# Late-time tails and mode coupling of linear waves on Kerr spacetimes

Yannis Angelopoulos<EMAIL_ADDRESS>The Division of Physics, Mathematics
and Astronomy, Caltech, 1200 E California Blvd, Pasadena CA 91125, USA
Stefanos Aretakis<EMAIL_ADDRESS>Department of Mathematics,
University of Toronto, 40 St George Street, Toronto, ON, Canada  Dejan Gajic
<EMAIL_ADDRESS><EMAIL_ADDRESS>Centre for Mathematical Sciences,
University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK  Department
of Mathematics, Radboud University, 6525 AJ Nijmegen, The Netherlands

(February 23, 2021)

###### Abstract

We provide a rigorous derivation of the precise late-time asymptotics for
solutions to the scalar wave equation on subextremal Kerr backgrounds,
including the asymptotics for projections to angular frequencies $\ell\geq 1$
and $\ell\geq 2$. The $\ell$-dependent asymptotics on Kerr spacetimes differ
significantly from the non-rotating Schwarzschild setting (“Price’s law”). The
main differences with Schwarzschild are slower decay rates for higher angular
frequencies and oscillations along the null generators of the event horizon.
We introduce a physical space-based method that resolves the following two
main difficulties for establishing $\ell$-dependent asymptotics in the Kerr
setting: 1) the coupling of angular modes and 2) a loss of ellipticity in the
ergoregion. Our mechanism identifies and exploits the existence of conserved
charges along null infinity via a time invertibility theory, which in turn
relies on new elliptic estimates in the full black hole exterior. This
framework is suitable for resolving the conflicting numerology in Kerr late-
time asymptotics that appears in the numerics literature.

###### Contents

  1. 1 Introduction
    1. 1.1 Main results and first remarks
    2. 1.2 Previous work
    3. 1.3 Analogues of Price’s law in Kerr
    4. 1.4 Main ideas and techniques
    5. 1.5 Outline of the paper
    6. 1.6 Acknowledgments
  2. 2 Preliminaries: geometry
    1. 2.1 The Kerr metric
    2. 2.2 Vector fields
    3. 2.3 Foliations and conformal coordinates
    4. 2.4 Additional notation
    5. 2.5 Systematic integration by parts
    6. 2.6 Spherical harmonic modes
    7. 2.7 Hardy inequalities along hyperboloidal hypersurfaces
  3. 3 Preliminaries: wave equation
    1. 3.1 Main equations
    2. 3.2 Propagation of regularity and limits at null infinity
    3. 3.3 Energy boundedness and integrated local energy decay
  4. 4 Newman–Penrose charges
  5. 5 Hierarchies of $r^{p}$-weighted energy estimates
    1. 5.1 Hierarchies of $r^{p}$-weighted energy estimates for $\phi^{(n)}$
    2. 5.2 Hierarchies of $r^{p}$-weighted energy estimates for $P_{0}$, $P_{1}$ and $P_{2}$
    3. 5.3 Additional hierarchies for higher-order derivatives
  6. 6 Energy decay estimates
    1. 6.1 Additional energy decay estimates
  7. 7 Hierarchies of $r^{-2k}$-weighted elliptic estimates
    1. 7.1 Spacelike redshift estimates
    2. 7.2 Elliptic estimates outside ergoregion
    3. 7.3 Application of the hierarchy of elliptic estimates
  8. 8 Pointwise decay estimates
  9. 9 Elliptic theory of time inversion
    1. 9.1 Inverting $\mathcal{L}$
    2. 9.2 Decay towards $\mathcal{I}^{+}$
    3. 9.3 Construction of time integral data
    4. 9.4 Time-inverted Newman–Penrose charges
  10. 10 Late-time polynomial tails: the $\ell=0$ projection
    1. 10.1 Asymptotics in $\mathcal{A}_{\gamma^{\alpha}}$
    2. 10.2 Asymptotics in $\mathcal{R}\setminus\mathcal{A}_{\gamma^{\alpha}}$
    3. 10.3 Asymptotics with vanishing Newman–Penrose charges
  11. 11 Late-time polynomial tails: the $\ell=1$ projection
    1. 11.1 Asymptotics in $\mathcal{A}_{\gamma^{\alpha}}$
    2. 11.2 Asymptotics in $\mathcal{R}\setminus\mathcal{A}_{\gamma^{\alpha}}$
    3. 11.3 Asymptotics with vanishing Newman–Penrose charges
  12. 12 Late-time polynomial tails: the $\ell=2$ projection
    1. 12.1 Asymptotics in $\mathcal{A}_{\gamma^{\alpha}}$
    2. 12.2 Asymptotics in $\mathcal{R}\setminus\mathcal{A}_{\gamma^{\alpha}}$
    3. 12.3 Asymptotics with vanishing Newman–Penrose charges
  13. A Weighted pointwise estimates
  14. B A basic interpolation inequality

##  1 Introduction

The Kerr spacetimes $(\mathcal{M}_{M,a},g_{M,a})$ constitute a 2-parameter
family of solutions to the Einstein vacuum equations

$\textnormal{Ric}[g]=0$

that are expected to describe all possible final states of a wide variety of
gravitational collapse scenarios in an astrophysical setting [Pen82]. A
preliminary step to describing the intricate dynamical properties of the
gravitational radiation that is emitted when black hole exteriors settle down
to Kerr solutions is to _verify_ that they indeed settle down to Kerr
solutions. In the context of the evolution of dynamical black holes arising
from “small perturbations” of Kerr initial data to the Einstein equations,
this is the black hole stability problem. There has been significant recent
progress towards addressing the question of linear and nonlinear stability;
see for example [DHR19b, KS20, Joh19, ABBM19, HHV21, SRdC20] and references
therein.

In this paper, we develop mathematical tools necessary for going _beyond_ the
question of stability, by addressing the _precise_ late-time behaviour of
gravitational radiation emitted in the evolution of perturbations of Kerr
initial data in the context of the model problem of the linear scalar wave
equation on a fixed subextremal $(|a|<M)$ Kerr spacetime background
$(\mathcal{M}_{M,a},g_{M,a})$:

$\square_{g_{M,a}}\psi=0.$ (1.1)

The evolution of the wave $\psi$ models the evolution of key dynamical
quantities for the Einstein equations. Addressing the precise late-time
dynamics of gravitational radiation is motivated by the exterior as well as
the interior of black holes:

  1. A)

The decay rates and leading-order coefficients of gravitational radiation
measured by observers “at infinity” are expected to encode information about
the initial perturbation and the final Kerr solution. Deducing information
purely from gravitational radiation is important since it can be detected and
analysed experimentally at gravitational wave observatories and forms a key
signature of astrophysical black hole processes, see also [AAG18a].

  2. B)

The precise late-time behaviour of gravitational radiation crossing the event
horizon, the boundary between black hole interior and exterior, plays a direct
role in uncovering the nature and strength of singularities that may be
present in black hole interiors. As such, it is important for addressing the
_strong cosmic censorship conjecture_ ; see the introduction of [DL17] for a
comprehensive overview of the black hole interior and the strong cosmic
censorship conjecture.

In this paper, we determine the precise leading-order late-time behaviour of
$\psi$ and its higher angular frequencies $\psi_{\ell\geq 1}$ and
$\psi_{\ell\geq 2}$ on Kerr black hole exteriors. We develop a physical space
mechanism that exploits the existence of conserved charges together with a
time inversion theory, in order to derive the presence of “tails” in the late-
time dynamics. Our mechanism deals with new difficulties in deriving late-time
asymptotics that are caused by the rotation of the black hole background: a
coupling of different angular frequencies and a loss of ellipticity of
operators relevant for the time inversion theory. We derive new phenomena that
arise due to rotation: slower decay rates at the level of higher angular
frequencies and oscillations along the event horizon.

We provide below an outline of the remainder of the introductory section of
the paper.

  * •

In Section 1.1, we present the main theorems of the paper.

  * •

In Section 1.2, we discuss previous work on late-time asymptotics for waves on
black holes.

  * •

In Section 1.3, we sketch an analogue of an angular frequency-dependent
Price’s law in the Kerr setting.

  * •

In Section 1.4, we outline the main new ideas and methods that appear in the
proofs of the theorems.

###  1.1 Main results and first remarks

In this section, we present the main results obtained in this paper and
provide some additional remarks.

We first introduce briefly the notation appearing in the statements of the
theorems below.

  * •

Let $\Sigma_{\tau}$ denote appropriate spacelike, asymptotically hyperboloidal
hypersurfaces in the Kerr manifold $\mathcal{M}_{M,a}$ that intersect the
future event horizon $\mathcal{H}^{+}$. The spacetime region of interest is
then the union $\bigcup_{\tau\in[0,\infty)}\Sigma_{\tau}$ which constitutes
the main region of interest in $\mathcal{M}_{M,a}$. See Figure 1 for a
pictorial representation.

  * •

The hypersurfaces $\Sigma_{\tau}$ are foliated by Boyer–Lindquist spheres
$S^{2}_{\tau,r}$, which we equip with angular coordinates
$(\theta,\varphi_{*})$. The label $r$ denotes the radial Boyer–Lindquist
coordinate, which takes the values $r\in[r_{+},\infty)$ along $\Sigma_{\tau}$,
with $r=r_{+}$ at $\mathcal{H}^{+}$.

Note that the components of the rescaled induced metric
$r^{-2}g_{S^{2}_{\tau,r}}$ approach the components of the metric on the unit
round sphere in standard spherical coordinates $(\theta,\varphi_{*})$ as
$r\to\infty$. In this sense, the Boyer–Lindquist spheres are asymptotically
round.

  * •

The notation $\psi_{\ell}$ and $\psi_{\geq\ell}$ indicates a projection of the
function $\psi$ to standard spherical harmonic modes on the spheres
$S^{2}_{\tau,r}$ equipped with the unit round metric, with angular frequencies
equal to $\ell$, and greater or equal to $\ell$, respectively. The spherical
harmonics here are defined with respect to the angular coordinates
$(\theta,\varphi_{*})$ and are denoted by $Y_{\ell,m}(\theta,\varphi_{*})$. We
denote with $\pi_{\ell}$ the operator that projects a function on
$S^{2}_{\tau,r}$ to the spherical harmonics with angular frequency $\ell$.

See Section 2 for a precise introduction of the spacetime geometry, the
Boyer–Lindquist coordinates and other notational conventions.

####  1.1.1 Main results

We state here the main theorems proved in the paper.

Figure 1: A 2-dimensional representation of the spacetime $\mathcal{M}_{M,a}$,
with the hypersurfaces $\Sigma_{\tau}$ and the shaded region depicting
$\bigcup_{\tau\in[0,\infty)}\Sigma_{\tau}$. Each point in the picture
represents a Boyer–Lindquist sphere $S^{2}_{\tau,r}$ and the hypersurface
$\mathcal{I}^{+}$, which represents the points
$(\tau,\infty,\theta,\varphi_{*})$ is depicted at a finite distance.

######  Theorem 1.1.

Let $\psi$ be a solution arising from smooth and compactly supported initial
data for (1.1). Then $\psi$ satisfies the following asymptotic behaviour along
$\\{r=r_{0}\\}$, for each $r_{0}\geq r_{+}$:

$\displaystyle\psi(\tau,r_{0},\theta,\varphi_{*})=$
$\displaystyle-8I_{0}^{(1)}(1+\tau)^{-3}+\mathcal{O}_{r_{0}}(\tau^{-3-\eta}),$
$\displaystyle r^{-1}\psi_{1}(\tau,r_{0},\theta,\varphi_{*})=$
$\displaystyle-\frac{32}{3}I_{1}^{(1)}(\theta,\varphi_{*})(1+\tau)^{-5}+\mathcal{O}_{r_{0}}(\tau^{-5-\eta}),$
$\displaystyle\psi_{\geq 2}(\tau,r_{0},\theta,\varphi_{*})=$
$\displaystyle-\frac{16}{3}\sqrt{\frac{\pi}{5}}a^{2}I_{0}^{(1)}Y_{2,0}(\theta)(1+\tau)^{-5}+\mathcal{O}_{r_{0}}(\tau^{-5-\eta}),$

where $\eta>0$, $I_{i}^{(1)}$ are functions on $\mathbb{S}^{2}$ and denote the
_time-inverted Newman–Penrose charges of $\psi$_ (see Section 1.1.2), and
$\mathcal{O}_{r_{0}}(\tau^{-p})$ denotes schematically terms that can be
bounded uniformly in $\tau$ by weighted initial data $L^{2}$-norms multiplied
by $(1+\tau)^{-p}$, with constants that may depend on $r_{0}$.

When $r\to\infty$ the “radiation field” $r\psi$ has the following asymptotic
behaviour:

$\displaystyle r\psi(\tau,\infty,\theta,\varphi_{*})=$
$\displaystyle-2I_{0}^{(1)}(1+\tau)^{-2}+\mathcal{O}(\tau^{-2-\eta}),$
$\displaystyle r\psi_{1}(\tau,\infty,\theta,\varphi_{*})=$
$\displaystyle-\frac{4}{3}I_{1}^{(1)}(\theta,\varphi_{*})(1+\tau)^{-3}+\mathcal{O}(\tau^{-3-\eta}),$
$\displaystyle r\psi_{\geq 2}(\tau,\infty,\theta,\varphi_{*})=$
$\displaystyle\left[-\frac{2}{5}I_{2}^{(1)}(\theta,\varphi_{*})+\frac{8}{3}\sqrt{\frac{\pi}{5}}a^{2}I_{0}^{(1)}Y_{2,0}(\theta)\right](1+\tau)^{-4}+\mathcal{O}(\tau^{-4-\eta}),$

where $\mathcal{O}(\tau^{-p})$ denote schematically terms that can be bounded
uniformly in $\tau$ by weighted initial data $L^{2}$-norms multiplied by
$(1+\tau)^{-p}$.

A more detailed version of Theorem 1.1 follows directly by combining
Propositions 10.7, 11.7 and 12.7. In these propositions we moreover obtain
global analogues of the expressions in Theorem 1.1 with additional derivatives
$\partial_{\tau}^{k}$ on both sides, for arbitrary $k\in\mathbb{N}_{0}$.

######  Theorem 1.2.

Let $\psi$ be a solution arising from initial data for (1.1) on $\Sigma_{0}$
that are smooth with respect to the differentiable structure on the conformal
compactification of $\Sigma_{0}$, with non-zero _Newman–Penrose charges_
$I_{\ell}$, $\ell=0,1,2$ (see Section 1.1.2). Then the rates and coefficients
in the late-time expansions presented in Theorem 1.1 and Corollary 1.3 are
modified according to the expressions displayed in Table 1.

See Section 2.3 for a precise definition of the conformal compactification of
$\Sigma_{0}$. We directly obtain a more detailed version of Theorem 1.1 by
combining Corollary 10.5 and Propositions 10.6, 11.5, 11.6, 12.5 and 12.6.

Proj | $r=r_{0}\geq r_{+}$ | $\mathcal{I}^{+}$
---|---|---
|  |
$\psi_{0}$ | $4I_{0}\cdot\frac{1}{\tau^{2}}$ | $2I_{0}\cdot\frac{1}{\tau}$
$\psi_{1}$ | $\frac{8r_{0}}{3}I_{1}(\theta,\varphi_{*})\cdot\frac{1}{\tau^{4}}$ | $\frac{2}{3}I_{1}(\theta,\varphi_{*})\cdot\frac{1}{\tau^{2}}$
$\psi_{\geq 2}$ | $\frac{4}{3}\sqrt{\frac{\pi}{5}}a^{2}I_{0}\cdot Y_{2,0}(\theta)\cdot\frac{1}{\tau^{4}}$ | $\left[\frac{2}{15}I_{2}(\theta,\varphi_{*})-\frac{8}{9}\sqrt{\frac{\pi}{5}}a^{2}I_{0}\cdot Y_{2,0}(\theta)\right]\cdot\frac{1}{\tau^{3}}$
Table 1: Late-time asymptotics of $\psi_{0},\psi_{1},\psi_{\geq 2}$ for non-
zero Newman–Penrose charges $I_{\ell}$

######  Remark 1.1 (Logarithmic next-to-leading order terms).

One can apply the arguments in [AAG19] to the setting of Kerr spacetimes, with
minimal modifications, to obtain the following higher-order extension of the
late asymptotics of $r\psi$ at infinity in Theorem 1.1:

$r\psi(\tau,\infty,\theta,\varphi_{*})=-2I_{0}^{(1)}(1+\tau)^{-2}+8MI_{0}^{(1)}\log(1+\tau)(1+\tau)^{-3}+\mathcal{O}(\tau^{-3}).$

See also [BVW18] for additional results pertaining to polyhomogeneity in late-
time expansions on asymptotically Minkowksi spacetimes.

######  Remark 1.2 (Initial data regularity).

The initial data in Theorem 1.1 are assumed to be compactly supported. This
assumption is made purely to simplify the expressions of the time-inverted
Newman–Penrose charges $I_{\ell}^{(1)}$ in terms of initial data for $\psi$;
see Section 9.4. The assumption can be weakened by merely assuming the data
have sufficient, finite regularity with respect to the differentiable
structure on the conformal compactification of the hypersurface $\Sigma_{0}$,
with moreover vanishing Newman–Penrose charges $I_{\ell}$. In fact, one can
consider even rougher data by modifying the powers of $r$ appearing in the
definitions of the Newman–Penrose charges, which will affect the decay rates;
see also the upcoming [Keh21] for an example of late-time asymptotics arising
from rougher initial data that naturally come up in scattering problems.

######  Remark 1.3 (Blow-up in the black hole interior).

The results in Theorems 1.1 and 1.2 can be applied to justify the assumptions
made along the event horizon in [LS16] in order to derive blow-up of the
$H^{1}_{\rm loc}$ norm of $\psi$ at the inner horizon, the future causal
boundary of the Kerr black hole interior for $a\neq 0$.

######  Remark 1.4 (Linearized Einstein equations).

The methods that are developed in this paper to prove Theorems 1.1 and 1.2
could be extended naturally to the setting of the spin-2 Teukolsky equations,
when combined with the integrated energy estimates in [DHR19a, SRdC20]. As the
Teukolsky equations dictate the dynamical behaviour of perturbations of Kerr
initial data in the context of the linearized Einstein equations, these
methods provide a clear strategy for deriving late-time tails in the evolution
of metric perturbations.

####  1.1.2 Newman–Penrose charges

The Newman–Penrose charges $I_{\ell}$ are the values of _asymptotic
quantities_ that are defined along the spheres foliating future null infinity
and are conserved in time. See point 3 of Section 1.4.1 for an illustration of
this conservation property and see Section 4 for the precise definition of the
Newman–Penrose charges. They were originally introduced in [NP68] on Minkowski
background spacetimes and more generally in the context of the nonlinear
Einstein equations.

In the Minkowski spacetime, the Newman–Penrose charges can be expressed as the
following weighted derivatives of the radiation field:

$I_{\ell}[\psi]=(2r^{2}L)^{\ell}(r\psi_{\ell})(\tau,r=\infty,\theta,\varphi_{*}),$

and they are independent of $\tau$. Here, $L$ denotes a standard outgoing null
vector field.

In Schwarzschild spacetimes, or Kerr spacetimes with mass $M$ and angular
parameter $a=0$, the above quantity is still conserved when $\ell=0$, but for
$\ell\geq 1$, it has to be modified as follows:

$I_{\ell}[\psi]=r^{2}L(2r^{2}w_{\ell}L(\ldots(2r^{2}w_{1}L(2r^{2}L(r\psi_{\ell})))\ldots)(\tau,r=\infty,\theta,\varphi_{*}),$

with $w_{i}$ denoting polynomials in $r^{-1}$ of degree $\ell+1-i$, such that
$\lim_{r\to\infty}w_{i}(r)=1$, which depend on the mass $M$. See also [AAG21]
for a more precise inductive definition of $I_{\ell}$ on Schwarzschild.
Analogous modified expressions can also be shown to hold in more general
spherically symmetric, asymptotical flat spacetimes.

In order to construct $I_{\ell}$ on general Kerr spacetimes, one has to deal
with the _coupling_ of two difficulties:

  1. 1.

A modification of the weighted vector fields $r^{2}L$ due to the presence of
mass $M$,

  2. 2.

The appearance of the spherical harmonic modes $\psi_{\ell-2}$ and
$\psi_{\ell+2}$ in the expression for $I_{\ell}$, due to the presence of
angular momentum $a\neq 0$ and the use of Boyer–Lindquist spheres.

Note that difficulty 2 is already present in Minkowski when considering a
foliation by oblate spheroids that are asymptotically round, rather than
exactly round spheres, i.e. by taking $M=0$ in the Kerr metric, which leads to
a spacetime that is isometric to Minkowski.111The eccentricity of the
spheroids is then given by $\frac{|a|}{\sqrt{r^{2}+a^{2}}}$. For example, the
charge $I_{0}$ then takes the form:

$I_{0}[\psi]=\left[2r^{2}L(r\psi_{0})-\frac{1}{2}a^{2}\pi_{0}(\sin^{2}\theta
T(r\psi))\right](\tau,r=\infty,\theta,\varphi_{*}),$

with $\pi_{0}$ and $\psi_{0}$ now denoting projections with respect to
ellipsoids, rather than round spheres. The presence of the term
$\pi_{0}(\sin^{2}\theta T(r\psi))$ above may be viewed as an effect of angular
mode coupling in Boyer–Lindquist coordinates at infinity. See Section 1.1.4
for more details.

In Kerr, $I_{0}$ takes the same form as in Minkowski with respect to oblate
spheroidal coordinates, but $I_{\ell}$ with $\ell\geq 1$ includes
modifications depending on $M$. We derive the Newman–Penrose charges
$I_{\ell}$ in Kerr, with $\ell=0,1,2$ in Section 4 and the same strategy may
be applied inductively to find $I_{\ell}$.

The time-inverted Newman–Penrose charges, which are the quantities
$I_{\ell}^{(1)}$ appearing in the leading-order terms in Theorem 1.1 can be
interpreted as the Newman–Penrose charges of the function:

$\partial_{\tau}^{-1}\psi(\tau,r,\theta,\varphi):=-\int_{\tau}^{\infty}\psi(\tau^{\prime},r,\theta,\varphi)\,d\tau^{\prime},$

the time integral of $\psi$. We moreover show that $I_{\ell}^{(1)}$ can be
expressed solely in terms of initial data for $\psi$ on $\Sigma_{0}$, for
example, for $\ell=0$:

$I_{0}^{(1)}=\frac{1}{4\pi}M(r_{+}^{2}+a^{2})\int_{\Sigma_{0}\cap\mathcal{H}^{+}}\psi\,d\omega+\frac{1}{4\pi}M\int_{\Sigma_{0}}\mathbf{n}_{0}(\psi)\,d\mu_{0},$

with $\mathbf{n}_{0}$ the normal vector field with respect to $\Sigma_{0}$ and
$d\mu_{0}$ the natural volume form with respect to the induced metric on
$\Sigma_{0}$. See Section 9.4 for the precise integral expressions of
$I_{\ell}^{(1)}$.

####  1.1.3 Horizon oscillations

As an immediate corollary of the late-time asymptotics presented in Theorem
1.1, we derive an oscillatory and decaying behaviour of the $\ell=1$ angular
mode of $\psi$ when measured along the null generators of the event horizon.

######  Corollary 1.3.

Let $\gamma_{\theta,\varphi_{\mathcal{H}^{+}}}(\tau)$ denote a null generator
of the event horizon $\mathcal{H}^{+}$ emanating from the point
$(\theta,\varphi_{\mathcal{H}^{+}})$ on the 2-sphere
$S^{2}_{0,r_{+}}=\Sigma_{0}\cap\mathcal{H}^{+}$, with time parameter $\tau$.
Then:

$\displaystyle\psi|_{\gamma_{(\theta,\varphi_{*})}(\tau)}=$
$\displaystyle-8I_{0}^{(1)}(1+\tau)^{-3}+\mathcal{O}(\tau^{-3-\eta}),$
$\displaystyle r^{-1}\psi_{1}|_{\gamma_{(\theta,\varphi_{*})}(\tau)}=$
$\displaystyle-\frac{32}{3}\sum_{m=-1}^{1}{I_{1m}^{(1)}}Y_{1,m}(\theta,\varphi_{\mathcal{H}^{+}})e^{im\upomega_{+}\tau}(1+\tau)^{-5}+\mathcal{O}(\tau^{-5-\eta}),$
$\displaystyle\psi_{\geq 2}|_{\gamma_{(\theta,\varphi_{*})}(\tau)}=$
$\displaystyle-\frac{16}{3}\sqrt{\frac{\pi}{5}}a^{2}I_{0}^{(2)}Y_{2,0}(\theta)(1+\tau)^{-5}+\mathcal{O}(\tau^{-5-\eta}),$

where $\upomega_{+}=\frac{a}{r_{+}^{2}+a^{2}}$ is the _angular velocity of the
black hole_ and $I_{\ell m}^{(1)}$ denotes the projection of $I_{\ell}^{(1)}$
to the $m$-th azimuthal mode.

We therefore see that the leading-order behaviour of the $\ell=1$ mode in time
_encodes_ the angular velocity of the black hole in the form of an oscillatory
factor $e^{im\upomega_{+}\tau}$. Corollary 1.3 provides the first rigorous
derivation of _horizon oscillations_ in Kerr spacetimes, which were originally
suggested by Barack–Ori in [BO99b].

The presence of horizon oscillations illustrates the different roles played by
the Killing vector fields $T$ and $K$. Here, $T$ denotes a suitably normalized
Killing vector field that is timelike for large values of $r$, but spacelike
near the event horizon, whereas $K$ denotes the Killing vector field that is
tangential to the null generators of the horizon and is timelike close to the
horizon, but spacelike far away from the horizon. Hence, $T$ provides a
“natural” choice of time direction far away from the horizon and $K$ provides
a natural choice of time direction close to the horizon. Note that in the
coordinate chart $(\tau,r,\theta,\varphi_{*})$ that we have chosen, we can
simply express $T=\partial_{\tau}$. The two Killing vector fields can be
related as follows:

$K=T+\upomega_{+}\Phi,$

where $\Phi$ denotes the Killing vector field that generates the axisymmetry
of the Kerr spacetime, and $K=T$ when $a=0$.

Horizon oscillations follow from the fact that when $a\neq 0$ the time
derivative $T\psi$ decays one power faster than $\psi$ to leading order,
whereas the time derivative $K\psi$ does not. In fact, using the above
relation between $T$ and $K$, we can see that the non-axisymmetric part of
$K\psi$ must agree precisely with $\upomega_{+}\Phi\psi$ to leading order in
time, which explains the oscillatory behaviour in the $\ell=1$ mode.

We note that it has been suggested that horizon oscillations will similarly
appear in the leading-order late-time asymptotics of general solutions to non-
zero integer spin Teukolsky equations, see [BO99a] and Remark 1.4. They are
therefore expected to play a leading-order role in the general late-time
horizon dynamics in the setting of linearized gravity and the Maxwell
equations on Kerr spacetimes.

####  1.1.4 Angular mode coupling

The spherical harmonic modes in Theorems 1.1 and 1.2 are coupled in their
evolution, in contrast with the Schwarzschild case $(a=0)$ where they evolve
independently.

Indeed, when defined with respect to the metric on the unit round sphere on
Boyer–Lindquist spheres, the angular mode coupling takes the following form:

$\rho^{2}\square_{g_{M,a}}\psi_{\ell}=a^{2}{T^{2}}(\pi_{\ell}(\sin^{2}\theta\psi)-\sin^{2}\theta\pi_{\ell}\psi),$
(1.2)

where we can schematically write the right-hand side as:

$a^{2}{T^{2}}(\pi_{\ell}(\sin^{2}\theta\psi)-\sin^{2}\theta\pi_{\ell}\psi)\approx
a^{2}(c_{-}T^{2}\psi_{\ell-2}+c_{0}T^{2}\psi_{\ell}+c_{+}T^{2}\psi_{\ell+2}),$
(1.3)

with $c_{0},c_{-},c_{+}$ constants, see Lemma 2.4 for a precise expression.

If the different angular modes were uncoupled, we would be able to show that
modes supported on _higher_ $\ell$ decay _faster_. This is precisely the case
when $a=0$, see [AAG21]. In view of (1.2) and (1.3), however, we can see
heuristically that the decay of $\psi_{\ell}$ is limited by the decay of
$T^{2}\psi_{\ell-2}$. We would therefore have the following schematic relation
for the late-time asymptotics of $\psi_{\ell}$:

$\displaystyle\psi_{\ell}\sim T^{2}\psi_{\ell-2}\sim\ldots\sim
T^{\ell}\psi_{0}\quad\textnormal{for even $\ell\geq 2$},$
$\displaystyle\psi_{\ell}\sim T^{2}\psi_{\ell-2}\sim\ldots\sim
T^{\ell-1}\psi_{1}\quad\textnormal{for odd $\ell\geq 3$}.$

So while the decay rate of $\psi_{\ell}$ is limited by the lower angular modes
$T^{\ell}\psi_{0}$ and $T^{\ell-1}\psi_{1}$, we can use that $T$-derivatives
decay faster, as mentioned in Section 1.1.3, together with the above
heuristics to predict that higher angular frequencies $\psi_{\ell}$ _do_ in
fact decay faster than lower angular frequencies, but they decay slower
compared to the uncoupled setting.

In this paper, we give a rigorous validation of the above heuristics in the
case of the late-time asymptotics of $\ell=2$. We also discuss the general
$\ell$ case in Section 1.3.

We conclude that while there is no _a priori_ natural choice of 2-spheres to
foliate $\mathcal{M}_{M,a}$, Theorems 1.1 and 1.2 imply that the choice of
Boyer–Lindquist spheres is natural from the point of view of late-time
asymptotics, as it has the favourable property that it results in a “late-time
decoupling” of the corresponding spherical harmonic modes $\psi_{\ell}$, i.e.
the terms in the expansion of $\psi$ in $\tau^{-1}$ at any fixed $r$ up to
order $N\in\mathbb{N}$ will be supported only on spherical harmonic modes with
angular frequencies $\ell\leq L(N)<\infty$, where we expect
$L(N)=\max\\{N-3,0\\}$.

###  1.2 Previous work

In this section we give an overview of some relevant previous results in the
literature on late-time tails and sharp decay estimates for waves on black
hole spacetimes.

####  1.2.1 Physics literature

The first discussion on inverse polynomial late-time tails for wave equations
on black hole background appeared in a paper of Price [Pri72], who provided a
heuristic argument for the presence of a $\tau^{-2\ell-3}$ tail in
Schwarzschild. Late-time tails on spherically symmetric backgrounds have since
been discussed frequently in the physics literature, using both heuristic and
numerical arguments; see the introduction of [AAG20] for an overview.

The first numerical discussion on late-time tails in the context of
subextremal Kerr spacetimes appeared in [KLPA97], which was followed by
heuristic analyses in [Hod99] and [BO99b]. The latter work suggested the power
laws for generic, compactly supported data that are stated in Section 1.3.1.
Subsequent numerical work on late-time tails focused on the case of initial
data supported on a single harmonic mode and has produced conflicting
numerology, caused by the difficulty of characterising the evolutionary
coupling and excitation of spherical harmonic modes (numerically). We refer to
[ZKB14, BK14] for the latest numerical results and corrected predictions of
late-time tails and also to the references therein for a more complete history
of the problem. As discussed in Section 1.3.2, the methods developed in the
present paper allow for a final resolution of the differences in the
numerology of late-time tails suggested in the literature, by relating the
numerology to the vanishing and non-vanishing of time-inverted Newman–Penrose
charges, which we expect to align with the numerical results of [ZKB14, BK14].

####  1.2.2 Mathematics literature

The first mathematically rigorous derivation of the leading-order late-time
behaviour of waves on black hole background was obtained in [AAG18c, AAG18b]
for a variety of spherically symmetric backgrounds. The presence of
$\ell$-dependent late-time tails on Schwarzschild and more general subextremal
Reissner–Nordström backgrounds is derived in [AAG21]. These results appeal to
important ideas and results that appeared in previous literature on
(integrated) energy decay estimates, most notably [DR09, DR10], see also
[DR13]. The methods of [AAG18c, AAG18b] have also been adapted to the setting
of the Dirac equation on Schwarzschild backgrounds [MZ20].

A proof of the existence of $\tau^{-3}$ tails on Kerr spacetimes was recently
established in [Hin20], using methods based in Fourier space.

We refer also to [Tat13, MTT12, Hin20] for sharp decay estimates for $\psi$ in
a linear setting and [DR05] in a spherically symmetric nonlinear setting. See
the also the results in [DSS11, DSS12], investigating the $\ell$-dependence in
decay estimates and [Ma20] for some sharp decay estimates in the context of
the Maxwell equations and a characterisation of the Newman–Penrose charges on
Schwarzschild in the Maxwell setting.

Finally, the results in the present paper appeal to the energy boundedness and
integrated energy decay estimates that have already been established in Kerr
spacetimes in [DRSR16], which capture the key geometric obstructions to wave
decay on Kerr caused by the presence of trapped null geodesics and an
ergoregion. We refer to the introduction of [DRSR16] for a complete overview
of the history of integrated energy decay estimates on Kerr spacetimes.

####  1.2.3 Extremal black hole spacetimes

The leading-order behaviour of waves on extremal black hole backgrounds
features additional interesting geometric phenomena. The first mathematically
rigorous proof of the existence and form of late-time tails on extremal
Reissner–Nordström spacetimes was obtained in [AAG20] and follows previous
numerical and heuristic results, see the introduction of [AAG20] for a
comprehensive overview. The late-time tails in this setting involve an
additional conserved charge that occurs along the event horizon of extremal
black hole spacetimes and was first discovered in [Are11]. It is connected to
the absence of the redshift effect along extremal black hole horizons and may
be related to the Newman–Penrose charges, see [LMRT13, BF13]. See also
[AAG18a] and the subsequent numerical work [BKS19, BKS21] for discussions on
the _signature_ of this conserved horizon charge at infinity.

While decay estimates have also been obtained in the setting of rotating
extremal black hole spacetimes, for the special case of axisymmetric solutions
[Are12], the late-time properties of non-axisymmetric linear waves on extremal
Kerr ($|a|=M$) remain open. In this setting, only mode stability has been
established in a mathematically rigorous setting in [dC20], but heuristics
have also been provided on the rates of late-time tails for fixed azimuthal
modes in [CGZ16].

###  1.3 Analogues of Price’s law in Kerr

####  1.3.1 Generic initial data

One can apply the methods developed in this paper to extend the results of
Theorem 1.1 (and Theorem 1.2) beyond $\ell=2$ and obtain the following late-
time asymptotics for higher spherical harmonic modes:

$\displaystyle\psi_{\ell=2k}(\tau,r_{0},\theta,\varphi_{*})=$
$\displaystyle\>c_{\ell,0}a^{k}I_{0}^{(1)}Y_{\ell,0}(\theta)(1+\tau)^{-\ell-3}+\mathcal{O}_{r_{0}}(\tau^{-\ell-3-\eta}),$
$\displaystyle r^{-1}\psi_{\ell=2k+1}(\tau,r_{0},\theta,\varphi_{*})=$
$\displaystyle\>\sum_{m=-1}^{1}c_{\ell,m}a^{k}I_{1m}^{(1)}Y_{\ell,m}(\theta,\varphi_{*})(1+\tau)^{-\ell-4}+\mathcal{O}_{r_{0}}(\tau^{-\ell-4-\eta}),$
$\displaystyle r\psi_{\ell=2k}(\tau,\infty,\theta,\varphi_{*})=$
$\displaystyle\>\sum_{k=0}^{\frac{\ell}{2}}\sum_{m=-2k}^{2k}c_{\ell,k,m}a^{\ell-2k}I_{2k\,m}^{(1)}Y_{\ell,m}(\theta,\varphi_{*})(1+\tau)^{-\ell-2}+\mathcal{O}(\tau^{-\ell-2-\eta}),$
$\displaystyle r\psi_{\ell=2k+1}(\tau,\infty,\theta,\varphi_{*})=$
$\displaystyle\>\sum_{k=0}^{\frac{\ell-1}{2}}\sum_{m=-(2k+1)}^{2k+1}c_{\ell,k,m}a^{\ell-(2k+1)}I_{2k+1\,m}^{(1)}Y_{\ell,m}(\theta,\varphi_{*})(1+\tau)^{-\ell-2}+\mathcal{O}(\tau^{-\ell-2-\eta}),$

where $c_{\ell,m}$ and $c_{\ell,k,m}$ are dimensionless constants. We do not
provide a proof of the above expressions in the present paper, but we refer to
the heuristics in Section 1.1.4 and leave them for the interested reader to
verify, equipped with the techniques and estimates that are developed in the
paper.

The above behaviour suggests in particular the following version of “Price’s
law” at fixed radius for $a\neq 0$:

$\displaystyle\psi_{\ell}\sim$
$\displaystyle\tau^{-\ell-3}\quad\textnormal{when $\ell$ is odd},$
$\displaystyle\psi_{\ell}\sim$
$\displaystyle\tau^{-\ell-4}\quad\textnormal{when $\ell$ is even},$

which differs from Price’s law when $a=0$: $\psi_{\ell}\sim\tau^{-2\ell-3}$,
see [AAG21]. We note that the above decay rates are consistent with the
heuristics in [BO99b].

####  1.3.2 Initial data restricted to higher angular frequencies and mode
excitation

Rather than considering generic initial data supported on all angular
frequencies, one can further investigate the mode-coupling mechanism by
considering restricted data, supported on angular frequencies
$\geq\ell^{\prime}$ and then investigate the late-time tails of $\psi_{\ell}$
for each $\ell\in\mathbb{N}_{0}$. This may be viewed as a study of “mode
excitation” to leading order in time.

For $|\ell-\ell^{\prime}|>2$, it follows easily that $I_{\ell}^{(1)}=0$. In
this case, one would have to apply the time inversion procedure $k$ times, for
suitable values of $k$, to obtain a non-vanishing _higher-order_ time-inverted
charge $I_{\ell}^{(k)}[\psi]:=I_{\ell}[\partial_{\tau}^{-k}\psi]$, which would
then be relevant for the late-time behaviour. It is straightforward to show
via the equation for the time inverse (9.30) that if the (smooth and compactly
supported) initial data are supported on the angular frequency
$\ell^{\prime}$, with $\ell^{\prime}\notin\\{0,1\\}$, the first (generically)
non-zero time-inverted charge for $\ell\neq\ell^{\prime}$ would be

$I_{\ell}^{(|\ell^{\prime}-\ell|-1)}[\psi]=I_{\ell}[\partial_{\tau}^{-|\ell^{\prime}-\ell|+1}\psi],$

if $\ell$ and $\ell^{\prime}$ are both either even or odd (note that
$I_{\ell}^{(k)}$ vanishes for all $k$ if $\ell$ and $\ell^{\prime}$ have
opposite parity). Moreover,
$I_{\ell^{\prime}}^{(1)}=I_{\ell^{\prime}}[\partial_{\tau}^{-1}\psi]\neq 0$,
generically. See also Section 9.2 of [AAG18b] for a discussion on higher-order
time inversions in the spherically symmetric context.

In light of the non-vanishing of the above higher-order time inverted charges,
we expect that, analogously to what is described in Section 1.3, one could
apply the methods in the present paper to obtain schematically the following
late-time tails for initial data supported on angular frequency
$\ell^{\prime}$: let $j=0$ when $\ell^{\prime}$ is even and $j=1$ when
$\ell^{\prime}$ is odd, then

$\displaystyle\psi_{\ell}|_{r=r_{0}}\sim$
$\displaystyle\sum_{\begin{subarray}{c}0\leq
k\leq\lfloor\frac{\max\\{\ell,\ell^{\prime}-2\\}}{2}\rfloor\end{subarray}}I_{2k+j}^{(|\ell^{\prime}-2k|-1)}\tau^{-\ell-\ell^{\prime}-1}\quad\textnormal{when}\,\,\ell^{\prime}\notin\\{0,1\\},$
(1.4) $\displaystyle\psi_{\ell}|_{r=r_{0}}\sim$ $\displaystyle
I_{\ell^{\prime}}^{(1)}\tau^{-\ell-\ell^{\prime}-3}\quad\textnormal{when}\,\,\ell^{\prime}\in\\{0,1\\},$
(1.5) $\displaystyle r\psi_{\ell}|_{\mathcal{I}^{+}}\sim$ $\displaystyle
I_{\ell}^{(\ell^{\prime}-\ell-1)}\tau^{-\ell^{\prime}}\quad\textnormal{when}\,\,\ell\leq\ell^{\prime}-2,$
(1.6) $\displaystyle r\psi_{\ell}|_{\mathcal{I}^{+}}\sim$
$\displaystyle\sum_{-1\leq
k\leq\min\\{1,\frac{\ell-\ell^{\prime}}{2}\\}}^{1}I_{\ell^{\prime}+2k}^{(1)}\tau^{-\ell-2}\quad\textnormal{when}\,\,\ell\geq\ell^{\prime},$
(1.7)

with $\ell^{\prime}$ and $\ell$ both either even or odd. Note in particular
that (1.5)–(1.7) follow directly from Theorem 1.1 and the expressions in
Section 1.3.1, whereas (1.4) requires a minor further extension of the methods
in the present paper.

The above power laws agree with an extrapolation of the numerical results of
[ZKB14], but deviate from earlier suggestions in the literature. See also
Section 1.2.

###  1.4 Main ideas and techniques

In this section, we introduce the main new ideas and techniques involved in
proving Theorems 1.1 and 1.2.

We make use of the following additional notation in this section:

  * •

We cover $\Sigma_{0}$ with the coordinates $(\uprho,\theta,\varphi_{*})$,
where $\uprho=r|_{\Sigma_{0}}$ and employ the chart
$(\tau,\uprho,\theta,\varphi_{*})$ to cover the spacetime region
$\bigcup_{\tau\in[0,\infty)}\Sigma_{\tau}$. Furthermore, we denote the
standard volume form on the unit round 2-sphere $\mathbb{S}^{2}$ by $d\omega$
and the natural volume form corresponding to the induced metric on
$\Sigma_{\tau}$ by $d\mu_{\tau}$.

  * •

We use the notation $T=\partial_{\tau}$ and $X=\partial_{\uprho}$, and we use
$L$ and $\underline{L}$ to denote the principal outgoing and ingoing null
vector fields in Kerr. We also use $\not{\nabla}_{\mathbb{S}^{2}}$ to denote
the covariant derivative on $\mathbb{S}^{2}$.

  * •

We the following rescaling of $\psi$:

$\phi=\sqrt{r^{2}+a^{2}}\psi,$

where the limit $\phi(\tau,\infty,\theta,\varphi_{*})$ is known as the
_Friedlander radiation field_. This is a natural quantity to consider for
scattering problems; see for example [DRSR18].

  * •

We use $\mathbb{T}$ to denote the stress energy momentum tensor corresponding
to (1.1):

$\mathbb{T}(V,W)=V\psi
W\psi-\frac{1}{2}g_{M,a}(V,W)(g^{-1}_{M,a})^{\alpha\beta}\partial_{\alpha}\psi\partial_{\beta}\psi.$

We refer to Section 2 for a more precise introduction to all the above
notation.

####  1.4.1 Step 1: Energy decay via new hierarchies of $r^{p}$-weighted
estimates

The first step towards determining the leading-order behaviour of $\psi$ is to
obtain time decay estimates for energy norms along the leaves of a suitable
spacetime foliation.

We consider a foliation by asymptotically hyperboloidal hypersurfaces
$\Sigma_{\tau}$ intersecting the event horizon, see Figure 1. We obtain energy
decay by using extensions of the original Dafermos–Rodnianski $r^{p}$-weighted
energy method [DR10]. See [DRSR18] for an application of the
Dafermos–Rodnianski $r^{p}$-weighted energy method in Kerr and [Mos16] for the
case of more general spacetimes.

  1. 1.

(Dafermos–Rodnianski hierarchy) Consider first the Dafermos–Rodnianski (D–R)
hierarchy, which takes the schematic form

$\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}r^{p-1}(L\phi)^{2}\,d\omega
d\uprho d\tau\lesssim\int_{\Sigma_{\tau_{1}}}r^{p}(L\phi)^{2}\,d\omega
d\uprho+\ldots,$ (1.8)

for $0<p\leq 2$, where we have omitted additional angular and $T$-derivatives
appearing on the left-hand side and $\ldots$ denotes terms arising from an
application of integrated energy (Morawetz) and energy boundedness estimates,
which we ignore in the current discussion.

Taking $p=1$ and applying the mean-value theorem in $\tau$ gives:

$\int_{\Sigma_{\tau}}(L\phi)^{2}\,d\omega
d\uprho\lesssim(1+\tau)^{-1}\int_{\Sigma_{\tau_{0}}}r(L\phi)^{2}\,d\omega
d\uprho+\ldots.$

Taking $p=2$ and applying the mean value theorem once again (along an
appropriate sequence of times) allows us to estimate the right-hand side
further and obtain the following uniform energy decay estimates:

$\int_{\Sigma_{\tau}}(L\phi)^{2}\,d\omega
d\uprho\lesssim(1+\tau)^{-2}\int_{\Sigma_{\tau_{0}}}r^{2}(L\phi)^{2}\,d\omega
d\uprho+\ldots.$

  2. 2.

(Extending the D–R hierarchy via $r^{2}L$ derivatives) The D–R hierarhcy is
limited by the restriction $p\leq 2$. To be able to get a decay rate of
$(1+\tau)^{-3}$ and higher, we need control over the left-hand side of (1.8)
for $p\geq 3$. By an application of a standard Hardy inequality, we can show
that:

$\begin{split}\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}r^{2}(L\phi)^{2}\,d\omega
d\uprho
d\tau=&\>\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}r^{-2}(r^{2}L\phi)^{2}\,d\omega
d\uprho d\tau\\\
\lesssim&\>\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}(L(r^{2}L\phi))^{2}\,d\omega
d\uprho d\tau+\ldots.\end{split}$

In the case where $\psi$ is supported on $\ell\geq 1$, we can further
estimate: for $0\leq p\leq 2$

$\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}r^{p-1}(L(r^{2}L\phi_{\geq
1}))^{2}\,d\omega d\uprho
d\tau\lesssim\int_{\Sigma_{\tau_{1}}}r^{p}(L(r^{2}L\phi_{\geq 1})^{2}\,d\omega
d\uprho+\boxed{a^{2}\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}r^{p-3}(r^{2}LT^{2}\phi)^{2}\,d\omega
d\uprho d\tau}\ldots.$

In fact, more generally, for all $n\leq\ell$ and $0\leq p\leq 2$:

$\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}r^{p-1}(L(r^{2}L)^{n}\phi_{\geq\ell})^{2}\,d\omega
d\uprho
d\tau\lesssim\int_{\Sigma_{\tau_{1}}}r^{p}(L(r^{2}L)^{n}\phi_{\geq\ell})^{2}\,d\omega
d\uprho+\boxed{a^{2}\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}r^{p-3}((r^{2}L)^{n}T^{2}\phi_{\geq\ell-2})^{2}\,d\omega
d\uprho d\tau}\ldots.$ (1.9)

When $a=0$, we can therefore continue applying the mean-value theorem together
with a Hardy inequality and apply (1.9) in order to obtain (at least)
$(1+\tau)^{-2-2\ell}$ energy decay for $\phi_{\geq\ell}$. This is done in
[AAG21].

However, when $a\neq 0$, we have to address the boxed term in (1.9), which
reflects the fact that we cannot treat $\phi_{\geq\ell}$ independently from
$\phi_{\leq\ell-1}$ (recall that the spherical harmonic modes are coupled).

For example, in the case $\ell=2$, we can rewrite the boxed term in (1.9) as
follows:

$a^{2}\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}r^{p-3}((r^{2}L)^{2}T^{2}\phi)^{2}\,d\omega
d\uprho d\tau\lesssim
a^{2}\sum_{k=0}^{1}\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}r^{p+3}(L(rL)^{k}T^{2}\phi)^{2}\,d\omega
d\uprho.$

By applying the wave equation (1.1) we can exchange $T$ derivatives for $L$
and angular derivatives to estimate further:

$a^{2}\sum_{k=0}^{1}\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}r^{p+3}(L(rL)^{k}T^{2}\phi)^{2}\,d\omega
d\uprho\lesssim a^{2}\sum_{0\leq k_{1}+k_{2}\leq
3}\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}r^{p-1}|\not{\nabla}_{\mathbb{S}^{2}}^{k_{1}}L(rL)^{k_{2}}\phi|^{2}\,d\omega
d\uprho.$

When then show that the term above can be controlled after showing the
estimate (1.8) also holds when $\phi$ is replaced with
$\not{\nabla}_{\mathbb{S}^{2}}^{k_{1}}(rL)^{k_{2}}\phi$. That is to say, _the
(extended) $r^{p}$-weighted hierarchies remain valid after commuting
arbitrarily many times with $rL$ and $\not{\nabla}_{\mathbb{S}^{2}}$_. This
favourable commutation property also plays an important role in [Mos16,
Sch13]. We stress moreover that this property contrasts strongly with
commutation with $r^{2}L$, which, as outlined above, is more delicate as it is
only valid after projecting to suitably higher angular frequencies.

We apply the above strategy for $\ell=1,2,3$ to obtain $(1+\tau)^{-2-2\ell}$
energy decay for $\phi_{\geq\ell}$ and moreover
$(1+\tau)^{-2-2\ell-2k}$ energy decay for $T^{k}\phi$ by exploiting the above
exchange of $T$-derivatives for $rL$-derivatives and angular derivatives to
extend the hierarchy even further.

See Section 5 for a precise discussion of the various $r^{p}$-weighted
hierarchies and Section 6 for the arguments that use these hierarchies to
obtain energy decay.

  3. 3.

(Maximal length hierarchies via conserved charges at infinity) In order to
obtain sharp energy decay rates, we complement the extended hierarchies above
with an additional extension that relies on the existence of conservation
charges at future null infinity.

We consider first $\phi_{0}$. In order to go beyond $p=2$ in the
Dafermos–Rodnianski hierarchy (1.8), we replace $L\phi_{0}$ in (1.8) with
$P_{0}$, which is defined as follows:

$P_{0}:=L\phi_{0}-\frac{1}{4}\frac{\Delta}{(r^{2}+a^{2})^{2}}a^{2}\pi_{0}(\sin^{2}\theta
T\phi).$

Since $P_{0}$ satisfies schematically

$\underline{L}P_{0}=O(r^{-3})(\phi_{0}+a^{2}rL\phi_{0}+a^{2}T\phi_{0}+a^{2}rL\phi_{2})$

it follows that $\lim_{r\to\infty}r^{2}{P_{0}}|_{\Sigma_{\tau}}$ is conserved
in $\tau$ and an analogue of (1.8) holds for the extended range $0<p<3$, with
$L\phi_{0}$ replaced by $P_{0}$. We use the constant $I_{0}$ to denote these
limits and refer to this quantity as the Newman–Penrose charge for $\ell=0$;
see Section 1.1.2.

We can similarly replace $(r^{2}L)\phi_{1}$ and $(r^{2}L)^{2}\phi_{2}$ by the
quantities $P_{1}$ and $P_{2}$ (defined precisely in Section 4) to extend
(1.9) with $n=\ell\in\\{0,1,2\\}$, to $p<3$. See also [AAG18c, AAG18b, AAG21]
for a similar extension of the hierarchy of $r^{p}$-weighted energy estimates
in the $a=0$ case. In this case, the corresponding limits
$\lim_{r\to\infty}r^{2}{P_{\ell}}|_{\Sigma_{\tau}}$, which are conserved in
$\tau$, are functions of $(\theta,\varphi_{*})$ and are denoted by
$I_{\ell}(\theta,\varphi_{*})$. We refer to them as the Newman–Penrose charges
for $\ell=1$ and $\ell=2$.

We therefore conclude the energy decay rates $(1+\tau)^{-3-2\ell-2k}$ for
$T^{k}\phi_{\ell}$, with $\ell=0,1,2$, which we can combine with the already
established energy decay rate $(1+\tau)^{-9-2k}$ for $\phi_{\geq 3}$.

The application of the quantities $P_{\ell}$ to $r^{p}$-weighted energy
estimates are derived in Section 5.2.

By a straightforward application of the fundamental theorem of calculus in
$\uprho$ (see Appendix A) together with standard Sobolev estimates on
$\mathbb{S}^{2}$, we can convert the energy decay estimates above into the
following pointwise estimates:

$\displaystyle|rT^{k}\psi_{0}|\lesssim$
$\displaystyle\>(1+\tau)^{-1+\epsilon},$
$\displaystyle|rT^{k}\psi_{1}|\lesssim$
$\displaystyle\>(1+\tau)^{-2-k+\epsilon},$
$\displaystyle|rT^{k}\psi_{2}|\lesssim$
$\displaystyle\>(1+\tau)^{-3-k+\epsilon},$ $\displaystyle|rT^{k}\psi_{\geq
3}|\lesssim$ $\displaystyle\>(1+\tau)^{-\frac{7}{2}-k+\epsilon}.$

Additional difficulties: In the above outline, we have suppressed the role of
the geometric phenomena of _redshift_ , _trapped null geodesics_ and
_ergoregion_ towards establishing energy boundedness and decay in Kerr
spacetimes. While these play an important role, their effect on integrated
energy estimates has already been dealt with in [DRSR16] and we appeal to
these estimates in the present paper. We refer to the introduction of [DRSR16]
for a comprehensive discussion.

Note however that the estimates in [DRSR16] do not distinguish between
different spherical harmonic modes $\psi_{\ell}$, so we need to modify them
slightly to obtain more refined integrated estimates for $\psi_{\geq\ell}$ to
be able to carry out the arguments described above. We give an overview of the
necessary energy boundedness and integrated energy decay estimates, including
the refined integrated estimates for $\psi_{\geq\ell}$, in Section 3.3.

####  1.4.2 Step 2: Improved pointwise decay via a hierarchy of elliptic
estimates

We subsequently improve the pointwise decay rates for $r\psi$ above further by
considering the $r$-rescalings: $\psi_{0},\psi_{2},\psi_{\geq 3}$ and and
$r^{-1}\psi_{1}$, and we apply in addition a hierarchy of elliptic estimates.

We first rewrite the wave equation (1.1) as follows;

$\mathcal{L}\psi=F[T\psi],$

with $\mathcal{L}$ defined as the differential operator:

$\mathcal{L}\psi=X(\Delta
X\psi)+2aX\Phi\psi+\not{\Delta}_{\mathbb{S}^{2}}\psi$

and $F[T\psi]$ containing only $T$-derivatives of $\psi$, see (7.2) for the
precise expression. In this section, we treat $F[T\psi]$ as an inhomogeneity.

The operator $\mathcal{L}$ is elliptic when $T$ is a timelike vector field.
This is the case outside the ergoregion, i.e. when
$r^{2}-2Mr+a^{2}\cos^{2}\theta>0$. The loss of ellipticity inside the
ergoregion prevents the use of a standard elliptic estimate to control $\psi$
in terms of $F$, which would involve integrating by parts the equation
$(\mathcal{L}\psi)^{2}=F^{2}$ on $\Sigma_{\tau}$.222In the $a=0$ case, where
ellipticity only fails at the event horizon, this strategy does work and
results in elliptic estimates that degenerate at the event horizon. See for
example [AAG18c, AAG21].

We establish nevertheless for $k\in(\frac{1}{2},\ell+\frac{1}{2})$:

$\sum_{n=0}^{\ell}\int_{\Sigma_{\tau}}r^{-2k}\left[(X(rX)^{n}\psi_{\ell})^{2}+r^{-2}|\not{\nabla}_{\mathbb{S}^{2}}(rX)^{n}\psi_{\ell}|^{2}\right]\,r^{2}d\omega
d\uprho\lesssim\sum_{n=0}^{\ell}\int_{\Sigma_{\tau}}r^{-2k}((rX)^{n}({\pi}_{\ell}F))^{2}\,d\omega
d\uprho$ (1.10)

by applying the following estimates:

  1. A.)

A “spacelike redshift multiplier estimate”, i.e. when $\ell=0$, we integrate
by parts on $\Sigma_{\tau}\cap\\{r\leq R_{0}\\}$ the product:

$X\psi_{0}\cdot\mathcal{L}\psi_{0}$

and more generally,

$X^{\ell+1}\psi_{\ell}\cdot X^{\ell}(\mathcal{L}\psi_{\ell}).$

In contrast with the standard redshift estimates [DR09, DR13], which involve
integrating by parts in spacetime with timelike vector field multipliers, and
which are only valid in a neighbourhood of the event horizon, the above
spacelike redshift estimates hold with arbitrarily large radial coordinate
$r$.

  2. B.)

An elliptic estimate in the region $r\geq R_{0}$, with $R_{0}\gg M$, which
follows from integrating

$r^{-2k}(X^{\ell}\mathcal{L}\psi)^{2}=r^{-2k}(X^{\ell}F)^{2}$

where we apply A.) to estimate the boundary terms on $r=R_{0}$. The cross
terms that arise when expanding the square on the left-hand side above are
estimated via a Poincaré inequality and the range of allowed values for $k$
depends therefore on the angular frequency $\ell$.

  3. C.)

Another spacelike redshift multiplier estimate to control lower-order
derivative terms, by integrating

$X^{n+1}\psi_{\ell}\cdot X^{n}(\mathcal{L}\psi_{\ell}),$

with $n\leq\ell-1$ and applying an induction argument, using A.) and B.).

Filling in the expression for $F[T\psi]$, we therefore obtain the following
_elliptic hierarchy_ of energy estimates:

$\begin{split}\sum_{n=0}^{\ell}\int_{\Sigma_{\tau}}r^{-2k}\mathbb{T}[(rX)^{n}\psi_{\ell}](T,\mathbf{n}_{\tau})\,d\mu_{\tau}\lesssim&\>\sum_{n=0}^{\ell}\int_{\Sigma_{\tau}}r^{-2k+2}\mathbb{T}[(rX)^{n}T\psi_{\ell}](T,\mathbf{n}_{\tau})\,d\mu_{\tau}\\\
&+a^{2}\int_{\Sigma_{\tau}}r^{-2k}\left\\{\mathbb{T}[(rX)^{n}T^{2}\psi_{\ell-2}](T,\mathbf{n}_{\tau})+\mathbb{T}[(rX)^{n}T\psi_{\ell+2}](T,\mathbf{n}_{\tau})\right\\}\,d\mu_{\tau},\end{split}$
(1.11)

for $k\in(\frac{1}{2},\ell+\frac{1}{2})$.

The elliptic hierarchy allows us to exchange negative $r$ weights for
additional $T$ derivatives, which decay faster in time. One may compare this
with extra time decay obtained via an extended hierarchy $r^{p}$-weighted
energy estimates, as described in Section 1.4.1, where additional time decay
follows instead by controlling an additional time integral on the left-hand
side and applying the mean-value theorem.

In the $a=0$ case, we can apply (1.11) starting from $k=\frac{1}{2}-\delta$ to
$k=\ell+\frac{1}{2}-\delta$ to obtain a hierarchy of estimates of length
$\ell$ and therefore estimate:

$\begin{split}\sum_{n=0}^{\ell}\int_{\Sigma_{\tau}}r^{-2\ell-1-\delta}\mathbb{T}[(rX)^{n}\psi_{\ell}](T,\mathbf{n}_{\tau})\,d\mu_{\tau}\lesssim&\>\sum_{n=0}^{\ell}\int_{\Sigma_{\tau}}r^{1-\delta}\mathbb{T}[(rX)^{n}T^{\ell}\psi_{\ell}](T,\mathbf{n}_{\tau})\,d\mu_{\tau},\end{split}$

and obtain the sharp decay rate of the weighted energy on the left-hand side,
together with the sharp decay rate of $r^{-\ell}\psi_{\ell}$.

When $a\neq 0$, the above strategy can be repeated for $\ell=0$ and $\ell=1$,
but it fails for $\ell=2$, due to the presence of the terms involving
$\psi_{0}$ on the right-hand side of (1.11) when $\ell=0$, which limits the
decay rate and hence the length of the elliptic hierarchy. We refer to Section
7 for a precise discussion of the above arguments.

When $a\neq 0$, we therefore conclude pointwise decay estimates with the
following weights:

$\displaystyle|T^{k}\psi_{0}|\lesssim$
$\displaystyle\>(1+\tau)^{-2+\epsilon},$
$\displaystyle|r^{-1}T^{k}\psi_{1}|\lesssim$
$\displaystyle\>(1+\tau)^{-4-k+\epsilon},$
$\displaystyle|T^{k}\psi_{2}|\lesssim$
$\displaystyle\>(1+\tau)^{-4-k+\epsilon},$ $\displaystyle|T^{k}\psi_{\geq
3}|\lesssim$ $\displaystyle\>(1+\tau)^{-\frac{9}{2}-k+\epsilon}.$

Note in particular that the decay rate for the $\ell=1$ and $\ell=2$ mode is
the same, in contrast with the $a=0$ case.

####  1.4.3 Step 3: Elliptic theory of time inversion

The pointwise decay rates stated in Sections 1.4.1 and 1.4.2 are (almost)
sharp in the case of initial data for which the conserved Newman–Penrose
charges $I_{\ell}$ are non-zero, with $\ell=0,1,2$. For initial data that
decay faster as $r\to\infty$, such that $I_{\ell}=0$, it is possible to gain
one power in the decay rate.

The idea is the following: we can show that $\psi$ can be thought of as being
a time derivative of another solution to (1.1), i.e.

$T\widetilde{\psi}=\psi,$

for some suitably regular function $\widetilde{\psi}$ such that
$\widetilde{\psi}_{\Sigma_{0}}\to 0$ as $r\to\infty$ and
$\square_{g_{M,a}}\widetilde{\psi}=0$. We denote
$T^{-1}\psi:=\widetilde{\psi}$ and refer to it as the _time inverse_ of
$\psi$.

When the charges $I_{\ell}[\psi]$ are zero, for example, when the initial data
for $\psi$ is compactly supported, it turns out that the charges corresponding
to $\widetilde{\psi}$ are finite, i.e.

$I_{\ell}[\widetilde{\psi}]<\infty$

or $\ell=0,1,2$ and, generically, $I_{\ell}[\widetilde{\psi}]\neq 0$.
Furthermore, we can express $I_{\ell}[\widetilde{\psi}]$ purely in terms of
initial data for $\psi$, see Section 1.1.2 and Section 9.4.

We construct the initial data leading to $T^{-1}\psi$ by solving the equation:

$\mathcal{L}\widetilde{\psi}=F[\psi].$

As explained in Section 1.4.2, the operator $\mathcal{L}$ is elliptic outside
the ergoregion. Constructing $\mathcal{L}^{-1}$ requires the use of spacelike
redshift multipliers and is reminiscent of the strategy for constructing
resolvent operators in [War15]. Note that in the $a=0$ setting, the spherical
modes $\psi_{\ell}$ can be treated independently, so obtaining
$\mathcal{L}^{-1}$ for a single mode amounts to solving a standard ODE, see
[AAG21]. When $a\neq 0$, however, it is neccessary to construct
$\mathcal{L}^{-1}$ acting on the full solution _before_ projecting to a fixed
spherical harmonic mode.

In Section 9, we construct $\mathcal{L}^{-1}$ and established $r$-decay
properties $T^{-1}\psi|_{\Sigma_{0}}$ required to apply the time decay
arguments sketched in the sections when $\psi$ is replaced with $T^{-1}\psi$.

We then obtain in particular the decay estimates:

$\displaystyle|\psi_{0}|=|T\widetilde{\psi}_{0}|\lesssim$
$\displaystyle\>(1+\tau)^{-3+\epsilon},\quad|r\psi_{0}|=|rT\widetilde{\psi}_{0}|\lesssim(1+\tau)^{-2+\epsilon}$
$\displaystyle|r^{-1}\psi_{1}|=|r^{-1}T\widetilde{\psi}_{1}|\lesssim$
$\displaystyle\>(1+\tau)^{-4-k+\epsilon},\quad|r\psi_{1}|=|rT\widetilde{\psi}_{1}|\lesssim(1+\tau)^{-3-k+\epsilon}$
$\displaystyle|\psi_{2}|=|T\widetilde{\psi}_{2}|\lesssim$
$\displaystyle\>(1+\tau)^{-5-k+\epsilon},\quad|r\psi_{2}|=|rT\widetilde{\psi}_{2}|\lesssim(1+\tau)^{-4-k+\epsilon}$
$\displaystyle|\psi_{\geq 3}|=|T\widetilde{\psi}_{\geq 3}|\lesssim$
$\displaystyle\>(1+\tau)^{-\frac{11}{2}+\epsilon},\quad|r\psi_{\geq
3}|=|rT\widetilde{\psi}_{\geq 3}|\lesssim(1+\tau)^{-\frac{9}{2}+\epsilon},$

which are sharp up to arbitrarily small $\epsilon$.

####  1.4.4 Step 4: Late-time asymptotics and inverse polynomial tails

In order to derive the precise leading order terms of $\psi$ and
$\psi_{\geq\ell}$ with $\ell=1,2$, we need to go beyond the _upper bound
estimates_ outlined in the previous sections.

We extend the method developed in [AAG18b] in a spherically symmetric setting
to the setting of Kerr spacetimes. The strategy is to derive the leading-order
asymptotics for $T^{k}\widetilde{\psi}$ (with arbitrary $k$) by: 1) extending
the conservation and non-vanishing property of $r^{2}P_{\ell}$ from $r=\ $ to
a suitable far-away spacetime region to obtain the late-time asymptotics for
$r\psi_{\ell}$ there, and 2) propagating these asymptotics to the rest of the
spacetime by using that

$X^{1+\ell}\widetilde{\psi}_{\ell}\quad\textnormal{and}\quad
TX^{\ell}\widetilde{\psi}_{\ell}$

both decay _faster_ in time than $X^{\ell}\widetilde{\psi}_{\ell}$ when
$\ell=0,1$, $X\widetilde{\psi}_{2}$ and $T\widetilde{\psi}_{2}$ decay faster
than $\widetilde{\psi}_{2}$, so we can simply integrate (multiply times) in
the $X$ direction. This allows us to prove Theorem 1.2. Theorem 1.1 then
follows simply from the observation that $\psi=T\widetilde{\psi}$.

We refer to a more expansive outline of a similar strategy in the spherically
symmetric setting in [AAG18b, AAG21]. The key new difficulty in Kerr in this
step is the coupling between spherical harmonic modes $\psi_{\ell}$. See
Section 1.1.4 for an outline of this phenomenon. More concretely, obtaining
late-time asymptotics in the far-away region for $\psi_{2}$ requires deriving
first the late-time asymptotics of $\psi_{0}$ and plugging those in suitably
in the derivation for $\psi_{2}$. A precise derivation of the late-time
asymptotics for the $\ell=0,1,2$ modes and the effects of mode coupling can be
found in Sections 10–12.

###  1.5 Outline of the paper

We provide below an outline of the remaining sections in the paper.

  * •

In Section 2 we introduce the Kerr geometry and the relevant vector fields and
spacetime foliations. We moreover state a systematic method for integrating by
parts, which we apply to derive all the energy estimates in the paper.
Finally, we state and derive some key properties regarding spherical
harmonics.

  * •

In Section 3 we provide various different forms of the wave equation 1.1 and
state the preliminary energy boundedness and integrated energy decay
estimates, which we utilise in the rest of the paper.

  * •

In Section 4 we define the Newman–Penrose charges for the $\ell=0,1,2$ modes
and derive their conservation properties.

  * •

In Section 5 we derive the main hierarchies of $r^{p}$-weighted energy
estimates.

  * •

In Section 6 we use the $r^{p}$-weighted energy estimates from Section 5 to
derive energy decay estimates.

  * •

In Section 7, we derive a hierarchy of weighted elliptic estimates.

  * •

In Section 8, we convert the (weighted) energy decay estimates into pointwise
decay estimates.

  * •

In Section 9, we construct the time inverse $T^{-1}\psi$ and establish its
decay properties along $\Sigma_{0}$. We moreover define the time-inverted
Newman–Penrose charges and derive explicit expressions for them in terms of
initial data for $\psi$ on $\Sigma_{0}$.

  * •

Finally, in Sections 10–12, we use the pointwise estimates from Section 8,
applied moreover to both $\psi$ and the time inverse $T^{-1}\psi$ constructed
in Section 9, to derive the precise late-time asymptotics for $\psi_{0}$,
$\psi_{1}$ and $\psi_{2}$ and hence determine the precise form of the
corresponding late-time tails.

###  1.6 Acknowledgments

The second author (S.A.) acknowledges support through the NSERC grant 502581
and the Ontario Early Researcher Award.

##  2 Preliminaries: geometry

In this section, we introduce the Kerr family of spacetimes and the spacetime
foliations of interest. We moreover derive a systematic method of integrating
by parts to derive energy estimates, and we prove some relevant properties of
spherical harmonic decompositions.

###  2.1 The Kerr metric

We introduce in this section the $2$-parameter family of Kerr black hole
exteriors $(\mathcal{M}_{M,a},g_{M,a})$ in _ingoing Kerr coordinates_.

Let
$\mathcal{M}_{M,a}=\mathbb{R}_{v}\times[r_{+},\infty)_{r}\times(\mathbb{S}^{2})_{\theta,\varphi_{*}}$
be a manifold-with-boundary equipped with a coordinate chart
$(v,r,\theta,\varphi_{*})$ that is global, excluding the standard degeneration
of spherical coordinates $(\theta,\varphi_{*})$ on $\mathbb{S}^{2}$. Let
$g_{M,a}$ denote the Lorentzian metric:

$\begin{split}g_{M,a}=&-\rho^{-2}\left(\Delta-a^{2}\sin^{2}\theta\right)dv^{2}+2dvdr-4Mar\rho^{-2}\sin^{2}\theta
dvd\varphi_{*}-2a\sin^{2}\theta drd\varphi_{*}+\rho^{2}d\theta^{2}\\\
&+\rho^{-2}((r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta)\sin^{2}\theta
d\varphi_{*}^{2},\end{split}$ (2.1)

where

$\displaystyle\Delta=$ $\displaystyle\>r^{2}-2Mr+a^{2}=(r_{+}-r)(r-r_{-}),$
$\displaystyle r_{+,-}=$ $\displaystyle\>M\pm\sqrt{M^{2}-a^{2}},$
$\displaystyle\rho^{2}=$ $\displaystyle\>r^{2}+a^{2}\cos^{2}\theta.$

We will restrict our considerations to sub-extremal Kerr spacetimes by
assuming that $|a|<M$, with $M>0$.333Extremal Kerr spacetimes correspond to
the subclass of spacetimes satisfying $|a|=M$.

We denote the boundary of $\mathcal{M}_{M,a}$ with
$\mathcal{H}^{+}:=\\{r=r_{+}\\}$ and will refer to $\mathcal{H}^{+}$ as the
_(future) event horizon_ of the spacetime. The level sets
$S^{2}_{v^{\prime},r^{\prime}}=\\{v=v^{\prime},r=r^{\prime}\\}$ are 2-surfaces
diffeomorphic to $\mathbb{S}^{2}$, which are known as the _Boyer–Lindquist
spheres_. Note that it is possible to extend the spacetimes
$(\mathcal{M}_{M,a},g_{M,a})$ smoothly by attaching a _black hole region_
where $r_{-}<r<r_{+}$. This extension will not be necessary in the present
paper.

On the manifold $\mathcal{M}_{M,a}\setminus\mathcal{H}^{+}$, one can
alternatively consider the _standard Boyer–Lindquist coordinates_
$(t,r,\theta,\varphi)$, where

$\displaystyle t=$ $\displaystyle\>v-r_{*},$ $\displaystyle\varphi=$
$\displaystyle\>\varphi_{*}+\int_{r}^{\infty}\frac{a}{\Delta}\,dr^{\prime}\mod
2\pi,$

where $r_{*}:(r_{+},\infty)\to\mathbb{R}$ is a solution to

$\frac{dr_{*}}{dr}=\frac{r^{2}+a^{2}}{\Delta},$

defined uniquely up to a constant.

We introduce also the function
$u:\mathcal{M}_{M,a}\setminus\mathcal{H}^{+}\to\mathbb{R}$, with

$u=v-2r_{*}=t-r_{*}.$

It will be convenient to state moreover the inverse metric $g^{-1}_{M,a}$:

$\begin{split}g^{-1}_{M,a}=&\>a^{2}\rho^{-2}\sin^{2}\theta\partial_{v}\otimes\partial_{v}+\rho^{-2}(r^{2}+a^{2})[\partial_{v}\otimes\partial_{r}+\partial_{r}\otimes\partial_{v}]+\Delta\rho^{-2}\partial_{r}\otimes\partial_{r}\\\
&+a\rho^{-2}[(\partial_{v}+\partial_{r})\otimes\partial_{\varphi_{*}}+\partial_{\varphi_{*}}\otimes(\partial_{v}+\partial_{r})]+\rho^{-2}[\partial_{\theta}\otimes\partial_{\theta}+\sin^{-2}\theta\partial_{\varphi_{*}}\otimes\partial_{\varphi_{*}}].\end{split}$
(2.2)

###  2.2 Vector fields

The following vector fields are Killing vector fields with respect to
$g_{M,a}$:

$\displaystyle T=$ $\displaystyle\>\partial_{v},$ $\displaystyle\Phi=$
$\displaystyle\>\partial_{\varphi_{*}},$ $\displaystyle K=$
$\displaystyle\>T+\upomega_{+}\Phi,$

where $\upomega_{+}=\frac{a}{r_{+}^{2}+a^{2}}$ may be interpreted as the
“angular velocity of the Kerr black hole”. Furthermore, $K$ is tangent to
$\mathcal{H}^{+}$.

It can easily be verified that the vector field $T$ is timelike for all
$(v,r,\theta,\varphi_{*})$ satisfying the condition:

$\Delta-a^{2}\sin^{2}\theta>0.$

The subset $\\{\Delta-a^{2}\sin^{2}\theta<0\\}$, where $T$ fails to be causal
is called the _ergoregion_. Note that

$\\{\Delta-a^{2}\sin^{2}\theta<0\\}\subset\\{r<2M\\}.$

We will introduce the following additional vector fields, which will play an
important role in later analysis:

$\displaystyle Y=$ $\displaystyle\>\partial_{r},$
$\displaystyle\underline{L}=$
$\displaystyle\>-\frac{\Delta}{2(r^{2}+a^{2})}Y,$ $\displaystyle L=$
$\displaystyle\>T+\frac{a}{r^{2}+a^{2}}\Phi-\underline{L}.$

The vector fields $L$ and $\underline{L}$ are null and define the _principal
null directions_.

Note moreover the following identities

$\displaystyle L(v)=$ $\displaystyle\>1,$ $\displaystyle\underline{L}(u)=$
$\displaystyle\>1,$ $\displaystyle g(L,\underline{L})=$
$\displaystyle-\frac{\Delta\rho^{2}}{2(r^{2}+a^{2})^{2}},$
$\displaystyle[\underline{L},L]=$
$\displaystyle\>\frac{ar\Delta}{(r^{2}+a^{2})^{3}}\Phi,$
$\displaystyle\underline{L}(\varphi_{*})=$ $\displaystyle\>0,$ $\displaystyle
L(\varphi_{*})=$ $\displaystyle\>\frac{a}{r^{2}+a^{2}}.$

We introduce moreover the _horizon azimuthal angle_

${\varphi}_{\mathcal{H}^{+}}=\varphi_{*}-\upomega_{+}v,$ (2.3)

which satisfies $K({\varphi}_{\mathcal{H}^{+}})=0$, i.e. it is constant along
the null generators of the future event horizon $\mathcal{H}^{+}$.

Finally, we introduce some notation for angular derivatives. Consider
$\mathbb{S}^{2}$ equipped with standard spherical coordinates
$(\theta,\varphi_{*})$. We denote with $\not{\nabla}_{\mathbb{S}^{2}}$ the
covariant derivative (Levi–Civita connection) on the unit round sphere
$\mathbb{S}^{2}$. The corresponding Laplacian $\not{\Delta}_{\mathbb{S}^{2}}$
takes the following form in $(\theta,\varphi_{*})$ coordinates:

$\not{\Delta}_{\mathbb{S}^{2}}(\cdot)=\frac{1}{\sin\theta}\partial_{\theta}(\sin\theta\partial_{\theta}(\cdot))+\frac{1}{\sin^{2}\theta}\partial_{\varphi_{*}}^{2}(\cdot).$

###  2.3 Foliations and conformal coordinates

It will be convenient to keep track of decay with respect to the coordinate
$r$ via the following notation: let $f:[r_{+},\infty)_{r}\to\mathbb{R}$ and
$k\in\mathbb{Z}$. We write $f=O(r^{-k})$ when there exists a constant $C>0$,
depending only on $M$ and $a$ such that $|f(r)|\leq Cr^{-k}$ for all
$r\in[r_{+},\infty)$. We write $f=O_{N}(r^{-k})$ when moreover
$\frac{d^{n}f}{dx^{n}}=O(r^{-k-n})$ for all $0\leq n\leq N$, where $f$ is
assumed to be suitably regular. Finally, we write $f=O_{\infty}(r^{-k})$ if
$f=O_{N}(r^{-k})$ for all $N\in\mathbb{N}_{0}$.

Consider the following _time function_ on $\mathcal{M}_{M,a}$:

$\tau=v-\int_{r_{+}}^{r}h(r^{\prime})\,dr^{\prime}-v_{0},$

where $v_{0}\in\mathbb{R}_{>0}$ and $h:[r_{+},\infty)\to\mathbb{R}$ is a
smooth, non-negative function, such that

$\displaystyle\frac{2(r^{2}+a^{2})}{\Delta}-h(r)=$ $\displaystyle\>O(r^{-2}),$
(2.4) $\displaystyle h(r)\left(\frac{2(r^{2}+a^{2})}{\Delta}-h(r)\right)>$
$\displaystyle\>a^{2}\Delta^{-1}\sin^{2}\theta.$ (2.5)

By construction, $T(\tau)=1$.

Let $\hat{h}:[0,\frac{1}{r_{+}}]_{x}\to\mathbb{R}$, with

$\hat{h}(x)=2(x^{-2}+a^{2})-h(x^{-1})(x^{-2}-2Mx^{-1}+a^{2}).$

Then $\hat{h}(x)=O(x^{0})$ by (2.4), where we take $O(x^{k})$, $O_{N}(x^{k})$
and $O_{\infty}(x^{k})$ to have the same meaning as above with $x$ replacing
$r$.

We will make the following additional assumptions for the sake of convenience:

$\displaystyle\hat{h}\in$ $\displaystyle\>C^{\infty}([0,r_{+}^{-1}]),$ (2.6)
$\displaystyle\hat{h}(0)=:$ $\displaystyle\>h_{0}>0.$ (2.7)

Smoothness of $\hat{h}$ is not a necessary assumption and the main results
below are still valid with a lower regularity assumption on $\hat{h}$, in
particular they hold also when $\hat{h}\in C^{N_{0}}([0,r_{+}^{-1}])$, with
$N_{0}\in\mathbb{N}_{0}$ chosen suitably large.

Define the following 1-parameter family of hypersurfaces:

$\Sigma_{\tau^{\prime}}:=\\{\tau=\tau^{\prime}\\}.$

We denote $\Sigma:=\Sigma_{0}$ and $\mathcal{R}=J^{+}(\Sigma)\cap
J^{-}(\mathcal{H}^{+})$.

Let $R>2M$ be an arbitrarily large radius. Then we introduce the notation:

$D_{\tau_{1}}^{\tau_{2}}:=\bigcup_{\tau_{1}\leq\tau\leq\tau_{2}}\Sigma_{\tau}\cap\\{r\geq
R\\}.$

We moreover denote $N_{\tau}=\Sigma_{\tau}\cap\\{r\geq R\\}$.

Let $\uprho=r|_{\Sigma}$. Then $(\uprho,\theta,\varphi_{*})$ defines a
coordinate chart on $\Sigma$ and $(\tau,\uprho,\theta,\varphi_{*})$ defines a
coordinate chart on $\mathcal{R}$. We moreover introduce the notation:

$X:=(\partial_{\uprho})_{\Sigma_{\tau}}=Y+hT.$

Note that we can express the metric $g_{M,a}$ as follows in
$(\tau,\uprho,\theta,\varphi_{*})$ coordinates:

$\begin{split}g_{M,a}=&-\rho^{-2}\left(\Delta-a^{2}\sin^{2}\theta\right)d\tau^{2}-2\left(\rho^{-2}h(\Delta-a^{2}\sin^{2}\theta)-1\right)d\tau
d\uprho+\left(2\rho^{2}-h(\Delta-a^{2}\sin^{2}\theta)\right)\rho^{-2}hd\uprho^{2}\\\
&-4Mar\rho^{-2}\sin^{2}\theta d\tau
d\varphi_{*}-2(a+2Mar\rho^{-2}h)\sin^{2}\theta d\uprho
d\varphi_{*}+\rho^{2}d\theta^{2}+\rho^{-2}((r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta)\sin^{2}\theta
d\varphi_{*}^{2}.\end{split}$

We further define the function

$s=v-2r_{*}+\int_{r_{+}}^{r}h(r^{\prime})\,dr^{\prime}+v_{0}=\tau-2r_{*}+2\int_{r_{+}}^{r}h(r^{\prime})\,dr^{\prime}+2v_{0}.$

and consider the corresponding level sets

$I_{s^{\prime}}:=\\{s=s^{\prime}\\}\cap\mathcal{R}.$

We can equip the level sets $I_{s}$ with the coordinate chart
$(\tau,\theta,\varphi_{*})$, and we express:

$\begin{split}(\partial_{\tau})_{I_{s}}=&\>T-\frac{1}{2}\frac{1}{h-\frac{r^{2}+a^{2}}{\Delta}}X,\\\
=&\>-\frac{1}{2}\frac{\frac{2(r^{2}+a^{2})}{\Delta}-h}{h-\frac{r^{2}+a^{2}}{\Delta}}T-\frac{1}{2}\frac{\Delta}{r^{2}+a^{2}+(\Delta
h-2(r^{2}+a^{2}))}Y.\end{split}$

See Figure 2 below for the relevant pictorial representations.

Figure 2: A 2-dimensional representation of the spacetime $\mathcal{M}_{M,a}$
and the hypersurfaces $\Sigma_{\tau_{i}}$, $I_{s_{i}}$, with $i=1,2$ and
$\tau_{1}<\tau_{2}$, $s_{1}<s_{2}$. Each point in the picture represents a
Boyer–Lindquist sphere $S^{2}_{v,r}$ and the hypersurface $\mathcal{I}^{+}$ is
depicted at a finite distance.

The lemma below establishes the key causual properties of the hypersurfaces
$\Sigma_{\tau}$, which are also represented pictorially in Figure 2.

######  Lemma 2.1.

The 1-parameter family $\\{\Sigma_{\tau}\\}_{\tau\geq 0}$ and the level sets
$I_{s}$ satisfy the following properties:

  1. 1.

$\Sigma_{\tau}$ and $I_{s}$ (if non-empty) are spacelike for all $\tau,s\geq
0$,

  2. 2.

$\Sigma_{\tau}$ is isometric to $\Sigma_{\tau^{\prime}}$ for all
$\tau,\tau^{\prime}\geq 0$,

  3. 3.

$\Sigma_{\tau}\cap\mathcal{H}^{+}=\\{v=\tau+v_{0}\\}\cap\mathcal{H}^{+}=S^{2}_{\tau+v_{0},r_{+}}$,

  4. 4.

For $v_{0}$ sufficiently large, depending on $h$, there exists $u_{0}>0$ such
that $\Sigma_{0}\subset J^{+}(\\{u=u_{0}\\})$.

###### Proof.

Property 3. follows directly from the definition of $\tau$. Property 2.
follows from the fact that each $\Sigma_{\tau}$ can be obtained from
$\Sigma_{0}$ along the flow corresponding to the Killing vector field $T$.

By (2.2), it follows that

$g^{-1}(d\tau,d\tau)=g^{-1}(dv-hdr,dv-
hdr)=a^{2}\rho^{-2}\sin^{2}\theta-\Delta\rho^{-2}h\left(\frac{2(r^{2}+a^{2})}{\Delta}-h\right),$

so by (2.5), Property 1. must also hold for $\Sigma_{\tau}$. The same
conclusion follows for $I_{s}$:

$g^{-1}(ds,ds)=g^{-1}(dv+(h-2(r^{2}+a^{2})\Delta^{-1})dr,dv+(h-2(r^{2}+a^{2})\Delta^{-1})dr)=a^{2}\rho^{-2}\sin^{2}\theta-\Delta\rho^{-2}h\left(\frac{2(r^{2}+a^{2})}{\Delta}-h\right)<0.$

In order to infer that property 4. holds, we first observe that

$u|_{\Sigma_{0}}(r)=v|_{\Sigma_{0}}(r)-2r_{*}(r)=v_{0}+\int_{r_{+}}^{r}h(r^{\prime})\,dr^{\prime}-2r_{*}(r).$

so
$u|_{\Sigma_{0}}(2r_{+})=v_{0}-\int_{r_{+}}^{2r_{+}}h(r^{\prime})\,dr^{\prime}-2r_{*}(2r_{+})>\frac{1}{2}v_{0}$
for $v_{0}$ suitably large. Then for $r\geq 2r_{+}$

$u|_{\Sigma_{0}}(r)=u|_{\Sigma_{0}}(2r_{+})+\int_{2r_{+}}^{r}(h-\frac{2(r^{\prime
2}+a^{2})}{\Delta})(r^{\prime})\,dr^{\prime}\to
u_{0}\in\mathbb{R}\quad\textnormal{as $r\to\infty$},$

as the integral above is well-defined by the asymptotics in (2.4). For $v_{0}$
suitably large, we can additionally ensure positivity of $u_{0}$. ∎

Let $\mathbf{n}_{\tau}$ denote the future-directed unit normal vector field
with respect to $\Sigma_{\tau}$. We may extend $\mathbf{n}_{\tau}$ as a vector
field on $\mathcal{R}$ and introduce the vector field $N$ as follows via a
smooth cut-off function:

$\displaystyle N=$ $\displaystyle\>\mathbf{n}_{\tau}$
$\displaystyle\textnormal{when $r\leq 3M$},$ (2.8) $\displaystyle N=$
$\displaystyle\>T$ $\displaystyle\textnormal{when $r\geq 4M$}.$ (2.9)

such that $N$ is timelike everywhere.444We used here that $g(T,T)<0$ for
$r>2M$, i.e. outside the ergoregion. It follows moreover that exists a
constant $c=c(M,a)>0$, such that $g(N,N)\leq-c$.

From Lemma 2.1 it follows that $\\{\Sigma_{\tau}\\}_{\tau\geq 0}$ defines a
foliation of $\mathcal{R}$ by isometric spacelike hypersurfaces. We can
express $d\mu$, the natural volume form with respect to $g_{M,a}$, as follows:

$d\mu:=\sqrt{\det g_{M,a}}dvdrd\theta d\varphi_{*}=\rho^{2}d\omega
dvdr=\rho^{2}d\omega d\tau d\uprho,$

with $d\omega=\sin\theta d\theta d\varphi_{*}$ the natural volume form on
$\mathbb{S}^{2}$.

Various estimates on the induced volume forms on $\Sigma_{\tau}$ and $I_{s}$
that are relevant in the context of divergence theorems are contained in
Section 2.1.2 of [DHR19a]. For the sake of completeness and notational
convention, we prove these estimates in Lemma 2.2 below.

######  Lemma 2.2.

The volume forms $d\mu_{\tau}:=\sqrt{\det g|_{\Sigma_{\tau}}}d\uprho d\theta
d\varphi_{*}$ and $d\mu_{I_{s}}:=\sqrt{\det g|_{I_{s}}}d\tau d\theta
d\varphi_{*}$ satisfy the following properties:

  1. (i)

We can express

$\displaystyle d\mu_{\tau}=$ $\displaystyle\>m_{1}(\uprho,\theta)rd\omega
d\uprho,$ $\displaystyle d\mu_{I_{s}}=$
$\displaystyle\>m_{2}(\uprho,\theta)rd\omega d\tau$
$\displaystyle\textnormal{in $\\{r\geq r_{+}\\}$},$

with $m_{i}:[r_{+},\infty)\times(0,\pi)\to\mathbb{R}$, $i=1,2$, smooth
functions satisfying:

$c_{h}\leq m_{i}\leq C_{h},$

for positive constants $c_{h},C_{h}$, depending on $M$, $a$ and the choice of
$h$.

  2. (ii)

We can moreover express:

$\displaystyle-g\left(L,\mathbf{n}_{\tau}\right)\sqrt{\det
g|_{\Sigma_{\tau}}}=$ $\displaystyle\>m_{L,1}(\uprho,\theta)\sin\theta,$
$\displaystyle-g\left(\frac{r^{2}+a^{2}}{\Delta}\underline{L},\mathbf{n}_{\tau}\right)\sqrt{\det
g|_{\Sigma_{\tau}}}=$
$\displaystyle\>m_{\underline{L},1}(\uprho,\theta)r^{2}\sin\theta,$
$\displaystyle-g\left(L,\mathbf{n}_{I_{s}}\right)\sqrt{\det g|_{I_{s}}}=$
$\displaystyle\>m_{L,2}(\uprho,\theta)r^{2}\sin\theta,$
$\displaystyle-g\left(\frac{r^{2}+a^{2}}{\Delta}\underline{L},\mathbf{n}_{I_{s}}\right)\sqrt{\det
g|_{I_{s}}}=$ $\displaystyle\>m_{\underline{L},2}(\uprho,\theta)\sin\theta,$

with $m_{L,i},m_{\underline{L},i}:[r_{+},\infty)\times(0,\pi)\to\mathbb{R}$,
$i=1,2$, smooth functions satisfying:

$c_{h}\leq m_{L,i},m_{\underline{L},i}\leq C_{h},$

for $c_{h},C_{h}>0$.

  3. (iii)

The following divergence identities hold:

$\displaystyle\nabla_{\alpha}\left(\rho^{-2}\frac{r^{2}+a^{2}}{\Delta}L^{\alpha}\right)=$
$\displaystyle\>0,$
$\displaystyle\nabla_{\alpha}\left(\rho^{-2}\frac{r^{2}+a^{2}}{\Delta}\underline{L}^{\alpha}\right)=$
$\displaystyle\>0.$

###### Proof.

Property (i) results from the observation that $g(X,X)\sim r^{-2}$ and
$g((\partial_{\tau})_{I_{s}},(\partial_{\tau})_{I_{s}})\sim r^{-2}$.

In order to obtain property (ii), note first that we can express:

$g^{-1}=g^{-1}(d\tau^{\sharp},d\tau^{\sharp})d\tau^{\sharp}\otimes
d\tau^{\sharp}+g^{-1}|_{\Sigma_{\tau}},$

so we have that with respect to $(\tau,\uprho,\theta,\varphi_{*})$
coordinates:

$\det g=\frac{\det g|_{\Sigma_{\tau}}}{g^{-1}(d\tau^{\sharp},d\tau^{\sharp})}$

and we must therefore have that

$-\sqrt{\det g|_{\Sigma_{\tau}}}\mathbf{n}_{{\tau}}=\rho^{2}\sin\theta
d\tau^{\sharp}.$ (2.10)

Furthermore,

$\displaystyle\rho^{2}d\tau^{\sharp}=$
$\displaystyle\>[a^{2}\sin^{2}\theta-(r^{2}+a^{2})h]T+[r^{2}+a^{2}-\Delta
h]Y+a(1-h)\Phi,$ $\displaystyle\rho^{2}ds^{\sharp}=$
$\displaystyle\>[a^{2}\sin^{2}\theta-(r^{2}+a^{2})\Delta^{-1}\hat{h}(r^{-1})]T+[r^{2}+a^{2}-\hat{h}(r^{-1})]Y+a(1-h)\Phi,$

where $\hat{h}(r^{-1})=2(r^{2}+a^{2})-\Delta h(r)$, so after a straightforward
computation, we obtain

$\displaystyle
g\left(\frac{r^{2}+a^{2}}{\Delta}\underline{L},\rho^{2}d\tau^{\sharp}\right)=$
$\displaystyle\>\frac{1}{2}(r^{2}+a^{2})h+(1-2h)a^{2}\sin^{2}\theta=r^{2}+O(r),$
$\displaystyle g\left(L,\rho^{2}d\tau^{\sharp}\right)=$
$\displaystyle\>\frac{1}{2}\hat{h}(1/r)+O(r^{-1}).$

Similar expressions involving $(ds)^{\sharp}$ follow after replacing $h(r)$
with $\Delta^{-1}\hat{h}(r^{-1})$ and vice versa. Property (ii) then follows
from the assumption that $\hat{h}(0)=h_{0}>0$.

Finally, we obtain property (iii) via a straightforward computation. ∎

We introduce the _conformal radial coordinate_ $x$, defined as follows in
$\mathcal{R}$:

$x=\frac{1}{\uprho}.$

We now define the conformally rescaled metric $\hat{g}_{M,a}$ in the region
$\mathcal{R}=[0,\infty)_{\tau}\times(0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2}$:

$\hat{g}_{M,a}:=r^{-2}{g}_{M,a}=x^{2}{g}_{M,a}.$

Note that

$\begin{split}\hat{g}_{M,a}=&-(x^{2}+O_{\infty}(x^{3}))d\tau^{2}+2(1+O_{\infty}(x))d\tau
dx+O_{\infty}(x^{0})dx^{2}+O_{\infty}(x^{5})\sin^{2}\theta d\tau
d\varphi_{*}-2(a^{2}+O_{\infty}(x))dxd\varphi_{*}\\\
&+(1+O_{\infty}(x^{2}))d\theta^{2}+(1+O_{\infty}(x))\sin^{2}\theta
d\varphi_{*}^{2},\end{split}$

and the above metric components are smooth functions with respect to
$(\tau,x,\theta,\varphi_{*})$. Furthermore, $\hat{g}_{M,a}$ can be smoothly
extended to the manifold-with-boundary:

$\widehat{\mathcal{R}}=[0,\infty)_{\tau}\times[0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2}.$

We refer to the boundary

$\mathcal{I}^{+}=[0,\infty)_{\tau}\times\\{0\\}_{x}\times\mathbb{S}^{2}$

as _future null infinity_. It follows immediately that $\mathcal{I}^{+}$ is a
null hypersurface with respect to $\hat{g}_{M,a}$. Furthermore, it is
straightforward to see that

$x|_{I_{s}}(\tau)\to 0$

as $s\to\infty$.

We moreover denote with

$\widehat{\Sigma}_{\tau}:=\\{\tau\\}_{\tau}\times[0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2}$

the extension of $\Sigma_{\tau}$ to $\widehat{\mathcal{R}}$.

###  2.4 Additional notation

We will denote with $H^{k}(\Sigma_{\tau})$, with $k\in\mathbb{N}_{0}$, the
standard Sobelev spaces with respect to the natural volume form corresponding
to the induced metric $g_{M,a}|_{\Sigma_{\tau}}$. Similarly, will denote with
$H^{k}(\widehat{\Sigma}_{\tau})$, with $k\in\mathbb{N}_{0}$, the standard
Sobelev spaces with respect to the natural volume form corresponding to the
induced conformal metric $\hat{g}_{M,a}|_{\Sigma_{\tau}}$.

Similarly, we use $H^{k}(\mathbb{S}^{2})$ to denote Sobolev spaces of
functions on $\mathbb{S}^{2}$ with respect to the standard volume form.

We will frequently use the notations $c$ and $C$ to indicate constants
appearing at the right-hand side of an inequality. When the notation $C$ or
$c$ appears in an inequality, we will make use of the following “algebra of
constants”:

$C+C=CC=C,\quad c+c=cc=c.$

in order to avoid the introduction of additional notation to denote different
constants in an estimate.

We will use the notation $f\sim g$, with $f,g$ two non-negative definite
expressions to mean:

$cg\leq f\leq Cg,$

with constants $0<c<C$ that depend only on $a$, $M$ and the function $h$ that
determines the foliation $\\{\Sigma_{\tau}\\}$.

###  2.5 Systematic integration by parts

We appeal to the properties of $\Sigma_{\tau}$ and $I_{s}$ established in
Lemma 2.2 to derive a systematic method of integrating by parts; see also
Remark 5.1 of [DHR19a].

######  Lemma 2.3.

Let $\mathcal{F}_{\underline{L}},\mathcal{F}_{L}$ be smooth functions on
$\mathcal{M}_{M,a}$ and let $\mathcal{F}_{\not{\nabla}}$ be a smooth vector
field on $S^{2}_{v,r}$ extended to $\mathcal{M}_{M,a}$, such that the
following identity holds:

$\underline{L}(\mathcal{F}_{\underline{L}})+L(\mathcal{F}_{L})+\not{\textnormal{div}}_{\mathbb{S}^{2}}\mathcal{F}_{\not{\nabla}}+\Phi(F_{\Phi})+\mathcal{J}=0,$
(2.11)

where $\not{\textnormal{div}}_{\mathbb{S}^{2}}$ denotes the divergence
operator with respect to the round metric on $\mathbb{S}^{2}$.

  1. (i)

Then

$\textnormal{div}\,\left[\left(\rho^{-2}\frac{r^{2}+a^{2}}{\Delta}\mathcal{F}_{\underline{L}}\right)\underline{L}+\left(\rho^{-2}\frac{r^{2}+a^{2}}{\Delta}\mathcal{F}_{L}\right)L\right]+\rho^{-2}\frac{r^{2}+a^{2}}{\Delta}\mathcal{J}+\rho^{-2}\frac{r^{2}+a^{2}}{\Delta}(\not{\textnormal{div}}_{\mathbb{S}^{2}}\mathcal{F}_{\not{\nabla}}+\Phi(\mathcal{F}_{\Phi}))=0.$

The following integral identity holds:

$\begin{split}\int_{\Sigma_{\tau_{2}}\cap\\{s\leq
S\\}}&\rho^{-2}\mathcal{F}_{\underline{L}}\sqrt{\det
g_{\Sigma_{\tau}}}g\left(\frac{r^{2}+a^{2}}{\Delta}\underline{L},-\mathbf{n}_{\Sigma_{\tau}}\right)+\rho^{-2}\mathcal{F}_{L}\sqrt{\det
g_{\Sigma_{\tau}}}g\left(\frac{r^{2}+a^{2}}{\Delta}L,-\mathbf{n}_{\Sigma_{\tau}}\right)\,d\uprho
d\theta d\varphi_{*}\\\
&+\int_{\mathcal{H}^{+}\cap\\{\tau_{1}\leq\tau\leq\tau_{2}\\}}\frac{1}{2}\mathcal{F}_{\underline{L}}\,d\omega
d\tau\\\
&+\int_{I_{S}\cap\\{\tau_{1}\leq\tau\leq\tau_{2}\\}}\rho^{-2}\mathcal{F}_{\underline{L}}\sqrt{\det
g_{I_{S}}}g\left(\frac{r^{2}+a^{2}}{\Delta}\underline{L},-\mathbf{n}_{I_{s}}\right)+\rho^{-2}\mathcal{F}_{L}\sqrt{\det
g_{I_{S}}}g\left(\frac{r^{2}+a^{2}}{\Delta}L,-\mathbf{n}_{I_{S}}\right)\,d\tau
d\theta d\varphi_{*}\\\ &+\int_{D^{\tau_{2}}_{\tau_{1}}\cap\\{s\leq
S\\}}\rho^{-2}\frac{r^{2}+a^{2}}{\Delta}\mathcal{J}\,d\mu\\\
=&\>\int_{\Sigma_{\tau_{1}}\cap\\{s\leq
S\\}}\rho^{-2}\mathcal{F}_{\underline{L}}\sqrt{\det
g_{\Sigma_{\tau}}}g\left(\frac{r^{2}+a^{2}}{\Delta}\underline{L},-\mathbf{n}_{\Sigma_{\tau}}\right)+\rho^{-2}\mathcal{F}_{L}\sqrt{\det
g_{\Sigma_{\tau}}}g\left(\frac{r^{2}+a^{2}}{\Delta}L,-\mathbf{n}_{\Sigma_{\tau}}\right)\,d\uprho
d\theta d\varphi_{*}.\end{split}$

  2. (ii)

If $\mathcal{F}_{\underline{L}},\mathcal{F}_{L},\mathcal{F}_{\not{\nabla}}$
are non-negative definite, then we have that:

$\begin{split}\int_{\Sigma_{\tau_{2}}\cap\\{s\leq
S\\}}&\left(\mathcal{F}_{\underline{L}}+\Delta^{-1}\mathcal{F}_{L}\right)\,d\omega
d\uprho+\int_{I_{S}\cap\\{\tau_{1}\leq\tau\leq\tau_{2}\\}}\left(\mathcal{F}_{L}+r^{-2}\mathcal{F}_{\underline{L}}\right)\,d\omega
d\tau+\int_{\tau_{1}}^{\tau_{2}}\left[\int_{\Sigma_{\tau}\cap\\{s\leq
S\\}}\frac{r^{2}+a^{2}}{\Delta}\mathcal{J}\,d\omega d\uprho\right]\,d\tau\\\
&+\int_{\mathcal{H}^{+}\cap\\{\tau_{1}\leq\tau\leq\tau_{2}\\}}\mathcal{F}_{\underline{L}}\,d\omega
d\tau\\\ \sim&\>\int_{\Sigma_{\tau_{1}}\cap\\{s\leq
S\\}}\left(\mathcal{F}_{\underline{L}}+\Delta^{-1}\mathcal{F}_{L}\right)\,d\omega
d\uprho.\end{split}$

###### Proof.

Part (i) follows directly from Stokes’ Theorem combined with (iii) of Lemma
2.2. Part (ii) follows by applying additionally the remaining identities in
Lemma 2.2. ∎

###  2.6 Spherical harmonic modes

Let $\ell\in\mathbb{N}_{0}$ and consider the following projection operators

$\displaystyle\pi_{\ell}:$ $\displaystyle L^{2}(\mathbb{S}^{2})\to
L^{2}(\mathbb{S}^{2}),$ $\displaystyle\pi_{\ell}f=$ $\displaystyle
f_{\ell}:=\sum_{m=-\ell}^{\ell}f_{\ell,m}Y_{\ell,m}(\theta,\varphi_{*}),$

with $f_{\ell,m}\in\mathbb{C}$ and $Y_{\ell,m}(\theta,\varphi_{*})$,
$m=-\ell,\ldots,\ell$ spherical harmonics with angular momentum $\ell$, with
respect to the polar angle $\theta$ and the azimuthal angle $\varphi_{*}$.

Note that

$\sum_{\ell=0}^{\infty}\int_{\mathbb{S}^{2}}f_{\ell}^{2}\,d\omega=\int_{\mathbb{S}^{2}}f^{2}\,d\omega.$

The operator $\pi_{\ell}$ is well-defined on the function space
$C^{\infty}(\Sigma_{\tau})$, where we interpret $\pi_{\ell}$ as acting on the
restrictions of functions in $C^{\infty}(\Sigma_{\tau})$ to functions on the
Boyer–Lindquist spheres foliating $\Sigma_{\tau}$, which we cover with angular
coordinates $(\theta,\varphi_{*})$. Since $\pi_{\ell}$ is a bounded linear
operator with respect to $||\cdot||_{L^{2}(\Sigma_{\tau})}$, the following
extension is also well-defined:

$\pi_{\ell}:L^{2}(\Sigma_{\tau})\to L^{2}(\Sigma_{\tau}).$

We will introduce the following additional notation: let $L\in\mathbb{N}_{0}$
and $f\in L^{2}(\Sigma_{\tau})$, then

$\displaystyle f_{\leq L}:=$ $\displaystyle\>\sum_{\ell=0}^{L}\pi_{\ell}f,$
$\displaystyle f_{\geq L+1}:=$ $\displaystyle\>f-f_{\leq L}.$

We will moreover need to investigate how the projection operators $\pi_{\ell}$
act on product functions of the form $\sin^{2}\theta f$, with $f\in
L^{2}(\mathbb{S}^{2})$.

######  Lemma 2.4.

Let $f\in L^{2}(\mathbb{S}^{2})$. Then there exist numerical constants
$N_{\ell,m,-2},N_{\ell,m,0},N_{\ell,m,+2}$, such that

$\pi_{\ell}(\sin^{2}\theta
f)=\sum_{m=-\ell}^{\ell}(N_{\ell,m,-2}f_{\ell-2,m}+N_{\ell,m,0}f_{\ell,m}+N_{\ell,m,+2}f_{\ell+2,m})Y_{\ell,m}(\theta,\varphi_{*}),$
(2.12)

with $N_{\ell,m,0}$ and $N_{\ell,m,+2}$ non-vanishing and $N_{\ell,m,-2}$ non-
vanishing if and only if $|m|\leq\ell-2$.

In particular,

$\pi_{2}(\sin^{2}\theta\pi_{0}(f))=-\frac{4}{3}\sqrt{\frac{\pi}{5}}\pi_{0}(f)Y_{2,0}(\theta).$
(2.13)

###### Proof.

We can express:

$\pi_{\ell}(\sin^{2}\theta
f)=\sum_{m=-\ell}^{\ell}\left[\sum_{\ell^{\prime}=|m|}^{\infty}f_{\ell^{\prime},m}\int_{\mathbb{S}^{2}}\sin^{2}\theta
Y_{\ell^{\prime},m}\overline{Y}_{\ell,m}\,\sin\theta d\theta
d\varphi\right]Y_{\ell,m}.$

Note that for given $\ell$ and $\ell^{\prime}$, the integral

$\int_{\mathbb{S}^{2}}\sin^{2}\theta
Y_{\ell^{\prime},m}\overline{Y}_{\ell,m}\,\sin\theta d\theta d\varphi=0$

if and only if

$\int_{-1}^{1}(1-x^{2})P^{m}_{\ell^{\prime}}(x)P^{m}_{\ell}(x)\,dx=0,$ (2.14)

where $P^{m}_{\ell}$ and $P^{m}_{\ell^{\prime}}$ denote associated Legendre
polynomials. By the definition of Legendre polynomials, it follows immediately
that the above integral vanishes if $\ell^{\prime}-\ell$ is odd and is non-
vanishing when $\ell=\ell^{\prime}$. It remains to consider the cases when
$\ell^{\prime}-\ell$ is even and non-zero. Without loss of generality, we can
assume that $0\leq m\leq\ell^{\prime}$ and $\ell\geq\ell^{\prime}+2$.

We use the following standard recursive relations between associated Legendre
polynomials (see for example Chapter 8 of [AS72]): let $l\neq 0$, then

$\displaystyle\sqrt{1-x^{2}}P_{l}^{m}=$
$\displaystyle-(2l+1)^{-1}[(l-m+1)(l-m+2)P^{m-1}_{l+1}-(l+m-1)(l+m)P^{m-1}_{l-1}]\quad\textnormal{for
$m\geq 1$},$ $\displaystyle\sqrt{1-x^{2}}P_{l}^{m}=$
$\displaystyle-(2l+1)^{-1}[P^{m+1}_{l+1}-P^{m+1}_{l-1}]\quad\textnormal{for
$m\leq l-2$}.$

to write

$\begin{split}(2\ell+1)(2\ell^{\prime}+1)(1-x^{2})P^{m}_{\ell^{\prime}}P^{m}_{\ell}=&\>\left[(\ell-m+1)(\ell-m+2)P^{m-1}_{\ell+1}-(\ell+m-1)(\ell+m)P^{m-1}_{\ell-1}\right]\\\
\cdot&\left[(\ell^{\prime}-m+1)(\ell^{\prime}-m+2)P^{m-1}_{\ell^{\prime}+1}-(\ell^{\prime}+m-1)(\ell^{\prime}+m)P^{m-1}_{\ell^{\prime}-1}\right],\end{split}$

when $\ell^{\prime}\neq 0$ and $m\geq 1$. For $\ell^{\prime}\neq 0$ and $m=0$,
we obtain instead

$(2\ell+1)(2\ell^{\prime}+1)(1-x^{2})P^{0}_{\ell^{\prime}}P^{0}_{\ell}=[P^{1}_{\ell+1}-P^{1}_{\ell-1}][P^{1}_{\ell^{\prime}+1}-P^{1}_{\ell^{\prime}-1}].$

The integral (2.14) corresponding to the above cases is non-vanishing if and
only if $\ell=\ell^{\prime}$ or $|\ell-\ell^{\prime}|=2$.

It only remains to consider the case: $\ell^{\prime}=m=0$ and $\ell\geq 2$,
for which it follows by integrating by parts that

$\begin{split}\int_{-1}^{1}(1-x^{2})P^{0}_{\ell}P^{0}_{0}\,dx=2^{-\ell}(\ell!)^{-1}\int_{-1}^{1}(1-x^{2})\frac{d^{\ell}}{dx^{\ell}}((x^{2}-1)^{\ell})\,dx\end{split}$

is non-vanishing if and only if $\ell=2$. The expression (2.12) then follows
and (2.13) can easily be computed explicitly by keeping track of the
appropriate normalization constants. ∎

In the lemma below, we state various Poincaré-type inequalities on
$\mathbb{S}^{2}$.

######  Lemma 2.5.

Let $f\in H^{2}(\mathbb{S}^{2})$. Then:

$\displaystyle\int_{\mathbb{S}^{2}}|\not{\nabla}_{\mathbb{S}^{2}}f_{\geq\ell}|^{2}\,d\omega\geq$
$\displaystyle\>\ell(\ell+1)\int_{\mathbb{S}^{2}}f_{\geq\ell}^{2}\,d\omega,$
(2.15)
$\displaystyle\int_{\mathbb{S}^{2}}|\not{\nabla}_{\mathbb{S}^{2}}f_{\ell}|^{2}\,d\omega=$
$\displaystyle\>\ell(\ell+1)\int_{\mathbb{S}^{2}}f_{\ell}^{2}\,d\omega,$
(2.16)
$\displaystyle\int_{\mathbb{S}^{2}}f_{\ell}^{2}+|\not{\nabla}_{\mathbb{S}^{2}}f_{\ell}|^{2}+|\not{\nabla}_{\mathbb{S}^{2}}^{2}f_{\ell}|^{2}\,d\omega\leq$
$\displaystyle\>(1+\ell(\ell+1)+\ell^{2}(\ell+1)^{2})\int_{\mathbb{S}^{2}}f_{\ell}^{2}\,d\omega.$
(2.17)

Furthermore, let $N\in\mathbb{N}$ and assume that $f\in
H^{\max\\{N,2\\}}(\mathbb{S}^{2})$. Then there exists constants $0<c<C$,
depending on $N$, such that

$c\int_{\mathbb{S}^{2}}|\not{\nabla}_{\mathbb{S}^{2}}^{s}\not{\Delta}_{\mathbb{S}^{2}}^{\frac{N-s}{2}}f|^{2}\,d\omega\leq\sum_{n=0}^{N}\int_{\mathbb{S}^{2}}|\not{\nabla}_{\mathbb{S}^{2}}^{n}f|^{2}\,d\omega\leq
C\int_{\mathbb{S}^{2}}|\not{\nabla}_{\mathbb{S}^{2}}^{s}\not{\Delta}_{\mathbb{S}^{2}}^{\frac{N-s}{2}}f|^{2}\,d\omega,$
(2.18)

where $s=0$ if $N$ is even and $s=1$ if $N$ is odd.

###### Proof.

The inequalities (2.15)–(2.17) follow by decomposing
$f_{\geq\ell}=\sum_{\ell^{\prime}=\ell}^{\infty}f_{\ell^{\prime}}$, using that
$\not{\Delta}_{\mathbb{S}^{2}}f_{\ell}=-\ell(\ell+1)f_{\ell}$ and integrating
by parts on $\mathbb{S}^{2}$.

We obtain (2.18) by integrating by parts with respect to covariant derivation
on $\mathbb{S}^{2}$. ∎

###  2.7 Hardy inequalities along hyperboloidal hypersurfaces

Throughout this paper we will frequently appeal to Hardy inequalities, in the
form of the estimates in Lemma 2.6 below, to estimate lower-order weighted
derivatives in terms of higher-order weighted derivatives.

######  Lemma 2.6.

Let $0\leq a<b$ and let $f\in C^{1}([a,b])$, then for
$p\in\mathbb{R}\setminus\\{-1\\}$:

$\displaystyle\int_{a}^{b}x^{p}f^{2}(x)\,dx\leq$
$\displaystyle\>4(p+1)^{-2}\int_{a}^{b}x^{p+2}\left(\frac{df}{dx}\right)^{2}(x)\,dx+2(p+1)^{-1}\left[(x^{p+1}f^{2})(b)-(x^{p+1}f^{2})(a)\right].$
(2.19)

Let $r_{+}\leq r_{1}<r_{2}$ and let $f\in C^{1}(\Sigma_{\tau})$, then for
$p\in\mathbb{R}\setminus\\{-1\\}$:

$\begin{split}\int_{\Sigma_{\tau}\cap\\{r_{1}\leq\uprho\leq
r_{2}\\}}r^{p}f^{2}\,d\omega
d\uprho\leq&\>4(p+1)^{-2}\int_{\Sigma_{\tau}\cap\\{r_{1}\leq\uprho\leq
r_{2}\\}}r^{p+2}(Xf)^{2}\,d\omega d\uprho\\\
&+2(p+1)^{-1}\left[\int_{S^{2}_{\tau,r_{2}}}r_{2}^{p+1}f^{2}\,d\omega-\int_{S^{2}_{\tau,r_{1}}}r_{1}^{p+1}f^{2}\,d\omega\right].\end{split}$
(2.20)

And for $r_{+}<r_{1}<r_{2}$ and $f\in C^{1}(\mathcal{R})$, there exists a
constant $C=C(M,a,r_{1})>0$ such that

$\begin{split}\int_{\Sigma_{\tau}\cap\\{r_{1}\leq\uprho\leq
r_{2}\\}}r^{p}f^{2}\,d\omega
d\uprho\leq&\>C(p+1)^{-2}\int_{\Sigma_{\tau}\cap\\{r_{1}\leq\uprho\leq
r_{2}\\}}r^{p+2}(Lf)^{2}+r^{p-2}(Tf)^{2}+r^{p-2}(\Phi f)^{2}\,d\omega
d\uprho\\\
&+2(p+1)^{-1}\left[\int_{S^{2}_{\tau,r_{2}}}r_{2}^{p+1}f^{2}\,d\omega-\int_{S^{2}_{\tau,r_{1}}}r_{1}^{p+1}f^{2}\,d\omega\right].\end{split}$
(2.21)

###### Proof.

The estimate (2.19) follows from integrating
$\frac{d}{dx}\left(x^{p+1}f^{2})\right)$ and applying the fundamental theorem
of calculus and then immediately implies (2.20). We conclude (2.21) from
(2.20) combined with the following relation between $L$ and $X$:

$L=\frac{\Delta}{2(r^{2}+a^{2})}X+O(r^{-2})T+O(r^{-2})\Phi.$

∎

##  3 Preliminaries: wave equation

We recall and derive in this section some preliminary estimates for the
geometric wave equation on Kerr spacetimes that will form important
ingredients for the estimates in subsequent sections.

###  3.1 Main equations

In the lemma below we represent the wave equation $\square_{g_{M,a}}\psi=0$ in
terms of the vector fields $T,Y,\Phi,X,L$ and $\underline{L}$.

######  Lemma 3.1.

Let $\psi\in C^{\infty}(\mathcal{R}\to\mathbb{C})$ be a solution to

$\square_{g_{M,a}}\psi:=\frac{1}{\sqrt{\det
g_{M,a}}}\partial_{\alpha}\left(\sqrt{\det
g_{M,a}}(g_{M,a}^{-1})^{\alpha\beta}\partial_{\beta}\psi\right)=0.$ (3.1)

Then $\psi$ satisfies the following equation:

$0=\>a^{2}\sin^{2}\theta T^{2}\psi+2(r^{2}+a^{2})TY\psi+Y(\Delta
Y\psi)+2aT\Phi\psi+2aY\Phi\psi+2rT\psi+\not{\Delta}_{\mathbb{S}^{2}}\psi.$
(3.2)

We can reformulate (3.2) as follows:

$\begin{split}X(\Delta
X\psi)+2aX\Phi\psi+\not{\Delta}_{\mathbb{S}^{2}}\psi=&\>2[h\Delta-(r^{2}+a^{2})]XT\psi+[(\Delta
h)^{\prime}-2r]T\psi\\\
&+[2h(r^{2}+a^{2})-h^{2}\Delta-a^{2}\sin^{2}\theta]T^{2}\psi+2a(h-1)T\Phi\psi.\end{split}$
(3.3)

Furthermore, the rescaled quantity $\phi:=\sqrt{r^{2}+a^{2}}\psi$ satisfies
the equations:

$\displaystyle 0=$ $\displaystyle\>a^{2}\sin^{2}\theta
T^{2}\phi+2aT\Phi\phi+2aY\Phi\phi+2(r^{2}+a^{2})TY\phi+(r^{2}+a^{2})Y\left(\frac{\Delta}{r^{2}+a^{2}}Y\phi\right)+\not{\Delta}_{\mathbb{S}^{2}}\phi$
(3.4)
$\displaystyle-2a\frac{r}{r^{2}+a^{2}}\Phi\phi-\sqrt{r^{2}+a^{2}}\frac{d}{dr}(r(r^{2}+a^{2})^{-3/2}\Delta)\phi,$
$\displaystyle 4\underline{L}L\phi=$
$\displaystyle\>a^{2}\sin^{2}\theta\frac{\Delta}{(r^{2}+a^{2})^{2}}T^{2}\phi+2a\frac{\Delta}{(r^{2}+a^{2})^{2}}T\Phi\phi+\frac{\Delta}{(r^{2}+a^{2})^{2}}\not{\Delta}_{\mathbb{S}^{2}}\phi+2a\frac{r\Delta}{(r^{2}+a^{2})^{3}}\Phi\phi$
(3.5)
$\displaystyle-\frac{1}{2r}\frac{d}{dr}\left(r^{2}\Delta^{2}(r^{2}+a^{2})^{-3}\right)\phi.$

###### Proof.

The above identities follow from straightforward computations that use the
expressions for $g_{M,a}$ and $g_{M,a}^{-1}$ in (2.1) and (2.2), respectively,
and the definitions of the vector fields $T,\Phi,Y,X,L,\underline{L}$ in
Section 2.2. ∎

By rewriting (3.3) in terms of the conformal coordinates
$(\tau,x,\theta,\varphi_{*})$, we obtain an equation for $\phi$ (defined in
the statement of Lemma 3.1) with smooth coefficients with respect to the
differentiable structure on $\widehat{\mathcal{R}}$:

######  Corollary 3.2.

Let $\psi\in C^{\infty}(\mathcal{R}\to\mathbb{C})$ be a solution to (3.1).
Then $\phi$ satisfies the following equation in $(\tau,x,\theta,\varphi_{*})$
coordinates:

$\begin{split}0=&\>(1+a^{2}x^{2})\partial_{x}\left(\frac{\Delta}{(r^{2}+a^{2})}x^{2}\partial_{x}\phi\right)+2[1+a^{2}x^{2}-x^{2}\hat{h}(x)]\partial_{\tau}\partial_{x}\phi+\left[-\frac{2(r^{2}+a^{2})-\hat{h}}{\Delta}\hat{h}+a^{2}\sin^{2}\theta\right]\partial_{\tau}^{2}\phi+\not{\Delta}_{\mathbb{S}^{2}}\phi\\\
&+2a\left[1-\frac{2(r^{2}+a^{2})-\hat{h}}{\Delta}\right]\partial_{\tau}\partial_{\varphi_{*}}\phi-2ax^{2}\partial_{x}\partial_{\varphi_{*}}\phi-(1+a^{2}x^{2})\frac{d}{dx}\left(\frac{\hat{h}x^{2}}{1+a^{2}x^{2}}\right)\partial_{\tau}\phi-\frac{2ax}{1+a^{2}x^{2}}\partial_{\varphi_{*}}\phi\\\
&+(1+a^{2}x^{2})^{\frac{1}{2}}\frac{d}{dx}\left(\frac{(1-2Mx+a^{2})}{(1+a^{2}x^{2})^{\frac{3}{2}}}\right)\phi.\end{split}$
(3.6)

######  Remark 3.1.

When viewed as a function on the extended manifold $\widehat{\mathcal{R}}$,
the restriction $\phi|_{\mathcal{I}^{+}}$ is known as _the (Friedlander)
radiation field_.

We introduce the following higher-order quantities in $\\{r>r_{+}\\}$:

$\phi^{(n)}=(2(r^{2}+a^{2})^{2}\Delta^{-1}L)^{n}\phi.$

Observe that we can alternatively express
$\phi^{(n)}=(-1)^{n}(\partial_{x})^{n}\phi$ in $(u,x,\theta,\varphi_{*})$
coordinates, so $\phi^{(n)}$ may be considered natural higher-order analogues
of $\phi$ from the point of view of the conformal spacetime
$(\widehat{\mathcal{R}},\hat{g}_{M,a})$.

######  Proposition 3.3.

Let $\psi\in C^{\infty}(\mathcal{R}\to\mathbb{C})$ be a solution to (3.1).
Then $\phi^{(n)}$ satisfies the following equation

$\begin{split}4\underline{L}L\phi^{(n)}=&\>a^{2}\sin^{2}\theta\frac{\Delta}{(r^{2}+a^{2})^{2}}T^{2}\phi^{(n)}+2a\frac{\Delta}{(r^{2}+a^{2})^{2}}T\Phi\phi^{(n)}+2(1+2n)a\frac{r\Delta}{(r^{2}+a^{2})^{3}}\Phi\phi^{(n)}\\\
&+\Delta(r^{2}+a^{2})^{-2}\not{\Delta}_{\mathbb{S}^{2}}\phi^{(n)}+\Delta(r^{2}+a^{2})^{-2}[n(n+1)+O_{\infty}(r^{-1})]\phi^{(n)}-[4nr^{-1}+O_{\infty}(r^{-2})]L\phi^{(n)}\\\
&+a\sum_{k=0}^{n-1}O_{\infty}(r^{-2})\Phi\phi^{(k)}+n\sum_{k=0}^{n-1}O_{\infty}(r^{-2})\phi^{(k)}.\end{split}$
(3.7)

###### Proof.

We will prove (3.7) by induction. First, note that the $n=0$ case follows from
(3.5). Now suppose (3.7) holds for some $n\in\mathbb{N}$. Then

$\begin{split}4\underline{L}L\phi^{(n+1)}=&\>4L\left(\underline{L}\left(2\Delta^{-1}(r^{2}+a^{2})^{2}L\phi^{(n)}\right)\right)+4[\underline{L},L]\phi^{(n+1)}\\\
=&\>8L\left(\Delta^{-1}(r^{2}+a^{2})^{2}\underline{L}L\phi^{(n)}\right)-4L\left(\frac{d}{dr}(\Delta^{-1}(r^{2}+a^{2})^{2})\Delta(r^{2}+a^{2})^{-1}L\phi^{(n)}\right)\\\
&+4\frac{ar\Delta}{(r^{2}+a^{2})^{3}}\Phi\phi^{(n+1)}.\end{split}$

By applying (3.7), we obtain

$\begin{split}8L\left(\Delta^{-1}(r^{2}+a^{2})^{2}\underline{L}L\phi^{(n)}\right)=&\>a^{2}\sin^{2}\theta\frac{\Delta}{(r^{2}+a^{2})^{2}}T^{2}\phi^{(n+1)}+2a\frac{\Delta}{(r^{2}+a^{2})^{2}}T\Phi\phi^{(n+1)}+\Delta(r^{2}+a^{2})^{-2}\not{\Delta}_{\mathbb{S}^{2}}\phi^{(n+1)}\\\
&+4(1-2n)a\frac{r\Delta}{(r^{2}+a^{2})^{2}}\Phi\phi^{(n+1)}+\Delta(r^{2}+a^{2})^{-2}[n(n+1)+O_{\infty}(r^{-1})]\phi^{(n+1)}\\\
&-[4nr^{-1}+O_{\infty}(r^{-2})]L\phi^{(n+1)}\\\
&+a\sum_{k=0}^{n}O_{\infty}(r^{-2})\Phi\phi^{(k)}+(n+1)\sum_{k=0}^{n}O_{\infty}(r^{-2})\phi^{(k)}.\end{split}$

Furthermore,

$\begin{split}-4L\left(\frac{d}{dr}(\Delta^{-1}(r^{2}+a^{2})^{2})\Delta(r^{2}+a^{2})^{-1}L\phi^{(n)}\right)=&\>[-4r^{-1}+O_{\infty}(r^{-2})]L\phi^{(n+1)}+[2r^{-2}+O_{\infty}(r^{-3})]\phi^{(n)}\end{split}$

By combining the above equations, we arrive at (3.7) with $n$ replaced by
$n+1$, which concludes the induction argument. ∎

###  3.2 Propagation of regularity and limits at null infinity

It will be convenient to appeal to a propagation of regularity result for the
wave equation (3.1) with respect to the differentiable structure on both
$\mathcal{R}$ and $\widehat{\mathcal{R}}$.

######  Proposition 3.4.

Let $k\in\mathbb{N}_{0}$.

  1. (i)

Let $(\Psi,\Psi^{\prime})\in H^{k+1}_{\rm loc}(\Sigma_{0})\times H^{k}_{\rm
loc}(\Sigma_{0})$. Then there exists a solution $\psi$ to (3.1) (in the
distributional sense), such that for all $T\geq 0$:

$\displaystyle\psi\in$ $\displaystyle\>C^{0}([0,T],H^{k+1}_{\rm
loc}(\Sigma_{0}))\cap C^{1}([0,T],H^{k}_{\rm loc}(\Sigma_{0})),$
$\displaystyle\psi|_{\Sigma_{0}}=$ $\displaystyle\>\Psi,$ $\displaystyle
T\psi|_{\Sigma_{0}}=$ $\displaystyle\>\Psi^{\prime}.$

  2. (ii)

Assume moreover that
$(\sqrt{r^{2}+a^{2}}\Psi,\sqrt{r^{2}+a^{2}}\Psi^{\prime})\in
H^{k+1}(\widehat{\Sigma}_{0})\times H^{k}(\widehat{\Sigma}_{0})$. Then, for
all $\tau\geq 0$,

$(\phi|_{\Sigma_{\tau}},T\phi|_{\Sigma_{\tau}})\in
H^{k+1}(\widehat{\Sigma}_{\tau})\times H^{k}(\widehat{\Sigma}_{\tau}).$ (3.8)

  3. (iii)

Let $(\Psi,\Psi^{\prime})\in C^{\infty}(\Sigma_{0})\times
C^{\infty}(\Sigma_{0})$. Assume moreover that

$(\sqrt{r^{2}+a^{2}}\Psi,\sqrt{r^{2}+a^{2}}\Psi^{\prime})\in
H^{N+1}(\widehat{\Sigma}_{0})\times H^{N}(\widehat{\Sigma}_{0})$

for some $N\in\mathbb{N}_{0}$. Then for all $\tau\geq 0$ and
$k,m,n\in\mathbb{N}_{0}$, such that $k+m+n\leq N$,

$\int_{\mathbb{S}^{2}}|\not{\nabla}_{\mathbb{S}^{2}}^{k}\partial_{\tau}^{m}\phi^{(n)}|^{2}(\tau,\uprho,\theta,\varphi)\,d\omega$

attains a finite limit as $\uprho\to\infty$.

###### Proof.

Part (i) follows from standard result for linear wave equations, see for
example [Sog08]. In order to obtain part (ii), we consider (3.6) and observe
that the coefficients in the equation are smooth functions of $(x,\theta)$
(note in particular the smoothness in $r$.) Therefore, (3.8) follows from
standard (finite-in-time) higher-order energy estimates with respect to
$(\tau,x,\theta,\varphi_{*})$ coordinates.

By the fundamental theorem of calculus it follows that for $0<x_{0}\leq
r_{+}^{-1}$ and $x\leq x_{0}$

$\begin{split}\int_{\mathbb{S}^{2}}&|\not{\nabla}_{\mathbb{S}^{2}}^{k_{1}}\partial_{\tau}^{k_{2}}\partial_{x}^{k_{3}}\phi|^{2}(\tau,x,\theta,\varphi)\,d\omega=\int_{\mathbb{S}^{2}}|\not{\nabla}_{\mathbb{S}^{2}}^{k_{1}}\partial_{\tau}^{k_{2}}\partial_{x}^{k_{3}}\phi|^{2}(\tau,x_{0},\theta,\varphi)\,d\omega\\\
&-2\int_{x}^{x_{0}}\not{\nabla}_{\mathbb{S}^{2}}^{k_{1}}\partial_{\tau}^{k_{2}}\partial_{x}^{k_{3}}\phi\cdot\not{\nabla}_{\mathbb{S}^{2}}^{k_{1}}\partial_{\tau}^{k_{2}}\partial_{x}^{k_{3}+1}\phi(\tau,x^{\prime},\theta,\varphi)\,d\omega
dx^{\prime}.\end{split}$

Observe that the first integral on the right-hand side is finite by smoothness
of $\psi$ with respect to the differentiable structure on $\mathcal{R}$, which
in turn follows from the smoothness of $\Psi$ and $\Psi^{\prime}$, together
with a standard propagation of regularity argument; see again [Sog08].

Then, after applying Cauchy–Schwarz to the second term on the right-hand side
of the equation above, together with (3.8) and (3.6), it follows that the
limit of the left-hand side at $x=0$ is well-defined provided $k_{1}+k_{2}\leq
N$. Since $\not{\nabla}_{\mathbb{S}^{2}}^{k}\partial_{\tau}^{m}\phi^{(n)}$,
with $k+m+n\leq N$, can be written as a linear combination of
$\not{\nabla}_{\mathbb{S}^{2}}^{k_{1}}\partial_{\tau}^{k_{2}}\partial_{x}^{k_{3}}\phi$
with $k_{1}+k_{2}+k_{3}\leq N$, we obtain part (iii). ∎

Throughout the remainder of the paper we will assume for the sake of
convenience that the initial data for (3.1) satisfy:

$(\sqrt{r^{2}+a^{2}}\Psi,\sqrt{r^{2}+a^{2}}\Psi^{\prime})\in
C^{\infty}(\widehat{\Sigma}_{0})\times C^{\infty}(\widehat{\Sigma}_{0}),$

though this assumption can be significantly weakened by a standard density
argument with respect to the (weighted) energy norms in the relevant
estimates.

###  3.3 Energy boundedness and integrated local energy decay

Two key ingredients towards deriving the results in the present article are
_energy boundedness_ and _integrated local energy decay_. In the context of
subextremal Kerr spacetimes with $|a|<M$, these were both derived in [DRSR16].
We summarize the relevant results as Theorem A below.

######  Theorem A (Dafermos–Rodnianski–Shlapentokh-Rothman, [DRSR16]).

Let $\psi$ be a solution to (3.1) arising from suitably regular initial data
with respect to $\widehat{\mathcal{R}}$. Then the following estimates hold:

  1. (i)

There exists a constant $C=C(M,a)>0$, such that for any
$\tau_{2}\geq\tau_{1}\geq 0$:

$\int_{\Sigma_{\tau_{2}}}\mathbf{J}^{N}[\psi]\cdot\mathbf{n}_{\tau_{2}}\,r^{2}d\omega
d\uprho\leq
C\int_{\Sigma_{\tau_{1}}}\mathbf{J}^{N}[\psi]\cdot\mathbf{n}_{\tau_{1}}\,r^{2}d\omega
d\uprho.$ (3.9)

  2. (ii)

Let $n\in\mathbb{N}_{0}$. For $R_{0}>0$ suitably large and $R_{1}>0$, there
exists a constant $C=C(n,R_{0},R_{1},M,a)>0$, such that for any
$\tau_{2}\geq\tau_{1}\geq 0$:

$\sum_{0\leq k_{1}+k_{2}+k_{3}\leq
n}\int_{\tau_{1}}^{\tau_{2}}\left[\int_{\Sigma_{\tau}\cap\\{R_{0}\leq r\leq
R_{1}\\}}|\not{\nabla}_{\mathbb{S}^{2}}^{k_{1}}T^{k_{2}}Y^{k_{3}}\psi|^{2}d\omega
d\uprho\right]\,d\tau\leq C\sum_{m\leq
n}\int_{\Sigma_{\tau_{1}}}\mathbf{J}^{N}[T^{m}\psi]\cdot\mathbf{n}_{\tau_{1}}\,r^{2}d\omega
d\uprho.$ (3.10)

Furthermore, for all $\delta>0$, there exists $C=C(n,M,a)>0$, such that

$\int_{D^{\tau_{2}}_{\tau_{1}}}r^{-1-\delta}\left[(\underline{L}\phi)^{2}+(L\phi)^{2}+r^{-2}\phi^{2}+r^{-2}|\not{\nabla}_{\mathbb{S}^{2}}\phi|^{2}\right]\,d\omega
d\rho d\tau\leq
C\int_{\Sigma_{\tau_{1}}}\mathbf{J}^{N}[\psi]\cdot\mathbf{n}_{\tau_{1}}\,r^{2}d\omega
d\uprho.$ (3.11)

  3. (iii)

Let $n\in\mathbb{N}_{0}$. For $R_{0}>0$ arbitrarily large, there exists a
constant $C=C(n,R_{0},M,a)>0$, such that for any $\tau_{2}\geq\tau_{1}\geq 0$:

$\sum_{0\leq k_{1}+k_{2}+k_{3}\leq
n}\int_{\tau_{1}}^{\tau_{2}}\left[\int_{\Sigma_{\tau}\cap\\{r\leq
R_{0}\\}}|\not{\nabla}_{\mathbb{S}^{2}}^{k_{1}}T^{k_{2}}Y^{k_{3}}\psi|^{2}d\omega
d\uprho\right]\,d\tau\leq C\sum_{k_{1}+k_{2}\leq
n+1}\int_{\Sigma_{0}}\mathbf{J}^{N}[T^{k_{1}}\Phi^{k_{2}}\psi]\cdot\mathbf{n}_{\tau}\,r^{2}d\omega
d\uprho.$ (3.12)

We note moreover that by definition of $\Sigma_{\tau}$ in Section 2.3, and the
choice $h_{0}\neq 0$ in particular, we can estimate

$\mathbf{J}^{N}[\psi]\cdot\mathbf{n}_{\tau}\sim(X\psi)^{2}+r^{-2}(T\psi)^{2}+r^{-2}|\not{\nabla}_{\mathbb{S}^{2}}\psi|^{2}.$

It follows immediately by combining (3.9) and (3.10) and applying (2.20) with
$p=0$ that for all $0\leq\tau_{1}\leq\tau_{2}$:

$\int_{N_{\tau_{2}}}(L\phi)^{2}+r^{-2}(\underline{L}\phi)^{2}+r^{-2}|\not{\nabla}_{\mathbb{S}^{2}}\phi|^{2}\,d\omega
d\rho+\int_{I_{s}\cap
D^{\tau_{2}}_{\tau_{1}}}(\underline{L}\phi)^{2}+r^{-2}(L\phi)^{2}+r^{-2}|\not{\nabla}_{\mathbb{S}^{2}}\phi|^{2}\,d\omega
d\rho\leq
C\int_{\Sigma_{\tau_{1}}}\mathbf{J}^{N}[\psi]\cdot\mathbf{n}_{\tau_{1}}\,r^{2}d\omega
d\uprho.$ (3.13)

In order to obtain more refined energy decay estimates for the projections
$\psi_{\geq\ell}$, we need additional integrated local energy decay estimates
for $\psi_{\geq\ell}$.

######  Proposition 3.5.

Let $\psi$ be a solution to (3.1) arising from suitably regular initial data
with respect to $\widehat{\mathcal{R}}$. Then the following estimates hold for
$\ell\in\mathbb{N}$:

  1. (i)

For $R_{0}>2M$ suitably large, there exists a constant $C=C(M,a,R_{0})>0$,
such that for any $\tau_{2}\geq\tau_{1}\geq 0$:

$\begin{split}\int_{\Sigma_{\tau_{2}}}&J^{N}[\psi_{\geq\ell}]\cdot\mathbf{n}_{\tau_{2}}\,r^{2}d\omega
d\uprho\\\ &+\int_{\tau_{1}}^{\infty}\left[\int_{\Sigma_{\tau}\cap\\{r\geq
R_{0}\\}}M^{\delta}r^{1-\delta}[(T\psi_{\geq\ell})^{2}+(Y\psi_{\geq\ell})^{2}]+r^{-1}|\not{\nabla}_{\mathbb{S}^{2}}\psi_{\geq\ell}|^{2}+M^{\delta}r^{-1-\delta}\psi_{\geq\ell}^{2}\,d\omega
d\uprho\right]d\tau\\\
\leq&\>Ca^{2}\int_{\tau_{1}}^{\infty}\left[\int_{\Sigma_{\tau}}M^{-\delta}r^{-1+\delta}\left[(T^{2}\psi_{\geq\max\\{\ell-2,0\\}})^{2}+M^{2}(T^{3}\psi_{\geq\max\\{\ell-2,0\\}})^{2}+(T^{2}\Phi\psi_{\geq\max\\{\ell-2,0\\}})^{2}\right]\,d\omega
d\uprho\right]d\tau\\\
&+C\int_{\Sigma_{\tau_{1}}}\left[\mathbf{J}^{N}[\psi_{\geq\ell}]\cdot\mathbf{n}_{\tau_{1}}+a^{2}\mathbf{J}^{N}[T^{2}\psi_{\geq\max\\{0,\ell-2\\}}]\cdot\mathbf{n}_{\tau_{1}}\right]\,r^{2}d\omega
d\uprho.\end{split}$ (3.14)

  2. (ii)

There exists a constant $C=C(M,a)>0$, such that for any
$\tau_{2}\geq\tau_{1}\geq 0$:

$\begin{split}\int_{\tau_{1}}^{\infty}&\left[\int_{\Sigma_{\tau}}M^{\delta}r^{1-\delta}[(T\psi_{\geq\ell})^{2}+(Y\psi_{\geq\ell})^{2}]+r^{-1}|\not{\nabla}_{\mathbb{S}^{2}}\psi_{\geq\ell}|^{2}+M^{\delta}r^{-1-\delta}\psi_{\geq\ell}^{2}\,d\omega
d\uprho\right]d\tau\\\ \leq&\>Ca^{2}\sum_{k_{1}+k_{2}\leq
2}\int_{\tau_{1}}^{\tau_{2}}\left[\int_{\Sigma_{\tau}}M^{-\delta}r^{-1+\delta}(T^{2+k_{1}}\Phi^{k_{2}}\psi_{\geq\max\\{\ell-2,0\\}})^{2}\,d\omega
d\uprho\right]d\tau\\\ &+C\sum_{k_{1}+k_{2}\leq
1}\int_{\Sigma_{\tau_{1}}}M^{k_{1}}\mathbf{J}^{N}[T^{k_{1}}\Phi^{k_{2}}\psi_{\geq\ell}]\cdot\mathbf{n}_{\tau_{1}}+a^{2}M^{k_{1}}\mathbf{J}^{N}[T^{2+k_{1}}\Phi^{k_{2}}\psi_{\geq\max\\{0,\ell-2\\}}]\cdot\mathbf{n}_{\tau_{1}}\,r^{2}d\omega
d\uprho.\end{split}$ (3.15)

  3. (iii)

Let $n\in\mathbb{N}_{0}$. For $R_{0}>2M$ suitably large and $R_{1}>R_{0}$,
there exists a constant $C=C(M,a,n,R_{0},R_{1})>0$, such that for any
$\tau_{1}\geq 0$:

$\begin{split}&\sum_{0\leq k_{1}+k_{2}+k_{3}\leq
n}\int_{\tau_{1}}^{\infty}\left[\int_{\Sigma_{\tau}\cap\\{R_{0}\leq r\leq
R_{1}\\}}|\not{\nabla}_{\mathbb{S}^{2}}^{k_{1}}T^{k_{2}}Y^{k_{3}}\psi_{\geq\ell}|^{2}d\omega
d\uprho\right]\,d\tau\\\
\leq&\>C\sum_{k=0}^{n}\left[\int_{\Sigma_{\tau_{1}}}J^{N}[T^{k}\psi_{\geq\ell}]\cdot\mathbf{n}_{\tau_{1}}\,r^{2}d\omega
d\uprho+\sum_{\begin{subarray}{c}1\leq l\leq\lceil\ell/2\rceil\\\ 0\leq
l_{1}+l_{2}\leq
l\end{subarray}}a^{2}\int_{\Sigma_{\tau_{1}}}J^{N}[T^{k+2l}T^{l_{1}}\Phi^{l_{2}}\psi_{\geq\max\\{\ell-2l,0\\}}\cdot\mathbf{n}_{\tau_{1}}\,r^{2}d\omega
d\uprho\right].\end{split}$ (3.16)

###### Proof.

Let $\psi$ be a solution to (3.1) arising from suitably regular initial data
with respect to $\widehat{\mathcal{R}}$. Then $\psi_{\geq\ell}$ satisfies the
following inhomogeneous equation:

$\rho^{2}\square_{g}\psi_{\geq\ell}=a^{2}T^{2}[\pi_{\geq\ell},\sin^{2}\theta]\psi=:F_{\geq\ell}.$

We can generalize the Morawetz estimate in Proposition 9.1.1 of [DRSR16] to
inhomogeneous wave equations where the inhomogeneity $F_{\geq\ell}$ is given
by the above expression and we consider a hyperboloidal foliation by
$\Sigma_{\tau}$ rather than an asymptotically flat foliation; see also
Proposition 9.8.1 of [DRSR16]. We obtain the following estimate after assuming
without loss of generality that $M=1$:

$\begin{split}\int_{\tau_{1}}^{\infty}&\left[\int_{\Sigma_{\tau}}r^{1-\delta}[\zeta(T\psi_{\geq\ell})^{2}+(Y\psi_{\geq\ell})^{2}]+r^{-1}\zeta|\not{\nabla}_{\mathbb{S}^{2}}\psi_{\geq\ell}|^{2}+r^{-1-\delta}\psi_{\geq\ell}^{2}\,d\omega
d\uprho\right]d\tau\\\
\leq&\>C\int_{\tau_{1}}^{\infty}\left[\int_{\Sigma_{\tau}}O_{\infty}(r^{0})F_{\geq\ell}\cdot(T\psi_{\geq\ell}+\Phi\psi_{\geq\ell}+Y\psi_{\geq\ell}+r^{-1}\psi_{\geq\ell})\,d\omega
d\uprho\right]d\tau+C\int_{\Sigma_{\tau_{1}}}\mathbf{J}^{N}[\psi_{\geq\ell}]\cdot\mathbf{n}_{0}\,r^{2}d\omega
d\uprho,\end{split}$

with $\delta>0$ and $\zeta$ a smooth cut-off function that vanishes in an
interval $[-s_{1,3M}+3M,3M+s_{2,3M}]$, with $-s_{1,3M}+3M>r_{+}$ and
$s_{2,3M}<\infty$ depending on $a$ and $M$.

After integrating by parts in $T$ and taking $\epsilon>0$ arbitrarily small,
we can further estimate:

$\begin{split}\int_{\tau_{1}}^{\infty}&\left[\int_{\Sigma_{\tau}}O_{\infty}(r^{0})F_{\geq\ell}\cdot(T\psi_{\geq\ell}+\Phi\psi_{\geq\ell}+Y\psi+\psi_{\geq\ell})\,d\omega
d\uprho\right]d\tau\\\
\leq&\>\epsilon\int_{\tau_{1}}^{\infty}\left[\int_{\Sigma_{\tau}}r^{1-\delta}[\zeta(T\psi_{\geq\ell})^{2}+(Y\psi_{\geq\ell})^{2}]+r^{-1}\zeta|\not{\nabla}_{\mathbb{S}^{2}}\psi_{\geq\ell}|^{2}+r^{-1-\delta}\psi_{\geq\ell}^{2}\,d\omega
d\uprho\right]d\tau\\\
&+C\epsilon^{-1}\int_{\tau_{1}}^{\infty}\left[\int_{\Sigma_{\tau}}r^{-1+\delta}F_{\geq\ell}^{2}\,d\omega
d\uprho\right]d\tau\\\
&+C\epsilon^{-1}\int_{\tau_{1}}^{\infty}\left[\int_{\Sigma_{\tau}\cap\\{-s_{3,3M}+3M\leq
r\leq 3M+s_{4,3M}\\}}O_{\infty}(r^{0})(TF_{\geq\ell}+\Phi
F_{\geq\ell})\cdot\psi_{\geq\ell}\,d\omega d\uprho\right]d\tau\\\
&+C\epsilon^{-1}\int_{\Sigma_{\tau_{1}}\cap\\{-s_{3,3M}+3M\leq r\leq
3M+s_{4,3M}\\}}|F_{\geq\ell}||\psi_{\geq\ell}|\,d\omega d\uprho,\end{split}$

with suitable $s_{1,3M}<s_{3,3M}<(3M-r_{+})$ and $s_{2,3M}<s_{4,3M}<\infty$.
We have that

$\int_{\mathbb{S}^{2}}F_{\geq\ell}^{2}\,d\omega\leq
C\int_{\mathbb{S}^{2}}(T^{2}\psi_{\geq\ell-2})^{2}\,d\omega.$

We obtain (3.14) by combining the above estimates. The estimate (3.15) follows
by additionally applying the above estimates with $\psi$ replaced by $T\psi$
or $\Phi\psi$ (using that $T$ and $\Phi$ are Killing vector fields), to remove
the degenerate factor $\zeta$ on the left-hand side. The estimate (3.16)
follows from standard higher-order elliptic estimates in the far-away region
$2M<R_{0}<r<R_{1}$, where $T$ is timelike, together with a repeated
application of (3.14), using moreover that all estimates apply to $T^{k}\psi$
replacing $\psi$. ∎

##  4 Newman–Penrose charges

In this section we derive conservation laws for weighted derivatives of $\phi$
along $\mathcal{I}^{+}$ by constructing Newman–Penrose charges in Kerr
spacetimes. These Newman–Penrose charges will play a key role in the analysis
in the remainder of the paper, both when deriving sharp decay estimates and
precise late-time asymptotics for solutions to (3.1).

We first introduce the following _renormalized derivatives_ of $\phi$:

$\displaystyle\check{\phi}^{(1)}:=$
$\displaystyle\>\left[1+(\alpha+\alpha_{\Phi}\Phi)r^{-1}+(\beta+\beta_{\Phi}\Phi+\beta_{\Phi^{2}}\Phi^{2})r^{-2}\right]\frac{2(r^{2}+a^{2})^{2}}{\Delta}L\phi,$
(4.1) $\displaystyle\check{\phi}^{(2)}:=$
$\displaystyle\>\left[1+(\gamma+\gamma_{\Phi}\Phi)r^{-1}\right]\frac{2(r^{2}+a^{2})^{2}}{\Delta}L\check{\phi}^{(1)},$
(4.2)

with

$\displaystyle\alpha=$ $\displaystyle\>-M,$
$\displaystyle\beta=-\frac{1}{3}M^{2}-\frac{1}{6}a^{2},$
$\displaystyle\gamma=-2M,$ $\displaystyle\alpha_{\Phi}=$ $\displaystyle\>a,$
$\displaystyle\beta_{\Phi}=0,\>\beta_{\Phi^{2}}=\frac{1}{3}a^{2}$
$\displaystyle\gamma_{\Phi}=0.$

It will be useful to consider a decompositions into _azimuthal modes_
$\psi=\sum_{m\in\mathbb{Z}}(\psi)_{m}$, where $(\psi)_{m}$ can be expressed
via the following spherical harmonic decomposition:

$(\psi)_{m}=\sum_{\ell=|m|}^{\infty}\psi_{\ell,m}Y_{\ell,m}.$

Note that $\Phi(\psi)_{m}=im(\psi)_{m}$ and the azimuthal modes $(\psi)_{m}$
are _decoupled_ , i.e. they independently satisfy (3.1):
$\square_{g_{M,a}}(\psi)_{m}=0$, so it is possible to consider each $m$-th
azimuthal mode independently.

We can then split

$\phi_{1}=\pi_{1}(\phi)=\sum_{m=-1}^{1}(\phi_{1})_{m}\quad\textnormal{and}\quad\phi_{2}=\pi_{2}(\phi)=\sum_{m=-2}^{2}(\phi_{2})_{m}.$

with $\Phi(\phi_{j})_{m}=im(\phi_{j})_{m}$ for $j=1,2$. When restricted to
fixed azimuthal modes, (4.1) and (4.2) take the form:

$\displaystyle(\check{\phi}^{(1)})_{m}:=$
$\displaystyle\>\left[1+(\alpha+im\alpha_{\Phi})r^{-1}+(\beta+im\beta_{\Phi}-m^{2}\beta_{\Phi^{2}})r^{-2}\right]\frac{2(r^{2}+a^{2})^{2}}{\Delta}L(\phi)_{m},$
$\displaystyle(\check{\phi}^{(2)})_{m}:=$
$\displaystyle\>\left[1+(\gamma+im\gamma_{\Phi})r^{-1}\right]\frac{2(r^{2}+a^{2})^{2}}{\Delta}L(\check{\phi}^{(1)})_{m}.$

In a slight abuse of notation, when restricting to fixed azimuthal modes with
azimuthal number $m$, we will therefore also use $\alpha$, $\beta$ and
$\gamma$ to denote also the complex numbers $\alpha+im\alpha_{\Phi}$,
$\beta+im\beta_{\Phi}+m^{2}\beta_{\Phi^{2}}$ and $\gamma+im\gamma_{\Phi}$,
respectively.

######  Proposition 4.1.

Let $\psi\in C^{\infty}(\mathcal{R}\to\mathbb{C})$ denote a solution to (3.1)
that is supported on a fixed azimuthal mode with azimuthal number $m$, i.e. it
satisfies $\Phi\psi=im\psi$. Then:

$\begin{split}4\frac{(r^{2}+a^{2})^{2}}{\Delta}\underline{L}L\check{\phi}^{(1)}_{1}=&\>4\frac{(r^{2}+a^{2})^{2}}{\Delta}\underline{L}L\pi_{1}(\check{\phi}^{(1)})=\left[a^{2}\pi_{1}(\sin^{2}\theta
T^{2}\check{\phi}^{(1)})+2aT\Phi\check{\phi}^{(1)}_{1}\right]\\\
&+[-\alpha-\alpha_{\Phi}\Phi+O_{\infty}(r^{-1})][a^{2}\pi_{1}(\sin^{2}\theta
T^{2}\phi)+2aT\Phi\phi_{1}]\\\
&+[-4r+O_{\infty}(1)]L\check{\phi}^{(1)}_{1}+O_{\infty}(r^{-1})\check{\phi}^{(1)}_{1}+O_{\infty}(r^{-1})\Phi\check{\phi}^{(1)}_{1}+O_{\infty}(r^{-1})\phi_{1}\end{split}$
(4.3)

and

$\begin{split}4\frac{(r^{2}+a^{2})^{2}}{\Delta}\underline{L}L\check{\phi}^{(2)}_{2}=&\>\left[a^{2}\pi_{2}(\sin^{2}\theta
T^{2}\check{\phi}^{(2)})+2aT\Phi\check{\phi}^{(2)}_{2}\right]\\\
&+[M-\gamma-a\Phi+O_{\infty}(r^{-1})][a^{2}\pi_{2}(\sin^{2}\theta
T^{2}\check{\phi}^{(1)})+2aT\Phi\check{\phi}^{(1)}_{2}]\\\
&+[2\beta+2\beta_{\Phi}\Phi+2\beta_{\Phi^{2}}\Phi^{2}+\gamma(a\Phi-M)+O_{\infty}(r^{-1})][a^{2}\pi_{2}(\sin^{2}\theta
T^{2}\phi)+2aT\Phi\phi_{2}]\\\
&+[-8r+O_{\infty}(r^{0})]L\check{\phi}^{(2)}_{2}+O_{\infty}(r^{-1})\check{\phi}^{(2)}_{2}+O_{\infty}(r^{-1})\Phi\check{\phi}^{(2)}_{2}\\\
&+O_{\infty}(r^{-1})\Phi\check{\phi}^{(1)}_{2}+O_{\infty}(r^{-1})\check{\phi}^{(1)}_{2}+O_{\infty}(r^{-1})\Phi\phi_{2}+O_{\infty}(r^{-1})\phi_{2},\end{split}$
(4.4)

where we allow the constants in the terms in $O_{\infty}(r^{-k})$, $k=0,1$, to
depend also on $m$.

###### Proof.

We have that

$\check{\phi}^{(1)}=(1+\alpha r^{-1}+\beta
r^{-2})\frac{2(r^{2}+a^{2})^{2}}{\Delta}L\phi,$

and therefore

$\begin{split}4\underline{L}\check{\phi}^{(1)}=&\>\overbrace{2\underline{L}\left((1+\alpha
r^{-1}+\beta r^{-2})\frac{2(r^{2}+a^{2})^{2}}{\Delta}\right)(1+\alpha
r^{-1}+\beta
r^{-2})^{-1}\frac{\Delta}{(r^{2}+a^{2})^{2}}\check{\phi}^{(1)}}^{=:T_{1}}\\\
&+\overbrace{(1+\alpha r^{-1}+\beta
r^{-2})\frac{2(r^{2}+a^{2})^{2}}{\Delta}4\underline{L}L\phi}^{=:T_{2}}.\end{split}$

We can write

$T_{1}=\frac{\Delta}{(r^{2}+a^{2})^{2}}[-4r+2(\alpha+2M)+O_{\infty}(r^{-1})]\check{\phi}^{(1)},$

where we allow the terms in $O_{\infty}(r^{-1})$ to depend on $\alpha$ and
$\beta$.

Hence,

$L(T_{1})=\frac{\Delta}{(r^{2}+a^{2})^{2}}[-4r+2(\alpha+2M)+O_{\infty}(r^{-1})]L\check{\phi}^{(1)}+\frac{\Delta}{(r^{2}+a^{2})^{2}}\left[2-(2\alpha+12M)r^{-1}+O_{\infty}(r^{-2})\right]\check{\phi}^{(1)}.$

Furthermore, we can rewrite (3.5) as follows:

$\begin{split}4\underline{L}L\phi=&\>\frac{\Delta}{(r^{2}+a^{2})^{2}}\left[a^{2}\sin^{2}\theta
T^{2}\phi+2aT\Phi\phi+\not{\Delta}_{\mathbb{S}^{2}}\phi+2a\frac{r}{r^{2}+a^{2}}\Phi\phi\right]\\\
&+\frac{\Delta}{(r^{2}+a^{2})^{2}}[-2Mr^{-1}-a^{2}r^{-2}+O_{\infty}(r^{-3})]\phi.\end{split}$

Therefore,

$T_{2}=2(1+\alpha r^{-1}+\beta r^{-2})\left[a^{2}\sin^{2}\theta
T^{2}\phi+2aT\Phi\phi+\not{\Delta}_{\mathbb{S}^{2}}\phi+2a\frac{r}{r^{2}+a^{2}}\Phi\phi+(-2Mr^{-1}-a^{2}r^{-2}+O_{\infty}(r^{-3}))\phi\right].$

We then obtain

$\begin{split}L(T_{2})=&\>\frac{\Delta}{(r^{2}+a^{2})^{2}}\left[a^{2}\sin^{2}\theta
T^{2}\check{\phi}^{(1)}+2aT\Phi\check{\phi}^{(1)}+2a\frac{r}{r^{2}+a^{2}}\Phi\check{\phi}^{(1)}+\not{\Delta}_{\mathbb{S}^{2}}\check{\phi}^{(1)}\right]\\\
&+\frac{\Delta}{(r^{2}+a^{2})^{2}}[-2Mr^{-1}-a^{2}r^{-2}+O_{\infty}(r^{-3})]\check{\phi}^{(1)}\\\
&+\frac{\Delta}{(r^{2}+a^{2})^{2}}[-\alpha-2\beta
r^{-1}+O_{\infty}(r^{-2})][a^{2}\sin^{2}\theta
T^{2}\phi+2aT\Phi\phi+\not{\Delta}_{\mathbb{S}^{2}}\phi]\\\
&+\frac{\Delta}{(r^{2}+a^{2})^{2}}[-2a-4a\alpha
r^{-1}+O_{\infty}(r^{-2})]\Phi\phi+\frac{\Delta}{(r^{2}+a^{2})^{2}}\left[2M+(2a^{2}+4\alpha
M)r^{-1}+O_{\infty}(r^{-2})\right]\phi\end{split}$

Combining the expressions for $L(T_{1})$ and $L(T_{2})$ and commuting $L$ and
$\underline{L}$, we obtain:

$\begin{split}4\frac{(r^{2}+a^{2})^{2}}{\Delta}\underline{L}L\check{\phi}^{(1)}=&\>\left[a^{2}\sin^{2}\theta
T^{2}\check{\phi}^{(1)}+2aT\Phi\check{\phi}^{(1)}+6a\frac{r}{r^{2}+a^{2}}\Phi\check{\phi}^{(1)}\right]\\\
&+[-4r+2(\alpha+2M)+O_{\infty}(r^{-1})]L\check{\phi}^{(1)}+\left[2+\not{\Delta}_{\mathbb{S}^{2}}\right]\check{\phi}^{(1)}\\\
&+[(-2\alpha-14M)r^{-1}+O_{\infty}(r^{-2})]\check{\phi}^{(1)}\\\
&+[-\alpha-2\beta r^{-1}+O_{\infty}(r^{-2})][a^{2}\sin^{2}\theta
T^{2}\phi+2aT\Phi\phi+\not{\Delta}_{\mathbb{S}^{2}}\phi]\\\ &+[-2a-4a\alpha
r^{-1}+O_{\infty}(r^{-2})]\Phi\phi+\left[2M+(2a^{2}+4\alpha
M)r^{-1}+O_{\infty}(r^{-2})\right]\phi\end{split}$ (4.5)

In order to guarantee that the right-hand side above either contains a
$T$-derivative or vanishes when we act with $\pi_{1}$ on both sides and take
$r\to\infty$, we use that

$\alpha=-M+iam.$

Now let

$\check{\phi}^{(2)}=(1+\gamma
r^{-1})\frac{2(r^{2}+a^{2})^{2}}{\Delta}L\check{\phi}^{(1)},$

so

$4\underline{L}\check{\phi}^{(2)}=\overbrace{2\underline{L}\left((1+\gamma
r^{-1})\frac{2(r^{2}+a^{2})^{2}}{\Delta}\right)(1+\gamma
r^{-1})^{-1}\frac{\Delta}{(r^{2}+a^{2})^{2}}\check{\phi}^{(2)}}^{=:T_{3}}+\overbrace{(1+\gamma
r^{-1})\frac{2(r^{2}+a^{2})^{2}}{\Delta}4\underline{L}L\check{\phi}^{(1)}}^{=:T_{4}}.$

Note that

$L(T_{3})=\frac{\Delta}{(r^{2}+a^{2})^{2}}[-4r+O_{\infty}(r^{0})]L\check{\phi}^{(2)}+\frac{\Delta}{(r^{2}+a^{2})^{2}}\left[2+O_{\infty}(r^{-1})\right]\check{\phi}^{(2)},$

where the terms in $O_{\infty}(r^{-1})$ depend on $\gamma$.

By (4.5) with $\alpha=-M+iam$, we have that:

$\begin{split}T_{4}=&\>2(1+\gamma r^{-1})[a^{2}\sin^{2}\theta
T^{2}\check{\phi}^{(1)}+2aT\Phi\check{\phi}^{(1)}+6ar(r^{2}+a^{2})^{-1}\Phi\check{\phi}^{(1)}]\\\
&+\Delta(r^{2}+a^{2})^{-2}[-4r+2(M+iam)+O_{\infty}(r^{-1})]\check{\phi}^{(2)}+2(1+\gamma
r^{-1})[2+\not{\Delta}_{\mathbb{S}^{2}}]\check{\phi}^{(1)}\\\ &+2(1+\gamma
r^{-1})[(-12M-2iam)r^{-1}+O_{\infty}(r^{-2})]\check{\phi}^{(1)}\\\
&+2(1+\gamma r^{-1})[M-iam-2\beta
r^{-1}+O_{\infty}(r^{-2})][a^{2}\sin^{2}\theta
T^{2}\phi+2aT\Phi\phi+\not{\Delta}_{\mathbb{S}^{2}}\phi]\\\ &+2(1+\gamma
r^{-1})[-2a+(4Ma-4ima^{2})r^{-1}+O_{\infty}(r^{-2})]\Phi\phi\\\ &+2(1+\gamma
r^{-1})[2M+(2a^{2}-4M^{2}+4iamM)r^{-1}+O_{\infty}(r^{-2})]\phi\end{split}$

and hence,

$\begin{split}\frac{(r^{2}+a^{2})^{2}}{\Delta}L(T_{4})=&\>[a^{2}\sin^{2}\theta
T^{2}\check{\phi}^{(2)}+2aT\Phi\check{\phi}^{(2)}+6ar(r^{2}+a^{2})^{-1}\Phi\check{\phi}^{(2)}]\\\
&+[-4r+O_{\infty}(r^{0})]L\check{\phi}^{(2)}+[2+\not{\Delta}_{\mathbb{S}^{2}}+O_{\infty}(r^{-1})]\check{\phi}^{(2)}\\\
&-[\gamma+O_{\infty}(r^{-1})][a^{2}\sin^{2}\theta
T^{2}\check{\phi}^{(1)}+2aT\Phi\check{\phi}^{(1)}]-[6a+O_{\infty}(r^{-1}]\Phi\check{\phi}^{(1)}\\\
&+[2+O_{\infty}(r^{-1})]\check{\phi}^{(2)}-[\gamma+O_{\infty}(r^{-1})](2+\not{\Delta}_{\mathbb{S}^{2}})\check{\phi}^{(1)}+[12M+2iam+O_{\infty}(r^{-1})]\check{\phi}^{(1)}\\\
&+[M-iam+O_{\infty}(r^{-1})](a^{2}\sin^{2}\theta
T^{2}\check{\phi}^{(1)}+2aT\Phi\check{\phi}^{(1)}+\not{\Delta}_{\mathbb{S}^{2}}\check{\phi}^{(1)})\\\
&+[-2a+O_{\infty}(r^{-1})]\Phi\check{\phi}^{(1)}+[2M+O_{\infty}(r^{-1})]\check{\phi}^{(1)}\\\
&+[-\gamma(M-iam)+2\beta+O_{\infty}(r^{-1})](a^{2}\sin^{2}\theta
T^{2}\phi+2aT\Phi\phi+\not{\Delta}_{\mathbb{S}^{2}}\phi)\\\
&+[2a\gamma-4Ma+4ima^{2}+O_{\infty}(r^{-1})]\Phi\phi\\\
&+[-2M\gamma-(2a^{2}-4M^{2}+4iamM)+O_{\infty}(r^{-1})]\phi.\end{split}$

Combining the above expression with the expression for $L(T_{3})$, using that
$\Phi\phi=im\phi$ (and commuting $L$ and $\underline{L}$), we obtain

$\begin{split}4\frac{(r^{2}+a^{2})^{2}}{\Delta}\underline{L}L\check{\phi}^{(2)}=&\>\left[a^{2}\sin^{2}\theta
T^{2}\check{\phi}^{(2)}+2aT\Phi\check{\phi}^{(2)}+10a\frac{r}{r^{2}+a^{2}}\Phi\check{\phi}^{(2)}\right]\\\
&+[-8r+O_{\infty}(r^{0})]L\check{\phi}^{(2)}+\left[6+\not{\Delta}_{\mathbb{S}^{2}}\right]\check{\phi}^{(2)}+O_{\infty}(r^{-1})\check{\phi}^{(2)}\\\
&+[M-aim-\gamma+O_{\infty}(r^{-1})][a^{2}\sin^{2}\theta
T^{2}\check{\phi}^{(1)}+2aT\Phi\check{\phi}^{(1)}]\\\
&+[-\gamma(2+\not{\Delta}_{\mathbb{S}^{2}})+14M-6iam+(M-iam)\not{\Delta}_{\mathbb{S}^{2}}+O_{\infty}(r^{-1})]\check{\phi}^{(1)}\\\
&+[2\beta-M\gamma+iam\gamma+O_{\infty}(r^{-1})](a^{2}\sin^{2}\theta
T^{2}\phi+2aT\Phi\phi)\\\
&+[(-M\gamma+iam\gamma+2\beta)\not{\Delta}_{\mathbb{S}^{2}}-2M\gamma-2a^{2}+4M^{2}-4iamM+2iam\gamma-4iamM\\\
&-4m^{2}a^{2}+O_{\infty}(r^{-1})]\phi\end{split}$ (4.6)

In order to guarantee that the terms involving $\check{\phi}^{(2)}$ and
$\check{\phi}^{(1)}$ on the right-hand side above either contains a
$T$-derivative or vanishes when we act with $\pi_{2}$ on both sides and take
$r\to\infty$, we use that:

$\gamma=-2M.$

With the above value of $\gamma$, it follows that if we moreover use that

$\beta=-\frac{1}{3}M^{2}-\frac{1}{6}a^{2}-\frac{1}{3}a^{2}m^{2},$

then all the terms involving $\phi$ on the right-hand side above either
contain a $T$-derivative or vanish when we act with $\pi_{2}$ on both sides
and take $r\to\infty$. ∎

We define the following special linear combinations of derivatives of
$\check{\phi}^{(j)}$, $j=0,1,2$:

$\displaystyle\frac{(r^{2}+a^{2})^{2}}{\Delta}P_{0}:=$
$\displaystyle\>\frac{(r^{2}+a^{2})^{2}}{\Delta}L\phi_{0}-\frac{1}{4}a^{2}\pi_{0}(\sin^{2}\theta
T\phi),$ (4.7) $\displaystyle\frac{(r^{2}+a^{2})^{2}}{\Delta}P_{1}:=$
$\displaystyle\>\frac{(r^{2}+a^{2})^{2}}{\Delta}L\check{\phi}_{1}^{(1)}-\frac{1}{4}\left[a^{2}\pi_{1}(\sin^{2}\theta
T\check{\phi}^{(1)})+2a\Phi\check{\phi}^{(1)}_{1}\right]$ (4.8)
$\displaystyle-\frac{1}{4}[-\alpha-\alpha_{\Phi}\Phi][a^{2}\pi_{1}(\sin^{2}\theta
T\phi)+2a\Phi\phi_{1}],$
$\displaystyle\frac{(r^{2}+a^{2})^{2}}{\Delta}P_{2}:=$
$\displaystyle\>\frac{(r^{2}+a^{2})^{2}}{\Delta}L\check{\phi}_{2}^{(2)}-\frac{1}{4}\left[a^{2}\pi_{2}(\sin^{2}\theta
T\check{\phi}^{(2)})+2a\Phi\check{\phi}^{(2)}_{2}\right]$ (4.9)
$\displaystyle-\frac{1}{4}[M-\gamma-a\Phi][a^{2}\pi_{2}(\sin^{2}\theta
T\check{\phi}^{(1)})+2a\Phi\check{\phi}^{(1)}_{2}]$
$\displaystyle-\frac{1}{4}[2\beta+2\beta_{\Phi^{2}}\Phi^{2}+\gamma(a\Phi-M)][a^{2}\pi_{2}(\sin^{2}\theta
T\phi)+2a\Phi\phi_{2}].$

######  Corollary 4.2.

Let $\psi\in C^{\infty}(\mathcal{R}\to\mathbb{C})$ denote a solution to (3.1)
that is supported on a fixed azimuthal mode with azimuthal number $m$, i.e. it
satisfies $\Phi\psi=im\psi$. Then:

$\displaystyle 4\underline{L}P_{0}=$
$\displaystyle\>O_{\infty}(r^{-3})\phi_{0}+a^{2}O_{\infty}(r^{-3})T\pi_{0}(\sin^{2}\theta\phi)+a^{2}O_{\infty}(r^{-2})LT\pi_{0}(\sin^{2}\theta\phi),$
(4.10) $\displaystyle 4\underline{L}P_{1}=$
$\displaystyle\>[-4r^{-1}+O_{\infty}(r^{-2})]P_{1}+O_{\infty}(r^{-3})[\check{\phi}^{(1)}_{1}+\phi_{1}]+a^{2}O_{\infty}(r^{-3})[T\pi_{1}(\sin^{2}\theta\check{\phi}^{(1)})+T\pi_{1}(\sin^{2}\theta\phi)]$
(4.11)
$\displaystyle+a^{2}O_{\infty}(r^{-2})[LT\pi_{1}(\sin^{2}\theta\check{\phi}^{(1)})+LT\pi_{1}(\sin^{2}\theta\phi)],$
$\displaystyle 4\underline{L}P_{2}=$
$\displaystyle\>[-8r^{-1}+O_{\infty}(r^{-2})]P_{2}+O_{\infty}(r^{-3})[\check{\phi}^{(2)}_{2}+\check{\phi}^{(1)}_{2}+\phi_{2}]$
(4.12)
$\displaystyle+a^{2}O_{\infty}(r^{-3})[T\pi_{2}(\sin^{2}\theta\check{\phi}^{(2)})+T\pi_{2}(\sin^{2}\theta\check{\phi}^{(1)})+T\pi_{2}(\sin^{2}\theta\phi)]$
$\displaystyle+a^{2}O_{\infty}(r^{-2})[LT\pi_{2}(\sin^{2}\theta\check{\phi}^{(2)})+LT\pi_{2}(\sin^{2}\theta\check{\phi}^{(1)})+LT\pi_{2}(\sin^{2}\theta\phi)],$

where we allow the constants in the terms in $O_{\infty}(r^{-k})$, $k=0,1$, to
depend also on $m$.

Furthermore, if $\phi\in C^{4}(\widehat{\mathcal{R}})$, then

$\displaystyle\lim_{\uprho\to\infty}\underline{L}(r^{2}P_{0})(\tau,\uprho)=$
$\displaystyle\>0,$ (4.13)
$\displaystyle\lim_{\uprho\to\infty}\underline{L}(r^{2}P_{1})(\tau,\uprho,\theta,\varphi_{*})=$
$\displaystyle\>0,$ (4.14)
$\displaystyle\lim_{\uprho\to\infty}\underline{L}(r^{2}P_{2})(\tau,\uprho,\theta,\varphi_{*})=$
$\displaystyle\>0.$ (4.15)

###### Proof.

By the rewriting the equations in Proposition 4.1 in terms of the quantities
$P_{i}$, $i=0,1,2$, using the definitions (4.7)–(4.9), we obtain
(4.10)–(4.12).

Now suppose $\phi\in C^{4}(\widehat{\mathcal{R}})$, then the limits on the
left-hand sides of (4.13)–(4.15) are well-defined. By applying (4.10)–(4.12),
we moreover show that these limits must vanish. ∎

######  Definition 4.1.

Let $\psi$ be a solution to such that $\phi=\sqrt{r^{2}+a^{2}}\psi\in
C^{4}(\widehat{\mathcal{R}})$. Then we define the _Newman–Penrose_ _charges_
as the following limits:

$\displaystyle I_{0}[\psi]=$
$\displaystyle\>\lim_{\uprho\to\infty}2r^{2}P_{0}(\tau,\uprho),$
$\displaystyle I_{1}[\psi](\theta,\varphi_{*})=$
$\displaystyle\>\lim_{\uprho\to\infty}2r^{2}P_{1}(\tau,\uprho,\theta,\varphi_{*}),$
$\displaystyle I_{2}[\psi](\theta,\varphi_{*})=$
$\displaystyle\>\lim_{\uprho\to\infty}2r^{2}P_{2}(\tau,\uprho,\theta,\varphi_{*}),$

which are well-defined and conserved in $\tau$ by (4.13)–(4.15).

##  5 Hierarchies of $r^{p}$-weighted energy estimates

In this section, we derive the key weighted hierarchies of estimates that are
involved in establishing sharp energy decay estimates.

We will use $R$ to denote the area radius that appears in the definitions of
$D^{\tau_{2}}_{\tau_{1}}$ and $N_{\tau}$ in Section 2.3. We will need to take
$R>r_{+}$ appropriately large for the estimates in the sections below to hold.

###  5.1 Hierarchies of $r^{p}$-weighted energy estimates for $\phi^{(n)}$

This section is concerned with establishing hierarchies of weighted energy
estimates for the quantities $\phi^{(n)}$, which were introduced in Section
3.1. In Proposition 5.1 below, we show that we can derive weighted energy
estimates for a larger range of $n\in\mathbb{N}_{0}$, provided we restrict to
$\phi^{(n)}_{\geq\ell}$ with $\ell$ appropriately large.

######  Proposition 5.1.

Let $n\in\mathbb{N}_{0}$ , $\ell\in\mathbb{N}_{0}$, such that $\ell\geq n$ and
$-4n<p\leq 2$. For $R>r_{+}$ suitably large, there exists a constant
$C=C(M,a,n,\ell,R,p)>0$, such that

$\begin{split}\int_{N_{\tau_{2}}}&r^{p}(L\phi^{(n)}_{\geq\ell})^{2}+r^{p-4}[|\not{\nabla}_{\mathbb{S}^{2}}\phi^{(n)}_{\geq\ell}|^{2}-n(n+1)(\phi_{\geq
n}^{(n)})^{2}]\,d\omega d\uprho\\\
&+\int_{\tau_{1}}^{\tau_{2}}\left[\int_{N_{\tau}}r^{p-1}(L\phi^{(n)}_{\geq\ell})^{2}+(2-p)r^{p-3}\left(|\not{\nabla}_{\mathbb{S}^{2}}\phi^{(n)}_{\geq\ell}|^{2}-n(n+1)(\phi_{\geq\ell}^{(n)})^{2}+\delta_{n0}\delta_{\ell
0}a^{2}\sin^{2}\theta(T\phi)^{2}\right)\,d\omega d\uprho\right]d\tau\\\
\leq&\>C\int_{N_{\tau_{1}}}r^{p}(L\phi^{(n)}_{\geq\ell})^{2}+r^{p-4}\left(|\not{\nabla}_{\mathbb{S}^{2}}\phi^{(n)}_{\geq\ell}|^{2}-n(n+1)(\phi_{\geq\ell}^{(n)})^{2}\right)\,d\omega
d\uprho\\\
&+C(n+\ell)\int_{\tau_{1}}^{\tau_{2}}\left[\int_{N_{\tau}}r^{p-3}a^{4}(T^{2}\phi^{(n)}_{\geq\max\\{\ell-2,0\\}})^{2}+r^{p-3}a^{2}(T\Phi\phi^{(n)}_{\geq\ell})^{2}\,d\omega
d\uprho\right]d\tau\\\
&+Cn\int_{\tau_{1}}^{\tau_{2}}\left[\int_{N_{\tau}}r^{p-1}(L\Phi\phi_{\geq\ell}^{(n-1)})^{2}+r^{p-3}(LT\phi_{\geq\ell}^{(n-1)})^{2}\,d\omega
d\uprho\right]d\tau\\\
&+Cn\sum_{m=0}^{n-1}\int_{\tau_{1}}^{\tau_{2}}\left[\int_{N_{\tau}}a^{2}r^{p-3}(\Phi\phi_{\geq\ell}^{(m)})^{2}+r^{p-3}(\phi_{\geq\ell}^{(m)})^{2}\,d\omega
d\uprho\right]d\tau\\\
&+C\sum_{m=0}^{n}\left[\int_{\Sigma_{\tau_{1}}}J^{N}[T^{m}\psi_{\geq\ell}]\cdot\mathbf{n}_{\tau_{1}}\,r^{2}d\omega