In this section, we prove Proposition 4.2 by proving the solvability of the linearised \(C^3\)-null gluing problem which we formulate as follows.

Proposition 5.1

(Linearised \(C^3\)-null gluing) Given

linearised \(C^3\)-sphere data on \(S_\), \(}}}_\in }}(S_)\),

linearised \(C^3\)-matching data on \(}_\subset \underline}}_\), \(\tilde}}_\in }}_}(}_)\),

linearised source functions \(}}}_}}\in }}_}}}\) on \(}}_\) for

$$\begin }\in \},\dot\chi )},}}}}},}},\dot}}})},}}}}}}}}, }}}}},}}}}}},}^2}}}}}, }}, }}}}},}}}}}}\}, \end$$

there exists linearised null data \(}}}}\in }}(}}_)\) and perturbation functions \(}\) and \(}\) such that

$$\begin }}}}\big \vert _} = }}}}_, \qquad }}(}}}}\big \vert _} + }}_(},})) = }}}_, \end$$

(5.1)

and \(}}}}\) solves the linearised null structure equations for the source functions

$$\begin }}}}_}}(}}}}) = }}}_}}}(}}}}). \end$$

(5.2)

Furthermore, \(}}}}\) and the perturbations are bounded as follows,

$$\begin&\Vert }}}}\Vert _}}(}}_)} + \Vert }\Vert _}}_}}+\Vert }\Vert _}}_q} + \Vert }}_(},})\Vert _}}(S_)},\nonumber \\&\quad \lesssim \Vert }}}}_\Vert _}}(S_)} + \Vert }}}_\Vert _}}_}}(}_)} + \Vert }}}_}}}\Vert _}}_}}}}. \end$$

(5.3)

Remark 5.2

We make the following remarks on Proposition 5.1.

(1)

In the linearised setting, the characteristic seed along \(}}_\) can be taken to be the pair \((}},}}}}})\) as a result of the linearised first variation equation, see the third equation of (B.1). Since we will construct the characteristic seed along \(}}_\) and \((}},}}}}})\) are given on \(S_1\) and \(S_2\), the gluing of the D derivative of these quantities is trivial. That is the quantities

$$\begin (}},D}},D^2}},}}}}},D}}}}}) \end$$

are glued at \(S_2\) given that the characteristic seed \((}},}}}}})\) matches the data given on \(S_2\).

(2)

The proof of Proposition 5.1 employs the analysis of the linearised \(C^2\)-null gluing problem of Theorem 4.1 of [3] as well as an analysis of the quantities novel in the \(C^3\)-sphere data. Therefore, we analyse the novel components of the \(C^3\)-sphere data

$$\begin (D^2}},D}}.}^2}}}}}, }}}), \end$$

and employ the linearised \(C^2\)-null gluing result of Theorem 4.1 in [3] to glue the linearised \(C^2\)-sphere data.

We will prove Proposition 5.1 by splitting the given data into its constituent \(C^2\)- and \(C^3\)-parts. The \(C^2\)-part has already been solved for in [3] and we collect the necessary formulas we need in Section 5.1. The gluing of the \(C^3\)-part is contained in the following proposition.

Proposition 5.3

Denote by \(}_1\) the \(C^3\)-part of the linearised sphere data

$$\begin }_1:= (D^2}}, }^2}}}}},}}},\widehat}}}}}}}) \end$$

on \(S_1\). Denote by \(\tilde}}_2\)the \(C^3\)-part of the matching data

$$\begin \tilde}}_2:=(D^\omega ,}^}}^, \tilde}}}_}^}}}}}^},},\widehat}}}}^) \end$$

on \(}_2\). Let \(}_1\), \(\tilde}}_2\) and the higher-order source functions \(}}}_}^2}}}}}}\) and \(}}}_}}}}}}}\) on \(}}_\) be given. Then there exists linearised null data \(}}}}\in }}(}}_)\) and perturbation functions \(}\) and \(}\) such that the restriction of \(}}}}\) to its \(C^3\)-part, \(}\), satisfies

$$\begin }\vert _} = }_1, \qquad }(}}}\vert _} + }}_(},})) = \tilde}}_, \end$$

(5.4)

and \(}}}}\) solves the linearised null structure equations for the source functions

$$\begin }}}}_}^2}}}}}}(}}}}) = }}}_}^2}}}}}}(}}}})\qquad }}}}_}}}}}}}(}}}}) = }}}_}}}}}}}(}}}}). \end$$

(5.5)

This section is organised as follows.

In Section 5.1, we present the necessary formulas and estimates from the linearised \(C^2\)-null gluing problem of [3] which we will need to prove Proposition 5.1

In Section 5.2 we derive transport equations and representation formulas for the novel \(C^3\) components of the sphere data, \(}\).

In Section 5.3, we analyse the novel conservation laws that appear in the representation formulas derived in Section 5.2.

In Section 5.4, we investigate the gauge dependence of the conserved quantities under the sphere perturbations of Section 2.3.

Finally, in Section 5.5, we apply the results of the preceding sections in order to construct a characteristic seed \((}},}}}}})\) and associated solution to the null structure equations along \(}}_\). The solution is constructed so that it satisfies the conditions (5.1) and (5.2). Moreover, we prove the bound (5.3) and conclude the proof of Proposition 5.1.

5.1 The Solution of the Linearised \(C^2\)-Null Gluing Problem

In this section, we collect the necessary results and estimates in order to apply the linearised \(C^2\)-null gluing problem of [3] as a black box. We may write the main result of their linear analysis (Theorem 4.1 of [3]) as follows.

Proposition 5.4

Given

linearised \(C^2\)-sphere data \(}}_1\in X^8(S_1)\),

linearised \(C^2\)-matching data \(}_2\in }}_}}(}_2)\),

linearised source functions \(}}}_}}\in }}_}}}\) on \(}}_\) for

$$\begin }\in \},\dot\chi )},}}}}},}},\dot}}})},}}}}}}}}, }}}}},}}}}}}, }}, }}}}}\}, \end$$

there exists linearised null data \(}}\in }}(}}_)\) and perturbation functions \(}\) and \(}\) such that

$$\begin }}\vert _} = }}_, \qquad (}}\vert _} + }}_(},})) = }_, \end$$

(5.6)

and \(}}\) solves the linearised null structure equations for the source functions

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

(5.7)

with \(}\in \},\dot\chi )},}}}}},}},\dot}}})},}}}}}}}}, }}}}},}}}}}}, }}, }}}}}\}\). Furthermore, \(}}\), \(\dot\) and \(\dot\) are bounded as follows,

$$\begin&\Vert }}\Vert _}}_)} + \Vert }\Vert _}}_}}+\Vert }\Vert _}}_q} + \Vert }}_(},})\Vert _)},\nonumber \\&\quad \lesssim \Vert }}_\Vert _)} + \Vert }_2\Vert _}}_}}(}_)} + \Vert }}}_}}}\Vert _}}_}}}}. \end$$

(5.8)

The proof of Proposition 5.4 can be found in Section 4 of [3]. We will make use of the following representation formula for \(}}}}}}\) from equation (4.40) in [3].

$$\begin \begin&\left[ }}}}}}-\frac \left( /\ }}-3\right) }}_}}})}}+ \frac }\,}}\hspace/\ }}}\,}}\hspace/\ }}}}_}}}}}}}}+\frac }\,}}\hspace/\ }}}\,}}\hspace/\ }}}}\hspace/\ \hspace_2^* /\,}}}_}}\right] _1^v\\&\qquad -\left[ \frac }\,}}\hspace/\ }}\left( /\,}}\,}}\hspace/\ }}-2 + }\,}}\hspace/\ }}}}_2^*\right) \left( \eta + \frac/\,}\left( \dot\chi )}-\frac}}\right) \right) \right] _1^v\\&\qquad + \left[ \frac }\,}}\hspace/\ }}/\,}}\,}}\hspace/\ }}}\,}}\hspace/\ }}/\ \hspace}_c}}+ \frac\left( \frac/\ }}/\ }}-\frac/\ }}+\frac}\,}}\hspace/\ }}}\,}}\hspace/\ }}}}\hspace/\ \hspace_2^* /\,}- 1\right) }}}_2\right] _1^v\\&\quad = \frac }\,}}\hspace/\ }}\left( 2-}\,}}\hspace/\ }}}}\hspace/\ \hspace_2^*\right) }\,}}\hspace/\ }}\left( \int _1^v \frac}}}}}dv'\right) +\int _1^v h_}}}}}}} dv'. \end \end$$

(5.9)

where \(h_}}}}}}}\) is a function of the source terms \(}}}_\).

5.2 Representation Formulas for Higher-Order Sphere Data

In this section, we derive representation formulas for \(}}}}}}\) and \(}^2}}}}}\). In order to derive the following representation formulas, consider some given sphere data \(}_\) on \(S_\) and linearised characteristic seed \((},}}}})\) on \(}}_\). The representation formulas construct the null data along \(}}_\) used to solve the linearised characteristic gluing problem. We will also make use of the linearised null structure equations (2.14) as source terms in the null structure equations. We therefore set \(}}}}_}}=}}}_}}\). For ease of notation, we evaluate at \(u=0\) and set \(}}=}}_\).

5.2.1 Analysis of \(}}}}}}\).

Recall from Lemma 2.16 that the linearised null constraint equation for \(}}}}}}\) is

$$\begin vD\left( \frac}}}}}}\right) +\frac}(2}}\hspace/\ \hspace_^}\,}}\hspace/\ }}-1)}}}}}-\frac}}\hspace/\ \hspace_^\left( \frac}}\,}}\hspace/\ }}}}}}}}}}-\frac/\,}\dot}}})}-\frac}}\right) =}}}_}}}}}}}. \end$$

(5.10)

Using the linearised null structure equations, we obtain the following lemma.

Lemma 5.5

(Representation formula of \(}}}}}}\)) The null structure equation (5.10) implies the following null transport equation for \(}}}}}}\)

$$\begin \begin&D\left( \frac}}}}}}+\frac}(1-2}}\hspace/\ \hspace_^}\,}}\hspace/\ }})}}}}}\right) \\&\qquad +D\left( \frac}}}_^(2-}\,}}\hspace/\ }}}}\hspace/\ \hspace_^)\left\}\,}}\hspace/\ }}}}_}}}}}}}}-\frac/\,}}}_}}})}} -2/\,}(/\ }}+2)}}_}}\right. \right. \\&\qquad -\left( }\,}}\hspace/\ }}}}\hspace/\ \hspace_2^* + 1 + /\,}}\,}}\hspace/\ }}\right) \left( }}+ \frac/\,}\left( \dot\chi )}-\frac}}\right) \right) \\&\qquad \left. \left. + \left( }\,}}\hspace/\ }}}}\hspace/\ \hspace_2^* + 1 + /\,}}\,}}\hspace/\ }}\right) }\,}}\hspace/\ }}/\ \hspace}_c}}- \frac\left( }\,}}\hspace/\ }}}}\hspace/\ \hspace_2^* + 1 + /\,}}\,}}\hspace/\ }}\right) /\,}}}}_\chi )}}\right\} \right) \\&\quad =\frac}}}\hspace/\ \hspace_^(2-}\,}}\hspace/\ }}}}\hspace/\ \hspace_^) \left( }\,}}\hspace/\ }}}}\hspace/\ \hspace_2^* + 1 + /\,}}\,}}\hspace/\ }}\right) }\,}}\hspace/\ }}}}}}}+ h_}}}}}}}, \end \end$$

(5.11)

where \(h_}}}}}}}\) is the source term

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

and the quantities \(}}_}}\), \(}}_}}})}}\) and \(}}_}}}}}}}}\) are defined in Appendix B.2. Integrating the transport equation yields the representation formula

$$\begin \begin&\left[ \frac}}}}}}+\frac}(1-2}}\hspace/\ \hspace_^}\,}}\hspace/\ }})}}}}}\right] _^ \\&\qquad +\left[ \frac}}}_^(2-}\,}}\hspace/\ }}}}\hspace/\ \hspace_^)\left\}\,}}\hspace/\ }}}}_}}}}}}}}-\frac/\,}}}_}}})}} -2/\,}(/\ }}+2)}}_}}\right. \right. \\&\qquad -\left( }\,}}\hspace/\ }}}}\hspace/\ \hspace_2^* + 1 + /\,}}\,}}\hspace/\ }}\right) \left( }}+ \frac/\,}\left( \dot\chi )}-\frac}}\right) \right) \\&\qquad \left. \left. + \left( }\,}}\hspace/\ }}}}\hspace/\ \hspace_2^* + 1 + /\,}}\,}}\hspace/\ }}\right) }\,}}\hspace/\ }}/\ \hspace}_c}}- \frac\left( }\,}}\hspace/\ }}}}\hspace/\ \hspace_2^* + 1 + /\,}}\,}}\hspace/\ }}\right) /\,}}}}_\chi )}}\right\} \right] ^v_\\&\quad =}}\hspace/\ \hspace_^(2-}\,}}\hspace/\ }}}}\hspace/\ \hspace_^) \left( }\,}}\hspace/\ }}}}\hspace/\ \hspace_2^* + 1 + /\,}}\,}}\hspace/\ }}\right) }\,}}\hspace/\ }}\left( \int _^\frac}}}}}}dv'\right) + \int _^ h_}}}}}}} dv'. \end \end$$

(5.12)

The linearised null structure equation further implies the estimate

$$\begin \Vert }}}}}}\Vert _}}^_}\lesssim \Vert }}}}}\Vert _}})}}+\Vert }}\Vert _}})}}+\Vert }}}_1 \Vert _}}^8(S_1)}+ \Vert (}}}_i)_ \Vert _}_}}}}}} \end$$

Remark 5.6

The quantities \(}}_}}\), \(}}_}}})}}\) and \(}}_}}}}}}}}\) are themselves conserved charges appearing in the \(C^2\)-linearised null gluing problem. In the notation of [3], they correspond to \(}}_1\), \(}}_2\) and \(}}_3\), respectively.

5.2.2 Analysis of \(}^}\)

Recall from Lemma 2.16 that the linearised null constraint equation for \(}^}}}}}\) is

$$\begin&D(}^})-\frac}\left( }}+\frac\dot}}})}-\frac\dot\chi )} +\frac}}\right) \nonumber \\&\quad = \frac}}\,}}\hspace/\ }}\left( \frac}}\,}}\hspace/\ }}}}}}}}}- \frac/\,}\dot}}})}-\frac}}\right) -\frac}}\,}}\hspace/\ }}}\,}}\hspace/\ }}}}}} +}}}_}^2}}}}}} \end$$

(5.13)

The null structure equations imply the following lemma.

Lemma 5.7

(Representation formula for \(}^}}\)) For the linearised null structure equation (5.13) for \(}^}}}}}\), we have the following transport equation

$$\begin \begin&D\left( }^}}}}}- \frac}}\,}}\hspace/\ }}}\,}}\hspace/\ }}}}}}}+ \frac}}\,}}\hspace/\ }}(8-}\,}}\hspace/\ }}}}\hspace/\ \hspace_^)}\,}}\hspace/\ }}}}_}}}}}}}}\right) \\&\qquad -D\left( \frac}\left( 2(/\ }}-3) +\frac/\ }}(/\ }}+2) \right) }}_}}})}}\right) \\&\qquad -D\left( \frac}}\,}}\hspace/\ }}\left( 8-}\,}}\hspace/\ }}}}\hspace/\ \hspace_^\right) /\,}(/\ }}+2)}}_}}\right) \\&\qquad -D\left( \frac}}\,}}\hspace/\ }}\left( 8}\,}}\hspace/\ }}}}\hspace/\ \hspace_^-4+8/\,}}\,}}\hspace/\ }}+\frac}\,}}\hspace/\ }}}}\hspace/\ \hspace_^\left( }\,}}\hspace/\ }}}}\hspace/\ \hspace_2^* + 1 + /\,}}\,}}\hspace/\ }}\right) \right) \dot}}\right) \\&\qquad +D\left( \frac}}\,}}\hspace/\ }}(8-}\,}}\hspace/\ }}}}\hspace/\ \hspace_^)\left( }\,}}\hspace/\ }}}}\hspace/\ \hspace_2^* + 1 + /\,}}\,}}\hspace/\ }}\right) }\,}}\hspace/\ }}/\ \hspace}_c}}\right) \\&\qquad +D\left( \frac}}\,}}\hspace/\ }}\left( 4}\,}}\hspace/\ }}}}\hspace/\ \hspace_^-2+4/\,}}\,}}\hspace/\ }}+\frac}\,}}\hspace/\ }}}}\hspace/\ \hspace_^\left( }\,}}\hspace/\ }}}}\hspace/\ \hspace_2^* + 1 + /\,}}\,}}\hspace/\ }}\right) \right) /\,}}}}_\chi )}}\right) \\&\quad =\frac}\,}}\hspace/\ }}\left( 36+2(8-}\,}}\hspace/\ }}}}\hspace/\ \hspace_^)\left( }\,}}\hspace/\ }}}}\hspace/\ \hspace_2^* + 1 + /\,}}\,}}\hspace/\ }}\right) \right) }\,}}\hspace/\ }}\left( \frac}}}\right) + h_}^}}}}}} \end \end$$

(5.14)

where \(\dot}}\) denotes the quantity

$$\begin \dot}}=v^2}}+ \frac/\,}\left( \dot\chi )}-\frac}}\right) \end$$

and the source term \(h_}^}}}}}}\) is

$$\begin&h_}^}}}}}}\\&\quad =}}}_}^2}}}}}}-\frac}}\,}}\hspace/\ }}}\,}}\hspace/\ }}}}}_}}}}}}+\frac}}\,}}\hspace/\ }}(8-}\,}}\hspace/\ }}}}\hspace/\ \hspace_^)}\,}}\hspace/\ }}(D}}_}}}}}}}})\\&\qquad -\frac}\left( 2(/\ }}-3) +\frac/\ }}(/\ }}+2) \right) (D}}_}}})}})\\&\qquad -\frac}}\,}}\hspace/\ }}\left( 8-}\,}}\hspace/\ }}}}\hspace/\ \hspace_^\right) /\,}(/\ }}+2)(D}}_}})\\&\qquad -\frac}}\,}}\hspace/\ }}\left( 8}\,}}\hspace/\ }}}}\hspace/\ \hspace_^-4+8/\,}}\,}}\hspace/\ }}+\frac}\,}}\hspace/\ }}}}\hspace/\ \hspace_^\left( }\,}}\hspace/\ }}}}\hspace/\ \hspace_2^* + 1 + /\,}}\,}}\hspace/\ }}\right) \right) \\&\qquad \left( }}}_}}}+/\,}}}}_}}}+\frac}/\,}}}}_\chi )}}\right) \\&\qquad +\frac}}\,}}\hspace/\ }}(8-}\,}}\hspace/\ }}}}\hspace/\ \hspace_^)\left( }\,}}\hspace/\ }}}}\hspace/\ \hspace_2^* + 1 + /\,}}\,}}\hspace/\ }}\right) }\,}}\hspace/\ }}}}}_}}}}}} \end$$

Integrating (5.14) yields the desired representation formula. The null structure equation (5.13) implies that the following estimate holds.

$$\begin \Vert }^}}}}}\Vert _}}^_}\lesssim \Vert }}}}}\Vert _}})}}+\Vert }}\Vert _}})}}+\Vert }}}_1 \Vert _}}^8(S_1)}+ \Vert (}}}_i)_ \Vert _}_}}}}}}. \end$$

5.3 Analysis of Novel Conservation Laws

By analysing the kernels of the differential operators on the right-hand sides of the representation formulae (5.11) and (5.14), we can derive conservation laws along \(}}_\). Moreover, quantities for which the \(v'\)-weight appearing in the \(}}}}}\) integrals of the right-hand sides of representation formula can be combined into conservation laws. Beginning with the right-hand side of (5.11), performing a spherical harmonic decomposition on the operator

$$\begin 2-}\,}}\hspace/\ }}}}\hspace/\ \hspace_2^*, \end$$

see D.2, we obtain that the kernel consists of the set of vectorfields

Projecting the representation formula (5.11) onto the \(l=2\) spherical harmonic modes yields the following conservation law.

Lemma 5.8

(Conservation law for \(}}}}}}\)) The charge \(}}_}}}}}}^}\) defined by

$$\begin }}_}}}}}}^} := \frac}}}}}}^ +\frac}(1-2}}\hspace/\ \hspace_^}\,}}\hspace/\ }})}}}}}^, \end$$

(5.15)

is conserved. It satisfies the equation

$$\begin D}}_}}}}}}^} = \frac}}}_}}}}}}}^+\frac}(1-2}}\hspace/\ \hspace_^}\,}}\hspace/\ }})}}}_}}}}}}^. \end$$

Remark 5.9

The charge \(}}_}}}}}}^}\) is precisely the linearisation the combination of linearisations of \(}\) and \(}\), i.e.

$$\begin }}_}}}}}}^} = \sum _ }^\psi ^+}^\phi ^ \end$$

Comparing the representation formulas for \(}}}}}}\) from (5.9) and for \(}}}}}}\) from (5.11), we see that the \(v'\)-weight appearing in both representation formulas is \(1/v'^4\). Thus, these representation formulas can be combined into a conservation law. The operator \(2-}\,}}\hspace/\ }}\hspace/\ \hspace_2^*\) on the right-hand side has a kernel, see Appendix D.2 and thus for the \(l=2\) modes there exists the conserved chargeFootnote 1

$$\begin }}_}}}}}}^}&:= }}}}}}^ +\frac }}_}}})}}^+ \frac }\,}}\hspace/\ }}}\,}}\hspace/\ }}}}_}}}}}}}}^ -\frac }}_}}^ \nonumber \\&\quad + \frac }\,}}\hspace/\ }}\left( \eta + \frac/\,}\left( \dot\chi )}-\frac}}\right) \right) ^ - }\,}}\hspace/\ }}}\,}}\hspace/\ }}/\ \hspace}_c}}}^. \end$$

(5.16)

Combining the representation formula (5.11) and (5.9) for modes \(l\ge 3\) yields the following lemma.

Lemma 5.10

(Conservation law for \(}}}}}}\) and \(}}}}}}\)) The charge \(}}_}}}}}}^_\psi }\) defined by

$$\begin \begin }}_}}}}}}^_\psi }&:=\frac}}}}}}_^+\frac}(1-2}}\hspace/\ \hspace_^}\,}}\hspace/\ }})}}}}}_^\\&\quad +2}}\hspace/\ \hspace_^\left( \frac\left( \frac}}\,}}\hspace/\ }}}}}}}}}}-\frac/\,}\dot}}})}-\frac}}\right) +}}\hspace/\ \hspace_^(}}}}}},0) \right) _^ \end \end$$

(5.17)

is conserved. It satisfies the equations

$$\begin D}}_}}}}}}^_\psi }&= \frac(}}}_}}}}}}})_^+/\,}(}}}_}}}}}}})_\psi ^+\frac}(1-2}}\hspace/\ \hspace_^}\,}}\hspace/\ }})(}}}_}}}}}})_^+\frac}}\,}}\hspace/\ }}(}}}_}}}}}}}}})_^\\&\quad -\frac/\,}(}}}_}}})}})_^-\frac}(}}}_}}})_^ \end$$

Proof

The conservation law for \(}}_}}}}}}^_\psi } \) is obtained by applying the null structure equations. Differentiating \(}}_}}}}}}^_\psi } \), we obtain (suppressing the notation \([\ge 3]\))

$$\begin \begin D}}_}}}}}}^_\psi }&= D\left( \frac}}}}}}_+\frac}(1-2}}\hspace/\ \hspace_^}\,}}\hspace/\ }})}}}}}_\right) \\&\quad +D\left( 2}}\hspace/\ \hspace_^\left( \frac\left( \frac}}\,}}\hspace/\ }}}}}}}}}}-\frac/\,}\dot}}})}-\frac}}\right) -/\,}}}}}}}\right) _\right) . \end \end$$

Applying the null structure equations, we have that

$$\begin&D\left( \frac}}}}}}_+\frac}(1-2}}\hspace/\ \hspace_^}\,}}\hspace/\ }})}}}}}_\right) \\&\quad =\frac(}}}_}}}}}}})_+\frac}(1-2}}\hspace/\ \hspace_^}\,}}\hspace/\ }})(}}}_}}}}}})_\\&\qquad +\frac}}}\hspace/\ \hspace_^(2-}\,}}\hspace/\ }}}}\hspace/\ \hspace_^)\left( \frac}}\,}}\hspace/\ }}}}}}}}}}-\frac/\,}\dot}}})}-\frac}}\right) , \end$$

while

$$\begin&D\left( 2}}\hspace/\ \hspace_^\left( \frac\left( \frac}}\,}}\hspace/\ }}}}}}}}}}-\frac}/\,}v^(\dot}}})})-\frac}(v^}})\right) -/\,}}}}}}}\right) _\right) \\&\quad =2}}\hspace/\ \hspace_^\left\}\left( \frac}}\,}}\hspace/\ }}}}}}}}}}-\frac/\,}\dot}}})}-\frac}}\right) \right) _\right. \\&\qquad +\frac\left( \frac}}}+\frac/\,}\dot}}})} -\frac}\,}}\hspace/\ }}}}}}}}}}\right) _+\frac}}\,}}\hspace/\ }}\left( \frac }}\hspace/\ \hspace_2^* \left( }}-2 /\,}}}\right) +\frac}}}_6\right) _\\&\qquad \left. -\frac}/\,}\left( - 2 }\,}}\hspace/\ }}\left( }}-2/\,}}}\right) + v^2 }}}_5\right) _-\frac}\left( \frac /\,}\left( \frac}}\right) +v/\,}}}+ v^2 }}}_4\right) _\right\} \\&\qquad -/\,}\left( \frac}\left( \dot}}})}\right) + \frac }\,}}\hspace/\ }}\left( \frac }\,}}\hspace/\ }}}}}}}}}}- \frac}}- \frac /\,}\dot}}})}\right) + }}}_9\right) _\psi . \end$$

Collecting terms in this formula and using that

$$\begin (}\,}}\hspace/\ }}}}\hspace/\ \hspace_^+1+\frac/\,}}\,}}\hspace/\ }})_V = 0 \end$$

for vectorfields V, yields the stated formula. \(\square \)

Finally, the operator on the right-hand side of (5.14) does not have a kernel for modes \(l\ge 2\), the related charge is obtained by projecting the representation formula onto the \(l\le 1\) spherical harmonic modes.

Lemma 5.11

(Conservation law for \(}^2}}}}}\)) Let \(}}_}^}}}}}^} \) denote the charge

$$\begin }}_}^}}}}}^} =}^}}}}}^-\frac}(/\ }}-3) }}_^+\frac}}\,}}\hspace/\ }}}}_ \end$$

(5.18)

Then, the charge satisfies the equation

$$\begin D(}}_}^}}}}}^} +\frac}}}}_\chi )}})=h_}^}}}}}}^ \end$$

The following lemma shows that the novel charges are bounded.

Lemma 5.12

(Charge estimate) Let \(}}}_\) denote linearised \(C^3\)-sphere data on a sphere \(S_\) with \(1\le v \le 2\). Then

$$\begin \Vert }}_}}}}}}^} \Vert _+ \Vert }}_}}}}}}^_\psi } \Vert _+\Vert }}_}^}}}}}^} \Vert _ \lesssim \Vert }}}_\Vert _}}^8(S_)}. \end$$

5.4 Gauge Dependence of Conserved Charges

In this section, we investigate the gauge-dependence of the novel charges of Lemmas 5.8, 5.10 and 5.11. That is, we prove the following proposition.

Proposition 5.13

The charge \(}}_}}}}}}^}\) is gauge-invariant, i.e. for all linearised perturbation functions \(}\) and \(}\),

$$\begin }}_}}}}}}^}(}}}}_(},}))=0. \end$$

Now, let \((}}_}^}}}}}^} )_\) be a scalar field and \(\left( }}_}}}}}}^_\psi }\right) _\) a symmetric tracefree 2-tensor satisfying

$$\begin (}}_}^}}}}}^} )_ = (}}_}^}}}}}^} )_^, \qquad \left( }}_}}}}}}^_\psi }\right) _=\left( \left( }}_}}}}}}^_\psi }\right) _\right) _\psi ^. \end$$

(5.19)

Then there exists linearised perturbation functions \(\dot\) and \(}\) at \(S_2\) such that

$$\begin }}_}^}}}}}^}\left( }}}}_(},})\right) =(}}_}^}}}}}^})_0 \qquad \left( }}_}}}}}}^_\psi }\right) \left( }}}}_(},}\right) =\left( }}_}}}}}}^_\psi }\right) _0. \end$$

Furthermore, under the assumption that

$$\begin \Vert (}}_}}}}}}^_\psi } )_0 \Vert _ +\Vert (Q_}^2}}}}}})_ \Vert _(S_)}<\infty , \end$$

the following bound holds for \(_u^3}\) and \(_u^4}\)

$$\begin \begin \Vert _^}^ \Vert _(S_)}+\Vert _^}\Vert _(S_)}&\lesssim \Vert (}}_}}}}}}^_\psi } )_0 \Vert _+\Vert (Q_}^2}}}}}})_ \Vert _(S_)}. \end \end$$

(5.20)

Proof

Direct substitution of the linearised perturbation map \(}}}}_(},})\) from Lemma 2.17 into the charges \(}}_}}}}}}^}\), \(}}_}}}}}}^_\psi }\) and \(}}_}^}}}}}^}\) yields

$$\begin \begin }}_}}}}}}^}&= 0,\\ }}_}}}}}}^_\psi }&= }}\hspace/\ \hspace_^}}\hspace/\ \hspace_^(_^},0)^_\psi ,\\ }}_}^}}}}}^}&= \frac_^}^. \end \end$$

(5.21)

We see immediately that \(}}_}}}}}}^}\) is gauge-invariant. Furthermore, by definition \(}}_}}}}}}^}\) is an \(l=2\) spherical harmonic mode consisting of an electric and magnetic part and therefore it is 10-dimensional. Thus, this gives the 10-dimensional space of novel gauge-invariant charges of the \(C^3\) linearised gluing problem.

It remains to show that for \(}}_}}}}}}^_\psi }\) and \(}}_}^}}}}}^}\) a suitable choice of \(}\) can be made. We proceed by constructing directly the functions \(_u^3}^\) and \(_u^4}\). For \(_^}^\), we have the equations from (5.21)

$$\begin }}\hspace/\ \hspace_^}}\hspace/\ \hspace_^\left( _u^3 },0\right) ^_\psi&=\left( }}_}}}}}}^_\psi }\right) _. \end$$

By (5.19), \(_^}\) is well defined and performing a spherical harmonic decomposition yields

$$\begin \left( _u^3 }\right) ^=-\frac l(l+1)-1}\sqrt}\left( \left( }}_}}}}}}^_\psi }\right) _\right) ^. \end$$

Putting this together we obtain the bound

$$\begin \Vert _^} ^\Vert _(S_)}\lesssim \Vert (}}_}}}}}}^_\psi } )_0 \Vert _ \end$$

For \(_^}\), we have the formula

$$\begin \begin _^}^&=2(Q_}^2}}}}}})_. \end \end$$

By letting \(_^}^=0\), \(_^}\) is well defined and bounded by

$$\begin \Vert _^}\Vert _(S_)}\lesssim \Vert (Q_}^2}}}}}})_ \Vert _(S_)}. \end$$

Thus, putting the two estimates together, we obtain the estimate (5.20). \(\square \)

Remark 5.14

Proposition 5.13 shows that the linearised perturbation function \(}\) is well defined for the novel \(C^3\) charges. However, we also need to show that \(}\) is well-defined for the charges arising from the \(C^2\)-null gluing problem. The charges appearing in the \(C^2\)-null gluing can be found in Appendix B.2. In Section 4 of [3] it is found, for example, that

$$\begin \begin }}_}}}}}}^}&= \frac _u^3 }^ , \\ }}_}}}}}}^}&=\frac _u^3 }^-3 }^.\\ \end \end$$

(5.22)

Proposition 4.4 in [3] shows that \(}\) is well defined for the charges appearing for the \(C^2\)-sphere data. Since the \(l\ge 3\) modes of \(_u^3}\) are not considered, this result can be combined with Proposition 5.13 and Lemma 2.18 to obtain the estimate

$$\begin \begin&\Vert } \Vert _}}_}} + \Vert }\Vert _}}_}+ \Vert }}}}_(\dot) \Vert _}(S_2)} +\Vert }}}}_(}) \Vert _}(S_2)}\\&\quad \lesssim \Vert (}}_}})_0 \Vert _+\Vert (}}_}}})}})_0 \Vert _+\Vert (}}_}}}}}}}})_0 \Vert _+\Vert (}}_}}_\psi })_0\Vert _\\&\qquad +\Vert (}}_}}}}}^})_0 \Vert _+\Vert (}}_}}}}}}^})_0 \Vert _+\Vert (}}_}}}}}}^})_0\Vert _\\&\qquad +\Vert (}}_}}}}}}^_\psi } )_0 \Vert _+\Vert (}}_}^}}}}}^} )_0 \Vert _. \end \end$$

(5.23)

where the charges on the right-hand side are defined analogously to (5.19).

5.5 Proof of Proposition 5.1

In this section, we use the results of sections 5.2, 5.3 and 5.4 to prove Proposition 5.1 and hence prove Proposition 4.2. Let

$$\begin }}}_\in }}(S_),\qquad }}}_\in }}_}(}_), \qquad }}}_}}\in }}_}}}, \end$$

be given linearised \(C^3\)-sphere data on \(S_\), linearised \(C^3\)-matching data on \(}_\subset \underline}}_\) and linearised source functions on \(}}_\), respectively. We prove Proposition 5.1 by showing the following.

In Section 5.5.1, we show that the matching map \(}_\) contains the complement of \((}},}},}},}},}},}})\) in the sphere data \(}_\).

In Section 5.5.2, we apply Proposition 5.13 to match the gauge-dependent charges of Lemmas 5.10 and 5.11 at \(S_2\).

In Section 5.5.3, we construct conditions on the linearised characteristic seed \((}},}}}}})\) such that the novel part of the \(C^3\)-null data \(}_\) given by the representation formulae of Section 5.2 satisfies (5.1).

Finally, in Section 5.5.4, we prove the estimate (5.3) and conclude the proof of Proposition 5.1.

5.5.1 \(}\) Complements \((}},}},}},}},}},}})\)

This section is concerned with the proof of the following lemma.

Lemma 5.15

(Matching all gluable quantities) Let \(}_\) and \(}'_\) be \(C^3\)-sphere data on \(S_\) such that for a real number \(\varepsilon >0\),

$$\begin \Vert }_ -} \Vert _}}(S_)} + \Vert }'_ -} \Vert _}}(S_)} < \varepsilon , \end$$

and

$$\begin }}(}_)=}}(}'_). \end$$

(5.24)

For \(\varepsilon >0\) sufficiently small, if the charges coincide, that is,

$$\begin (},}, }, },},})(}_) = (},}, }, },},})(}'_), \end$$

(5.25)

then the following holds,

Remark 5.16

Before stating the proof, we make the following remarks.

(1)

The quantities \(\tilde}}}_}}}\), \(\tilde}}}_}}}}\) and \(\tilde}}}_}^2}}}\) appearing in the matching map, see Definition 2.19, linearise to the conserved charges \(}}_}}}}}}\), \(}}_}}}}}}}\) and \(}}_}^}}}}}^}\).

(2)

The proof for the \(C^2\) part of the matching data \(m_\) is contained in Lemma 2.12 of [3] and therefore, we only need to prove Lemma 5.15 for the novel \(C^3\) part, \(}\), consisting of the quantities

$$\begin (D^\omega ,}^}}^, \tilde}}}_}^}}}}}^},},\widehat}}}}^). \end$$

Proof of Lemma 5.15

The matching of \(}\) and \(}\) in (5.25) can be combined to write

$$\begin \frac\widehat}}}}^(}_)+\frac}(1+\nabla \hspace/\ \widehat}\,}}\hspace/\ )}}^(}_)=\frac\widehat}}}}^(}'_)+\frac}(1+\nabla \hspace/\ \widehat}\,}}\hspace/\ )}}^(}'_). \end$$

By the matching of \(}}\) and \(\phi \) in (5.24), we directly obtain that

$$\begin \widehat}}}}^(}_)=\widehat}}}}^(}'_). \end$$

Consider now the quantity \(}^}}\). On the one hand \(}^}}^\) is matched by (5.24). On the other hand, the quantity

$$\begin \begin \tilde}}}_}^}}}}}}&:= \left( }^}}\right) ^ -\frac (/\ }}-3) \left( \frac}\Omega }}}}- \frac(/\ }}+2)\phi \right) ^ \\&\quad + \frac} \left( /\ }}/\ }}+ 2/\ }}-3\right) \left( \Omega \chi }-\frac\Omega \right) ^ -\frac }\,}}\hspace/\ }}\eta ^ \end \end$$

is matched by (5.24). In turn, this implies that for modes \(l\le 1\),

$$\begin }^}}^(}_)= }^}}^(}'_). \end$$

\(\square \)

5.5.2 Gluing of Gauge-Dependent Charges

We proceed by showing that linearised perturbations of sphere data can be added to the charges

$$\begin (}}_}}}}}}^_\psi }, }}_}^}}}}}^}) \end$$

in order to match the charges at \(S_\) with the charges coming from \(S_\). Firstly, let \(^}}_}}}}}}^_\psi }\) and \(^ }}_}^}}}}}^}\) be solutions to transport equations from Lemmas 5.10 and