Peeling for tensorial wave equations on Schwarzschild spacetime

PHAM Truong Xuan111Faculty of Pedagogy, VNU University of Education, Vietnam
National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam
Email<EMAIL_ADDRESS>
Abstract. In this paper, we establish the asymptotic behaviour along outgoing
and incoming radial geodesics, i.e., the peeling property for the tensorial
Fackrell-Ipser and spin $\pm 1$ Teukolsky equations on Schwarzschild
spacetime. Our method combines a conformal compactification with vector field
techniques to prove the two-side estimates of the energies of tensorial fields
through the future and past null infinity ${\mathscr{I}}^{\pm}$ and the
initial Cauchy hypersurface $\Sigma_{0}=\left\\{t=0\right\\}$ in a
neighbourhood of spacelike infinity $i_{0}$ far away from the horizon and
future timelike infinity. Our results obtain the optimal initial data which
guarantees the peeling at all orders.

Keywords. Peeling, vector field techniques, black holes, tensorial Fackerell-
Ipser equations, spin $\pm 1$ Teukolsky equations, Schwarzschild metric, null
infinity, Penrose’s 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’s conformal compactification
    2. 2.2 Neighbourhood of spacelike infinity
    3. 2.3 Maxwell, spin $\pm 1$ Teukolsky and tensorial Fackerell-Ipser equations
  3. 3 Basic formulae
    1. 3.1 Approximate conservation laws
    2. 3.2 Energy fluxes
  4. 4 Peeling
    1. 4.1 Peeling for tensorial Fackerell-Ipser equations
    2. 4.2 Peeling for Teukolsky equations

##  1 Introduction

The Peeling is a type of asymptotic behaviour of zero rest-mass fields
initially discovered by R. Sachs [44, 45]. Its initial formulation involved an
expansion of the field in powers of $1/r$ along a null geodesic going out to
infinity, and the alignment of a certain number of principal null directions
of each term in the expansion along the null geodesic considered. Penrose
introduced the conformal technique in the early 1960s [37, 38] and used it to
establish that the peeling property is equivalent to the mere continuity of
the rescaled field at null infinity [40].

The peeling for linearized gravity and for full gravity has been studied
intensively (see Friedrich [15], Christodoulou-Klainerman [5], Corvino [9],
Chrusciel and Delay [7, 8], Corvino-Schoen [10] and Klainerman-Nicolò [23, 24,
25]) and is now fairly well understood, at least in the flat case. However, it
is not yet clear, given an asymptotically flat spacetime, which class of
initial data yields solutions that admit the peeling property at a given
order, and whether these classes are smaller than in Minkowski spacetime or
not.

The works of Mason and Nicolas [32, 33] were precisely aimed at answering this
last question for the Schwarzschild metric, for scalar, Dirac and Maxwell
fields. Their method combines the Penrose compactification of the spacetime
and geometric energy estimates. By working in a neighbourhood of spacelike
infinity on the compactified spacetime, one obtains energy estimates at all
orders for the rescaled field, which control weighted Sobolev norms on
${\mathscr{I}}$ in terms of similar norms on a Cauchy hypersurface and vice
versa. The finiteness of the norms up to order $k$ (where $k\in\mathbb{N}$) at
${\mathscr{I}}$ defines the peeling of order $k$. By the completion of smooth
compactly supported data on the Cauchy hypersurface in the corresponding
norms, one obtains the optimal classes of data ensuring that the associated
solution peels at order $k$. The result does not strictly refer to the
regularity near spacelike infinity $i_{0}$. Indeed, if the regularity is
controlled in a neighbourhood of $i_{0}$ and on the full initial data
hypersurface, it can be extended to the whole of ${\mathscr{I}}$ by standard
results for hyperbolic equations. The works [32, 33] put further a program of
peeling on the asymptotic flat spacetimes. Continuing this program Nicolas and
Pham [35, 42] established explicitly the peeling for linear (or semilinear)
scalar wave and Dirac equations on Kerr spacetime which is not static and not
symmetric spherical such as Schwarzschild spacetime.

On the other hand, various dynamical constructions of spacetimes appear in
physical reality and violate Sachs peeling property of gravitational
radiation. These spacetimes do not possess a smooth null infinity
${\mathscr{I}}$. In particular, Christodoulou [6] assumed that the Bondi mass
along ${\mathscr{I}}^{+}$ decays with the rate predicted by the quadrupole
approximation for a system of $N$ infalling masses coming from past infinity,
combined with the assumption that there be no incoming radiation from past
null infinity ${\mathscr{I}}^{-}$, then the evolutions of Einstein-Vacuum
equations (see [5] for the detailed expression of these evolutions) does not
admit a smooth conformal compactification, i.e., Penrose’s proposal of smooth
conformal compactification of spacetime (or smooth null infinity) fails for
this case. Recently, motivated by [6], Kehrberger [20] has constructed
counter-examples to smooth null infinity by considering the solution of
spherically symmetric Einstein-Scalar field system with positive Hawking mass
and the polynomially decaying data on a timelike boundary (or on an ingoing
null hypersurface) and no incoming radiation from past null infinity
${\mathscr{I}}^{-}$ (see theorems 2.1, 2.2 and 2.4 in [20]). Moreover,
Kehrberger [21] has also shown the failure of Sachs peeling in a neighbourhood
of $i_{0}$ by proving the logarithmically modified Price’s law asymptotics
near ${\mathscr{I}}^{+}$ for the linear scalar wave equation $\Box_{g}\phi=0$
on Schwarzschild spacetime. In particular, he has considered conformally
smooth initial data on an ingoing null hypersurface and vanishing data on
${\mathscr{I}}^{-}$ for $\Box_{g}\phi=0$, then he has obtained precise
asymptotics of the solution $\phi$ with logarithmic terms (see theorems 6.1
and 7.1 in [21]). This also shows that the non-smoothness of null infinity
near $i_{0}$ propagates and translates into logarithmic tails near $i^{+}$
(see also [22] for the early-time asymptotics (logarithmically modified
Price’s law) for fixed-frequency solutions $\phi_{l}$ to the wave equation
$\Box_{g}\phi_{l}=0$ with no incoming radiation condition on
${\mathscr{I}}^{-}$ and polynomially decaying data $r\phi_{l}\simeq t^{-1}$ as
$t\to-\infty$).

Comeback to our work, we will explore the method in [32, 35, 42] to establish
the peeling for tensorial Fackerell-Ipser and spin $\pm 1$ Teukolsky equations
on Schwarzschild spacetime. First, we recall briefly some results about these
equations: the spin $\pm 1$ Teukolsky equations are satisfied by the extreme
components of Maxwell equation (see [2, 36]) and the spin $\pm 2$ Teukolsky
equations arise from the linear and nonlinear stability problems of black hole
spacetimes. We refer the reader to [1, 12, 17] for the linear stability and
[14, 18, 26, 27, 19] for the nonlinear stability. Moreover, the spin $\pm 1$
Teukolsky equations have been studied in some recent works by Giorgi [16], Ma
[29] and Pasqualotto [36]. On the other hand, the spin $\pm 2$ Teukolsky
equations have been studied by Dafermos et al. [13] and Ma [30]. In
particular, the authors in [13, 16, 29, 30, 36] have used $r^{p}$-method (see
[11]) to establish the boundedness of energy and study time decays of the
associated solutions of Teukolsky equations on Schwarzschild, Reissner-
Nordström and Kerr spacetimes. The Price’s law decays for Teukolsky equations
on Kerr spacetime has been studied by Ma and Zhang [31], where they have
established the global sharp decay for the spin $\pm s$ (where $s=0,1,2$)
components, which are solutions to Teukolsky equations, in the black hole
exterior and on the event horizon of a slowly rotating Kerr spacetime. On the
other hand, the scattering theories for spin $\pm 1$ and $\pm 2$ Teukolsky
equations on Schwarzschild spacetime have been studied by Pham [43] and
Masaood [28], respectively.

The tensorial Fackerell-Ipser equations are obtained by commuting the spin
$\pm 1$ Teukolsky equations with the projected covariant derivatives
$\not{\nabla}_{L}$ 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 (see Subsection
2.3). Both the tensorial Fackerell-Ipser and spin $\pm 1$ Teukolsky equations
are expressed under tensorial forms. We consider these tensorial equations on
a neighbourhood of spacelike infinity $\Omega_{u_{0}}^{+}$ which is foliated
by a family of spacelike hypersurfaces
$\left\\{\mathcal{H}_{s}\right\\}_{0\leq s\leq 1}$. Using the stress-energy
tensor for the tensorial linear Klein-Gordon equation we obtain the
approximate conservation laws for the Fackerell-Ipser and Teukolsky equations
and calculate energy fluxes of the associated solutions through
$\mathcal{H}_{s}$ (see Lemma 2). In order to define the peeling we establish
the two-side estimates of the energies through the future null infinity
${\mathscr{I}}^{+}$ and the initial Cauchy hypersurface $\Sigma_{0}$ inside
$\Omega_{u_{0}}^{+}$ (see theorems 1 and 6). These estimates are obtained at
all order $k$ (where $k\in\mathbb{N}$) of the projected covariant derivatives
(see theorems 5 and 10). As a consequence, we can define the peeling at all
order $k$ as well as the finiteness of energy norms of the solutions through
the spacelike infinity ${\mathscr{I}}^{+}$ inside $\Omega_{u_{0}}^{+}$.
Moreover, we can also give the optimal initial data endowed Sobolev norm on
$\Sigma_{0}$ which guarantees this definition.

The paper is organized as follows: Section 2 presents Schwarzschild spacetime
and its Penrose’s conformal compactification, the neighbourhood of the
spacelike infinity $\Omega_{u_{0}}^{+}$ and its foliation, the Maxwell,
tensorial Fackerell-Ipser and spin $\pm 1$ Teukolsky equations; Section 3
contains the approximate conservation laws and energy fluxes of the associated
solutions; Section 4 relates the peeling for equations.

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
$\Lambda^{1}(\mathcal{B})$.
$\bullet$ We denote the orthogonal projection of covariant derivative $\nabla$
on $\mathcal{B}$ by $\not{\nabla}$.
$\bullet$ The space of $1$-forms on the unit $2$-sphere is denoted by
$\Lambda^{1}(\mathbb{S}^{2})$. The basic frame of
$\Lambda^{1}(\mathbb{S}^{2})$ is denoted by
$(\not{\nabla}_{\partial_{\theta}},\not{\nabla}_{\partial_{\varphi}})$.
$\bullet$ We denote the space of smooth compactly supported scalar functions
on $\mathcal{M}$ (a smooth manifold without boundary) by
$\mathcal{C}_{0}^{\infty}(\mathcal{M})$. The space of smooth compactly
supported $1$-forms in $\Lambda^{1}(\mathbb{S}^{2})$ on $\mathcal{M}$ is
denoted by
$\mathcal{C}_{0}^{\infty}(\Lambda^{1}(\mathbb{S}^{2})|_{\mathcal{M}})$.
$\bullet$ Let $f(x)$ and $g(x)$ be two real functions. We write $f\lesssim g$
if there exists a constant $C\in(0,+\infty)$ which is independent of the
functions $f,\,g,$ and such that $f(x)\leq Cg(x)$ for all parameter $x$. We
write $f\simeq g$ if both $f\lesssim g$ and $g\lesssim f$ are valid.
Acknowledgements. The author would like to thank the referee for his or her
careful reading of the manuscript, give useful comments and related references
which help us to improve this paper. This work is supported by the Vietnam
Institute for Advanced Study in Mathematics (VIASM) 2023.

##  2 Geometrical and analytical setting

###  2.1 Schwarzschild metric and Penrose’s conformal compactification

We consider the region outside the Schwarzschild black-hole
$({\cal{M}}=\mathbb{R}_{t}\times]2M,+\infty[_{r}\times\mathbb{S}^{2},g)$,
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.

We recall that the Regge-Wheeler coordinate $r_{*}=r+2M\log(r-2M)$ satisfies
$\mathrm{d}r=F\mathrm{d}r_{*}$. In the coordinates $(t,r_{*},\theta,\varphi)$,
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

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

respectively.

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

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

The 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 $(v,\,R=1/r,\,\theta,\,\varphi)$, the
rescaled metric $\hat{g}$ takes the form

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

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

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

Penrose’s conformal compactification of $\mathcal{M}$ is

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

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

Note that the compactified spacetime $\bar{\mathcal{M}}$ is not compact. There
are three ”points” missing to the boundary: $i^{+}$, or future timelike
infinity, defined as the limit point of uniformly timelike curves as
$t\to+\infty$, $i^{-}$, past timelike infinity, symmetric of $i^{+}$ in the
distant past, and $i_{0}$, spacelike infinity, the limit point of uniformly
spacelike curves as $r\to+\infty$. These ”points”, that can be described as
$2$-spheres, are singularities of the rescaled metric $\hat{g}$.

In the retarded coordinates $(u,\,R,\,\theta,\,\varphi)$ we have the following
relation

$\partial_{R}=-\frac{r^{2}}{F}(\partial_{t}+\partial_{r_{*}})=-\frac{r^{2}}{F}L.$

In the advanced coordinates $(v,\,R,\,\theta,\,\varphi)$ we have the following
relation

$\partial_{R}=-\frac{r^{2}}{F}(\partial_{t}-\partial_{r_{*}})=-\frac{r^{2}}{F}\underline{L}.$

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

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

The volume form associated with the rescaled metric $\hat{g}$ are

$\mathrm{dVol}_{\hat{g}}=-\mathrm{d}u\wedge\mathrm{d}R\wedge\mathrm{d}^{2}\omega=-\mathrm{d}v\wedge\mathrm{d}R\wedge\mathrm{d}^{2}\omega,$

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

###  2.2 Neighbourhood of spacelike infinity

By symmetry, we will study the peeling properties of Fackerell-Ipser and spin
$\pm 1$ Teukolsky equations in domain $\left\\{t\geq 0\right\\}$ of
$\bar{\mathcal{M}}$, the peeling in domain $\left\\{t\leq 0\right\\}$ is done
similarly. Following [32, 33], we work on a future neighbourhood
$\Omega_{u_{0}}^{+}\,(\hbox{where }u_{0}\ll-1)$, of spacelike infinity $i_{0}$
that is sufficiently far away from the black hole and singularities. The
neighbourhood $\Omega_{u_{0}}^{+}$ is bounded by a part of the Cauchy
hypersurface $\Sigma_{0}=\left\\{t=0\right\\}$, a part of future null infinity
${\mathscr{I}}^{+}$ and the following null hypersurface

$\mathcal{S}_{u_{0}}=\left\\{u=u_{0},\,t\geq 0\right\\}\text{ for
}{u_{0}}\ll-1.$

The neighbourhood $\Omega_{u_{0}}^{+}$ can be given precisely as

$\Omega^{+}_{u_{0}}={\cal I}^{-}(\mathcal{S}_{u_{0}})\cap\\{t\geq 0\\}$

in the compactification domain $\bar{\mathcal{M}}$. We foliate
$\Omega_{u_{0}}^{+}$ (the gray domain in Figure 1) by the following spacelike
hypersurfaces

${\cal H}_{s}\,=\,\left\\{u=-sr_{*};\,u\leq u_{0}\right\\},\,\,\,\,0\leq s\leq
1,\text{ for a given }u_{0}\ll-1.$

We explain the boundaries of $\Omega_{u_{0}}^{+}$ as follows (see Figure 1):

  * •

If $s=1$, we have the first boundary ${\cal H}_{1}=\left\\{t=0,\,u\leq
u_{0}\right\\}$, which is a part of $\Sigma_{0}$ inside $\Omega_{u_{0}}^{+}$.

  * •

If $s=0$, we have that the second boundary ${\cal H}_{0}$ is the limit of
${\cal H}_{s}$, when $s$ tends to zero. If $u$ is fixed and $s$ tends to $0$,
then $r_{*}$ tends to $+\infty$ and $R$ tends to $0$. Therefore, we obtain
that ${\cal H}_{0}=\left\\{R=0,\,u\leq u_{0}\right\\}$, which is a part of
${\mathscr{I}}^{+}$ inside $\Omega_{u_{0}}^{+}$. We also denote ${\cal H}_{0}$
by ${\mathscr{I}}^{+}_{u_{0}}$.

  * •

The third boundary is null hypersurface
$\mathcal{S}_{u_{0}}=\left\\{u=u_{0},\,t\geq 0\right\\}$. Given $0\leq
s_{1}<s_{2}\leq 1$, we denote by $\mathcal{S}_{u_{0}}^{s_{1},s_{2}}$ the part
of $\mathcal{S}_{u_{0}}$ between $\mathcal{H}_{s_{1}}$ and
$\mathcal{H}_{s_{2}}$.

Figure 1: Foliation $\left\\{\mathcal{H}_{s}\right\\}_{0\leq s\leq 1}$ of
neighbourhood $\Omega_{u_{0}}^{+}$.

With the foliation $\left\\{{\cal H}_{s}\right\\}_{0\leq s\leq 1}$, we choose
an identifying vector field $\nu$ that satisfies $\nu(s)=1$ as follows

$\nu=r_{*}^{2}R^{2}(1-2MR)|u|^{-1}\partial_{R}.$

The $4-$volume measure $\mathrm{dVol}_{\hat{g}}$ can be decomposed into the
product of $\mathrm{d}s$ along the integral lines of $\nu^{a}$ and the
$3$-volume measure

$\nu{\begin{picture}(0.833,0.8)\put(0.15,0.08){\line(1,0){0.35}}
\put(0.5,0.08){\line(0,1){0.5}} \end{picture}}\mathrm{dVol}_{\hat{g}}|_{{\cal
H}_{s}}=-r_{*}^{2}R^{2}(1-2MR)|u|^{-1}\mathrm{d}u\mathrm{d}^{2}\omega|_{{\cal
H}_{s}}$

on each slice ${\cal H}_{s}$.

We need the following lemma to establish simpler equivalent expressions in the
next sections (see Lemma 2.1 in [33]):

######  Lemma 2.1.

Let $\varepsilon>0$, then for $u_{0}\ll-1$, $|u_{0}|$ large enough, in
$\Omega^{+}_{u_{0}}$, we have

$r<r_{*}<(1+\varepsilon)r,\,0<|u|R<1+\varepsilon,\,1<r_{*}R<1+\varepsilon\,\,\,\mbox{and}\,\,\,1-\varepsilon<1-2MR<1.$

The factor $r_{*}^{2}R^{2}(1-2MR)|u|^{-1}$ appearing in the $3$-volume measure
$\nu{\begin{picture}(0.833,0.8)\put(0.15,0.08){\line(1,0){0.35}}
\put(0.5,0.08){\line(0,1){0.5}} \end{picture}}\mathrm{dVol}|_{{\cal H}_{s}}$
satisfies that

$\frac{1+\varepsilon}{|u|}<r_{*}^{2}R^{2}(1-2MR)|u|^{-1}<\frac{(1+\varepsilon)^{2}}{|u|}.$

###  2.3 Maxwell, spin $\pm 1$ Teukolsky and tensorial Fackerell-Ipser
equations

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

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

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

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

The system can be reformulated as follows

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

where the square brackets denote the antisymmetrization of indices.

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

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

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

Let $\mathcal{F}$ be in $\Lambda^{2}(\mathcal{M})$ such that $\mathcal{F}$
satisfies the Maxwell equation on $\mathcal{M}$. Then, we have the following
formulas (see [36, Proposition 3.6]):

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

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 $\Lambda^{1}(\mathcal{B})$:

$\phi_{a}:=\frac{r^{2}}{F}\not{\nabla}_{\underline{L}}(r\alpha_{a}),\,\,\,\underline{\phi}_{a}:=\frac{r^{2}}{F}\not{\nabla}_{L}(r\underline{\alpha}_{a}).$
(1)

Moreover, the extreme components $\alpha_{a}$ and $\underline{\alpha}_{a}$
satisfy the spin $\pm 1$ Teukolsky equations respectively (see original proof
in [2] and recent [36, Proposition 3.6]):

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

where $F=1-2MR$, $\not{\nabla}$ and
$\not{\Delta}=\frac{1}{r^{2}}\Delta_{\mathbb{S}^{2}}$ 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 (see [36, Proposition 3.7]):

######  Proposition 1.

Suppose that $(\alpha_{a},\underline{\alpha}_{a},\rho,\sigma)$ satisfy the
Maxwell equation, then the $1$-forms $\phi_{a}$ and $\underline{\phi}_{a}$
satisfy the following tensorial Fackerell-Ipser equations

$\not{\Box}_{g}(\phi_{a})+\frac{1}{r^{2}}\phi_{a}=0,$ (4)
$\not{\Box}_{g}(\underline{\phi}_{a})+\frac{1}{r^{2}}\underline{\phi}_{a}=0,$
(5)

where we denote the tensorial wave operator by

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

######  Remark 1.

The tensorial Fackerell-Ipser operator has the same form as the rescaled
tensorial wave operator by multiplying the factor $r^{2}$ due to

$\displaystyle\not{\Box}_{\hat{g}}=\frac{r^{2}}{F}\not{\nabla}_{L}\not{\nabla}_{\underline{L}}-\not{\Delta}_{\mathbb{S}^{2}}=$
$\displaystyle-2\not{\nabla}_{v}\not{\nabla}_{R}-\not{\nabla}_{R}R^{2}(1-2MR)\not{\nabla}_{R}-\not{\Delta}_{\mathbb{S}^{2}}$
$\hbox{in }(v,\,R,\,\theta,\,\varphi),$ (6a)
$\displaystyle-2\not{\nabla}_{u}\not{\nabla}_{R}-\not{\nabla}_{R}R^{2}(1-2MR)\not{\nabla}_{R}-\not{\Delta}_{\mathbb{S}^{2}}$
$\displaystyle$\hbox{in }(u,\,R,\,\theta,\,\varphi).$$

In the advanced coordinates $(v,\,R,\,\theta,\,\varphi)$ the tensorial
Fackerell-Ipser and spin $+1$ Teukolsky equations (4) and (2) have the
following forms

$-2\not{\nabla}_{v}\not{\nabla}_{R}\phi_{a}-\not{\nabla}_{R}R^{2}(1-2MR)\not{\nabla}_{R}\phi_{a}-\not{\Delta}_{\mathbb{S}^{2}}\phi_{a}+\phi_{a}=0$
(7)

and

$-2\not{\nabla}_{v}\not{\nabla}_{R}\widehat{\alpha}_{a}-\not{\nabla}_{R}R^{2}(1-2MR)\not{\nabla}_{R}\widehat{\alpha}_{a}-\not{\Delta}_{\mathbb{S}^{2}}\widehat{\alpha}_{a}-2R(1-3MR)\not{\nabla}_{R}\widehat{\alpha}_{a}+\widehat{\alpha}_{a}=0,\hbox{
}\widehat{\alpha}_{a}=r\alpha_{a}$ (8)

respectively.

In the retarded coordinates $(u,\,R,\,\theta,\,\varphi)$ the tensorial
Fackerell-Ipser and spin $-1$ Teukolsky equations (5) and (3) have the
following forms

$-2\not{\nabla}_{u}\not{\nabla}_{R}\underline{\phi}_{a}-\not{\nabla}_{R}R^{2}(1-2MR)\not{\nabla}_{R}\underline{\phi}_{a}-\not{\Delta}_{\mathbb{S}^{2}}\underline{\phi}_{a}+\underline{\phi}_{a}=0$
(9)

and

$-2\not{\nabla}_{u}\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}-\not{\nabla}_{R}R^{2}(1-2MR)\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}-\not{\Delta}_{\mathbb{S}^{2}}\widehat{\underline{\alpha}}_{a}+2R(1-3MR)\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}+\widehat{\underline{\alpha}}_{a}=0,\hbox{
}\widehat{\underline{\alpha}}_{a}=r\underline{\alpha}_{a}$ (10)

respectively.

In the rest of this paper, we will establish the peeling for the tensorial
Fackerell-Ipser and spin $-1$ Teukolsky equations (9) and (10) in the
neighbourhood $\Omega_{u_{0}}^{+}$, i.e., the asymptotic behaviours of the
associated solutions along outgoing radial geodesics in coordinates
$(u,\,R,\,\theta,\,\varphi)$. The constructions for the equations (7) and (8)
are done similarly in coordinates $(v,\,R,\,\theta,\,\varphi)$.

##  3 Basic formulae

###  3.1 Approximate conservation laws

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

$\xi^{a}=\xi_{b}g_{\mathbb{S}^{2}}^{ab},\hbox{
}\not{\nabla}_{\underline{L}}\xi^{a}=\not{\nabla}_{\underline{L}}\xi_{b}g_{\mathbb{S}^{2}}^{ab}\hbox{
and }\not{\nabla}_{L}\xi^{a}=\not{\nabla}_{L}\xi_{b}g_{\mathbb{S}^{2}}^{ab}.$

Putting

$|\xi_{a}|^{2}:=\xi_{a}\xi^{a},\,|\not{\nabla}_{u}\xi_{a}|^{2}:=\not{\nabla}_{u}{\xi}_{a}\not{\nabla}_{u}{\xi}^{a}\hbox{
and
}|\not{\nabla}_{R}\xi_{a}|^{2}:=\not{\nabla}_{R}\xi_{a}\not{\nabla}_{R}\xi^{a},$

where the scalar product depends on the metric $g_{\mathbb{S}^{2}}$.

We define the stress-energy tensor for the tensorial linear Klein-Gordon
equation $\not{\Box}_{\hat{g}}\xi_{a}+\xi_{a}=0$ as follows

$\mathbb{T}_{cd}(\xi_{a})=\mathbb{T}_{(cd)}(\xi_{a})=\not{\nabla}_{c}\xi_{a}\not{\nabla}_{d}\xi^{a}-\frac{1}{2}\left<\not{\nabla}\xi_{a},\not{\nabla}\xi^{a}\right>_{\hat{g}}\hat{g}_{cd}+\frac{1}{2}|\xi_{a}|^{2}\hat{g}_{cd}.$
(11)

In order to obtain the approximate conservation laws for (9) and (10) we use
the Morawetz vector field

$T^{c}\partial_{c}=u^{2}\partial_{u}-2(1+uR)\partial_{R}.$

By using Lie derivative of the rescaled Schwarzschild metric $\hat{g}$ follows
the Morawetz vector field $T$, we can derive that (see [32, 35]):

$\not{\nabla}_{c}T_{d}=\not{\nabla}_{(c}T_{d)}=4MR^{2}(3+uR)\mathrm{d}u^{2}.$
(12)

For a solution $\underline{\phi}_{a}$ of the tensorial Fackerell-Ipser
equation (9), we have

$T^{d}\not{\nabla}^{c}\mathbb{T}_{cd}(\underline{\phi}_{a})=\left(\not{\Box}_{\hat{g}}\underline{\phi}_{a}+\underline{\phi}_{a}\right)\not{\nabla}_{T}\underline{\phi}^{a}=0.$
(13)

For a solution $\widehat{\underline{\alpha}}_{a}$ of the Teukolsky equation
(10), we have

$T^{d}\not{\nabla}^{c}\mathbb{T}_{cd}(\widehat{\underline{\alpha}}_{a})=\left(\not{\Box}_{\hat{g}}\widehat{\underline{\alpha}}_{a}+\widehat{\underline{\alpha}}_{a}\right)\not{\nabla}_{T}\widehat{\underline{\alpha}}^{a}=-2R(1-3MR)\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}\not{\nabla}_{T}\widehat{\underline{\alpha}}^{a}.$
(14)

Setting

$\displaystyle
J_{c}(\underline{\phi}_{a}):=T^{d}\mathbb{T}_{cd}(\underline{\phi}_{a})\hbox{
and
}J_{c}(\widehat{\underline{\alpha}}_{a}):=T^{d}\mathbb{T}_{cd}(\widehat{\underline{\alpha}}_{a}).$
(15)

From (13) and (14) the nonlinear energy currents $J_{c}(\underline{\phi}_{a})$
and $J_{c}(\widehat{\underline{\alpha}}_{a})$ satisfy the following
approximate conservation laws

$\not{\nabla}^{c}J_{c}(\underline{\phi}_{a})=(\not{\nabla}_{c}T_{d})\mathbb{T}^{cd}(\underline{\phi}_{a})=4MR^{2}(3+uR)|\not{\nabla}_{R}\underline{\phi}_{a}|^{2}.$
(16)

and

$\displaystyle\not{\nabla}^{c}J_{c}(\widehat{\underline{\alpha}}_{a})$
$\displaystyle=$
$\displaystyle-2R(1-3MR)\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}\not{\nabla}_{T}\widehat{\underline{\alpha}}^{a}+(\not{\nabla}_{c}T_{d})\mathbb{T}^{cd}(\widehat{\underline{\alpha}}_{a})$
(17) $\displaystyle=$
$\displaystyle-2R(1-3MR)\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}\left(u^{2}\not{\nabla}_{u}\widehat{\underline{\alpha}}^{a}-2(1+uR)\not{\nabla}_{R}\widehat{\underline{\alpha}}^{a}\right)$
(19)
$\displaystyle+4MR^{2}(3+uR)|\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}|^{2}$

respectively.

Because of the symmetries of Schwarzschild spacetime, we have for any
$k\in\mathbb{N}$:

$\not{\Box}_{\hat{g}}\not{\nabla}^{k}_{u}\underline{\phi}_{a}+\not{\nabla}^{k}_{u}\underline{\phi}_{a}=0,\,\not{\Box}_{\hat{g}}\not{\nabla}^{k}_{u}\widehat{\underline{\alpha}}_{a}+2R(1-3MR)\not{\nabla}_{R}\not{\nabla}_{u}^{k}\widehat{\underline{\alpha}}_{a}+\not{\nabla}^{k}_{u}\widehat{\underline{\alpha}}_{a}=0$

and

$\not{\Box}_{\hat{g}}\not{\nabla}^{k}_{\mathbb{S}^{2}}\underline{\phi}_{a}+\not{\nabla}^{k}_{\mathbb{S}^{2}}\underline{\phi}_{a}=0,\,\not{\Box}_{\hat{g}}\not{\nabla}^{k}_{\mathbb{S}^{2}}\widehat{\underline{\alpha}}_{a}+2R(1-3MR)\not{\nabla}_{R}\not{\nabla}_{\mathbb{S}^{2}}^{k}\widehat{\underline{\alpha}}_{a}+\not{\nabla}^{k}_{\mathbb{S}^{2}}\widehat{\underline{\alpha}}_{a}=0$

Therefore, the approximate conservation laws (16) and (17) are valid at high
order for $\not{\nabla}_{u}^{k}$ and $\not{\nabla}_{\mathbb{S}^{2}}^{k}$.

We commute the operator $\not{\nabla}_{R}$ into the equations (9) and (10) to
obtain

$\not{\Box}_{\hat{g}}\not{\nabla}_{R}\underline{\phi}_{a}+\not{\nabla}_{R}\underline{\phi}_{a}=2(1-3M)R\not{\nabla}_{R}^{2}\underline{\phi}_{a}-2(1-6MR)\not{\nabla}_{R}\underline{\phi}_{a}.$

and

$\not{\Box}_{\hat{g}}\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}+\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}=6MR(R-1)\not{\nabla}^{2}_{R}\widehat{\underline{\alpha}}_{a}-4(1-6MR)\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}.$

Therefore, we obtain the approximate conservation laws for
$\not{\nabla}_{R}\underline{\phi}_{a}$ and
$\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}$ as follows

$\displaystyle\not{\nabla}^{c}J_{c}(\not{\nabla}_{R}\underline{\phi}_{a})$
$\displaystyle=$
$\displaystyle\left(2(1-3M)R\not{\nabla}_{R}^{2}\underline{\phi}_{a}-2(1-6MR)\not{\nabla}_{R}\underline{\phi}_{a}\right)\not{\nabla}_{T}\not{\nabla}_{R}\underline{\phi}^{a}+(\not{\nabla}_{c}T_{d})\mathbb{T}^{cd}(\not{\nabla}_{R}\underline{\phi}_{a})$
(20) $\displaystyle=$
$\displaystyle\left(2(1-3M)R\not{\nabla}_{R}^{2}\underline{\phi}_{a}-2(1-6MR)\not{\nabla}_{R}\underline{\phi}_{a}\right)\left(u^{2}\not{\nabla}_{u}\not{\nabla}_{R}\underline{\phi}^{a}-2(1+uR)\not{\nabla}_{R}^{2}\underline{\phi}^{a}\right)$
(22)
$\displaystyle+4MR^{2}(3+uR)|\not{\nabla}^{2}_{R}\underline{\phi}_{a}|^{2}.$

and

$\displaystyle\not{\nabla}^{c}J_{c}(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a})$
$\displaystyle=$
$\displaystyle\left(6MR(R-1)\not{\nabla}_{R}^{2}\widehat{\underline{\alpha}}_{a}-4(1-6MR)\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}\right)\not{\nabla}_{T}\not{\nabla}_{R}\widehat{\underline{\alpha}}^{a}+(\not{\nabla}_{c}T_{d})\mathbb{T}^{cd}(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a})$
(23) $\displaystyle=$
$\displaystyle\left(6MR(R-1)\not{\nabla}_{R}^{2}\widehat{\underline{\alpha}}_{a}-4(1-6MR)\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}\right)\left(u^{2}\not{\nabla}_{u}\not{\nabla}_{R}\widehat{\underline{\alpha}}^{a}-2(1+uR)\not{\nabla}_{R}^{2}\widehat{\underline{\alpha}}^{a}\right)$
(25)
$\displaystyle+4MR^{2}(3+uR)|\not{\nabla}^{2}_{R}\widehat{\underline{\alpha}}_{a}|^{2}.$

By the same way we can obtain the approximate conservation laws for
$\not{\nabla}_{R}^{k}\underline{\phi}_{a}$ and
$\not{\nabla}_{R}^{k}\widehat{\underline{\alpha}}_{a}$ for all
$k\in\mathbb{N}$.

###  3.2 Energy fluxes

Moreover, we follow the convention used by Penrose and Rindler [41] about the
Hodge dual of a $1$-form $\alpha$ on a spacetime $({\cal M},g)$ (i.e. a
$4-$dimensional Lorentzian manifold that is oriented and time-oriented):

$(*\alpha)_{abcd}=e_{abcd}{\alpha}^{d},$

where $e_{abcd}$ is the volume form on $({\cal M},g)$, denoted simply by
$\mathrm{dVol}_{g}$. We shall use the following differential operator of the
Hodge star

$\mathrm{d}*\alpha=-\frac{1}{4}(\nabla_{a}\alpha^{a})\mathrm{dVol}^{4}_{g}.$

If $\mathcal{S}$ is the boundary of a bounded open set $\Omega$ and has
outgoing orientation, using Stokes theorem, we have

$-4\int_{\mathcal{S}}*\alpha=\int_{\Omega}(\nabla_{a}\alpha^{a})\mathrm{dVol}^{4}_{g}.$
(26)

Let $\underline{\phi}_{a}$ and $\widehat{\underline{\alpha}}_{a}$ be solutions
of (9) and (10) with smooth and compactly supported initial data on the
rescaled spacetime $(\bar{\mathcal{M}},\hat{g})$. By using (26) we define the
rescaled energy fluxes associated with the Morawertz vector field $T$, through
an oriented hypersurface $\mathcal{S}$ as follows

$\mathcal{E}_{\mathcal{S}}(\underline{\phi}_{a})=-4\int_{\mathcal{S}}*J_{c}(\underline{\phi}_{a})\mathrm{d}x^{c}=\int_{\mathcal{S}}J_{c}(\underline{\phi}_{a})\hat{N}^{c}\hat{L}{\begin{picture}(0.833,0.8)\put(0.15,0.08){\line(1,0){0.35}}
\put(0.5,0.08){\line(0,1){0.5}} \end{picture}}\mathrm{dVol}_{\hat{g}}$ (27)

and

$\mathcal{E}_{\mathcal{S}}(\widehat{\underline{\alpha}}_{a})=-4\int_{\mathcal{S}}*J_{c}(\widehat{\underline{\alpha}}_{a})\mathrm{d}x^{c}=\int_{\mathcal{S}}J_{c}(\widehat{\underline{\alpha}}_{a})\hat{N}^{c}\hat{L}{\begin{picture}(0.833,0.8)\put(0.15,0.08){\line(1,0){0.35}}
\put(0.5,0.08){\line(0,1){0.5}} \end{picture}}\mathrm{dVol}_{\hat{g}}$ (28)

where $\hat{L}$ is a transverse vector to $\mathcal{S}$ and $\hat{N}$ is the
normal vector field to $\mathcal{S}$ such that $\hat{L}^{a}\hat{N}_{a}=1$.

We recall the following Poincaré-type inequality (see [32, Lemma 4.2]):

######  Lemma 1.

Given $u_{0}<0$, there exists a constant $C>0$ such that for any $f\in
C_{0}^{\infty}(\mathbb{R})$ such that

$\int_{-\infty}^{u_{0}}(f(u))^{2}\mathrm{d}u\leq
C\int_{-\infty}^{u_{0}}|u|^{2}(f^{\prime}(u))^{2}\mathrm{d}u.$

Using Lemma 1, we can give the simpler equivalent expressions of energy fluxes
for equations (9) and (10) across the leaves of the foliation
$\mathcal{H}_{s}$ of $\Omega_{u_{0}}^{+}$:

######  Lemma 2.

For $|u_{0}|$ large enough, the energy fluxes of $\underline{\phi}_{a}$ and
$\widehat{\underline{\alpha}}_{a}$ through the hypersurface
$\mathcal{H}_{s},\,0<s\leq 1$ and $H_{0}={\mathscr{I}}_{u_{0}}^{+}$ have the
following simpler equivalent expressions

$\mathcal{E}_{\mathcal{H}_{s}}(\underline{\phi}_{a})\simeq\int_{\mathcal{H}_{s}}\left(|u|^{2}|\not{\nabla}_{u}\underline{\phi}_{a}|^{2}+\frac{R}{|u|}|\not{\nabla}_{R}\underline{\phi}_{a}|^{2}+|\not{\nabla}_{\mathbb{S}^{2}}\underline{\phi}_{a}|^{2}+|\underline{\phi}_{a}|^{2}\right)\mathrm{d}u\mathrm{d}^{2}\omega,$
$\mathcal{E}_{\mathcal{H}_{s}}(\widehat{\underline{\alpha}}_{a})\simeq\int_{\mathcal{H}_{s}}\left(|u|^{2}|\not{\nabla}_{u}\widehat{\underline{\alpha}}_{a}|^{2}+\frac{R}{|u|}|\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}|^{2}+|\not{\nabla}_{\mathbb{S}^{2}}\widehat{\underline{\alpha}}_{a}|^{2}+|\widehat{\underline{\alpha}}_{a}|^{2}\right)\mathrm{d}u\mathrm{d}^{2}\omega$

and

$\mathcal{E}_{{\mathscr{I}}_{u_{0}}^{+}}(\underline{\phi}_{a})\simeq\int_{\mathcal{H}_{s}}\left(|u|^{2}|\not{\nabla}_{u}\underline{\phi}_{a}|^{2}+|\not{\nabla}_{\mathbb{S}^{2}}\underline{\phi}_{a}|^{2}+|\underline{\phi}_{a}|^{2}\right)\mathrm{d}u\mathrm{d}^{2}\omega,$
$\mathcal{E}_{{\mathscr{I}}_{u_{0}}^{+}}(\widehat{\underline{\alpha}}_{a})\simeq\int_{\mathcal{H}_{s}}\left(|u|^{2}|\not{\nabla}_{u}\widehat{\underline{\alpha}}_{a}|^{2}+|\not{\nabla}_{\mathbb{S}^{2}}\widehat{\underline{\alpha}}_{a}|^{2}+|\widehat{\underline{\alpha}}_{a}|^{2}\right)\mathrm{d}u\mathrm{d}^{2}\omega.$

Here, we denote that

$|\not{\nabla}_{\mathbb{S}^{2}}\underline{\phi}_{a}|^{2}=|\not{\nabla}_{\partial_{\theta}}\underline{\phi}_{a}|^{2}+\frac{1}{\sin^{2}\theta}|\not{\nabla}_{\partial_{\varphi}}\underline{\phi}_{a}|^{2},\,\,|\not{\nabla}_{\mathbb{S}^{2}}\widehat{\underline{\alpha}}_{a}|^{2}=|\not{\nabla}_{\partial_{\theta}}\widehat{\underline{\alpha}}_{a}|^{2}+\frac{1}{\sin^{2}\theta}|\not{\nabla}_{\partial_{\varphi}}\widehat{\underline{\alpha}}_{a}|^{2}.$

###### Proof.

The proof is similar to the case of scalar wave equation using formulas (27),
(28) and Lemma 1 (see Lemma 4.1 in [32] or Lemma 4.1 in [35]). ∎

##  4 Peeling

###  4.1 Peeling for tensorial Fackerell-Ipser equations

Integrating the approximate conservation law (16) on the domain

$\Omega_{u_{0}}^{s_{1},s_{2}}:=\Omega_{u_{0}}^{+}\cap\left\\{s_{1}\leq s\leq
s_{2}\right\\}\hbox{ with }0\leq s_{1}<s_{2}\leq 1,$ (29)

we get

$\displaystyle\left|\mathcal{E}_{\mathcal{H}_{s_{1}}}(\underline{\phi}_{a})+\mathcal{E}_{\mathcal{S}^{s_{1},s_{2}}_{u_{0}}}(\underline{\phi}_{a})-\mathcal{E}_{\mathcal{H}_{s_{2}}}(\underline{\phi}_{a})\right|$
(30) $\displaystyle\simeq$
$\displaystyle\left|\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}4MR^{2}(3+uR)|\not{\nabla}_{R}\underline{\phi}_{a}|^{2}\frac{1}{|u|}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s\right|$
(31) $\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}4MR^{2}(3+uR)|\not{\nabla}_{R}\underline{\phi}_{a}|^{2}\frac{1}{|u|}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
(32) $\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}\frac{R^{2}}{|u|}|\not{\nabla}_{R}\underline{\phi}_{a}|^{2}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
(33) $\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}\frac{1}{\sqrt{s}}\frac{R}{|u|}|\not{\nabla}_{R}\underline{\phi}_{a}|^{2}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
(34) $\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\frac{1}{\sqrt{s}}\mathcal{E}_{\mathcal{H}_{s}}(\underline{\phi}_{a})\mathrm{d}s,$
(35)

Here, we used the fact that on $\mathcal{H}_{s}$, we have $u=-sr_{*}$, then
$\dfrac{R^{2}}{|u|}\simeq\dfrac{s}{|u|}\dfrac{R}{|u|}\leqslant\dfrac{1}{\sqrt{s}}\dfrac{R}{|u|}$
for $u\leq u_{0}\ll-1$ and $R$ large enough. The inequality (30) leads to

$\displaystyle\mathcal{E}_{\mathcal{H}_{s_{1}}}(\underline{\phi}_{a})\lesssim\mathcal{E}_{\mathcal{H}_{1}}(\underline{\phi}_{a})+\int_{s_{1}}^{1}\frac{1}{\sqrt{s}}\mathcal{E}_{\mathcal{H}_{s}}(\underline{\phi}_{a})\mathrm{d}s,$
(36)
$\displaystyle\mathcal{E}_{\mathcal{H}_{s_{2}}}(\underline{\phi}_{a})\lesssim\mathcal{E}_{\mathcal{S}^{+}_{u_{0}}}(\underline{\phi}_{a})+\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\underline{\phi}_{a})+\int_{0}^{s_{2}}\frac{1}{\sqrt{s}}\mathcal{E}_{\mathcal{H}_{s}}(\underline{\phi}_{a})\mathrm{d}s$
(37)

for all $0\leq s_{1}<s_{2}\leq 1$. Note that, the factor $\dfrac{1}{\sqrt{s}}$
is not really necessary here because we can estimate
$\dfrac{R^{2}}{|u|}\leq\dfrac{R}{|u|}$ in inequality (30), and then
inequalities $\eqref{0}$ and $\eqref{1}$ have not the factor
$\dfrac{1}{\sqrt{s}}$. However, we still present $\dfrac{1}{\sqrt{s}}$ to
serve the estimates at higher order below (see inequality (54)).

Since the function $\dfrac{1}{\sqrt{s}}$ is integrable on $[0,\,1]$, we use
Gronwall’s inequality for inequalities (36) and (37) with the scalar function
$y(s)=\mathcal{E}_{\mathcal{H}_{s}}(\underline{\phi}_{a})$ and get the
following result for energy estimates at zero order.

######  Theorem 1.

For $u_{0}<0$ and $|u_{0}|$ large enough and for any smooth compactly
supported initial data at $\Sigma_{0}=\left\\{t=0\right\\}$, the associated
solutions $\underline{\phi}_{a}$ of (9) satisfying that

$\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\underline{\phi}_{a})\lesssim\mathcal{E}_{\mathcal{H}_{1}}(\underline{\phi}_{a}),$
$\mathcal{E}_{\mathcal{H}_{1}}(\underline{\phi}_{a})\lesssim\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\underline{\phi}_{a})+\mathcal{E}_{{\mathcal{S}^{+}_{u_{0}}}}(\underline{\phi}_{a}).$

Since the approximate conservation law (16) is valid for
$\not{\nabla}_{u}^{k}\underline{\phi}_{a}$ and
$\not{\nabla}_{\mathbb{S}^{2}}^{k}\underline{\phi}_{a}$ with $k\in\mathbb{N}$,
we have the following estimates at higher order for all projected covariant
derivatives.

######  Theorem 2.

For $u_{0}<0$ and $|u_{0}|$ large enough and for any smooth compactly
supported initial data at $\Sigma_{0}=\left\\{t=0\right\\}$, the associated
solutions $\underline{\phi}_{a}$ of (9) satisfying that

$\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}_{u}^{k}\not{\nabla}_{\mathbb{S}^{2}}^{l}\underline{\phi}_{a})\lesssim\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{u}^{k}\not{\nabla}_{\mathbb{S}^{2}}^{l}\underline{\phi}_{a}),$
$\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{u}^{k}\not{\nabla}_{\mathbb{S}^{2}}^{l}\underline{\phi}_{a})\lesssim\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}_{u}^{k}\not{\nabla}_{\mathbb{S}^{2}}^{l}\underline{\phi}_{a})+\mathcal{E}_{\mathcal{H}_{\mathcal{S}^{+}_{u_{0}}}}(\not{\nabla}_{u}^{k}\not{\nabla}_{\mathbb{S}^{2}}^{l}\underline{\phi}_{a}),$

where $k,l\in\mathbb{N}$.

Since on the hypersurface $\mathcal{H}_{s}$, we have
$s=-\dfrac{u}{r_{*}}\simeq-uR$, then we obtain the equivalence

$\frac{1}{|u|}\simeq\frac{1}{\sqrt{s}}\sqrt{\frac{R}{|u|}}.$ (38)

Now integrating the approximate conservation law (20) and using (38) we obtain

$\displaystyle\left|\mathcal{E}_{\mathcal{H}_{s_{1}}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{\mathcal{S}^{s_{1},s_{2}}_{u_{0}}}(\not{\nabla}_{R}\underline{\phi}_{a})-\mathcal{E}_{\mathcal{H}_{s_{2}}}(\not{\nabla}_{R}\underline{\phi}_{a})\right|$
(39) $\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}\left|\left(2(1-3M)R\not{\nabla}_{R}^{2}\underline{\phi}_{a}-2(1-6MR)\not{\nabla}_{R}\underline{\phi}_{a}\right)\right.$
(42) $\displaystyle\hskip
113.81102pt\times\left.\left(u^{2}\not{\nabla}_{u}\not{\nabla}_{R}\underline{\phi}^{a}-2(1+uR)\not{\nabla}_{R}^{2}\underline{\phi}^{a}\right)\right|\frac{1}{|u|}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle+\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}4MR^{2}(3+uR)|\not{\nabla}^{2}_{R}\underline{\phi}_{a}|^{2}\frac{1}{|u|}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}\left(2|1-3M|s\not{\nabla}_{R}^{2}\underline{\phi}_{a}\right)\left(|u|\not{\nabla}_{u}\not{\nabla}_{R}\underline{\phi}^{a}\right)\frac{1}{\sqrt{s}}\sqrt{\frac{R}{|u|}}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
(47)
$\displaystyle+\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}4|(1-3M)(1+uR)||\not{\nabla}_{R}^{2}\underline{\phi}_{a}|^{2}\frac{1}{\sqrt{s}}\frac{R}{|u|}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle+\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}2|1-6MR|\left(\not{\nabla}_{R}\underline{\phi}_{a}\right)\left(|u|\not{\nabla}_{u}\not{\nabla}_{R}\underline{\phi}^{a}\right)\frac{1}{\sqrt{s}}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle+\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}4|(1-6MR)(1+uR)|\left(\not{\nabla}_{R}\underline{\phi}_{a}\right)\left(\not{\nabla}^{2}_{R}\underline{\phi}^{a}\right)\frac{1}{\sqrt{s}}\sqrt{\frac{R}{|u|}}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle+\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}\frac{1}{\sqrt{s}}\frac{R}{|u|}|\not{\nabla}^{2}_{R}\underline{\phi}_{a}|^{2}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\frac{1}{\sqrt{s}}\int_{\mathcal{H}_{s}}\left(\frac{R}{|u|}|\not{\nabla}_{R}^{2}\underline{\phi}_{a}|^{2}+|u|^{2}|\not{\nabla}_{u}\not{\nabla}_{R}\underline{\phi}_{a}|^{2}\right)\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
(52)
$\displaystyle+\int_{s_{1}}^{s_{2}}\frac{1}{\sqrt{s}}\int_{\mathcal{H}_{s}}\frac{R}{|u|}|\not{\nabla}_{R}^{2}\underline{\phi}_{a}|^{2}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle+\int_{s_{1}}^{s_{2}}\frac{1}{\sqrt{s}}\int_{\mathcal{H}_{s}}\left(|\not{\nabla}_{R}\underline{\phi}_{a}|^{2}+|u|^{2}\not{\nabla}_{u}\not{\nabla}_{R}\underline{\phi}_{a}|^{2}\right)\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle+\int_{s_{1}}^{s_{2}}\frac{1}{\sqrt{s}}\int_{\mathcal{H}_{s}}\left(|\not{\nabla}_{R}\underline{\phi}_{a}|^{2}+\frac{R}{|u|}|\not{\nabla}^{2}_{R}\underline{\phi}_{a}|^{2}\right)\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle+\int_{s_{1}}^{s_{2}}\frac{1}{\sqrt{s}}\int_{\mathcal{H}_{s}}\frac{R}{|u|}|\not{\nabla}^{2}_{R}\underline{\phi}_{a}|^{2}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\frac{1}{\sqrt{s}}\left(\mathcal{E}_{\mathcal{H}_{s}}(\underline{\phi}_{a})+\mathcal{E}_{\mathcal{H}_{s}}(\not{\nabla}_{R}\underline{\phi}_{a})\right)\mathrm{d}s.$
(53)

Note that we can remove the factor $\dfrac{1}{\sqrt{s}}$ by changing variable
$\tau=-2(\sqrt{s}-1)$, hence
$\dfrac{\mathrm{d}\tau}{\mathrm{d}s}=-\dfrac{1}{\sqrt{s}}$ and $\tau$ varies
from $2$ to $0$, when $s$ varies from $0$ to $1$. Therefore, we have

$\displaystyle\left|\mathcal{E}_{\mathscr{H}_{\tau(s_{1})}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{\mathcal{S}^{s_{1},s_{2}}_{u_{0}}}(\not{\nabla}_{R}\underline{\phi}_{a})-\mathcal{E}_{\mathscr{H}_{\tau(s_{2})}}(\not{\nabla}_{R}\underline{\phi}_{a})\right|$
$\displaystyle\lesssim$
$\displaystyle\int_{\tau(s_{1})}^{\tau(s_{2})}\left(\mathcal{E}_{\mathscr{H}_{\tau(s)}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{\mathscr{H}_{\tau(s)}}(\underline{\phi}_{a})\right)\mathrm{d}s,$

where $\mathscr{H}_{\tau(s)}=\mathcal{H}_{s}$ (see also for scalar wave
equations [32, Appendix A4 and A6, pages 24-25]).

Combining inequalities (39) and (30), we can establish that

$\displaystyle\left|\left(\mathcal{E}_{\mathcal{H}_{s_{1}}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{\mathcal{H}_{s_{1}}}(\underline{\phi}_{a})\right)+\left(\mathcal{E}_{\mathcal{S}^{s_{1},s_{2}}_{u_{0}}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{\mathcal{S}^{s_{1},s_{2}}_{u_{0}}}(\underline{\phi}_{a})\right)-\left(\mathcal{E}_{\mathcal{H}_{s_{2}}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{\mathcal{H}_{s_{2}}}(\underline{\phi}_{a})\right)\right|$
(54) $\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\frac{1}{\sqrt{s}}\left(\mathcal{E}_{\mathcal{H}_{s}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{\mathcal{H}_{s}}(\underline{\phi}_{a})\right)\mathrm{d}s.$
(55)

This inequality leads to

$\displaystyle\mathcal{E}_{\mathcal{H}_{s_{1}}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{\mathcal{H}_{s_{1}}}(\underline{\phi}_{a})\lesssim\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{\mathcal{H}_{1}}(\underline{\phi}_{a})$
(56) $\displaystyle\hskip
142.26378pt+\int_{s_{1}}^{1}\frac{1}{\sqrt{s}}\left(\mathcal{E}_{\mathcal{H}_{s}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{\mathcal{H}_{s}}(\underline{\phi}_{a})\right)\mathrm{d}s,$
(57)
$\displaystyle\mathcal{E}_{\mathcal{H}_{s_{2}}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{\mathcal{H}_{s_{2}}}(\underline{\phi}_{a})\lesssim\mathcal{E}_{\mathcal{S}^{+}_{u_{0}}}(\underline{\phi}_{a})+\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\underline{\phi}_{a})$
(58) $\displaystyle\hskip
142.26378pt+\int_{0}^{s_{2}}\frac{1}{\sqrt{s}}\left(\mathcal{E}_{\mathcal{H}_{s}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{\mathcal{H}_{s}}(\underline{\phi}_{a})\right)\mathrm{d}s$
(59)

for all $0\leq s_{1}<s_{2}\leq 1$. Since the function $\dfrac{1}{\sqrt{s}}$ is
integrable on $[0,\,1]$, we use Gronwall’s inequality for inequalities (57)
and (59) with the scalar function
$z(s)=\mathcal{E}_{\mathcal{H}_{s}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{\mathcal{H}_{s}}(\underline{\phi}_{a})$,
and get the following result of energy estimates at first order.

######  Theorem 3.

For $u_{0}<0$ and $|u_{0}|$ large enough and for any smooth compactly
supported initial data at $\Sigma_{0}=\left\\{t=0\right\\}$, the associated
solutions $\underline{\phi}_{a}$ of (9) satisfying that

$\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\underline{\phi}_{a})\lesssim\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{\mathcal{H}_{1}}(\underline{\phi}_{a}),$
$\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{\mathcal{H}_{1}}(\underline{\phi}_{a})\lesssim\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\underline{\phi}_{a})++\mathcal{E}_{{\mathcal{S}^{+}_{u_{0}}}}(\not{\nabla}_{R}\underline{\phi}_{a})+\mathcal{E}_{{\mathcal{S}^{+}_{u_{0}}}}(\underline{\phi}_{a}).$

By the same way we have the higher order estimates for
$\not{\nabla}_{R}\underline{\phi}_{a}$ in the following theorem.

######  Theorem 4.

For $u_{0}<0$ and $|u_{0}|$ large enough and for any smooth compactly
supported initial data at $\Sigma_{0}=\left\\{t=0\right\\}$, the associated
solutions $\underline{\phi}_{a}$ of (9) satisfying that

$\sum_{p=0}^{k}\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}_{R}^{p}\underline{\phi}_{a})\lesssim\sum_{p=0}^{k}\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{R}^{p}\underline{\phi}_{a}),$
$\sum_{p=0}^{k}\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{R}^{p}\underline{\phi}_{a})\lesssim\sum_{p=0}^{k}\left(\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}^{p}_{R}\underline{\phi}_{a})+\mathcal{E}_{{\mathcal{S}^{+}_{u_{0}}}}(\not{\nabla}^{p}_{R}\underline{\phi}_{a})\right)$

for all $k\in\mathbb{N}$.

Combining the two theorems 2 and 4 we obtain the two-side estimates of the
energies for all covariant derivatives of tensorial fields.

######  Theorem 5.

For $u_{0}<0$ and $|u_{0}|$ large enough and for any smooth compactly
supported initial data at $\Sigma_{0}=\left\\{t=0\right\\}$, the associated
solutions $\underline{\phi}_{a}$ of (9) satisfying that

$\sum_{m+n+p\leq
k}\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}_{u}^{m}\not{\nabla}_{R}^{n}\not{\nabla}_{\mathbb{S}^{2}}^{p}\underline{\phi}_{a})\lesssim\sum_{m+n+p\leq
k}\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{u}^{m}\not{\nabla}_{R}^{n}\not{\nabla}_{\mathbb{S}^{2}}^{p}\underline{\phi}_{a}),$
$\sum_{m+n+p\leq
k}\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{u}^{m}\not{\nabla}_{R}^{n}\not{\nabla}_{\mathbb{S}^{2}}^{p}\underline{\phi}_{a})\lesssim\sum_{m+n+p\leq
k}\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}_{u}^{m}\not{\nabla}_{R}^{n}\not{\nabla}_{\mathbb{S}^{2}}^{p}\underline{\phi}_{a})+\mathcal{E}_{{\mathcal{S}^{+}_{u_{0}}}}(\not{\nabla}_{u}^{m}\not{\nabla}_{R}^{n}\not{\nabla}_{\mathbb{S}^{2}}^{p}\underline{\phi}_{a}),$

where $k,m,n,p\in\mathbb{N}$.

Now we give the definition of the peeling at order $k\in\mathbb{N}$:

######  Definition 1.

A solution $\underline{\phi}_{a}$ of the tensorial Fackerell equation (9) in
$\Omega_{u_{0}}^{+}$ peels at order $k\in\mathbb{N}$ if $\underline{\phi}_{a}$
satisfies

$\sum_{m+n+p\leq
k}\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}_{u}^{m}\not{\nabla}_{R}^{n}\not{\nabla}_{\mathbb{S}^{2}}^{p}\underline{\phi}_{a})<+\infty.$

Theorem 5 gives us a characterization of the class of initial data on
$\mathcal{H}_{1}$ that guarantees that the corresponding solution peels at a
given order $k\in\mathbb{N}$; it is the completion of smooth compactly
supported data on $\mathcal{H}_{1}$ in the norm

$\left(\sum_{m+n+p\leq
k}\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{u}^{m}\not{\nabla}_{R}^{n}\not{\nabla}_{\mathbb{S}^{2}}^{p}\underline{\phi}_{a})\right)^{1/2}.$

###  4.2 Peeling for Teukolsky equations

Integrating the approximate conservation law (17) on the domain
$\Omega_{u_{0}}^{s_{1},s_{2}}$ given by (29) we obtain

$\displaystyle\left|\mathcal{E}_{\mathcal{H}_{s_{1}}}(\widehat{\underline{\alpha}}_{a})+\mathcal{E}_{\mathcal{S}^{s_{1},s_{2}}_{u_{0}}}(\widehat{\underline{\alpha}}_{a})-\mathcal{E}_{\mathcal{H}_{s_{2}}}(\widehat{\underline{\alpha}}_{a})\right|$
(60) $\displaystyle\simeq$
$\displaystyle\left|\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}-2R(1-3MR)\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}\left(u^{2}\not{\nabla}_{u}\widehat{\underline{\alpha}}^{a}-2(1+uR)\not{\nabla}_{R}\widehat{\underline{\alpha}}^{a}\right)\frac{1}{|u|}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s\right|$
(62)
$\displaystyle+\left|\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}4MR^{2}(3+uR)|\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}|^{2}\frac{1}{|u|}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s\right|$
$\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}2R|1-3MR|\left(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}\right)\left(|u|\not{\nabla}_{u}\widehat{\underline{\alpha}}^{a}\right)\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
(65)
$\displaystyle+\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}4|(1-3MR)(1+uR)||\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}|^{2}\frac{R}{|u|}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle+\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}4MR^{2}(3+uR)|\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}|^{2}\frac{1}{|u|}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\frac{1}{\sqrt{s}}\int_{\mathcal{H}_{s}}\left(R^{2}|\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}|^{2}+|u|^{2}|\not{\nabla}_{u}\widehat{\underline{\alpha}}^{a}|^{2}\right)\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
(68)
$\displaystyle+\int_{s_{1}}^{s_{2}}\frac{1}{\sqrt{s}}\int_{\mathcal{H}_{s}}\frac{R}{|u|}|\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}|^{2}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle+\int_{s_{1}}^{s_{2}}\frac{1}{\sqrt{s}}\int_{\mathcal{H}_{s}}\frac{R}{|u|}|\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}|^{2}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\frac{1}{\sqrt{s}}\int_{\mathcal{H}_{s}}\left(\frac{sR}{|u|}|\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}|^{2}+|u|^{2}|\not{\nabla}_{u}\widehat{\underline{\alpha}}^{a}|^{2}\right)\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
(70)
$\displaystyle+\int_{s_{1}}^{s_{2}}\frac{1}{\sqrt{s}}\mathcal{E}_{\mathcal{H}_{s}}(\widehat{\underline{\alpha}}_{a})\mathrm{d}s$
$\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\frac{1}{\sqrt{s}}\mathcal{E}_{\mathcal{H}_{s}}(\widehat{\underline{\alpha}}_{a})\mathrm{d}s.$
(71)

Similar to the energy estimate for the Fackerell-Ipser equation at zero order,
the factor $\dfrac{1}{\sqrt{s}}$ is presented to serve the higher order
estimates.

Since the function $\dfrac{1}{\sqrt{s}}$ is integrable on $[0,\,1]$,
Gronwall’s inequality entails the following result for energy estimates at
zero order (by the same way in Subsection 4.1).

######  Theorem 6.

For $u_{0}<0$ and $|u_{0}|$ large enough and for any smooth compactly
supported initial data at $\Sigma_{0}=\left\\{t=0\right\\}$, the associated
solutions $\underline{\phi}_{a}$ of (9) satisfying that

$\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\widehat{\underline{\alpha}}_{a})\lesssim\mathcal{E}_{\mathcal{H}_{1}}(\widehat{\underline{\alpha}}_{a}),$
$\mathcal{E}_{\mathcal{H}_{1}}(\widehat{\underline{\alpha}}_{a})\lesssim\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\widehat{\underline{\alpha}}_{a})+\mathcal{E}_{\mathcal{H}_{\mathcal{S}^{+}_{u_{0}}}}(\widehat{\underline{\alpha}}_{a}).$

Since the approximate conservation law (17) is valid for
$\not{\nabla}_{u}^{k}\widehat{\underline{\alpha}}_{a}$ and
$\not{\nabla}_{\mathbb{S}^{2}}^{k}\widehat{\underline{\alpha}}_{a}$ with
$k\in\mathbb{N}$, we have the following theorem.

######  Theorem 7.

For $u_{0}<0$ and $|u_{0}|$ large enough and for any smooth compactly
supported initial data at $\Sigma_{0}=\left\\{t=0\right\\}$, the associated
solutions $\underline{\phi}_{a}$ of (9) satisfying that

$\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}_{u}^{k}\not{\nabla}_{\mathbb{S}^{2}}^{l}\widehat{\underline{\alpha}}_{a})\lesssim\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{u}^{k}\not{\nabla}_{\mathbb{S}^{2}}^{l}\widehat{\underline{\alpha}}_{a}),$
$\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{u}^{k}\not{\nabla}_{\mathbb{S}^{2}}^{l}\widehat{\underline{\alpha}}_{a})\lesssim\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}_{u}^{k}\not{\nabla}_{\mathbb{S}^{2}}^{l}\widehat{\underline{\alpha}}_{a})+\mathcal{E}_{\mathcal{H}_{\mathcal{S}^{+}_{u_{0}}}}(\not{\nabla}_{u}^{k}\not{\nabla}_{\mathbb{S}^{2}}^{l}\widehat{\underline{\alpha}}_{a}),$

where $k,l\in\mathbb{N}$.

Now integrating the approximate conservation law (20) and using (38) we obtain

$\displaystyle\left|\mathcal{E}_{\mathcal{H}_{s_{1}}}(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a})+\mathcal{E}_{\mathcal{S}^{s_{1},s_{2}}_{u_{0}}}(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a})-\mathcal{E}_{\mathcal{H}_{s_{2}}}(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a})\right|$
(72) $\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}\left|\left(6MR(R-1)\not{\nabla}_{R}^{2}\widehat{\underline{\alpha}}_{a}-4(1-6MR)\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}\right)\right.$
(75) $\displaystyle\hskip
113.81102pt\times\left.\left(u^{2}\not{\nabla}_{u}\not{\nabla}_{R}\widehat{\underline{\alpha}}^{a}-2(1+uR)\not{\nabla}_{R}^{2}\widehat{\underline{\alpha}}^{a}\right)\right|\frac{1}{|u|}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle+\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}4MR^{2}(3+uR)|\not{\nabla}^{2}_{R}\widehat{\underline{\alpha}}_{a}|^{2}\frac{1}{|u|}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}\left(6M|R-1|s\not{\nabla}_{R}^{2}\widehat{\underline{\alpha}}_{a}\right)\left(|u|\not{\nabla}_{u}\not{\nabla}_{R}\widehat{\underline{\alpha}}^{a}\right)\frac{1}{\sqrt{s}}\sqrt{\frac{R}{|u|}}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
(80)
$\displaystyle+\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}12|M(R-1)(1+uR)||\not{\nabla}_{R}^{2}\widehat{\underline{\alpha}}_{a}|^{2}\frac{1}{\sqrt{s}}\frac{R}{|u|}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle+\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}4|1-6MR|\left(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}\right)\left(|u|\not{\nabla}_{u}\not{\nabla}_{R}\widehat{\underline{\alpha}}^{a}\right)\frac{1}{\sqrt{s}}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle+\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}4|(1-6MR)(1+uR)|\left(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}\right)\left(\not{\nabla}^{2}_{R}\widehat{\underline{\alpha}}^{a}\right)\frac{1}{\sqrt{s}}\sqrt{\frac{R}{|u|}}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle+\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}\frac{1}{\sqrt{s}}\frac{R}{|u|}|\not{\nabla}^{2}_{R}\widehat{\underline{\alpha}}_{a}|^{2}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}\frac{1}{\sqrt{s}}\left(\frac{R}{|u|}|\not{\nabla}_{R}^{2}\widehat{\underline{\alpha}}_{a}|^{2}+|u|^{2}|\not{\nabla}_{u}\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}|^{2}\right)\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
(85)
$\displaystyle+\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}\frac{1}{\sqrt{s}}\frac{R}{|u|}|\not{\nabla}_{R}^{2}\widehat{\underline{\alpha}}_{a}|^{2}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle+\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}\frac{1}{\sqrt{s}}\left(|\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}|^{2}+|u|^{2}\not{\nabla}_{u}\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}|^{2}\right)\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle+\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}\frac{1}{\sqrt{s}}\left(|\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}|^{2}+\frac{R}{|u|}|\not{\nabla}^{2}_{R}\widehat{\underline{\alpha}}_{a}|^{2}\right)\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle+\int_{s_{1}}^{s_{2}}\int_{\mathcal{H}_{s}}\frac{1}{\sqrt{s}}\frac{R}{|u|}|\not{\nabla}^{2}_{R}\widehat{\underline{\alpha}}_{a}|^{2}\mathrm{d}u\mathrm{d}^{2}\omega\mathrm{d}s$
$\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\frac{1}{\sqrt{s}}\left(\mathcal{E}_{\mathcal{H}_{s}}(\widehat{\underline{\alpha}}_{a})+\mathcal{E}_{\mathcal{H}_{s}}(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a})\right)\mathrm{d}s.$
(86)

Combining inequalities (60) and (72), we can establish that

$\displaystyle\left|\left(\mathcal{E}_{\mathcal{H}_{s_{1}}}(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a})+\mathcal{E}_{\mathcal{H}_{s_{1}}}(\widehat{\underline{\alpha}}_{a})\right)+\left(\mathcal{E}_{\mathcal{S}^{s_{1},s_{2}}_{u_{0}}}(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a})+\mathcal{E}_{\mathcal{S}^{s_{1},s_{2}}_{u_{0}}}(\widehat{\underline{\alpha}}_{a})\right)-\left(\mathcal{E}_{\mathcal{H}_{s_{2}}}(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a})+\mathcal{E}_{\mathcal{H}_{s_{2}}}(\widehat{\underline{\alpha}}_{a})\right)\right|$
(87) $\displaystyle\lesssim$
$\displaystyle\int_{s_{1}}^{s_{2}}\frac{1}{\sqrt{s}}\left(\mathcal{E}_{\mathcal{H}_{s}}(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a})+\mathcal{E}_{\mathcal{H}_{s}}(\widehat{\underline{\alpha}}_{a})\right)\mathrm{d}s.$
(88)

Since the function $\dfrac{1}{\sqrt{s}}$ is integrable on $[0,\,1]$,
Gronwall’s inequality entails the following result (by the same way in
Subsection 4.1).

######  Theorem 8.

For $u_{0}<0$ and $|u_{0}|$ large enough and for any smooth compactly
supported initial data at $\Sigma_{0}=\left\\{t=0\right\\}$, the associated
solutions $\underline{\phi}_{a}$ of (10) satisfying that

$\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a})+\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\widehat{\underline{\alpha}}_{a})\lesssim\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a})+\mathcal{E}_{\mathcal{H}_{1}}(\widehat{\underline{\alpha}}_{a}),$
$\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a})+\mathcal{E}_{\mathcal{H}_{1}}(\widehat{\underline{\alpha}}_{a})\lesssim\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a})+\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\widehat{\underline{\alpha}}_{a})+\mathcal{E}_{{\mathcal{S}^{+}_{u_{0}}}}(\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a})+\mathcal{E}_{{\mathcal{S}^{+}_{u_{0}}}}(\widehat{\underline{\alpha}}_{a}).$

By the same way, we have the higher order estimates for
$\not{\nabla}_{R}\widehat{\underline{\alpha}}_{a}$ in the following theorem.

######  Theorem 9.

For $u_{0}<0$ and $|u_{0}|$ large enough and for any smooth compactly
supported initial data at $\Sigma_{0}=\left\\{t=0\right\\}$, the associated
solutions $\underline{\phi}_{a}$ of (10) satisfying that

$\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}_{R}^{k}\widehat{\underline{\alpha}}_{a})\lesssim\sum_{p=0}^{k}\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{R}^{p}\widehat{\underline{\alpha}}_{a}),$
$\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{R}^{k}\widehat{\underline{\alpha}}_{a})\lesssim\sum_{p=0}^{k}\left(\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}^{p}_{R}\widehat{\underline{\alpha}}_{a})+\mathcal{E}_{{\mathcal{S}^{+}_{u_{0}}}}(\not{\nabla}^{p}_{R}\widehat{\underline{\alpha}}_{a})\right)$

for all $k,m,n,p\in\mathbb{N}$.

Combining the two theorems 7 and 9 we obtain the two-side estimates of the
energies for all covariant derivatives of tensorial fields.

######  Theorem 10.

For $u_{0}<0$ and $|u_{0}|$ large enough and for any smooth compactly
supported initial data at $\Sigma_{0}=\left\\{t=0\right\\}$, the associated
solutions $\underline{\phi}_{a}$ of (10) satisfying that

$\sum_{m+n+p\leq
k}\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}_{u}^{m}\not{\nabla}_{R}^{n}\not{\nabla}_{\mathbb{S}^{2}}^{p}\widehat{\underline{\alpha}}_{a})\lesssim\sum_{m+n+p\leq
k}\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{u}^{m}\not{\nabla}_{R}^{n}\not{\nabla}_{\mathbb{S}^{2}}^{p}\widehat{\underline{\alpha}}_{a}),$
$\sum_{m+n+p\leq
k}\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{u}^{m}\not{\nabla}_{R}^{n}\not{\nabla}_{\mathbb{S}^{2}}^{p}\widehat{\underline{\alpha}}_{a})\lesssim\sum_{m+n+p\leq
k}\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}_{u}^{m}\not{\nabla}_{R}^{n}\not{\nabla}_{\mathbb{S}^{2}}^{p}\widehat{\underline{\alpha}}_{a})+\mathcal{E}_{{\mathcal{S}^{+}_{u_{0}}}}(\not{\nabla}_{u}^{m}\not{\nabla}_{R}^{n}\not{\nabla}_{\mathbb{S}^{2}}^{p}\widehat{\underline{\alpha}}_{a}),$

where $k,m,n,p\in\mathbb{N}$.

Now we give the definition of the peeling at order $k\in\mathbb{N}$:

######  Definition 2.

A solution $\underline{\phi}_{a}$ of the tensorial Fackerell equation (10) in
$\Omega_{u_{0}}^{+}$ peels at order $k\in\mathbb{N}$ if
$\widehat{\underline{\alpha}}_{a}$ satisfies

$\sum_{m+n+p\leq
k}\mathcal{E}_{{\mathscr{I}}^{+}_{u_{0}}}(\not{\nabla}_{u}^{m}\not{\nabla}_{R}^{n}\not{\nabla}_{\mathbb{S}^{2}}^{p}\widehat{\underline{\alpha}}_{a})<+\infty.$

Theorem 10 gives us a characterization of the class of initial data on
$\mathcal{H}_{1}$ that guarantees that the corresponding solution peels at a
given order $k\in\mathbb{N}$; it is the completion of smooth compactly
supported data on $\mathcal{H}_{1}$ in the norm

$\left(\sum_{m+n+p\leq
k}\mathcal{E}_{\mathcal{H}_{1}}(\not{\nabla}_{u}^{m}\not{\nabla}_{R}^{n}\not{\nabla}_{\mathbb{S}^{2}}^{p}\widehat{\underline{\alpha}}_{a})\right)^{1/2}.$

######  Remark 2.

By the same way we can establish the peeling for the tensorial Fackerell-Ipser
and spin $+1$ equations (7) and (8) respectively. This means that we establish
the asymptotic behaviours of associated solutions of (7) and (8) along the
incoming radial geodesics in coordinates $(v,\,R,\,\theta,\,\varphi)$.

It seems that the results in this paper can be extended to the tensorial
Regge-Wheeler and spin $\pm 2$ Teukolsky equations on spherically symmetric
black hole spacetimes such as Schwarzschild and Reissner-Nordström de Sitter
spacetimes, where some recent works [12, 17, 28] can be useful.

The extension of peeling for tensorial Fackerell-Ipser, tensorial Regge-
Wheeler, spin $\pm 1$ and spin $\pm 2$ Teukolsky equations on Kerr spacetime
is an interesting question, where our work [35] can be useful. We hope to
treat this problem in a forthcoming paper.

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