilde}}\vert _}\).

In Sect. A.1, we derive explicit expressions for the sphere data \(}}_(\tilde}})\). In Sect. A.2, we prove Proposition 2.21; that is, we analyze \(}}_(\tilde}})\).

1.1 Explicit Formulas for Transversal Sphere Perturbations

In the following, we rigorously set up transversal perturbations \(}}_\) 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 \(}}\) be a spacelike 2-sphere in a spacetime \((}},\textbf)\). Let \((},},\tilde^1,\tilde^2)\) be a local double null coordinate system around \(}}\), that is,

$$\begin \begin \textbf&= - 4 \tilde^2 d} d} + \tilde/\ \hspace}}_ (d\tilde^C - }^C d})(d\tilde^D - }^D d}), \end \end$$

(A.1)

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

The geodesic null vectorfields are defined by

$$\begin \begin \tilde}}}' := -2 \textbf}, \,\, }' := -2 \textbf}, \end \end$$

(A.2)

where \(\textbf\) denotes the covariant derivative on \((}},\textbf)\).

The normalized null vectorfields are defined by

$$\begin \begin \widehat}} := \Omega }', \,\, \widehat}}}} := \Omega \tilde}}}'. \end \end$$

The equivariant null vectorfields are defined by

$$\begin \begin }} := \tilde^2 L' , \,\, }}}} := \tilde^2 }}'. \end \end$$

(A.3)

The Ricci coefficients are defined with respect to the above vectorfields as follows,

$$\begin \begin \tilde_&:= \textbf(\textbf_}}} \widehat}}},_}}),&\tilde}}}_&:= \textbf(\textbf_}} \widehat}}}}},_}}),&\tilde_A&:= \frac\textbf(\textbf_}} \widehat}}},\widehat}}}}}), \\ }&:= } + /\,}}} \log },&}&:= }} \log },&\tilde}}&:= \tilde}}} \log }, \end \end$$

(A.4)

where \(\tilde/\,}}\) denotes the exterior derivative on spheres \(}_\).

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

Lemma A.1

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

(1)

The inverse \(\textbf^\) of (A.1) is given by

$$\begin \begin \textbf^&= -\frac^2} \left( _}}} \otimes _}}} + _}}} \otimes _}}}\right) -\frac}^C}^2 } \left( _}}} \otimes _}}} + _}}} \otimes _}}}\right) + \tilde/\ \hspace}}^ _}}} \otimes _}}}. \end \end$$

(A.5)

Specifically,

$$\begin \begin \textbf^}} }}}=\textbf^}} }}} =0. \end \end$$

(A.6)

(2)

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

$$\begin \begin }} = _}}} + }^A _}^A}, \,\, }}}} = _}}}. \end \end$$

(A.7)

(3)

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

$$\begin \begin _}}} }^A = 4\tilde^2 \tilde^A. \end \end$$

(A.8)

(4)

It holds that

$$\begin \begin \Gamma ^}}}_}} }}}&= \Gamma ^}}}_}} }}} = \Gamma ^}}}_}} }}} =0,&\Gamma ^}}}_}} }}}&= _}}} \log } - \tilde_A - \frac}} \tilde}}}_ }^B,&\Gamma ^}}}_}} }}}&= \frac}} \tilde}}}_, \end \end$$

(A.9)

where the Christoffel symbols are defined by \(\Gamma ^\gamma _:= \frac\textbf^} \left( _\mu \textbf_\nu }+_\nu \textbf_\mu }\right. \left. -_\textbf_\right) \).

1.1.2 Definition of u on \(\tilde}}}}_2\) and Analysis of Foliation Geometry

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

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

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

(A.10)

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

$$\begin \begin _u = \left( 1+_u f\right) _}}}, \,\, _ = _}^A} + (_ f) _}}}, \,\, _ f =\left( 1+_u f\right) _}^A} f . \end \end$$

(A.11)

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

$$\begin \begin }}:= _u, \,\, }}' := -2 \textbf} = \tilde}}}', \end \end$$

(A.12)

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

$$\begin \begin }}= ^2 }}'. \end \end$$

(A.13)

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

(1)

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

$$\begin \begin }}= \left( 1+_u f\right) }}}} = \left( 1+_u f\right) \tilde^\tilde}}}' = \left( 1+_u f\right) \tilde^ }}}', \end \end$$

(A.14)

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

$$\begin \begin \Omega ^2 = \tilde^2 \left( 1+_u f\right) . \end \end$$

(A.15)

(2)

By (A.11), it follows that the induced metric \(/\ \hspace}\) on level sets of u on \(\tilde}}}_2\) is given for \(A,B=1,2\) by

$$\begin \begin /\ \hspace}_ := \textbf\left( _A, _B\right) = \textbf\left( _}}}, _}}}\right) = \tilde/\ \hspace}}_. \end \end$$

(A.16)

This implies further that

$$\begin \begin /\ \hspace}^ = \tilde/\ \hspace}}^. \end \end$$

(A.17)

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

$$\begin \begin \Omega ^2(u,\theta ^1,\theta ^2)&= \left( 1+(_u f)(u,\theta ^1,\theta ^2)\right) \tilde^2(u+f(u,\theta ^1,\theta ^2),\theta ^1,\theta ^2),\\ /\ \hspace}_(u,\theta ^1,\theta ^2)&= \tilde/\ \hspace}}_\left( u+f(u,\theta ^1,\theta ^2),\theta ^1,\theta ^2\right) . \end \end$$

(3)

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

$$\begin \begin \textbf\left( L,}}\right) = -2 \Omega ^2, \,\, \textbf\left( L,_\right) =\textbf\left( L,_\right) =0. \end \end$$

(A.18)

An explicit calculation shows that L is given by

$$\begin \begin L&= \left( \tilde^2\vert \nabla \hspace/\ f \vert _/\ \hspace}^2 \right) \tilde}}} + } + \left( 2\tilde^2 \tilde/\ \hspace}}^_C f \right) _}^A}. \end \end$$

(A.19)

where \(\vert \nabla \hspace/\ f \vert _/\ \hspace}^2:= \tilde/\ \hspace}}^_Af _B f\). We define further \(}:= \Omega ^ L\).

1.1.3 Analysis of Ricci Coefficients on \(\tilde}}}_2\)

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

$$\begin \begin \chi _&:= \textbf(\textbf_ },_B),&}}_&:= \textbf(\textbf_A \widehat}}},_B),&\zeta _A&:= \frac\textbf(\textbf_A },\widehat}}}), \\ \eta&:= \zeta + /\,}\log \Omega ,&\omega&:= L \log \Omega ,&}&:= }}\log \Omega . \end \end$$

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

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

$$\begin \begin }:= }}\log \Omega = \Omega ^ _u \Omega = \frac _u \left( \Omega ^2\right) . \end \end$$

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

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

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

$$\begin \begin }&= \frac \left( 2\tilde _}}} \tilde \left( 1+_u f\right) ^2 + \tilde^2 _u^2 f\right) = \tilde}} \left( 1+_u f\right) + \frac\frac^2} _u^2 f. \end \end$$

(A.20)

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

$$\begin \begin }}_ := \textbf(\textbf_A \widehat}}},_B) =\Omega ^ \left( 1+_u f\right) \tilde \tilde}}}_, \end \end$$

(A.21)

where we used that

$$\begin \begin \textbf\left( \textbf__}}}} _}}}, _}}}\right) = \tilde^4 \textbf\left( \textbf_}}}'} \tilde}}}', _}}}\right) = 0. \end \end$$

We can separate (A.21) into

$$\begin \begin \Omega }}}= \left( 1+_u f\right) \tilde\tilde}}}}, \,\, }}}}_ = \frac} \left( 1+_u f\right) \tilde}}}}}_, \end \end$$

(A.22)

that is, in explicit notation,

$$\begin \begin \left( \Omega }}}\right) (u,\theta ^1,\theta ^2)&= \left( 1+_u f(u,\theta ^1,\theta ^2)\right) \tilde \textrm\tilde}}}(u+f(u,\theta ^1,\theta ^2),\theta ^1,\theta ^2),\\ }}}}_(u,\theta ^1,\theta ^2)&=\frac}(u+f(u,\theta ^1,\theta ^2),\theta ^1,\theta ^2) \left( 1+_u f(u,\theta ^1,\theta ^2)\right) \\&\quad \tilde}}}}}_(u+f(u,\theta ^1,\theta ^2),\theta ^1,\theta ^2). \end \end$$

Analysis of \(\omega \). We calculate

$$\begin \begin \omega := L \log \Omega \end \end$$

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

$$\begin \begin L'(}}) = \Omega ^. \end \end$$

(A.23)

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

$$\begin \begin L'\left( \Omega ^\right) = L'\left( L'(}}) \right) =\textbf_\left( \textbf_ }}\right) = \textbf__L'}_} }} + \textbf_ \textbf_ }}= \Omega ^ \textbf_L \textbf_L }}, \end \end$$

(A.24)

where we note that \(\textbf\textbf}}\) is the covariant Hessian.

Second, we have the algebraic relation

$$\begin \begin L'\left( \Omega ^\right) = -\frac L' \left( \Omega \right) . \end \end$$

(A.25)

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

$$\begin \begin \omega = L \left( \log \Omega \right) = \Omega L' \left( \Omega \right) = -\frac L' \left( \Omega ^\right) = -\frac \textbf_\textbf_L }}. \end \end$$

(A.26)

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

$$\begin \begin \omega&= -\frac \left( \tilde^2 \vert \nabla \hspace/\ f \vert ^2_/\ \hspace}\right) ^2 \textbf_}} }\textbf_}}} }} \underbrace \textbf_}}} \textbf_}}} }}_}} -2\tilde^4 \tilde/\ \hspace}}^ \tilde/\ \hspace}}^(_B f)(_C f) \textbf_}}} \textbf_}}} }} \\&- \left( \tilde^2 \vert \nabla \hspace/\ f \vert ^2_/\ \hspace}\right) \textbf_}}} \textbf__}}} +}^C _}}} } }} - \left( \tilde^2 \vert \nabla \hspace/\ f \vert ^2_/\ \hspace}\right) \left( 2\tilde^2 \tilde/\ \hspace}}^_B f \right) \textbf_}}} \textbf_}}} }} \\&- \left( 2\tilde^2 \tilde/\ \hspace}}^_B f \right) \textbf__}}} +}^C _}}}}\textbf_}}} }}. \end \end$$

(A.27)

Here, the Hessian \(\textbf\textbf}}\) is given in coordinates \(\mu , \nu \in \}}, }}, }^1, }^2\} \) by

$$\begin \begin \textbf_\mu \textbf_\nu }}&= _\mu _\nu }} - \Gamma ^_ _ }} = - \Gamma ^}}}_. \end \end$$

(A.28)

From (A.9) and (A.28), we conclude that \(\textbf_}}}\textbf_}}}}}= \textbf_}}} \textbf_}}}}} = \textbf_}}} \textbf_}}}}} =0\) and

$$\begin \begin \textbf_}}} \textbf_}}} }}= - _}}} \log } + \tilde_A + \frac}} \tilde}}}_ }^B, \,\, \textbf_}}} \textbf_}}} }}= - \frac}} \tilde}}}_. \end \end$$

(A.29)

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

$$\begin \begin \omega&= } +\tilde^3 \tilde}}}^(_A f)(_B f) +\left( 2\tilde^2 \tilde/\ \hspace}}^_B f \right) \left( _}}} \log } - \tilde_A \right) . \end \end$$

(A.30)

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

$$\begin \begin&\zeta _A := \frac\textbf(\textbf_A }, \widehat}}}) \\ &\quad = \frac^2} _A_u f - \frac \left( 1+_u f\right) \textbf\left( L, \textbf_A _}}}\right) - _A \log \Omega . \end \end$$

(A.31)

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

$$\begin \begin \textbf\left( L, \textbf_A _}}}\right)&= -2 \tilde^2 \left( \tilde_A + _}}} \log \tilde + 2 (_A f) }}} - \tilde\tilde/\ \hspace}}^(_Bf) \tilde}}}_\right) . \end \end$$

(A.32)

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

$$\begin \begin \zeta _A&= - _A \log \Omega + \frac^2} _A_u f \\&+ \frac^2} \left( 1+_u f\right) \left( \tilde_A + _}}} \log \tilde + 2 (_A f) }}} - \tilde\tilde/\ \hspace}}^(_Bf) \tilde}}}_\right) \end \end$$

(A.33)

We conclude from the above that

$$\begin \begin \eta _A&:= \zeta _A +_A \log \Omega \\&= \frac^2} _A_u f + \frac^2} \left( 1+_u f\right) \left( \tilde_A + _}}} \log \tilde + 2 (_A f) }}} - \tilde\tilde/\ \hspace}}^(_Bf) \tilde}}}}}_\right) \\&- \frac^3} \left( 1+_u f\right) (_Af) \textrm\tilde}}}. \end \end$$

(A.34)

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

$$\begin \begin \chi _ := \textbf(\textbf_ },_B)&= \frac^3} \vert \nabla \hspace/\ f \vert _/\ \hspace}^2 \tilde}}}_ +\frac}}\tilde_ + \Omega ^ (_A f) \textbf\left( \textbf_}}}}} }}, _}}}\right) \\&+ \Omega ^ (_A f)(_Bf) \textbf\left( \textbf_}}}}}}}, }}}}\right) +\Omega ^(_Bf)\textbf\left( \textbf_}}} }}, }}}}\right) \\&+ \Omega ^\textbf\left( \textbf_A \left( \left( 2\tilde^2 \tilde/\ \hspace}}^_C f \right) _}}}\right) ,_B\right) . \end \end$$

(A.35)

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

$$\begin \begin \textbf\left( \textbf_}}}}} }}, _}}}\right)&=2 \tilde^2 \tilde_B, \,\, \textbf\left( \textbf_}}} }}, }}}}\right) = 2\tilde^2 \left( \tilde_A - 2 _}}} \log \tilde\right) , \end \end$$

as well as

$$\begin \begin \textbf\left( \textbf_}}}}}}}, }}}}\right)&= \tilde^2 \textbf\left( \textbf_}}}}} }}, \tilde}}}'\right) = -\tilde^2 \textbf\left( }}, \textbf_}}}}} \tilde}}}'\right) = -\tilde^4 \textbf(}}, \underbrace_}}}'} \tilde}}}'}_) =0, \end \end$$

and

$$\begin&\textbf\left( \textbf_A \left( 2\tilde \tilde/\ \hspace}}^ (_C f) _}}} \right) , _B\right) = _A \left( 2\tilde^2\right) _B f\\&\quad + \left( 2\tilde^2\right) \left( (_A_B f) + (_Cf) \textbf\left( \textbf_A \left( \tilde/\ \hspace}}^_}}} \right) , _B\right) \right) , \end$$

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

$$\begin \begin \textbf\left( \textbf_A \left( \tilde/\ \hspace}}^_}}} \right) , _B\right) = -\tilde^C_ - (_Af) \tilde \tilde}}}_ \tilde/\ \hspace}}^- (_Bf) \tilde/\ \hspace}}^ \tilde \tilde}}}_, \end \end$$

yielding that

$$\begin \begin&\textbf\left( \textbf_A \left( 2\tilde \tilde/\ \hspace}}^ (_C f) _}}} \right) , _B\right) \\&= _A \left( 2\tilde^2\right) _B f + \left( 2\tilde^2\right) \left( (_A_B f) - (_Cf) \tilde^C_\right) \\&- \left( 2\tilde^2\right) \left( (_Af)\tilde\tilde}}}_\tilde/\ \hspace}}^(_Cf) + (_Bf)\tilde\tilde}}}_\tilde/\ \hspace}}^(_Cf)\right) . \end \end$$

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

$$\begin \chi _&= \frac^3} \vert \nabla \hspace/\ f \vert _/\ \hspace}^2 \tilde}}}_ +\frac}}\tilde_ \nonumber \\&+\frac^2} \left( (_A f)\tilde_B + (_B f)\tilde_A\right) +\frac^2} \left( _A_B f - \tilde^C_ _C f\right) \nonumber \\&- \frac^2} \left( (_Af)\tilde\tilde}}}_\tilde/\ \hspace}}^(_Cf) + (_Bf)\tilde\tilde}}}_\tilde/\ \hspace}}^(_Cf)\right) . \end$$

(A.36)

Analysis of \(}}\). We have by explicit computation that

$$\begin \begin }}:= _u \left( _u \log \Omega \right) = \frac_u^3 f}_u f\right) } - \frac_u^2 f\right) ^2}_u f\right) ^2}+ \tilde}} }}} \left( 1+_uf\right) ^2 + }}} _u^2 f. \end \end$$

(A.37)

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

$$\begin \begin L'\left( \Omega \right) = -\frac L'\left( \Omega ^\right) = -\frac L'\left( L'(}})\right) , \end \end$$

we have that

$$\begin \begin D\omega&= L \left( L \log \Omega \right) = \Omega ^2 L' \left( \Omega L' \left( \Omega \right) \right) = \Omega ^2 L' \left( -\fracL'\left( L'(}})\right) \right) \\&= 4 \omega ^2 - \frac L'\left( L' \left( L'(}})\right) \right) . \end \end$$

Using that \(\textbf_L' =0\), it follows further that

$$\begin \begin D\omega =&4 \omega ^2 - \frac \textbf_ \textbf_ \textbf_ }}= 4\omega ^2 - \frac \textbf_L \textbf_L \textbf_L }}. \end \end$$

By (A.19), we can furthermore expand \(\textbf_L \textbf_L \textbf_L }}\) (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

$$\begin \begin 2 \left( 2\tilde^2 \tilde/\ \hspace}}^(_Af)\right) \textbf_}}} \textbf_}}} \textbf_}}} }} \text 2\tilde^2 \tilde/\ \hspace}}^(_E f) \textbf_}}} \textbf_}}} \textbf_}}} }}, \end \end$$

and we note that at Minkowski,

$$\begin \begin \textbf_}}}\textbf_}}}\textbf_}}}}}= \textbf_}}} \textbf_}}} \textbf_}}} }} = \textbf_}}} \textbf_}}} \textbf_}}} }} =0. \end \end$$

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

1.1.4 Calculation of Null Curvature Components on \(\tilde}}}_2\)

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

$$\begin \begin \alpha _&:= }(_A,}, _B, }),&\beta _A&:= \frac}(_A, },\widehat}}},}),&&:= \frac }(\widehat}}}, }, \widehat}}}, }), \\ \sigma \in \hspace/\ \hspace_&:= \frac}(_A,_B,\widehat}}}, }),&}_A&:= \frac}(_A, \widehat}}},\widehat}}},}),&}_&:= }(_A,\widehat}}}, _B, \widehat}}}). \end \end$$

(A.38)

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

$$\begin \begin _&= _}^A} + (_ f) _}}},&\Omega ^2&= \tilde^2 \left( 1+_u f\right) , \\ }}&= \left( 1+_u f\right) }}}},&L&= \left( \tilde^2 \vert \nabla \hspace/\ f\vert ^2_/\ \hspace}\right) \tilde}}} + } + \left( 2\tilde^2 \tilde/\ \hspace}}^_C f \right) _}^A}, \end \end$$

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

1.2 Proof of Proposition 2.21

In this section, we prove Proposition 2.21; that is, we discuss the mapping \(}}_(\tilde}})\) and prove estimates.

First, recall from (2.56) that

$$\begin \begin }}_(\tilde}}):= }}_}}_(\tilde}}), \end \end$$

and that in Sects. A.1.1-A.1.4 we discussed the explicit formulas for \(}}_(\tilde}})\).

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

$$\begin \begin \Vert q \Vert _}_q} \le \varepsilon _0, \end \end$$

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

$$\begin \begin \Vert \Phi _1(q)^(T) - T \Vert _)} \le&C_(S_)}} \Vert q \Vert _}_q}, \\ \Vert \Phi _1(q)^(T) \Vert _)} \le&\left( 1+ \Vert q \Vert _}_q}\right) \Vert T \Vert _(S_)}. \end \end$$

(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 \(q\in }_q\) is a smooth mapping from tensors in \(H^m(S_)\) to tensors in \(H^m(S_)\) for integers \(0\le m \le 6\).

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

$$\begin \begin x_ := }}_\left( \tilde}}\right) = }}_}}_(\tilde}}). \end \end$$

In the following, we prove that

$$\begin \begin \Vert }}_(\tilde}}) - \tilde}}_ \Vert _}(S_)} \lesssim \Vert f \Vert _}_} + \Vert q \Vert _}_q} + \Vert \tilde}}-\underline}} \Vert _}^+(\tilde}}}_)}, \end \end$$

(A.40)

where we recall from Definition 2.5 the sphere data norm

$$\begin \begin \Vert x_ \Vert _}(S_)}&:= \Vert \Omega \Vert _(S_)} +\Vert /\ \hspace}\Vert _(S_)} + \Vert \Omega \chi }\Vert _(S_)} + \Vert }}\Vert _(S_)}\\&+ \Vert \Omega }}}\Vert _(S_)} + \Vert }}}}\Vert _(S_)} + \Vert \eta \Vert _(S_)} \\&+ \Vert \omega \Vert _(S_)}+ \Vert D\omega \Vert _(S_)}+\Vert }\Vert _(S_)} + \Vert }}\Vert _(S_)} \\&+ \Vert \Vert _(S_)} +\Vert }\Vert _(S_)}, \end \end$$

and from Definition 2.20 the sphere perturbation function norms

$$\begin \begin \Vert f \Vert _}_f}&:= \Vert f(0) \Vert _}^2)} + \Vert _u f(0) \Vert _}^2)}+\Vert _u^2 f(0) \Vert _}^2)}+\Vert _u^3 f(0) \Vert _}^2)}, \\ \Vert q \Vert _}_q}&:= \Vert q_1 \Vert _}^2)} + \Vert q_2 \Vert _}^2)}. \end \end$$

Indeed, the proof is based on three ingredients. First, we work with a higher regularity \(\tilde}}\) along \(\tilde}}}_\) and thus higher derivatives falling on \(\tilde}}\) 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 \(\tilde}}\)) 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,

$$\begin \begin }}(u,\theta ^1,\theta ^2)&= \left( \frac_u^3 f}_u f\right) } - \frac_u^2 f\right) ^2}_u f\right) ^2}+ \tilde}} }}} \left( 1+_uf\right) ^2 + }}} _u^2 f\right) \\&\quad \left( u+f(u,\theta ^1,\theta ^2), \theta ^1,\theta ^2\right) , \end \end$$

and that by definition of \(}}_\),

$$\begin \begin }}:= }}\left( }}_(\tilde}})\right) \circ \Phi _1(q), \end \end$$

where \(}}\left( }}_(\tilde}})\right) \) denotes the component \(}}\) of \(}}_(\tilde}})\), we estimate

$$\begin \begin&\Vert }}- \tilde}}\tilde}}\Vert _)}\\&\quad \le \Vert \tilde}}\tilde}}\circ \Phi _1(q) - \tilde}}\tilde}}\Vert _)}\\&\qquad + \left\| \left( \frac_u^3f}_uf)} - \frac_u^2f)^2}_uf)^2} + (2_uf + (_uf)^2)\tilde}}\tilde}}+\tilde}}_u^2f\right) \circ \Phi _1(q)\right\| _)}\\&\quad \lesssim \Vert \tilde}}-\underline}} \Vert _}^+(\tilde}}}_)} + \Vert q \Vert _}_q} + \Vert f \Vert _}_f}\\&\qquad + (1+\Vert q \Vert _}_q}) \left( \Vert f \Vert _}_f} + \Vert \tilde}}-\underline}} \Vert _}^+(\tilde}}}_)}\right) , \end \end$$

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 \(}}\) along \(}}\). In Sect. B.2, we derive the linearized null transport equations of Lemma 4.14 for \(}}}, }}}}\) and \(}}}\).

1.1 Derivation of Null Transport Equation for \(}}\)

In this section, we prove the transport Eq. (2.20) for \(}}\) along \(}}\). We remark that in case of a geodesic foliation on \(}}=}}_\), that is, \(\Omega \equiv 1\) on \(}}\), 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,

$$\begin }D W - D }W = \mathcal \hspace/\ \hspace_ W. \end$$

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

$$\begin D }= \Omega ^2\left( 2 (\eta ,}}) - \vert \eta \vert ^2 -\right) . \end$$

(B.1)

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

$$\begin \begin D}}&= -4\Omega ^2 \zeta (}) + }D}\\&= -4\Omega ^2 \zeta (})+ }\left( \Omega ^2 \left( 2(\eta ,}})-\vert \eta \vert ^2 - \right) \right) \\&=-4\Omega ^2 \zeta (}) +2 \Omega ^2 }\left( 2(\eta ,}})- \vert \eta \vert ^2 - \right) \\&\quad +\Omega ^2\left( 4\Omega }}(\eta ,}}) + 2 \left( -\Omega \left( }}\cdot \eta + }\right) + 2 /\,}}, }}\right) +2 \left( \eta , \Omega \left( }}\cdot \eta + }\right) \right) \right) \\&\quad + \Omega ^2 \left( -2\Omega }}(\eta , \eta ) - 2\left( -\Omega \left( }}\cdot \eta + }\right) + 2/\,}}, \eta \right) \right) \\&\quad + \Omega ^2 \left( \frac\Omega }}}\rho +\Omega \left( }\,}}\hspace/\ }+(2\eta -\zeta ,})+\frac(}},})\right) \right) \\&= -12 \Omega ^2 (\eta -/\,}\log \Omega ,/\,}})+2\Omega ^2}\left( (\eta ,-3\eta +4/\,}\log \Omega )-\rho \right) \\&\quad +4\Omega ^3}}(\eta ,/\,}\log \Omega ) +\Omega ^3 \left( },7\eta -3/\,}\log \Omega \right) \\ &\quad + \frac \Omega ^3}}}+ \Omega ^3 }\,}}\hspace/\ }+ \frac (}}, }). \end \end$$

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

1.2 Derivation of Transport Equations for \(}}}\), \(}}}\) and \(}}}}\)

In this section, we prove Lemma 4.14. To simplify notation, we use that in Minkowski on \(}}=}}_\) it holds that \(r=v\). First we recall from (4.5) that

$$\begin \begin }}_1&:=\frac \left( \dot\chi )}-\frac}\right) + \frac}}, \\ }}_2&:= v^2 \dot}})}-\frac}\,}}\hspace/\ }}\left( v^2}+\frac/\,}\left( \dot\chi )}-\frac}\right) \right) -v^2 \left( \dot\chi )}-\frac}\right) +2v^3 }, \\ }}_3&:= \frac}}}}}}} -\frac\left( }}\hspace/\ \hspace_2^*}\,}}\hspace/\ }}+1\right) /\ \hspace}_c}}+ }}\hspace/\ \hspace_2^*\left( }+ \frac/\,}\left( \dot\chi )}-\frac}\right) \right) - v }}\hspace/\ \hspace_2^*/\,}\left( \dot\chi )}-\frac}\right) . \end \end$$

and that by Lemmas 4.8 and 4.11,

$$\begin \begin D}}_1&=}}_1-D\left( \frac }}_2\right) , \\ D}}_2&= v^2 }}_5 -2v }\,}}\hspace/\ }}}}_4 -2v(/\ }}+1)}}_1+ D\left( (/\ }}+1)}}_2\right) \\ &\quad -\frac(/\ }}+2) }}_2+ \frac}\,}}\hspace/\ }}}\,}}\hspace/\ }}}}_3, \\ D}}_3&= \frac}}_6 -\frac\left( }}\hspace/\ \hspace_2^*}\,}}\hspace/\ }}+1\right) }}_3}+ }}\hspace/\ \hspace_2^*}}_4-}}\hspace/\ \hspace_2^*/\,}}}_1 +D\left( \frac}}\hspace/\ \hspace_2^*/\,}}}_2\right) \\ &\quad + \frac}}\hspace/\ \hspace_2^*/\,}}}_2. \end \end$$

Transport equation for \(}}}\). We have by using (2.16) that

$$\begin \begin&D\left( }}}+\frac}}_2 +\frac }\,}}\hspace/\ }}\left( }+\frac/\,}\left( \dot\chi )}-\frac}\right) \right) \right) \\&= }}_7 + }+ \frac \dot}})}-\frac \dot\chi )}+ \frac }+ \fracD}}_2 \\&\quad -\frac \left( v^2\dot}})}-\frac}\,}}\hspace/\ }}\left( v^2}+\frac/\,}\left( \dot\chi )}-\frac}\right) \right) -v^2 \left( \dot\chi )}-\frac}\right) +2v^3 }\right) \\&\quad -\frac }\,}}\hspace/\ }}\left( }+\frac/\,}\left( \dot\chi )}-\frac}\right) \right) \\&\quad +\frac }\,}}\hspace/\ }}\left( }\,}}\hspace/\ }}}}}}+v^2}}_4+v^2 /\,}}}_1+\frac/\,}}}_2-D\left( \frac/\,}}}_2\right) \right) \\&= }}_7+ \fracD}}_2+\frac }\,}}\hspace/\ }}\left( }\,}}\hspace/\ }}}}}}+v^2}}_4+v^2 /\,}}}_1+\frac/\,}}}_2-D\left( \frac/\,}}}_2\right) \right) \\&= }}_7+ \fracD}}_2+\frac }\,}}\hspace/\ }}\left( }\,}}\hspace/\ }}}}}}+v^2}}_4+v^2 /\,}}}_1 - /\,}}}_2\right) -D\left( \frac/\ }}}}_2\right) \\&= }}_7+ \frac}}_5 + \frac }\,}}\hspace/\ }}}\,}}\hspace/\ }}}}_3 - \frac }\,}}\hspace/\ }}}}_4-\frac(/\ }}+3) }}_1 -\frac /\ }}}}_2 + \frac }\,}}\hspace/\ }}}\,}}\hspace/\ }}}}}}. \end \end$$

Transport equation for \(}}}\). Using the above definition of \(}}_1\), \(}}_2\) and \(}}_3\), we have by the system (4.2),

$$\begin&D\left( \frac}}}} +\frac}}\hspace/\ \hspace_2^*}\,}}\hspace/\ }}}}_3 - \frac }}\hspace/\ \hspace_2^*/\,}}}_2 - \frac }}\hspace/\ \hspace_2^*/\,}\left( /\ }}+2\right) }}_1 \right) \\&= \frac}}_8 +\frac}}\hspace/\ \hspace_2^*\left( \frac}\,}}\hspace/\ }}}}}}}}-\frac/\,}\dot}})}-\frac}\right) -\frac}}\hspace/\ \hspace_2^*}\,}}\hspace/\ }}}}_3 +\frac}}\hspace/\ \hspace_2^*}\,}}\hspace/\ }}\left( D}}_3\right) \\&\quad +\frac}}\hspace/\ \hspace_2^*/\,}}}_2 -\frac}}\hspace/\ \hspace_2^*/\,}\left( D}}_2\right) +\frac}}\hspace/\ \hspace_2^*/\,}\left( /\ }}+2\right) }}_1 -\frac}}\hspace/\ \hspace_2^*/\,}\left( /\ }}+2\right) \left( D}}_1\right) \\&= \frac}}_8 +\frac}}\hspace/\ \hspace_2^*\left( \frac}\,}}\hspace/\ }}}}}}}}-\frac/\,}\dot}})}-\frac}\right) \\&\quad +\frac}}\hspace/\ \hspace_2^*}\,}}\hspace/\ }}\left( D}}_3\right) -\frac}}\hspace/\ \hspace_2^*/\,}\left( D}}_2\right) -\frac}}\hspace/\ \hspace_2^*/\,}\left( /\ }}+2\right) \left( D}}_1\right) \\&\quad -\frac}}\hspace/\ \hspace_2^*}\,}}\hspace/\ }}\left( \frac}}}}}}} -\frac\left( }}\hspace/\ \hspace_2^*}\,}}\hspace/\ }}+1\right) /\ \hspace}_c}}+ }}\hspace/\ \hspace_2^*\left( }+ \frac/\,}\left( \dot\chi )}-\frac}\right) \right) \right. \\ &\left. - v }}\hspace/\ \hspace_2^*/\,}\left( \dot\chi )}-\frac}\right) \right) \\&\quad +\frac}}\hspace/\ \hspace_2^*/\,}\left( v^2 \dot}})}-\frac}\,}}\hspace/\ }}\left( v^2}+\frac/\,}\left( \dot\chi )}-\frac}\right) \right) \right. \\ &\left. -v^2 \left( \dot\chi )}-\frac}\right) +2v^3 }\right) \\&\quad +\frac}}\hspace/\ \hspace_2^*/\,}\left( /\ }}+2\right) \left( \frac \left( \dot\chi )}-\frac}\right) + \frac}}\right) , \end$$

and

$$\begin \begin&-\frac}}\hspace/\ \hspace_2^*\left( }\,}}\hspace/\ }}}}\hspace/\ \hspace_2^*+1 + /\,}}\,}}\hspace/\ }}\right) D\left( \frac\left( v^2}+\frac/\,}\left( \dot\chi )}-\frac}\right) \right) \right) \\&\quad =\frac }}\hspace/\ \hspace_2^*\left