Conformal scattering theory for a tensorial Fackerell-Ipser equation on the
Schwarzschild spacetime

PHAM Truong Xuan111Departement of informatics, faculty of mathematics, Thuyloi
university, 175 Tay Son, Dong Da, Ha Noi, Viet Nam. Email<EMAIL_ADDRESS>or<EMAIL_ADDRESS>
Abstract. In this paper, we prove that the existence of the energy and
pointwise decays for the fields satisfying the tensorial Frackerell-Ipser
equations (which are obtained from the Maxwell and spin $\pm 1$ Teukolsky
equations) on the Schwarzschild spacetime is sufficient to obtain a conformal
scattering theory. This work is the continuation of the recent work [45] on
the conformal scattering theory for the Regge-Wheeler and Zerilli equations
arising from the linearized gravity fields and the spin $\pm 2$ Teukolsky
equations.

Keywords. Conformal scattering, Goursat problem, black holes, tensorial
Fackerell-Ipser equation, Teukolsky equation, Schwarzschild metric, null
infinity, Penrose conformal compactification.

Mathematics subject classification. 35L05, 35P25, 35Q75, 83C57.

###### Contents

  1. 1 Introduction
  2. 2 Geometrical and analytical setting
    1. 2.1 Schwarzschild metric and Penrose conformal compactification
    2. 2.2 The original equations
    3. 2.3 The rescaled equations
  3. 3 Energies of the tensorial Fackerell-Ipser field
    1. 3.1 Energy conservation for the original field
    2. 3.2 Energy conservation for the rescaled field
    3. 3.3 Tensorial field space of initial data and Cauchy problem
    4. 3.4 Energy identity up to $i^{+}$ and trace operator
  4. 4 Conformal scattering theory
    1. 4.1 Generalisation of L. Hörmander’s result for tensorial wave equations
    2. 4.2 Goursat problem and conformal scattering operator

##  1 Introduction

The idea of the conformal compactification structure of spacetimes was posed
initially by Penrose [42] in 1960’s. After that this has played an important
role in the study of peeling and conformal scattering, the two aspects of
conformal asymptotic analysis. In particular, the conformal scattering theory
has been studied extensively from the early works by Friedlander [15, 16, 17,
18, 19], Baez et al. [6], Hörmander [25] to recent ones by Mason and Nicolas
[34], Joudioux [28, 29], Nicolas [40], Mokdad [37] and Taujanskas [49].

Normally, a conformal scattering theory on the asymptotic flat spacetimes such
as Schwarzschild and Kerr ones consists of three steps. First, we solve the
Cauchy problem of the rescaled equations on the rescaled spacetime, then we
define and extend the trace operators $\mathcal{T}^{\pm}$. Second, we need to
show that the extension of trace operator is injective by proving the energy
identity up to the future timelike infinity $i^{+}$. Third, we solve the
Goursat problem with the initial data on the conformal boundaries consisting
of the future (resp. past) horizon and future (resp. past) null infinity, then
as a consequence, we obtain that the extension of the trace operator is
subjective.

Therefore, the extended trace operator is an isometry between the space of the
initial data and the space of the scattering data. As a consequence we define
the conformal scattering operator
$S:=\mathcal{T}^{+}\circ(\mathcal{T}^{-})^{-1}$ that is an isometry which maps
the past scattering data to the future scattering data.

Nicolas [40] and Mokdad [37] constructed explicitly the conformal scattering
theories for the scalar wave and Maxwell fields on the static and symmetric
spherical spcatimes such as Schwarzschild and Reissner-Nordström de Sitter
spacetimes respectively. In their works, they used the results about the
bounded energy, Morawertz estimate and pointwise decay of the fields to prove
the energy identity up to $i^{+}$ in the second step of the theory.

There are some related works which use also the energy and pointwise decay
results to construct the scattering theory. We refer the reader to the works
about the scattering theories for the scalar wave equation on the interior of
Reissner-Nordström de Sitter of Keller et al. [31], on the extremal Reissner-
Nordström of Angelopoulos et al. [4], and on slowly Kerr spacetimes of
Dafermos et al. [12].

The uniformly bounded energy, Morawertz’s estimate, energy and pointwise
decays are obtained naturally in the program to prove linear and nonlinear
stability of black hole spacetimes and the related problems (see [3, 11, 12,
13, 22, 23, 30, 29, 47, 26]). The method of $r^{p}$-theory of Dafermos and
Rodnianski [10] is an essential tool of the proof in a lot of work later.

In the recent work [45] we have constructed conformal scattering theories for
the scalar Reeger-Wheeler and Zerelli equations which arised from the
linearized gravity fields and the spin $\pm 2$ Teukolsky equations. This is a
first step to obtain the conformal scattering theory for the linearized
gravity fields on the Schwarzschild spacetime.

In the present paper we explore the method in [40, 37, 45] to construct the
conformal scattering theory for the tensorial Fackerell-Ipser equations in the
Schwarzschild spacetime. There are two ways to establish the tensorial
Fackerell-Ipser equations. The first one is obtained by commuting the spin
$\pm 1$ Teukolsky equations with the projected covariant derivetives
$\not{\nabla}_{L}$ and $\not{\nabla}_{\underline{L}}$ on the $2$-sphere
$\mathbb{S}^{2}_{(t,r)}$ at $(t,r)$, where $L$ and $\underline{L}$ are
outgoing and incoming principal null directions respectively. The second one
is obatined by commuting the scalar wave equation with the angular derivatives
$r\not{\nabla}$. We will use the latter construction to derive the rescaled
tensorial equation in the conformal compactification domain (see Section 2.3).
We can see that the potentials in the scalar Regger-Wheeler and Zerelli
equations [45] and also the scalar (real or complex) Fackerell-Ipser equations
in [1, 8] decay as $r^{-3}$, whence the potentials of the tensorial Fackerell-
Ipser equations decay as $r^{-2}$.

In Sections 3.2 and 3.3 we will establish the conservation laws for the
tensorial Fackerell-Ipser equations (resp. rescaled equations) that depends
only on the derivatives with respect to $L$ and $\underline{L}$ (or
$\partial_{t}$ and $\partial_{r_{*}}$, here $r_{*}$ is the Regge-Wheeler
variable). Integrating these conservation laws, we will obtain the energy
equalities between the initial hypersurface $\Sigma_{0}=\left\\{t=0\right\\}$
and $\Sigma_{T}=\left\\{t=T\right\\}$ for $T>0$. By combining these equalities
and Leray’s theorem for the hyperbolic differential equations, we can
establish the well-posedness of Cauchy problem for the original equations
(resp. rescaled equations).

In order to prove the energy identity up to the timelike infinity $i^{+}$, we
need to use the energy and pointwise decays previously obtained in the
literature. The decays of the solution of the tensorial Fackerell-Ipser
equations can be established from the decays of the scalar Fackerell-Ipser
solutions. There are some works on the decay of the solutions of the scalar
Fackerell-Ipser equations in Schwarzschild spacetime such as [8, 20, 36].
However, in this work we will use the energy and pointwise decays which are
obtained in the recent work of Pasqualotto [46]. Using these results, we will
show that the energy across the hypersurfaces
$\mathcal{H}_{u_{T}}=\left\\{u=u_{T},\,v_{T}\leq v\leq 2v_{T}\right\\}$ and
$\mathcal{I}_{v_{T}}=\left\\{u_{T}\leq u\leq 2u_{T},\,v=v_{T}\right\\}$ tends
to zero as $u_{T}$ and $v_{T}$ tend to infinity, where $\mathcal{H}_{u_{T}}$
and $\mathcal{I}_{v_{T}}$ are parallel to the future horizon
$\mathfrak{H}^{+}$ and the future null infinity ${\mathscr{I}}^{+}$
respectively. This leads to the energy indentity up to $i^{+}$, i.e the energy
of the rescaled tensorial fields restricted on the hypersurface
$\Sigma_{0}=\left\\{t=0\right\\}$ is equal to the sum of energies of the
rescaled tensorial fields restricted on $\mathfrak{H}^{+}$ (resp.
$\mathfrak{H}^{-}$) and on ${\mathscr{I}}^{+}$ (resp. ${\mathscr{I}}^{-}$).
From that, we can define the trace operator $\mathcal{T}^{+}$ that maps the
smoothtly supported compact initial data to the trace of rescaled tensorial
solution on the conformal boundaries $\mathfrak{H}^{+}\cup{\mathscr{I}}^{+}$.
This trace operator is extended by density. As a consequence of the energy
indentity up to $i^{+}$, the trace operator is injective (see Section 3.4).

In Section 4 we prove that the trace operator is subjective, this corresponds
to solve the Goursat problem with the smoothtly supported compact initial data
on the conformal boundaries. This work is done by developing the works in [25,
39, 40] for the tensorial wave equations in Schwarzschild spacetime. We
project the tensorial Fackerell-Ipser equations on the basic frame of the
$2$-sphere $\mathbb{S}^{2}$, we get a system which consists redof scalar wave
equations with potentials at first order of derivatives. Then we re-write the
projected system under the $2\times 2$-matrix wave form. The well-posedness of
the Goursat problem consists of two parts: in the first one, we apply the
results in [25, 39] to solve the Goursat problem of the matrix wave equation
in the future of a spacelike hypersurface $\mathcal{S}$ whose intersection
with the horizon is the crossing sphere and which crosses ${\mathscr{I}}^{+}$
strictly in the past of the support of the data; in the second one, we extend
the solution obtained in the first part down to $\Sigma_{0}$, where the method
is developed from [40]. The well-posedness of Goursat problem shows that the
trace operator is subjective, hence an isometry. Therefore, we can define the
conformal scattering operator that maps the past scattering data to the future
scattering data, i.e.

$S:=\mathcal{T}^{+}\circ(\mathcal{T}^{-})^{-1}.$

Notation.
$\bullet$ We denote the bundle tangent to each $2$-sphere
$\mathbb{S}^{2}(t,r)$ at $(t,r)$ by $\mathcal{B}$ and the vector space of all
smooth sections of $\mathcal{B}$ by $\Gamma(\mathcal{B})$. The space of all
$1$-forms on $\mathbb{S}^{2}_{(t,r)}$ is denoted by $\Lambda^{1}(\mathcal{B})$
and a basic frame is denoted by $(\not{\nabla}_{a},\not{\nabla}_{b})$.
$\bullet$ We denote local coordinates for the unit $2$-sphere $\mathbb{S}^{2}$
by $(\theta^{a},\theta^{b})$ and the associated vector fields to
$(\theta^{a},\theta^{b})$ by $\partial_{\theta^{a}}$ and
$\partial_{\theta^{b}}$. Normaly, we have
$(\theta^{a},\theta^{b})=(\theta,\varphi)$.
$\bullet$ We follow the convention used by Penrose and Rindler [48] about the
Hodge dual of a $2$-form $\beta$ on a spacetime $({\cal M},g)$ (i.e. a
$4-$dimensional Lorentzian manifold that is oriented and time-oriented)

$(*\beta)_{ab}=\frac{1}{2}e_{abcd}\beta^{cd},$

where $e_{abcd}$ is the volume form on $({\cal M},g)$, denoted simply as
$\mathrm{dVol}_{g}$.
$\bullet$ We denote the space of smooth compactly supported scalar functions
on $\mathcal{M}$ (a smooth manifold without boundary) by
$\mathcal{C}_{0}^{\infty}(\mathcal{M})$ and the space of distributions on
$\mathcal{M}$ by $\mathcal{D}^{\prime}(\mathcal{M})$. The space of smooth
compactly supported $1$-forms in $\Lambda^{1}(\mathcal{B})$ on $\mathcal{M}$
is denoted by
$\mathcal{C}_{0}^{\infty}(\Lambda^{1}(\mathcal{B})|_{\mathcal{M}})$.
$\bullet$ Let $f(x)$ and $g(x)$ be two real functions. We write $f\lesssim g$
if there exists a constant $C\in(0,+\infty)$ such that $f(x)\leq Cg(x)$ for
all $x$, and write $f\simeq g$ if both $f\lesssim g$ and $g\lesssim f$ are
valid.

##  2 Geometrical and analytical setting

###  2.1 Schwarzschild metric and Penrose conformal compactification

Let $({\cal{M}}=\mathbb{R}_{t}\times]0,+\infty[_{r}\times\mathbb{S}^{2},g)$ be
a $4$-dimensional Schwarzschild manifold, equipped with the Lorentzian metric
$g$ given by

$g=F\mathrm{d}t^{2}-F^{-1}\mathrm{d}r^{2}-r^{2}\mathrm{d}\mathbb{S}^{2},\,F=F(r)=1-\mu,\,\mu=\frac{2M}{r},$

where $\mathrm{d}\mathbb{S}^{2}$ is the Euclidean metric on the unit
$2$-sphere $\mathbb{S}^{2}$, and $M>0$ is the mass of the black hole. In this
paper, we work on the exterior of the black hole
$\mathcal{B}_{I}=\left\\{r>2M\right\\}$.

Recall the Regge-Wheeler coordiante $r_{*}=r+2M\log(r-2M)$, such that
$\mathrm{d}r=F\mathrm{d}r_{*}$. In the variables
$(t,r_{*},\theta^{a},\theta^{b})$, the Schwarzschild metric takes the form

$g=F(\mathrm{d}t^{2}-\mathrm{d}r_{*}^{2})-r^{2}\mathrm{d}\mathbb{S}^{2}.$

The retarded and advanced Eddington-Finkelstein coordinates $u$ and $v$ are
defined by

$u=t-r_{*},\,v=t+r_{*}.$

The outgoing and incoming principal null directions are respectively

$L=\partial_{v}=\partial_{t}+\partial_{r_{*}},\,\underline{L}=\partial_{u}=\partial_{t}-\partial_{r_{*}}.$

Putting $\Omega=1/r$ and $\hat{g}=\Omega^{2}g$. We obtain a conformal
compactification of the exterior domain in the retarded variables
$(u,\,R=1/r,\,\theta^{a},\,\theta^{b})$ that is
$\left(\mathbb{R}_{u}\times\left[0,\dfrac{1}{2M}\right]\times\mathbb{S}^{2},\hat{g}\right)$
with the rescaled metric

$\hat{g}=R^{2}(1-2MR)\mathrm{d}u^{2}-2\mathrm{d}u\mathrm{d}R-\mathrm{d}\mathbb{S}^{2}.$
(1)

Future null infinity ${\mathscr{I}}^{+}$ and the past horizon
$\mathfrak{H}^{-}$ are null hypersurfaces of the rescaled spacetime

${\mathscr{I}}^{+}=\mathbb{R}_{u}\times\left\\{0\right\\}_{R}\times\mathbb{S}^{2},\,\mathfrak{H}^{-}=\mathbb{R}_{u}\times\left\\{1/2M\right\\}_{R}\times\mathbb{S}^{2}.$

If we use the advanded variables $(v,\,R=1/r,\,\theta^{a},\,\theta^{b})$, the
rescaled metric $\hat{g}$ takes the form

$\hat{g}=R^{2}(1-2MR)\mathrm{d}v^{2}+2\mathrm{d}v\mathrm{d}R-\mathrm{d}\mathbb{S}^{2}.$
(2)

The Past null infinity ${\mathscr{I}}^{-}$ and the future horizon
$\mathfrak{H}^{+}$ are described as the null hypersurfaces

${\mathscr{I}}^{-}=\mathbb{R}_{v}\times\left\\{0\right\\}_{R}\times\mathbb{S}^{2},\,\mathfrak{H}^{+}=\mathbb{R}_{v}\times\left\\{1/2M\right\\}_{R}\times\mathbb{S}^{2}.$

The Penrose conformal compactification of $\mathcal{B}_{I}$ is

$\bar{\mathcal{B}_{I}}=\mathcal{B}_{I}\cup{\mathscr{I}}^{+}\cup\mathfrak{H}^{+}\cup{\mathscr{I}}^{-}\cup\mathfrak{H}^{-}\cup
S_{c}^{2},$

where $S_{c}^{2}$ is the crossing sphere.

We have the following relations between the rescaled coordinates and the
original ones

$\hat{L}=r^{2}L,\,\hat{\underline{L}}=\underline{L},\,\partial_{{\theta}^{a}}=r\partial_{x^{a}},$

where $(x^{a},x^{b})$ denote local coordinates for the $2$-sphere
$\mathbb{S}^{2}_{(t,r)}$ at $(t,r)$.

The scalar curvature of the rescaled metric $\hat{g}$ is

$\mathrm{Scal}_{\hat{g}}=12MR.$

The volume forms associated with the Schwarzschild metric $g$ and the rescaled
metric $\hat{g}$ are

$\mathrm{dVol}_{g}=r^{2}F\mathrm{d}t\wedge\mathrm{d}r_{*}\wedge\mathrm{d}\mathbb{S}^{2}\hbox{
and
}\mathrm{dVol}_{\hat{g}}=R^{2}F\mathrm{d}t\wedge\mathrm{d}r_{*}\wedge\mathrm{d}\mathbb{S}^{2}=\frac{R^{2}F}{2}\mathrm{d}u\wedge\mathrm{d}v\wedge\mathrm{d}\mathbb{S}^{2}$

respectively, where $\mathrm{d}\mathbb{S}^{2}$ is the euclidean area element
on unit $2$-sphere $\mathbb{S}^{2}$.

###  2.2 The original equations

Let $F$ be an antisymmetric $2$-form on the exterior of Schwarzschild black
hole $\mathcal{B}_{I}$. The Maxwell equations take the form

$\mathrm{d}F=0,\,\,\,\mathrm{d}*F=0,$

where $*$ denotes the Hodge dual operator of $2$-form, i.e,

$(*F)_{\mu\nu}=\frac{1}{2}e_{\mu\nu\gamma\delta}F^{\gamma\delta}.$

The system can be expressed as

$\nabla_{[\mu}F_{\kappa\lambda]}=0,\,\,\,\nabla^{\mu}F_{\mu\nu}=0,$

where the square brackets denote antisymmetrization of indices.

The Maxwell field $F$ can be decomposed into $1$-forms
$\alpha_{a},\,\underline{\alpha}_{a}\in\Lambda^{1}(\mathcal{B})$ and
$\rho,\,\sigma\in C^{\infty}(\mathcal{B}_{I})$ which are defined as follows

$\displaystyle\alpha(V):=F(V,L),\,\underline{\alpha}(V):=F(V,\underline{L})\hbox{
for all }V\in\Gamma(\mathcal{B}),$
$\displaystyle\rho:=\frac{1}{2}\left(1-\frac{2M}{r}\right)^{-1}F(\underline{L},L),\,\sigma:=\frac{1}{2}e^{cd}F_{cd},$

where $e_{cd}\in\Lambda^{2}(\mathcal{B})$ is the volume form of $2$-sphere
$\mathbb{S}^{2}_{(t,r)}$ at $(t,r)$.

Let $F$ be in $\Lambda^{2}(\mathcal{B}_{I})$ such that $F$ satisfies the
Maxwell equation on $\mathcal{B}_{I}$. Then, we have the following formulas
(see Proposition 3.6 [46]):

$\frac{1}{r}\not{\nabla}_{L}(r\underline{\alpha}_{a})=-(1-\mu)(\not{\nabla}_{a}\rho-
e_{ab}\not{\nabla}^{b}\sigma)$

and

$\frac{1}{r}\not{\nabla}_{\underline{L}}(r\alpha_{a})=(1-\mu)(\not{\nabla}_{a}\rho+e_{ab}\not{\nabla}^{b}\sigma).$

From this, we can define the $1$-forms in $\Lambda^{1}(\mathcal{B})$:

$\phi_{a}:=\frac{r^{2}}{F}\not{\nabla}_{\underline{L}}(r\alpha_{a}),\,\,\,\underline{\phi}_{a}:=\frac{r^{2}}{F}\not{\nabla}_{L}(r\underline{\alpha}_{a}).$

Moreover, the extreme components $\alpha_{a}$ and $\underline{\alpha}_{a}$
satisfy the spin $\pm 1$ Teukolsky equations respectively (see original proof
in [7] and recent Proposition 3.6 [46]):

$\not{\nabla}_{L}\not{\nabla}_{\underline{L}}(r\alpha_{a})+\frac{2}{r}\left(1-\frac{3M}{r}\right)\not{\nabla}_{\underline{L}}(r\alpha_{a})-F\not{\Delta}(r\alpha_{a})+\frac{F}{r^{2}}r\alpha_{a}=0,$
(3)
$\not{\nabla}_{L}\not{\nabla}_{\underline{L}}(r\underline{\alpha}_{a})+\frac{2}{r}\left(1-\frac{3M}{r}\right)\not{\nabla}_{L}(r\underline{\alpha}_{a})-F\not{\Delta}(r\underline{\alpha}_{a})+\frac{F}{r^{2}}r\underline{\alpha}_{a}=0,$
(4)

where $F=1-2MR$ and $\not{\nabla}$ and $\not{\Delta}$ are the orthogonal
projection of covariant derivative $\nabla$ and covariant Laplacian operator
on the bundle tangent $\mathcal{B}$ of $2$-sphere $\mathbb{S}^{2}_{(t,r)}$
respectively.

The tensorial Fackerell-Ipser equations are established from the spin $\pm 1$
Teukolsky equations by the following proposition.

######  Proposition 1.

Suppose that $(\alpha_{a},\underline{\alpha}_{a},\rho,\sigma)$ satisfy the
Maxwell equation, then the one forms $\phi_{a}$ and $\underline{\phi}_{a}$
satisfy the following tensorial Fackerell-Ipser equations

$\not{\nabla}_{L}\not{\nabla}_{\underline{L}}(\phi_{a})-F\not{\Delta}(\phi_{a})+\frac{F}{r^{2}}\phi_{a}=0,$
(5)
$\not{\nabla}_{L}\not{\nabla}_{\underline{L}}(\underline{\phi}_{a})-F\not{\Delta}(\underline{\phi}_{a})+\frac{F}{r^{2}}\underline{\phi}_{a}=0.$
(6)

###### Proof.

We treat the equation for $\underline{\phi}_{a}$. We do the proof for the
equation of $\underline{\phi}_{a}$, the one for $\phi_{a}$ is obtained
similarly. A straightforward calculation gives

$\underline{L}\left(\frac{r^{2}}{F}\right)=-\frac{r^{2}}{F}\frac{2}{r}\left(1-\frac{3M}{r}\right),$

the Teukolsky equation (4) is equivalent to

$\frac{F}{r^{2}}\not{\nabla}_{\underline{L}}\left(\frac{r^{2}}{F}\not{\nabla}_{L}(r\underline{\alpha}_{a})\right)-F\not{\Delta}(r\underline{\alpha}_{a})+\frac{F}{r^{2}}r\underline{\alpha}_{a}=0.$

Therefore

$\not{\nabla}_{\underline{L}}\left(\frac{r^{2}}{F}\not{\nabla}_{L}(r\underline{\alpha}_{a})\right)-r^{2}\not{\Delta}(r\underline{\alpha}_{a})+r\underline{\alpha}_{a}=0.$

By applying $\not{\nabla}_{L}$ to the above equation and by using that
$[\not{\nabla}_{L},r^{2}\not{\Delta}]=0$ and
$[\not{\nabla}_{L},\not{\nabla}_{\underline{L}}]=0$, we get the Fackerell-
Ipser equation for $\underline{\phi}_{a}$ which is given by,

$\not{\nabla}_{L}\not{\nabla}_{\underline{L}}(\underline{\phi}_{a})-F\not{\Delta}(\underline{\phi}_{a})+\frac{F}{r^{2}}\underline{\phi}_{a}=0.$

∎

We define the tensorial wave operator by

$\not{\Box}_{g}=\frac{1}{F}\not{\nabla}_{L}\not{\nabla}_{\underline{L}}-\not{\Delta},$

then we have

$\not{\Box}_{g}(\phi_{a})+\frac{1}{r^{2}}\phi_{a}=0,$ (7)
$\not{\Box}_{g}(\underline{\phi}_{a})+\frac{1}{r^{2}}\underline{\phi}_{a}=0.$
(8)

######  Remark 1.

We have the following relations

$\phi_{a}=r^{3}(\not{\nabla}_{a}\rho+e_{ab}\not{\nabla}^{b}\sigma),\,\,\,\underline{\phi}_{a}=r^{3}(-\not{\nabla}_{a}\rho+e_{ab}\not{\nabla}^{b}\sigma).$

Moreover $r^{2}\rho$ and $r^{2}\sigma$ satisfy the scalar Fackerell-Ipser
equation, in precisely a scalar wave equation

$\Box_{g}\psi=\frac{1}{F}L\underline{L}\psi-\not{\Delta}\psi=0.$ (9)

Since $r\not{\nabla}_{a}=\not{\nabla}_{\partial_{\theta}^{a}}$, we have the
commutators
$[r\not{\nabla},\not{\nabla}_{L}]=[r\not{\nabla},\not{\nabla}_{\underline{L}}]=0$.
Applying the projected covariant angular derivative $r\not{\nabla}$ on the
unit $2$-sphere $\mathbb{S}^{2}$ to the scalar wave equation (9), we get a new
equation i.e the tensorial Fackerell-Ipser equation with the appearance of the
additional zeroth order term since
$[r\not{\nabla},\not{\Delta}]=\dfrac{1}{r^{2}}$. Therefore, we can also obtain
the vectorial Fackerell-Ipser equations (7) and (8) from the scalar wave
equation (9) (see Remark 2.10 [46] and also Remark 7.1 [13]).

We can see that the potentials of the scalar Regge-Wheeler and Zerilli
equations (see [45]) decay as $r^{-3}$, whence the ones of (7) and (8) decay
as $r^{-2}$.

###  2.3 The rescaled equations

In order to obtain the conformal scattering theory for the tensorial
Fackerell-Ipser equations (7) and (8) we need to establish their conformal
rescaled forms in the conformal compactification domain
$\bar{\mathcal{B}_{I}}$.

Since $r^{2}\rho$ and $r^{2}\sigma$ satisfy the scalar wave equation (9) and
this equation is conformal invariant under the following formula

$({\Box}_{g}+\frac{1}{6}\mathrm{Scal}_{g})\psi=\Omega^{3}({\Box}_{\hat{g}}+\frac{1}{6}\mathrm{Scal}_{\hat{g}})\Omega^{-1}\psi,$

then $r(r^{2}\rho)$ and $r(r^{2}\sigma)$ satisfy the rescaled scalar wave
equation

$\frac{r^{2}}{F}L\underline{L}\hat{\psi}-r^{2}\not{\Delta}\hat{\psi}+2MR\hat{\psi}=0\hbox{
where }\hat{\psi}=r\psi.$ (10)

Putting

$\hat{\phi}_{a}=r\phi_{a}=r[\not{\nabla}_{a}r(r^{2}\rho)+e_{ab}\not{\nabla}^{b}r(r^{2}\sigma)].$

We commute (10) with the projected covariant angular derivative
$r\not{\nabla}$ by the same way in Remark 1, then we establish that
$\hat{\phi}_{a}$ is a solution of the following equation

$\frac{1}{F}{\not{\nabla}}_{\hat{L}}{\not{\nabla}}_{\hat{\underline{L}}}\hat{\phi}_{a}-\not{\Delta}_{\mathbb{S}^{2}}\hat{\phi}_{a}+(2MR+1)\hat{\phi}_{a}=0,$
(11)

where $\hat{L}=r^{2}L$, $\hat{\underline{L}}=\underline{L}$ and
$\not{\Delta}_{\mathbb{S}^{2}}=r^{2}\not{\Delta}$. Therefore, the tensorial
equation (7) has the following rescaled form in the compactification domain
$\bar{\mathcal{B}}_{I}$:

$\not{\Box}_{\hat{g}}\hat{\phi}_{a}+(2MR+1)\hat{\phi}_{a}=0,$ (12)

where

$\not{\Box}_{\hat{g}}=\frac{1}{F}{\not{\nabla}}_{\hat{L}}{\not{\nabla}}_{\hat{\underline{L}}}-\not{\Delta}_{\mathbb{S}^{2}}.$

By the same way Equation (8) has the rescaled form as

$\not{\Box}_{\hat{g}}\underline{\hat{\phi}}_{a}+(2MR+1)\underline{\hat{\phi}}_{a}=0,$
(13)

where $\underline{\hat{\phi}}_{a}=r\underline{\phi}_{a}$.

For the convenience in the next sections we multiply Equations (12) and (13)
by a factor $FR^{2}$ and we get the following forms respectively

${\not{\nabla}}_{L}{\not{\nabla}}_{{\underline{L}}}\hat{\phi}_{a}-FR^{2}\not{\Delta}_{\mathbb{S}^{2}}\hat{\phi}_{a}+(2MR+1)FR^{2}\hat{\phi}_{a}=0,$
(14)
${\not{\nabla}}_{L}{\not{\nabla}}_{{\underline{L}}}\underline{\hat{\phi}}_{a}-FR^{2}\not{\Delta}_{\mathbb{S}^{2}}\hat{\phi}_{a}+(2MR+1)FR^{2}\underline{\hat{\phi}}_{a}=0.$
(15)

######  Remark 2.

Since the tensorial Equations (7) and (8) have the rescaled forms (12) and
(13) respectively, we can see that they are not conformal invariance. The
reason is due to the appearance of the potential $\dfrac{1}{r^{2}}\phi$ which
arises from the commutator $[r\not{\nabla},\not{\Delta}]$. However, this
potential decay is fast enough to ensure that the energy and pointwise decays
of the solutions lead to the energy identity up to the future null infinity
$i^{+}$. Therefore, we can still obtain conformal scattering theories for
Equations (7) and (8) (see Section 3).

Equations (14) and (15) (and also the original Equations (7) and (8)) have
tensorial forms, precisely $2$-forms on the $2$-sphere
$\mathbb{S}^{2}_{(t,r)}$. Taking their decompositions on the basic frame
$(\not{\nabla}_{a},\not{\nabla}_{b})$ of $\Lambda^{1}{(\mathcal{B})}$ we get

$P\hat{\Phi}+L_{1}\hat{\Phi}=0\,,$ (16)

where

$P=\left(\begin{matrix}\Box_{g}&&0\\\ 0&&\Box_{g}\end{matrix}\right)$

is the $2\times 2$-matrix diagram with
$\Box_{g}=\frac{1}{F}L\underline{L}-\not{\Delta}$,

$\hat{\Phi}=\left(\begin{matrix}\hat{\phi}_{1}\\\
\hat{\phi}_{2}\end{matrix}\right)$

is the components of $\hat{\phi}_{a}$ on the basic frame
$(\not{\nabla}_{a},\not{\nabla}_{b})$ of $\Lambda^{1}(\mathcal{B})$ and

$L_{1}=\left(\begin{matrix}L_{1}^{11}&&L_{1}^{12}\\\
L_{1}^{21}&&L_{1}^{22}\end{matrix}\right)$

is the $2\times 2$-matrix where the components are the operators that have the
smooth coefficients

$L_{1}^{ij}=b_{0}^{ij}\partial_{t}+b_{a}^{ij}\partial_{a}+c^{ij}.$

##  3 Energies of the tensorial Fackerell-Ipser field

###  3.1 Energy conservation for the original field

Let $\not{g}_{ab}$ be the restriction of the Schwarzschild metric $g$ to
$\mathbb{S}^{2}_{(t,r)}$. We have $\not{g}_{ab}=r^{2}g_{\mathbb{S}^{2}}$ and
the following identities hold

$\not{\nabla}_{L}\not{g}^{ab}=\not{\nabla}_{\underline{L}}\not{g}^{ab}=0.$

Indeed, by a straightforward calculation we have
$\not{\nabla}_{L}\partial_{\theta^{a}}=FR\partial_{\theta^{a}}$ then

$\not{\nabla}_{L}\not{g}^{ab}=(\not{\nabla}_{L}r^{-2})g_{\mathbb{S}^{2}}^{-1}+r^{2}\not{\nabla}_{L}g_{\mathbb{S}^{2}}^{-1}=-2FR^{3}g_{\mathbb{S}^{2}}^{-1}+2FR^{3}g_{\mathbb{S}^{2}}^{-1}=0.$

Therefore, for a $1$-form $\phi_{a}\in\Lambda^{1}(\mathcal{B})$ on the
$2$-sphere $\mathbb{S}^{2}_{(t,r)}$ we can define

${\phi}^{a}={\phi}_{b}\not{g}^{ab},\hbox{
}\not{\nabla}_{\underline{L}}{\phi}^{a}=\not{\nabla}_{\underline{L}}{\phi}_{b}\not{g}^{ab}\hbox{
and }\not{\nabla}_{L}{\phi}^{a}=\not{\nabla}_{L}{\phi}_{b}\not{g}^{ab}.$

Putting

$|\Phi|^{2}:={\phi}_{a}{\phi}^{a},\,|\not{\nabla}_{\underline{L}}{\Phi}|^{2}:=\not{\nabla}_{\underline{L}}{\phi}_{a}\not{\nabla}_{\underline{L}}{\phi}^{a}\hbox{
and
}|\not{\nabla}_{L}\Phi|^{2}:=\not{\nabla}_{L}{\phi}_{a}\not{\nabla}_{L}{\phi}^{a},$

where the scalar product depends on the metric $\not{g}_{ab}$.

The Fackerell-Ipser Equation (5) leads to

$\displaystyle
4\left(\not{\nabla}_{\underline{L}}\not{\nabla}_{L}{\phi}_{a}-F\not{\Delta}{\phi}_{a}+R^{2}F{\phi}_{a}\right)\not{\nabla}_{L}{\phi}^{a}$
$\displaystyle+4\left(\not{\nabla}_{\underline{L}}\not{\nabla}_{L}{\phi}_{a}-F\not{\Delta}{\phi}_{a}+R^{2}F{\phi}_{a}\right)\not{\nabla}_{\underline{L}}{\phi}^{a}=0.$

This equation is equivalent to

$\displaystyle
2\not{\nabla}_{\underline{L}}\left(|\not{\nabla}_{L}{\Phi}|^{2}+F|R\not{\nabla}_{\mathbb{S}^{2}}{\Phi}|^{2}+R^{2}F|{\Phi}|^{2}\right)$
$\displaystyle+2\not{\nabla}_{L}\left(|\not{\nabla}_{\underline{L}}{\Phi}|^{2}+F|R\not{\nabla}_{\mathbb{S}^{2}}{\Phi}|^{2}+R^{2}F|{\Phi}|^{2}\right)=0,$
(17)

here $\not{\nabla}_{\mathbb{S}^{2}}$ is the angular derivative on the unit
$2$-sphere $\mathbb{S}^{2}$. Since
${g}^{-1}=4F^{-1}\partial_{u}\partial_{v}-r^{-2}g^{-1}_{\mathbb{S}^{2}}$ we
have that (17) is equivalent to

$\displaystyle
8F^{-1}\not{\nabla}^{L}\left(|\not{\nabla}_{L}\hat{\Phi}|^{2}+F|R\not{\nabla}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)$
$\displaystyle+8F^{-1}\not{\nabla}^{\underline{L}}\left(|\not{\nabla}_{\underline{L}}\hat{\Phi}|^{2}+F|R\not{\nabla}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)=0.$
(18)

Equation (17) is also equivalent to

$\displaystyle({\not{\nabla}}_{L}+\not{\nabla}_{\underline{L}})\left(|\not{\nabla}_{L}{\Phi}|^{2}+|\not{\nabla}_{\underline{L}}{\Phi}|^{2}+2F|R\not{\nabla}_{\mathbb{S}^{2}}{\Phi}|^{2}+R^{2}F|{\Phi}|^{2}\right)$
$\displaystyle-(\not{\nabla}_{L}-\not{\nabla}_{\underline{L}})\left(|\not{\nabla}_{L}{\Phi}|^{2}-|\not{\nabla}_{\underline{L}}{\Phi}|^{2}\right)=0\,.$

Hence, in the coordinates $(t,\,r_{*},\,\omega)$ Equation (17) becomes

$\displaystyle
2\not{\nabla}_{\partial_{t}}\left(|\not{\nabla}_{L}{\Phi}|^{2}+|\not{\nabla}_{\underline{L}}{\Phi}|^{2}+2F|R\not{\nabla}_{\mathbb{S}^{2}}{\Phi}|^{2}+R^{2}F|{\Phi}|^{2}\right)$
$\displaystyle+2\not{\nabla}_{\partial_{r_{*}}}\left(|\not{\nabla}_{L}{\Phi}|^{2}-|\not{\nabla}_{\underline{L}}{\Phi}|^{2}\right)=0.$

Since
${g}^{-1}=F^{-1}(\partial_{t}^{2}-\partial_{r_{*}}^{2})-r^{-2}g^{-1}_{\mathbb{S}^{2}}$,
the above equation can be written as

$\displaystyle
2F^{-1}\not{\nabla}^{\partial_{t}}\left(|\not{\nabla}_{L}{\Phi}|^{2}+|\not{\nabla}_{\underline{L}}{\Phi}|^{2}+2F|R\not{\nabla}_{\mathbb{S}^{2}}{\Phi}|^{2}+R^{2}F|{\Phi}|^{2}\right)$
$\displaystyle+2F^{-1}\not{\nabla}^{\partial_{r_{*}}}\left(|\not{\nabla}_{L}{\Phi}|^{2}-|\not{\nabla}_{\underline{L}}{\Phi}|^{2}\right)=0.$
(19)

Equations (18) and (19) are the expressions of the tensorial conservation law
for the tensorial Fackerell-Ipser Equation (5) in the coordinates
$(u,\,v,\,\omega)$ and $(t,\,r_{*},\omega)$ respectively. Using (18) we can
calculate the energy fluxes which across the null hypersurfaces. On the other
hand, the energy flux across the spacelike hypersurface such as
$\left\\{t=0\right\\}$ can be computed from (19). We can re-write (18) and
(19) by a generalized form as

$g(r,F)\not{\nabla}^{a}C(\Phi)=0,$ (20)

where
$\partial_{a}\in\left\\{L,\,\underline{L},\,\partial_{t},\,\partial_{r_{*}}\right\\}$
and $g(r,F)=8F^{-1}$ in coordinates $(u,\,v,\,\omega)$ and $g(r,F)=2F^{-1}$ in
coordinates $(t,\,r_{*},\,\omega)$.

Suppose $\phi_{a}$ is a solution of Equation (5) with the smooth compactly
supported initial data. We integrate the conservation law (20) of ${\phi}_{a}$
on the domain $O$ bounded by the hypersurfaces $\Sigma_{0}$ and $\Sigma_{t}$
in $(\mathcal{B}_{I},g)$ and then by using Stokes’s formula we get

$\int_{O}g(r,F)\not{\nabla}^{a}C(\Phi)\mathrm{dVol}_{g}=\int_{\partial
O}C(\Phi)g(r,F)\partial_{a}{\begin{picture}(0.833,0.8)\put(0.15,0.08){\line(1,0){0.35}}
\put(0.5,0.08){\line(0,1){0.5}} \end{picture}}\mathrm{dVol}_{g},$ (21)

where
$\mathrm{dVol}_{g}=r^{2}F\mathrm{d}t\mathrm{d}r_{*}\mathrm{d}\mathbb{S}^{2}$.
This implies that

$\mathcal{E}_{\Sigma_{t}}({\Phi})=\mathcal{E}_{\Sigma_{0}}({\Phi}),$ (22)

where

$\mathcal{E}_{\Sigma_{0}}(\Phi)=4\int_{\Sigma_{0}}\left(|\not{\nabla}_{\partial_{t}}{\Phi}|^{2}+|\not{\nabla}_{\partial_{r_{*}}}{\Phi}|^{2}+R^{2}F|{\not{\nabla}}_{\mathbb{S}^{2}}{\Phi}|^{2}+R^{2}F|{\Phi}|^{2}\right)r^{2}\mathrm{d}r_{*}\mathrm{d}\mathbb{S}^{2}$

and

$\mathcal{E}_{\Sigma_{t}}(\Phi)=4\int_{\Sigma_{t}}\left(|\not{\nabla}_{\partial_{t}}{\Phi}|^{2}+|\not{\nabla}_{\partial_{r_{*}}}{\Phi}|^{2}+R^{2}F|{\not{\nabla}}_{\mathbb{S}^{2}}{\Phi}|^{2}+R^{2}F|{\Phi}|^{2}\right)r^{2}\mathrm{d}r_{*}\mathrm{d}\mathbb{S}^{2}.$

We consider the domain $\Omega$ which is bounded by the following five
hypersurfaces

$\bar{\Sigma}_{0}=\left\\{t=0,\,u\leq u_{2T},\,v\leq
v_{2T}\right\\},\,\mathcal{H}^{+}_{2u_{T}}=\left\\{u=2u_{T},v\leq
v_{T}\right\\},\,\mathcal{I}^{+}_{2v_{T}}=\left\\{v=2v_{T},u\leq
u_{T}\right\\},$ $\mathcal{H}_{u_{T}}=\left\\{u=u_{T},v_{T}\leq v\leq
2v_{T}\right\\},\,\mathcal{I}_{v_{T}}=\left\\{v=v_{T},u_{T}\leq u\leq
2u_{T}\right\\}.$

Figure 1: The domain $\Omega$ in the Penrose’s conformal compactification
$\bar{\mathcal{B}}_{I}$.

By integrating the conservation law (20) of ${\phi}_{a}$ on the domain
$\Omega$ in $(\mathcal{B}_{I},g)$ and by using again Stokes’s formula we get

$\int_{\Omega}g(r,F)\not{\nabla}^{a}C(\Phi)\mathrm{dVol}_{g}=\int_{\partial\Omega}C(\Phi)g(r,F)\partial_{a}{\begin{picture}(0.833,0.8)\put(0.15,0.08){\line(1,0){0.35}}
\put(0.5,0.08){\line(0,1){0.5}} \end{picture}}\mathrm{dVol}_{g},$ (23)

where
$\mathrm{dVol}_{g}=\dfrac{r^{2}F}{2}\mathrm{d}u\mathrm{d}v\mathrm{d}\mathbb{S}^{2}=r^{2}F\mathrm{d}t\mathrm{d}r_{*}\mathrm{d}\mathbb{S}^{2}$.
This implies that

$\mathcal{E}_{\mathcal{H}^{+}_{2u_{T}}}({\Phi})+\mathcal{E}_{\mathcal{I}_{v_{T}}}({\Phi})+\mathcal{E}_{\mathcal{H}_{u_{T}}}({\Phi})+\mathcal{E}_{\mathcal{I}_{2v_{T}}^{+}}({\Phi})=\mathcal{E}_{\bar{\Sigma}_{0}}(\Phi),$
(24)

where

$\mathcal{E}_{\bar{\Sigma}_{0}}(\Phi)=4\int_{\bar{\Sigma}_{0}}\left(|\not{\nabla}_{\partial_{t}}{\Phi}|^{2}+|\not{\nabla}_{\partial_{r_{*}}}{\Phi}|^{2}+R^{2}F|{\not{\nabla}}_{\mathbb{S}^{2}}{\Phi}|^{2}+R^{2}F|{\Phi}|^{2}\right)r^{2}\mathrm{d}r_{*}\mathrm{d}\mathbb{S}^{2},$
$\mathcal{E}_{\mathcal{I}^{+}_{2v_{T}}}({\Phi})=4\int_{-\infty}^{u_{T}}\int_{\mathbb{S}^{2}}\left(|\not{\nabla}_{\underline{L}}{\Phi}|^{2}+R^{2}F|\not{\nabla}_{\mathbb{S}^{2}}{\Phi}|^{2}+R^{2}F|{\Phi}|^{2}\right)r^{2}\mathrm{d}u\mathrm{d}\mathbb{S}^{2},$
$\mathcal{E}_{\mathcal{I}_{v_{T}}}({\Phi})=4\int_{u_{T}}^{2u_{T}}\int_{\mathbb{S}^{2}}\left(|\not{\nabla}_{\underline{L}}{\Phi}|^{2}+R^{2}F|\not{\nabla}_{\mathbb{S}^{2}}{\Phi}|^{2}+R^{2}F|{\Phi}|^{2}\right)r^{2}\mathrm{d}u\mathrm{d}\mathbb{S}^{2}.$
$\mathcal{E}_{\mathcal{H}^{+}_{2u_{T}}}({\Phi})=4\int^{v_{T}}_{-\infty}\int_{\mathbb{S}^{2}}\left(|\not{\nabla}_{L}{\Phi}|^{2}+R^{2}F|\not{\nabla}_{\mathbb{S}^{2}}{\Phi}|^{2}+R^{2}F|{\Phi}|^{2}\right)r^{2}\mathrm{d}v\mathrm{d}\mathbb{S}^{2},$
$\mathcal{E}_{\mathcal{H}_{u_{T}}}({\Phi})=4\int^{2v_{T}}_{v_{T}}\int_{\mathbb{S}^{2}}\left(|\not{\nabla}_{L}{\Phi}|^{2}+R^{2}F|\not{\nabla}_{\mathbb{S}^{2}}{\Phi}|^{2}+R^{2}F|{\Phi}|^{2}\right)r^{2}\mathrm{d}v\mathrm{d}\mathbb{S}^{2}.$

###  3.2 Energy conservation for the rescaled field

Since the rescaled field $\hat{\phi}_{a}$ has the decomposition on the basic
frame of the $2$-sphere $\mathbb{S}^{2}_{(t,r)}$ at $(t,r)$ as follows

$\hat{\phi}_{a}=r\phi_{a}=r[\not{\nabla}_{a}r(r^{2}\rho)+e_{ab}\not{\nabla}^{b}r(r^{2}\sigma)],$

we can define the following norms by the same way for the original field

$|\hat{\Phi}|^{2}:=\hat{\phi}_{a}\hat{\phi}_{b}\not{g}^{ab},\,|\not{\nabla}_{L}\hat{\Phi}|^{2}:=\not{\nabla}_{L}\hat{\phi}_{a}(\not{\nabla}_{L}\hat{\phi}_{b})\not{g}^{ab}\hbox{
and
}|\not{\nabla}_{\underline{L}}\hat{\Phi}|^{2}:=\not{\nabla}_{\underline{L}}\hat{\phi}_{a}(\not{\nabla}_{\underline{L}}\hat{\phi}_{b})\not{g}^{ab},$

where the scalar product depends on the projected metric $\not{g}_{ab}$.

######  Remark 3.

Since we have

$\hat{\phi}_{a}=r\phi_{a}=\not{\nabla}_{{\theta^{a}}}r(r^{2}\rho)+e_{\theta^{a}\theta^{b}}\not{\nabla}^{{\theta^{b}}}r(r^{2}\sigma),$

the norms of the rescaled field $\hat{\phi}_{a}$ can be also defined on the
unit $2$-sphere $\mathbb{S}^{2}$ as

$|\hat{\Phi}|_{\mathbb{S}^{2}}^{2}=[\not{\nabla}_{{\theta^{a}}}r(r^{2}\rho)+e_{\theta^{a}\theta^{b}}\not{\nabla}^{{\theta^{b}}}r(r^{2}\sigma)][\not{\nabla}_{{\theta^{a}}}r(r^{2}\rho)+e_{\theta^{a}\theta^{b}}\not{\nabla}^{{\theta^{b}}}r(r^{2}\sigma)]{g}_{\mathbb{S}^{2}}^{\theta^{a}\theta^{b}}.$

Clearly, we have

$\displaystyle|\hat{\Phi}|_{\mathbb{S}^{2}}^{2}$ $\displaystyle=$
$\displaystyle
r^{2}[\not{\nabla}_{{\theta^{a}}}(r(r^{2}\rho))+e_{\theta^{a}\theta^{b}}\not{\nabla}^{{\theta^{b}}}(r(r^{2}\sigma))][\not{\nabla}_{{\theta^{a}}}(r(r^{2}\rho))+e_{\theta^{a}\theta^{b}}\not{\nabla}^{{\theta^{b}}}(r(r^{2}\sigma))]\frac{1}{r^{2}}{g}_{\mathbb{S}^{2}}^{\theta^{a}\theta^{b}}$
$\displaystyle=$ $\displaystyle r^{2}|\hat{\Phi}|.$

The rescaled Equation (14) leads to

$\displaystyle
4\left(\not{\nabla}_{\underline{L}}\not{\nabla}_{L}\hat{\phi}_{a}-FR^{2}\not{\Delta}_{\mathbb{S}^{2}}\hat{\phi}_{a}+(2MR+1)FR^{2}\hat{\phi}_{a}\right)(\not{\nabla}_{L}\hat{\phi}_{b})\not{g}^{ab}$
$\displaystyle+4\left(\not{\nabla}_{\underline{L}}\not{\nabla}_{L}\hat{\phi}_{a}-FR^{2}\not{\Delta}_{\mathbb{S}^{2}}\hat{\phi}_{a}+(2MR+1)FR^{2}\hat{\phi}_{a}\right)(\not{\nabla}_{\underline{L}}\hat{\phi}_{b})\not{g}^{ab}=0.$

By using
$\not{\nabla}_{L}\not{g}^{ab}=\not{\nabla}_{\underline{L}}\not{g}^{ab}=0$ we
see that this equation is equivalent to

$\displaystyle
2\not{\nabla}_{\underline{L}}\left(|\not{\nabla}_{L}\hat{\Phi}|^{2}+F|R\not{\nabla}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)$
$\displaystyle+2\not{\nabla}_{L}\left(|\not{\nabla}_{\underline{L}}\hat{\Phi}|^{2}+F|R\not{\nabla}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)=0.$
(25)

If we use the norms $|.|_{\mathbb{S}^{2}}$ on the unit $2$-sphere
$\mathbb{S}^{2}$ we will get the same equality as (26) due to

$L(r^{2})+\underline{L}(r^{2})=0.$

Raising the index of (25) by
$\hat{g}^{-1}=4r^{2}F^{-1}\partial_{u}\partial_{v}-g^{-1}_{\mathbb{S}^{2}}$,
we get

$\displaystyle
8r^{2}F^{-1}\not{\nabla}^{L}\left(|\not{\nabla}_{L}\hat{\Phi}|^{2}+F|R\not{\nabla}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)$
$\displaystyle+8r^{2}F^{-1}\not{\nabla}^{\underline{L}}\left(|\not{\nabla}_{\underline{L}}\hat{\Phi}|^{2}+F|R\not{\nabla}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)=0.$
(26)

Equation (25) is also equivalent to

$\displaystyle({\not{\nabla}}_{L}+\not{\nabla}_{\underline{L}})\left(|\not{\nabla}_{L}\hat{\Phi}|^{2}+|\not{\nabla}_{\underline{L}}\hat{\Phi}|^{2}+2F|R\not{\nabla}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+2(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)$
$\displaystyle-(\not{\nabla}_{L}-\not{\nabla}_{\underline{L}})\left(|\not{\nabla}_{L}\hat{\Phi}|^{2}-|\not{\nabla}_{\underline{L}}\hat{\Phi}|^{2}\right)=0.$

Hence, in the coordinates $(t,\,r_{*},\,\omega)$ Equation (25) becomes

$\displaystyle
2\not{\nabla}_{\partial_{t}}\left(|\not{\nabla}_{L}\hat{\Phi}|^{2}+|\not{\nabla}_{\underline{L}}\hat{\Phi}|^{2}+2F|R\not{\nabla}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+2(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)$
$\displaystyle+2\not{\nabla}_{\partial_{r_{*}}}\left(|\not{\nabla}_{L}\hat{\Phi}|^{2}-|\not{\nabla}_{\underline{L}}\hat{\Phi}|^{2}\right)=0.$

Since
$\hat{g}^{-1}=r^{2}F^{-1}(\partial_{t}^{2}-\partial_{r_{*}}^{2})-g^{-1}_{\mathbb{S}^{2}}$,
we obtain that

$\displaystyle
2r^{2}F^{-1}\not{\nabla}^{\partial_{t}}\left(|\not{\nabla}_{L}\hat{\Phi}|^{2}+|\not{\nabla}_{\underline{L}}\hat{\Phi}|^{2}+2F|R\not{\nabla}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+2(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)$
$\displaystyle+2r^{2}F^{-1}\not{\nabla}^{\partial_{r_{*}}}\left(|\not{\nabla}_{L}\hat{\Phi}|^{2}-|\not{\nabla}_{\underline{L}}\hat{\Phi}|^{2}\right)=0.$
(27)

Equations (26) and (27) are the expressions of the rescaled tensorial
conservation law for (14) in the coordinates $(u,\,v,\,\omega)$ and
$(t,\,r_{*},\omega)$ respectively. We can re-write (26) and (27) by a
generalized form as

$f(r,F)\not{\nabla}^{a}C(\hat{\Phi})=0,$ (28)

where
$\partial_{a}\in\left\\{L,\,\underline{L},\,\partial_{t},\,\partial_{r_{*}}\right\\}$
and $f(r,F)=8r^{2}F^{-1}$ in coordinates $(u,\,v,\,\omega)$ and
$f(r,F)=2r^{2}F^{-1}$ in coordinates $(t,\,r_{*},\,\omega)$.

Suppose $\hat{\phi}_{a}$ is a solution of the rescaled tensorial Equation (14)
with the smooth compactly supported initial data. By integrating the
conservation law (28) of $\hat{\phi}_{a}$ on the domain $O$ bounded by the
hypersurfaces $\Sigma_{0}$ and $\Sigma_{t}$ in
$(\bar{\mathcal{B}}_{I},\hat{g})$ and by using Stokes’s formula we get

$\int_{O}f(r,F)\not{\nabla}^{a}C(\hat{\Phi})\mathrm{dVol}_{\hat{g}}=\int_{\partial
O}C(\hat{\Phi})f(r,F)\partial_{a}{\begin{picture}(0.833,0.8)\put(0.15,0.08){\line(1,0){0.35}}
\put(0.5,0.08){\line(0,1){0.5}} \end{picture}}\mathrm{dVol}_{\hat{g}},$ (29)

where
$\mathrm{dVol}_{\hat{g}}=R^{2}F\mathrm{d}t\mathrm{d}r_{*}\mathrm{d}\mathbb{S}^{2}$.
This implies that

$\mathcal{E}_{\Sigma_{0}}(\hat{\Phi})=\mathcal{E}_{\Sigma_{t}}(\hat{\Phi}),$
(30)

where

$\mathcal{E}_{\Sigma_{0}}(\hat{\Phi})=4\int_{\Sigma_{0}}\left(|\not{\nabla}_{\partial_{t}}\hat{\Phi}|^{2}+|\not{\nabla}_{\partial_{r_{*}}}\hat{\Phi}|^{2}+R^{2}F|{\not{\nabla}}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)\mathrm{d}r_{*}\mathrm{d}\mathbb{S}^{2}.$
(31)

and

$\mathcal{E}_{\Sigma_{t}}(\hat{\Phi})=4\int_{\Sigma_{t}}\left(|\not{\nabla}_{\partial_{t}}\hat{\Phi}|^{2}+|\not{\nabla}_{\partial_{r_{*}}}\hat{\Phi}|^{2}+R^{2}F|{\not{\nabla}}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)\mathrm{d}r_{*}\mathrm{d}\mathbb{S}^{2}.$

We recall that $\Omega$ is the domain bounded by the following five
hypersurfaces

$\bar{\Sigma}_{0}=\left\\{t=0\right\\},\,\mathcal{H}^{+}_{2u_{T}}=\left\\{u=2u_{T},v\leq
v_{T}\right\\},\,\mathcal{I}^{+}_{2v_{T}}=\left\\{v=2v_{T},u\leq
u_{T}\right\\},$ $\mathcal{H}_{u_{T}}=\left\\{u=u_{T},v_{T}\leq v\leq
2v_{T}\right\\},\,\mathcal{I}_{v_{T}}=\left\\{v=v_{T},u_{T}\leq u\leq
2u_{T}\right\\}.$

By integrating the conservation law (28) of $\hat{\phi}_{a}$ on the domain
$\Omega$ in $(\bar{\mathcal{B}_{I}},\hat{g})$ and by using Stokes’s formula we
get

$\int_{\Omega}f(r,F)\not{\nabla}^{a}C(\hat{\Phi})\mathrm{dVol}_{\hat{g}}=\int_{\partial\Omega}C(\hat{\Phi})f(r,F)\partial_{a}{\begin{picture}(0.833,0.8)\put(0.15,0.08){\line(1,0){0.35}}
\put(0.5,0.08){\line(0,1){0.5}} \end{picture}}\mathrm{dVol}_{\hat{g}},$ (32)

where
$\mathrm{dVol}_{\hat{g}}=\dfrac{R^{2}F}{2}\mathrm{d}u\mathrm{d}v\mathrm{d}\mathbb{S}^{2}=R^{2}F\mathrm{d}t\mathrm{d}r_{*}\mathrm{d}\mathbb{S}^{2}$.
This implies that

$\mathcal{E}_{\mathcal{H}^{+}_{2u_{T}}}(\hat{\Phi})+\mathcal{E}_{\mathcal{I}_{v_{T}}}(\hat{\Phi})+\mathcal{E}_{\mathcal{H}_{u_{T}}}(\hat{\Phi})+\mathcal{E}_{\mathcal{I}_{2v_{T}}^{+}}(\hat{\Phi})=\mathcal{E}_{\bar{\Sigma}_{0}}(\hat{\Phi}),$
(33)

where

$\mathcal{E}_{\bar{\Sigma}_{0}}(\hat{\Phi})=4\int_{\bar{\Sigma}_{0}}\left(|\not{\nabla}_{\partial_{t}}\hat{\Phi}|^{2}+|\not{\nabla}_{\partial_{r_{*}}}\hat{\Phi}|^{2}+R^{2}F|{\not{\nabla}}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)\mathrm{d}r_{*}\mathrm{d}\mathbb{S}^{2},$
$\mathcal{E}_{\mathcal{I}^{+}_{2v_{T}}}(\hat{\Phi})=4\int_{-\infty}^{u_{T}}\int_{\mathbb{S}^{2}}\left(|\not{\nabla}_{\underline{L}}\hat{\Phi}|^{2}+R^{2}F|\not{\nabla}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)\mathrm{d}u\mathrm{d}\mathbb{S}^{2},$
$\mathcal{E}_{\mathcal{I}_{v_{T}}}(\hat{\Phi})=4\int_{u_{T}}^{2u_{T}}\int_{\mathbb{S}^{2}}\left(|\not{\nabla}_{\underline{L}}\hat{\Phi}|^{2}+R^{2}F|\not{\nabla}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)\mathrm{d}u\mathrm{d}\mathbb{S}^{2}.$
$\mathcal{E}_{\mathcal{H}^{+}_{2u_{T}}}(\hat{\Phi})=4\int^{v_{T}}_{-\infty}\int_{\mathbb{S}^{2}}\left(|\not{\nabla}_{L}\hat{\Phi}|^{2}+R^{2}F|\not{\nabla}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)\mathrm{d}v\mathrm{d}\mathbb{S}^{2},$
$\mathcal{E}_{\mathcal{H}_{u_{T}}}(\hat{\Phi})=4\int^{2v_{T}}_{v_{T}}\int_{\mathbb{S}^{2}}\left(|\not{\nabla}_{L}\hat{\Phi}|^{2}+R^{2}F|\not{\nabla}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)\mathrm{d}v\mathrm{d}\mathbb{S}^{2}.$

######  Remark 4.

We observe that the hypersurfaces $\mathcal{H}_{u_{T}}$ and
$\mathcal{I}_{v_{T}}$ will tend to the future horizon $\mathfrak{H}^{+}$ and
future null infinity ${\mathscr{I}}^{+}$ as $u_{T}$ and $v_{T}$ (as well as
$T$) tend to infinity respectively. Therefore, the energy on the future
horizon and null infinity are defined as follows

$\mathcal{E}_{\mathfrak{H}_{T}^{+}}(\hat{\Phi})=4\int_{\mathfrak{H}^{+}_{T}}|\not{\nabla}_{L}\hat{\Phi}|_{\mathfrak{H}_{T}^{+}}|^{2}\mathrm{d}v\mathrm{d}\mathbb{S}^{2},$
$\mathcal{E}_{{\mathscr{I}}_{T}^{+}}(\hat{\Phi})=4\int_{{\mathscr{I}}^{+}_{T}}|\not{\nabla}_{\underline{L}}\hat{\Phi}|_{{\mathscr{I}}_{T}^{+}}|^{2}\mathrm{d}u\mathrm{d}\mathbb{S}^{2}.$

As a direct consequence of the energy equality (33) we have the following
result.

######  Proposition 2.

Consider the smooth and compactly supported initial data on $\Sigma_{0}$, we
can define the energy fluxes of the rescaled solution $\hat{\phi}_{a}$ of (14)
across the null conformal boundary $\mathfrak{H}^{+}\cup{\mathscr{I}}^{+}$ by
${\mathcal{E}}_{{\mathscr{I}}^{+}}(\hat{\Phi})+{\mathcal{E}}_{\mathfrak{H}^{+}}(\hat{\Phi}):=\lim_{T\rightarrow\infty}{\mathcal{E}}_{{\mathscr{I}}_{T}^{+}}(\hat{\Phi})+{\mathcal{E}}_{\mathfrak{H}_{T}^{+}}(\hat{\Phi})$.
Moreover, we have

${\mathcal{E}}_{{\mathscr{I}}^{+}}(\hat{\Phi})+{\mathcal{E}}_{\mathfrak{H}^{+}}(\hat{\Phi})\leq{\mathcal{E}}_{\Sigma_{0}}(\hat{\Phi}),$

the equality holds if and only if

$\lim_{u_{T},v_{T}\rightarrow\infty}\left({\mathcal{E}}_{\mathcal{H}_{u_{T}}}(\hat{\Phi})+{\mathcal{E}}_{\mathcal{I}_{v_{T}}}(\hat{\Phi})\right)=0.$

By the same manner for $\hat{\phi}_{a}$, we can also establish the energy
fluxes of $\hat{\underline{\phi}}_{a}$ and then we obtain the energy equality
as follows.

######  Proposition 3.

Consider the smooth and compactly supported initial data on $\Sigma_{0}$, we
can define the energy fluxes of the rescaled solution
$\hat{\underline{\phi}}_{a}$ of (15) across the null conformal boundary
$\mathfrak{H}^{+}\cup{\mathscr{I}}^{+}$ by
${\mathcal{E}}_{{\mathscr{I}}^{+}}(\hat{\underline{\Phi}})+{\mathcal{E}}_{\mathfrak{H}^{+}}(\hat{\underline{\Phi}}):=\lim_{T\rightarrow\infty}{\mathcal{E}}_{{\mathscr{I}}_{T}^{+}}(\hat{\underline{\Phi}})+{\mathcal{E}}_{\mathfrak{H}_{T}^{+}}(\hat{\underline{\Phi}})$.
Moreover, we have

${\mathcal{E}}_{{\mathscr{I}}^{+}}(\hat{\underline{\Phi}})+{\mathcal{E}}_{\mathfrak{H}^{+}}(\hat{\underline{\Phi}})\leq{\mathcal{E}}_{\Sigma_{0}}(\hat{\underline{\Phi}}),$

the equality holds if and only if

$\lim_{u_{T},v_{T}\rightarrow\infty}\left({\mathcal{E}}_{\mathcal{H}_{u_{T}}}(\hat{\underline{\Phi}})+{\mathcal{E}}_{\mathcal{I}_{v_{T}}}(\hat{\underline{\Phi}})\right)=0.$

Here we define

$\mathcal{E}_{\Sigma_{0}}(\hat{\underline{\Phi}})=4\int_{\Sigma_{0}}\left(|\not{\nabla}_{\partial_{t}}\hat{\underline{\Phi}}|^{2}+|\not{\nabla}_{\partial_{r_{*}}}\hat{\underline{\Phi}}|^{2}+R^{2}F|{\not{\nabla}}_{\mathbb{S}^{2}}\hat{\underline{\Phi}}|^{2}+(2MR+1)R^{2}F|\hat{\underline{\Phi}}|^{2}\right)\mathrm{d}r_{*}\mathrm{d}\mathbb{S}^{2}.$
$\mathcal{E}_{\mathcal{I}_{v_{T}}}(\hat{\underline{\Phi}})=4\int_{u_{T}}^{2u_{T}}\int_{\mathbb{S}^{2}}\left(|\not{\nabla}_{\underline{L}}\hat{\underline{\Phi}}|^{2}+R^{2}F|\not{\nabla}_{\mathbb{S}^{2}}\hat{\underline{\Phi}}|^{2}+(2MR+1)R^{2}F|\hat{\underline{\Phi}}|^{2}\right)\mathrm{d}u\mathrm{d}\mathbb{S}^{2}.$
$\mathcal{E}_{\mathcal{H}_{u_{T}}}(\hat{\underline{\Phi}})=4\int^{2v_{T}}_{v_{T}}\int_{\mathbb{S}^{2}}\left(|\not{\nabla}_{L}\hat{\underline{\Phi}}|^{2}+R^{2}F|\not{\nabla}_{\mathbb{S}^{2}}\hat{\underline{\Phi}}|^{2}+(2MR+1)R^{2}F|\hat{\underline{\Phi}}|^{2}\right)\mathrm{d}v\mathrm{d}\mathbb{S}^{2}.$

###  3.3 Tensorial field space of initial data and Cauchy problem

Since the energy of the rescaled field $\hat{\phi}_{a}$ across the
hypersurface $\Sigma_{0}$ is given by (31), we define the finite energy space
as follows.

######  Definition 1.

We denote by $\mathcal{H}$ the completion of
$\mathcal{C}_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{\Sigma_{0}})\times\mathcal{C}_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{\Sigma_{0}})$
in the norm

$\left\|(\hat{\Phi}_{0},\hat{\Phi}_{1})\right\|_{\mathcal{H}}=2\left(\int_{\Sigma_{0}}\left(|\not{\nabla}_{\partial_{t}}\hat{\Phi}|^{2}+|\not{\nabla}_{\partial_{r_{*}}}\hat{\Phi}|^{2}+R^{2}F|\not{\nabla}_{\mathbb{S}^{2}}\hat{\phi}|^{2}+(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)\mathrm{d}r_{*}\mathrm{d}\mathbb{S}^{2}\right)^{1/2}.$

Before stating the theorem of the well-posedness of Cauchy problem, we need
the following definition of the Sobolev spaces for tensorial fields that are
defined on open sets (see Definition 2 [40] for the same definition for scalar
fields):

######  Definition 2.

Let $s\in[0,+\infty)$, a tensorial field $u_{a}$ on
$\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{B}_{I}}$ is said to belong to
$H^{s}_{loc}(\Lambda^{1}(\mathbb{S}^{2})|_{\bar{\mathcal{B}_{I}})}$ if for any
local chart $(\Omega,\zeta)$, such that $\Omega\subset\mathcal{B}_{I}$ is an
open set with smooth compact boundary in $\bar{\mathcal{B}_{I}}$ (note that
this excludes neighbourhoods of either $i^{\pm}$ or $i^{0}$ but allows open
sets whose boundary contains parts of the conformal boundary) and $\zeta$ is a
smooth diffeomorphism from $\Omega$ onto a bounded open set
$U\subset\mathbb{R}^{4}$ with smooth compact boundary, we have
$u_{a}\circ\zeta^{-1}\in H^{s}(\Lambda^{1}(\mathbb{S}^{2})|_{U})$.

By Leray’s theorem for the hyperbolic differential equations and the energy
identities (22) and (30), we can establish the well-posedness of Cauchy
problem for Equation (7) and the rescaled Equation equation (12) as follows

######  Proposition 4.

The Cauchy problem for (7) on
$\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{B}_{I}})$ (and for (12) on
$\Lambda^{1}(\mathbb{S}^{2})|_{\bar{\mathcal{B}_{I}}}$) is well-posed in
$\mathcal{H}$. This means that for any
$(\hat{\phi}_{a0},\hat{\phi}_{a1})\in\mathcal{H}$, there exists a unique
solution
$\phi_{a}\in\mathcal{D}^{\prime}(\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{B}_{I})}$
of (7) such that

$(r\phi_{a},r\not{\nabla}_{\partial_{t}}\phi_{a})\in\mathcal{C}(\mathbb{R}_{t};\,\mathcal{H}):\,r\phi_{a}|_{t=0}=\hat{\phi}_{a0};\,r\not{\nabla}_{\partial_{t}}\phi_{a}|_{t=0}=\hat{\phi}_{a1}.$

Moreover, $\hat{\phi}_{a}=r\phi_{a}$ belongs to
$H^{1}_{loc}(\Lambda^{1}(\mathbb{S}^{2})|_{\bar{{\mathcal{B}}_{I}}})$.

###  3.4 Energy identity up to $i^{+}$ and trace operator

In this section, we will show that
$\lim_{u_{T},v_{T}\rightarrow\infty}\left({\mathcal{E}}_{\mathcal{H}_{u_{T}}}(\hat{\Phi})+{\mathcal{E}}_{\mathcal{I}_{v_{T}}}(\hat{\Phi})\right)=0$
and then we can obtain the energy equality

${\mathcal{E}}_{\Sigma_{0}}(\hat{\Phi})={\mathcal{E}}_{\mathfrak{H}^{+}}(\hat{\Phi})+{\mathcal{E}}_{{\mathscr{I}}^{+}}(\hat{\Phi}).$

We shall use the following results about the energy and pointwise decays of
the tensorial field $\phi_{a}$ which satisfies Equation (7). These results are
a direct consequence of Theorems 4.2 and Lemma 5.8 in [46]).

######  Theorem 1.

There exist a positive number $(r_{FH})_{*}$ and positive constants $C$ and
$D$ depending only on $M$ and $(r_{FH})_{*}$ such that the following
assertions hold

  * i)

(Energy Decay). Let $\phi_{a}$ be a smooth tensorial solution to the
Fackerell-Ipser Equations (7) on $\left\\{u\geq u_{0},\,v\geq v_{0}\right\\}$.
We have the following decay energy for the original field $\phi_{a}$ on
$\left\\{u=u_{T},v\geq v_{T}\right\\}\cup\left\\{u\geq u_{T},v=v_{T}\right\\}$
(where $u_{T}\geq u_{0}$ and $v_{T}=u_{T}+2(r_{FH})_{*}$):

$\mathcal{E}_{u=u_{T},v\geq v_{T}}(\Phi)+\mathcal{E}_{u\geq
u_{T},v=v_{T}}(\Phi)\leq Cu^{-2}.$ (34)

  * ii)

(Pointwise Decay). In the region
$\left\\{r_{*}\leq(r_{FH})_{*}\right\\}\cap\left\\{u\geq u_{0},\,v\geq
v_{0}\right\\}$ the solution $\phi_{a}$ of (7) decays as $v^{-1}$. In the
region $\left\\{r_{*}\geq(r_{FH})_{*}\right\\}\cap\left\\{u\geq u_{0},\,v\geq
v_{0}\right\\}$, it decays as $r^{-1}(|u|+1)^{-1/2}$ or $r^{-1/2}(|u|+1)^{-1}$
i.e

$|\Phi|^{2}\leq\frac{D}{r(|u|+1)^{1/2}}\hbox{ and }\frac{D}{r^{1/2}(|u|+1)}.$

######  Remark 5.

Assertion i) is an exact consequence of Lemma 5.8 in [46]. On the other hand,
Assertion ii) is obtained by combining Remark 1 and Theorem 4.2 in [46].

Using these decay results we prove the following theorem.

######  Theorem 2.

Let $\hat{\phi}_{a}$ (resp. $\hat{\underline{\phi}}_{a}$)be the solution of
the rescaled tensorial Fackerell-Ipser Equation (14) (resp. (15)). The energy
of $\hat{\phi}_{a}$ (resp. $\hat{\underline{\phi}}_{a}$) across the
hypersurface $\mathcal{H}_{T}$ tends to zero as $T$ tends to infinity

$\lim_{u_{T},v_{T}\rightarrow\infty}\left({\mathcal{E}}_{\mathcal{H}_{u_{T}}}(\hat{\Phi})+{\mathcal{E}}_{\mathcal{I}_{v_{T}}}(\hat{\Phi})\right)=0$
$(\hbox{resp.
}\lim_{u_{T},v_{T}\rightarrow\infty}\left({\mathcal{E}}_{\mathcal{H}_{u_{T}}}(\hat{\underline{\Phi}})+{\mathcal{E}}_{\mathcal{I}_{v_{T}}}(\hat{\underline{\Phi}})\right)=0.)$

As a consequence, the equality of the energies remains true

${\mathcal{E}}_{\Sigma_{0}}(\hat{\Phi})={\mathcal{E}}_{\mathfrak{H}^{+}}(\hat{\Phi})+{\mathcal{E}}_{{\mathscr{I}}^{+}}(\hat{\Phi}),$
(35) $(\hbox{resp.
}{\mathcal{E}}_{\Sigma_{0}}(\hat{\underline{\Phi}})={\mathcal{E}}_{\mathfrak{H}^{+}}(\hat{\underline{\Phi}})+{\mathcal{E}}_{{\mathscr{I}}^{+}}(\hat{\underline{\Phi}}).)$

###### Proof.

In the region $\left\\{r_{*}\geq(r_{FH})_{*}\right\\}$, we have

$\displaystyle{\mathcal{E}}_{\mathcal{I}_{v_{T}}}(\hat{\Phi})$
$\displaystyle=$ $\displaystyle
4\int_{v_{T}}^{2v_{T}}\int_{\mathbb{S}^{2}}\left(|\not{\nabla}_{L}\hat{\Phi}|^{2}+R^{2}F|\not{\nabla}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)\mathrm{d}v\mathrm{d}\mathbb{S}^{2}$
$\displaystyle=$ $\displaystyle
4\int_{v_{T}}^{2v_{T}}\int_{\mathbb{S}^{2}}\left(r^{2}|\not{\nabla}_{L}{\Phi}|^{2}+F|\not{\nabla}_{\mathbb{S}^{2}}{\Phi}|^{2}+F^{2}|\Phi|^{2}+(2MR+1)F|\Phi|^{2}\right)\mathrm{d}v\mathrm{d}\mathbb{S}^{2}$
$\displaystyle=$ $\displaystyle
4\int_{v_{T}}^{2v_{T}}\int_{\mathbb{S}^{2}}\left(|\not{\nabla}_{L}{\Phi}|^{2}+F|R\not{\nabla}_{\mathbb{S}^{2}}{\Phi}|^{2}+FR^{2}|\Phi|^{2}\right)r^{2}\mathrm{d}v\mathrm{d}\mathbb{S}^{2}$
$\displaystyle+4\int_{v_{T}}^{2v_{T}}\int_{\mathbb{S}^{2}}\left((F^{2}-F)|\Phi|^{2}+(2MR+1)F|{\Phi}|^{2}\right)\mathrm{d}v\mathrm{d}\mathbb{S}^{2}$
$\displaystyle\lesssim$
$\displaystyle\mathcal{E}_{\mathcal{I}_{v_{T}}}(\Phi)+\int_{v_{T}}^{2v_{T}}\int_{\mathbb{S}^{2}}|\Phi|^{2}\mathrm{d}v\mathrm{d}\mathbb{S}^{2}.$

Using the decay results in Theorem 1 we get

$\displaystyle\mathcal{E}_{\mathcal{I}_{v_{T}}}(\hat{\Phi})$
$\displaystyle\lesssim$ $\displaystyle
Cu_{T}^{-2}+\int_{v_{T}}^{2v_{T}}\frac{4\pi D}{r(u_{T}+1)^{2}}\mathrm{d}v$
$\displaystyle\lesssim$ $\displaystyle
Cu_{T}^{-2}+\frac{v_{T}}{(r_{FH})_{*}(u_{T}+1)^{2}}=cu_{T}^{-2}+4\pi
D\frac{u_{T}+2(r_{FH})_{*}}{(r_{FH})_{*}(u_{T}+1)^{2}}\longrightarrow 0\hbox{
as }u_{T}\rightarrow\infty.$

On the other hand, in the region $\left\\{r_{*}\leq(r_{FH})_{*}\right\\}$ we
have

$\displaystyle{\mathcal{E}}_{\mathcal{H}_{u_{T}}}(\hat{\Phi})$
$\displaystyle=$ $\displaystyle
4\int_{u_{T}}^{2u_{T}}\int_{\mathbb{S}^{2}}\left(|\not{\nabla}_{\underline{L}}\hat{\Phi}|^{2}+R^{2}F|\not{\nabla}_{\mathbb{S}^{2}}\hat{\Phi}|^{2}+(2MR+1)R^{2}F|\hat{\Phi}|^{2}\right)\mathrm{d}u\mathrm{d}\mathbb{S}^{2}$
$\displaystyle=$ $\displaystyle
4\int_{u_{T}}^{2u_{T}}\int_{\mathbb{S}^{2}}\left(r^{2}|\not{\nabla}_{\underline{L}}{\Phi}|^{2}+F|\not{\nabla}_{\mathbb{S}^{2}}{\Phi}|^{2}+F^{2}|\Phi|^{2}+(2MR+1)F|{\phi}|^{2}\right)\mathrm{d}u\mathrm{d}\mathbb{S}^{2}$
$\displaystyle=$ $\displaystyle
4\int_{u_{T}}^{2u_{T}}\int_{\mathbb{S}^{2}}\left(|\not{\nabla}_{\underline{L}}{\Phi}|^{2}+F|\not{\nabla}_{\mathbb{S}^{2}}{\Phi}|^{2}+FR^{2}|\Phi|^{2}\right)r^{2}\mathrm{d}u\mathrm{d}\mathbb{S}^{2}$
$\displaystyle+4\int_{u_{T}}^{2u_{T}}\int_{\mathbb{S}^{2}}\left((F^{2}-F)|\Phi|^{2}+(2MR+1)F|{\Phi}|^{2}\right)\mathrm{d}u\mathrm{d}\mathbb{S}^{2}$
$\displaystyle\lesssim$
$\displaystyle\mathcal{E}_{\mathcal{H}_{u_{T}}}(\Phi)+\int_{u_{T}}^{\infty}\int_{\mathbb{S}^{2}}|\Phi|^{2}\mathrm{d}v\mathrm{d}\mathbb{S}^{2}.$

Using again Theorem 1 we get

$\displaystyle{\mathcal{E}}_{\mathcal{H}_{u_{T}}}(\hat{\Phi})$
$\displaystyle\lesssim$ $\displaystyle
Cu_{T}^{-2}+\int_{u_{T}}^{2u_{T}}\frac{D}{v_{T}^{2}}\mathrm{d}u$
$\displaystyle\lesssim$ $\displaystyle
Cu_{T}^{-2}+\frac{u_{T}}{v_{T}^{2}}=Cu_{T}^{-2}+4\pi
D\frac{v_{T}-2(r_{FH})_{*}}{v_{T}^{2}}\longrightarrow 0\hbox{ as
}{u_{T},\,v_{T}}\rightarrow\infty.$

Therefore we establish that

$\lim_{u_{T},v_{T}\rightarrow\infty}\left({\mathcal{E}}_{\mathcal{H}_{u_{T}}}(\hat{\Phi})+{\mathcal{E}}_{\mathcal{I}_{v_{T}}}(\hat{\Phi})\right)=0.$

Together with (33), this gives the equality of the energies (35) for
$\hat{\phi}_{a}$. The equality energy for $\hat{\underline{\phi}}_{a}$ is
established in the same way. Our proof is then completed. ∎

The energy equality leads us to define the trace operator on the conformal
boundary as follows.

######  Definition 3.

(Trace operator). Let
$(\hat{\phi}_{a0},\hat{\phi}_{a1})\in\mathcal{C}_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{\Sigma_{0}})\times\mathcal{C}_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{\Sigma_{0}})$.
Consider the solution of the rescaled Equations (12), and let
$\hat{\phi}_{a}\in\mathcal{C}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{\bar{\mathcal{B}_{I}}})$
such that

$\hat{\phi}_{a}|_{\Lambda^{1}(\mathbb{S}^{2})|_{\Sigma_{0}}}=\hat{\phi}_{a0},\,\not{\nabla}_{t}\hat{\phi}_{a}|_{\Lambda^{1}(\mathbb{S}^{2})|_{\Sigma_{0}}}=\hat{\phi}_{a1}.$

We define the trace operator $\mathcal{T}^{+}$ from
$\mathcal{C}_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{\Sigma_{0}})\times\mathcal{C}_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{\Sigma_{0}})$
to
$\mathcal{C}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{\mathfrak{H}^{+}})\times\mathcal{C}_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{{\mathscr{I}}^{+}})$
by

$\mathcal{T}^{+}(\hat{\phi}_{a0},\hat{\phi}_{a1})=(\hat{\phi}_{a}|_{\mathfrak{H}^{+}},\hat{\phi}_{a}|_{{\mathscr{I}}^{+}}).$

Then we can extend the function space for scattering data by density as in the
following definition.

######  Definition 4.

The function space for scattering data $\mathcal{H}^{+}$ is the completion of
$\mathcal{C}_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{\mathfrak{H}^{+}})\times\mathcal{C}_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{{\mathscr{I}}^{+}})$
in the norm

$\left\|(\xi_{a},\zeta_{a})\right\|_{\mathcal{H}^{+}}=\left(\int_{\mathfrak{H}^{+}}|\not{\nabla}_{L}\xi|^{2}\mathrm{d}v\mathrm{d}^{2}\omega+\int_{{\mathscr{I}}^{+}}|\not{\nabla}_{\underline{L}}\zeta|^{2}\mathrm{d}u\mathrm{d}^{2}\omega\right)^{1/2}.$

This means that

$\mathcal{H}^{+}\simeq\dot{H}^{1}(\mathbb{R}_{v};\,L^{2}(\Lambda^{1}(\mathbb{S}^{2})))\times\dot{H}^{1}(\mathbb{R}_{u};\,L^{2}(\Lambda^{1}(\mathbb{S}^{2}))).$

As a direct consequence of the equality energy (35), we have the following
theorem.

######  Theorem 3.

The trace operator extends uniquely as a bounded linear map from $\mathcal{H}$
to $\mathcal{H}^{+}$. The extended operator is a partial isometry i.e for any
$(\hat{\phi}_{a0},\hat{\phi}_{a1})\in\mathcal{H}$,

$\left\|\mathcal{T}^{+}(\hat{\phi}_{a0},\hat{\phi}_{a1})\right\|_{\mathcal{H}^{+}}=\left\|(\hat{\phi}_{a0},\hat{\phi}_{a1})\right\|_{\mathcal{H}}.$

##  4 Conformal scattering theory

###  4.1 Generalisation of L. Hörmander’s result for tensorial wave equations

Hörmander [25] proved the well-posedness of the Goursat problem for the second
order scalar wave equations with regular first order potentials in the
spatially compact spacetime. Nicolas [39] extended the results of Hörmander
with very slightly more regular metric and potential, precisely a
$\mathcal{C}^{1}$ metric and a potential with continuous coefficients of the
first order terms and locally $L^{\infty}$ coefficients for the terms of order
$0$. Here we will solve the Goursat problem for the tensorial wave equations
with regular first order tensorial potentials. More precisely, we will show
how we can apply the results of Hörmander and Nicolas for the tensorial wave
Equation (14) (or (15)) with the smooth compactly supported initial data on
the conformal boundary i.e,
$(\xi_{a},\zeta_{a})\in\mathcal{C}_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{\mathfrak{H}^{+}})\times\mathcal{C}_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{{\mathscr{I}}^{+}})$
in Schwarzschild background.

By the same manner in Appendix B [40], the conformal compactification domain
$\bar{\mathcal{B}}_{I}$ is embedded partly in a cylindrical globally
hyperbolic spacetime by the following technique: let $\mathcal{S}$ be a
spacelike hypersurface on $\bar{\mathcal{B}}_{I}$ whose intersection with the
horizon is the crossing sphere and which crosses ${\mathscr{I}}^{+}$ strictly
in the past of the support of the data. We cut off the future $\mathcal{V}$ of
a point in $\bar{\mathcal{B}}_{I}$ lying in the future of the past of the
support of the Goursat data and get a spacetime denoted by $\mathfrak{M}$.
Then we extend $\mathfrak{M}$ as a cylindrical globally hyperbolic spacetime
$(\mathbb{R}_{t}\times S^{3},\mathfrak{g})$. The conformal boundary
$\mathfrak{H}^{+}\cup{\mathscr{I}}^{+}$ is extended inside
$\mathcal{I}^{+}(\mathcal{S})/\mathcal{V}$ as a null hypersurface
$\mathcal{C}$, that is the graph of a Lipschitz function over $S^{3}$ and the
data by zero on the rest of the extended hypersurface.

Figure 2: The cutting off $\mathcal{V}$ and extension of the future
$\mathcal{I}^{+}(\mathcal{S})$.

By following Remark 2 we have the decomposition of the rescaled tensorial
Equation (14) on the basic frame $(\not{\nabla}_{a},\not{\nabla}_{b})$ of the
$2$-sphere $\mathbb{S}^{2}_{(t,r)}$ obtained as follows

$P\hat{\Phi}+L_{1}\hat{\Phi}=0\,,$ (36)

with

$P=\left(\begin{matrix}\Box_{g}&&0\\\ 0&&\Box_{g}\end{matrix}\right)$

is the $2\times 2$-matrix diagram with
$\Box_{g}=\frac{1}{F}L\underline{L}-\not{\Delta}$,

$\hat{\Phi}=\left(\begin{matrix}\hat{\phi}_{1}\\\
\hat{\phi}_{2}\end{matrix}\right),$

where $\hat{\phi}_{i}\,(i=1,2)$ are the components of $\hat{\phi}_{a}$
decomposed on the basic frame $(\not{\nabla}_{a},\not{\nabla}_{b})$ of
$\Lambda^{1}(\mathcal{B})$ and

$L_{1}=\left(\begin{matrix}L_{1}^{11}&&L_{1}^{12}\\\
L_{1}^{21}&&L_{1}^{22}\end{matrix}\right)$

is the $2\times 2$-matrix where the components are the operators that have the
smooth coefficients

$L_{1}^{ij}=b_{0}^{ij}\partial_{t}+b_{a}^{ij}\partial_{a}+c^{ij}.$

In the spacetime $(\mathbb{R}_{t}\times S^{3},\mathfrak{g})$, Equation (36)
becomes

$P_{\mathfrak{g}}\hat{\Phi}+L_{1}\hat{\Phi}=0\,,$ (37)

where

$P_{\mathfrak{g}}=\left(\begin{matrix}\Box_{\mathfrak{g}}&&0\\\
0&&\Box_{\mathfrak{g}}\end{matrix}\right).$

If we define the norm of the vector by the maximal element’s norm, we have
that

$0=\sum_{i=1}^{2}\partial_{t}\bar{\hat{\phi}}_{i}\left(P\hat{\Phi}+L_{1}\hat{\Phi}\right)_{i}=\sum_{i=1}^{2}\partial_{t}\bar{\hat{\phi}}_{i}\left(P\hat{\phi}_{i}+(L_{1}\hat{\Phi})_{i}\right)$

and

$\sum_{i=1}^{2}\partial_{t}\bar{\hat{\phi}}_{i}(L_{1}\hat{\Phi})_{i}\leq\left\|\partial_{t}\hat{\Phi}\right\|\left\|L_{1}\hat{\Phi}\right\|\leq\left\|\partial_{t}\hat{\Phi}\right\|\left(\left\|\nabla\hat{\Phi}\right\|+\left\|\partial_{t}\hat{\Phi}\right\|+\left|\hat{\Phi}\right\|\right).$

The above estimates are used to control the energies for the matrix wave
equation (37) by the same method for the scalar wave equation in [39]. In
addition the metric $\mathfrak{g}$ and the coefficients in $L_{1}$ operator
satisfy the conditions ${\bf H}_{1}$ and ${\bf H}_{2}$ in [39]. Therefore, The
Goursat problem for the system (37) is solved by the same manner for the
scalar wave equation with regular first order potential in [39].

By local uniqueness and causality, using in particular the fact that as a
consequence of the finite propagation speed, the solution $\Phi$ vanishes in
$\mathcal{I}^{+}(\mathcal{S})/\bar{\mathcal{B}}_{I}$, the Goursat problem of
the equation (36) has a unique smooth solution in the future of $\mathcal{S}$,
that is the restriction of $\Phi$ to $\mathfrak{M}$. Therefore, we have
established the well-posedness of the Goursat problem of (36), hence (14) in
the following theorem.

######  Theorem 4.

With the initial data
$(\xi,\,\zeta)\in{\mathcal{C}}_{0}^{\infty}(\mathfrak{H}^{+})\times{\mathcal{C}}_{0}^{\infty}({\mathscr{I}}^{+})$
where $\xi=(\xi_{1},\,\xi_{2})$ and $\zeta=(\zeta_{1},\,\zeta_{2})$, Equation
(36) has a unique solution $\hat{\Phi}=(\hat{\phi}_{1},\,\hat{\phi}_{2})$
satisfying

$\hat{\phi}_{i}\in{\mathcal{C}}(\mathbb{R};\,H^{1}(\mathcal{S}))\cap\mathcal{C}^{1}(\mathbb{R};\,L^{2}(\mathcal{S})),$

for all $i=1,\,2.$

In the term of tensor: with the initial data
$(\xi_{a},\zeta_{a})\in\mathcal{C}_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{\mathfrak{H}^{+}})\times\mathcal{C}_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{{\mathscr{I}}^{+}})$,
the rescaled tensorial equation (14) has a unique solution $\hat{\phi}_{a}$
such that

$\hat{\phi}_{a}\in{\mathcal{C}}(\mathbb{R};\,H^{1}(\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{S}}))\cap\mathcal{C}^{1}(\mathbb{R};\,L^{2}(\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{S}})).$

######  Remark 6.

We have solved partly the Goursat problem for the tensorial Equation (14) in
the future $\mathcal{I}^{+}(\mathcal{S})$, the problem will be solved
completely by extending this solution down to $\Sigma_{0}$. This will be done
in the next section.

###  4.2 Goursat problem and conformal scattering operator

In order to obain the conformal scattering operator, we need to show that the
trace operator is subjective. This corresponds to solve the Goursat problem
for the rescaled Equation (14) with the initial data on the conformal
boundaries $\mathfrak{H}^{+}\cup{\mathscr{I}}^{+}$ (resp.
$\mathfrak{H}^{-}\cup{\mathscr{I}}^{-}$) in the conformal compactification
domain $\bar{\mathcal{B}}_{I}$. In the previous section, we have solved the
Goursat problem in the future of the spacelike hypersurface $\mathcal{S}$. The
following theorem gives the full solution of the problem in
$\bar{\mathcal{B}}_{I}$.

######  Theorem 5.

The Goursat problem of the rescaled tensorial Equation (14) is well-posed.
This means that for the initial data $(\xi,\zeta)\in
C_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{\mathfrak{H}^{+}})\times
C_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{{\mathscr{I}}^{+}})$, there
exists a unique solution of (14) satisfying

$(\hat{\phi}_{a},\not{\nabla}_{t}\hat{\phi}_{a})\in\mathcal{C}(\mathbb{R}_{t};\,\mathcal{H})\hbox{
and
}\mathcal{T}^{+}(\hat{\phi}_{a}|_{\Lambda^{1}(\mathbb{S}^{2})|_{\Sigma_{0}}},\not{\nabla}_{t}\hat{\phi}_{a}|_{\Lambda^{1}(\mathbb{S}^{2})|_{\Sigma_{0}}})=(\xi_{a},\zeta_{a}).$

###### Proof.

Following Theorem 4 there exists a unique solution $\hat{\phi}_{a}$ of (14)
which satisfies the following properties.

  * $\bullet$

$\hat{\phi}_{a}\in
H^{1}(\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{I}^{+}(\mathcal{S}}))$, where
$\mathcal{I}^{+}(\mathcal{S})$ is the causal future of $\mathcal{S}$ in
$\bar{\mathcal{B}_{I}}$. Since the support of the initial data is compact,the
solution vanishes in the neigbourhood of $i^{+}$. Then we do not need to
distinguish between $H^{1}(\mathcal{I}^{+}(\mathcal{S}))$ and
$H^{1}_{loc}(\mathcal{I}^{+}(\mathcal{S}))$.

  * $\bullet$

$\hat{\phi}_{a}$ is continuous in $\tau$ with values in $H^{1}$ of the slices
and $C^{1}$ in $\tau$ with values in $L^{2}$ of the slices.

  * $\bullet$

$\hat{\phi}_{a}|_{{\mathscr{I}}^{+}}=\zeta_{a},\,\hat{\phi}_{a}|_{\mathfrak{H}^{+}}=\xi_{a}$.

Figure 3: Extending the solution $\hat{\phi}_{a}$ down to $\Sigma_{0}$.

We need to extend the solution down to $\Sigma_{0}$ in a manner that avoids
the singularity at $i_{0}$. Since the hypersurface $\mathcal{S}$ intersects
the horizon at the crossing sphere and intersects ${\mathscr{I}}^{+}$ strictly
in the past of the support of the data, we have that the restriction of
$\hat{\phi}_{a}$ to $\mathcal{S}$ is in
$H^{1}(\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{S}})$ and its trace on
$\mathcal{S}\cap{\mathscr{I}}^{+}$ is also the trace of $\zeta_{a}$ on
$\mathcal{S}\cap{\mathscr{I}}^{+}$. Therefore,
$\hat{\phi}_{a}|_{\Lambda^{1}(\mathcal{S}^{2})|_{\mathcal{S}}}$ can be
approached by a sequence
$\left\\{\hat{\phi}^{n}_{a0,\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{S}}}\right\\}_{n\in\mathbb{N}}$
of the smooth tensorial fields on $\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{S}}$
supported away from ${\mathscr{I}}^{+}$ that converges towards
$\hat{\phi}_{a}|_{\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{S}}}$ in
$H^{1}(\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{S}})$. Moreover,
$\not{\nabla}_{\partial_{t}}\hat{\phi}_{a}|_{\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{S}}}$
can be approached by a sequence
$\left\\{\hat{\phi}^{n}_{a1,\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{S}}}\right\\}_{n\in\mathbb{N}}$
of the smooth tensorial fields on $\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{S}}$
supported away from ${\mathscr{I}}^{+}$ that converges towards
$\not{\nabla}_{\partial_{t}}\hat{\phi}_{a}|_{\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{S}}}$
in $L^{2}(\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{S}})$. Consider
$\hat{\psi}_{a}^{n}$, the smooth tensorial solution of (14) on
$\Lambda^{1}(\mathbb{S}^{2})|_{\bar{\mathcal{B}_{I}}}$ with data
$(\hat{\phi}^{n}_{a0,\Lambda^{1}{\mathbb{S}^{2}}|_{\mathcal{S}}},\hat{\phi}^{n}_{a1,\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{S}}})$
on $\mathcal{S}$. This tensorial solution vanishes in the neighbourhood of
$i_{0}$ and we can establish energy estimates for $\hat{\psi}_{a}^{n}$ between
$\mathcal{S}$ and $\Sigma_{0}$ as follows

$\mathcal{E}^{\partial_{t}}_{\mathcal{S}}(\hat{\psi}_{a}^{n})=\mathcal{E}^{\partial_{t}}_{\Sigma_{0}}(\hat{\psi}_{a}^{n}).$
(38)

By the same way we have energy identities between $\mathcal{S}$ and the
hypersurfaces $\Sigma_{t}\hbox{ for }t>0$. Therefore, the sequence
$(\hat{\psi}_{a}^{n},\not{\nabla}_{\partial_{t}}\hat{\psi}_{a}^{n})$ converges
towards $(\hat{\psi}_{a},\not{\nabla}_{\partial_{t}}\hat{\psi}_{a})$ in
$\mathcal{C}(\mathbb{R}_{t},\mathcal{H})$, where $\hat{\psi}_{a}$ is a
solution of (14). By local uniqueness $\hat{\psi}_{a}$ coincides with
$\hat{\phi}_{a}$ in the future of $\mathcal{S}$. Therefore, if we denote

$\hat{\phi}_{a0}=\hat{\phi}_{a}|_{\Lambda^{1}(\mathbb{S}^{2})|_{\Sigma_{0}}},\,\,\hat{\phi}_{a1}=\not{\nabla}_{\partial_{t}}\hat{\phi}_{a}|_{\Lambda^{1}(\mathbb{S}^{2})|_{\Sigma_{0}}},$

we have

$(\hat{\phi}_{a0},\hat{\phi}_{a1})\in\mathcal{H}\hbox{ and
}(\xi_{a},\zeta_{a})=\mathcal{T}^{+}(\hat{\phi}_{a0},\hat{\phi}_{a1}).$

Therefore, the range of $\mathcal{T}^{+}$ contains
$\mathcal{C}_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{\mathfrak{H}^{+}})\times\mathcal{C}_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{{\mathscr{I}}^{+}})$.
∎

######  Remark 7.

Theorem 5 shows that the trace operator
$\mathcal{T}^{+}:\mathcal{H}\to\mathcal{H}^{+}$ is subjective. Combining with
Theorem 3 we obtain that the trace operator
$\mathcal{T}^{+}:\mathcal{H}\to\mathcal{H}^{+}$ is an isometry. Similarly, we
can establish the past trace operator $\mathcal{T}^{-}$ and the space
$\mathcal{H}^{-}$ of past scattering data on the past horizon and the past
null infinity.

We now define the conformal scattering operator for the tensorial Fackerell-
Ipser Equation (7) (which has the rescaled form (12)) as follows.

######  Definition 5.

The conformal scattering operator $S$ of Equation (7) is an isometry that maps
the past scattering data to the future scattering data, i.e.

$S:=\mathcal{T}^{+}\circ(\mathcal{T}^{-})^{-1}.$

######  Remark 8.

By the same manner we can construct the conformal scattering operator for
Equation (8).

This work can be extended to construct a conformal scattering theory for the
Fackerell-Ipser equations on the other symmetric spherical spacetimes (such as
Reissner-Nordström-de Sitter back hole) can be done by the same way: first, by
extending the work [46] to obtain the decay results (where the results of
Giorgi [21] can be useful) and then using these results to the construction of
the theory, where it remains useful to use the time-like Killing vector field
$\partial_{t}$ to obtain the energy of the fields.

Conformal scattering theories for the spin $\pm 1$ or $\pm 2$ Teukolsky
equations can be seem also to construct firstly in Schwarzschild spacetime.

The extension of the conformal scattering theory on the stationary and non
symmetric spherical spacetimes such as Kerr spacetimes is unknown so far and
remains an open problem. In Kerr spacetimes, the existence of the orbiting
null geodesics and the fact that the vector field $\partial_{t}$ is no longer
global time-like in the exterior of the black hole lead an issue that the
energies on the Cauchy hypersurfaces $\Sigma_{T}=\left\\{t=T\right\\}$ are not
defined as usual.

The peeling property of the rescaled Equations (12) is also an interesting
question where the method can be done by the same manner for the linear scalar
fields on Schwarzschild spacetimes in [35] and extended to Kerr spacetimes in
[41, 44].

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