# On the Relation Between Asymptotic Charges,
the Failure of Peeling and Late-time Tails

Dejan Gajic<EMAIL_ADDRESS>Radboud University, Department of Mathematics,
Huygens Building, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
Leonhard M. A. Kehrberger<EMAIL_ADDRESS>University of Cambridge, Department
of Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge
CB3 0WA, United Kingdom

###### Abstract

The last few years have seen considerable mathematical progress concerning the
asymptotic structure of gravitational radiation in dynamical, astrophysical
spacetimes.

In this paper, we distil some of the key ideas from recent works and assemble
them in a new way in order to make them more accessible to the wider general
relativity community. We also announce some new physical findings in this
process.

First, we introduce the conserved $f(r)$-modified Newman–Penrose charges on
asymptotically flat spacetimes, and we show that these charges provide a
dictionary that relates asymptotics of massless, general spin fields in
different regions: Asymptotic behaviour near $i^{+}$ (“late-time tails”) can
be read off from asymptotic behaviour towards $\mathcal{I}^{+}$, and,
similarly, asymptotic behaviour towards $\mathcal{I}^{+}$ can be read off from
asymptotic behaviour near $i^{-}$ or $\mathcal{I}^{-}$.

Using this dictionary, we then explain how: (I) the quadrupole approximation
for a system of $N$ infalling masses from $i^{-}$ causes the “peeling property
towards $\mathcal{I}^{+}$” to be violated, and (II) this failure of peeling
results in deviations from the usual predictions for tails in the late-time
behaviour of gravitational radiation: Instead of the Price’s law rate
$r\Psi^{[4]}|_{\mathcal{I}^{+}}\sim u^{-6}$ as $u\to\infty$, we predict that
$r\Psi^{[4]}|_{\mathcal{I}^{+}}\sim u^{-4}$, with the coefficient of this
latter decay rate being a multiple of the monopole and quadrupole moments of
the matter distribution in the infinite past.

###### Contents

  1. 1 Introduction
  2. 2 Coordinates, foliations and conventions
  3. 3 The $f(r)$-modified Newman–Penrose charges
  4. 4 Deriving asymptotics towards $i^{+}$ from asymptotics towards $\mathcal{I}^{+}$
  5. 5 Deriving asymptotics towards $\mathcal{I}^{+}$ from physical data near $i^{-}$
  6. 6 Completing the dictionary and extending to gravitational perturbations
  7. 7 Further directions
  8. 8 Conclusion

##  1 Introduction

Over the last few years, there has been significant progress in the
mathematical study of general relativity concerning our understanding of how
certain asymptotic conservation laws—related to the Newman–Penrose charges
[NP65, NP68]—can serve as a mechanism for deriving statements about
asymptotics of gravitational waves on black hole spacetimes. This mechanism
was first explored in [AAG18b] and has since lead to new, potentially
physically measurable predictions [AAG18a, BKS19, Keh21a, Keh21b]. The present
paper aims to distil and generalise some of the main ideas behind this
mechanism, and to provide a physical and more accessible interpretation
thereof. We also preview several upcoming results.

The ideas presented in this note generally pertain to the linearised equations
of gravity around a Kerr black hole background $(\mathcal{M}_{M,a},g_{M,a})$,
which are, at a fundamental level, governed by the spin-$\pm 2$ Teukolsky
equations111In fact, the ideas discussed in the present paper apply to the
Teukolsky equations for any spin $s\in\frac{1}{2}\mathbb{N}_{0}$. [Teu73]
satisfied by the Newman–Penrose scalars $\Psi^{[0]}$ and $\Psi^{[4]}$:

$\mathcal{T}_{g_{M,a}}^{[s]}\Psi^{[|s|-s]}=0,\quad s=\pm 2,$ (1.1)

with $\mathcal{T}_{g_{M,a}}^{[s]}$ a differential operator similar to the wave
operator $\Box_{g_{M,a}}$ on Kerr, and with $\Psi^{[0]}$ and $\Psi^{[4]}$ the
gauge-invariant, extremal components of the perturbed Weyl tensor in the
Newman–Penrose formalism.

To keep the presentation as clear as possible, however, we will restrict most
of the discussion to the simpler case of $s=0$, in which (1.1) in fact equals
the scalar wave equation

$\Box_{g_{M,a}}\psi=0.$ (1.2)

We will moreover restrict to the subcase with specific angular momentum $a=0$,
where $g_{M,a}$ reduces to the Schwarzschild metric $g_{M}$. Extensions to
non-zero $s$ and $a$ will be discussed at the end of the paper.

This paper will answer the following two questions (see Fig. 1):

  * A)

How can we read off late-time asymptotics of $\psi$ near future timelike
infinity $i^{+}$—in particular, along future null infinity $\mathcal{I}^{+}$
and the event horizon $\mathcal{H}^{+}$—from asymptotics towards
$\mathcal{I}^{+}$?

  * B)

How can we derive asymptotics towards $\mathcal{I}^{+}$ from physically
motivated scattering data assumptions modelling a system of $N$ infalling
masses from the infinite past $i^{-}$ and excluding incoming radiation from
past null infinity $\mathcal{I}^{-}$?

Figure 1: Depiction of the problems A) and B) described in the paragraph
above.

###  1.1 Motivation and background

Let us first motivate problems A) and B) individually:

A) The study of the dynamics of (1.1) at late times is motivated by the
ambitious final state conjecture, which in particular asserts that two
inspiralling black holes will settle down to a Kerr solution outside the
horizon [Pen69],222In the restricted case of spacetimes arising from small
perturbations of subextremal Kerr initial data, this conjecture is known as
the Kerr stability conjecture, see for example [DHRT21, KS21] and references
therein for recent mathematical progress towards a proof thereof. and by the
fact that, mathematically, a large part of the radiation emitted as dynamical
spacetimes settle down to a Kerr spacetime is encoded precisely in the
solutions $\Psi^{[4]}$ and $\Psi^{[0]}$ to eq. (1.1).

Hence, in the idealisation of an isolated gravitational system, where
gravitational wave observatories operate at $\mathcal{I}^{+}$, the
mathematical late-time behaviour (of appropriate rescalings) of the Teukolsky
variables $\Psi^{[4]}$ and $\Psi^{[0]}$ along $\mathcal{I}^{+}$ offers
predictions for the late-time part of signals measured by actual gravitational
wave detectors; see for example the analysis of the late-time parts of
gravitational signals coming from recent black hole mergers in [A+16, A+21].
This gives rise to several interesting points:

  1. 1)

Suppose one manages to measure the decay rates and coefficients in the late-
time asymptotics of the Teukolsky variables, and interprets these as
asymptotics towards $i^{+}$ along $\mathcal{I}^{+}$. Given a good mathematical
understanding of these late-time tails, one may then extract from the
measurements information about both the nature of the final Kerr black hole
state as well as the asymptotic properties of the initial state. Late-time
tails therefore serve as important signatures of black holes in dynamical
astrophysical processes.

  2. 2)

In addition, the late-time behaviour along $\mathcal{I}^{+}$ is
_mathematically strongly correlated_ with the late-time behaviour of
appropriate renormalisations of the Teukolsky variables along
$\mathcal{H}^{+}$.

  3. 3)

In turn, the late-time behaviour along $\mathcal{H}^{+}$ is related to the
strength of the null singularity that is expected to exist generically inside
dynamical black holes; see for example [Daf05, LO19]. A sufficiently precise
understanding of the behaviour of the Teukolsky variables along
$\mathcal{H}^{+}$ is therefore necessary for resolving the _Strong Cosmic
Censorship conjecture_ (see [DL17] and references therein) in the setting of
dynamical black hole spacetimes. Hence, via 2), the information measured along
$\mathcal{I}^{+}$ would also relate to behaviour in the black hole interior!

Of course, the late-time asymptotics near $i^{+}$ will depend on the
assumptions made on the initial state, i.e. the choice of initial data
hypersurface and the prescribed initial data. For instance, if the initial
data are posed on an asymptotically hyperboloidal hypersurface $\Sigma$ (see
Fig. 1), we will see that the leading-order asymptotics near $i^{+}$ are, in
many cases, completely determined by how the data for $\psi$ behave near
$\mathcal{I}^{+}$, i.e. by the asymptotics of $\psi$ along $\Sigma$ towards
$\mathcal{I}^{+}$. But what should these early-time asymptotics towards
$\mathcal{I}^{+}$ be?

B) In large (but not all) parts of the literature, there have been two
predominant data assumptions on the asymptotics towards $\mathcal{I}^{+}$
along $\Sigma$: It has been assumed (e.g. in the original heuristic work on
late-time asymptotics [Pri72]) that these asymptotics are either trivial, i.e.
that the data are of compact support along $\Sigma$ and therefore vanish
identically near $\mathcal{I}^{+}$, or—typically justified by Penrose’s
concept of smooth conformal compactification of spacetime (a.k.a. smooth null
infinity) [Pen65]—it has been assumed that the initial data satisfy “peeling”
[Sac61, Sac62], i.e. that they have an asymptotic expansion in powers of $1/r$
and exhibit certain leading-order decay towards $\mathcal{I}^{+}$ (e.g.
$\Psi^{[0]}=O(r^{-5}),\Psi^{[4]}=\mathcal{O}(r^{-1})$).

Now, we would argue that the former assumption is incompatible with the model
of an isolated system, as any such system will have radiated for all times
and, therefore, will not have hypersurfaces $\Sigma$ of compact radiation
content, see Fig. 1.

The assumption of peeling becomes similarly questionable for physically
relevant systems, as will be explained in this paper. Indeed, the approach we
take here is to not make assumptions on the asymptotics along $\Sigma$ towards
$\mathcal{I}^{+}$, but to instead derive them from physical principles: We
shall consider a scattering data setup as in [Keh21a] that a) has no incoming
radiation from $\mathcal{I}^{-}$ and that b) attempts to capture the
gravitational radiation of $N$ masses—approaching each other from infinitely
far away in the infinite past—by imposing data on some null cone $\mathcal{C}$
(to be thought of as enclosing these masses) that are predicted by post-
Newtonian arguments (such as the quadrupole approximation) [WW79, Dam86,
Chr02]. From this scattering setup, we will then dynamically derive the
asymptotics towards $\mathcal{I}^{+}$, see Fig. 2.

###  1.2 The main result

Let us here already give an outline of the results we obtain from the
scattering data setup described above, focussing first on the simpler case
$s=0$. See also Fig. 2.

  * •

We start by assuming that, if $\psi_{0}$ denotes the spherically symmetric
part of $\psi$, and if $\phi_{0}:=r\psi_{0}$, then
$\phi_{0}|_{\mathcal{C}}\sim u^{-1}$ as $u\to-\infty$ along some ingoing null
hypersurface $\mathcal{C}$. This assumption is motivated by post-Newtonian
arguments for systems of $N$ infalling masses from $i^{-}$, see §5.

  * •

Combining this assumption with the condition of no incoming radiation from
$\mathcal{I}^{-}$, we then prove that $\partial_{v}\phi_{0}\sim r^{-3}\log r$
towards $\mathcal{I}^{+}$, so the peeling property fails. In fact, we prove
that the limit $\lim_{\mathcal{I}^{+}}\frac{r^{3}}{\log
r}\partial_{v}\phi_{0}$ is conserved along $\mathcal{I}^{+}$.

  * •

Finally, if one smoothly, but arbitrarily, extends the data along
$\mathcal{C}$ towards $\mathcal{H}^{+}$, then this failure of peeling will
lead to the following late-time asymptotics along $\mathcal{I}^{+}$:
$\phi_{0}|_{\mathcal{I}^{+}}\sim u^{-2}\log u$ as $u\to\infty$. This should be
contrasted with the Price’s law333Price’s law is the statement that the
following asymptotic behaviour should hold given sufficiently rapidly decaying
data: $\psi_{\ell}\sim t^{-3-2\ell}$ along curves of constant $r$ as
$t\to\infty$, and $\phi_{\ell}|_{\mathcal{I}^{+}}\sim u^{-2-\ell}$ as
$u\to\infty$. These asymptotics were first predicted from heuristic arguments
in [Pri72] and [Lea86], respectively, and proved using mathematically rigorous
arguments in [AAG18b, Hin20, AAG21b]. rate one obtains for Cauchy data of
compact support: $\phi_{0}|_{\mathcal{I}^{+}}\sim u^{-2}$.

Figure 2: Summary of the main results of the paper, focussing on the
spherically symmetric ($\ell=0$) part of the scalar field ($s=0$).

For the scalar field ($s=0$), the measurable rate along $\mathcal{I}^{+}$ thus
only differs by a logarithm from the Price’s law rate. This difference is much
less subtle in the case of gravitational perturbations ($|s|=2$): There, under
the same setup, we expect the following: $\Psi^{[0]}$ violates the peeling
property towards $\mathcal{I}^{+}$, $\Psi^{[0]}\sim\mathcal{O}(r^{-4})$, and
this leads to $r\Psi^{[4]}|_{\mathcal{I}^{+}}$ decaying like $u^{-4}$ along
$\mathcal{I}^{+}$ as $u\to\infty$, as opposed to the $u^{-6}$-rate that one
obtains in the case of compactly supported Cauchy data and the $u^{-5}$-rate
that one obtains for data consistent with the peeling property!

The proof of this expectation is work in progress [KM22], but §6 of the
present paper already contains a preview of the argument.

###  1.3 Structure

The geometry, coordinates and foliations of the Schwarzschild spacetime are
set up in §2. We then give the definitions of the $f(r)$-modified
Newman–Penrose charges and their associated conservation laws in §3. In §4, we
use these conservation laws to explain how to translate various asymptotics
towards $\mathcal{I}^{+}$ into late-time asymptotics along $\mathcal{I}^{+}$
and near $i^{+}$. In §5, we then construct a simple model capturing a system
of $N$ infalling masses from $i^{-}$ and discuss the asymptotics towards
$\mathcal{I}^{+}$ exhibited by this model. In §6, we combine the results of §5
and §4 to obtain a complete dictionary translating asymptotics near
$\mathcal{I}^{-}$ to asymptotics near $i^{+}$. In particular, this gives
predictions on the (in principle) measurable late-time asymptotics along
$\mathcal{I}^{+}$.

We briefly touch upon some extensions of the methods to different settings in
§7, and we conclude in §8.

##  2 Coordinates, foliations and conventions

We consider the Schwarzschild black hole exterior spacetimes, denoted
$(\mathring{\mathcal{M}}_{M},g_{M})$, where
$\mathring{\mathcal{M}}_{M}=\mathbb{R}_{t}\times(2M,\infty)_{r}\times\mathbb{S}^{2}_{(\theta,\varphi)}$,
and where

$\displaystyle
g_{M}=-D\mathop{}\\!\mathrm{d}t^{2}+D^{-1}\mathop{}\\!\mathrm{d}r^{2}+r^{2}(\mathop{}\\!\mathrm{d}\theta^{2}+\sin^{2}\theta\mathop{}\\!\mathrm{d}\varphi^{2}),\quad
D=1-\tfrac{2M}{r}.$

Note that the constant-$t$ slices foliating $\mathring{\mathcal{M}}_{M}$ are
_asymptotically flat_ and, in the Kruskal extension of the spacetime, approach
the bifurcation sphere $\mathcal{B}$ (see Fig. 3). To capture radiative
properties in the spacetime, it will be more convenient to introduce the
following $\tau$-slicing by _asymptotically hyperboloidal_ hypersurfaces that
penetrate the event horizon strictly to the future of $\mathcal{B}$: Consider
the new time function

$\displaystyle\tau=t+r_{*}-2(r-2M)-4M\log\left(\tfrac{r}{2M}\right)\quad\textnormal{with}\quad
r_{*}=r+2M\log\left(\tfrac{r-2M}{M}\right).$

It may easily be verified that we have the following expression for $g_{M}$ in
$(\tau,r,\theta,\varphi)$ coordinates:

$g_{M}=-(1-\tfrac{2M}{r})\mathop{}\\!\mathrm{d}\tau^{2}-2(1-\tfrac{8M^{2}}{r^{2}})\mathop{}\\!\mathrm{d}\tau\mathop{}\\!\mathrm{d}r+\tfrac{16M^{2}}{r^{2}}(1+\tfrac{2M}{r})\mathop{}\\!\mathrm{d}r^{2}+r^{2}(\mathop{}\\!\mathrm{d}\theta^{2}+\sin^{2}\theta\mathop{}\\!\mathrm{d}\varphi^{2}).$

In fact, this metric is well-defined on the manifold-with-boundary
$\mathcal{M}_{M}=\mathbb{R}_{\tau}\times[2M,\infty)_{r}\times\mathbb{S}^{2}_{(\theta,\varphi)}$,
which may be thought of as an extension of $\mathring{\mathcal{M}}_{M}$ that
includes the future event horizon $\mathcal{H}^{+}$ as the level set
$\mathcal{H}^{+}=\\{r=2M\\}$. We denote with $\Sigma_{\tau}$ the
constant-$\tau$ level sets:

$\Sigma_{\tau_{0}}=\\{\tau=\tau_{0}\\}.$

We will moreover make use of the double null functions

$\displaystyle v=\>t+r_{*},$ $\displaystyle u=\>t-r_{*},$ (2.1)

and we will consider double null coordinates $(u,v,\theta,\varphi)$ on
$\mathring{\mathcal{M}}_{M}$. The level sets of constant $u$ or $v$ are null
hypersurfaces, and we may formally444One can also view these sets as conformal
boundaries of the spacetime. represent future and past null infinity,
$\mathcal{I}^{+}$ and $\mathcal{I}^{-}$, as the level sets

$\displaystyle\mathcal{I}^{+}=\\{v=\infty\\},$
$\displaystyle\mathcal{I}^{-}=\\{u=-\infty\\}.$

Figure 3: A Penrose diagrammatic depiction of $\mathcal{M}_{M}$ with the
asymptotically hyperboloidal foliation by $\Sigma_{\tau}$, constant-$t$ level
sets and $u$\- and $v$-null hypersurfaces.

A final word on conventions: The letter $C$ appearing in inequalities will
always denote a positive constant whose precise value is not relevant and
which is allowed to change from line to line. In other words, $C$ obeys the
“algebra of constants”: $C\cdot C=C+C=C\dots$. We will occasionally omit $C$
and just write $f\lesssim g$ or $f\gtrsim g$ if the quantities $f,g$ satisfy
$f\leq Cg$ or $f\geq Cg$, respectively. If both $f\lesssim g$ and $f\gtrsim
g$, we write $f\sim g$.

If we talk about constants whose specific values do matter, we will typically
use either the letter $A$ or $B$, the former in the case when $A$ is just a
constant and nonvanishing multiple of certain data quantities, the latter when
$B$ is a more complicated expression (that can in principle vanish).

##  3 The $f(r)$-modified Newman–Penrose charges

In double null coordinates (2.1), one can derive from (1.2) the following wave
equation for the rescaled quantity $\phi:=r\psi$:

$4\partial_{u}\partial_{v}\phi=\frac{D}{r^{2}}{\not{\Delta}_{\mathbb{S}^{2}}\phi}-\frac{2MD}{r^{3}}\phi,$
(3.1)

where $\not{\Delta}_{\mathbb{S}^{2}}$ is the spherical Laplacian with
eigenvalues ${-\ell(\ell+1)}$, $\ell\in\mathbb{N}_{0}$.

To develop an intuition for the $f(r)$-modified Newman–Penrose (N–P) charges,
let us first consider the case $M=0$: Projecting onto spherical harmonics
$Y_{\ell m}$, $\phi_{\ell m}:=\langle\phi,Y_{\ell
m}\rangle_{L^{2}(\mathbb{S}^{2})}$ (and suppressing the $m$-index), it is
straightforward to derive from (3.1) the following infinite set of exact
conservation laws:

$\partial_{u}(r^{-2\ell}\partial_{v}(r^{2}\partial_{v})^{\ell}\phi_{\ell})=0.$
(3.2)

If $M\neq 0$, we no longer have global conservation laws, but we can still
derive asymptotic conservation laws. Focussing first on the $\ell=0$-mode
$\phi_{0}$, the analogue of (3.2) reads

$4\partial_{u}\partial_{v}\phi_{0}=-\frac{2MD}{r^{3}}\phi_{0}.$ (3.3)

Clearly, $\partial_{v}\phi_{0}$ is no longer globally conserved; however, the
RHS exhibits a good $r^{-3}$-weight. Therefore, for any function $f$ with

$(f(r))^{-1}=o(r^{3}),$ (3.4)

the $f(r)$-modified Newman–Penrose charge

$I_{0}^{f}[\psi](u):=\lim_{v\to\infty}(f(r))^{-1}\partial_{v}\phi_{0}(u,v)$
(3.5)

will be conserved in $u$ (so long as it exists for some value of $u$), which
follows from commuting (3.3) with $f(r)^{-1}$.

While the charges $I_{0}^{f}[\psi]$ have first been defined for $f(r)=r^{-2}$
[NP68, AAG21b], we will see in the present paper that other choices of $f$ are
equally important; see also [Kro00, Kro01, Keh21b, Keh21c].

Notice however the restriction $f(r)^{-1}=o(r^{3})$ (also a consequence of the
$r^{-3}$-weight in (3.3)). In particular, if $\partial_{v}\phi_{0}=O(r^{-3})$
initially, then all ($\ell=0$) $f(r)$-modified N–P charges vanish, and one
cannot directly associate a non-zero asymptotic conserved charge to (3.3).

#### Generalising (3.3) to higher $\ell$:

Consider the rescaled null vector field
$\boldsymbol{\widehat{L}}:=D^{-1}r^{2}\partial_{v}$, and replace
$(r^{2}\partial_{v})^{\ell}\phi_{\ell}$ in (3.2) with

$\Phi_{\ell}:=\boldsymbol{\widehat{L}}^{\ell}\phi_{\ell}+\sum_{i=1}^{\ell}x_{i}^{(\ell)}\cdot
M^{i}\cdot\boldsymbol{\widehat{L}}^{\ell-i}\phi_{\ell},\quad\text{with}\quad
x_{i}^{(\ell)}=\frac{1}{i!}\frac{(2\ell-i)!}{(2\ell)!}\left(\frac{\ell!}{(\ell-i)!}\right)^{3}.$
(3.6)

A lengthy computation then shows that

$\partial_{u}(D^{\ell}r^{-2\ell}\partial_{v}\Phi_{\ell})=\frac{D^{\ell+1}M}{r^{2\ell+3}}\sum_{i=0}^{\ell}\left(y_{i}^{(\ell)}+\frac{M}{r}z_{i}^{(\ell)}\right)\cdot
M^{i}\cdot\boldsymbol{\widehat{L}}^{\ell-i}\phi_{\ell},$ (3.7)

where $\\{y_{i}^{(\ell)},z_{i}^{(\ell)}\\}$ is a set of numerical constants.
As before, we can now associate, for any $f$ with $(f(r))^{-1}=o(r^{3})$, the
following $f(r)$-modified N–P charges to (3.7):

$I^{f}_{\ell}[\psi](u):=\lim_{v\to\infty}f(r)^{-1}\partial_{v}\Phi_{\ell}(u,v).$
(3.8)

These $I^{f}_{\ell}[\psi](u)$ will again be conserved if finite for some value
of $u$.

Importantly, in addition to being conserved, the charges $I^{f}_{\ell}[\psi]$
also provide a measure of conformal regularity, i.e. regularity in the
variable $x=1/r$, of the field $\psi$.555Note that in coordinates
$(u,x=1/r,\theta,\varphi)$, one has $2\boldsymbol{\widehat{L}}=-\partial_{x}$.
In order to illustrate this point, we consider the following example: Given
data $\psi$ on a hyperboloidal hypersurface $\Sigma_{0}$ that only have a
finite asymptotic expansion in powers of $1/r$, say,

$r\psi_{\ell}=\sum_{i=0}^{\lceil
p_{\ell}\rceil}\frac{C_{i}}{r^{i}}+C^{\ast}\frac{\log^{q_{\ell}}r}{r^{p_{\ell}+1}}+\dots\quad\text{for
some }q_{\ell}\in\mathbb{N},p_{\ell}\in\mathbb{R}_{+},$

then the $f(r)$-modified N–P charges associated to $\psi$ can be read off from
Table 1 below:

Values of $p_{\ell},q_{\ell}$ |  $p_{\ell}<\ell$ or $(p_{\ell}=\ell,q_{\ell}>0)$: |  $(p_{\ell}=\ell,q_{\ell}=0)$: |  $p_{\ell}>\ell$:
---|---|---|---
$\partial_{v}\Phi_{\ell}=$ | $\frac{A\log^{q_{\ell}}r}{r^{2-(\ell-p_{\ell})}}+\dots$ | $\frac{B}{r^{2}}+\dots$ | $\frac{B}{r^{2}}+\frac{A\log^{q_{\ell}}r}{r^{2-(\ell-p_{\ell})}}+\dots$
Table 1: The letter $A$ is a placeholder for a nonvanishing constant multiple
of $C^{*}$, whereas $B$ stands for a linear combination of the $C_{i}$ and may
therefore be vanishing. The second row of the table indicates for which $f$
the $f(r)$-modified N–P charge is finite and nonvanishing. For instance, if
$p_{\ell}<\ell$, then we can take $f(r)=r^{\ell-p_{\ell}-2}\log^{q_{\ell}}r$
and $I_{\ell}^{f}[\psi]=A$.

#### The $f(r)$-modified N–P charges for higher spin $s$:

In a similar fashion, one can define conserved charges for more general spin.
For instance, if $s=\pm 2$, then the generalisation of the Minkowskian
identities (3.2) is given by

$\partial_{u}(r^{-2\ell-2s}\partial_{v}(r^{2}\partial_{v})^{\ell-s}(r^{|s|+s+1}\Psi^{[|s|-s]}_{\ell}))=0,$
(3.9)

where $\Psi^{[|s|-s]}_{\ell}$ is the projection of $\Psi^{[|s|-s]}$ onto the
spin-$s$ weighted spherical harmonics ${}_{s}Y_{\ell m}$, which are defined
for $\ell\geq|s|$ (here, we again suppressed the $m$-index). From (3.9), one
can then derive an equation similar to (3.7) in order to derive the relevant
$f(r)$-modified N–P charges for $M\neq 0,|s|=2$. Comparing (3.2) with (3.9),
one thus finds that, roughly speaking, the $\ell$-th mode of $r^{4}\Psi^{[0]}$
behaves like the $(\ell-2)$-mode of $\psi$ would. Similarly, the $\ell$-th
mode of $\Psi^{[4]}$ behaves like the $(\ell+2)$-mode of $\psi$ would.

##  4 From asymptotics towards $\mathcal{I}^{+}$ to asymptotics towards
$i^{+}$

We will now sketch the derivation of the leading-order late-time asymptotics
in time $\tau$ as $\tau\to\infty$ of solutions to the wave equation (1.2)
starting from initial data on the hyperboloidal initial hypersurface
$\Sigma_{0}$. The arguments in this section generalise arguments from [AAG18c,
AAG18b, AAG21b, Keh21b].

We first consider data for which there exists $f_{\ell}$ such that
$I_{\ell}^{f_{\ell}}[\psi]\neq 0$ in §4.1. As we will see, the late-time tails
for $\psi_{\ell}$ are directly encoded in the value of
$I_{\ell}^{f_{\ell}}[\psi]\neq 0$ in this case. In §4.2, we then consider data
for which $I_{\ell}^{f_{\ell}}[\psi]=0$ for any choice of $f_{\ell}$, and
reduce this case to that of §4.1. The analyses of §4.1 and §4.2 produce the
late-time tails for fixed angular frequency solutions $\psi_{\ell}$. We
comment on general solutions in §4.3.

###  4.1 The case of nonvanishing N–P charge $I_{\ell}^{f}[\psi]$

We take $f_{\ell}$ to be a function of $r$ with the following general form:

$f_{\ell}(r)=r^{-p^{\prime}_{\ell}}(\log r)^{q_{\ell}},$ (4.1)

where $1-\ell<p^{\prime}_{\ell}<3$ and $q_{\ell}\geq 0$,
$p_{\ell}^{\prime}\in\mathbb{R}$, $q_{\ell}\in\mathbb{N}$. Given $f_{\ell}$,
we make the following assumptions on the $r$-asymptotics of the initial data
on $\Sigma_{0}$: The $\ell$-th spherical harmonic mode $\psi_{\ell}$ satisfies

$\displaystyle\begin{split}\partial_{v}\Phi_{\ell}|_{\Sigma_{0}}=&\>I_{\ell}^{f_{\ell}}[\psi]f_{\ell}(r)+O(r^{-p^{\prime}_{\ell}})\quad\textnormal{if\leavevmode\nobreak\
$q_{\ell}>0$},\\\
\partial_{v}\Phi_{\ell}|_{\Sigma_{0}}=&\>I_{\ell}^{f_{\ell}}[\psi]f_{\ell}(r)+O(r^{-p^{\prime}_{\ell}-\beta})\quad\textnormal{if\leavevmode\nobreak\
$q_{\ell}=0$},\end{split}$ (4.2)

with $I_{\ell}^{f_{\ell}}[\psi]\neq 0$, $\beta>0$, and with $\Phi_{\ell}$
defined in (3.6).

In Steps 0–3 below, we outline how we can translate the above initial data
$r$-asymptotics to the following late-time $\tau$-asymptotics and
$u$-asymptotics:

$\displaystyle\psi_{\ell}|_{r=r_{0}}(\tau)=$
$\displaystyle\>A_{\ell}w_{\ell}(r_{0})I_{\ell}^{f_{\ell}}[\psi]f_{\ell}(\tau)\tau^{-2\ell}+\ldots\quad(\tau\to\infty),$
(4.3) $\displaystyle r\psi_{\ell}|_{\mathcal{I}^{+}}(u)=$
$\displaystyle\>A_{\ell}I_{\ell}^{f_{\ell}}[\psi]f_{\ell}(u)u^{1-\ell}+\ldots\quad(u\to\infty),$
(4.4)

with $A_{\ell}\in\mathbb{R}$ constants that depend only on $p,q,\ell$, and
$w_{\ell}(r_{0})$ also depending on $r_{0}$ (see (4.12) for the precise
$r_{0}$-dependence), and where $\ldots$ schematically denote terms that
contribute as higher-order terms in $\tau^{-1}$ or $u^{-1}$.

#### Step 0:

The mechanism for deriving late-time tails relies on the following type of
_upper bound time-decay estimate_ :

$|r^{-\ell}\psi_{\ell}|\leq
B_{\epsilon}\tau^{1-p^{\prime}_{\ell}-\ell+\epsilon}(\tau+r)^{-\ell-1},\\\ $
(4.5)

with $\epsilon>0$ arbitrarily small and $B_{\epsilon}>0$ an appropriately
large constant depending on $L^{2}$-type initial data norms and diverging as
$\epsilon\to 0$. In light of the expected time-decay that can be read off from
(4.3) and (4.4), the estimate (4.5) can be thought of as an _almost sharp_
time-decay estimate.

We will moreover make use of the fact that, in $(\tau,r)$-coordinates, when
acting with the vector fields $\partial_{\tau}$, $r\partial_{r}$ and
$r^{2}\partial_{r}$ on $r\psi_{\ell}$, the $\tau$-decay rate in (4.5) changes
according to Table 2:

Table 2: Change in decay rate when acting with weighted vector fields on $\psi_{\ell}$. Vector field | Change in power of $\tau$-factor in (4.5)
---|---
$\partial_{\tau}$ | -1
$r\partial_{r}$ | +0
$r^{2}\partial_{r}$ | +1

The estimate (4.5) and the properties of Table 2 are slight generalisations of
what is derived in [AAG18c, AAG18b, AAG21b]. The methods used to derive (4.5)
build on the vast literature on uniform boundedness and decay estimates for
linear waves on black hole spacetime backgrounds and involve geometric
properties of Schwarzschild like the trapping of null geodesics and the red-
shift effect; see [DRSR16] for a comprehensive overview of this literature. A
derivation of these almost-sharp estimates lies beyond the scope of the
present paper, so we will view them simply as _black box assumptions_ in a
self-contained derivation of late-time tails.

In Steps 1–3 below, we will outline the derivation of the precise leading-
order late-time asymptotics and late-time tails for the spherically symmetric
$\ell=0$-mode. We will briefly describe the generalization to $\ell\geq 1$
afterwards.

#### Step 1:

Let $\alpha\in(0,1)$. We first restrict to the region $\\{v\geq
v_{\gamma}(u)\\}$. Here, $\gamma$ is a timelike curve along which

$r=u^{\alpha}+\ldots=v^{\alpha}+\ldots,$

with $\ldots$ denoting terms that are higher order in $u^{-1}$ and $v^{-1}$,
and $v_{\gamma}(u)$ denotes the unique $v$ such that $(u,v)\in\gamma$. In this
region, we apply (3.3) and (4.2) to obtain:

$\displaystyle\partial_{u}\partial_{v}\phi_{0}=O(r^{-3})\phi_{0}+\ldots,\quad\textnormal{with
}\partial_{v}\phi_{0}|_{\Sigma_{0}}(v)=I_{0}^{f_{0}}[\psi]f_{0}\left(\tfrac{v}{2}\right)+\ldots,$

with $\ldots$ again denoting terms that contribute to sub-leading order in the
argument. Here, we used moreover that $r=\frac{1}{2}(v-u)+\ldots$ to leading
order in $v-u$, so $r|_{\Sigma_{0}}=\frac{v}{2}+\ldots$.

Figure 4: Step 1: integrating $\partial_{u}\partial_{v}\phi_{0}$ in $u$.

Since $\alpha>0$, $r$ _grows_ along $\gamma$, and the factor $r^{-3}$ on the
right-hand side of $\partial_{u}\partial_{v}\phi_{0}$ _decays_ in the region
$\\{v\geq v_{\gamma}(u)\\}$. Therefore, we can integrate the equation for
$\partial_{u}\partial_{v}\phi_{0}$ in the $u$-direction (see Fig. 4) and plug
in the almost-sharp estimate (4.5) on the right-hand side to propagate the
asymptotics of $\partial_{v}\phi_{0}$ from $\Sigma_{0}$ to the rest of the
region $\\{v\geq v_{\gamma}(u)\\}$:

$\displaystyle\partial_{v}\phi_{0}(u,v)=$
$\displaystyle\>I_{0}^{f_{0}}[\psi]f_{0}\left(\tfrac{v}{2}\right)+\ldots.$
(4.6)

Note that (4.6) in particular implies the following modified N–P conservation
law along $\mathcal{I}^{+}$: The limit
$\lim_{r\to\infty}\frac{1}{f_{0}(r)}\partial_{v}\phi_{0}(u,r)$ is conserved in
$u$ along $\mathcal{I}^{+}$ and is equal to $I_{0}^{f_{0}}[\psi]$.

#### Step 2:

In this step, we integrate (4.6) in the $v$-direction in the region $\\{v\geq
v_{\gamma}(u)\\}$, now starting from $\gamma$ (see Fig. 5):

$\phi_{0}(u,v)=\phi_{0}|_{\gamma}(u)+\int_{v_{\gamma}(u)}^{v}\partial_{v}\phi_{0}(u,v^{\prime})\mathop{}\\!\mathrm{d}v^{\prime}.$

The curve $\gamma$ is chosen such that $\phi_{0}|_{\gamma}$ only contributes
to higher order. Indeed, using our definition of $\gamma$ ($r_{\gamma}\sim
u^{\alpha}$ along $\gamma$), we can split

$|\phi_{0}|_{\gamma}|=r_{\gamma}|\psi_{0}|_{\gamma}|\sim
u^{\alpha}|\psi_{0}|_{\gamma}|.$

Figure 5: Step 2: integrating $\partial_{v}\phi_{0}$ in $v$ from $\gamma$.

Since $\alpha<1$, and since $u\sim\tau$ along $\gamma$, we can then apply the
estimate (4.5) with $\epsilon>0$ suitably small to conclude that
$\phi_{0}|_{\gamma}$ in fact decays _faster_ than $u^{-p_{0}^{\prime}+1}$.
Hence,

$\begin{split}\phi_{0}(u,v)=\int_{v_{\gamma}(u)}^{v}\partial_{v}\phi_{0}(u,v^{\prime})\mathop{}\\!\mathrm{d}v^{\prime}+\ldots=I_{0}^{f_{0}}[\psi]\int_{v_{\gamma}(u)}^{v}f_{0}(\tfrac{v^{\prime}}{2})\mathop{}\\!\mathrm{d}v^{\prime}+\ldots.\end{split}$

Given the form of $f_{0}$ assumed in (4.1), we can evaluate the above integral
to obtain:

$\begin{split}\phi_{0}(u,v)=&\>I_{0}^{f_{0}}[\psi]\int_{v_{\gamma}(u)}^{v}(\tfrac{v^{\prime}}{2})^{-p_{0}^{\prime}}\log^{q_{0}}(\tfrac{v^{\prime}}{2})\mathop{}\\!\mathrm{d}v^{\prime}+\ldots\\\
=&\>\tfrac{2^{p_{0}^{\prime}}}{p_{0}^{\prime}-1}I_{0}^{f_{0}}[\psi]\cdot\left(\log^{q_{0}}(\tfrac{v_{\gamma}(u)}{2})(v_{\gamma}(u)^{-p_{0}^{\prime}+1}-v^{-p_{0}^{\prime}+1})+v^{-p_{0}^{\prime}+1}\log^{q_{0}}(\tfrac{v_{\gamma}(u)}{v})\right)\ldots\end{split}$

everywhere in $\\{v\geq v_{\gamma}(u)\\}$, where … here denote terms that will
contribute as higher order terms in $u^{-1}$ or $\tau^{-1}$ below. In
particular, restricting to a smaller region $\\{v\geq
v_{\tilde{\gamma}}(u)\\}$, where $\tilde{\gamma}$ is a timelike curve along
which $r=u^{\tilde{\alpha}}+\ldots=v^{\tilde{\alpha}}+\ldots$, for
$\alpha<\tilde{\alpha}<1$ with $\tilde{\alpha}$ suitably close to 1, we
obtain:

$\phi_{0}(u,v)=\tfrac{2^{p_{0}^{\prime}}}{p-1}I_{0}^{f_{0}}[\psi](\log^{q_{0}}(u)(u^{-p_{0}^{\prime}+1}-v^{-p_{0}^{\prime}+1})+\ldots.$
(4.7)

Taking the limit $v\to\infty$, we thus obtain

$\displaystyle\phi_{0}|_{\mathcal{I}^{+}}(u)=$
$\displaystyle\>\tfrac{2^{p_{0}^{\prime}}}{p_{0}^{\prime}-1}I_{0}^{f_{0}}[\psi]u^{-p_{0}^{\prime}+1}\log^{q_{0}}u+\ldots,$

which proves (4.4) for $\ell=0$. Moreover, by Taylor expanding the expression
$\frac{2}{v_{\tilde{\gamma}}(u)-u}(u^{-p+1}-v_{\tilde{\gamma}}(u)^{-p+1})$ in
$\frac{v_{\tilde{\gamma}}(u)-u}{u}=u^{\tilde{\alpha}-1}$ around 0, we can also
obtain the late-time asymptotics of $\psi_{0}$ along $\tilde{\gamma}$:

$\displaystyle\psi_{0}|_{\tilde{\gamma}}(u)=\tfrac{2\phi_{0}|_{\tilde{\gamma}}(u)}{v_{\tilde{\gamma}}(u)-u}+\ldots=$
$\displaystyle\>2^{p_{0}^{\prime}+1}I_{0}^{f_{0}}[\psi]u^{-p_{0}^{\prime}}\log^{q_{0}}u+\ldots.$
(4.8)

#### Step 3:

Figure 6: Step 3: integrating $\partial_{r}(Dr^{2}\partial_{r}\psi_{0})$ along
$\Sigma_{\tau}$ from $r=2M$ and then $\partial_{r}\psi_{0}$ from
$\tilde{\gamma}$.

To also conclude (4.3), we _propagate_ the late-time asymptotics (4.8) derived
along the curve $\tilde{\gamma}$ in Step 2 all the way to the event horizon at
$r=2M$. We use that, in $(\tau,r,\theta,\varphi)$ coordinates, the wave
equation for $\psi_{0}$ takes the following schematic form:

$\partial_{r}((r^{2}-2Mr)\partial_{r}\psi_{0})=O(r)\cdot\left(\partial_{\tau}\psi_{0}+r\partial_{r}\partial_{\tau}\psi_{0}+\partial_{\tau}^{2}\psi_{0}\right).$
(4.9)

We integrate eq. (4.9) from $r=2M$ (see Fig. 6) to obtain everywhere in
$2M\leq r\leq r_{\tilde{\gamma}}(\tau)$:

$(r^{2}-2Mr)|\partial_{r}\psi_{0}|(\tau,r)\lesssim\int_{2M}^{r}r\left(|\partial_{\tau}\psi_{0}|+|r\partial_{r}\partial_{\tau}\psi_{0}|+|\partial_{\tau}^{2}\psi_{0}|\right)\mathop{}\\!\mathrm{d}r^{\prime}.$

Now, we can exploit the fact that the right-hand side above involves terms
with at least one $\partial_{\tau}$-derivative: Indeed, the estimate (4.5)
together with Table 2 tells us that the integrand on the RHS goes like
$r\cdot\tau^{-p_{0}^{\prime}-1+\epsilon}$. Doing the integral, we thus obtain,
for all $r\leq r_{\tilde{\gamma}}(\tau)$:

$|\partial_{r}\psi_{0}|\leq C(1+\tau)^{-p_{0}^{\prime}-1+\epsilon}.$ (4.10)

Finally, for any $r\in[2M,r_{\tilde{\gamma}}(\tau)]$, we write
$\psi_{0}(\tau,r)=\psi_{0}(\tau,r_{\tilde{\gamma}}(\tau))+\int\partial_{r}\psi_{0}(\tau,r^{\prime})\mathop{}\\!\mathrm{d}r^{\prime}$,
and combine estimates (4.8) and (4.10). Again, the integral term only
contributes at higher order, so we obtain

$\psi_{0}(\tau,r)=\psi_{0}(\tau,r_{\tilde{\gamma}}(\tau))+\ldots=2^{p_{0}^{\prime}+1}I_{0}^{f_{0}}[\psi]\cdot\tau^{-p_{0}^{\prime}}(\log\tau)^{q_{0}}+\ldots$
(4.11)

everywhere in $2M\leq r\leq r_{\tilde{\gamma}}(\tau)$. This concludes the
derivation of (4.3) in the case $\ell=0$.

#### Treatment of $\ell\geq 1$ modes:

The argument for higher $\ell$-modes runs similarly, with the role of
$\phi_{0}$ now being played by $\Phi_{\ell}$. We start Step 1 by integrating
(3.7) in $u$, which schematically reads

$\displaystyle\partial_{u}(D^{\ell}r^{-2\ell}\partial_{v}\Phi_{\ell})=$
$\displaystyle\>O(r^{-3-2\ell})\Phi_{\ell}+\ldots.$

This proves the analogue of (4.6), giving us a sharp estimate for
$\partial_{v}\Phi_{\ell}$ and the conservation of $I_{\ell}^{f_{\ell}}[\psi]$.
Then, in Step 2, we integrate $\partial_{v}\Phi_{\ell}$ in $v$ to first obtain
the late-time behaviour of $\Phi_{\ell}$, and we subsequently use that

$\Phi_{\ell}(u,v)=(D^{-1}r^{2}\partial_{v})^{\ell}(r\psi_{\ell})+\ldots,$

with $\ldots$ terms that contribute as higher-order terms, to obtain the late-
time behaviour of $\phi_{\ell}$ after another $\ell$ integrations in $v$.
Since each of these integrations picks up an extra $1/u$-term from a suitably
chosen $\gamma$, this thus proves (4.4), as well as the analogue of (4.8).

Finally, in Step 3, we consider the weighted quantity
$w_{\ell}^{-1}(r)\psi_{\ell}$, with $w_{\ell}$ a smooth, non-zero radial
weight function that satisfies

$\displaystyle\square_{g}(w_{\ell}Y_{\ell m})=0\quad\textnormal{for
all\leavevmode\nobreak\ $|m|\leq\ell$},\quad\textnormal{and}\quad
w_{\ell}(r)\sim$
$\displaystyle\>r^{\ell}\quad\textnormal{as\leavevmode\nobreak\
$r\to\infty$}.$ (4.12)

The product $w_{\ell}^{-1}(r)\psi_{\ell}$ then satisfies the equation:

$\partial_{r}((r^{2}-2Mr)w_{\ell}^{2}\partial_{r}(w_{\ell}^{-1}\psi_{\ell}))=O(r^{\ell+1})\cdot\left(\partial_{\tau}\psi_{\ell}+r\partial_{r}\partial_{\tau}\psi_{\ell}+\partial_{\tau}^{2}\psi_{\ell}\right),$
(4.13)

which involves only terms with $\partial_{\tau}$-derivatives on the right-hand
side. We integrate this equation to show that
$\partial_{r}(w_{\ell}^{-1}\psi_{\ell})$ is higher-order in $\tau^{-1}$. One
more integration in $r$ then results in the global late-time asymptotics for
$w_{\ell}^{-1}\psi_{\ell}$ and, in particular, proves (4.3).

###  4.2 Vanishing $I_{\ell}^{f}[\psi]$

When $I_{\ell}^{f_{\ell}}[\psi]=0$ in the initial data assumptions (4.2), the
steps outlined in Section 4.1 cannot be applied directly. We now explain how
to proceed in this case. Let $f_{\ell}(r)=r^{-p^{\prime}_{\ell}}(\log
r)^{q_{\ell}}$ with $p^{\prime}_{\ell}\geq 2$, and assume the following
asymptotics on data:

$\displaystyle\partial_{v}\Phi_{\ell}|_{\Sigma_{0}}=$
$\displaystyle\>J_{\ell}r^{-1}f_{\ell}(r)+O(r^{-p^{\prime}_{\ell}-1})\quad\textnormal{if\leavevmode\nobreak\
$q_{\ell}>0$},$ (4.14) $\displaystyle\partial_{v}\Phi_{\ell}|_{\Sigma_{0}}=$
$\displaystyle
J_{\ell}r^{-1}f_{\ell}(r)+O(r^{-p^{\prime}_{\ell}-1-\beta})\quad\textnormal{if\leavevmode\nobreak\
$q_{\ell}=0$},$ (4.15)

for some $J_{\ell}\in\mathbb{R}$. Assumption (4.15) implies that
$I_{\ell}^{f}[\psi]=0$ for any choice of $f$. Furthermore, since we allow
$J_{\ell}$ to be zero, these assumptions include initial data of compact
support.

The object that is key to deriving late-time tails for these rapidly
$r$-decaying initial data is the _time integral_ :

$\partial_{\tau}^{-1}\psi_{\ell}(\tau,r):=-\int_{\tau}^{\infty}\psi_{\ell}(\tau^{\prime},r)\mathop{}\\!\mathrm{d}\tau^{\prime}.$
(4.16)

By the decay estimates (4.5) in Step 0, $\partial_{\tau}^{-1}\psi_{\ell}$ is
well-defined and regular at the event horizon. Moreover, by definition,
$\partial_{\tau}(\partial_{\tau}^{-1}\psi_{\ell})=\psi_{\ell}$, and, by the
symmetries of Schwarzschild, $\partial_{\tau}^{-1}\psi_{\ell}$ still solves
(1.2) projected to $\ell$. Lastly, we can determine the value of
$\partial_{\tau}^{-1}\psi_{\ell}|_{\Sigma_{0}}$ via simple integration of:

$\partial_{r}((r^{2}-2Mr)w_{\ell}^{2}\partial_{r}(w_{\ell}^{-1}\partial_{\tau}^{-1}\psi_{\ell}))|_{\Sigma_{0}}=O(r^{\ell+1})\left[\psi_{\ell}+r\partial_{r}\psi_{\ell}+\partial_{\tau}\psi_{\ell}\right]|_{\Sigma_{0}},$
(4.17)

using that
$(r^{2}-2Mr)w_{\ell}^{2}\partial_{r}(w_{\ell}^{-1}\partial_{\tau}^{-1}\psi_{\ell})$
vanishes at $r=2M$ and $w_{\ell}^{-1}\partial_{\tau}^{-1}\psi_{\ell}$ vanishes
as $r\to\infty$.

It then follows that the asymptotics of
$\partial_{v}\partial_{\tau}^{-1}\Phi_{\ell}$ can be grouped into two cases:

$\displaystyle\partial_{v}\partial_{\tau}^{-1}\Phi_{\ell}|_{\Sigma_{0}}=$
$\displaystyle(I_{\ell}^{(1)})^{f_{\ell}}[\psi]f_{\ell}(r)+O(r^{-p^{\prime}_{\ell}})\quad\textnormal{if\leavevmode\nobreak\
$q_{\ell}>0$ \text@underline{and}\leavevmode\nobreak\ $p^{\prime}_{\ell}=2$
\text@underline{and}\leavevmode\nobreak\ $J_{\ell}\neq 0$},$
$\displaystyle\partial_{v}\partial_{\tau}^{-1}\Phi_{\ell}|_{\Sigma_{0}}=$
$\displaystyle(I_{\ell}^{(1)})^{r^{-2}}[\psi]r^{-2}+O(r^{-2-\beta^{\prime}})\quad\textnormal{with\leavevmode\nobreak\
$\beta^{\prime}>0$, if\leavevmode\nobreak\ $q_{\ell}=0$
\text@underline{or}\leavevmode\nobreak\ $p^{\prime}_{\ell}>2$
\text@underline{or}\leavevmode\nobreak\ $J_{\ell}=0$},$

where $(I_{\ell}^{(1)})^{f}[\psi]$ is the $f(r)$-modified N–P charge of
$\partial_{\tau}^{-1}\psi$, which can be expressed either as a constant
multiple of $J_{\ell}$ if $q_{\ell}>0$, $p^{\prime}_{\ell}=2$ and
$J_{\ell}\neq 0$, or in terms of an integral of initial data for $\psi$ along
$\Sigma_{0}$ otherwise. In the latter case, $(I_{\ell}^{(1)})^{r^{-2}}[\psi]$
is _generically nonvanishing if $M\neq 0$_.666Here, “generic” can be given a
precise meaning: $(I_{\ell}^{(1)})^{f}[\psi]=0$ only for a codimension-1
subset of data in any suitably $r$-weighted function space. In fact, within
this codimension-1 subset, we can consider _another_ time integral, so we
apply time integration twice. Using multiple time inversions, we can conclude
that the set of data leading to solutions that do not behave inverse
polynomially in time is therefore of infinite codimension.

We can now apply the arguments of §4.1 to obtain the late-time asymptotics of
$\partial_{\tau}^{-1}\psi_{\ell}$ (cf. (4.3), (4.4)) and finally take a
$\partial_{\tau}$-derivative to deduce the asymptotics for $\psi$ itself. We
obtain

$\displaystyle\psi_{\ell}|_{r=r_{0}}(\tau,\theta,\varphi)=$
$\displaystyle\>A_{\ell}w_{\ell}(r_{0})(I_{\ell}^{(1)})^{f}[\psi]\frac{d}{d\tau}(f(\tau)\tau^{-2\ell})+\ldots\quad(\tau\to\infty),$
(4.18) $\displaystyle r\psi_{\ell}|_{\mathcal{I}^{+}}(u,\theta,\varphi)=$
$\displaystyle\>A_{\ell}(I_{\ell}^{(1)})^{f}[\psi]\frac{d}{du}(f(u)u^{1-\ell})+\ldots\quad(u\to\infty),$
(4.19)

with $f(x)$ either given by $f_{\ell}(x)$ or by $x^{-2}$. Thus, for rapidly
decaying initial data, the $\tau$-decay rates in the corresponding late-time
tails are encoded in the $r$-decay of the initial data of the time integral
$\partial_{\tau}^{-1}\psi$!

The results of §4.1 and §4.2 are summarised in the table below, where
$f_{\ell}=r^{-p_{\ell}^{\prime}}\log^{q_{\ell}}r$ with
$p^{\prime}_{\ell}>1-\ell$ and $q_{\ell}\geq 0$:

|  $p_{\ell}^{\prime}<3$ or $p_{\ell}^{\prime}=3$ and $q_{\ell}>0$: |  $p_{\ell}^{\prime}>3$ or $p_{\ell}^{\prime}=3$ and $q_{\ell}=0$:
---|---|---
$\partial_{v}\Phi_{\ell}|_{\Sigma_{0}}$ as $r\to\infty$: |  $I_{\ell}^{f_{\ell}}[\psi]\cdot f_{\ell}(r)+\dots$ |  $J_{\ell}\cdot f_{\ell}(r)+\ldots$
$\phi_{\ell}|_{\mathcal{I}^{+}}$ as $u\to\infty$: | $A_{\ell}I_{\ell}^{f_{\ell}}[\psi]\cdot f_{\ell}(u)u^{1-\ell}+\dots$ | $A_{\ell}\cdot(I_{\ell}^{(1)})^{r^{-2}}[\psi]u^{-2-\ell}+\dots$
$\psi_{\ell}|_{r=r_{0}}$ as $\tau\to\infty$: | $A_{\ell}w_{\ell}(r_{0})I_{\ell}^{f_{\ell}}[\psi]\cdot f_{\ell}(\tau)\tau^{-2\ell}+\dots$ | $A_{\ell}w_{\ell}(r_{0})\cdot(I_{\ell}^{(1)})^{r^{-2}}[\psi]\cdot\tau^{-3-2\ell}+\dots$
Table 3: Summary of the results of §4. The $A_{\ell}$ are placeholders for
nonvanishing numerical constants. The function $w(r_{0})$ is defined in
(4.12). The rates in the second column are independent of the mass $M$,
whereas the constant $(I_{\ell}^{(1)})^{r^{-2}}[\psi]$ appearing in the third
column does depend on $M$.

###  4.3 Summing over $\ell$

In §4.1 and §4.2, we considered the late-time asymptotics of spherical
harmonic modes of fixed $\ell$. When choosing initial data such that higher
$\ell$-modes have more regularity in $\frac{1}{r}$ as $r\to\infty$, and, in
particular, when considering smooth, compactly supported data, higher modes
will decay faster. In such cases, the estimate (4.5) can be extended to

$|r^{-\ell}\psi_{\geq\ell}|\leq
C\tau^{1-p^{\prime}_{\ell}-\ell+\epsilon}(\tau+r)^{-\ell-1}.$ (4.20)

Then, by (4.20), it follows that when splitting
$\psi_{\geq\ell}=\psi_{\ell}+\psi_{\geq\ell+1}$, the $\psi_{\geq\ell+1}$
contributes at higher order in $\tau^{-1}$ and $u^{-1}$, so the late-time
asymptotics of $\psi_{\geq\ell}$ agree with the late-time asymptotics of
$\psi_{\ell}$, which are given by (4.3) and (4.4), or (4.18) and (4.19).
However, initial data on $\Sigma_{0}$ for which higher modes do not have
higher regularity at infinity may arise naturally in scattering problems, see
§1.3.3 of [Keh21c]. There, it is conjectured that compactly supported
scattering data lead to solutions where all modes contribute to the late-time
asymptotics at the same order. See already the fourth row, second column of
Table 5.

##  5 Deriving asymptotics towards $\mathcal{I}^{+}$ from physical data near
$i^{-}$

In §4, we derived late-time asymptotics near $i^{+}$ from given asymptotics
along some hyperboloidal initial data hypersurface $\Sigma_{0}$, i.e. from
asymptotics towards $\mathcal{I}^{+}$. Naturally, to decide what the “correct”
predictions for late-time asymptotics are, one thus needs a way to decide what
the “correct” asymptotics towards $\mathcal{I}^{+}$ are. As explained in the
introduction, instead of assuming, say, vanishing or “peeling” asymptotics
near $\mathcal{I}^{+}$, we will dynamically derive the asymptotics towards
$\mathcal{I}^{+}$ from a scattering data setup that a) has no incoming
radiation from $\mathcal{I}^{-}$ and b) resembles—in some sense—a system of
$N$ infalling masses following unbound Keplerian orbits near the infinite
past. This section follows the works [Keh21a, Keh21b, Keh21c].

###  5.1 The data setup

In the context of the scalar wave equation on a fixed Schwarzschild
background, one simple model with data that realise a) and b) is depicted in
Fig. 7: In order to satisfy a) on $\mathcal{I}^{-}$, we demand
$\partial_{v}(r\psi)|_{\mathcal{I}^{-}}\equiv 0$ to be vanishing identically.
This corresponds to a vanishing energy flux along $\mathcal{I}^{-}$. Realising
b), on the other hand, is less straightforward. The idea is as follows: While
it may, for now, be too ambitious to try and analytically treat a system of
$N$ infalling masses, we can instead, for sufficiently large negative retarded
times $u$, consider an ingoing null cone $\mathcal{C}$ from $\mathcal{I}^{-}$,
to be thought of as enclosing the $N$ infalling masses, and impose data on
$\mathcal{C}$ that capture the structure of the radiation emitted by these $N$
infalling masses, see Fig. 7.

Figure 7: Depiction of the data setup: We aim to capture the radiation of the
infalling masses by imposing the quadrupole approximation on $\mathcal{C}$.
The depicted spacetime can be thought of as part of the spacetime depicted in
Fig. 1.

The heuristic tool that allows us to take this step is the quadrupole
approximation, which predicts that (gravitational) radiation decays
polynomially along $\mathcal{C}$ with a certain rate.

Identifying this radiation with scalar radiation $\psi$, we now give a brief
sketch of where this polynomial decay comes from: The quadrupole approximation
for $N$ infalling masses predicts that the loss of gravitational energy along
$\mathcal{I}^{+}$ is given by

$\frac{dE_{\text{grav}}}{dt}\sim-\dddot{\mathbf{Q}}_{ij}^{TT}\dddot{\mathbf{Q}}_{ij}^{TT},$

the $\mathbf{Q}_{ij}$ denoting the quadrupole moment of the mass distribution.
In the case of hyperbolic orbits777We could do a similar analysis for
parabolic orbits, which would give the exponent $10/3$ instead of 4 in (5.1).
(i.e. if the relative velocities of the masses tend to nonzero constants in
the infinite past), one thus gets that (see [Chr02] for a derivation)

$\frac{dE_{\text{grav}}}{dt}\sim-C|u|^{-4}+\dots\quad\textnormal{as\leavevmode\nobreak\
$u\to-\infty$}.$ (5.1)

We can now identify this gravitational energy along $\mathcal{I}^{+}$ with the
scalar field energy, namely the flux of the Noether current associated to
$T=\partial_{t}$:

$\frac{dE_{\text{grav}}}{dt}\sim\frac{dE_{\text{scalar}}}{dt}=-\int_{\mathbb{S}^{2}}(\partial_{u}\phi|_{\mathcal{I}^{+}})^{2}{\mathop{}\\!\mathrm{d}\sigma_{\mathbb{S}^{2}}}.$
(5.2)

In this sense, the quadrupole approximation “predicts” that
$|\partial_{u}\phi|_{\mathcal{I}^{+}}|\sim|u|^{-2}$, and hence
$|\phi|_{\mathcal{I}^{+}}|\sim|u|^{-1}$.

One can make a more sophisticated argument at the level of gravitational (i.e.
not scalar) perturbations ($s=\pm 2$) that also allows one to obtain such a
rate on an ingoing null hypersurface at a finite $v$-distance $\mathcal{C}$ as
opposed to $\mathcal{I}^{+}$ [WW79, Dam86]. This justifies the assumption that
$\phi|_{\mathcal{C}}=Q|u|^{-p}+\dots$ for $p=1$, $Q\neq 0$.

In fact, a more detailed perturbative analysis would result in different
predictions on the exponent $p$ for each angular mode $\ell$. For the purpose
of this expository note, we will simply make the assumption that

$\phi_{\ell}|_{\mathcal{C}}=Q_{\ell}|u|^{-p_{\ell}}+\dots$ (5.3)

for some general $p_{\ell}\in\mathbb{R}_{+}$,
$Q_{\ell}\in\mathbb{R}\setminus\\{0\\}$, keeping in mind that for $\ell=0$,
$p_{0}=1$ is the “physically relevant” exponent.888Of course, we can only
speak of physical relevance in the case of gravitational radiation, i.e. for
$s=\pm 2$, which we discuss in §6. However, since the quadrupole approximation
gives a prediction for the lowest angular modes of $\Psi^{[0]}$ and
$\Psi^{[4]}$, namely the $\ell=2$ modes, and the lowest angular mode of $\psi$
is given by $\ell=0$, we here take $p_{0}=1$ to be physically relevant
exponent for the scalar field. See also the arguments relating the analysis of
the $s=0$ case to the $s=\pm 2$ case in §6 (cf. Footnote 11).

We will now complete our dictionary by explaining how these asymptotics along
$\mathcal{C}$ translate into asymptotics towards $\mathcal{I}^{+}$, first for
$\ell=0$ in §5.2, then for general $\ell$ in §5.3. We will give an outlook on
the case of $s=\pm 2$ in §6.

###  5.2 Analysis of the spherically symmetric part $\phi_{0}$

We now explain how to treat solutions $\psi_{0}$ arising from a data setup as
in Fig. 7, i.e. with data $\phi_{0}|_{\mathcal{C}}(u)=Q_{0}|u|^{-p_{0}}+\dots$
on $\mathcal{C}$ and $\partial_{v}\phi_{0}|_{\mathcal{I}^{-}}\equiv 0$ on
$\mathcal{I}^{-}$. These data are to be understood as scattering data.
Recently developed scattering theory [Nic16, DRSR18, Mas20] ensures the unique
existence of a solution $\psi_{0}$ that attains the prescribed data along
$\mathcal{C}$ and $\mathcal{I}^{-}$, and moreover provides us with the (far
from optimal) global bound that (see also [Keh21a])

$|r\psi_{0}|\leq C\sqrt{r}.$ (5.4)

To obtain asymptotic estimates, the analysis will roughly follow two steps:

  * (I)

We first insert the weak estimate $|\phi_{0}|\lesssim\sqrt{r}$ into (3.3) and
integrate (3.3) from $\mathcal{I}^{-}$ to obtain a weak estimate on
$\partial_{v}\phi_{0}$.

  * (II)

We can then integrate this estimate for $\partial_{v}\phi_{0}$ from
$\mathcal{C}$ to obtain an estimate for $\phi_{0}$.

The crucial observation then is that, as a consequence of the strong
$r^{-3}$-weight appearing on the right-hand side of (3.3), this new estimate
on $\phi_{0}$ will be improved compared to the original one. One can then
iterate steps (I) and (II) until one obtains a sharp estimate. Now, the
details:

(I): Inserting $|\phi_{0}|\leq\sqrt{r}$ into (3.3), and integrating (3.3) from
$\mathcal{I}^{-}$, we find

$\partial_{v}\phi_{0}(u,v)=\int_{-\infty}^{u}\partial_{u}\partial_{v}\phi_{0}\mathop{}\\!\mathrm{d}u^{\prime}\leq\int_{-\infty}^{u}\frac{MCD}{2r^{5/2}}\mathop{}\\!\mathrm{d}u^{\prime}\leq
Cr^{-3/2}.$ (5.5)

Here, we used the no-incoming-radiation condition
$\partial_{v}\phi_{0}|_{\mathcal{I}^{-}}\equiv 0$.

(II): Integrating (5.5) from $\mathcal{C}$ (where $2r|_{\mathcal{C}}\sim|u|$),
we then find

$|\phi_{0}(u,v)-\phi|_{\mathcal{C}}(u)|\leq
C\int_{1}^{v}r^{-3/2}\mathop{}\\!\mathrm{d}v^{\prime}\leq C|u|^{-1/2}.$ (5.6)

This clearly improves the initial estimate $|\phi_{0}|\leq C\sqrt{r}$ to
$|\phi_{0}|\leq C|u|^{-\min(p_{0},1/2)}$.

One can now insert this improved estimate back into (5.5) and iteratively
repeat the procedure of steps (I) and (II), i.e. of (5.5), (5.6), to
eventually obtain the estimate

$|\phi_{0}(u,v)-\phi_{0}|_{\mathcal{C}}(u)|\leq C|u|^{-p_{0}-1}.$ (5.7)

Since $\phi_{0}|_{\mathcal{C}}(u)=Q_{0}|u|^{-p_{0}}+\dots$, (5.7) implies that
the leading-order term in the asymptotics of $\phi_{0}$ as $u\to-\infty$ is
$Q_{0}|u|^{-p_{0}}$.

Finally, in order to also obtain an asymptotic estimate for
$\partial_{v}\phi_{0}$ as $v\to\infty$, we once again repeat the calculation
(5.5), this time equipped with the asymptotic estimate (5.7). This gives

$\partial_{v}\phi_{0}(u,v)=\int_{-\infty}^{u}\partial_{u}\partial_{v}\phi_{0}\mathop{}\\!\mathrm{d}u^{\prime}=\int_{-\infty}^{u}-\frac{MDQ_{0}|u|^{-p_{0}}}{2r^{3}}\mathop{}\\!\mathrm{d}u^{\prime}+\dots.$
(5.8)

The integral on the RHS can be computed by writing
$2r=v-u+\mathcal{O}(\log(v-u))$. Let us here give the concrete computation
only in the case $p_{0}=1$:

$\int_{-\infty}^{u}\frac{1}{(v-u^{\prime})^{3}u^{\prime}}\mathop{}\\!\mathrm{d}u^{\prime}=\int_{-\infty}^{u}\frac{1}{v^{3}}\left(\frac{1}{u^{\prime}}+\frac{1}{v-u^{\prime}}+\frac{v}{(v-u^{\prime})^{2}}+\frac{v^{2}}{(v-u^{\prime})^{3}}\right)\mathop{}\\!\mathrm{d}u^{\prime}\\\
=\frac{\log|u|-\log(v-u)}{v^{3}}+\frac{3v-2u}{2v^{2}(v-u)^{2}}.$ (5.9)

In particular, we conclude that, along $u=const$ (where $v\sim r$), we have

$\partial_{v}\phi_{0}(u,v)=-\frac{MQ_{0}}{2}\frac{\log
r-\log|u|}{r^{3}}+\mathcal{O}(r^{-3}).$ (5.10)

Similar computations can be done for general $p_{0}$, see Table 4 below:

$\phi_{0}|_{\mathcal{C}}=Q_{0}|u|^{-p_{0}}+\dots$ |  $p_{0}\in(0,1)$: |  $p_{0}=1$: |  $p_{0}\in(1,2)$: |  $p_{0}=2$: | $\dots$
---|---|---|---|---|---
$\partial_{v}\phi_{0}(u=const,v)=$ | $\frac{AM}{r^{2+p_{0}}}+$ | $\frac{AM\log r}{r^{3}}+$ | $\frac{h(u)}{r^{3}}+\frac{AM}{r^{2+p_{0}}}+$ | $\frac{h(u)}{r^{3}}+\frac{AM\log r}{r^{4}}+$ | $\dots$
Table 4: Relation between decay of $\phi_{0}$ as $u\to-\infty$ and asymptotics
for $\partial_{v}\phi_{0}$ as $v\to\infty$. Here, $h(u)$ is a function
decaying in $u$ as $u\to-\infty$, and $A$ stands for a constant multiple of
$Q_{0}$.

In particular, if $p_{0}=1$, then (5.10) implies that
$I_{0}^{f_{0}}[\psi]=-MQ_{0}/2$ for $f_{0}=r^{-3}\log r$. Thus, combining the
findings of the present section and of §4 (cf. Table 3), we can in particular
conclude the following [Keh21a, Keh21b]:

######  Theorem 5.1.

Consider data on $\mathcal{C}$ and $\mathcal{I}^{-}$ that satisfy
$\phi_{0}|_{\mathcal{C}}=Q_{0}|u|^{-1}+\mathcal{O}(|u|^{-1-\epsilon})$ as
$u\to-\infty$ and $\partial_{v}\phi_{0}|_{\mathcal{I}^{-}}\equiv 0$. Then,
along any outgoing null hypersurface of constant $u$,
$2\partial_{v}\phi_{0}=-MQ_{0}r^{-3}\log r+\dots$ as $v\to\infty$, so
$I_{0}^{{r^{-3}\log r}}[\psi]\equiv-MQ_{0}/2$.

Moreover, if one smoothly extends the data along $\mathcal{C}$ to
$\mathcal{H}^{+}$ as in Fig. 8, one obtains the following late-time
asymptotics near $i^{+}$:999Note that, owing to the conservation of
$I_{0}^{r^{-3}\log r}[\phi]$ along $\mathcal{I}^{+}$, the leading-order
asymptotics (5.11) are independent of the extension of the data towards
$\mathcal{H}^{+}$.

$\displaystyle\psi_{0}|_{\mathcal{H}^{+}}=-4MQ_{0}v^{-3}\log v+\dots$
$\displaystyle\phi_{0}|_{\mathcal{I}^{+}}=-2MQ_{0}u^{-2}\log u+\dots.$ (5.11)

Figure 8: Depiction of the resulting asymptotics in various regions if
$\phi|_{\mathcal{C}}\sim u^{-1}$ initially.

###  5.3 Higher $\ell$-modes $\psi_{\ell}$

We now turn our attention to higher $\ell$-modes: We consider data such that
$r\psi_{\ell}|_{\mathcal{C}}(u)=Q_{\ell}|u|^{-p_{\ell}}+\dots$ on
$\mathcal{C}$ and such that
$\partial_{v}(r\psi_{\ell})|_{\mathcal{I}^{-}}\equiv 0$ along
$\mathcal{I}^{-}$.

The analysis of higher $\ell$-modes will again be be similar to the analysis
of the $\ell=0$ mode, with the relevant equation now being (3.7) instead of
(3.3). For simplicity, let us work with a simplified version of (3.7), namely
(recall $\boldsymbol{\widehat{L}}:=D^{-1}r^{2}\partial_{v}$):

$\partial_{u}(D^{\ell}r^{-2\ell}\partial_{v}\boldsymbol{\widehat{L}}^{\ell}\phi_{\ell})=\frac{MD^{\ell+1}y_{0}^{(\ell)}}{r^{2\ell+3}}\boldsymbol{\widehat{L}}^{\ell}\phi_{\ell}.$
(5.12)

This simplification corresponds to setting $x_{i}^{(\ell)}=0=y_{i}^{(\ell)}$
for all $i>0$ and $z_{i}^{(\ell)}=0$ for all $i\geq 0$ in (3.7), but still
allows us to capture the main ideas of the proof: Essentially, we can now
repeat a procedure almost identical to steps (I) and (II) of §5.2, with
$\phi_{0}$ replaced by
$\Phi_{\ell}:=\boldsymbol{\widehat{L}}^{\ell}\phi_{\ell}$. The main difference
is that, in order to get an estimate similar to (5.7), one first needs to
compute the values of $\Phi_{\ell}$ along $\mathcal{C}$. This is achieved by
inductively integrating the equations satisfied by
$\partial_{u}(r^{-2N}\partial_{v}\boldsymbol{\widehat{L}}^{N}\phi_{\ell})$ for
$N<\ell$ from $u=-\infty$, with these equations in turn being obtained by
simply commuting the wave equation (3.1) with $\boldsymbol{\widehat{L}}^{N}$.
This gives:

$\boldsymbol{\widehat{L}}^{N}\phi_{\ell}|_{\mathcal{C}}(u)=A_{N}^{(\ell)}|u|^{-p_{\ell}+N}+\dots$
(5.13)

for some constants $A_{N}^{(\ell)}$ that are nonvanishing multiples of
$Q_{\ell}$ for $N\leq\ell$.

Equipped with this estimate for $\Phi_{\ell}|_{\mathcal{C}}$, we can now,
similarly to how we showed (5.7), show that

$|\Phi_{\ell}(u,v)-\Phi_{\ell}|_{\mathcal{C}}(u)|\leq
C\begin{cases}r^{\ell-1-p_{\ell}}&\text{ if }p_{\ell}<\ell-1\\\ \log
r-\log|u|&\text{ if }p_{\ell}=\ell-1\\\ |u|^{\ell-1-p_{\ell}}&\text{ if
}p_{\ell}>\ell-1\end{cases}$ (5.14)

(Note that the first two cases in (5.14) did not appear for $\ell=0$ since we
assumed $p_{\ell}$ to be positive. Formally, however, the calculations here
are valid for any value of $p_{\ell}\in\mathbb{R}$.) As with the
$\ell=0$-case, we can now insert this estimate into (5.12) to obtain an
asymptotic estimate on $\partial_{v}\Phi_{\ell}$:

$\partial_{v}\Phi_{\ell}(u,v)=r^{2\ell}\int_{-\infty}^{u}\frac{MDy_{0}^{(\ell)}}{r^{2\ell+3}}\Phi_{\ell}|_{\mathcal{C}}\mathop{}\\!\mathrm{d}u^{\prime}+\ldots=\begin{cases}Ar^{\ell-
p_{\ell}-2}+\dots&\text{ if }p_{\ell}<\ell+1\\\ Ar^{-3}(\log
r-\log|u|)+\dots&\text{ if }p_{\ell}=\ell+1\\\
A|u|^{\ell+1-p_{\ell}}r^{-3}+\dots&\text{ if }p_{\ell}>\ell+1\end{cases}$
(5.15)

Notice that if $p_{\ell}\leq\ell+1$, then the RHS of (5.15) gives the relevant
$f(r)$-modified N–P charge and shows that it is conserved as well. For
instance, for $p_{\ell}<\ell+1$ and $f_{\ell}=r^{\ell-p_{\ell}-2}$, we have
$I_{\ell}^{f_{\ell}}[\psi]=A$. On the other hand, if $p_{\ell}>\ell+1$, then
all $f(r)$-modified N–P charges vanish.

Finally, by integrating (5.15) $\ell+1$ times from $\mathcal{C}$ (where
$2r=|u|+\ldots$), each time picking up a term on $\mathcal{C}$ that is given
by (5.13), we can now also obtain an estimate for $\phi_{\ell}$ itself.101010
We resort to an example in order to schematically explain this. Consider the
$\ell=1$-mode $\psi_{1}$ with $p_{1}>0$. From the estimate (5.14), we obtain
that
$r^{2}\partial_{v}\phi_{1}(u,v)=r^{2}\partial_{v}\phi_{1}|_{\mathcal{C}}(u)+\dots$.
We further compute from (3.1) that
$r^{2}\partial_{v}\phi_{1}|_{\mathcal{C}}(u)=A_{1}^{(1)}\cdot|u|^{-p_{1}+1}+\dots$,
with $A_{1}^{(1)}=-Q_{1}/(2(p_{1}+1))$ (cf. (5.13)). Therefore, we can
schematically compute $\phi_{1}(u,v)$ via
$\phi_{1}(u,v)=\phi_{1}|_{\mathcal{C}}(u)+\int\partial_{v}\phi_{1}(u,v)\mathop{}\\!\mathrm{d}v=\frac{Q_{1}}{|u|^{p_{1}}}+\frac{A_{1}^{(1)}}{|u|^{p_{1}-1}}\int
r^{-2}\mathop{}\\!\mathrm{d}v+\ldots=\frac{Q_{1}}{|u|^{p_{1}}}-\frac{Q_{1}}{(p+1)|u|^{p_{1}-1}}\left(\frac{1}{r|_{\mathcal{C}}}-\frac{1}{r}\right)+\ldots.$
In particular, $\lim_{v\to\infty}\phi_{1}(u,v)$ decays faster in $u$ than
$\phi_{1}|_{\mathcal{C}}(u)$ does initially iff $p_{1}=1$. More generally, if
$\phi_{\ell}|_{\mathcal{C}}$ decays like $|u|^{-p_{\ell}}$ initially, then
$\phi_{\ell}|_{\mathcal{I}^{+}}$ will decay faster than $|u|^{-p_{\ell}}$ if
and only if $p_{\ell}\in\\{1,\dots,\ell\\}$. A more detailed discussion of
this can be found in [Keh21c].

##  6 Completing the dictionary and extending to gravitational perturbations

#### Completing the dictionary:

We can finally summarise our findings and combine the results of §5 with those
of §4. Suppose that we have data for $\psi_{\ell}$ such that
$\phi_{\ell}|_{\mathcal{C}}=Q_{\ell}|u|^{-p_{\ell}}+\dots$ near
$\mathcal{I}^{-}$ and such that
$\partial_{v}\phi_{\ell}|_{\mathcal{I}^{-}}\equiv 0$. We can then read off the
relevant choice of $f(r)$-modified N–P charge from (5.15). Thus, combining
this with the results of §4, cf. Table 3 (and again smoothly but arbitrarily
extending the data towards $\mathcal{H}^{+}$), we can directly connect the
behaviour of $\phi_{\ell}$ near $\mathcal{I}^{-}$ to its behaviour near
$i^{+}$. This is done in Table 5 below.

$\phi_{\ell}|_{\mathcal{C}}=Q_{\ell}|u|^{-p_{\ell}}+\dots$ as $u\to-\infty$: |  $p_{\ell}<\ell+1$: |  $p_{\ell}=\ell+1$: |  $p_{\ell}>\ell+1$:
---|---|---|---
$\phi_{\ell}|_{\mathcal{I}^{+}}$ as $u\to-\infty$: |  $A|u|^{-p_{\ell}}$ unless $p_{\ell}\in\\{1,\dots,\ell\\}$
$\partial_{v}\Phi_{\ell}$ as $u=const,v\to\infty$: | $AMr^{-2+\ell-p_{\ell}}$ | $AMr^{-3}\log r$ | $Br^{-3}$
$\phi_{\ell}|_{\mathcal{I}^{+}}$ as $u\to\infty$: | $AMu^{-1-p_{\ell}}$ | $AMu^{-2-\ell}\log u$ | $Bu^{-2-\ell}$
$\psi_{\ell}|_{\mathcal{H}^{+}}$ as $v\to\infty$: | $AMv^{-2-\ell-p_{\ell}}$ | $AMv^{-2\ell-3}\log v$ | $Bv^{-2\ell-3}$
Table 5: The full thesaurus: The letter $A$ is always a placeholder for a
constant, nonvanishing multiple of $Q_{\ell}$ (that is different from cell to
cell), whereas $B$ stands for a constant that also depends on the extension of
the data along $\mathcal{C}$ towards $\mathcal{H}^{+}$ and is only generically
nonvanishing.

#### Extending to gravitational perturbations ($s=\pm 2$):

So far, we have focussed mostly on $s=0$. The more realistic case of
gravitational perturbations will be discussed in detail in upcoming work
[KM22], but we shall already list the main points here: Post-Newtonian
arguments [WW79, Dam86] predict the following rates along $\mathcal{C}$ at the
level of quadrupolar radiation (i.e. for $\ell=2$): $\Psi^{[0]}\sim|u|^{-3}$
and $\Psi^{[4]}\sim|u|^{-4}$ as $u\to-\infty$.111111The reader could have
loosely guessed these rates using the same simplistic heuristics as we
presented for $s=0$: The rate $r\Psi^{[4]}|_{\mathcal{C}}\sim|u|^{-3}$ as
$u\to-\infty$ on data will remain the same on $\mathcal{I}^{+}$, where
$r\Psi^{[4]}=\partial_{u}N$ gives the rate of change of the News function $N$.
Furthermore, we have the Bondi mass loss formula [BVdBM62, Sac62, CK93] along
$\mathcal{I}^{+}$:
$\frac{dE_{\text{grav}}}{dt}=-\frac{1}{4\pi}\int_{\mathbb{S}^{2}}|N|^{2}{\mathop{}\\!\mathrm{d}\sigma_{\mathbb{S}^{2}}}$.
The rate for $r\Psi^{[4]}$ thus comes from the quadrupole approximation
prediction that $\frac{dE_{\text{grav}}}{dt}\sim-|u|^{-4}$. The rate for
$\Psi^{[0]}$ is then enforced by the Teukolsky–Starobinsky identites [TP74]
and the no-incoming-radiation condition.  Combining this with the no-incoming-
radiation condition, a preliminary analysis then gives:

  1. $\upalpha)$

In addition to $\Psi^{[4]}$ violating the peeling rate near $\mathcal{I}^{-}$
(which states that $\Psi^{[4]}=\mathcal{O}(r^{-5})$), the peeling rate of
$\Psi^{[0]}$ near $\mathcal{I}^{+}$ is also violated: Instead of the peeling
rate $\Psi^{[0]}=\mathcal{O}(r^{-5})$, one will obtain that $\Psi^{[0]}\sim
r^{-4}$ towards $\mathcal{I}^{+}$. Motivations for this rate have appeared
before in [Dam86, Chr02].

  2. $\upbeta)$

While the radiation field for $\Psi^{[0]}$,
$\lim_{\mathcal{I}^{+}}r^{5}\Psi^{[0]}$, thus blows up, the radiation field
for $\Psi^{[4]}$, $\lim_{\mathcal{I}^{+}}r\Psi^{[4]}$, is still defined, and
we conjecture that the failure of peeling (i.e. of conformal regularity) in
$\upalpha)$ translates into the following late-time decay rate along
$\mathcal{I}^{+}$: $r\Psi^{[4]}|_{\mathcal{I}^{+}}\sim u^{-4}$ as
$u\to\infty$. This should be contrasted with the Price’s law rate, which
predicts that $r\Psi^{[4]}|_{\mathcal{I}^{+}}\sim u^{-6}$; see [MZ21a] for a
derivation of the Price’s law rate for $|s|=2$.

In fact, at a heuristic level, the reader can already guess the rates in
$\upalpha)$ and $\upbeta)$ by the observation from §3 that the following
correspondences hold:

$\displaystyle r^{5}\Psi^{[0]}_{\ell=2}\leftrightarrow\phi_{\ell=0},$
$\displaystyle r\Psi^{[4]}_{\ell=2}\leftrightarrow\phi_{\ell=4}.$

Therefore, our rates on $\mathcal{C}$ read:

$\displaystyle|u|^{2}\sim
r^{5}\Psi^{[0]}_{\ell=2}|_{\mathcal{C}}\leftrightarrow$
$\displaystyle\>\phi_{\ell=0}|_{\mathcal{C}}\quad\implies p_{0}=-2,$
$\displaystyle|u|^{-3}\sim r\Psi^{[4]}_{\ell=2}|_{\mathcal{C}}\leftrightarrow$
$\displaystyle\>\phi_{\ell=4}|_{\mathcal{C}}\quad\implies p_{4}=3.$

By applying Table 5 with these correspondences and values for $p_{\ell}$, we
obtain in particular:

$\displaystyle\partial_{v}(r^{5}\Psi^{[0]}_{\ell=2})|_{u={const}}\leftrightarrow\partial_{v}\phi_{\ell=0}|_{u={const}}\overset{p_{0}=-2}{\sim}r^{-2+0-(-2)}=r^{0}\quad$
$\displaystyle\text{as }r\to\infty$
$\displaystyle\overset{\textnormal{integrating}}{\implies}\Psi^{[0]}_{\ell=2}|_{u={const}}\sim
r^{-4}\quad$ $\displaystyle\text{as }r\to\infty,$ (6.1) $\displaystyle
r\Psi^{[4]}_{\ell=2}|_{\mathcal{I}^{+}}\leftrightarrow\phi_{\ell=4}|_{\mathcal{I}^{+}}\overset{p_{4}=3}{\sim}u^{-1-3}=u^{-4}\quad$
$\displaystyle\text{as }u\to\infty.$ (6.2)

The numerology presented above is rooted in the assumption of hyperbolic
Keplerian orbits in the infinite past and (5.1). Using similar arguments, the
reader can also find the numerology in the case of parabolic orbits, cf.
Footnote 7.

##  7 Further directions

We append the main body of the paper with some remarks on extensions of the
presented methods.

#### From Schwarzschild to subextremal Kerr:

For solutions to the scalar wave equation on subextremal Kerr arising from
conformally regular or compactly supported initial data, the leading-order
late-time asymptotics feature the same rates as in Schwarzschild [Hin20,
AAG21a]. See also [SRTdC20, MZ21b] for related recent results in the setting
of the Teukolsky equations and [BO99a, BO99b] for a heuristic analysis of
late-time tails in Kerr spacetimes.

The effects of the non-zero angular momentum of Kerr do, however, affect the
decay rates of higher angular modes. Since Kerr is not spherically symmetric,
there is no _a priori_ canonical definition of spherical harmonics and
corresponding angular modes,121212Note however that in phase space, the
Boyer–Lindquist time-frequency-dependent spheroidal harmonics form a natural
choice of angular modes, since they are involved in the separability of the
wave equation after taking a Fourier transform in time. and, whichever
definition is chosen, obtaining late-time tails for each mode will involve the
difficulty of _mode coupling_.

This difficulty was addressed and studied in [AAG21a], where it was shown that
the choice of spherical harmonics with respect to Boyer–Lindquist spheres at
infinity allows for a modified analogue of Price’s law in Kerr. The topic of
generalising Price’s law on Kerr spacetimes has a long history featuring
various conflicting predictions. See [ZKB14, BK14] and references therein for
an overview of this problem and for the latest numerical results, which are in
alignment with the mathematically rigorous results derived in [AAG21a].

We expect that the techniques outlined in the present paper will also be
applicable in combination with the methods developed in [AAG21a] to study the
effects that a violation of peeling has on late-time tails for the Teukolsky
equations on subextremal Kerr backgrounds.

#### Extremal black holes:

Extremal black holes feature several additional fascinating phenomena that
have an effect on the rates of decay in late-time tails and are inherently
connected to the degeneracy of extremal event horizons. For instance, the wave
equation on extremal Reissner–Nordström black holes and the axisymmetric wave
equation on extremal Kerr black holes possess additional conserved charges
along the event horizon [Are15] that lead to different decay rates from the
subextremal setting [AAG20] and are connected to the presence of asymptotic
instabilities known in the literature as the _Aretakis instabilities_ [Are15];
see also earlier heuristics in [Ori13, Sel16]. In fact, in extremal
Reissner–Nordström, these additional conservation laws can be related to the
Newman–Penrose charges by applying the Couch–Torrence conformal isometry that
maps null infinity to the event horizon [CT84, BF13, LMRT13].

In upcoming work (see [Gaj21]), it is shown that the late-time tails of non-
axisymmetric solutions to the wave equation on extremal Kerr exhibit even
stronger deviations from the subextremal case, as well as stronger
instabilities, consistent with the heuristics in [GA01, CGZ16]. Due to the
lack of conserved charges along the event horizon in the non-axisymmetric
setting, we moreover need a different mechanism for deriving late-time tails
from the one presented in the present paper.131313A similar lack of conserved
charges occurs also in the model problems of scalar fields with an inverse-
square potential on Schwarzschild, and a new mechanism is developed in [Gaj22]
to overcome this.

Late-time tails in gravitational radiation are of particular observational
relevance in the extremal setting since they have been predicted to form the
dominant part of gravitational wave signals at much earlier stages in the
ringdown process than in the subextremal setting [YZZ+13]. They therefore
provide a promising observational signature of (near)-extremality of black
holes. Since extremal late-time tails also decay much slower, and since they
are a phenomenon associated to regularity at the future event horizon and not
at future null infinity, we do not expect the failure of peeling to have an
effect on decay rates, in contrast to the subextremal setting.

#### Moving beyond linear perturbations:

The results of §5 have been extended to the coupled Einstein-scalar field
system under the assumption of spherical symmetry in [Keh21a]. The effects of
nonlinearities on Price’s law (i.e. including backreaction) have been
investigated heuristically and numerically in [BCR09, BR10], where deviations
to Price’s law have been predicted for higher spherical harmonic modes arising
from compactly supported data. See also the recently announced results of
Luk–Oh on late-time tails on fixed, but _dynamical_ black hole spacetime
backgrounds, where mathematically rigorous methods are applied to obtain
similar deviations [Luk21]. The above works suggest that in the full nonlinear
theory, compactly supported Cauchy data should lead to the following late-time
tails along $\mathcal{I}^{+}$:

$r\Psi^{[4]}|_{\mathcal{I}^{+}}\sim u^{-5}.$

While this is slower than the $u^{-6}$-tail expected from linear theory (for
compactly supported data), this late-time tail still decays _faster_ than the
$u^{-4}$-tail that we predicted above as a consequence of the failure of
peeling. In light of this, we expect that, even in the fully nonlinear theory,
the dominant late-time behaviour will be $u^{-4}$, provided that we consider
the physically motivated scattering data of the present paper.

Finally, in view of the impressive techniques that have been developed to
study the dynamics of gravitational radiation in the setting of the full
system of the nonlinear Einstein vacuum equations [CK93, DHRT21], we expect a
mathematically rigorous investigation of the above prediction to be within
reach.

##  8 Conclusion

We want to conclude with the following points:

  * $\diamond$

If one has initial data on some hyperboloidal hypersurface $\Sigma_{0}$ for
which one can define nonvanishing $f(r)$-modified N–P charges, then the late-
time asymptotics towards $i^{+}$ can be read off from these N–P charges
according to Table 3. The crucial question then is: What is the right choice
of $f(r)$-modified N–P charge?

  * $\diamond$

If one poses polynomially decaying data on some ingoing null hypersurface
emanating from $\mathcal{I}^{-}$ and excludes radiation coming in from
$\mathcal{I}^{-}$, then, because of the nonvanishing background mass $M$ near
spatial infinity $i^{0}$, the backscatter of gravitational radiation at early
times will lead to $\mathcal{I}^{+}$ not being smooth, and this failure of
smoothness will determine the choice of $f(r)$-modified N–P charge and,
therefore, the late-time tails of gravitational radiation near $i^{+}$, see
Table 5. The smoothness of $\mathcal{I}^{-}$ plays no role here.

  * $\diamond$

The assumption of polynomial decay towards $\mathcal{I}^{-}$, in turn, comes
from post-Newtonian arguments and the assumption that the system under
consideration, e.g. two infalling masses, follows approximately hyperbolic
Keplerian orbits in the infinite past.

We record again that it is frequently assumed throughout large parts of the
literature that one has spatially compact support on $\Sigma_{0}$, or that
gravitational radiation has only started radiating at some fixed, finite time
(which of course implies the former). However, in the context of an isolated
system describing an astrophysical process, we believe the assumptions of the
present paper, i.e. that the system under consideration has radiated for all
times, to be more natural.

Independently of the above considerations, we also hope to have convinced the
reader that, even from a purely theoretical point of view, the assumption of
smooth null infinity might be too rigid, and that by avoiding this assumption,
one can perform many more general arguments that give new and deeper insights
into the nature of general relativity!

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