# Physical-space estimates for axisymmetric waves on extremal Kerr spacetime

Elena<EMAIL_ADDRESS>Jingbo<EMAIL_ADDRESS>
###### Abstract

We study axisymmetric solutions to the wave equation $\square_{g}\psi=0$ on
extremal Kerr backgrounds and obtain integrated local energy decay (or
Morawetz estimates) through an analysis exclusively in physical-space.
Boundedness of the energy and Morawetz estimates for axisymmetric waves in
extremal Kerr were first obtained by Aretakis [11] through the construction of
frequency-localized currents used in particular to express the trapping
degeneracy. Here we extend to extremal Kerr a method introduced by Stogin [60]
in the sub-extremal case, simplifying Aretakis’ derivation of Morawetz
estimates through purely classical currents.

##  1 Introduction

The study of the Cauchy problem for the wave equation

$\displaystyle\square_{g}\psi=0,$ (1)

where $g$ is given by a black hole solution to the Einstein equation, is a
topic that has been extensively studied in the past two decades. One of the
most important black hole solutions is the vacuum Kerr family, a 2-parameter
family of solutions $(\mathcal{N}_{M,a},g_{M,a})$ with $|a|\leq M$,
representing a stationary and rotating black hole. Boundedness and decay
properties for solutions to the wave equation on Kerr have been obtained in
numerous works in the past two decades, see already Section 1.1 for an
overview of previous results. Numerous such works rely on the derivation of
integrated local energy decay estimates, or Morawetz estimates, through an
analysis in physical- or frequency-space, or the use of pseudo-differential
operators.

We consider here the case of axially symmetric solutions to the wave equation
on extremal Kerr backgrounds, corresponding to $|a|=M$. Even though
instability properties hold for solutions to the wave equations in extremal
black holes as shown by Aretakis [9][10][12], boundedness of the energy as
well as integrated local energy and pointwise decay estimates have been
obtained by Aretakis [11] for axially symmetric waves in extremal Kerr. In
[11], Aretakis adapted the method used by Dafermos-Rodnianski in [26] relying
on the separability of the wave equation to construct frequency-localized
currents. For general solutions to the wave equation, it is in fact not
possible to obtain positive definite spacetime estimates through classical
energy currents, or in physical-space, as shown by Alinhac [1]. This is
related to the complicated structure of trapped null geodesics for $|a|\neq
0$, whose (Boyer-Lindquist) constant $r$-value covers an open range of values.
On the other hand, for axially symmetric solutions the trapping degeneracy
collapses to a hypersurface in physical-space and, as observed by Aretakis in
the introduction of [11], “the obstruction uncovered by Alinhac [1] does not
apply to the axisymmetric case and thus one could in principle expect to
derive integrated decay for the full range $|a|\leq M$ using purely classical
currents; this remains however an open problem.”

In this paper, we address such problem by deriving integrated local energy
estimates for axially symmetric solutions to the wave equation on extremal
Kerr exclusively through a physical-space analysis. Here by physical-space
estimates we refer to an analysis of the wave equation which does not require
a mode or frequency decomposition and involves only differential operators.

Recall that in extremal Kerr the event horizon lies at $r=M$ and the effective
photon sphere lies at $r_{trap}=(1+\sqrt{2})M$. The degenerate energy for
solutions to (1) is given by

$\displaystyle E^{(T)}[\psi](0)$ $\displaystyle=$
$\displaystyle\int_{\Sigma_{0}}|\partial_{t}\psi|^{2}+\left(1-\frac{M}{r}\right)^{2}|\partial_{r}\psi|^{2}+|\nabla\mkern-13.0mu/\,\psi|^{2},$

where
$|\nabla\mkern-13.0mu/\,\psi|^{2}=\frac{1}{r^{2}}|\nabla\mkern-13.0mu/\,_{\mathbb{S}^{2}}\psi|^{2}$
with $|\nabla\mkern-13.0mu/\,_{\mathbb{S}^{2}}\psi|^{2}$ the norm of the
gradient of $\psi$ on the unit sphere with respect to the standard metric, and
in what follows $(t,r,\theta,\phi)$ denote the Boyer-Lindquist coordinates. We
prove the following.

######  Theorem 1.1.

Let $\psi$ be a sufficiently regular axisymmetric solution to the wave
equation in extremal Kerr spacetime with initial data on a spacelike
hypersurface $\Sigma_{0}$ which decays sufficiently fast. Then the following
Morawetz estimates on the exterior region:

$\displaystyle\int_{\mathcal{M}}\frac{1}{r^{3}}\left(1-\frac{M}{r}\right)^{2}|\partial_{r}\psi|^{2}+\frac{1}{r}\left(1-\frac{r_{trap}}{r}\right)^{2}\left(\frac{1}{r^{2}}(\partial_{t}\psi)^{2}+|\nabla\mkern-13.0mu/\,\psi|^{2}\right)+\frac{1}{r^{4}}\left(1-\dfrac{M}{r}\right)^{2}|\psi|^{2}\leq
CE^{(T)}[\psi](0),$ (2)

where $C$ only depending on $M$, can be obtained through exclusively a
physical-space analysis.

As a corollary of our main result we recover the following bound that appeared
as the crucial Proposition 12.5.1 in [11], which summarized the results
involving frequency decomposition, reflecting the fact that the
microlocalization in [11] was only needed in a spatially compact region
located away from the horizon.

######  Corollary 1.2.

Let $\psi$ be a sufficiently regular axisymmetric solution to the wave
equation in the extremal Kerr spacetime with initial data on $\Sigma_{0}$
which decays sufficiently fast and let $R_{e}>r_{e}>M$. Then the following
Morawetz estimates:

$\displaystyle\int_{r_{e}\leq r\leq
R_{e}}\Big{(}|\partial_{r*}\psi|^{2}+|\psi|^{2}+\big{(}r-r_{trap}\big{)}^{2}\big{(}|\nabla\mkern-13.0mu/\,\psi|^{2}+|\partial_{t}\psi|^{2}\big{)}\Big{)}\leq
CE^{(T)}[\psi](0),$ (3)

where $r^{*}=\int\frac{r^{2}+a^{2}}{\Delta}$ and $C$ only depends on $M$,
$r_{e}$ and $R_{e}$, can be obtained through exclusively a physical-space
analysis.

The proof of Proposition 12.5.1 in [11] relied on the separation of variables
for the solution of (1) and an analysis in frequency-space relying on a series
of involved microlocal currents tailored to the different regions of validity
in frequency space. On the other hand, we obtain Theorem 1.1, and consequently
Corollary 1.2, through the definition of one physical-space current, resulting
in a considerable simplification of the construction.

In [11], Aretakis combined the above result as stated in Corollary 1.2 with
positive-definite currents near null infinity and near the horizon and with a
uniform boundedness statement of energy, both of which were obtained in [11]
in physical-space, to deduce a complete integrated local energy decay.
Finally, by applying Dafermos-Rodnianski’s $r^{p}$-method [24], Aretakis [11]
improved the decay towards null infinity of the integrated local energy decay
and used the improved decay to obtain pointwise decay for the solution. Since
these proofs were obtained in [11] through exclusively a physical-space
analysis we will not rederive them here, with the exception of the boundedness
of the degenerate energy in Section 3.4, and refer to [11] for details. In
particular, by combining Theorem 1.1 with the above mentioned steps obtained
by Aretakis in [11], one can recover the full results of pointwise and power-
law energy decay for axially symmetric waves in extremal Kerr exclusively in
physical-space.

###  1.1 Previous works

We recall here the main results and techniques used in the analysis of the
wave equation (1) and related stability problems in black hole solutions.

Stability results for the wave equation on Schwarzschild spacetime,
corresponding to the case of $a=0$, have been first obtained by Kay-Wald [46],
who derived a statement of energy boundedness. In the following decades, such
statement has been refined to include local energy decay estimates, also known
as Morawetz estimates [56], which give control over a positive-definite
spacetime norm through the use of a current associated to the radial
vectorfield, as in Blue-Soffer [14][15], Blue-Sterbenz [16], Dafermos-
Rodnianski [25], Marzuola-Metcalfe-Tataru-Tohaneanu [53]. The estimates in
this case are obtained as a modified version of the classical Morawetz
integral energy decay estimate through the use of a vectorfield of the form
$\mathcal{F}(r)\partial_{r}$, with $\mathcal{F}$ vanishing at $r=3M$, which is
the location of orbital null geodesics in Schwarzschild called the photon
sphere. Also, in [25] Dafermos-Rodnianski introduced a vectorfield estimate
which captures the so-called redshift effect, allowing for pointwise estimates
along the event horizon.

In the case of Kerr spacetime with $|a|\neq 0$, the orbital null geodesics are
not confined to a hypersurface in physical-space, but cover an open region in
(Boyer-Lindquist) $r$-value which depends on the energy and angular momentum
of the geodesics. Moreover, the stationary Killing vectorfield $\partial_{t}$
fails to be timelike in an ergoregion, and therefore its associated conserved
energy is not positive definite, in a phenomenon called superradiance. The
analysis of the wave equation is complicated by the presence of the ergoregion
and the dependence of the trapping region on the frequency of the solution, as
the the high frequency obstruction to decay given by the trapping region
cannot be described by the classical vectorfield method as shown by Alinhac
[1]. For this reason, the derivation of a Morawetz estimate in this case
requires a more refined analysis involving both the vectorfield method and
mode decompositions or pseudo-differential operators.

The mode decomposition refers to the analysis of mode solutions of the
separated form

$\psi(r,t,\theta,\phi)=e^{-i\omega
t}e^{im\phi}R(r)S(\theta),\qquad\omega\in\mathbb{R},\qquad m\in\mathbb{Z}$ (4)

which is related to the Fourier transform of the solution with respect to the
symmetries of the spacetime, and corresponds to its frequency decomposition.
The presence of an additional hidden symmetry of the spacetime, known as the
Carter tensor [17], allows to reduce the study of the wave equation to the
respective radial and angular ODEs for the functions $R(r)$ and $S(\theta)$.
Such frequency-analysis has been developed by Dafermos-Rodnianski [26] and
Dafermos-Rodnianski-Shlapentokh-Rothman [30] in sub-extremal Kerr, where
frequency-dependent multiplier currents for the separated solutions are
carefully constructed using the structure of trapping (which stays localized
in frequency-space) and the fact that superradiant frequencies are not
trapped. This allows for the construction of a frequency-space analogue of the
current $\mathcal{F}(r)\partial_{r}$ which vanishes at a different $r_{trap}$
for each set of trapped frequencies [27][28][29]. Remarkably, the frequency-
space analysis in [30] is the only one among the techniques mentioned here
which holds in the full sub-extremal range $|a|<M$. Observe that to justify
the separation of general solutions into (4) through a Fourier transform, one
needs to require square integrability in time, which can be proved to hold
through a continuity argument in $a$.

The use of pseudo-differential operators appeared in the work of Tataru-
Tohaneanu [61], where they made use of a pseudo-differential modification of
the vectorfield $\mathcal{F}(r)\partial_{r}$ which was differential in
$\partial_{t}$, and with a kernel supported in a small neighborhood of $r=3M$.
The pseudo-differential operator is constructed perturbatively from the
choices of vectorfield and functions in Schwarzschild given in [53], and
therefore yields local energy decay estimates for slowly rotating Kerr
spacetime only.

Despite Alinhac’s obstruction [1], Andersson-Blue [4] obtained integrated
local energy estimates for the equation in slowly rotating Kerr spacetime
exclusively in physical space by generalizing the vectorfield method.
Andersson-Blue’s method makes use of the Carter hidden symmetry in Kerr as a
physical-space commutator to the wave equation. This allows to obtain a local
energy decay identity at the level of three derivatives of the solution which
degenerate near $r=3M$, as trapped null geodesics lie within $O(|a|)$ of the
photon sphere $r=3M$ for slowly rotating Kerr black holes. Such physical-space
estimates have the usual advantages of the classical vectorfield method, such
as being robust with respect to perturbations of the metric, see [39][38].

The geometry of the extremal Kerr spacetime satisfying $|a|=M$ (or extremal
Reissner-Nordström with $|Q|=M$) exhibits several qualitative differences from
the sub-extremal case, most notably the degeneration of the red-shift effect
at the horizon due to the vanishing surface gravity. In extremal Kerr for
generic solutions to the wave equation certain higher order derivatives
asymptotically blow up along the event horizon as a consequence of
conservation laws discovered by Aretakis [9][10][12], in what is now known as
the Aretakis instability. This generic blow up is unrelated to superradiance
and holds even for axially symmetric solutions. See also [5][6][7].

Axially symmetric solutions to the wave equation, both in the sub-extremal and
the extremal case, present two major simplifications: superradiance is
effectively absent and the trapping region collapses to a physical-space
hypersurface. The conserved energy associated to $\partial_{t}$ is positive-
definite for axially symmetric solutions even though the energy is
degenerate333There is however a way to capture in a quantitative way the
degenerate redshift close to the event horizon in extremal Kerr as shown by
Aretakis [11]. along the event horizon (see Section 3.4). Also, in axial
symmetry the orbital null geodesics all asymptote towards a unique
hypersurface $\\{r=r_{trap}\\}$ in physical-space, where $r_{trap}$ is defined
as the unique root in the exterior region of the polynomial

$\mathcal{T}:=r^{3}-3Mr^{2}+a^{2}r+Ma^{2}.$ (5)

In this case, the construction of the current $\mathcal{F}(r)\partial_{r}$
simplifies (see [26]) and can in principle be performed in physical-space. In
[60], Stogin constructed a current in physical space which yields positivity
of the local integrated energy estimates in the full sub-extremal range
$|a|<M$. Notice that to obtain positivity of the zero-th order term, Stogin
uses the non-degenerate redshift effect which is absent in the extremal case.

In the case of axially symmetric solutions in extremal Kerr, Aretakis [11]
proved integrated local energy decay, energy and pointwise uniform boundedness
of solutions and power-law energy and pointwise decay of solutions, all of
them up to and including the event horizon. The derivation of the integrated
local energy decay in [11] uses an adaptation of the frequency-analysis of
Dafermos-Rodnianski [26], which require novel microlocal currents allowing to
decouple the Morawetz estimates from the (degenerate) redshift. As in [26], to
justify the Fourier transform in time, cut-off in time are needed which create
error terms that have to be controlled by auxiliary microlocal currents in
addition to novel classical vectorfields, resulting in intricate constructions
to obtain positivity of the spacetime energy.

Finally, we remark here that the advances developed for the study of the wave
equation have been used in the analysis of the Einstein equation in various
settings, see [18][51][43] for perturbations of Minkowski spacetime, see
[20][44][45][47][22][13] for perturbations of Schwarzschild spacetime, see
[33][34][35][36] for perturbations of sub-extremal Reissner-Nordström and the
recent [8] for perturbations of extremal Reissner-Nordström, see
[63][58][52][21][2][62][3][40][48][49][59][50][57][39] for perturbations of
Kerr, see [19][37][38] for perturbations of Kerr-Newman. In the case of
positive cosmological constant, see [23][42][41][54][55].

###  1.2 Overview of the result

We give here an overview of the proof of Theorem 1.1. We apply the vectorfield
method to the current associated to a vectorfield $X$, a scalar function $w$
and a one-form $J$:

$\displaystyle\mathcal{P}_{\mu}^{(X,w,J)}[\psi]$ $\displaystyle=$
$\displaystyle\mathcal{Q}[\psi]_{\mu\nu}X^{\nu}+\frac{1}{2}w\psi{\partial}_{\mu}\psi-\frac{1}{4}({\partial}_{\mu}w)|\psi|^{2}+\frac{1}{4}J_{\mu}|\psi|^{2},$

where $\mathcal{Q}[\psi]_{\mu\nu}$ is the energy-momentum tensor associated to
a solution to the wave equation

$\displaystyle\mathcal{Q}[\psi]_{\mu\nu}$ $\displaystyle=$
$\displaystyle{\partial}_{\mu}\psi{\partial}_{\nu}\psi-\frac{1}{2}{\bf
g}_{\mu\nu}{\partial}_{\lambda}\psi{\partial}^{\lambda}\psi.$

In order to derive Morawetz estimates, we use the vectorfield
$X=\mathcal{F}(r)\partial_{r}$, scalar function $w$ and one-form $J$ given by

$\displaystyle\mathcal{F}=zu,\qquad w=z\partial_{r}u,\qquad J=v\partial_{r},$

where $z(r)$, $u(r)$, and $v(r)$ are well-chosen functions of $r$, so that the
divergence of the current can be written as

$\displaystyle|q|^{2}{\bf D}^{\mu}\mathcal{P}_{\mu}^{(X,w,J)}[\psi]$
$\displaystyle=$
$\displaystyle\mathcal{A}|{\partial}_{r}\psi|^{2}+\mathcal{U}^{{\alpha}{\beta}}({\partial}_{\alpha}\psi)({\partial}_{\beta}\psi)+\mathcal{V}|\psi|^{2}+\frac{1}{4}|q|^{2}{\mbox{d}iv\,}(J|\psi|^{2}),$
(6)

where $|q|^{2}=r^{2}+a^{2}\cos^{2}\theta$ and $\partial_{\alpha}$,
$\partial_{\beta}$ indicate $\partial_{t}$, $\partial_{\theta}$,
$\partial_{\phi}$, see Lemma 3.2 for the expression of the coefficients. The
axial symmetry of the solution crucially allows to simplify further the
principal term
$\mathcal{U}^{{\alpha}{\beta}}({\partial}_{\alpha}\psi)({\partial}_{\beta}\psi)$,
which for $z=\frac{\Delta}{(r^{2}+a^{2})^{2}}$ is given by

$\displaystyle\mathcal{U}^{{\alpha}{\beta}}({\partial}_{\alpha}\psi)({\partial}_{\beta}\psi)$
$\displaystyle=$
$\displaystyle\frac{u{\mathcal{T}}}{(r^{2}+a^{2})^{3}}\,|q|^{2}|\nabla\psi|^{2},$

where $|\nabla\psi|^{2}$ is defined by (12).

For the choice of functions $z$, $u$, $w$, we adapt a construction introduced
by Stogin [60] in sub-extremal Kerr for $|a|<M$ (also subsequently used and
adapted in [47][33][36]). In [60] the function $u$ is defined in terms of $w$
using the relation $w=z\partial_{r}u$ and imposed to vanish at $r_{trap}$ the
root of polynomial ${\mathcal{T}}$, while $z$ and $w$ are given respectively
by the geodesic potential and a differentiable function defined piecewise. In
the sub-extremal case, such construction implies the non-negativity of the
first three coefficients on the right hand side of (6), but still presents
remaining issues. In particular, the vectorfield $X=zu\partial_{r}$ blows up
logarithmically towards the horizon and the coefficient of the zero-th order
term $\mathcal{V}$ vanishes in an interval of $r$ outside the event horizon.
Such issues are solved in [60] by relying on the use of the redshift
vectorfield in sub-extremal Kerr: the vectorfield $X$ and function $w$ are
modified close to the event horizon to obtain a vectorfield which is regular
up to the event horizon (see also [53]) but such modification introduces a
negative contribution in the zero-th order term close to the event horizon.
The use of the redshift vectorfield as in [25] is then used to fix the
degeneracy of the $|\partial_{r}\psi|^{2}$ at the event horizon, which is then
used in an integrated local Hardy estimate to obtain positivity of the zero-th
order term.

Here in extremal Kerr, we set

$\displaystyle z=\frac{(r-M)^{2}}{(r^{2}+M^{2})^{2}},$

and explicitly define the differentiable function $w$ defined piecewise, see
(36). We show that in this case both the vectorfield $X$ and the function $w$
are regular up to the horizon, so the first issue appearing in the sub-
extremal case is not present here. Nevertheless, we still have the vanishing
of the zero-th order term in an interval of $r$, for which a non-degenerate
redshift estimate cannot be used as is absent in extremality. We rely instead
on a global pointwise Hardy inequality in $r\geq r_{e}>M$ which degenerates as
$r_{e}\to M$, capturing the degeneracy of the redshift. The Hardy inequality
is based on the use of the one-form $J=v\partial_{r}$ for an explicit function
$v$, see (39), solution to an ODE, that is used to obtain positivity of the
zero-th order term $\mathcal{V}$, fixing the second issue. We finally also add
a trapped control on the time derivative by using the Lagrangian of the wave
equation to prove Theorem 1.1. The above construction gives a simple
alternative proof of Aretakis’ result in [11] in physical-space, bypassing the
frequency decomposition and addressing the open problem raised by Aretakis
[11].

This paper is organized as follows: in Section 2 we recall the main properties
of the extremal Kerr spacetime and in Section 3 we review preliminary
computations on the wave equation and the vectorfield method. Finally, in
Section 4 we extend Stogin’s method [60] to derive the Morawetz estimates for
axisymmetric waves in extremal Kerr through a physical-space analysis.

Acknowledgements. The first author acknowledges the support of NSF Grant No.
DMS-2128386 and of a grant of the Simons Foundation (825870, EG).

##  2 Extremal Kerr spacetime

We recall here the main properties of the extremal Kerr spacetime which are
relevant to this paper. In Section 2.1 we introduce the metric in Boyer-
Lindquist and Eddington-Finkelstein coordinates and define the differential
structure of the manifold. In Section 2.2 we define the Killing vectorfields
of the metric and in Section 2.3 we recall the properties of trapped null
geodesics on extremal Kerr. For a more detailed presentation of properties of
extremal black holes see [9][10][11][12].

###  2.1 The manifold and the metric

The Kerr metric in Boyer-Lindquist coordinates $(t,r,\theta,\phi)$ takes the
form

$\displaystyle\begin{split}{\bf
g}_{M,a}&=-\frac{\Delta-a^{2}\sin^{2}\theta}{|q|^{2}}dt^{2}-\frac{2a\sin^{2}\theta}{|q|^{2}}\left((r^{2}+a^{2})-\Delta\right)dtd\phi+\frac{|q|^{2}}{\Delta}dr^{2}+|q|^{2}d\theta^{2}\\\
&+\frac{\sin^{2}\theta}{|q|^{2}}\left((r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right)d\phi^{2},\end{split}$ (7)

where

$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle
r^{2}-2Mr+a^{2}=(r-r_{+})(r-r_{-}),\qquad|q|^{2}=r^{2}+a^{2}\cos^{2}\theta.$

and $r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$.

The Kerr metric represent a stationary and rotating black hole of mass $M$ and
angular momentum $Ma$. For $|a|<M$ the metric describes the sub-extremal Kerr
spacetime, for $|a|=M$ the extremal Kerr and for $|a|>M$ the spacetime
contains a naked singularity. If $a=0$ we obtain the Schwarzschild solution.

If $|a|\leq M$, to remove the coordinate singularity at $\Delta=0$ describing
the black hole event horizon, one can define the functions

$\displaystyle
r^{*}=\int\frac{r^{2}+a^{2}}{\Delta},\qquad\phi^{*}=\phi+\int\frac{a}{\Delta},\qquad
v=t+r^{*}$

and obtain the Kerr metric in the ingoing Eddington-Finkelstein coordinates
$(v,r,\theta,\phi^{*})$

$\begin{split}{\bf
g}_{M,a}&=-\frac{\Delta-a^{2}\sin^{2}\theta}{|q|^{2}}dv^{2}+2dvdr-\frac{2a\sin^{2}\theta\left((r^{2}+a^{2})-\Delta\right)}{|q|^{2}}dvd\phi^{*}\\\
&-2a\sin^{2}\theta
drd\phi^{*}+|q|^{2}d\theta^{2}+\frac{\sin^{2}\theta}{|q|^{2}}\left((r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right)(d\phi^{*})^{2},\end{split}$ (8)

which is regular at the horizon.

From the form of the Kerr metric in Boyer-Lindquist coordinates given by (7),
one can deduce [4] that its conformal inverse $|q|^{2}{\bf g}_{M,a}^{-1}$ can
be written as

$\displaystyle|q|^{2}{\bf g}_{M,a}^{{\alpha}{\beta}}$ $\displaystyle=$
$\displaystyle\Delta\partial_{r}^{\alpha}\partial_{r}^{\beta}+\frac{1}{\Delta}\mathcal{R}^{{\alpha}{\beta}}$
(9)

where

$\displaystyle\mathcal{R}^{{\alpha}{\beta}}$ $\displaystyle=$
$\displaystyle-(r^{2}+a^{2})^{2}\partial_{t}^{\alpha}\partial_{t}^{\beta}-2a(r^{2}+a^{2})\partial_{t}^{({\alpha}}\partial_{\phi}^{{\beta})}-a^{2}\partial_{\phi}^{\alpha}\partial_{\phi}^{\beta}+\Delta
O^{{\alpha}{\beta}},$ (10) $\displaystyle O^{{\alpha}{\beta}}$
$\displaystyle=$
$\displaystyle\partial_{\theta}^{\alpha}\partial_{\theta}^{\beta}+\frac{1}{\sin^{2}\theta}\partial_{\phi}^{\alpha}\partial_{\phi}^{\beta}+2a\partial_{t}^{({\alpha}}\partial_{\phi}^{{\beta})}+a^{2}\sin^{2}\theta\partial_{t}^{\alpha}\partial_{t}^{\beta},$
(11)

where $O^{{\alpha}{\beta}}$ is related to the hidden Carter symmetry of the
Kerr spacetime. We denote

$\displaystyle
O^{{\alpha}{\beta}}({\partial}_{\alpha}\psi)({\partial}_{\beta}\psi)=|\partial_{\theta}\psi|^{2}+\big{|}\frac{1}{\sin\theta}\partial_{\phi}\psi+a\sin\theta\partial_{t}\psi\big{|}^{2}=:|q|^{2}|\nabla\psi|^{2}.$
(12)

We now describe the differential structure of the metric. Given standard
spherical coordinates $(\theta,\phi^{*})$ on the sphere $\mathbb{S}^{2}$ and
$(v,r)$ global coordinates system on $\mathbb{R}^{2}$, the ambient manifold is
defined to be
$\mathcal{N}=\\{(v,r,\theta,\phi^{*})\in\mathbb{R}\times\mathbb{R}\times\mathbb{S}^{2}\setminus\\{\mathbb{R}\times\\{0\\}\times
S_{eq}\\}\\}$, where $S_{eq}=\mathbb{S}^{2}\cap\\{\theta=\frac{\pi}{2}\\}$
denotes the equator of the sphere.

In the case of extremal Kerr spacetimes, we have

$\displaystyle\Delta=(r-M)^{2}$ (13)

and the roots of $\Delta=0$ degenerate to $r_{+}=r_{-}=M$. The event horizon
is defined by $\mathcal{H}^{+}=\mathcal{N}\cap\\{r=M\\}$, the black hole
region corresponds to $\mathcal{N}\cap\\{r<M\\}$ and the exterior region
(covered by the Boyer-Lindquist coordinates) is given by
$\mathcal{D}=\mathcal{N}\cap\\{r>M\\}$.

###  2.2 The Killing vectorfields

The coordinate vectorfields $T=\partial_{v}$ and $Z=\partial_{\phi^{*}}$
coincide with the coordinate vectorfields $\partial_{t}$ and $\partial_{\phi}$
in Boyer-Lindquist coordinates, which are manifestly Killing for the metric
(7). The stationary Killing vectorfield $T=\partial_{t}$ is asymptotically
timelike as $r\to\infty$, and spacelike close to the horizon, in the
ergoregion $\\{\Delta-a^{2}\sin^{2}\theta<0\\}$.

The vectorfield
$\widehat{T}:={\partial}_{t}+\frac{a}{r^{2}+a^{2}}{\partial}_{\phi}$
satisfies, see for example Proposition 3.2.2 of [39],

$\displaystyle{\bf g}_{M,a}(\widehat{T},\widehat{T})$ $\displaystyle=$
$\displaystyle-\frac{\Delta|q|^{2}}{(r^{2}+a^{2})^{2}},$ (14)

and is therefore timelike in the exterior region $\mathcal{D}$ and null on the
horizon $\mathcal{H}^{+}$. In particular, its restriction to the event
horizon, also called the Hawking vectorfield

$\displaystyle\widehat{T}_{\mathcal{H}}:=\partial_{t}+\omega_{\mathcal{H}}\partial_{\phi},\qquad\text{with}\quad\omega_{\mathcal{H}}=\frac{a}{r_{+}^{2}+a^{2}},$

is a Killing vectorfield which is null and normal to the horizon.

In the extremal case, the angular velocity $\omega_{\mathcal{H}}$ of the
horizon is given by $\omega_{\mathcal{H}}=\frac{1}{2M}$ and we have
$\nabla_{\widehat{T}_{\mathcal{H}}}\widehat{T}_{\mathcal{H}}=\kappa\widehat{T}_{\mathcal{H}}=0$
along the horizon, where $\kappa=\frac{r_{+}-r_{-}}{2(r_{+}^{2}+a^{2})}$ is
the surface gravity, which is positive in the sub-extremal range and vanishes
in the extremal case.

###  2.3 Trapped null geodesics

In Kerr spacetime there exist orbital null geodesics, i.e. geodesics for which
the radial coordinate $r$ remains constant. Because of the integrability of
the geodesic flow due to the presence of the Carter tensor [17], we can give
the following characterization of trapped null geodesics in Kerr spacetime.

######  Lemma 2.1 (Lemma 3.8.3 in [39]).

Let $\gamma(\lambda)$ be a null geodesics in Kerr spacetime whose constant of
motions

$\displaystyle{\bf e}:=-{\bf
g}(\dot{\gamma},\partial_{t}),\qquad{\bf\ell_{z}}:=-{\bf
g}(\dot{\gamma},\partial_{\phi})$

denote its energy and azimuthal angular momentum respectively. Then $\gamma$
is an orbital null geodesic if it satisfies

$\displaystyle{\mathcal{T}}_{{\bf
e},{\bf\ell_{z}}}:=\big{(}r^{3}-3Mr^{2}+a^{2}r+Ma^{2}\big{)}{\bf
e}-(r-M)a{\bf\ell_{z}}=0.$ (15)

The orbital null geodesics obtained above are trapped, i.e. neither cross the
event horizon nor terminate at null infinity. From (15) we can see that for
$a=0$ trapped null geodesics all concentrate at $\\{r=3M\\}$, which is the
photon sphere of Schwarzschild spacetime. On the other hand, for $|a|\neq 0$
there are null geodesics with constant $r$ for an open range of $r$.
Nevertheless, if ${\bf\ell_{z}}=0$, i.e. for trapped null geodesics orthogonal
to the axial Killing vectorfield, the trapped region defined by (15) reduces
to a hypersurface defined by

$\displaystyle{\mathcal{T}}:=r^{3}-3Mr^{2}+a^{2}r+Ma^{2}=0.$ (16)

Observe that the polynomial ${\mathcal{T}}$ has a unique single root in the
exterior of the black hole region, and we denote it by $r_{trap}$. Trapped
null geodesics constitute an obstruction to decay for the high frequency limit
of solutions to the wave equation. For axisymmetric waves, the trapping
obstruction simplifies as it concentrates on the hypersurface
${\mathcal{T}}=0$ in physical-space, becoming an effective photon sphere.

In the extremal Kerr for $|a|=M$, the trapping hypersurface becomes

$\displaystyle{\mathcal{T}}=r^{3}-3Mr^{2}+M^{2}r+M^{3}=(r-M)(r^{2}-2Mr-M^{2})$

which vanishes at $r_{trap}=(1+\sqrt{2})M$.

##  3 Preliminaries

We recall here some preliminaries concerning the wave equation. In Section 3.1
we introduce the wave equation operator and the foliation in extremal Kerr and
in Section 3.2 we recall the main notations of the vectorfield method. In
Section 3.3 we collect preliminaries computations for the derivation of the
Morawetz estimates obtained in Section 4.

###  3.1 The wave equation

The wave operator for a scalar function $\psi$ on a Lorentzian manifold is
given by

$\displaystyle\square_{\bf g}\psi=\frac{1}{\sqrt{-\det{\bf
g}}}\partial_{\alpha}((\sqrt{-\det{\bf g}}){\bf
g}^{{\alpha}{\beta}}\partial_{\beta}\psi).$

From the expression for the inverse metric (9), we deduce that the wave
operator for the Kerr metric in Boyer-Lindquist coordinates
$(t,r,\theta,\phi)$ is given by

$\begin{split}|q|^{2}\square_{{\bf
g}_{M,a}}&={\partial}_{r}(\Delta{\partial}_{r})+\frac{1}{\Delta}\Big{(}-(r^{2}+a^{2})^{2}{\partial}^{2}_{t}-2a(r^{2}+a^{2}){\partial}_{t}{\partial}_{\phi}-a^{2}{\partial}_{\phi}^{2}\Big{)}\\\
&+\frac{1}{\sin\theta}{\partial}_{\theta}(\sin\theta{\partial}_{\theta})+\frac{1}{\sin^{2}\theta}{\partial}^{2}_{\phi}+2a\partial_{t}\partial_{\phi}+a^{2}\sin^{2}\theta{\partial}^{2}_{t}.\end{split}$
(17)

In ingoing Eddington-Finkelstein coordinates $(v,r,\theta,\phi^{*})$, the wave
opeator is given by

$\displaystyle\begin{split}|q|^{2}\square_{{\bf
g}_{M,a}}&={\partial}_{r}(\Delta{\partial}_{r})+2(r^{2}+a^{2}){\partial}_{v}{\partial}_{r}+2a{\partial}_{r}{\partial}_{\phi^{*}}+2r{\partial}_{v}\\\
&+\frac{1}{\sin\theta}{\partial}_{\theta}(\sin\theta{\partial}_{\theta})+\frac{1}{\sin^{2}\theta}{\partial}^{2}_{\phi^{*}}+2a{\partial}_{v}{\partial}_{\phi^{*}}+a^{2}\sin^{2}\theta{\partial}_{v}^{2}.\end{split}$
(18)

Let $\Sigma_{0}$ be a closed connected axisymmetric spacelike hypersurface in
$({\mathcal{D}}\cup\mathcal{H}^{+})$ which crosses the event horizon
$\mathcal{H}^{+}$ and terminates at null infinity. We define the region
$\mathcal{M}=J^{+}(\Sigma_{0})\cap({\mathcal{D}}\cup\mathcal{H}^{+})$, and
consider the foliation $\Sigma_{\tau}=\phi_{\tau}^{T}(\Sigma_{0})$, where
$\phi_{\tau}^{T}$ is the flow of $T$. Since $T$ is Killing, the hypersurfaces
$\Sigma_{\tau}$ are all isometric to $\Sigma_{0}$. We denote by
$n_{\Sigma_{\tau}}$ the future directed unit vector field normal to
$\Sigma_{\tau}$. By convention, along the event horizon $\mathcal{H}^{+}$ we
choose $n_{\mathcal{H}^{+}}=\widehat{T}_{\mathcal{H}}$. We define the regions
$\mathcal{M}(0,\tau)=\cup_{0\leq\tilde{\tau}\leq\tau}\Sigma_{\tilde{\tau}}$,
$\mathcal{H}^{+}(0,\tau)=\mathcal{H}^{+}\cap\mathcal{M}(0,\tau)$ and
$\mathcal{I}^{+}(0,\tau)=\mathcal{I}^{+}\cap\mathcal{M}^{+}(0,\tau)$.

In what follows we consider axisymmetric solutions to the wave equation in
extremal Kerr, i.e.

$\displaystyle\square_{{\bf g}}\psi=0,\qquad\partial_{\phi}\psi=0,$

where ${\bf g}$ denotes the metric of the extremal Kerr spacetime. We consider
the Cauchy problem for the wave equation in $\mathcal{M}$ with axisymmetric
initial data prescribed on $\Sigma_{0}$,

$\displaystyle\psi|_{\Sigma_{0}}=\psi_{0}\in H^{k}_{loc}(\Sigma_{0}),\qquad
n_{\Sigma_{0}}\psi|_{\Sigma_{0}}=\psi_{1}\in H^{k-1}_{loc}(\Sigma_{0}),$

for $k\geq 2$ and assuming that $\lim_{x\to\mathscr{I}^{+}}r\psi^{2}(x)=0$.
Standard results imply well-posedness for the above Cauchy problem.

###  3.2 The vectorfield method

The vectorfield method is based on applying the divergence theorem in a causal
domain such as $\mathcal{M}(0,\tau)$, to certain energy currents, which are
constructed from the energy momentum tensor. The energy-momentum tensor
associated to the wave equation $\square_{\bf g}\psi=0$ is given by

$\displaystyle\mathcal{Q}[\psi]_{\mu\nu}$ $\displaystyle=$
$\displaystyle{\partial}_{\mu}\psi{\partial}_{\nu}\psi-\frac{1}{2}{\bf
g}_{\mu\nu}{\partial}_{\lambda}\psi{\partial}^{\lambda}\psi.$ (19)

If $\square_{\bf g}\psi=0$, the energy-momentum $\mathcal{Q}[\psi]_{\mu\nu}$
is divergence free.

Let $X$ be a vectorfield, $w$ be a scalar function and $J$ a one-form. The
current associated to $(X,w,J)$ is defined as

$\displaystyle\mathcal{P}_{\mu}^{(X,w,J)}[\psi]$ $\displaystyle=$
$\displaystyle\mathcal{Q}[\psi]_{\mu\nu}X^{\nu}+\frac{1}{2}w\psi{\partial}_{\mu}\psi-\frac{1}{4}({\partial}_{\mu}w)|\psi|^{2}+\frac{1}{4}J_{\mu}|\psi|^{2}.$
(20)

The energy associated to $(X,w,J)$ on the hypersurface $\Sigma_{\tau}$ is

$\displaystyle E^{(X,w,J)}[\psi](\tau)$ $\displaystyle=$
$\displaystyle\int_{\Sigma_{\tau}}\mathcal{P}^{(X,w,J)}_{\mu}[\psi]n_{\Sigma_{\tau}}^{\mu},$

where $n_{\Sigma_{\tau}}$ denotes the future directed timelike unit normal to
$\Sigma_{\tau}$.

A standard computation gives the following divergence of $\mathcal{P}$ for a
solution to the wave equation $\square_{g}\psi=0$, see for example [47][39],

$\displaystyle{\bf
D}^{\mu}\mathcal{P}_{\mu}^{(X,w,J)}[\psi]=\frac{1}{2}\mathcal{Q}[\psi]\cdot\,^{(X)}\pi-\frac{1}{4}\square_{\bf
g}w|\psi|^{2}+\frac{1}{2}w({\partial}_{\lambda}\psi{\partial}^{\lambda}\psi)+\frac{1}{4}{\mbox{d}iv\,}(J|\psi|^{2}),$
(21)

where $\,{}^{(X)}\pi_{\mu\nu}={\bf D}_{(\mu}X_{\nu)}$ is the deformation
tensor of the vectorfield $X$. Recall that if $X$ is a Killing vectorfield,
then $\,{}^{(X)}\pi=0$.

By applying the divergence theorem to $\mathcal{P}_{\mu}^{(X,w,J)}$ to
$\mathcal{M}(0,\tau)$ one obtains the associated energy identity:

$\displaystyle
E[\psi](\tau)+\int_{\mathcal{H}^{+}(0,\tau)}\mathcal{P}_{\mu}[\psi]n_{\mathcal{H}^{+}}^{\mu}+\int_{\mathcal{I}^{+}(0,\tau)}\mathcal{P}_{\mu}[\psi]n_{\mathcal{I}^{+}}^{\mu}+\int_{\mathcal{M}(0,\tau)}{\bf
D}^{\mu}\mathcal{P}_{\mu}[\psi]=E[\psi](0),$ (22)

where we suppressed the superscript $(X,w,J)$ and the induced volume forms are
to be understood. By convention, along the event horizon we choose
$n_{\mathcal{H}^{+}}=T+\frac{a}{M^{2}+a^{2}}Z$.

###  3.3 Preliminary computations for the Morawetz estimates

In deriving Morawetz estimates for the wave equation we make use of the
vectorfield $X=\mathcal{F}(r){\partial}_{r}$, for a well chosen function
$\mathcal{F}$. We collect here some relevant computations (see also
[4][60][39]) which will be used in the next section.

######  Lemma 3.1.

For $X=\mathcal{F}(r){\partial}_{r}$, we have

$\,{}^{(X)}\pi^{{\alpha}{\beta}}=|q|^{-2}\Big{(}2\Delta^{3/2}{\partial}_{r}\big{(}\frac{\mathcal{F}}{\Delta^{1/2}}\big{)}{\partial}_{r}^{\alpha}{\partial}_{r}^{\beta}-\mathcal{F}{\partial}_{r}\big{(}\frac{1}{\Delta}\mathcal{R}^{{\alpha}{\beta}}\big{)}\Big{)}+|q|^{-2}X\big{(}|q|^{2}\big{)}{\bf
g}^{{\alpha}{\beta}},$ (23)

and therefore

$\begin{split}|q|^{2}\mathcal{Q}[\psi]\cdot\,^{(X)}\pi&=2\Delta^{3/2}{\partial}_{r}\big{(}\frac{\mathcal{F}}{\Delta^{1/2}}\big{)}|{\partial}_{r}\psi|^{2}-\mathcal{F}{\partial}_{r}\big{(}\frac{1}{\Delta}\mathcal{R}^{{\alpha}{\beta}}\big{)}{\partial}_{\alpha}\psi{\partial}_{\beta}\psi\\\
&+\Big{(}X\big{(}|q|^{2}\big{)}-|q|^{2}({\mbox{d}iv\,}X)\Big{)}{\partial}_{\lambda}\psi{\partial}^{\lambda}\psi.\end{split}$
(24)

###### Proof.

Using the expression for the inverse metric (9), we compute

$\displaystyle\mathcal{L}_{X}(|q|^{2}{\bf g}^{{\alpha}{\beta}})$
$\displaystyle=$
$\displaystyle\mathcal{L}_{X}\big{(}\Delta{\partial}_{r}^{\alpha}{\partial}_{r}^{\beta}\big{)}+\mathcal{L}_{X}\big{(}\frac{1}{\Delta}\mathcal{R}^{{\alpha}{\beta}}\big{)}=X(\Delta){\partial}_{r}^{\alpha}{\partial}_{r}^{\beta}+\Delta[X,{\partial}_{r}]^{\alpha}{\partial}_{r}^{\beta}+\Delta{\partial}_{r}^{\alpha}[X,{\partial}_{r}]^{\beta}+\mathcal{L}_{X}\big{(}\frac{1}{\Delta}\mathcal{R}^{{\alpha}{\beta}}\big{)}.$

For $X=\mathcal{F}\partial_{r}$, we obtain

$\displaystyle\mathcal{L}_{X}(|q|^{2}{\bf g}^{{\alpha}{\beta}})$
$\displaystyle=$
$\displaystyle\mathcal{F}({\partial}_{r}\Delta){\partial}_{r}^{\alpha}{\partial}_{r}^{\beta}+\Delta[\mathcal{F}{\partial}_{r},{\partial}_{r}]^{\alpha}{\partial}_{r}^{\beta}+\Delta{\partial}_{r}^{\alpha}[\mathcal{F}{\partial}_{r},{\partial}_{r}]^{\beta}+\mathcal{F}\mathcal{L}_{{\partial}_{r}}\big{(}\frac{1}{\Delta}\mathcal{R}^{{\alpha}{\beta}}\big{)}$
$\displaystyle=$
$\displaystyle\mathcal{F}({\partial}_{r}\Delta){\partial}_{r}^{\alpha}{\partial}_{r}^{\beta}-2\Delta({\partial}_{r}\mathcal{F}){\partial}_{r}^{\alpha}{\partial}_{r}^{\beta}+\mathcal{F}\partial_{r}\big{(}\frac{1}{\Delta}\mathcal{R}^{{\alpha}{\beta}}\big{)}$
$\displaystyle=$
$\displaystyle-2\Delta^{3/2}{\partial}_{r}\big{(}\frac{\mathcal{F}}{\Delta^{1/2}}\big{)}{\partial}_{r}^{\alpha}{\partial}_{r}^{\beta}+\mathcal{F}\partial_{r}\big{(}\frac{1}{\Delta}\mathcal{R}^{{\alpha}{\beta}}\big{)}.$

By writing

$\,{}^{(X)}\pi^{{\alpha}{\beta}}$ $\displaystyle=$
$\displaystyle-\mathcal{L}_{X}\big{(}|q|^{-2}|q|^{2}{\bf
g}^{{\alpha}{\beta}}\big{)}=-|q|^{-2}\mathcal{L}_{X}\big{(}|q|^{2}{\bf
g}^{{\alpha}{\beta}}\big{)}-|q|^{2}\mathcal{L}_{X}\big{(}|q|^{-2}\big{)}{\bf
g}^{{\alpha}{\beta}}$

we obtain the stated expressions for the deformation tensors.

Finally we write

$\displaystyle\mathcal{Q}[\psi]\cdot\,^{(X)}\pi$ $\displaystyle=$
$\,{}^{(X)}\pi^{{\alpha}{\beta}}{\partial}_{\alpha}\psi{\partial}_{\beta}\psi-({\mbox{d}iv\,}X){\partial}_{\lambda}\psi{\partial}^{\lambda}\psi$

since ${\bf g}_{\mu\nu}\,^{(X)}\pi^{\mu\nu}={\bf g}_{\mu\nu}{\bf
D}^{(\mu}X^{\nu)}=2{\mbox{d}iv\,}X$.

∎

######  Lemma 3.2.

Let $z(r)$, $u(r)$, $v(r)$ be functions of $r$. Then for

$\displaystyle X=\mathcal{F}\partial_{r},\qquad\quad\mathcal{F}=zu,\qquad\quad
w=z{\partial}_{r}u,\qquad\quad J=v\partial_{r}$ (25)

the divergence of $\mathcal{P}_{\mu}^{(X,w,J)}[\psi]$ satisfies

$\displaystyle|q|^{2}{\bf D}^{\mu}\mathcal{P}_{\mu}^{(X,w,J)}[\psi]$
$\displaystyle=$
$\displaystyle\mathcal{A}|{\partial}_{r}\psi|^{2}+\mathcal{U}^{{\alpha}{\beta}}({\partial}_{\alpha}\psi)({\partial}_{\beta}\psi)+\mathcal{V}|\psi|^{2}+\frac{1}{4}|q|^{2}{\mbox{d}iv\,}(J|\psi|^{2}),$
(26)

where

$\displaystyle\mathcal{A}$ $\displaystyle=$ $\displaystyle
z^{1/2}\Delta^{3/2}\partial_{r}\left(\frac{z^{1/2}u}{\Delta^{1/2}}\right),$
(27) $\displaystyle\mathcal{U}^{{\alpha}{\beta}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}u{\partial}_{r}\left(\frac{z}{\Delta}\mathcal{R}^{{\alpha}{\beta}}\right),$
(28) $\displaystyle\mathcal{V}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}{\partial}_{r}\big{(}\Delta{\partial}_{r}w\big{)}=-\frac{1}{4}{\partial}_{r}\big{(}\Delta{\partial}_{r}\big{(}z{\partial}_{r}u\big{)}\big{)},$
(29) $\displaystyle\frac{1}{4}|q|^{2}{\mbox{d}iv\,}(J|\psi|^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{4}|q|^{2}\Big{(}2v\psi\cdot\nabla_{r}\psi+\big{(}{\partial}_{r}v+\frac{2r}{|q|^{2}}v\big{)}|\psi|^{2}\Big{)}.$
(30)

###### Proof.

Using (21) and (24) we compute for $J=0$

$\displaystyle|q|^{2}{\bf
D}^{\mu}\mathcal{P}_{\mu}^{(\mathcal{F}\partial_{r},w,J=0)}[\psi]$
$\displaystyle=$
$\displaystyle\Delta^{3/2}{\partial}_{r}\big{(}\frac{\mathcal{F}}{\Delta^{1/2}}\big{)}|{\partial}_{r}\psi|^{2}-\frac{1}{2}\mathcal{F}{\partial}_{r}\big{(}\frac{1}{\Delta}\mathcal{R}^{{\alpha}{\beta}}\big{)}{\partial}_{\alpha}\psi{\partial}_{\beta}\psi-\frac{1}{4}|q|^{2}\square_{\bf
g}w|\psi|^{2}$
$\displaystyle+\frac{1}{2}\Big{(}X\big{(}|q|^{2}\big{)}-|q|^{2}({\mbox{d}iv\,}X)+|q|^{2}w\Big{)}{\partial}_{\lambda}\psi{\partial}^{\lambda}\psi.$

By defining an intermediate function $w_{int}$ as

$\displaystyle\frac{1}{2}\Big{(}X\big{(}|q|^{2}\big{)}-|q|^{2}{\mbox{d}iv\,}X+|q|^{2}w\Big{)}=\frac{1}{2}|q|^{2}\Big{(}|q|^{-2}X\big{(}|q|^{2}\big{)}-{\mbox{d}iv\,}X+w\Big{)}=:-\frac{1}{2}|q|^{2}w_{int},$

and using (9) to write

$\displaystyle|q|^{2}{\partial}_{\lambda}\psi{\partial}^{\lambda}\psi$
$\displaystyle=$ $\displaystyle|q|^{2}{\bf
g}^{\lambda\mu}{\partial}_{\lambda}\psi{\partial}_{\nu}\psi=\Delta|{\partial}_{r}\psi|^{2}+\frac{1}{\Delta}\mathcal{R}^{{\alpha}{\beta}}{\partial}_{\alpha}\psi{\partial}_{\beta}\psi,$

we obtain

$\displaystyle|q|^{2}{\bf
D}^{\mu}\mathcal{P}_{\mu}^{(\mathcal{F}\partial_{r},w,J=0)}[\psi]$
$\displaystyle=$
$\displaystyle\mathcal{A}|{\partial}_{r}\psi|^{2}+\mathcal{U}^{{\alpha}{\beta}}({\partial}_{\alpha}\psi)({\partial}_{\beta}\psi)+\mathcal{V}|\psi|^{2},$

where

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\Delta^{3/2}{\partial}_{r}\big{(}\frac{\mathcal{F}}{\Delta^{1/2}}\big{)}-\frac{1}{2}w_{int}\Delta$
$\displaystyle\mathcal{U}^{{\alpha}{\beta}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\mathcal{F}{\partial}_{r}\left(\frac{1}{\Delta}\mathcal{R}^{{\alpha}{\beta}}\right)-\frac{1}{2}w_{int}\frac{1}{\Delta}\mathcal{R}^{{\alpha}{\beta}}$
$\displaystyle\mathcal{V}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}|q|^{2}\square_{\bf g}w.$

Now the above can be written as

$\displaystyle\mathcal{U}^{{\alpha}{\beta}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\mathcal{F}z^{-1}\partial_{r}\left(\frac{z}{\Delta}\mathcal{R}^{{\alpha}{\beta}}\right)+\frac{1}{2}\left(\mathcal{F}z^{-1}\partial_{r}z-w_{int}\right)\frac{\mathcal{R}^{{\alpha}{\beta}}}{\Delta}.$

Setting $\mathcal{F}=zu$ for a function $u$, and choosing
$w_{int}=\mathcal{F}z^{-1}\partial_{r}z=u{\partial}_{r}z$, we deduce the
stated expression for $\mathcal{U}^{{\alpha}{\beta}}$ in (28). With such
choices for $\mathcal{F}$ and $w_{int}$, we compute

$\displaystyle w$ $\displaystyle=$ $\displaystyle|q|^{2}{\bf
D}_{\alpha}\big{(}|q|^{-2}\mathcal{F}{\partial}_{r}^{\alpha}\big{)}-w_{int}=|q|^{2}{\partial}_{r}\big{(}|q|^{-2}\mathcal{F}\big{)}+\mathcal{F}({\bf
D}_{\alpha}{\partial}_{r}^{\alpha})-u{\partial}_{r}z$ $\displaystyle=$
$\displaystyle|q|^{2}{\partial}_{r}\big{(}|q|^{-2}zu\big{)}+zu|q|^{-2}{\partial}_{r}\big{(}|q|^{2}\big{)}-u{\partial}_{r}z={\partial}_{r}\big{(}zu\big{)}-u{\partial}_{r}z=z{\partial}_{r}u,$

where we used that ${\bf
D}_{\alpha}{\partial}_{r}^{\alpha}=\frac{1}{\sqrt{|{\bf
g}|}}{\partial}_{r}\big{(}\sqrt{|{\bf
g}|}\big{)}=\frac{1}{|q|^{2}}{\partial}_{r}\big{(}|q|^{2}\big{)}$. We also
compute

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\partial_{r}\left(\frac{\mathcal{F}}{\Delta^{1/2}}\right)\Delta^{3/2}-\frac{1}{2}\Delta
w_{int}=\partial_{r}\left(\frac{zu}{\Delta^{1/2}}\right)\Delta^{3/2}-\frac{1}{2}\Delta(\partial_{r}z)u$
$\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{r}z\frac{u}{\Delta^{1/2}}\Delta^{3/2}+z^{1/2}\partial_{r}\left(\frac{z^{1/2}u}{\Delta^{1/2}}\right)\Delta^{3/2}-\frac{1}{2}\Delta(\partial_{r}z)u=z^{1/2}\Delta^{3/2}\partial_{r}\left(\frac{z^{1/2}u}{\Delta^{1/2}}\right),$

and

$\displaystyle|q|^{2}\square_{\bf
g}w={\partial}_{r}\big{(}\Delta{\partial}_{r}w\big{)}={\partial}_{r}\big{(}\Delta{\partial}_{r}\big{(}z{\partial}_{r}u\big{)}\big{)},$

as stated. Finally, for $J=v\partial_{r}$ we compute

$\displaystyle{\bf D}^{\mu}(|\psi|^{2}J_{\mu})$ $\displaystyle=$
$\displaystyle 2v\psi\cdot\nabla_{r}\psi+|\psi|^{2}{\mbox{d}iv\,}J.$

Using that
${\mbox{d}iv\,}J=\frac{1}{|q|^{2}}{\partial}_{r}\big{(}|q|^{2}v)={\partial}_{r}v+\frac{2r}{|q|^{2}}v$,
we obtain the stated identity.

∎

###  3.4 Boundedness of the energy

We show here how to obtain boundedness of the energy associated to $T$ for
axially symmetric solutions to the wave equation in extremal Kerr. Such
statement already appeared in [11] and in axial symmetry can be proved
independently of the Morawetz estimates.

Even though the Killing vectorfield $T$ fails to be everywhere timelike and as
a consequence the energy $E^{(T)}[\psi]$ associated to it fails to be non-
negative definite, superradiance is effectively absent for axially symmetric
solutions. In fact, let $n$ be a vector orthogonal to $Z$. Then for an axially
symmetric $\psi$ we have

$\displaystyle
E^{(Z)}[\psi](\tau)=\int_{\Sigma_{\tau}}\mathcal{Q}[\psi]_{\mu\nu}Z^{\nu}n^{\mu}_{\Sigma_{\tau}}=\int_{\Sigma_{\tau}}Z(\psi)n_{\Sigma_{\tau}}(\psi)-\frac{1}{2}{\bf
g}(Z,n_{\Sigma_{\tau}}){\partial}_{\lambda}\psi{\partial}^{\lambda}\psi=0.$

On the other hand, the Hawking vectorfield $\widehat{T}$ is causal everywhere
in the exterior and using that
$\mathcal{Q}[\psi]_{\mu\nu}V_{1}^{\mu}V_{2}^{\nu}$ is non-negative if $V_{1}$,
$V_{2}$ are causal this implies

$\displaystyle E^{(T)}[\psi](\tau)=E^{(\widehat{T})}[\psi](\tau)\geq 0,$

and similarly
$\int_{\mathcal{H}^{+}(0,\tau)}\mathcal{P}^{(T)}_{\mu}[\psi]n_{\mathcal{H}^{+}}^{\mu},\int_{\mathcal{I}^{+}(0,\tau)}\mathcal{P}^{(T)}_{\mu}[\psi]n_{\mathcal{I}^{+}}^{\mu}\geq
0$.

Working in the $(v,r,\theta,\varphi^{*})$ coordinates, if
$n_{\Sigma_{\tau}}=n^{v}T+n^{r}Y+n^{\varphi}Z$, then for axially symmetric
solutions,

$\displaystyle E^{(T)}[\psi](\tau)$ $\displaystyle=$
$\displaystyle\int_{\Sigma_{\tau}}\mathcal{Q}[\psi]_{\mu\nu}T^{\nu}n^{\mu}_{\Sigma_{\tau}}=\int_{\Sigma_{\tau}}T(\psi)n_{\Sigma_{\tau}}(\psi)-\frac{1}{2}{\bf
g}(T,n_{\Sigma_{\tau}}){\partial}_{\lambda}\psi{\partial}^{\lambda}\psi$
$\displaystyle=$
$\displaystyle\int_{\Sigma_{\tau}}n^{v}|T(\psi)|^{2}+n^{r}T(\psi)Y(\psi)-\frac{1}{2}\big{(}n^{v}{\bf
g}(T,T)+n^{r}{\bf g}(T,Y)+n^{\varphi}{\bf
g}(T,Z)\big{)}{\partial}_{\lambda}\psi{\partial}^{\lambda}\psi,$

where from (8) we deduce that
${\partial}_{\lambda}\psi{\partial}^{\lambda}\psi=\frac{1}{|q|^{2}}\big{(}a^{2}\sin^{2}\theta|T\psi|^{2}+\Delta|Y\psi|^{2}+2(r^{2}+a^{2})T(\psi)Y(\psi)\big{)}+|\nabla\mkern-13.0mu/\,\psi|^{2}$.
Since the only contribution for $|Y\psi|^{2}$ comes from the term in
${\partial}_{\lambda}\psi{\partial}^{\lambda}\psi$, which vanishes at the
horizon, to have positivity of the energy we need the coefficient of
$T(\psi)Y(\psi)$ to vanish at the horizon too. We therefore obtain

$\displaystyle E^{(T)}[\psi](\tau)$ $\displaystyle\sim$
$\displaystyle\int_{\Sigma_{\tau}}|T\psi|^{2}+\left(1-\frac{M}{r}\right)^{2}|Y\psi|^{2}+|\nabla\mkern-13.0mu/\,\psi|^{2}.$

From the energy identity (22) applied to $X=T$, since
$\mathcal{E}^{(T,0)}[\psi]=0$ we then obtain

$\displaystyle
E^{(T)}[\psi](\tau)+\int_{\mathcal{H}^{+}(0,\tau)}\mathcal{P}^{(T)}_{\mu}[\psi]n_{\mathcal{H}^{+}}^{\mu}+\int_{\mathcal{I}^{+}(0,\tau)}\mathcal{P}^{(T)}_{\mu}[\psi]n_{\mathcal{I}^{+}}^{\mu}\leq
CE^{(T)}[\psi](0).$ (31)

As a consequence of the vanishing of the surface gravity of the Hawking
vectorfield at the horizon, the redshift that takes place there degenerates in
the extremal case. In particular, as shown in [11], there is no time invariant
timelike vectorfield $N$ such that $\mathcal{E}^{(N,0)}[\psi]$ is non-negative
on the horizon. However, one can still quantitatively capture the degenerate
redshift close to the horizon by using a current first introduced in [9], and
obtain a non-degenerate energy boundedness statement.

##  4 Morawetz estimates

We provide here the proof of our main result. In Section 4.1 we recall the
method introduced by Stogin [60] to construct the relevant functions in the
estimates and extend it to the extremal case and in Section 4.2 we complete
the construction with a new adapted global pointwise Hardy inequality and an
added trapped control on the time derivative of the solution.

###  4.1 Stogin’s construction

Recall Lemma 3.2, according to which for functions $z,u,v$ chosen as in (25),
the divergence of $\mathcal{P}_{\mu}^{(X,w,J)}[\psi]$ is given by

$\displaystyle|q|^{2}{\bf D}^{\mu}\mathcal{P}_{\mu}^{(X,w,J)}[\psi]$
$\displaystyle=$
$\displaystyle\mathcal{A}|{\partial}_{r}\psi|^{2}+\mathcal{U}^{{\alpha}{\beta}}({\partial}_{\alpha}\psi)({\partial}_{\beta}\psi)+\mathcal{V}|\psi|^{2}+\frac{1}{4}|q|^{2}{\mbox{d}iv\,}(J|\psi|^{2}),$

where $\mathcal{A}$, $\mathcal{U}$ and $\mathcal{V}$ are given as in (27),
(28), (29).

Following a standard choice in derivation of Morawetz estimates (see
[27][29][60][4][39]), we choose the function $z$ so that the coefficient of
$|\partial_{t}\psi|^{2}$ vanishes and the coefficient of $|\nabla\psi|^{2}$
degenerates at trapping. From (28) and (10), we have for axially symmetric
solutions

$\displaystyle\mathcal{U}^{{\alpha}{\beta}}({\partial}_{\alpha}\psi)({\partial}_{\beta}\psi)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}u{\partial}_{r}\left(\frac{z}{\Delta}\mathcal{R}^{{\alpha}{\beta}}\right)({\partial}_{\alpha}\psi)({\partial}_{\beta}\psi)$
$\displaystyle=$
$\displaystyle\frac{1}{2}u{\partial}_{r}\left(\frac{z}{\Delta}(r^{2}+M^{2})^{2}\right)|\partial_{t}\psi|^{2}-\frac{1}{2}u({\partial}_{r}z)\,O^{{\alpha}{\beta}}({\partial}_{\alpha}\psi)({\partial}_{\beta}\psi).$

So we set

$\displaystyle z=\frac{(r-M)^{2}}{(r^{2}+M^{2})^{2}},$

and obtain

$\displaystyle\mathcal{U}^{{\alpha}{\beta}}({\partial}_{\alpha}\psi)({\partial}_{\beta}\psi)$
$\displaystyle=$
$\displaystyle\frac{u{\mathcal{T}}}{(r^{2}+M^{2})^{3}}\,O^{{\alpha}{\beta}}({\partial}_{\alpha}\psi)({\partial}_{\beta}\psi)$
$\displaystyle=$
$\displaystyle\frac{u(r-M)(r^{2}-2Mr-M^{2})}{(r^{2}+M^{2})^{3}}\,|q|^{2}|\nabla\psi|^{2}.$

Observe that from (12) we have

$\displaystyle
O^{{\alpha}{\beta}}({\partial}_{\alpha}\psi)({\partial}_{\beta}\psi)=|q|^{2}|\nabla\psi|^{2}=(\partial_{\theta}\psi)^{2}+M^{2}\sin^{2}\theta(\partial_{t}\psi)^{2}.$

Using (27) and (29) we deduce for such choice of $z$,

$\displaystyle\mathcal{A}=\frac{(r-M)^{4}}{(r^{2}+M^{2})}\partial_{r}\big{(}\frac{u}{r^{2}+M^{2}}\big{)},\qquad\mathcal{V}=-\frac{1}{4}{\partial}_{r}\Big{(}(r-M)^{2}{\partial}_{r}w\Big{)}$
(32)

with

$\displaystyle w=\frac{(r-M)^{2}}{(r^{2}+M^{2})^{2}}{\partial}_{r}u.$ (33)

The main goal here is to choose the functions $u$, $w$ and $v$ so that the
divergence of $\mathcal{P}_{\mu}^{(X,w,J)}[\psi]$ is positive definite. For
the choice of functions $u$ and $w$ we make use of a construction due to
Stogin in the sub-extremal Kerr spacetime, see Lemma 5.2.6 in [60], also used
in [47] [36] [38]. In what follows, we adapt Stogin’s construction to the case
of extremal Kerr. Stogin’s construction fails to obtain a positive definite
term for $|\psi|^{2}$ in the entire exterior region and makes use of the
redshift estimate and local integrated Hardy inequality to fix this
deficiency. In the extremal case, because of the degenerate redshift estimate,
we need a new adapted Hardy inequality that we derive in Section 4.2.

In Stogin’s construction [60], the relation between $u$ and $w$ in (33) is
used to define $u$ in terms of $w$ and then treat $w$ as the free variable. In
order to have the coefficient of $|\nabla\psi|^{2}$ to be non-negative, the
function $u$ has to change sign at $r=r_{trap}$ and therefore we impose
following Stogin [60]:

$\displaystyle
u(r)=\int_{r_{trap}}^{r}\frac{(s^{2}+M^{2})^{2}}{(s-M)^{2}}w(s)ds.$ (34)

Further imposing the positivity of the function $w$ we obtain that $u$ is
increasing everywhere and changing sign at $r_{trap}$, which implies that
$\mathcal{U}^{{\alpha}{\beta}}({\partial}_{\alpha}\psi)({\partial}_{\beta}\psi)$
is non-negative.

Following Stogin, we now choose the function $w$ in order to have positivity
of $\mathcal{A}$, i.e. positivity of
$\partial_{r}\big{(}\frac{u}{r^{2}+M^{2}}\big{)}$. By defining
$\widetilde{\mathcal{A}}:=\frac{(r^{2}+M^{2})^{2}}{2r}\partial_{r}\left(\frac{u}{r^{2}+M^{2}}\right)$,
a straightforward computation shows that

$\displaystyle\partial_{r}\widetilde{\mathcal{A}}=(r^{2}+M^{2})\partial_{r}\left(\frac{w(r^{2}+M^{2})^{2}}{2r(r-M)^{2}}\right).$
(35)

Defining $r_{*}:=(2+\sqrt{3})M$ to be the point attaining the maximum of the
function $\frac{2r(r-M)^{2}}{(r^{2}+M^{2})^{2}}$, we define $w$ as the
positive $C^{1}$ function

$\displaystyle w=\begin{cases}\frac{1}{4M}\ \qquad&r\leq r_{*}\\\
\frac{2r(r-M)^{2}}{(r^{2}+M^{2})^{2}}\ \qquad&r>r_{*}.\end{cases}$ (36)

Since $r_{*}$ also attains the minimum of the function
$\frac{(r^{2}+M^{2})^{2}}{2r(r-M)^{2}}$, the above construction implies that
the function $\frac{w(r^{2}+M^{2})^{2}}{2r(r-M)^{2}}$ is constant for $r\geq
r_{*}$ and decreasing for $r\leq r_{*}$. From (35), one can deduce the same
behavior for $\widetilde{\mathcal{A}}$. We now show that the constant value of
this function is positive. We have

$\displaystyle\widetilde{\mathcal{A}}(r_{*})$ $\displaystyle=$
$\displaystyle\frac{(r_{*}^{2}+M^{2})^{2}}{2r_{*}}\partial_{r}\left(\frac{u}{r^{2}+M^{2}}\right)\Big{|}_{r=r_{*}}=\frac{(r_{*}^{2}+M^{2})}{2r_{*}}\partial_{r}u\Big{|}_{r=r_{*}}-u(r_{*})$
$\displaystyle=$
$\displaystyle\frac{(r_{*}^{2}+M^{2})^{3}}{2r_{*}(r_{*}-M)^{2}}w(r_{*})-w(r_{*})\int_{r_{trap}}^{r_{*}}\frac{(r^{2}+M^{2})^{2}}{(r-M)^{2}}dr$

Observe that since the function $\frac{(r^{2}+M^{2})^{2}}{(r-M)^{2}}$ is
increasing between $r_{trap}$ and $r_{*}$, we can bound the above by

$\displaystyle\widetilde{\mathcal{A}}(r_{*})$ $\displaystyle>$
$\displaystyle\frac{(r_{*}^{2}+M^{2})^{3}}{2r_{*}(r_{*}-M)^{2}}w(r_{*})-\frac{(r_{*}^{2}+M^{2})^{2}}{(r_{*}-M)^{2}}w(r_{*})(r_{*}-r_{trap})$
$\displaystyle=$
$\displaystyle\frac{(r_{*}^{2}+M^{2})^{2}}{2r_{*}(r_{*}-M)^{2}}w(r_{*})\Big{(}(r_{*}^{2}+M^{2})-2r_{*}(r_{*}-r_{trap})\Big{)}$
$\displaystyle=$
$\displaystyle\frac{(r_{*}^{2}+M^{2})^{2}}{4(r_{*}-M)^{2}}\big{(}1+\sqrt{2}-\sqrt{3}\big{)}=2r_{*}\big{(}1+\sqrt{2}-\sqrt{3}\big{)}M=c_{0}M^{2},$

where $c_{0}>0$ is a positive constant, explicitly given by
$c_{0}=2(2+\sqrt{3})\big{(}1+\sqrt{2}-\sqrt{3}\big{)}$, and where we used that
$r_{*}^{2}+M^{2}=4r_{*}M$, $(r_{*}-M)^{2}=2Mr_{*}$ and
$r_{*}-r_{trap}=(1+\sqrt{3}-\sqrt{2})M$.

Since
$\mathcal{A}=\frac{2r(r-M)^{4}}{(r^{2}+M^{2})^{3}}\widetilde{\mathcal{A}}$,
the above implies that $\mathcal{A}$ is non-negative, and more precisely:

$\displaystyle\mathcal{A}(r)\geq\frac{2r(r-M)^{4}}{(r^{2}+M^{2})^{3}}\widetilde{\mathcal{A}}(r_{*})\geq\frac{2c_{0}M^{2}r(r-M)^{4}}{(r^{2}+M^{2})^{3}}.$
(37)

Finally, we are left to analyze the positivity of $\mathcal{V}$. With the
choice of $w$ in (36) we compute explicitly

$\displaystyle\partial_{r}\Big{(}(r-M)^{2}\partial_{r}w\Big{)}=$
$\displaystyle\begin{cases}0\ &r\leq r_{*}\\\
-\frac{12M(r-M)^{2}(r^{4}-6M^{2}r^{2}+M^{4})}{(r^{2}+M^{2})^{4}}\
&r>r_{*}.\end{cases}$

Observe that the polynomial $r^{4}-6M^{2}r^{2}+M^{4}$ is positive for
$r>(1+\sqrt{2})M$, and since $r_{*}>(1+\sqrt{2})M$ the above is non-negative
everywhere in the exterior region.

Integrating the relation $u^{\prime}=\frac{(r^{2}+M^{2})^{2}}{(r-M)^{2}}w$
from (34) and using (36) we deduce a closed form for $u$:

$\displaystyle
u=\begin{cases}-\frac{M^{3}}{r-M}+\frac{5Mr}{4}+\frac{r^{2}}{4}+\frac{r^{3}}{12M}+2M^{2}\log(r-M)+C_{1},\
&r\leq r_{*}\\\ r^{2}+C_{2}\ &r\geq r_{*},\end{cases}$

where $C_{1}$, $C_{2}$ are suitable constants, such that $u(r_{trap})=0$ and
$u$ is continuous at $r_{*}$.

For $r$ close to $r_{+}$, we write
$u\approx-\frac{M^{3}}{r-M}+2M^{2}\log(r-M)+O(1)$, from which we deduce

$\displaystyle\frac{(r-M)^{4}}{(r^{2}+M^{2})}\partial_{r}\Big{(}\frac{u}{r^{2}+M^{2}}\Big{)}\approx$
$\displaystyle\frac{(r-M)^{4}}{(r^{2}+M^{2})}\left(\frac{M}{2}\frac{1}{(r-M)^{2}}+O(\frac{1}{r-M})\right)$
$\displaystyle\approx$ $\displaystyle\frac{1}{4M}(r-M)^{2}+O(r-M)^{3},$

which gives a degeneracy of multiplicity $2$ for the coefficient of
$|\partial_{r}\psi|^{2}$ at the horizon. We can deduce that for $J=0$

$\displaystyle\begin{split}|q|^{2}{\bf
D}^{\mu}\mathcal{P}_{\mu}^{(X,w,J=0)}[\psi]&\gtrsim\dfrac{1}{r}\left(1-\dfrac{M}{r}\right)^{2}|{\partial}_{r}\psi|^{2}+\dfrac{1}{r}\left(1-\dfrac{r_{trap}}{r}\right)^{2}|q|^{2}|\nabla\psi|^{2}+\frac{1}{r^{2}}1_{r>r_{*}}|\psi|^{2}.\end{split}$
(38)

Observe that, in view of the behavior of the function $u$ close to the
horizon, the vectorfield $X$ and the function $w$ are both regular up to the
horizon, in contrast with the sub-extremal case, where the vectorfield $X$
diverges like $\log$ approaching the horizon [60]. On the other hand, the main
issue with the estimate (38) is the vanishing of the coefficient for the zero-
th order for $r\leq r_{*}$. We will now fix this issue with a Hardy inequality
adapted to the extremal case.

###  4.2 The global Hardy inequality and trapped control of the time
derivative

Here we make use of the one-form $J$ to obtain positivity of the zero-th order
term in the entire exterior region. From (37) and since for $r>M$ the function
$\frac{r(r-M)^{4}}{(r^{2}+M^{2})^{3}}$ achieves its maximum at
$(3+2\sqrt{2})M>r_{*}=(2+\sqrt{3})M$, we define for any $r_{e}\in(M,r_{*})$
the following minimum $c_{1}:=\min\limits_{r\in[r_{e},r_{*}]}\mathcal{A}(r)$.
Observe that because of the bound (37), we have that $c_{1}>0$, with
$c_{1}\downarrow 0$ as $r_{e}\to M$. Then for $r\in[r_{e},r_{*}]$ we can use
the bound $\mathcal{A}\geq c_{1}$ and (30) to obtain

$\displaystyle\mathcal{A}|{\partial}_{r}\psi|^{2}+\frac{1}{4}|q|^{2}{\mbox{d}iv\,}(J|\psi|^{2})$
$\displaystyle=$
$\displaystyle\mathcal{A}|\partial_{r}\psi|^{2}+\frac{1}{4}|q|^{2}\left(2v\psi\partial_{r}\psi+\left(\partial_{r}v+\frac{2r}{|q|^{2}}v\right)|\psi|^{2}\right)$
$\displaystyle=$
$\displaystyle\mathcal{A}\left(\partial_{r}\psi+\frac{|q|^{2}}{4\mathcal{A}}v\psi\right)^{2}-\frac{|q|^{4}v^{2}}{16\mathcal{A}}|\psi|^{2}+\frac{1}{4}|q|^{2}\left(\partial_{r}v+\frac{2r}{|q|^{2}}v\right)|\psi|^{2}$
$\displaystyle\geq$
$\displaystyle\frac{1}{4}|q|^{2}\left(\partial_{r}v+\frac{2r}{|q|^{2}}v-\frac{|q|^{2}v^{2}}{4\mathcal{A}}\right)|\psi|^{2}$
$\displaystyle\geq$
$\displaystyle\frac{1}{4}|q|^{2}\left(\partial_{r}v+\frac{2r}{|q|^{2}}v-\frac{|q|^{2}v^{2}}{4c_{1}}\right)|\psi|^{2}$
$\displaystyle=$
$\displaystyle\frac{c_{1}}{4}|q|^{2}\left(\left(\frac{v}{c_{1}}\right)^{\prime}+\frac{2r}{|q|^{2}}\left(\frac{v}{c_{1}}\right)-\frac{|q|^{2}}{4}\left(\frac{v}{c_{1}}\right)^{2}\right)|\psi|^{2}.$

We want to find a function $y(r)$ that for $r\in[r_{e},r_{*}]$

$\displaystyle y^{\prime}+\frac{2r}{|q|^{2}}y-\frac{|q|^{2}}{4}y^{2}>0.$

Using that $r^{2}\leq|q|^{2}\leq r^{2}+M^{2}$ we observe that for a negative
function $y(r)<0$ we can bound

$\displaystyle y^{\prime}+\frac{2r}{|q|^{2}}y-\frac{|q|^{2}}{4}y^{2}\geq
y^{\prime}+\frac{2}{r}y-\frac{r^{2}+M^{2}}{4}y^{2}.$

In particular we will look for a negative function in $r\in[r_{e},r_{*}]$
satisfying $y^{\prime}+\frac{2}{r}y-\frac{r^{2}+M^{2}}{4}y^{2}>0$. A
straightforward computation shows that $y_{0}(r)=-\frac{4}{r(r+M)(r-M)}$ is a
negative solution in $r\in[r_{e},r_{*}]$ to the ODE
$y_{0}^{\prime}+\frac{2}{r}y_{0}-\frac{r^{2}+M^{2}}{4}y_{0}^{2}=0$. Let
$y(r)=\lambda y_{0}(r)$ be a rescaling of $y_{0}$ for any constant
$0<\lambda<1$, then

$\displaystyle y^{\prime}+\frac{2}{r}y-\frac{r^{2}+M^{2}}{4}y^{2}$
$\displaystyle=$
$\displaystyle\lambda\big{(}y_{0}+\frac{2}{r}y_{0}-\frac{r^{2}+M^{2}}{4}y_{0}^{2}\big{)}+\lambda(1-\lambda)\frac{r^{2}+M^{2}}{4}y_{0}^{2}$
$\displaystyle=$
$\displaystyle\lambda(1-\lambda)\frac{r^{2}+M^{2}}{4}y_{0}^{2}>0.$

In particular, for $y(r)=\frac{1}{2}y_{0}(r)=-\frac{2}{r(r+M)(r-M)}<0$, we
have in $r\in[r_{e},r_{*}]$

$\displaystyle y^{\prime}+\frac{2r}{|q|^{2}}y-\frac{|q|^{2}}{4}y^{2}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}(1-\frac{1}{2})\frac{r^{2}+M^{2}}{4}y_{0}^{2}=\frac{r^{2}+M^{2}}{r^{2}(r+M)^{2}(r-M)^{2}},$

and therefore for

$\displaystyle v(r)=c_{1}y(r)=-\frac{2c_{1}}{r(r+M)(r-M)},$ (39)

we have

$\displaystyle\mathcal{A}|{\partial}_{r}\psi|^{2}+\frac{1}{4}|q|^{2}{\mbox{d}iv\,}(J|\psi|^{2})$
$\displaystyle\geq$
$\displaystyle\frac{c_{1}}{4}\frac{r^{2}+M^{2}}{(r+M)^{2}(r-M)^{2}}|\psi|^{2}.$

To conclude, combining the above Hardy inequality with the bound (38) we can
improve it to

$\displaystyle|q|^{2}{\bf D}^{\mu}\mathcal{P}_{\mu}^{(X,w,J)}[\psi]$
$\displaystyle\gtrsim$
$\displaystyle\frac{1}{r}\left(1-\frac{M}{r}\right)^{2}|{\partial}_{r}\psi|^{2}+\frac{1}{r}\left(1-\dfrac{r_{trap}}{r}\right)^{2}|q|^{2}|\nabla\psi|^{2}+\frac{1}{r^{2}}1_{r>r_{e}}|\psi|^{2},$
(40)

for $r_{e}>M$.

The only term that is missing from the above right hand side to give the
integral appearing in (3) of Theorem 1.1 is the trapped control on the time
derivative. For a function of $r$ $w_{T}$, we have from (21)

$\displaystyle|q|^{2}{\bf D}^{\mu}\mathcal{P}_{\mu}^{(X=0,w_{T},J=0)}[\psi]$
$\displaystyle=$ $\displaystyle-\frac{1}{4}|q|^{2}\square_{\bf
g}w_{T}|\psi|^{2}+\frac{1}{2}w_{T}|q|^{2}({\partial}_{\lambda}\psi{\partial}^{\lambda}\psi)$
$\displaystyle=$ $\displaystyle-\frac{1}{4}|q|^{2}\square_{\bf
g}w_{T}|\psi|^{2}+\frac{1}{2}w_{T}\big{(}\Delta|{\partial}_{r}\psi|^{2}+\frac{1}{\Delta}\mathcal{R}^{{\alpha}{\beta}}{\partial}_{\alpha}\psi{\partial}_{\beta}\psi\big{)}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}w_{T}\frac{(r^{2}+M^{2})^{2}}{(r-M)^{2}}|\partial_{t}\psi|^{2}+\frac{1}{2}w_{T}(r-M)^{2}|{\partial}_{r}\psi|^{2}+\frac{1}{2}w_{T}O^{{\alpha}{\beta}}{\partial}_{\alpha}\psi{\partial}_{\beta}\psi$
$\displaystyle-\frac{1}{4}|q|^{2}\square_{\bf g}w_{T}|\psi|^{2}.$

We choose $w_{T}$ to be given by

$\displaystyle w_{T}=-\frac{(r-M)^{2}(r-r_{trap})^{2}}{r^{7}},$

and we have

$\displaystyle\begin{split}|q|^{2}{\bf
D}^{\mu}\mathcal{P}_{\mu}^{(X=0,w_{T},J=0)}[\psi]&=\frac{1}{2}\frac{(r-r_{trap})^{2}(r^{2}+M^{2})^{2}}{r^{7}}|\partial_{t}\psi|^{2}-\frac{1}{2}\frac{(r-M)^{4}(r-r_{trap})^{2}}{r^{7}}|{\partial}_{r}\psi|^{2}\\\
&-\frac{1}{2}\frac{(r-M)^{2}(r-r_{trap})^{2}}{r^{7}}O^{{\alpha}{\beta}}{\partial}_{\alpha}\psi{\partial}_{\beta}\psi-\frac{1}{4}|q|^{2}\square_{\bf
g}w_{T}|\psi|^{2}.\end{split}$ (41)

We explicitly compute

$\displaystyle-\frac{1}{4}|q|^{2}\square_{\bf g}w_{T}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}{\partial}_{r}((r-M)^{2}\partial_{r}w_{T})$
$\displaystyle=$
$\displaystyle\frac{(r-M)^{2}}{2r^{9}}\Big{[}3r^{4}-3\left(9+4\sqrt{2}\right)Mr^{3}+\left(93+68\sqrt{2}\right)M^{2}r^{2}$
$\displaystyle-7\left(21+16\sqrt{2}\right)M^{3}r+(56\sqrt{2}+84)M^{4}\Big{]}$
$\displaystyle=$
$\displaystyle\frac{3(r-M)^{2}}{2r^{9}}\big{(}r-x_{1}M\big{)}\big{(}r-x_{2}M\big{)}\big{(}r-x_{3}M\big{)}\big{(}r-x_{4}M\big{)},$

where $1<x_{1}<x_{2}<x_{3}<x_{4}$ are four roots of

$\displaystyle
3x^{4}-3\left(9+4\sqrt{2}\right)x^{3}+\left(93+68\sqrt{2}\right)x^{2}-7\left(21+16\sqrt{2}\right)x+56\sqrt{2}+84=0.$

Even though $-\frac{1}{4}|q|^{2}\square_{\bf g}w_{T}$ can be negative for
$r\in[x_{1}M,x_{4}M]$, it must have a finite negative lower bound there. In
particular, by choosing $r_{e}\in(M,\min(x_{1}M,r_{*}))$, there exists a
sufficiently small $\delta_{T}>0$ such that

$\displaystyle\frac{1}{r^{2}}1_{r>r_{e}}-\frac{1}{4}\delta_{T}|q|^{2}\square_{\bf
g}w_{T}\geq\frac{1}{r^{2}}\left(1-\dfrac{M}{r}\right)^{2}.$

Finally combining (40) and (41) we deduce

$\displaystyle{\bf D}^{\mu}\mathcal{P}_{\mu}^{(X,w+\delta_{T}w_{T},J)}[\psi]$
$\displaystyle\gtrsim$
$\displaystyle\frac{1}{r^{3}}\left(1-\frac{M}{r}\right)^{2}|\partial_{r}\psi|^{2}+\frac{1}{r}\left(1-\frac{r_{trap}}{r}\right)^{2}\Big{(}\frac{1}{r^{2}}(\partial_{t}\psi)^{2}+|\nabla\psi|^{2}\Big{)}$
$\displaystyle+\frac{1}{r^{4}}\left(1-\dfrac{M}{r}\right)^{2}|\psi|^{2}.$

Observe that the degeneracy at the horizon for the $\partial_{r}$ derivative
is consistent with the conservation laws along the event horizon [12] which
implies non-decay of the transversal derivative along the event horizon.

We are finally set to apply the divergence theorem to the current
$\mathcal{P}_{\mu}^{(X,w+\delta_{T}w_{T},J)}[\psi]$. Observe that, by making
use of the simple Hardy estimate

$\displaystyle\int_{0}^{\infty}|\psi|^{2}dx\lesssim\int_{0}^{\infty}x^{2}|\partial_{x}\psi|^{2}dx$

with $x=r-r_{+}$, see also [4], to obtain bounds for the zero-th order term,
one can estimate the boundary terms
$\int_{\Sigma_{\tau}}\mathcal{P}_{\mu}^{(X,w+\delta_{T}w_{T},J)}[\psi]n^{\mu}_{\Sigma_{\tau}}$
by a large constant times a positive definite energy current, such as

$\displaystyle E^{(T)}[\psi](\tau)$ $\displaystyle=$
$\displaystyle\int_{\Sigma_{\tau}}|T\psi|^{2}+\big{(}1-\frac{M}{r}\big{)}^{2}|\partial_{r}\psi|^{2}+|\nabla\mkern-13.0mu/\,\psi|^{2}.$

We therefore deduce

$\displaystyle\int_{0}^{\tau}\int\frac{1}{r^{3}}\left(1-\frac{M}{r}\right)^{2}|\partial_{r}\psi|^{2}+\frac{1}{r}\left(1-\frac{r_{trap}}{r}\right)^{2}\big{(}\frac{1}{r^{2}}(\partial_{t}\psi)^{2}+|\nabla\mkern-13.0mu/\,\psi|^{2}\big{)}+\frac{1}{r^{4}}\left(1-\dfrac{M}{r}\right)^{2}|\psi|^{2}$
$\displaystyle\leq$ $\displaystyle
C\Big{(}E^{(T)}[\psi](\tau)+\int_{\mathcal{H}^{+}(0,\tau)}\mathcal{P}^{(T)}_{\mu}[\psi]n_{\mathcal{H}^{+}}^{\mu}+E^{(T)}[\psi](0)\Big{)}.$

Using the boundedness of the energy statement given in (31) we conclude the
proof of Theorem 1.1.

## References

  * [1] Alinhac S., Energy multipliers for perturbations of the Schwarzschild metric. Commun. Math. Phys. 288, 199-224 (2009)
  * [2] Andersson L., Bäckdahl T., Blue P., Ma S. Stability for linearized gravity on the Kerr spacetime. arXiv:1903.03859 (2019)
  * [3] Andersson L., Bäckdahl T., Blue P., Ma S. Nonlinear radiation gauge for near Kerr spacetimes, arXiv:2108.03148.
  * [4] Andersson L., Blue P. Hidden symmetries and decay for the wave equation on the Kerr spacetime. Ann. of Math. (2) 182, no.3, 787-853 (2015)
  * [5] Angelopoulos Y., Aretakis S., Gajic D.  A vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary spacetimes. Annals of PDE, 4:15 (2018)
  * [6] Angelopoulos Y., Aretakis S., Gajic D. Late-time asymptotics wave equation on spherically symmetric, stationary spacetimes. Adv. in Math. 323, 529–621 (2018)
  * [7] Angelopoulos Y., Aretakis S., Gajic D. Late-time asymptotics for the wave equation on extremal Reissner–Nordström backgrounds. Adv. in Math. 375, 107363 (2020).
  * [8] Apetroaie M., Instability of gravitational and electromagnetic perturbations of extremal Reissner-Nordström spacetime. arXiv:2211.09182 (2022)
  * [9] Aretakis S. Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations I. Communications in mathematical physics 307 (1), 17-63 (2011)
  * [10] Aretakis S. Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations II. Annales Henri Poincaré 12 (8), 1491-1538 (2011)
  * [11] Aretakis S. Decay of axisymmetric solutions of the wave equation on extreme Kerr backgrounds. J. Funct. Anal. 263, 2770 (2012)
  * [12] Aretakis S. Horizon instability of extremal black holes. Advances in Theoretical and Mathematical Physics 19. 507-530 (2015)
  * [13] Benomio G., A new gauge for gravitational perturbations of Kerr spacetimes II: The linear stability of Schwarzschild revisited. arXiv:2211.00616. (2022)
  * [14] Blue P., Soffer A., Semilinear wave equations on the Schwarzschild manifold. I. Local decay estimates. Adv. Differ. Equ. 8(5), 595–614 (2003)
  * [15] Blue P., Soffer A., Errata for “Global existence and scattering for the nonlinear Schrödinger equation on Schwarzschild manifolds” , “Semilinear wave equations on the Schwarzschild manifold I: Local Decay Estimates”, and “ The wave equation on the Schwarzschild metric II: Local Decay for the spin 2 Regge Wheeler equation” , gr-qc/0608073, 6 pages.
  * [16] Blue P., Sterbenz J., Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space, Comm. Math. Phys. 268 (2006), 481–504.
  * [17] Carter B., Global structure of the Kerr family of gravitational fields, Physical Review. 174 (5): 1559–1571 (1968)
  * [18] Christodoulou D., Klainerman S., The global nonlinear stability of the Minkowski space, Princeton University Press (1993).
  * [19] Civin D., Stability at charged rotating black holes for linear scalar perturbations, Ph.D. thesis, University of Cambridge. Available at http://www.repository cam.ac.uk/handle/1810/247 (2014)
  * [20] Dafermos M., Holzegel G., Rodnianski I. The linear stability of the Schwarzschild solution to gravitational perturbations. Acta Mathematica, 222: 1-214 (2019)
  * [21] Dafermos M., Holzegel G., Rodnianski I. Boundedness and decay for the Teukolsky equation on Kerr spacetimes I: the case $|a|\ll M$. Ann. PDE, 5, 2 (2019)
  * [22] Dafermos M., Holzegel G., Rodnianski I., Taylor M., The non-linear stability of the Schwarzschild family of black holes, arXiv:2104.0822.
  * [23] Dafermos M., Rodnianski I. The wave equation on Schwarzschild-de Sitter spacetimes. arXiv:0709.2766. (2007)
  * [24] Dafermos M., Rodnianski I. A new physical-space approach to decay for the wave equation with applications to black hole spacetimes. XVIth International Congress on Mathematical Physics, 421-433, (2009)
  * [25] Dafermos M., Rodnianski I. The red-shift effect and radiation decay on black hole spacetimes. Comm. Pure Appl. Math, 62:859-919 (2009)
  * [26] Dafermos M., Rodnianski I. Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: The cases $|a|\ll M$ or axisymmetry. arXiv:1010.5132 (2010)
  * [27] Dafermos M., Rodnianski I., The black hole stability problem for linear scalar perturbations, Proceedings of the Twelfth Marcel Grossmann Meeting on General Relativity, T. Damour et al (ed.) (2011).
  * [28] Dafermos M., Rodnianski I., A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds, Invent. Math. 185(3), 467–559 (2011)
  * [29] Dafermos M., Rodnianski I. Lectures on black holes and linear waves. Evolution equations, Clay Mathematics Proceedings, Vol. 17, pages 97-205 Amer. Math. Soc. (2013)
  * [30] Dafermos M., Rodnianski I., Shlapentokh-Rothman Y. Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case $|a|<M$. Annals of Math. 183, no. 3, 787-913 (2016)
  * [31] Fang A., Nonlinear stability of the slowly-rotating Kerr-de Sitter family. arXiv:2112.07183 (2021)
  * [32] Fang A., Linearized stability of the slowly-rotating Kerr-de Sitter family. arXiv:2207.07902 (2022)
  * [33] Giorgi E. Boundedness and decay for the Teukolsky system of spin $\pm 2$ on Reissner-Nordström spacetime: the case $|Q|\ll M$. Ann. Henri Poincaré, 21, 2485 - 2580 (2020)
  * [34] Giorgi E. Boundedness and decay for the Teukolsky system of spin $\pm 1$ on Reissner–Nordström spacetime: the $\ell=1$ spherical mode. Class. Quantum Grav. 36, 205001 (2019)
  * [35] Giorgi E. The linear stability of Reissner-Nordström spacetime for small charge. Annals of PDE, 6, 8 (2020)
  * [36] Giorgi E. The linear stability of Reissner-Nordström spacetime: the full sub-extremal range $|Q|<M$. Commun. Math. Phys. 380, 1313–1360 (2020)
  * [37] Giorgi E., Electromagnetic-gravitational perturbations of Kerr-Newman spacetime: the Teukolsky and Regge-Wheeler equations J. Hyperbolic Differ. Equ., Vol. 19, No. 01, pp. 1-139 (2022)
  * [38] Giorgi E., The Carter tensor and the physical-space analysis in perturbations of Kerr-Newman spacetime preprint arXiv:2105.14379, to appear in J. Differential Geom. (2022)
  * [39] Giorgi E., Klainerman S., Szeftel J., Wave equation estimates and the nonlinear stability of slowly rotating Kerr black holes. arXiv:2205.14808 (2022), 917 pages
  * [40] Häfner D., Hintz P., Vasy A. Linear stability of slowly rotating Kerr black holes. Invent. Math., 223 (2021), 1227–1406.
  * [41] Hintz P. Non-linear stability of the Kerr-Newman-de Sitter family of charged black holes. Annals of PDE, 4(1):11 (2018)
  * [42] Hintz P., Vasy A. The global non-linear stability of the Kerr-de Sitter family of black holes. Acta Math., 220:1-206 (2018)
  * [43] Hintz P., Vasy A., Stability of Minkowski space and polyhomogeneity of the metric, Ann. PDE 6, 2 (2020).
  * [44] Hung P.-K. , Keller J. , Wang M.-T. Linear Stability of Schwarzschild Spacetime: The Cauchy Problem of Metric Coefficients. J. Differential Geom. 116 (3) 481 - 541 (2020)
  * [45] Johnson T. The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge. Ann. PDE 5, 13 (2019)
  * [46] Kay, B.S., Wald, R.M., Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation two sphere. Class. Quantum Grav. 4, 893–898 (1987)
  * [47] Klainerman S., Szeftel J. Global Non-Linear Stability of Schwarzschild Spacetime under Polarized Perturbations. Annals of Math Studies, 210. Princeton University Press, Princeton NJ, 2020, xviii+856 pp.
  * [48] Klainerman S., Szeftel J., Construction of GCM spheres in perturbations of Kerr, arXiv:1911.00697, to appear in Annals of PDE.
  * [49] Klainerman S., Szeftel J., Effective results in uniformization and intrinsic GCM spheres in perturbations of Kerr, arXiv:1912.12195. To appear in Annals of PDE.
  * [50] Klainerman S., Szeftel J., Kerr stability for small angular momentum, arXiv:2104.11857.
  * [51] Lindblad H., Rodnianski, I. Global existence in the Einstein Vacuum equations in wave co-ordinates. Comm. Math. Phys., 256 (2005), 43–110.
  * [52] Ma S. Uniform energy bound and Morawetz estimate for extreme component of spin fields in the exterior of a slowly rotating Kerr black hole II: linearized gravity. Commun. Math. Phys. 377, 2489–2551 (2020)
  * [53] J. Marzuola, J. Metcalfe, D. Tataru and M. Tohaneanu, Strichartz estimates on Schwarzschild black hole backgrounds, Comm. Math. Phys. 293 (2010), 37–83.
  * [54] Mavrogiannis G. Morawetz estimates without relative degeneration and exponential decay on Schwarzschild de Sitter spacetimes. arXiv:2111.09494 (2021)
  * [55] Mavrogiannis G. Quasilinear wave equations on Schwarzschild de Sitter. arXiv:2111.09495 (2021)
  * [56] C. Morawetz,  Decay of solutions of the exterior initial boundary value problem for the wave equation. Comm. Pure and App. Math. 14 (1961), 561–568.
  * [57] Shen D., Construction of GCM hypersurfaces in perturbations of Kerr, arXiv:2205.12336.
  * [58] Shlapentokh-Rothman Y., Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime, Ann. Henri Poincaré. 16 (2015), 289–345.
  * [59] Shlapentokh-Rothman Y., Teixeira da Costa R. Boundedness and decay for the Teukolsky equation on Kerr in the full subextremal range $|a|<M$: frequency space analysis, arXiv preprint arXiv:2007.07211.
  * [60] Stogin J., Nonlinear wave dynamics in black hole spacetimes, Ph.D. thesis, Princeton University. Available at http://arks.princeton.edu/ark:/88435/dsp01p5547t983 (2017)
  * [61] Tataru D., Tohaneanu M. A local energy estimate on Kerr black hole background. IMRN no.2, 248-292 (2011)
  * [62] Teixeira da Costa R. Mode stability for the Teukolsky equation on extremal and subextremal Kerr spacetimes, Commun. Math. Phys., 378(1), 705-781 (2020)
  * [63] Whiting B. F. Mode stability of the Kerr black hole. J. Math. Phys., 30 (6):1301-1305 (1989)


# Beautiful mixing and CP violation at LHCb

Philippe d’Argent on behalf of the LHCb collaboration  CERN, Geneva,
Switzerland

###### Abstract

Precision measurements of beauty hadron decays are sensitive probes of the
Standard Model and a promising way to look for new physics phenomena far
beyond the energy scale accessible for direct production searches. This
article reviews recent measurements of mixing and CP violation in beauty
decays performed at the LHCb experiment that have been presented at the
$55^{th}$ Rencontres de Moriond QCD conference.

##  1 The Standard Model and beyond

In the framework of the Standard Model of particle physics, the charge-parity
(CP) symmetry between quarks and antiquarks is broken by a single complex
phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix. The
unitarity of this matrix leads to the condition
$V_{ud}^{\phantom{*}}V^{*}_{ub}+V_{cd}^{\phantom{*}}V^{*}_{cb}+V_{td}^{\phantom{*}}V^{*}_{tb}=0$,
where $V_{ij}$ are the complex elements of the CKM matrix. This equation can
be visualised as a triangle in the complex plane with angles $\alpha$, $\beta$
and $\gamma$. A key consistency test of the Standard Model is to verify the
unitarity conditions by over-constraining the CKM matrix with various
independent measurements sensitive to distinct combinations of matrix
elements. While the magnitudes of the CKM matrix elements can be determined
from the decay rates of respective flavour-changing transitions, measurements
of CP asymmetries generally permit determining the CKM phases. Here, the angle
$\gamma\equiv\rm{arg}[-(V_{ud}^{\phantom{*}}V_{ub}^{*})/(V_{cd}^{\phantom{*}}V_{cb}^{*})]$
has particularly interesting features as it is the only CKM angle that can be
measured in tree-level decays. In such decays, the interpretation of physical
observables (rates and CP asymmetries) in terms of the underlying CKM
parameters is subject to negligible theoretical uncertainties. Hence, a
precision measurement of $\gamma$ provides a Standard Model benchmark, to be
compared with indirect determinations from other CKM matrix observables which
are more susceptible to new physics phenomena beyond the Standard Model.

##  2 Direct CP violation in beauty decays

The most stringent constraints on the CKM angle $\gamma$ come from
measurements of direct CP violation in $B^{\mp}\to DK^{\mp}$ decays, where $D$
represents an admixture of the $D^{0}$ and $\bar{D}^{0}$ flavour states. While
the $B^{-}\to D^{0}K^{-}$ decay proceeds via a $b\to c\bar{u}s$ quark-level
transition, a $b\to u\bar{c}s$ transition leads to the
$B^{-}\to\bar{D}^{0}K^{-}$ decay. Provided that the charm meson decays into a
final state, $f$, which is accessible for both flavour states, phase
information can be determined from the interference between these two decay
paths. The relative phase between the corresponding decay amplitudes has both
CP-violating ($\gamma$) and CP-conserving ($\delta_{B}^{DK}$) contributions. A
measurement of the decay rate asymmetry between $B^{+}$ and $B^{-}$ mesons
thus gives sensitivity to $\gamma$. The sensitivity is driven by the size of
$r_{B}^{DK}$, the ratio of the magnitudes of the $B^{-}\to\bar{D}^{0}K^{-}$
and $B^{-}\to D^{0}K^{-}$ amplitudes. Similar interference effects also occur
in $B^{\mp}\to D\pi^{\mp}$ decays, albeit with a significantly reduced
sensitivity to the phases due to additional Cabibbo-suppression
($r_{B}^{DK}\approx 20\,r_{B}^{D\pi}$). Two recent measurements of direct CP
violation in $B^{\mp}\to Dh^{\mp}$ ($h\in\\{K,\pi\\}$) decays study two-body
($D\to h^{\pm}h^{\mp}$) and self-conjugate three-body ($D\to K^{0}_{\rm
s}h^{\pm}h^{\mp}$) charm decays, respectively. Both analyses use data
accumulated with the LHCb detector over the period from 2011 to 2018 in $pp$
collisions at energies of $\sqrt{s}=7,8$ and $13$ TeV, corresponding to a
total integrated luminosity of approximately $9\,\rm{fb}^{-1}$.

The first analysis [1] considers the CP-eigenstates $D\to\pi^{\pm}\pi^{\mp}$
and $D\to K^{\pm}K^{\mp}$ as well as $D\to K^{+}\pi^{-}$, where the $D^{0}\to
K^{+}\pi^{-}$ and $\bar{D}^{0}\to K^{+}\pi^{-}$ decays are related by the
amplitude magnitude $r_{D}^{K\pi}$ and the strong-phase difference
$\delta_{D}^{K\pi}$. For the latter case, the similar magnitude of
$r_{B}^{DK}$ and $r_{D}^{K\pi}$ leads to significant interference between the
two decay paths (favoured B decay followed by suppressed D decay, and
suppressed B decay followed by favoured D decay). As is evident from the
invariant-mass distributions shown in Fig. 2, this results in a huge asymmetry
between the $B^{-}$ and $B^{+}$ decay rates. Moreover, the analysis includes
partially reconstructed $B^{\mp}\to D^{*}h^{\mp}$ decays, in which the vector
$D^{*}$ meson decays to either the $D\pi^{0}$ or $D\gamma$ final state. In
total 28 observables (CP asymmetries and decay rate ratios) are measured. The
combined information allows deriving tight constrains on the underlying
physics parameters $r_{B}^{X},\delta_{B}^{X},r_{D}^{f},\delta_{D}^{f}$ and
$\gamma$
($X\in\\{DK,D\pi,D^{*}K,D^{*}\pi\\},f\in\\{K^{\pm}\pi^{\mp},K^{+}K^{-},\pi^{+}\pi^{-}\\}$)
as displayed in Fig. 2 for the ($r_{B}^{DK},\gamma$) plane.

Similarly, the analysis of $D\to K^{0}_{\rm s}h^{\pm}h^{\mp}$ decays [2]
investigates differences in the phase-space distributions of $B^{+}$ and
$B^{-}$ meson decays. To interpret the result in terms of the physical
observables, knowledge of the the strong-phase variation over the Dalitz plot
of the $D$ decay is required. A model-unbiased approach is employed that uses
direct measurements of the strong-phase difference between $D^{0}$ and
$\bar{D}^{0}$ decays, averaged over regions of the phase space. These strong-
phase differences have been measured by the CLEO and the BESIII collaborations
using quantum correlated pairs of $D$ mesons produced in decays of
$\psi(3770)$. The Dalitz-plot binning scheme is optimized with the help of an
amplitude model. With this procedure, the CKM angle $\gamma$ is determined to
be $\gamma=(68.7^{+5.2}_{-5.1})^{\circ}$, the most precise single measurement
to date. The results are in good agreement with the $D\to h^{\pm}h^{\mp}$
analysis and are crucial to resolve the remaining ambiguities in the parameter
space, see Fig. 2.

Figure 1: Invariant-mass distribution of $B^{-}\to[K^{+}\pi^{-}]_{D}K^{-}$
(left) and $B^{+}\to[K^{-}\pi^{+}]_{D}K^{+}$ (right) candidates with the fit
projections overlaid. The signal component (red peak) and show a huge
asymmetry. Partially reconstructed decays are visible at low invariant mass.

Figure 2:  Confidence region in the ($r_{B}^{DK},\gamma$) plane for both
$B^{\mp}\to D^{(*)}h^{\mp}$ analyses.

The family of $B\to K\pi$ decays receives significant contributions from loop-
level transitions providing a powerful probe for new physics phenomena.
Measurements of direct CP violation in these channels have revealed
significant deviations from the expected isospin symmetry, an anomaly known as
the $K\pi$ puzzle. The reconstruction of the $B^{+}\to K^{+}\pi^{0}$ decay is
particularly challenging at a hadron collider as no B-meson decay vertex can
be reconstructed from a single charged track. Charged kaons that are
inconsistent with originating from the primary collision point but consistent
with the reconstructed trajectory of the b-meson candidate are selected from a
data sample corresponding to a luminosity of $5.4\rm{fb}^{-1}$. The CP
asymmetry between $B^{-}$ and $B^{+}$ decay rates is found to be [3]
$A_{CP}(B^{+}\to K^{+}\pi^{0})=0.025\pm 0.015\pm 0.006\pm 0.003$, where the
uncertainties are statistical, systematic and due to external inputs. This
result is consistent with the world average and exceeds its precision. It
confirms and significantly enhances the observed anomalous difference between
the direct CP asymmetries of the $B^{+}\to K^{+}\pi^{0}$ and $B^{+}\to
K^{+}\pi^{-}$ decays.

##  3 Mixing-induced CP violation in beauty decays

Neutral $B_{s}^{0}$ mesons can oscillate into their antiparticle counterparts
via quantum loop processes opening additional mechanisms for CP symmetry
breaking. The frequency of this oscillation, $\Delta m_{s}$, is an important
parameter of the Standard Mode and provides powerful constraints in global CKM
fits. The mixing from $B_{s}^{0}$ to $\bar{B}_{s}^{0}$ occurs about three
million million times per second, making it a major experimental challenge to
resolve it. Due to the excellent decay vertex resolution and track momentum
resolution, the LHCb detector is ideally suited for this task. Two recent
measurements of $\Delta m_{s}$ use flavour specific $B_{s}^{0}\to
D_{s}^{-}\pi^{+}\pi^{-}\pi^{+}$ ($9\,\rm{fb}^{-1}$) [4] and $B_{s}^{0}\to
D_{s}^{-}\pi^{+}$ ($6\,\rm{fb}^{-1}$) [5] decays, respectively. To determine
if a neutral meson oscillated into its antiparticle, knowledge of the
initially created flavour eigenstate is required. This is achieved by using a
combination of several flavour-tagging algorithms that exploit different
features of the b-hadron production process. Figure 3 shows the oscillation
pattern of signal decays having the same flavour at the production and decay,
and those, for which the flavour has changed. Both measurements of the
oscillation frequency are significantly more precise than the current world-
average value. Their combination with previous LHCb measurements yields
$\Delta m_{s}=17.7656\pm 0.0057\,\rm{ps}^{-1}$, a crucial legacy measurement
of the original LHCb detector.

Figure 3: Decay-time distribution of flavour-tagged $B_{s}^{0}\to
D_{s}^{-}\pi^{+}\pi^{-}\pi^{+}$ signal decays (left) and mixing asymmetry for
$B_{s}^{0}\to D_{s}^{\mp}K^{\pm}\pi^{\pm}\pi^{\mp}$ signal decays folded into
one oscillation period (right). The fit projections are overlaid (lines).

Interference between the amplitudes of a $B_{s}^{0}$ meson directly decaying
through $b\to c\bar{c}s$ into a CP eigenstate or after oscillation to a
$\bar{B}_{s}^{0}$ meson gives rise to the CP violating phase
$\phi_{s}\approx-2\beta_{s}$ with
$\beta_{s}\equiv\rm{arg}[-(V_{ts}^{\phantom{*}}V_{tb}^{*})/(V_{cs}^{\phantom{*}}V_{cb}^{*})]$.
The precise measurement of this phase is of high interest because of its
potential sensitivity to new particles altering the mixing amplitudes. A time-
dependent angular analysis of $B_{s}^{0}\to J/\psi\phi$ decays with $J/\psi\to
e^{+}e^{-}$ and $\phi\to K^{+}K^{-}$ determines the mixing phase to be
$\phi_{s}=0.00\pm 0.28\pm 0.05\,\rm{rad}$ ($3\,\rm{fb}^{-1}$) [6]. This is the
first measurement of $\phi_{s}$ with an electron pair in the final state. The
result shows no evidence of CP violation and is consistent with previous
measurements and the Standard Model prediction. It also constitutes an
important cross-check for the results with muons in the final state with
independent systematic uncertainties.

Complementary to the $\gamma$ measurements in charged b-hadron decays, mixing-
induced CP violation in $B_{s}^{0}\to D_{s}^{\mp}K^{\pm}\pi^{\pm}\pi^{\mp}$
decays provides sensitivity to the weak phase $\gamma-2\beta_{s}$. This is
studied for the first time using $9\,\rm{fb}^{-1}$ of $pp$ collision data
recorded by the LHCb detector [4]. Due to the multi-body final state, the
hadronic parameters vary across the five dimensional phase space of the decay.
A time-dependent amplitude analysis is performed to disentangle the various
intermediate-state components contributing via $b\to c$ or $b\to u$ quark-
level transitions. The prominent contributions are found to be the cascade
decays $B_{s}^{0}\to D_{s}^{\mp}K_{1}(1270)^{\pm}$ and $B_{s}^{0}\to
D_{s}^{\mp}K_{1}(1400)^{\pm}$ with $K_{1}(1270)^{\pm}\to
K^{*}(892)^{0}\pi^{\pm},K^{\pm}\rho(770)^{0},K^{*}_{0}(1430)^{0}\pi^{\pm}$ as
well as $K_{1}(1400)\to K^{*}(892)^{0}\pi^{\pm}$. Figure 3 shows the mixing
asymmetry for final state $f=D_{s}^{-}K^{+}\pi^{+}\pi^{-}$, defined as
$A_{\text{mix}}^{f}(t)=(N_{f}(t)-\bar{N}_{f}(t))/(N_{f}(t)+\bar{N}_{f}(t))$,
where $N_{f}(t)$ ($\bar{N}_{f}(t)$) denote the number of initially produced
$B_{s}^{0}$ ($\bar{B}_{s}^{0}$) mesons. The equivalent mixing asymmetry for
the CP-conjugate process, $A_{\text{mix}}^{\bar{f}}(t)$, shows a phase shift
related to the weak phase $\gamma-2\beta_{s}$ signifying time-dependent CP
violation. The CKM angle $\gamma$ is determined to be $\gamma=(44\pm
12)^{\circ}$ by taking the mixing phase $\beta_{s}$ as external input. An
alternative model-independent measurement, integrating over the phase space of
the decay, gives a consistent result, $\gamma=(44^{+20}_{-13})^{\circ}$, with
reduced statistical precision but free of model uncertainties related to the
amplitude parameterization.

The CP asymmetries of charmless $B^{0}_{(s)}$ decays to two-body charged final
states provide access to the CKM angles $\alpha$ and $\gamma$ and the $B^{0}$
and $B_{s}^{0}$ mixing phases. In contrast to the tree-level measurements from
$B^{\mp}\to Dh^{\mp}$ and $B_{s}^{0}\to D_{s}^{\mp}K^{\pm}\pi^{\pm}\pi^{\mp}$
decays, the sensitivity to the CKM angles originates from the interference of
the $b\to u$ tree-level with the $b\to d$ or $b\to s$ loop-level transitions.
Figure 5 shows the decay-time distribution of flavour-tagged $B_{s}^{0}\to
K^{+}K^{-}$ signal decays using a data sample corresponding to a luminosity of
$1.9\rm{fb}^{-1}$. The CP observables describing the decay-time distribution
are measured with world-best precision [7]. Combined with previous LHCb
results, the first observation of time-dependent CP violation in the
$B_{s}^{0}$ system is reported. This is an important milestone for flavour
physics.

Figure 4:  Decay-time distribution of flavour-tagged $B_{s}^{0}\to K^{+}K^{-}$
signal decays.

Figure 5:  Profile-likelihood scan of 1-CL ($p$-value) for the LHCb $\gamma$
combination.

##  4 Outlook

The LHCb collaboration continues to push the frontier of heavy flavour
physics. New measurements of the $B_{s}^{0}-\bar{B}_{s}^{0}$ mixing frequency
have reached unprecedented precision. While time-dependent CP violation in the
$B_{s}^{0}$ system has now been observed for the first time, the breaking of
CP symmetry has still not been observed in the baryon sector. With the first
amplitude analysis of any b-baryon decay mode allowing for CP-violation
effects, the LHCb collaboration is also pioneering in this field. Within the
current precision, no significant CP asymmetries have been observed for the
amplitude components contributing to $\Xi_{b}^{-}\to pK^{-}K^{-}$ decays [8].

Thanks to the combination of plenty of decay modes and advanced analysis
techniques, the LHCb collaboration achieved an impressive overall precision on
the CKM angle $\gamma$ as shown in Fig. 5. Including the new results presented
here, the LHCb average [9] yields $\gamma=(67\pm 4)^{\circ}$. This is in
excellent agreement with global CKM fits. With the upcoming Run 3 data-taking
period, the combination of LHCb results will enter the high precision region
where discrepancies between direct measurement and indirect CKM prediction may
be observed. An ultimate precision at the sub-degree level will be achievable
in the high luminosity LHC era. It remains thrilling to see whether new
phenomena beyond the established theory can be uncovered. The anomaly observed
in $B\to K\pi$ decays, strengthened by recent LHCb results, might already
point to internal inconsistencies of the Standard Model.

## References

## References

  * [1] LHCb collaboration, R. Aaij et al et al, JHEP 04, 081 (2021).
  * [2] LHCb collaboration, R. Aaij et al et al, JHEP 02, 169 (2021).
  * [3] LHCb collaboration, R. Aaij et al et al, Phys. Rev. Lett. 126, 091802 (2021).
  * [4] LHCb collaboration, R. Aaij et al et al, JHEP 03, 137 (2021).
  * [5] LHCb collaboration, R. Aaij et al et al, arXiv:2104.04421.
  * [6] LHCb collaboration, R. Aaij et al et al, LHCB-PAPER-2020-042.
  * [7] LHCb collaboration, R. Aaij et al et al, JHEP 03, 075 (2021).
  * [8] LHCb collaboration, R. Aaij et al et al, arXiv:2104.15074.
  * [9] LHCb collaboration, R. Aaij et al et al, LHCb-CONF-2020-003.