# Is it possible to construct asymptotically flat initial data using the
evolutionary forms of the constraints?

Károly Csukás 111E-mail<EMAIL_ADDRESS>Wigner RCP, H-1121
Budapest, Konkoly Thege Miklós út 29-33, Hungary The University of
Mississippi, University, Mississippi 38677, USA  István Rácz 222E-mail
<EMAIL_ADDRESS>Wigner RCP, H-1121 Budapest, Konkoly Thege
Miklós út 29-33, Hungary

###### Abstract

Near-Kerr black hole initial datasets are constructed by applying either the
parabolic-hyperbolic or the algebraic-hyperbolic form of the constraints. In
both cases, strongly and weakly asymptotically flat initial datasets with
desirable falloff rates are produced by controlling only the monopole part of
one of the freely specifiable variables. The viability of the applied method
is verified by numerically integrating the evolutionary forms of the
constraint equations in the case of various near-Kerr configurations.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Foliation based decompositions
    2. 2.2 Spin-weighted variables
    3. 2.3 The parabolic-hyperbolic equations
    4. 2.4 The algebraic-hyperbolic equations
    5. 2.5 Foliations of Kerr-Schild time slices
    6. 2.6 Asymptotic flatness
  3. 3 The main results: applying the new method
    1. 3.1 The outline of the applied numerical scheme
    2. 3.2 Asymptotics of strictly near-Kerr initial datasets
    3. 3.3 Applying the new method to the parabolic-hyperbolic system
      1. 3.3.1 Outline in the spherically symmetric setup
      2. 3.3.2 Strongly asymptotically flat numerical solution
      3. 3.3.3 Weakly asymptotically flat numerical solution
    4. 3.4 Applying the new method to the algebraic-hyperbolic system
      1. 3.4.1 Outline in the spherically symmetric setup
      2. 3.4.2 Strongly asymptotically flat numerical solution
      3. 3.4.3 Weakly asymptotically flat numerical solution
  4. 4 Summary
  5. 5 Acknowledgments

##  1 Introduction

For more than seven decades, the conformal or elliptic method, introduced by
Lichnerowicz and York [1, 2, 3, 4], was almost the only 333There have been
attempts to find other means of solving the constraints (see e.g. [5, 6]), but
none of them has proved to be a viable alternative to the conformal method.
means to solve the constraints of general relativity. Recently, in a series of
papers [7, 8, 9], two alternative evolutionary formulations of the Einstein
constraint equations were also introduced. It is well known that by specifying
a suitable boundary value problem within the elliptic method, it is
straightforward to arrange a setup capable of investigating the existence of
asymptotically flat solutions to the constraint equations. By contrast, it is
not evident if there is a way to guarantee the existence of asymptotically
flat solutions to the constraint equations when either of the novel parabolic-
hyperbolic or algebraic-hyperbolic forms of the constraints is used. The main
issue is whether controlling merely the freely specifiable part of the data,
along with the “initial data” relevant for the constraint variables—which are
given only on one of the level sets foliating the conventional initial data
surface—is sufficient to guarantee appropriate falloff behavior of all the
geometric fields involved in the constraints.

By their inherent nature, the constraints involve more variables than
equations; thereby, they always form an underdetermined system. For instance,
there are only four constrained variables, regardless of the method applied,
concerning the geometrical content of a three-dimensional initial dataset in
general relativity. These are the ones restricted by the Hamiltonian and
momentum constraints [10, 11]. In contrast, the remaining eight variables are
freely specifiable. Whenever one is looking for asymptotically flat solutions
located in a small neighborhood of a known one, it is very tempting to choose
all the freely specifiable variables to coincide with the corresponding
variables of the known asymptotically flat solution. This strategy was applied
by Beyer et al. [12, 13] in their pioneering studies of the asymptotic
behavior of solutions to the evolutionary form of vacuum constraints. More
concretely, they used data on a time slice of the Kerr-Schild form of a single
or superposed binary Schwarzschild black holes, and they altered only the
“initial data”—these were also read off on one of the foliating level
surfaces—for some of the constrained variables [12, 13]. Their investigations
demonstrated that for this choice of the freely specifiable data—we referred
to these in [14] as “strictly near Schwarzschild” configurations—the solutions
to the evolutionary form of the constraints, apart from the seed solution,
cannot be asymptotically flat. More concretely, in the strictly near-
Schwarzschild setup, neither the algebraic-hyperbolic formulation in the
single Schwarzschild case [12] nor the parabolic-hyperbolic formulation in the
single and binary Schwarzschild black hole case [13] allow suitable falloff
for one of the constrained variables, $\mathbf{K}$, that is nothing but the
trace of the tensorial projection of the extrinsic curvature [for its
definition see Eq. (6) below].444Strictly speaking, $\mathbf{K}$ is not the
only variable that violates the desired decay rate, but given the spherically
symmetric constraint equations, it can be argued that the slow decay rate of
the other problematic variables can always be traced back to the behavior of
$\mathbf{K}$. These findings were confirmed in [14]. Note also that our mode-
by-mode (numerical) analysis also allowed us to conclude that only the
monopole part of $\mathbf{K}$ violates the desired decay rate [14].
Accordingly, this monopole part is the only mode that gets in the way of
obtaining asymptotically flat solutions to the evolutionary form of the
constraints in the strictly near-Schwarzschild case [14].

It is important to emphasize that in the above-discussed investigations, the
enormous freedom we have in choosing the freely specifiable part of the data
was ignored entirely. Based on this observation and to ensure a suitable
falloff rate for the constrained variable, $\mathbf{K}$, Beyer et al. [15]
invented a method that applies to the parabolic-hyperbolic form of the
constraints, by setting $\boldsymbol{\kappa}$ —which is one of the freely
specifiable variables, and which is the scalar part of the extrinsic curvature
[see Eq. (5) below]—to be proportional to $\mathbf{K}$. Note that by assuming
$\boldsymbol{\kappa}=\mathcal{R}\,\mathbf{K}$, the principal part of the
constraints’ parabolic-hyperbolic form is also affected as these equations
contain tangential derivatives of $\boldsymbol{\kappa}$. Nevertheless,
provided that $\mathcal{R}>-\tfrac{1}{2}$ holds, the obtained modified
parabolic-hyperbolic system is guaranteed to be well posed, and, more
importantly, the initial data that was yielded by this method was also found
to be strongly asymptotically flat. The viability of this proposal of Beyer et
al. [15] was confirmed in our follow-up paper [14], where we also proposed
altering part of the freely specifiable data in case of the algebraic-
hyperbolic system. It is fair to admit, however, that the success of the
latter proposal was limited.

Several questions remained open for further study even after this significant
progress reported in [15]. For example, are there other suitable modifications
of the free part of the data? Are there modifications that do not affect the
principal parts of the evolutionary forms of the constraints? Could the more
general nonstrictly Kerr initial data configurations be asymptotically flat?
Is it possible to control the rate of decay of the initial data to produce
weakly or strongly asymptotically flat solutions?

The results reported in this paper were inspired by curiosity about the above
questions. Our motivation also benefited from a recent work by Beyer and
Ritchie [16] demonstrating that the original parabolic-hyperbolic
system—without applying any alteration—is suitable to yield asymptotically
hyperboloidal initial datasets. Analogous support was provided by the success
of constructing initial data—even though these are relevant only for the
strong field regime—using the parabolic-hyperbolic formulation of the
constraints, for single or binary rotating Kerr black hole configurations in
[17, 18].

The objectives set out in the present paper are many and varied. For the first
time, we investigate the asymptotic behavior of the space of initial datasets
in a small neighborhood of the initial data obtained on a Kerr-Schild time
slice of a rotating Kerr black hole. We do this by modifying only the monopole
part of some freely specifiable variables. This, on the one hand, guarantees
that no alteration of the principal parts of the evolutionary forms of the
constraints will occur. Hence the well posedness of the relevant partial
differential equation (PDE) systems is expected to hold, based on the results
covered in [9]. On the other hand, while the method proposed in [15] applies
only to the parabolic-hyperbolic system, our proposal can also be used in
solving the algebraic-hyperbolic system by controlling the monopole part of
the freely specifiable secondary lapse, $\widehat{N}$. In addition, we also
introduce a straightforward method, applicable to both of the evolutionary
forms of the constraints, that allows us to control the falloff rate of the
monopole part of the constrained variable $\mathbf{K}$, thereby generating
weakly and strongly asymptotically flat initial data with any desired falloff
rate. The viability of the introduced new methods is verified by integrating
the evolutionary forms of the constraint equations numerically in the case of
various near-Kerr initial data configurations.

The structure of the paper is as follows: In Sec. 2, we recall the notions and
notations utilized in this paper. Section 2.2 focuses on defining the most
relevant spin-weighted variables, while Secs. 2.3 and 2.4 recall the exact
form of the equations and some of their properties concerning the parabolic-
hyperbolic and algebraic-hyperbolic form of the constraints, respectively. In
Sec. 2.5 we select a suitable (Kerr-Schild) time slice foliation to the Kerr
black hole background. As a closing to Sec. 2, in Sec. 2.6, we recall the
notion of strong and weak asymptotic flatness. Our main results are presented
in Sec. 3. Since in all the former studies, exclusively the asymptotics of
near-Schwarzschild initial data was investigated, in Sec. 3.2, we start by
examining the asymptotics of strictly near-Kerr black hole initial data
configurations. Although this does not yield the desired falloff behavior, we
still find it beneficial to provide this for comparison and to verify that
even in this more complicated case, it is indeed only the monopole part of
$\mathbf{K}$ that requires suitable control. Sections 3.3 and 3.4 present our
proposal applied to the parabolic-hyperbolic and the algebraic-hyperbolic
formulations, respectively. These sections start with motivating our novel
approach by inspecting the analytic solutions in spherical symmetry and then
presenting the corresponding numerical results in the case of near-Kerr
initial data configurations. We close this paper with our final remarks in
Sec. 4. 555Unless indicated otherwise, our conventions and notations are the
same as in [11]. In particular, we use geometric units.

##  2 Preliminaries

The geometric content of an initial dataset is represented by a pair of
symmetric tensor fields $(h_{ab},K_{ab})$, where $h_{ab}$ is a Riemannian
metric on a three-dimensional manifold $\Sigma$. Once $\Sigma$ is embedded
into a four-dimensional spacetime $(M,g_{ab})$ the fields $h_{ab}$ and
$K_{ab}$ get to be the induced metric and extrinsic curvature of $\Sigma$ in
$(M,g_{ab})$. The fields $(h_{ab},K_{ab})$ are subject to the constraints
which, in the vacuum case, read as

$\displaystyle{}^{\scriptscriptstyle(\\!3\\!)}\\!R-K_{ab}K^{ab}+K^{2}$
$\displaystyle=0,$ (1) $\displaystyle D_{b}K^{b}{}_{a}-D_{a}K$
$\displaystyle=0\,,$ (2)

where $D_{a}$ and ${}^{\scriptscriptstyle(\\!3\\!)}\\!R$ are the covariant
derivative operator and Ricci scalar associated with $h_{ab}$, respectively,
and $K$ denotes the trace of the extrinsic curvature.

###  2.1 Foliation based decompositions

The geometric construction underlying the evolutionary interpretation of the
constraints lies on the assumption that the three-dimensional initial data
surface, $\Sigma$, can be foliated by a one-parameter family of $2$-surfaces,
$\mathcal{S}_{\varrho}$, which are the $\varrho=const$ level surfaces of a
function $\varrho:\Sigma\rightarrow\mathbf{R}$ [8, 7, 9].

The unit normal to the $\mathcal{S}_{\varrho}$ surfaces is
$\widehat{n}_{a}=\widehat{N}D_{a}\varrho$, such that
$\widehat{n}^{a}\widehat{n}_{a}=1$. Then the induced metric on
$\mathcal{S}_{\varrho}$ reads as

$\widehat{\gamma}_{ab}=h_{ab}-\widehat{n}_{a}\widehat{n}_{b}\,,$ (3)

whereas the operator projecting fields defined on $\Sigma$ onto
$\mathcal{S}_{\varrho}$ is $\widehat{\gamma}^{a}{}_{b}=h^{ae}{\gamma}_{eb}$.
The covariant derivative operator associated with $\widehat{\gamma}_{ab}$ is
denoted by $\widehat{D}_{a}$, whereas the corresponding Ricci scalar and the
extrinsic curvature of the $2$-surfaces by $\widehat{R}$ and
$\widehat{K}_{ab}$, respectively. To uncover implicit involvements of the
lapse, $\widehat{K}_{ab}$ will be replaced by the product
$\widehat{N}^{-1}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}_{ab}$,
where
$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}_{ab}=\tfrac{1}{2}\,\mathscr{L}_{[\rho-\widehat{N}]}\widehat{\gamma}_{ab}$.

A radial flow vector field, $\rho^{a}$, is also chosen such that its integral
curves intersect each $\mathcal{S}_{\varrho}$ level surface precisely once,
and it is normalized such that $\rho^{a}D_{a}\varrho=1$. Then the lapse and
shift associated with $\rho^{a}$ are $\widehat{N}=\rho^{a}\widehat{n}_{a}$ and
$\widehat{N}^{a}=\widehat{\gamma}^{a}{}_{b}\rho^{b}$, respectively. Using
these variables the geometric content of the metric, $h_{ab}$, can be uniquely
represented by the triplet
$(\widehat{N},\widehat{N}^{a},\widehat{\gamma}_{ab})$ [14].

The extrinsic curvature, $K_{ab}$, can also be decomposed as

$K_{ab}=\boldsymbol{\kappa}\,\widehat{n}_{a}\widehat{n}_{b}+\mathbf{k}_{a}\widehat{n}_{b}+\mathbf{k}_{b}\widehat{n}_{a}+\mathbf{K}_{ab},$
(4)

where

$\boldsymbol{\kappa}=K_{ab}\,\widehat{n}^{a}\widehat{n}^{b}\,,\quad\mathbf{k}_{a}=K_{bc}\,\widehat{\gamma}^{b}{}_{a}\widehat{n}^{c}\,,\quad\mathbf{K}_{ab}=K_{cd}\,\widehat{\gamma}^{c}{}_{a}\widehat{\gamma}^{d}{}_{b}\,.$
(5)

The tensorial projection of the extrinsic curvature can further be split into
its trace and trace-free parts, which are given as

$\mathbf{K}=\mathbf{K}_{ab}\widehat{\gamma}^{ab}\quad{\rm
and}\quad\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{ab}=\mathbf{K}_{ab}-\tfrac{1}{2}\,\mathbf{K}\,\widehat{\gamma}_{ab}\,.$
(6)

Note finally that the septet
$(\widehat{N},\widehat{N}^{a},\widehat{\gamma}_{ab};\boldsymbol{\kappa},\mathbf{k}_{a},\mathbf{K},\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{ab})$
is algebraically equivalent to the geometric content represented by the pair
$(h_{ab},K_{ab})$ on $\Sigma$.

###  2.2 Spin-weighted variables

As argued in [19, 20, 21], it is rewarding to use spin-weighted variables. In
doing so, one first introduces a complex null dyad,
$\\{q_{a},\overline{q}_{a}\\}$, where bar denotes complex conjugate, on one of
the $\mathcal{S}_{\varrho}$ level surfaces, say on $\mathcal{S}_{\varrho_{0}}$
such that $q_{ab}=q_{a}\overline{q}_{b}$ is required to be the unit sphere
metric on $\mathcal{S}_{\varrho_{0}}$. Then the dyad
$\\{q_{a},\overline{q}_{a}\\}$ is automatically normalized as
$q^{ab}q_{a}\overline{q}_{b}=2$, whereas the indices of the dyad are raised
and lowered by $q^{ab}$ and $q_{ab}$, respectively. To get a complex null
dyad, $\\{q_{a},\overline{q}_{a}\\}$ on each of the $\varrho=const$ level
surfaces the dyad fixed on $\mathcal{S}_{\varrho_{0}}$ is Lie-dragged with
respect to $\rho^{a}$ onto $\Sigma$. Note that then the action of the
tangential derivative $\widehat{D}_{a}$ can be rephrased using the Newman-
Penrose $\eth$ and $\overline{\eth}$ operators [21]. The basic variables we
use in the succeeding subsections, including the spin-weighted ones, are
listed in Table 1.

Notation | Definition |  Spin-weight
---|---|---
$\mathbf{a}$ | $\tfrac{1}{2}\,q^{i}\,\overline{q}^{j}\,\widehat{\gamma}_{ij}$ |  $0$
$\mathbf{b}$ | $\tfrac{1}{2}\,q^{i}q^{j}\,\widehat{\gamma}_{ij}$ |  $2$
$\mathbf{d}$ | $\mathbf{a}^{2}-\mathbf{b}\,\overline{\mathbf{b}}$ |  $0$
$\mathbf{A}$ | $q^{a}q^{b}{C^{e}}{}_{ab}\,\overline{q}_{e}=\mathbf{d}^{-1}\left\\{\mathbf{a}\left[2\,\eth\,\mathbf{a}-\,\overline{\eth}\,\mathbf{b}\right]-\,\overline{\mathbf{b}}\,\eth\,\mathbf{b}\right\\}$ |  $1$
$\mathbf{B}$ | $\,\overline{q}^{a}q^{b}{C^{e}}{}_{ab}\,q_{e}=\mathbf{d}^{-1}\left\\{\mathbf{a}\,\overline{\eth}\,\mathbf{b}-\mathbf{b}\,\eth\,\overline{\mathbf{b}}\right\\}$ |  $1$
$\mathbf{C}$ | $q^{a}q^{b}{C^{e}}{}_{ab}\,q_{e}=\mathbf{d}^{-1}\left\\{\mathbf{a}\,\eth\,\mathbf{b}-\mathbf{b}\left[2\,\eth\,\mathbf{a}-\,\overline{\eth}\,\mathbf{b}\right]\right\\}$ |  $3$
$\,\widehat{R}$ | $\tfrac{1}{2}\,{\mathbf{a}}^{-1}\left(2\,\mathbf{R}-\left\\{\,\eth\,\overline{\mathbf{B}}-\overline{\eth}\,\mathbf{A}-\tfrac{1}{2}\,\left[\,\mathbf{C}\,\overline{\mathbf{C}}-\mathbf{B}\,\overline{\mathbf{B}}\,\right]\,\right\\}\,\right)$ |  $0$
${\mathbf{N}}$ | $q_{i}\widehat{N}^{i}$ |  $1$
$\mathbf{k}$ | $q^{i}{\rm\bf k}{}_{i}$ |  $1$
$\mathbf{K}$ | $\widehat{\gamma}^{kl}\,{\rm\bf K}{}_{kl}$ |  $0$
$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}{}_{qq}$ | $q^{k}q^{l}\,\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}{}_{kl}$ |  $2$
$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}{}_{q\overline{q}}$ | $q^{k}\,\overline{q}^{l}\,\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}{}_{kl}=(2\,\mathbf{a})^{-1}[\,\mathbf{b}\,\overline{\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}{}_{qq}}+\overline{\mathbf{b}}\,\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}{}_{qq}\,]$ |  $0$
$\,\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}$ | ${\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}}=\widehat{\gamma}^{ij}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}_{ij}$ |  $0$
$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}{}_{qq}$ | $q^{i}q^{j}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}_{ij}=\tfrac{1}{2}\,\left\\{2\,\partial_{\rho}\mathbf{b}-2\,\eth\,\mathbf{N}+{\mathbf{C}}\,\overline{\mathbf{N}}+\mathbf{A}\,{\mathbf{N}}\,\right\\}$ |  $2$
$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}{}_{q\overline{q}}$ | $q^{k}\,\overline{q}^{l}\,\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}{}_{kl}={\mathbf{a}}^{-1}\\{\,\mathbf{d}\cdot\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}\\}+\tfrac{1}{2}\,[\,\mathbf{b}\,\overline{\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}}{}_{qq}+\overline{\mathbf{b}}\,\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}{}_{qq}\,]\,\\}$ |  $0$
Table 1: The variables applied in providing the evolutionary form of the
constraints.

Note that, in virtue of
$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{ab}\widehat{\gamma}^{ab}=0$,
for the contraction
$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{q\overline{q}}=\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{ab}q^{a}\overline{q}^{b}$,
as indicated in Table 1,

$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{q\overline{q}}=(2\mathbf{a})^{-1}\big{[}\mathbf{b}\overline{\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{qq}}+\overline{\mathbf{b}}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{qq}\big{]}$
(7)

holds. Note also that $\widehat{\gamma}_{ab}$ can be given, in terms of the
dyad $\\{q_{a},\overline{q}_{a}\\}$, and the variables $\mathbf{a}$ and
$\mathbf{b}$, as

$\widehat{\gamma}_{ab}=\mathbf{a}\,q_{ab}+\tfrac{1}{2}[\mathbf{b}\,\overline{q}_{a}\overline{q}_{b}+\overline{\mathbf{b}}\,q_{a}q_{b}]\,.$
(8)

Note, finally, that the geometric content of $(h_{ab},K_{ab})$ can also be
represented by the octet
$(\widehat{N},\mathbf{\mathbf{N}},\mathbf{a},\mathbf{b};\boldsymbol{\kappa},\mathbf{k},\mathbf{K},\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{qq})$.
As shown in [9, 19, 20] by choosing $\widehat{N}$, $\mathbf{k}$, and
$\mathbf{K}$ as constrained variables, we arrive at the parabolic-hyperbolic
formulation while selecting $\boldsymbol{\kappa}$, $\mathbf{k}$, and
$\mathbf{K}$ as constrained variables we arrive at the algebraic-hyperbolic
formulation of the constraints.

###  2.3 The parabolic-hyperbolic equations

Selecting $\widehat{N}$, $\mathbf{k}$, $\mathbf{K}$ as constrained variables,
the Hamiltonian and momentum constraints read as [21]

$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}\left[\partial_{r}\widehat{N}-\tfrac{1}{2}\,\mathbf{N}\,\overline{\eth}\widehat{N}-\tfrac{1}{2}\,\overline{\mathbf{N}}\,\eth\widehat{N}\right]\\\
-\tfrac{1}{2}\,\mathbf{d}^{-1}\widehat{N}^{2}\left[\,\mathbf{a}\left\\{\eth\overline{\eth}\widehat{N}-\mathbf{B}\,\overline{\eth}\widehat{N}\right\\}-\mathbf{b}\left\\{\overline{\eth}^{2}\\!\widehat{N}-\tfrac{1}{2}\,\overline{\mathbf{A}}\,\overline{\eth}\widehat{N}-\tfrac{1}{2}\,\overline{\mathbf{C}}\,\eth\widehat{N}\right\\}+cc.\right]\\\
-\mathcal{A}\,\widehat{N}-\mathcal{B}\,\widehat{N}^{\,3}=0\,,$ (9)
$\partial_{r}\mathbf{k}-\tfrac{1}{2}\,\mathbf{N}\,\overline{\eth}\mathbf{k}-\tfrac{1}{2}\,\overline{\mathbf{N}}\,\eth\mathbf{k}-\tfrac{1}{2}\,\widehat{N}\,\eth\mathbf{K}+\mathbf{f}_{PH}=0\,,$
(10)
$\partial_{r}\mathbf{K}-\tfrac{1}{2}\,\mathbf{N}\,\overline{\eth}\mathbf{K}-\tfrac{1}{2}\,\overline{\mathbf{N}}\,\eth\mathbf{K}-\tfrac{1}{2}\,\widehat{N}\,\mathbf{d}^{-1}\Big{\\{}\,\mathbf{a}(\eth\overline{\mathbf{k}}+\overline{\eth}\mathbf{k})-\mathbf{b}\,\overline{\eth}\overline{\mathbf{k}}-\overline{\mathbf{b}}\,\eth\mathbf{k}\,\Big{\\}}+\mathbf{F}_{PH}=0\,,$
(11)

where the coefficients $\mathcal{A}$, $\mathcal{B}$, and the source terms
$\mathbf{f}_{PH}$, $\mathbf{F}_{PH}$ in (9), (10), and (11), are given as

$\mathcal{A}=\partial_{r}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}-\tfrac{1}{2}\,\mathbf{N}\,\overline{\eth}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}-\tfrac{1}{2}\overline{\mathbf{N}}\,\eth\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}+\tfrac{1}{2}\,\Big{[}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}{}^{2}+\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}_{kl}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}{}^{kl}\Big{]}\,,$
(12)
$\mathcal{B}=-\tfrac{1}{2}\,\Big{[}\widehat{R}+2\,\boldsymbol{\kappa}\,\mathbf{K}+\tfrac{1}{2}\,\mathbf{K}^{2}-\mathbf{d}^{-1}[2\,\mathbf{a}\,\mathbf{k}\,\overline{\mathbf{k}}-\mathbf{b}\,\overline{\mathbf{k}}^{2}-\overline{\mathbf{b}}\,\mathbf{k}^{2}]-\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}{}_{kl}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}{}^{kl}\Big{]}\,,$
(13)
$\mathbf{f}_{PH}=-\tfrac{1}{2}\,\Big{[}\mathbf{k}\,\eth\overline{\mathbf{N}}+\overline{\mathbf{k}}\,\eth\mathbf{N}\Big{]}-\left[\boldsymbol{\kappa}-\tfrac{1}{2}\,\mathbf{K}\right]\eth\widehat{N}+\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}\,\mathbf{k}-\widehat{N}\Big{[}\eth\boldsymbol{\kappa}+q^{i}\accentset{\smash{\raisebox{-0.27126pt}{$\scriptstyle{\mathbin{\mathchoice{\hbox
to2.53569pt{\hfil\raise 0.0pt\hbox{\scalebox{0.5}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to2.53569pt{\hfil\raise
0.0pt\hbox{\scalebox{0.5}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to2.11308pt{\hfil\raise
0.0pt\hbox{\scalebox{0.5}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to1.83131pt{\hfil\raise
0.0pt\hbox{\scalebox{0.5}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}$}}}{\widehat{n}}\rule{0.0pt}{8.91248pt}{}^{l}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}{}_{li}-q^{i}\widehat{D}^{l}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}{}_{li}\Big{]}\,,$
(14)
$\displaystyle\mathbf{F}_{PH}=\tfrac{1}{4}\widehat{N}\,\mathbf{d}^{-1}\Big{\\{}2\,\mathbf{a}\,\mathbf{B}\,\overline{\mathbf{k}}-\mathbf{b}(\,\overline{\mathbf{C}}\,\mathbf{k}+\overline{\mathbf{A}}\,\overline{\mathbf{k}})+cc.\Big{\\}}$
$\displaystyle-\mathbf{d}^{-1}\Big{[}(\mathbf{a}\,\overline{\mathbf{k}}-\overline{\mathbf{b}}\,\mathbf{k})\,\eth\widehat{N}+cc.\Big{]}$
$\displaystyle+\Big{[}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{ij}{\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}}{}^{ij}-\left(\boldsymbol{\kappa}-\tfrac{1}{2}\,\mathbf{K}\right)\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}\Big{]}\,$
(15)

with $cc.$ denoting the complex conjugate of the preceding terms, while the
explicit form of the terms, such as
$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}_{ij}{\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}}{}^{ij}$,
$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{ij}{\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}}{}^{ij}$,
$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{ij}{\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}}{}^{ij}$,
$q^{i}\accentset{\smash{\raisebox{-0.27126pt}{$\scriptstyle{\mathbin{\mathchoice{\hbox
to2.53569pt{\hfil\raise 0.0pt\hbox{\scalebox{0.5}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to2.53569pt{\hfil\raise
0.0pt\hbox{\scalebox{0.5}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to2.11308pt{\hfil\raise
0.0pt\hbox{\scalebox{0.5}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to1.83131pt{\hfil\raise
0.0pt\hbox{\scalebox{0.5}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}$}}}{\widehat{n}}\rule{0.0pt}{8.91248pt}{}^{l}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}{}_{li}$,
$q^{i}\widehat{D}^{l}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}{}_{li}$,
can be found in [21].

As shown in [9], (9) is a Bernoulli-type parabolic equation, while (10) and
(11) form a first-order symmetric hyperbolic system. This coupled parabolic-
hyperbolic system always gets to be well posed in those subregions of $\Sigma$
where the positivity of
$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}$
can be guaranteed. As
$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}$
is determined by the freely specifiable variables $\mathbf{a}$, $\mathbf{b}$,
and $\mathbf{\mathbf{N}}$, the coupled parabolic-hyperbolic system can always
be guaranteed to be well posed at least on a one-sided neighborhood of the
initial data surface $\mathcal{S}_{\varrho_{0}}$ 666The initial data for the
evolutionary form of the constraints, specified on one of the
${\varrho}=const$ level sets in $\Sigma$ should not be confused with an
initial data set $(\Sigma;h_{ab},K_{ab})$ relevant for the time-evolution part
of the Einstein’s equations. in $\Sigma$. Note also that besides the freely
specifiable eight real functions, represented by the variables
$(\mathbf{\mathbf{N}},\mathbf{a},\mathbf{b};\boldsymbol{\kappa},\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{qq})$,
on $\Sigma$, we also have the freedom to choose initial data
$(\widehat{N}|_{\varrho_{0}},\mathbf{k}|_{\varrho_{0}},\mathbf{K}|_{\varrho_{0}})$
to the parabolic-hyperbolic equations (9)-(11) on $\mathcal{S}_{\varrho_{0}}$.

###  2.4 The algebraic-hyperbolic equations

Choosing $\mathbf{K}$, $\mathbf{k}$, and $\boldsymbol{\kappa}$ as basic
variables the constraints read as [21]

$\partial_{r}\mathbf{K}-\tfrac{1}{2}\,\mathbf{N}\,\overline{\eth}\mathbf{K}-\tfrac{1}{2}\,\overline{\mathbf{N}}\,\eth\mathbf{K}-\tfrac{1}{2}\,\widehat{N}\,\mathbf{d}^{-1}\Big{\\{}\mathbf{a}(\eth\overline{\mathbf{k}}+\overline{\eth}\mathbf{k})-\mathbf{b}\,\overline{\eth}\overline{\mathbf{k}}-\overline{\mathbf{b}}\,\eth\mathbf{k}\Big{\\}}+\mathbf{F}_{AH}=0\,,$
(16)
$\partial_{r}\mathbf{k}-\tfrac{1}{2}\,\mathbf{N}\,\overline{\eth}\mathbf{k}-\tfrac{1}{2}\,\overline{\mathbf{N}}\,\eth\mathbf{k}+\widehat{N}\,\mathbf{K}^{-1}\Big{\\{}\boldsymbol{\kappa}\,\eth\mathbf{K}-\mathbf{d}^{-1}\big{[}(\mathbf{a}\mathbf{k}-\mathbf{b}\overline{\mathbf{k}})\,\eth\overline{\mathbf{k}}+(\mathbf{a}\overline{\mathbf{k}}-\overline{\mathbf{b}}\mathbf{k})\,\eth\mathbf{k}\big{]}\Big{\\}}+\mathbf{f}_{AH}=0\,,$
(17)
$\boldsymbol{\kappa}=\tfrac{1}{2}\mathbf{K}^{-1}\Big{[}\mathbf{d}^{-1}\big{(}2\mathbf{a}\,\mathbf{k}\overline{\mathbf{k}}-\mathbf{b}\,\overline{\mathbf{k}}^{2}-\overline{\mathbf{b}}\,\mathbf{k}^{2}\big{)}-\tfrac{1}{2}\mathbf{K}^{2}-\boldsymbol{\kappa}_{0}\Big{]}\,,$
(18)

with

$\boldsymbol{\kappa}_{0}={}^{\scriptscriptstyle(\\!3\\!)}\\!R-\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{kl}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}{}^{kl}\,,$
(19)

where the pertinent source terms $\mathbf{F}_{AH}$, $\mathbf{f}_{AH}$ are
given as

$\displaystyle\mathbf{F}_{AH}=\tfrac{1}{4}\,\widehat{N}\,\mathbf{d}^{-1}\Big{\\{}2\,\mathbf{a}\,\mathbf{B}\,\overline{\mathbf{k}}-\mathbf{b}(\,\overline{\mathbf{C}}\,\mathbf{k}+\overline{\mathbf{A}}\,\overline{\mathbf{k}})+cc.\Big{\\}}$
$\displaystyle-\mathbf{d}^{-1}\Big{[}(\mathbf{a}\overline{\mathbf{k}}-\overline{\mathbf{b}}\mathbf{k})\,\eth\widehat{N}+cc.\Big{]}$
$\displaystyle+\Big{[}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{ij}{\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}}{}^{ij}-\left(\boldsymbol{\kappa}-\tfrac{1}{2}\,\mathbf{K}\right)\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}\Big{]}\,,$
(20) $\displaystyle\mathbf{f}_{AH}=$
$\displaystyle-\tfrac{1}{2}\,\Big{[}\mathbf{k}\,\eth\overline{\mathbf{N}}+\overline{\mathbf{k}}\,\eth\mathbf{N}\Big{]}$
$\displaystyle+\tfrac{1}{2}\,\widehat{N}\,(\mathbf{d}\cdot\mathbf{K})^{-1}\Big{[}(\mathbf{a}\,\mathbf{k}-\mathbf{b}\,\overline{\mathbf{k}})(\overline{\mathbf{B}}\,\mathbf{k}+\mathbf{B}\,\overline{\mathbf{k}})+(\mathbf{a}\,\overline{\mathbf{k}}-\overline{\mathbf{b}}\,\mathbf{k})(\mathbf{C}\,\overline{\mathbf{k}}+\mathbf{A}\,\mathbf{k})\Big{]}$
$\displaystyle-\left[\boldsymbol{\kappa}-\tfrac{1}{2}\,\mathbf{K}\right]\eth\widehat{N}+\widehat{N}\Big{[}\tfrac{1}{2}\mathbf{K}^{-1}\eth\boldsymbol{\kappa}_{0}+\widehat{N}^{-1}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}\,\mathbf{k}-q^{i}\accentset{\smash{\raisebox{-0.27126pt}{$\scriptstyle{\mathbin{\mathchoice{\hbox
to2.53569pt{\hfil\raise 0.0pt\hbox{\scalebox{0.5}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to2.53569pt{\hfil\raise
0.0pt\hbox{\scalebox{0.5}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to2.11308pt{\hfil\raise
0.0pt\hbox{\scalebox{0.5}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to1.83131pt{\hfil\raise
0.0pt\hbox{\scalebox{0.5}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}$}}}{\widehat{n}}\rule{0.0pt}{8.91248pt}{}{}^{l}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{li}+q^{i}\widehat{D}^{l}\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{li}\Big{]}\,.$
(21)

As shown in [9], (16)-(17) form a first-order Friedrichs symmetrizable
hyperbolic system for the vector valued variable $(\mathbf{K},\mathbf{k})$
provided that the inequality $\boldsymbol{\kappa}\cdot\mathbf{K}<0$ holds.
Notice that as this latter condition refers to the dependent variables, its
validity has to be checked during the integration of the underlying system.
Note also that in doing so the fields
$(\widehat{N},\mathbf{\mathbf{N}},\mathbf{a},\mathbf{b};\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{qq})$
can be specified at will throughout $\Sigma$, along with the Cauchy data for
the system (16)-(17) constituted by the pair
$(\mathbf{K}|_{\varrho_{0}},\mathbf{k}|_{\varrho_{0}})$ given on the
$\varrho=\varrho_{0}$ initial data level set.

###  2.5 Foliations of Kerr-Schild time slices

As indicated in the Introduction, one of our principal aims is to explore the
asymptotic behavior of at least a small neighborhood of the initial datasets
deduced from a rotating Kerr black hole on a Kerr-Schild time slice.
Accordingly, such as in all the former investigations [19, 12, 13, 15, 16,
14], we start by deducing a distinguished initial dataset, represented by
$(\widehat{N},\mathbf{\mathbf{N}},\mathbf{a},\mathbf{b};\boldsymbol{\kappa},\mathbf{k},\mathbf{K},\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{qq})$,
relevant for a Kerr spacetime, with mass and rotation parameters $M$ and $a$,
respectively, on a $t_{KS}=const$ Kerr-Schild time slice. These time slices
are preferable as they connect the black hole interior with spacelike
infinity.

The Kerr-Schild form

$g_{ab}=\eta_{ab}+2Hl_{a}l_{b}\,,$ (22)

with

$H=\frac{Mr^{3}}{r^{4}+a^{2}\tilde{z}^{2}}\,,\quad
l_{a}=\left(1,\frac{r\tilde{x}+a\tilde{y}}{r^{2}+a^{2}},\frac{r\tilde{y}-a\tilde{x}}{r^{2}+a^{2}},\frac{\tilde{z}}{r}\right)\,,$
(23)

is probably one of the simplest and most widely used representation of the
Kerr black hole spacetime, with spin pointing to the positive $\tilde{z}$
direction. In (22) $\eta_{ab}$ stands for the metric of an auxiliary Minkowski
spacetime with inertial coordinates
$(\tilde{t},\tilde{x},\tilde{y},\tilde{z})$. The spatial part of these
coordinates $(\tilde{x},\tilde{y},\tilde{z})$ and the Boyer-Lindquist radial
coordinate, $r$, are related via the implicit relation

$\frac{\tilde{x}^{2}+\tilde{y}^{2}}{r^{2}+a^{2}}+\frac{\tilde{z}^{2}}{r^{2}}=1\,.$
(24)

The $t_{KS}=const$ Kerr-Schild time slices of a Kerr spacetime are nothing but
$\tilde{t}=const$ time-level surfaces of the auxiliary Minkowski spacetime. As
$(\tilde{x},\tilde{y},\tilde{z})$ are Cartesian coordinates on the
$\tilde{t}=const$ time-level surfaces, it is tempting to foliate the Kerr-
Schild time slices by the $\tilde{x}^{2}+\tilde{y}^{2}+\tilde{z}^{2}=const$
level sets. Nevertheless, however controversial it may sound, the coordinates
$(\tilde{x},\tilde{y},\tilde{z})$ do not form a suitable admissible asymptotic
system for the Kerr solution, which (as we shall see in the following
subsection) plays a central role in setting up the precise formulation of
asymptotic flatness of the data specified on $t_{KS}=const$ Kerr-Schild time
slices.

For this purpose the spherical Kerr-Schild coordinates, introduced by Chen et
al. [22], turned out to be more appropriate. The spherical Kerr-Schild
coordinates, $(x,y,z)$, are related to $(\tilde{x},\tilde{y},\tilde{z})$ via
the relations

$x=\frac{r\tilde{x}}{\sqrt{r^{2}+a^{2}}},\qquad
y=\frac{r\tilde{y}}{\sqrt{r^{2}+a^{2}}},\qquad z=\tilde{z}\,.$ (25)

As noted in [22], while in the original Kerr-Schild coordinates
$(\tilde{x},\tilde{y},\tilde{z})$, and with respect to the Euclidean metric,
$\delta_{ab}$, the Boyer-Lindquist $r=const$ level sets are, in virtue of
(24), oblate spheroids, the same level sets, at least in an affine sense, are
the $x^{2}+y^{2}+z^{2}=r^{2}$ spheres with respect to the spherical Kerr-
Schild coordinates. Note that the
$\tilde{x}^{2}+\tilde{y}^{2}+\tilde{z}^{2}=const$ and
$x^{2}+y^{2}+z^{2}=const$ level sets do coincide in the $a\rightarrow 0$
Schwarzschild limit.

The payback of using the spherical Kerr-Schild coordinates $(x,y,z)$ is that
they form admissible coordinates on the $t_{KS}=const$ Kerr-Schild time
slices. This gets more transparent when one is applying the spherical
coordinates $(r,\vartheta,\varphi)$ defined via the (standard) implicit
relations 777It is important to emphasize that the
$(\tilde{x},\tilde{y},\tilde{z})$ oblate spheroidal coordinates can be given
in terms of the Boyer-Lindquist coordinates $(r,\theta,\phi)$ as
$\tilde{x}=\sqrt{r^{2}+a^{2}}\,\cos\phi\,\sin\theta\,,\quad\tilde{y}=\sqrt{r^{2}+a^{2}}\,\sin\phi\,\sin\theta\,,\quad\tilde{z}=r\,\cos\theta\,.$
(26) Note that the Boyer-Lindquist angular coordinates $(\theta,\phi)$ and
those in (27) are related as $\theta=\vartheta$ and
$\phi=\varphi-\arctan(a/r)$. It is indeed the very last relation that explains
why the $(x,y,z)$ spherical Kerr-Schild coordinates could be suitable, and, in
contrast, the $(\tilde{x},\tilde{y},\tilde{z})$ oblate spheroidal coordinates
cannot form asymptotically admissible coordinates on Kerr-Schild time slices.

$x=r\,\cos\varphi\,\sin\vartheta\,,\quad
y=r\,\sin\varphi\,\sin\vartheta\,,\quad z=r\,\cos\vartheta\,.$ (27)

Utilizing these spherical coordinates $(r,\vartheta,\varphi)$ the data
$(\widehat{N},\mathbf{\mathbf{N}},\mathbf{a},\mathbf{b};\boldsymbol{\kappa},\mathbf{k},\mathbf{K},\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{qq})$,
relevant for the considered Kerr spacetime on $\Sigma$, i.e., on the $t=0$
Kerr-Schild time slice, foliated by the $r=const$ level sets and with respect
to the flow $\rho^{a}=(\partial_{r})^{a}$, can be given as

$\widehat{N}=\sqrt{\frac{(1+2H)\Upsilon}{\Xi}}\,,\qquad\mathbf{N}=\mathrm{i}\,\frac{2aMr\sin\vartheta}{(a^{2}+r^{2})\,\Xi}\,,$
(28)
$\mathbf{a}=a^{2}+r^{2}-\tfrac{1}{2}a^{2}(1-2H)\sin^{2}\vartheta\,,\qquad\mathbf{b}=-\tfrac{1}{2}a^{2}(1+2H)\sin^{2}\vartheta\,,$
(29)
$\boldsymbol{\kappa}=\frac{2a^{2}\sqrt{1+2H}\sin^{2}\vartheta\,\partial_{r}H}{\Upsilon}-\frac{2\,(1+H)\,\Xi\,\partial_{r}H}{\Upsilon\,(1+2H)^{3/2}}\\\
-\frac{2a^{2}H\sqrt{1+2H}\sin^{2}\vartheta\,(r+a^{2}\sin^{2}\vartheta\,\partial_{r}H)}{\Upsilon\,\Xi}\,,$
(30)
$\mathbf{k}=-\frac{a^{2}\sin^{2}\vartheta\,\partial_{\vartheta}H}{\sqrt{\Upsilon\,\Xi}}+\sqrt{\frac{\Xi}{\Upsilon}}\,\frac{\partial_{\vartheta}H}{1+2H}\\\
-\mathrm{i}\left(a\sin\vartheta\,\sqrt{\frac{\Xi}{\Upsilon}}\,\partial_{r}H-\frac{2aH\sin\vartheta\,(r+a^{2}\sin^{2}\vartheta\,\partial_{r}H)}{\sqrt{\Upsilon\,\Xi}}\right)\,,$
(31)
$\mathbf{K}=-\frac{H-r\,\partial_{r}H}{r\sqrt{1+2H}}-\frac{2H(r+a^{2}\sin^{2}\vartheta\,\partial_{r}H)}{\sqrt{1+2H}\,\Xi}\,,$
(32)
$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{qq}=\frac{\Upsilon+\Xi}{\sqrt{1+2H}}\left(\frac{H(r+a^{2}\sin^{2}\vartheta\,\partial_{r}H)}{\Xi}-\frac{H-r\,\partial_{r}H}{2r}\right)+\mathrm{i}\,\frac{2a\sin\vartheta\,\partial_{\vartheta}H}{\sqrt{1+2H}}\,,$
(33)

where $\Upsilon=r^{2}+a^{2}\cos^{2}\vartheta$, $H=Mr/\Upsilon$ and
$\Xi=r^{2}+a^{2}+2a^{2}H\sin^{2}\vartheta$.

Note that due to the symmetries of the Kerr spacetime, $\mathbf{N}$ is purely
imaginary, and $\mathbf{b}$ is real, although, in general, they are complex-
valued functions. Note also that in the $a\rightarrow 0$ Schwarzschild limit
$\mathbf{N}$, $\mathbf{b}$, $\mathbf{k}$, and
$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{qq}$
all vanish, and that the variables
$(\widehat{N},\mathbf{\mathbf{N}},\mathbf{a},\mathbf{b};\boldsymbol{\kappa},\mathbf{k},\mathbf{K},\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{qq})$
reduce to the formulas applied in [14].

To avoid potential numerical inaccuracies in evaluating some of the more
complex geometric quantities, for instance,
${}^{\scriptscriptstyle(\\!3\\!)}\\!R$, it turned out to be rewarding to
inspect if subterms of the same magnitude with opposite signs are involved. In
practice, it was profitable to simplify the corresponding expressions using
symbolic computer algebra and evaluate the yielded formulas numerically, in
accordance with similar experiences reported in [12].

###  2.6 Asymptotic flatness

An initial dataset $(\Sigma,h_{ab},K_{ab})$ is considered to be strongly
asymptotically flat if the complement of a compact set in $\Sigma$ can be
mapped by an admissible coordinate system $(x,y,z)$ diffeomorphically onto the
complement of a closed ball in $\mathbf{R}^{3}$ such that in these coordinates

$\displaystyle h_{ab}$
$\displaystyle=\left(1+\tfrac{C}{r}\right)\delta_{\alpha\beta}+\mathcal{O}(r^{-2}),$
(34) $\displaystyle K_{ab}$ $\displaystyle\sim\mathcal{O}(r^{-2})$ (35)

hold as $r=\sqrt{x^{2}+y^{2}+y^{2}}$ goes to infinity [23], where
$\delta_{ab}$ denotes the components of the flat three-metric in the
admissible Cartesian coordinates $(x,y,z)$. These conditions guarantee that
the ADM mass, momentum, and angular momentum of the initial dataset are well
defined.

A weaker notion of asymptotic flatness is also frequently used. An initial
dataset $(\Sigma,h_{ab},K_{ab})$ will be referred to as weakly asymptotically
flat if the milder falloff conditions

$\displaystyle h_{ab}$
$\displaystyle=\left(1+\tfrac{C}{r}\right)\delta_{ab}+\mathcal{O}(r^{-3/2-\epsilon}),$
(36) $\displaystyle K_{ab}$ $\displaystyle\sim\mathcal{O}(r^{-3/2-\epsilon})$
(37)

hold, for some small positive number $\epsilon$. Note that these weaker
conditions guarantee that the ADM mass and linear momentum are well defined
[24].

It was shown in [14] that by transforming an admissible coordinate system
$(x,z,y)$ to spherical coordinates $(r,\vartheta,\varphi)$, by applying the
relation in (27), one can reformulate the falloff conditions (34) and (36) to
those introduced in Sec. 2.1. The corresponding falloff conditions for the
metric components in the case of strong asymptotic flatness are [14]

$\widehat{N}-1\sim\mathcal{O}(r^{-1})\,,\qquad\widehat{N}^{a}\sim\mathcal{O}(r^{-3})\,,\qquad\widehat{\gamma}_{ab}\sim
r^{2}q_{ab}+\mathcal{O}(r^{0}).$ (38)

Since the dyadic components are independent of $r$, the spin-weighted
variables inherit the falloff conditions of the tensor quantities from which
they are derived. Thus we have $\mathbf{N}\sim\mathcal{O}(r^{-3})$,
$\mathbf{a}\sim\mathcal{O}(r^{2})$, and $\mathbf{b}\sim\mathcal{O}(r^{2})$.
Looking at (8), however, we can derive a stronger condition on $\mathbf{b}$ by
realizing that the part of $\widehat{\gamma}_{ab}$ proportional to $q_{ab}$ is
entirely represented by $\mathbf{a}$, so $\mathbf{b}$ must satisfy
$\mathbf{b}\sim\mathcal{O}(r^{0})$. Similarly, in the case of strong
asymptotic flatness, the components of the extrinsic curvature must satisfy
the falloff conditions [14],

$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{ab}\sim\mathcal{O}(r^{0})\,,\qquad\mathbf{K}\sim\mathcal{O}(r^{-2})\,,\qquad\mathbf{k}_{a}\sim\mathcal{O}(r^{-1})\,,\qquad\boldsymbol{\kappa}\sim\mathcal{O}(r^{-2}),$
(39)

which results in
$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{qq}\sim\mathcal{O}(r^{0})$
and $\mathbf{k}\sim\mathcal{O}(r^{-1})$.

The background variables (28)-(33) have the asymptotic behavior

$\displaystyle\widehat{N}$ $\displaystyle\sim
1+\tfrac{M}{r}+\mathcal{O}(r^{-2})\,,$ $\displaystyle\mathbf{N}$
$\displaystyle\sim\tfrac{2\mathrm{i}aM\sin\vartheta}{r^{3}}+\mathcal{O}(r^{-4})\,,$
(40) $\displaystyle\mathbf{a}$ $\displaystyle\sim r^{2}+\mathcal{O}(r^{0})\,,$
$\displaystyle\mathbf{b}$
$\displaystyle\sim-\tfrac{1}{2}a^{2}\sin^{2}\vartheta+\mathcal{O}(r^{-1})\,,$
(41) $\displaystyle\boldsymbol{\kappa}$
$\displaystyle\sim\tfrac{2M}{r^{2}}+\mathcal{O}(r^{-3})\,,$
$\displaystyle\mathbf{k}$
$\displaystyle\sim\tfrac{3\mathrm{i}aM\sin\vartheta}{r^{2}}+\mathcal{O}(r^{-3})\,,$
(42) $\displaystyle\mathbf{K}$
$\displaystyle\sim-\tfrac{4M}{r^{2}}+\mathcal{O}(r^{-3})\,,$
$\displaystyle\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\circ$}}}{\mathbf{K}}\rule{0.0pt}{8.91248pt}_{qq}$
$\displaystyle\sim-\tfrac{2a^{2}M\sin^{2}\vartheta}{r^{2}}+\mathcal{O}(r^{-3})$
(43)

which correspond to the derived falloff conditions.

Since this work is concerned by their asymptotic behavior, we recall that the
falloff properties of the constrained spin-weighted variables read as

$\widehat{N}-1\sim\mathcal{O}(r^{-1}),\qquad\boldsymbol{\kappa}\sim\mathcal{O}(r^{-2}),\qquad\mathbf{k}\sim\mathcal{O}(r^{-1}),\qquad\mathbf{K}\sim\mathcal{O}(r^{-2})$
(44)

for strongly asymptotically flat initial data, while as

$\widehat{N}-1\sim\mathcal{O}(r^{-1}),\quad\boldsymbol{\kappa}\sim\mathcal{O}(r^{-3/2-\epsilon}),\quad\mathbf{k}\sim\mathcal{O}(r^{-1/2-\epsilon}),\quad\mathbf{K}\sim\mathcal{O}(r^{-3/2-\epsilon})$
(45)

for weakly asymptotically flat data.

##  3 The main results: applying the new method

This section introduces the new method that allows us to control the falloff
properties of the solutions to the evolutionary forms of the constraint
equations. In doing so, all the variables are expanded using spin-weighted
spherical harmonics, as done in [14]. The replacement of a spin-weight $s$
variable ${}^{(s)}\mathbf{V}$ by the expansion

${}^{(s)}\mathbf{V}(r,\vartheta,\varphi)=\sum_{\ell=|s|}^{\ell_{max}}\sum_{m=-\ell}^{\ell}\mathbf{V}_{\ell}{}^{m}(r)\cdot{}_{s}{Y_{\ell}}{}^{m}(\vartheta,\varphi)\,,$
(46)

where ${}_{s}{Y_{\ell}}{}^{m}$ denotes the spin-weight $s$ spherical
harmonics, allows us to evaluate its angular derivatives analytically as they
can always be related to the $\eth$ and $\overline{\eth}$ operators.

###  3.1 The outline of the applied numerical scheme

In this study, we utilize the same numerical solver as applied in [14], which
is based on spin-weighted spherical harmonics expansion in the angular sector
and a fourth-order accurate adaptive Runge-Kutta-Fehlberg (RKF) method in
solving the resulting ordinary differential equations for the expansion
coefficients.888Although the code itself is not yet open source, its
documentation is available to the public [25]. To ensure higher accuracy for
our numerical solver, in each case, instead of solving (9), (10) and
(16)-(18), we solve the equations which can be deduced from them for the
nonlinear perturbations
${}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}=\widehat{N}-{}^{\scriptscriptstyle(\\!0\\!)}\widehat{N}$,
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}=\mathbf{K}-{}^{\scriptscriptstyle(\\!0\\!)}\mathbf{K}$,
and
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}=\mathbf{k}-{}^{\scriptscriptstyle(\\!0\\!)}\mathbf{k}$,
in the parabolic-hyperbolic case, and for
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}$ and
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}$, in the algebraic-hyperbolic
case. Here ${}^{\scriptscriptstyle(\\!0\\!)}\widehat{N}$,
${}^{\scriptscriptstyle(\\!0\\!)}\mathbf{k}$, and
${}^{\scriptscriptstyle(\\!0\\!)}\mathbf{K}$ signify the background quantities
given by Eqs. (28), (31), and (32), respectively. The main payback of using
the formulation based on deviations is that the corresponding dominant
background fields do not hide the asymptotic behavior of the higher modes of
the constraint fields. Note that the derivation of the equations governing the
evolution of the nonlinear perturbations is straightforward. As they are
slightly altered versions of the equations provided in Appendix A of [14],
they will be omitted here.

The input parameters in our numerical simulations are as follows: The
integration of the deviation equations starts at $r=1$, that is, inside the
black hole region. The non-negligible initial data there is

${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}|_{r=1}=-Y_{2}{}^{0}\,,$ (47)

which is the same order of magnitude as the dominant monopole part
${}^{\scriptscriptstyle(\\!0\\!)}\mathbf{K}_{0}{}^{0}|_{r=1}=-5.7581$ of the
background ${}^{\scriptscriptstyle(\\!0\\!)}\mathbf{K}$ at $r=1$. As both the
initial data in (47) and the background fields are axially symmetric, only the
$m=0$ axisymmetric modes of the constrained fields get excited. Furthermore
due to the parity symmetry of Kerr and the functional form of the dyad
components, (47) excites only even $\ell$ modes of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}$,
${}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}$, $\boldsymbol{\kappa}$,
$\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}]$, and only odd
$\ell$ modes of
$\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}]$.

The integration of the deviation equations goes out until reaching $r=10^{9}$,
with a single exception (see Fig. 4 below), where it goes out to $r=10^{12}$.
The mass and rotation parameters of the Kerr background take the values $M=1$
and $a=1/2$. Starting from the strong field region necessitates angular
resolution with minimal cutoff $\ell_{max}=10$. This resolution allows us to
compute the square roots with desired relative accuracy of $5\cdot 10^{-18}$,
which is the same accuracy as we demand in performing division by using
Neumann series expansion. The error tolerance of RKF for each mode of a
variable $f$ is set to $(f_{\ell}{}^{m}+\partial_{r}f_{\ell}{}^{m})\epsilon$
with $\epsilon=10^{-5}$.

Note also that the use of the `WIGXJPF` library of Johansson and Forssén [26]
yielded notable improvements in the performance of our code. All the analytic
calculations were derived or verified using Mathematica 12.3. In particular,
the notebook we used to determine the background variables (28)-(33) (which
serve as the main input to our code) is available as Supplemental Material.

###  3.2 Asymptotics of strictly near-Kerr initial datasets

As was mentioned in the Introduction, in the strictly near-Schwarzschild
setup, neither the algebraic-hyperbolic formulation in the single
Schwarzschild case [12] nor the parabolic-hyperbolic formulation in the single
and binary Schwarzschild black hole case [13] allow suitable falloff for
$\mathbf{K}$.

For instance, using the parabolic-hyperbolic formulation, Beyer et al. [13]
reported that besides that the lapse, $\widehat{N}$, does not tend to the
desired asymptotic value $1$, in general, the falloff rate of $\mathbf{K}$ is
$\mathcal{O}(r^{-1})$ that is also far too slow. In our follow-up mode-by-mode
investigations, both of these observations were confirmed. Nevertheless, we
also found that apart from the limiting value of $\widehat{N}$ and the
$\ell=0,m=0$ monopole mode of $\mathbf{K}$, all the other modes of the
involved variables falloff with a rate compatible with conditions in strong
asymptotic flatness [14].

Analogously, in the algebraic-hyperbolic case, Beyer et al. [12] found that
the falloff rate for $\mathbf{K}$ is $\mathcal{O}(r^{-3/2})$ which is now
strictly at the borderline not to allow weak asymptotic flatness.
Surprisingly, the mode-by-mode investigations of similar strictly near-
Schwarzschild initial datasets demonstrated that all the other modes of the
involved variables falloff with a rate compatible with conditions in strong
asymptotic flatness [14].

Though in light of the results summarized above, one does not expect better
asymptotic behavior of the constraint variables in case of strictly near-Kerr
initial datasets, we devote this section to a short inspection of these
configurations that still appears to be beneficial.

It is rewarding to first glance at the asymptotic behavior of the background
fields ${}^{\scriptscriptstyle(\\!0\\!)}\widehat{N}$,
${}^{\scriptscriptstyle(\\!0\\!)}\mathbf{K}$, and
${}^{\scriptscriptstyle(\\!0\\!)}\mathbf{k}$. Compatible with the strong
asymptotic flatness of the Kerr solution,
${}^{\scriptscriptstyle(\\!0\\!)}\widehat{N}_{0}{}^{0}$ tends to the value $1$
with the rate $\mathcal{O}(r^{-1})$ while all the higher $\ell$ modes fall off
as $\mathcal{O}(r^{-\ell-1})$. Analogously, all the
${}^{\scriptscriptstyle(\\!0\\!)}\mathbf{k}_{\ell}{}^{0}$ modes fall off with
the rate $\mathcal{O}(r^{-\ell-1})$, whereas the
${}^{\scriptscriptstyle(\\!0\\!)}\mathbf{K}_{\ell}{}^{0}$ modes as
$\mathcal{O}(r^{-\ell-2})$.

In proceeding, note first that on all of the figures included in this paper,
to be able to use log-log scales and thereby demonstrate the claimed falloff
rates, we plot the absolute values of the nonlinear perturbations, along with
some auxiliary lines helping the comparison with the expected rates.
Specifically, in Fig. 1, various modes of the constrained fields, obtained by
integrating the parabolic-hyperbolic equations, in the strictly near-Kerr case
are plotted. While the monopole part
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}$ of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}$ decays as
$\mathcal{O}(r^{-1})$, all the higher $\ell$-modes fall off as
$\mathcal{O}(r^{-2})$. The modes comprising the real part of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}$ fall off as
$\mathcal{O}(r^{-1})$, whereas the modes in the imaginary part of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}$ decay with the rate
$\mathcal{O}(r^{-2})$. In Fig. 1(d) it is transparent that instead of its
proper limit value $0$, the monopole part
${}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{0}{}^{0}$ tends to
$3.4114\cdot 10^{-6}$. By contrast, all the other modes of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}$ fall off as
$\mathcal{O}(r^{-1})$. These observations underline that likewise, it occurred
in the strictly near-Schwarzschild case, the asymptotic behavior of either
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}$ or
${}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{0}{}^{0}$, respectively,
prevents even the weak asymptotic flatness of the corresponding strictly near-
Kerr initial dataset.

\begin{overpic}[width=433.62pt]{kerr-sph-KK20-1e0-dph_xlong_KKt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,69.0){\scriptsize$1$} \put(-6.0,60.0){\scriptsize$10^{-5}$}
\put(-8.0,51.0){\scriptsize$10^{-10}$} \put(-8.0,42.0){\scriptsize$10^{-15}$}
\put(-8.0,33.0){\scriptsize$10^{-20}$} \put(-8.0,24.0){\scriptsize$10^{-25}$}
\put(-8.0,15.0){\scriptsize$10^{-30}$} \put(-8.0,6.0){\scriptsize$10^{-35}$}
\put(18.0,50.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}|$}}
\put(18.0,44.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{2}{}^{0}|$}}
\put(18.0,38.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{4}{}^{0}|$}
\put(18.0,32.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{6}{}^{0}|$}
\put(18.0,26.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{8}{}^{0}|$}
\put(18.0,21.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{10}{}^{0}|$}
\put(18.0,15.0){\scriptsize$r^{-2}$} \put(18.0,9.0){\scriptsize$r^{-1}$}
\end{overpic} (a) Falloff rates of
$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{\ell}{}^{0}|$

\begin{overpic}[width=433.62pt]{kerr-sph-KK20-1e0-dph_xlong_Rekkt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,67.5){\scriptsize$1$} \put(-6.0,51.0){\scriptsize$10^{-5}$}
\put(-8.0,35.0){\scriptsize$10^{-10}$} \put(-8.0,19.0){\scriptsize$10^{-15}$}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{2}{}^{0}]|$}}
\put(18.0,32.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{4}{}^{0}]|$}}
\put(18.0,26.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{6}{}^{0}]|$}
\put(18.0,21.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{8}{}^{0}]|$}
\put(18.0,15.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{10}{}^{0}]|$}
\put(18.0,9.0){\scriptsize$r^{-1}$} \end{overpic} (b) Falloff rates of
$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]|$

\begin{overpic}[width=433.62pt]{kerr-sph-KK20-1e0-dph_xlong_Imkkt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,71.0){\scriptsize$1$} \put(-6.0,62.0){\scriptsize$10^{-5}$}
\put(-8.0,51.0){\scriptsize$10^{-10}$} \put(-8.0,41.0){\scriptsize$10^{-15}$}
\put(-8.0,29.0){\scriptsize$10^{-20}$} \put(-8.0,18.0){\scriptsize$10^{-25}$}
\put(-8.0,7.0){\scriptsize$10^{-30}$}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{1}{}^{0}]|$}}
\put(18.0,32.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{3}{}^{0}]|$}}
\put(18.0,26.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{5}{}^{0}]|$}
\put(18.0,21.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{7}{}^{0}]|$}
\put(18.0,15.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{9}{}^{0}]|$}
\put(18.0,9.0){\scriptsize$r^{-2}$} \end{overpic} (c) Falloff rates of
$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]|$

\begin{overpic}[width=433.62pt]{kerr-sph-KK20-1e0-dph_xlong_Nht.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(-6.0,62.0){\scriptsize$10^{-5}$} \put(-8.0,51.0){\scriptsize$10^{-10}$}
\put(-8.0,39.0){\scriptsize$10^{-15}$} \put(-8.0,28.0){\scriptsize$10^{-20}$}
\put(-8.0,16.0){\scriptsize$10^{-25}$} \put(-8.0,5.0){\scriptsize$10^{-30}$}
\put(18.0,44.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{0}{}^{0}|$}}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{2}{}^{0}|$}}
\put(18.0,32.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{4}{}^{0}|$}
\put(18.0,26.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{6}{}^{0}|$}
\put(18.0,21.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{8}{}^{0}|$}
\put(18.0,15.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{10}{}^{0}|$}
\put(18.0,9.0){\scriptsize$r^{-1}$} \end{overpic} (d) Falloff rates of
$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{\ell}{}^{0}|$

Figure 1: The falloff rates of the only nontrivial modes of the constrained
variables obtained by integrating the parabolic-hyperbolic system for strictly
near-Kerr initial data are depicted. It is transparent that the falloff rate
of ${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}$ or that of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{0}{}^{0}$, individually,
are capable of excluding the conditions of even the weak asymptotic flatness
to hold.

Analogously, in Fig. 2, various modes of the constrained variables, obtained
by integrating the algebraic-hyperbolic system, in the strictly near-Kerr case
are shown. It is transparent that
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}$ decays with the
rate $\mathcal{O}(r^{-3/2})$. Notably, all the $\mathbf{K}_{\ell}{}^{0}$
higher $\ell$ modes fall off somewhat faster than $\mathcal{O}(r^{-2})$, with
a rate close to $\mathcal{O}(r^{-2.2})$. Similarly, all the modes comprising
the real part of ${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}$ also decay
faster than $\mathcal{O}(r^{-1})$, with a rate close to
$\mathcal{O}(r^{-1.2})$, whereas the modes in the imaginary part of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}$ fall off with the rate
$\mathcal{O}(r^{-2})$. As a consequence of the slow decay rate of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}$, the solution to the
algebraic Hamiltonian constraint, $\boldsymbol{\kappa}$, also has the slow
$\mathcal{O}(r^{-3/2})$ decay rate for the monopole,
$\boldsymbol{\kappa}_{0}{}^{0}$, with $\ell>0$ modes decaying slightly faster
than $\mathcal{O}(r^{-2})$. Accordingly, likewise in the strictly near-
Schwarzschild case, the only modes violating both strong and weak asymptotic
flatness are ${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}$ and
$\boldsymbol{\kappa}_{0}{}^{0}$.

\begin{overpic}[width=433.62pt]{kerr-sph-KK20-1e0-dah_xlong_KKt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,68.0){\scriptsize$1$} \put(-6.0,55.0){\scriptsize$10^{-5}$}
\put(-8.0,42.0){\scriptsize$10^{-10}$} \put(-8.0,29.0){\scriptsize$10^{-15}$}
\put(-8.0,16.0){\scriptsize$10^{-20}$}
\put(18.0,50.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}|$}}
\put(18.0,44.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{2}{}^{0}|$}}
\put(18.0,38.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{4}{}^{0}|$}
\put(18.0,32.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{6}{}^{0}|$}
\put(18.0,26.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{8}{}^{0}|$}
\put(18.0,21.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{10}{}^{0}|$}
\put(18.0,15.0){\scriptsize$r^{-2}$} \put(18.0,9.0){\scriptsize$r^{-3/2}$}
\end{overpic} (a) Falloff rates of
$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{\ell}{}^{0}|$

\begin{overpic}[width=433.62pt]{kerr-sph-KK20-1e0-dah_xlong_Rekkt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,67.0){\scriptsize$1$} \put(-6.0,59.0){\scriptsize$10^{-2}$}
\put(-6.0,51.0){\scriptsize$10^{-4}$} \put(-6.0,43.0){\scriptsize$10^{-6}$}
\put(-6.0,35.0){\scriptsize$10^{-8}$} \put(-8.0,28.0){\scriptsize$10^{-10}$}
\put(-8.0,20.0){\scriptsize$10^{-12}$} \put(-8.0,12.0){\scriptsize$10^{-14}$}
\put(-8.0,5.0){\scriptsize$10^{-16}$}
\put(18.0,38.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{2}{}^{0}]|$}
\put(18.0,32.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{4}{}^{0}]|$}
\put(18.0,26.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{6}{}^{0}]|$}
\put(18.0,21.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{8}{}^{0}]|$}
\put(18.0,15.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{10}{}^{0}]|$}
\put(18.0,9.0){\scriptsize$r^{-1}$} \end{overpic} (b) Falloff rates of
$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]|$

\begin{overpic}[width=433.62pt]{kerr-sph-KK20-1e0-dah_xlong_Imkkt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(-6.0,70.0){\scriptsize$10^{-5}$} \put(-8.0,56.0){\scriptsize$10^{-10}$}
\put(-8.0,42.0){\scriptsize$10^{-15}$} \put(-8.0,28.0){\scriptsize$10^{-20}$}
\put(-8.0,13.0){\scriptsize$10^{-25}$}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{1}{}^{0}]|$}}
\put(18.0,32.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{3}{}^{0}]|$}
\put(18.0,26.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{5}{}^{0}]|$}
\put(18.0,21.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{7}{}^{0}]|$}
\put(18.0,15.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{9}{}^{0}]|$}
\put(18.0,9.0){\scriptsize$r^{-2}$} \end{overpic} (c) Falloff rates of
$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]|$

\begin{overpic}[width=433.62pt]{kerr-sph-KK20-1e0-dah_xlong_vart.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,68.5){\scriptsize$1$} \put(-6.0,56.0){\scriptsize$10^{-5}$}
\put(-8.0,43.0){\scriptsize$10^{-10}$} \put(-8.0,30.0){\scriptsize$10^{-15}$}
\put(-8.0,17.5){\scriptsize$10^{-20}$} \put(-8.0,5.0){\scriptsize$10^{-25}$}
\put(18.0,50.0){\scriptsize\hbox{\pagecolor{white}$|\boldsymbol{\kappa}_{0}{}^{0}|$}}
\put(18.0,44.0){\scriptsize\hbox{\pagecolor{white}$|\boldsymbol{\kappa}_{2}{}^{0}|$}}
\put(18.0,38.0){\scriptsize$|\boldsymbol{\kappa}_{4}{}^{0}|$}
\put(18.0,32.0){\scriptsize$|\boldsymbol{\kappa}_{6}{}^{0}|$}
\put(18.0,26.0){\scriptsize$|\boldsymbol{\kappa}_{8}{}^{0}|$}
\put(18.0,21.0){\scriptsize$|\boldsymbol{\kappa}_{10}{}^{0}|$}
\put(18.0,15.0){\scriptsize$r^{-2}$} \put(18.0,9.0){\scriptsize$r^{-3/2}$}
\end{overpic} (d) Falloff rates of $|\boldsymbol{\kappa}_{\ell}{}^{0}|$

Figure 2: The falloff rates of the only nontrivial modes of the constrained
variables yielded by the algebraic-hyperbolic equations for strictly near-Kerr
initial data are shown. It is visible that apart from the monopole part
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}$ of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}$ and
$\boldsymbol{\kappa}_{0}{}^{0}$ of $\boldsymbol{\kappa}$, each of the other
modes falls off at a rate that is even faster than required by conditions of
strong asymptotic flatness.

Our investigation reported in the panels of Figs. 1 and 2 confirm that, in the
case of the strictly near-Kerr initial datasets, apart from the monopole parts
of ${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}$ and
${}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}$, in case of the parabolic-
hyperbolic equations, and apart from the monopole parts of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}$ and $\boldsymbol{\kappa}$ in
case of the algebraic-hyperbolic equations, respectively, all the other modes
decay with a rate that would allow even strong asymptotic flatness.

###  3.3 Applying the new method to the parabolic-hyperbolic system

As was mentioned in the Introduction, Beyer et al. [15] invented a method that
applies to the constraints’ parabolic-hyperbolic formulation, allowing them to
produce strongly asymptotically flat near-Schwarzschild initial datasets. This
was done by setting the freely specifiable variable $\boldsymbol{\kappa}$ to
be proportional to $\mathbf{K}$. Though using the relation
$\boldsymbol{\kappa}=\mathcal{R}\,\mathbf{K}$ allows one to produce strongly
asymptotically flat near-Schwarzschild initial datasets, it also affects the
principal part of the parabolic-hyperbolic system as it also contains
tangential derivatives of $\boldsymbol{\kappa}$. The inequality
$\mathcal{R}>-\tfrac{1}{2}$ had to be imposed on the proportionality factor to
ensure well posedness of the system proposed in [15].

It was also mentioned earlier that one of our aims is to introduce a method
that can be applied on equal footing to the parabolic-hyperbolic and the
algebraic-hyperbolic systems such that no change in the principal parts
occurs. This idea was motivated by our observations in [14] concerning
strictly near-Schwarzschild initial datasets and by the results covered in the
previous section relevant to strictly near-Kerr initial datasets. Both of
these mode-by-mode investigations pointed to the fact that they are indeed the
monopole parts of ${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}$ and
${}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}$ in the case of the
parabolic-hyperbolic equations, and it is the monopole part of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}$ in the case of the
algebraic-hyperbolic equations that are needed to be adjusted. This
immediately raises the question if it is possible to get the desired falloff
rates by restricting only the monopole parts of some of the freely specifiable
variables such that for both the parabolic-hyperbolic and the algebraic-
hyperbolic systems, we can get asymptotically flat near-Kerr initial data
configurations.

It turned out that essentially by demanding the simple variant of the choice
of Beyer et al. [15] given below helps not only in getting strongly
asymptotically flat solutions to both of the alternative evolutionary forms of
the constraints but also we can have a more flexible control on the falloff
rates, and, in particular, we can produce weakly asymptotically flat initial
data with any desired decay rate.

In the parabolic hyperbolic case we restrict only the monopole part of the
freely specifiable field $\boldsymbol{\kappa}$ by setting

$\boldsymbol{\kappa}_{0}{}^{0}=\alpha\left[\mathscr{R}\,\mathbf{K}\right]_{0}{}^{0}\,,\quad{\rm
with}\quad\mathscr{R}={}^{\scriptscriptstyle(\\!0\\!)}\boldsymbol{\kappa}/{}^{\scriptscriptstyle(\\!0\\!)}\mathbf{K}\,,$
(48)

where $\alpha$ is a positive real number. As we shall see below, the role of
the parameter $\alpha$ is to fine-tune the falloff rate of the monopole part
of $\mathbf{K}$. Note also that the choice
$\mathscr{R}={}^{\scriptscriptstyle(\\!0\\!)}\boldsymbol{\kappa}/{}^{\scriptscriptstyle(\\!0\\!)}\mathbf{K}$
guarantees recovering the Kerr-Schild data on $\Sigma$ with choosing
$\alpha=1$ and trivial initial data for the nonlinear perturbations of the
constrained variables.

In the parabolic-hyperbolic case, we also assume that, apart from the monopole
part of $\boldsymbol{\kappa}$, all the other modes remain intact, i.e.,

$\boldsymbol{\kappa}_{\ell}{}^{0}\big{|}_{\ell>0}={}^{\scriptscriptstyle(\\!0\\!)}\boldsymbol{\kappa}_{\ell}{}^{0}\,.$
(49)

One of our primary motivations for controlling only the monopole part of the
freely specifiable variable $\boldsymbol{\kappa}$ by using (48) was that for
the tangential derivatives of $\boldsymbol{\kappa}$ the relation
$\eth\boldsymbol{\kappa}=\eth{}^{\scriptscriptstyle(\\!0\\!)}\boldsymbol{\kappa}$
holds. Accordingly, the principal part of the parabolic-hyperbolic equations,
(9), (10), and (11), remains intact, so it is reasonable to argue, that the
local well posedness of the system follows from results covered in [9].

####  3.3.1 Outline in the spherically symmetric setup

In spherical symmetry,999As in the spherically symmetric case, the variables
have only monopole parts; the $\ell,m$ indices are suppressed. using the
Schwarzschild background, the parabolic-hyperbolic equations (9) and (11)
simplify to the system

$\displaystyle\tfrac{2}{r}\tfrac{\mathrm{d}\widehat{N}}{\mathrm{d}r}$
$\displaystyle=\tfrac{1}{r^{2}}\,\widehat{N}-\tfrac{1}{2}\,\widehat{N}^{3}\Big{(}\tfrac{2}{r^{2}}+2\,\boldsymbol{\kappa}\,\mathbf{K}+\tfrac{1}{2}\,\mathbf{K}^{2}\Big{)}\,,$
(50) $\displaystyle\tfrac{\mathrm{d}\mathbf{K}}{\mathrm{d}r}$
$\displaystyle=\tfrac{2}{r}\,\Big{(}\boldsymbol{\kappa}-\tfrac{1}{2}\mathbf{K}\Big{)}\,.$
(51)

Substituting (48) with $\mathscr{R}=-(r+M)/(2\,(r+2\,M))$ (relevant for the
Schwarzschild background) immediately gives the solution to (51) as

$\mathbf{K}=\frac{C_{1}}{r^{1+\alpha}(1+2M/r)^{\alpha/2}}\,,$ (52)

possessing the asymptotic expansion

$\mathbf{K}\sim\frac{C_{1}}{r^{1+\alpha}}-\frac{C_{1}M\alpha}{r^{2+\alpha}}+\mathcal{O}\left(r^{-(3+\alpha)}\right)\,,$
(53)

where $C_{1}$ is a constant of integration. By virtue of (53) the parameter
$\alpha$, applied in (48), allows a straightforward fine-tuning of the
asymptotic falloff rate of $\mathbf{K}$. To guarantee $\mathbf{K}$ to fit the
requirements in weak asymptotic flatness, $\alpha>1/2$ should hold. If
$\alpha=1$, the falloff rate in (53) is compatible with the requirements in
strong asymptotic flatness. In the spherically symmetric case, we could also
choose faster falloff rates by setting $\alpha>1$. Note, however, that such a
rapid decay does not occur in the nonspherically symmetric case because there
are intimate couplings of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}$ and many of the
background fields with falloff rate $\mathcal{O}(r^{-2})$. Thereby, in the
generic case, one should not expect faster than $\mathcal{O}(r^{-2})$ falloff
rate for ${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}$.

Substituting (52) into (50) we get

$\widehat{N}=\left[1+\tfrac{C_{2}}{r}+\tfrac{C_{1}{}^{2}}{4}r^{-2\alpha}\left(1+\tfrac{2M}{r}\right)\right]^{-1/2}$
(54)

with the desired asymptotic value $1$, where $C_{2}$ is another constant of
integration. It is transparent that if $\alpha$ was smaller than $1/2$, then
the term $-\tfrac{C_{1}{}^{2}}{8r^{2\alpha}}$ could dominate, and whence
$\widehat{N}$ would fail to fall off with the rate $\mathcal{O}(r^{-1})$ that
is required by the weak and strong forms of asymptotic flatness in (45) and
(44), respectively. While the choice of the borderline value $\alpha=1/2$
would fit $\widehat{N}$, it is only the interval $\alpha>1/2$ that ensures the
decay rates are, simultaneously for $\mathbf{K}$ and $\widehat{N}$, compatible
with the weak and strong forms of asymptotic flatness in (45) and (44),
respectively.

####  3.3.2 Strongly asymptotically flat numerical solution

In Fig. 3 the falloff rates of the constraint fields
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K},{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}$,
and ${}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}$, yielded by integrating
the parabolic-hyperbolic equations (9), (10), and (11) for near-Kerr initial
data, and applying (48) with $\alpha=1$, are depicted. In this case each of
the ${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{\ell}{}^{0}$ modes falls
off as $\mathcal{O}(r^{-2})$,
$\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]$
as $\mathcal{O}(r^{-1})$ with
$\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]$
going as $\mathcal{O}(r^{-2})$. Finally, as it is desired,
${}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{0}{}^{0}$ tends to $0$ with
the rate $\mathcal{O}(r^{-1})$, whereas all the higher $\ell$ modes decay as
$\mathcal{O}(r^{-2})$. Note that all of these falloff rates, apart from that
of ${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}$, are the same as reported
in [14], relevant for near-Schwarzschild initial data using the method
proposed in [15]. In that case
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}$ was purely real and each
mode decayed at the rate $\mathcal{O}(r^{-2})$. In contrast, in the present
case only the imaginary part retains this fast decay rate, whereas the decay
of the real part is slower in consequence of using (48) instead of the
proposal in [15]. Note however that each of the falloff rates indicated in
Fig. 3 are compatible with the conditions of strong asymptotic flatness in
(44) as we desired to show.

\begin{overpic}[width=433.62pt]{kerrSKSlm10-KK20-1e0-dphb1-a1_xlong_KKt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,68.0){\scriptsize$1$} \put(-6.0,57.0){\scriptsize$10^{-5}$}
\put(-8.0,45.0){\scriptsize$10^{-10}$} \put(-8.0,34.0){\scriptsize$10^{-15}$}
\put(-8.0,23.0){\scriptsize$10^{-20}$} \put(-8.0,12.0){\scriptsize$10^{-25}$}
\put(18.0,44.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}|$}}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{2}{}^{0}|$}}
\put(18.0,32.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{4}{}^{0}|$}
\put(18.0,26.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{6}{}^{0}|$}
\put(18.0,21.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{8}{}^{0}|$}
\put(18.0,15.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{10}{}^{0}|$}
\put(18.0,9.0){\scriptsize$r^{-2}$} \end{overpic} (a) Falloff rates of
$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{\ell}{}^{0}|$

\begin{overpic}[width=433.62pt]{kerrSKSlm10-KK20-1e0-dphb1-a1_xlong_Rekkt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,67.5){\scriptsize$1$} \put(-6.0,52.0){\scriptsize$10^{-5}$}
\put(-8.0,36.0){\scriptsize$10^{-10}$} \put(-8.0,21.0){\scriptsize$10^{-15}$}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{2}{}^{0}]|$}}
\put(18.0,32.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{4}{}^{0}]|$}}
\put(18.0,26.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{6}{}^{0}]|$}
\put(18.0,21.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{8}{}^{0}]|$}
\put(18.0,15.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{10}{}^{0}]|$}
\put(18.0,9.0){\scriptsize$r^{-1}$} \end{overpic} (b) Falloff rates of
$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]|$

\begin{overpic}[width=433.62pt]{kerrSKSlm10-KK20-1e0-dphb1-a1_xlong_Imkkt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,71.0){\scriptsize$1$} \put(-6.0,62.0){\scriptsize$10^{-5}$}
\put(-8.0,51.0){\scriptsize$10^{-10}$} \put(-8.0,41.0){\scriptsize$10^{-15}$}
\put(-8.0,29.0){\scriptsize$10^{-20}$} \put(-8.0,18.0){\scriptsize$10^{-25}$}
\put(-8.0,7.0){\scriptsize$10^{-30}$}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{1}{}^{0}]|$}}
\put(18.0,32.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{3}{}^{0}]|$}}
\put(18.0,26.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{5}{}^{0}]|$}
\put(18.0,21.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{7}{}^{0}]|$}
\put(18.0,15.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{9}{}^{0}]|$}
\put(18.0,9.0){\scriptsize$r^{-2}$} \end{overpic} (c) Falloff rates of
$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]|$

\begin{overpic}[width=433.62pt]{kerrSKSlm10-KK20-1e0-dphb1-a1_xlong_Nht.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(-6.0,64.0){\scriptsize$10^{-5}$} \put(-8.0,55.0){\scriptsize$10^{-10}$}
\put(-8.0,46.0){\scriptsize$10^{-15}$} \put(-8.0,37.0){\scriptsize$10^{-20}$}
\put(-8.0,28.0){\scriptsize$10^{-25}$} \put(-8.0,19.0){\scriptsize$10^{-30}$}
\put(-8.0,10.0){\scriptsize$10^{-35}$}
\put(18.0,50.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{0}{}^{0}|$}}
\put(18.0,44.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{2}{}^{0}|$}}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{4}{}^{0}|$}}
\put(18.0,32.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{6}{}^{0}|$}
\put(18.0,26.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{8}{}^{0}|$}
\put(18.0,21.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{10}{}^{0}|$}
\put(18.0,15.0){\scriptsize$r^{-1}$} \put(18.0,9.0){\scriptsize$r^{-2}$}
\end{overpic} (d) Falloff rates of
$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{\ell}{}^{0}|$

Figure 3: The falloff rates of the only nontrivial modes of the constrained
variables yielded by the parabolic-hyperbolic equations for near-Kerr initial
data, and applying (48) with $\alpha=1$, are plotted. Note that each mode
falls off in accordance with the conditions in strong asymptotic flatness.

####  3.3.3 Weakly asymptotically flat numerical solution

In order to demonstrate that weakly asymptotically flat initial data can also
be produced by integrating the parabolic-hyperbolic equations in the near-Kerr
case, we determined the falloff rates of various modes of the constrained
variables using (48) with $\alpha=0.7$. These are depicted in Fig. 4. The
monopole mode ${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}$
falls off, as expected, with the rate $\mathcal{O}(r^{-1.7})$, whereas all the
other ${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{\ell}{}^{0}|_{\ell>0}$
modes decay with the rate $\mathcal{O}(r^{-2})$. The falloff behavior of the
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}$ modes follows
the rule observed in the $\alpha=1$ case. As
${}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{0}{}^{0}$ changed its sign
close to $r=10^{5}$ to determine the falloff rate’s precise value, we
integrated the parabolic-hyperbolic equations on a longer interval. The
observed falloff rate of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{0}{}^{0}$ is, as expected,
$\mathcal{O}(r^{-1})$. Note that changing the sign of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{0}{}^{0}$ does not affect
the sign of the ADM mass since
${}^{\scriptscriptstyle(\\!0\\!)}\widehat{N}_{0}{}^{0}-1$, also plotted in
Fig. 4(d) (see top curve), is always larger, so the sign of
$\widehat{N}_{0}{}^{0}-1$ does not change. Note also that all the higher
$\ell$ modes of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{\ell}{}^{0}$ fall off with
the rate $\mathcal{O}(r^{-1.7})$. It is important to emphasize that all the
reported falloff rates are compatible with conditions of weak asymptotic
flatness in (45).

\begin{overpic}[width=433.62pt]{kerrSKSlm10-KK20-1e0-dphb1-a07_xlong_KKt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,67.0){\scriptsize$1$} \put(-6.0,58.0){\scriptsize$10^{-5}$}
\put(-8.0,49.0){\scriptsize$10^{-10}$} \put(-8.0,41.0){\scriptsize$10^{-15}$}
\put(-8.0,32.0){\scriptsize$10^{-20}$} \put(-8.0,23.0){\scriptsize$10^{-25}$}
\put(-8.0,14.0){\scriptsize$10^{-30}$} \put(-8.0,5.0){\scriptsize$10^{-35}$}
\put(18.0,50.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}|$}}
\put(18.0,44.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{2}{}^{0}|$}}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{4}{}^{0}|$}}
\put(18.0,32.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{6}{}^{0}|$}
\put(18.0,26.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{8}{}^{0}|$}
\put(18.0,21.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{10}{}^{0}|$}
\put(18.0,15.0){\scriptsize$r^{-1.7}$} \put(18.0,9.0){\scriptsize$r^{-2}$}
\end{overpic} (a) Falloff rates of
$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{\ell}{}^{0}|$

\begin{overpic}[width=433.62pt]{kerrSKSlm10-KK20-1e0-dphb1-a07_xlong_Rekkt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,67.5){\scriptsize$1$} \put(-6.0,52.0){\scriptsize$10^{-5}$}
\put(-8.0,36.0){\scriptsize$10^{-10}$} \put(-8.0,21.0){\scriptsize$10^{-15}$}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{2}{}^{0}]|$}}
\put(18.0,32.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{4}{}^{0}]|$}}
\put(18.0,26.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{6}{}^{0}]|$}}
\put(18.0,21.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{8}{}^{0}]|$}
\put(18.0,15.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{10}{}^{0}]|$}
\put(18.0,9.0){\scriptsize$r^{-1}$} \end{overpic} (b) Falloff rates of
$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]|$

\begin{overpic}[width=433.62pt]{kerrSKSlm10-KK20-1e0-dphb1-a07_xlong_Imkkt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,71.0){\scriptsize$1$} \put(-6.0,60.0){\scriptsize$10^{-5}$}
\put(-8.0,49.0){\scriptsize$10^{-10}$} \put(-8.0,38.0){\scriptsize$10^{-15}$}
\put(-8.0,27.0){\scriptsize$10^{-20}$} \put(-8.0,16.0){\scriptsize$10^{-25}$}
\put(-8.0,5.0){\scriptsize$10^{-30}$}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{1}{}^{0}]|$}}
\put(18.0,32.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{3}{}^{0}]|$}}
\put(18.0,26.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{5}{}^{0}]|$}
\put(18.0,21.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{7}{}^{0}]|$}
\put(18.0,15.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{9}{}^{0}]|$}
\put(18.0,9.0){\scriptsize$r^{-2}$} \end{overpic} (c) Falloff rates of
$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]|$

\begin{overpic}[width=433.62pt]{kerrSKSlm10-KK20-1e0-dphb1-a07_xlong_Nht.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,69.5){\scriptsize$1$} \put(-6.0,61.0){\scriptsize$10^{-5}$}
\put(-8.0,52.5){\scriptsize$10^{-10}$} \put(-8.0,44.0){\scriptsize$10^{-15}$}
\put(-8.0,36.0){\scriptsize$10^{-20}$} \put(-8.0,27.0){\scriptsize$10^{-25}$}
\put(-8.0,19.0){\scriptsize$10^{-30}$} \put(-8.0,10.0){\scriptsize$10^{-35}$}
\put(18.0,56.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!0\\!)}\widehat{N}_{0}{}^{0}-1|$}}
\put(18.0,50.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{0}{}^{0}|$}}
\put(18.0,44.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{2}{}^{0}|$}}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{4}{}^{0}|$}}
\put(18.0,32.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{6}{}^{0}|$}
\put(18.0,26.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{8}{}^{0}|$}
\put(18.0,21.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{10}{}^{0}|$}
\put(18.0,15.0){\scriptsize$r^{-1}$} \put(18.0,9.0){\scriptsize$r^{-1.7}$}
\put(32.0,7.0){\scriptsize$10^{6}$} \put(42.0,7.0){\scriptsize$10^{7}$}
\put(52.0,7.0){\scriptsize$10^{8}$} \put(62.0,7.0){\scriptsize$10^{9}$}
\put(72.0,7.0){\scriptsize$10^{10}$} \put(82.0,7.0){\scriptsize$10^{11}$}
\put(92.0,7.0){\scriptsize\hbox{\pagecolor{white}$10^{12}$}} \end{overpic} (d)
Falloff rates of
$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{\ell}{}^{0}|$

Figure 4: The falloff rates of the only nontrivial modes of the constrained
variables yielded by the parabolic-hyperbolic equations for near-Kerr initial
data, and applying (48) with $\alpha=0.7$, are plotted. As
${}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{0}{}^{0}$ changed its sign
close to $r=10^{5}$ to determine the precise value of the falloff rate, we
integrated the parabolic-hyperbolic equations of a longer interval. Note that
changing the sign of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\widehat{N}_{0}{}^{0}$ does not affect
the sign of the ADM mass since
${}^{\scriptscriptstyle(\\!0\\!)}\widehat{N}_{0}{}^{0}-1$, also plotted in
4(d), is always larger, so the sign of $\widehat{N}_{0}{}^{0}-1$ does not
change. Note that each mode falls off according to the conditions in weak
asymptotic flatness.

As the plots in Figs. 3 and 4 demonstrate, the choice we made for the monopole
part of $\boldsymbol{\kappa}$ imposing (48) allowed us to produce both weakly
and strongly asymptotically flat near-Kerr initial data configurations.
Another preferable consequence of this choice was that it left intact the
principal part of the parabolic-hyperbolic equations. Although the implicit
averaging used to determine the monopole part of $\boldsymbol{\kappa}$ may
raise doubts about the nature of the system, the results in [9] on the
principal symbol of the equations, together with the observed convergence
properties, support our belief in the well posedness of the corresponding
initial value problem.

###  3.4 Applying the new method to the algebraic-hyperbolic system

In this section, using the algebraic-hyperbolic form of the constraint
equations, we impose a condition only on the monopole part of the lapse
$\widehat{N}$. In contrast, all the higher $\ell$ modes $\widehat{N}$ retain
their background form. In particular, $\widehat{N}_{0}{}^{0}$ is subject to
the ODE derived from (9), which is as follows:

$\displaystyle\frac{\mathrm{d}\widehat{N}_{0}{}^{0}}{\mathrm{d}r}=\big{[}(\partial_{r}\widehat{N})+\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}{}^{-1}\big{\\{}\widetilde{\mathcal{B}}-\mathcal{B}\big{\\}}\widehat{N}^{3}\big{]}_{0}{}^{0}\,.$
(55)

Here $(\partial_{r}\widehat{N})$ denotes the right-hand side of the equation
obtained by solving (9) for $\partial_{r}\widehat{N}$, whereas

$\widehat{N}=\widehat{N}_{0}{}^{0}+[{}^{\scriptscriptstyle(\\!0\\!)}\widehat{N}-{}^{\scriptscriptstyle(\\!0\\!)}\widehat{N}_{0}{}^{0}]\,,$
(56)

i.e., $\widehat{N}$ differs from ${}^{\scriptscriptstyle(\\!0\\!)}\widehat{N}$
only in its monopole part, implying
$\eth\widehat{N}=\eth{}^{\scriptscriptstyle(\\!0\\!)}\widehat{N}$. Note also
that the monopole part of the right-hand side (55) is taken,
$\widetilde{\mathcal{B}}$ is obtained from $\mathcal{B}$, as given in (13), by
replacing 101010Note that this step is exactly the same procedure as in Sec.
3.3. $\boldsymbol{\kappa}$ with

$\alpha\left[\mathscr{R}\,\mathbf{K}\right]_{0}{}^{0}+[{}^{\scriptscriptstyle(\\!0\\!)}\boldsymbol{\kappa}-{}^{\scriptscriptstyle(\\!0\\!)}\boldsymbol{\kappa}_{0}{}^{0}]\,,\quad{\rm
with}\
\mathscr{R}={}^{\scriptscriptstyle(\\!0\\!)}\boldsymbol{\kappa}/{}^{\scriptscriptstyle(\\!0\\!)}\mathbf{K}\,.$
(57)

Note that this choice of $\boldsymbol{\kappa}$ only affects the monopole part
of $\widehat{N}$ via (55). Note also that the true physical value of
$\boldsymbol{\kappa}$ is determined (as it should be) by the algebraic form of
the Hamiltonian constraint (18). As we will see below, the use of the positive
parameter $\alpha$ in (55) allows fine-tuning of the decay rate of the
monopole part of $\mathbf{K}$. Note also that (55) is able to reproduce the
Kerr limit by choosing $\alpha=1$ and trivial excitations for the fields
$\mathbf{K},\mathbf{k}$ at $r=1$. It is also straightforward to derive the
spherical limit by applying the relations
$\widehat{N}={}^{\scriptscriptstyle(\\!0\\!)}\widehat{N}=\widehat{N}_{0}{}^{0}$.

####  3.4.1 Outline in the spherically symmetric setup

Note that there is not much earned by restricting considerations to the
spherically symmetric case as the algebraic-hyperbolic system is essentially
the same as it was in the parabolic-hyperbolic case. More concretely, the
algebraic form of the Hamiltonian constraint is nothing but solving (50) for
$\boldsymbol{\kappa}$ or $2\,\boldsymbol{\kappa}\,\mathbf{K}$, whereas Eq.
(51) for $\mathbf{K}$ remains intact. Accordingly, the corresponding spherical
symmetric system with Schwarzschild background reads as

$\displaystyle 2\,\boldsymbol{\kappa}\,\mathbf{K}$
$\displaystyle=-\tfrac{2}{r^{2}}-\tfrac{1}{2}\mathbf{K}^{2}+\tfrac{2}{r^{2}\widehat{N}^{2}}-\tfrac{4}{r\widehat{N}^{3}}\tfrac{\mathrm{d}\widehat{N}}{\mathrm{d}r}\,,$
(58) $\displaystyle\frac{\mathrm{d}\mathbf{K}}{\mathrm{d}r}$
$\displaystyle=\tfrac{2}{r}\left(\boldsymbol{\kappa}-\tfrac{1}{2}\mathbf{K}\right)\,.$
(59)

Note that by eliminating the $r$ derivative of $\widehat{N}$ from (58) by
applying the spherical symmetric form of (55) one recovers (48). In light of
this observation, the analytic solutions to (59) must possess the form (52)
with asymptotic behavior as given in (53). Accordingly, the parameter
$\alpha$, applied in (55), allows a straightforward fine-tuning of the
asymptotic falloff rate of $\mathbf{K}$. As previously, in the spherically
symmetric parabolic-hyperbolic case, to guarantee $\mathbf{K}$ to fit the
requirements in weak asymptotic flatness, the inequality $\alpha>1/2$ must
hold. In the $\alpha=1$ case, the falloff rate in (53) allows the spherically
symmetric solution to be strongly asymptotically flat.

####  3.4.2 Strongly asymptotically flat numerical solution

In Fig. 5 the falloff rates of the constraint fields
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}$,
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}$, and $\boldsymbol{\kappa}$,
yielded by the integration of the algebraic-hyperbolic equations, (16)-(17),
for near-Kerr initial data, and applying (55) with $\alpha=1$, are depicted.
In this case each of the
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{\ell}{}^{0}$ modes falls off
as $\mathcal{O}(r^{-2})$,
$\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]$
as $\mathcal{O}(r^{-1})$ with
$\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]$
going as $\mathcal{O}(r^{-2})$. Note that using the algebraic-hyperbolic
equations, we had to perform several divisions by $\mathbf{K}$, which, because
of the uniform decay rate of all the
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{\ell}{}^{0}$ modes,
$\ell=0,2,4,6,8,10$, required a careful application of the division scheme
based on the Neumann series expansion. Note also the falloff of the modes of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}$ follow the same pattern
observed in Sec. 3.3. Since all modes of $\boldsymbol{\kappa}$ also fall off
with the rate $\mathcal{O}(r^{-2})$, the initial data obtained is
asymptotically flat in the strong sense. Note that although
$\widehat{N}_{0}{}^{0}$ also deviates from its background form, it falls at
the rate $\mathcal{O}(r^{-1})$ to $1$ as expected, so we do not include its
graph here.

\begin{overpic}[width=433.62pt]{kerrSKSlm10-KK20-1e0-dahm-a1SOBOLEV_KKt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,66.0){\scriptsize$1$} \put(-6.0,52.0){\scriptsize$10^{-5}$}
\put(-8.0,39.0){\scriptsize$10^{-10}$} \put(-8.0,26.5){\scriptsize$10^{-15}$}
\put(-8.0,13.0){\scriptsize$10^{-20}$}
\put(18.0,44.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}|$}}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{2}{}^{0}|$}}
\put(18.0,32.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{4}{}^{0}|$}
\put(18.0,26.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{6}{}^{0}|$}
\put(18.0,21.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{8}{}^{0}|$}
\put(18.0,15.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{10}{}^{0}|$}
\put(18.0,9.0){\scriptsize$r^{-2}$} \end{overpic} (a) Falloff rates of
$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{\ell}{}^{0}|$

\begin{overpic}[width=433.62pt]{kerrSKSlm10-KK20-1e0-dahm-a1SOBOLEV_Rekkt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,66.5){\scriptsize$1$} \put(-6.0,58.0){\scriptsize$10^{-2}$}
\put(-6.0,49.0){\scriptsize$10^{-4}$} \put(-6.0,40.0){\scriptsize$10^{-6}$}
\put(-6.0,31.0){\scriptsize$10^{-8}$} \put(-8.0,22.0){\scriptsize$10^{-10}$}
\put(-8.0,14.0){\scriptsize$10^{-12}$} \put(-8.0,5.0){\scriptsize$10^{-14}$}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{2}{}^{0}]|$}}
\put(18.0,32.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{4}{}^{0}]|$}}
\put(18.0,26.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{6}{}^{0}]|$}
\put(18.0,21.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{8}{}^{0}]|$}
\put(18.0,15.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{10}{}^{0}]|$}
\put(18.0,9.0){\scriptsize$r^{-1}$} \end{overpic} (b) Falloff rates of
$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]|$

\begin{overpic}[width=433.62pt]{kerrSKSlm10-KK20-1e0-dahm-a1SOBOLEV_Imkkt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(-6.0,61.0){\scriptsize$10^{-5}$} \put(-8.0,48.5){\scriptsize$10^{-10}$}
\put(-8.0,36.5){\scriptsize$10^{-15}$} \put(-8.0,24.5){\scriptsize$10^{-20}$}
\put(-8.0,12.0){\scriptsize$10^{-25}$}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{1}{}^{0}]|$}}
\put(18.0,32.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{3}{}^{0}]|$}}
\put(18.0,26.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{5}{}^{0}]|$}
\put(18.0,21.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{7}{}^{0}]|$}
\put(18.0,15.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{9}{}^{0}]|$}
\put(18.0,9.0){\scriptsize$r^{-2}$} \end{overpic} (c) Falloff rates of
$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]|$

\begin{overpic}[width=433.62pt]{kerrSKSlm10-KK20-1e0-dahm-a1SOBOLEV_vart.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(-6.0,55.5){\scriptsize$10^{-5}$} \put(-8.0,42.0){\scriptsize$10^{-10}$}
\put(-8.0,29.0){\scriptsize$10^{-15}$} \put(-8.0,16.0){\scriptsize$10^{-20}$}
\put(2.0,68.5){\scriptsize$1$}
\put(18.0,44.0){\scriptsize\hbox{\pagecolor{white}$|\boldsymbol{\kappa}_{0}{}^{0}|$}}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|\boldsymbol{\kappa}_{2}{}^{0}|$}}
\put(18.0,32.0){\scriptsize$|\boldsymbol{\kappa}_{4}{}^{0}|$}
\put(18.0,26.0){\scriptsize$|\boldsymbol{\kappa}_{6}{}^{0}|$}
\put(18.0,21.0){\scriptsize$|\boldsymbol{\kappa}_{8}{}^{0}|$}
\put(18.0,15.0){\scriptsize$|\boldsymbol{\kappa}_{10}{}^{0}|$}
\put(18.0,9.0){\scriptsize$r^{-2}$} \end{overpic} (d) Falloff rates of
$|\boldsymbol{\kappa}_{\ell}{}^{0}|$

Figure 5: The falloff rates of the only nontrivial modes of the constrained
variables yielded by the algebraic-hyperbolic equations for near-Kerr initial
data, and applying (55) with $\alpha=1$, are plotted. Note that each mode
falls off in accordance with the conditions in strong asymptotic flatness.

####  3.4.3 Weakly asymptotically flat numerical solution

In order to demonstrate that weakly asymptotically flat initial data can also
be produced by integrating the algebraic-hyperbolic equations in the near-Kerr
case, we determined the falloff rates of various modes of the constrained
variables using (55) with $\alpha=0.7$. These are depicted in Fig. 6. The
monopole mode ${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}$
falls off, as expected, with the rate $\mathcal{O}(r^{-1.7})$, whereas all the
other ${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{\ell}{}^{0}|_{\ell>0}$
modes decay—likewise it happened in the strictly near-Kerr case in Sec.
3.2—somewhat slower than $\mathcal{O}(r^{-2.2})$ but noticeably faster than
$\mathcal{O}(r^{-2})$. As for the asymptotic behavior of the modes
$\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]$
also somewhat slower than $\mathcal{O}(r^{-1.2})$ but noticeably faster than
$\mathcal{O}(r^{-1})$ falloff rate is observed. The falloff rate for the
$\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]$
modes is as usual $\mathcal{O}(r^{-2})$. Similar to
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}$,
$\boldsymbol{\kappa}_{0}{}^{0}$ decays as $\mathcal{O}(r^{-1.7})$, while
slightly faster than $\mathcal{O}(r^{-2})$ decay rates are observed for
$\ell>0$ modes of $\boldsymbol{\kappa}$. It is important to emphasize that all
the observed falloff rates are compatible with conditions of weak asymptotic
flatness in (45), i.e., the resulted initial data is asymptotically flat in
the weaker sense.

\begin{overpic}[width=433.62pt]{kerrSKSlm10-KK20-1e0-dahm-a07SOBOLEV_KKt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,66.0){\scriptsize$1$} \put(-6.0,54.0){\scriptsize$10^{-5}$}
\put(-8.0,41.0){\scriptsize$10^{-10}$} \put(-8.0,29.0){\scriptsize$10^{-15}$}
\put(-8.0,17.0){\scriptsize$10^{-20}$} \put(-8.0,5.0){\scriptsize$10^{-25}$}
\put(18.0,50.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}|$}}
\put(18.0,44.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{2}{}^{0}|$}}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{4}{}^{0}|$}}
\put(18.0,32.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{6}{}^{0}|$}
\put(18.0,26.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{8}{}^{0}|$}
\put(18.0,21.0){\scriptsize$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{10}{}^{0}|$}
\put(18.0,15.0){\scriptsize$r^{-1.7}$} \put(18.0,9.0){\scriptsize$r^{-2}$}
\end{overpic} (a) Falloff rates of
$|{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{\ell}{}^{0}|$

\begin{overpic}[width=433.62pt]{kerrSKSlm10-KK20-1e0-dahm-a07SOBOLEV_Rekkt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,66.5){\scriptsize$1$} \put(-6.0,58.0){\scriptsize$10^{-2}$}
\put(-6.0,50.0){\scriptsize$10^{-4}$} \put(-6.0,42.0){\scriptsize$10^{-6}$}
\put(-6.0,34.0){\scriptsize$10^{-8}$} \put(-8.0,25.0){\scriptsize$10^{-10}$}
\put(-8.0,17.0){\scriptsize$10^{-12}$} \put(-8.0,9.0){\scriptsize$10^{-14}$}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{2}{}^{0}]|$}}
\put(18.0,32.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{4}{}^{0}]|$}}
\put(18.0,26.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{6}{}^{0}]|$}
\put(18.0,21.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{8}{}^{0}]|$}
\put(18.0,15.0){\scriptsize$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{10}{}^{0}]|$}
\put(18.0,9.0){\scriptsize$r^{-1}$} \end{overpic} (b) Falloff rates of
$|\mathfrak{Re}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]|$

\begin{overpic}[width=433.62pt]{kerrSKSlm10-KK20-1e0-dahm-a07SOBOLEV_Imkkt.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(-6.0,59.0){\scriptsize$10^{-5}$} \put(-8.0,47.5){\scriptsize$10^{-10}$}
\put(-8.0,36.0){\scriptsize$10^{-15}$} \put(-8.0,24.0){\scriptsize$10^{-20}$}
\put(-8.0,12.0){\scriptsize$10^{-25}$}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{1}{}^{0}]|$}}
\put(18.0,32.0){\scriptsize\hbox{\pagecolor{white}$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{3}{}^{0}]|$}}
\put(18.0,26.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{5}{}^{0}]|$}
\put(18.0,21.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{7}{}^{0}]|$}
\put(18.0,15.0){\scriptsize$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{9}{}^{0}]|$}
\put(18.0,9.0){\scriptsize$r^{-2}$} \end{overpic} (c) Falloff rates of
$|\mathfrak{Im}[{}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{k}_{\ell}{}^{0}]|$

\begin{overpic}[width=433.62pt]{kerrSKSlm10-KK20-1e0-dahm-a07SOBOLEV_vart.png}
\put(4.0,1.0){\scriptsize$1$} \put(12.0,1.0){\scriptsize$10^{1}$}
\put(22.0,1.0){\scriptsize$10^{2}$} \put(32.0,1.0){\scriptsize$10^{3}$}
\put(42.0,1.0){\scriptsize$10^{4}$} \put(52.0,1.0){\scriptsize$10^{5}$}
\put(62.0,1.0){\scriptsize$10^{6}$} \put(72.0,1.0){\scriptsize$10^{7}$}
\put(82.0,1.0){\scriptsize$10^{8}$} \put(92.0,1.0){\scriptsize$10^{9}$}
\put(2.0,68.5){\scriptsize$1$} \put(-6.0,56.0){\scriptsize$10^{-5}$}
\put(-8.0,43.0){\scriptsize$10^{-10}$} \put(-8.0,31.0){\scriptsize$10^{-15}$}
\put(-8.0,18.0){\scriptsize$10^{-20}$} \put(-8.0,5.0){\scriptsize$10^{-25}$}
\put(18.0,50.0){\scriptsize\hbox{\pagecolor{white}$|\boldsymbol{\kappa}_{0}{}^{0}|$}}
\put(18.0,44.0){\scriptsize\hbox{\pagecolor{white}$|\boldsymbol{\kappa}_{2}{}^{0}|$}}
\put(18.0,38.0){\scriptsize\hbox{\pagecolor{white}$|\boldsymbol{\kappa}_{4}{}^{0}|$}}
\put(18.0,32.0){\scriptsize$|\boldsymbol{\kappa}_{6}{}^{0}|$}
\put(18.0,26.0){\scriptsize$|\boldsymbol{\kappa}_{8}{}^{0}|$}
\put(18.0,21.0){\scriptsize$|\boldsymbol{\kappa}_{10}{}^{0}|$}
\put(18.0,15.0){\scriptsize$r^{-2}$} \put(18.0,9.0){\scriptsize$r^{-1.7}$}
\end{overpic} (d) Falloff rates of $|\boldsymbol{\kappa}_{\ell}{}^{0}|$

Figure 6: The falloff rates of the only nontrivial modes of the constrained
variables yielded by integrating the algebraic-hyperbolic equations for near-
Kerr initial data, and applying (55) with $\alpha=0.7$, are plotted. Note that
each mode falls off according to the conditions in weak asymptotic flatness.

The plots in Figs. 5 and 6 demonstrate that the choice we made for the
monopole part of $\widehat{N}$ imposing (55) allowed us to produce both weakly
and strongly asymptotically flat near-Kerr initial data configurations by
integrating the algebraic-hyperbolic system. As it was emphasized several
times, restricting the monopole part of $\widehat{N}$ using (55) has the
preferable consequence that it does not affect the principal part of the
algebraic-hyperbolic equations. Note also that the parameter $\alpha$ in (55)
could be used to control the falloff rate of
${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}$ in the desired
way.

As expected, the solutions to the evolutionary form of the constraints
significantly depend on whether (48) or (55) is applied. To demonstrate this,
it is rewarding to compare the falloff behavior of the corresponding variables
yielded by using the same initial excitation as specified in (47). The falloff
rates of the $\ell>0,m=0$ modes of the variables
$\mathbf{K},\mathfrak{Re}[\mathbf{k}],\mathfrak{Im}[\mathbf{k}],\widehat{N},\boldsymbol{\kappa}$
are collected in Table 2 relevant for the weakly asymptotically flat solutions
to the parabolic-hyperbolic and algebraic-hyperbolic systems corresponding to
the choice $\alpha=0.7$.

$\alpha=0.7$; $\ell>0,m=0$ | $\mathbf{K}_{\ell}{}^{0}$ | $\mathfrak{Re}[\mathbf{k}_{\ell}{}^{0}]$ | $\mathfrak{Im}[\mathbf{k}_{\ell}{}^{0}]$ | $\widehat{N}_{\ell}{}^{0}$ | $\boldsymbol{\kappa}_{\ell}{}^{0}$
---|---|---|---|---|---
Parabolic-hyperbolic | $\mathcal{O}(r^{-2})$ | $\mathcal{O}(r^{-1})$ | $\mathcal{O}(r^{-2})$ | $\mathcal{O}(r^{-1.7})$ | $\mathcal{O}(r^{-\ell-2})$
Algebraic-hyperbolic | $\mathcal{O}(r^{-2.2+\epsilon})$ | $\mathcal{O}(r^{-1.2+\epsilon})$ | $\mathcal{O}(r^{-2})$ | $\mathcal{O}(r^{-\ell-1})$ | $\mathcal{O}(r^{-2.2+\epsilon})$
Table 2: The falloff rates of the $\ell>0,m=0$ modes of $\mathbf{K}$,
$\mathfrak{Re}[\mathbf{k}]$, $\mathfrak{Im}[\mathbf{k}]$, $\widehat{N}$,
$\boldsymbol{\kappa}$ for the weakly asymptotically flat solutions to the
parabolic-hyperbolic and algebraic-hyperbolic systems with $\alpha=0.7$, and
for some small $\epsilon>0$.

##  4 Summary

As mentioned earlier, one of the preferable aspects of using the elliptic (or
conformal) method is that one may set up a boundary value problem by
implementing suitable falloff requirements at infinity. This cannot be done
while solving the evolutionary form of the constraints. Nevertheless, in the
latter case, a delicate compensation arises. While one has to fix once and for
all throughout $\Sigma$ each of the freely specifiable variables in advance of
solving the elliptic problem, one has the freedom of specifying in the interim
of the “time integration” process the freely specifiable fields while solving
either of the evolutionary forms of the constraints. The only limitation is
that this in-flight setting of the freely specifiable fields should preserve
the well posedness of the system. Our novel proposal to relate only the
monopole part of a freely specifiable variable and a constrained field leaves
the principal symbol of the system intact. Due to the nonlocality, introduced
by the averaging in the proposed treatment of the monopole part, the well
posedness of the system is not self-evident. Nevertheless, since the principal
parts of the evolutionary systems remain intact, the observed convergence
rates suggest that the proposed method leads to well-posed problems in both
evolutionary formulations of the constraints. Note, however, that a rigorous
verification of this conjecture is beyond the scope of the present paper.

Concerning the question raised by the title and some others raised in the
Introduction, the results reported in this paper allow us to answer them in
the following way. Yes, it is possible to construct asymptotically flat
initial data by applying either of the evolutionary forms of the constraints.
Above this, we could introduce a novel method that allows us to explore the
asymptotic behavior of initial datasets located in a neighborhood of the data
that can be deduced on a Kerr-Schild time slice of a rotating Kerr black hole
spacetime. In applying our new method, we modify only the monopole part of the
freely specifiable variables $\boldsymbol{\kappa}$ or $\widehat{N}$ in the
parabolic-hyperbolic or algebraic-hyperbolic form of the constraint,
respectively. On the one hand, this left the principal parts of the
evolutionary forms of the constraints intact. Thus, the well posedness of
evolutionary systems seems to be plausible in line with the results in [9].

On the other hand, our proposal can be applied to the alternative evolutionary
systems on an equal footing. In contrast, the method proposed in [15] applies
only to the parabolic-hyperbolic system with the additional cost of modifying
the principal part of the pertinent PDEs. Another favorable aspect of the
novel method proposed in this paper, which applies to both of the evolutionary
forms of the constraints, is that we have direct control of the falloff rate
of the monopole part of the constrained variable $\mathbf{K}$. This allows us
to generate weakly and strongly asymptotically flat initial data
configurations with any desired falloff rate. The viability of the proposed
methods was demonstrated by integrating both of the alternative evolutionary
forms of the constraint equations numerically. As the applied initial data
deformations at $r=1$ were significant, the time evolution of the yielded
near-Kerr initial data configurations could be suitable to study the
gravitational wave output of highly excited Kerr black holes.

To indicate the notable differences between applying (48) and the choice made
in [15], it is rewarding to compare the solutions to the relevant evolutionary
forms of the constraint equations with setting $\alpha=1.5$. As for
$\mathcal{R}=\alpha\cdot\mathscr{R}=\alpha\cdot{}^{\scriptscriptstyle(\\!0\\!)}\boldsymbol{\kappa}_{0}{}^{0}/{}^{\scriptscriptstyle(\\!0\\!)}\mathbf{K}_{0}{}^{0}$
the inequality $\mathcal{R}>-1/2$ fails to hold, and, thereby, the
hyperbolicity condition relevant for “modified parabolic-hyperbolic equations”
in [15] is violated, the corresponding solution blows up. In contrast,
integrating the parabolic-hyperbolic equations by applying our proposal (48)
with setting $\alpha=1.5$, yields—due to the fact that the principal part was
retained, and also to the intimate coupling of various modes of the involved
variables—a completely regular strongly asymptotically flat initial data such
that ${}^{\scriptscriptstyle(\\!\Delta\\!)}\mathbf{K}_{0}{}^{0}$ decays with
the rate $\mathcal{O}(r^{-2})$.

Note also that there is a significant conceptual difference between using the
constraints’ parabolic-hyperbolic and algebraic-hyperbolic forms. Whereas the
parabolic-hyperbolic system can only be solved in general only on that side of
the $\varrho=\varrho_{0}$ initial data surface where
$\accentset{\smash{\raisebox{-0.32549pt}{$\scriptstyle\star$}}}{K}\rule{0.0pt}{8.91248pt}$
is guaranteed to be positive, the algebraic-hyperbolic form of the constraints
can always be solved on both sides of the initial data surface, provided that
$\boldsymbol{\kappa}\,\mathbf{K}<0$. From this point of view, it is notable
that (55) is deduced from the parabolic form of the Hamiltonian constraint
(9). Despite its parabolic origin, (55) is an ordinary differential equation
for $\widehat{N}_{0}{}^{0}$. Thereby, it can be integrated on both sides of
the $\varrho=\varrho_{0}$ initial data surface if it was aimed to solve the
algebraic-hyperbolic system on both sides of $\mathscr{S}_{\varrho_{0}}$.
Since, in this paper, the integration starts at $r=1$, and it always happens
towards infinity, the above-mentioned favorable aspects of the use of (55)
remain to economize in future investigations.

The results covered by the present paper provide significant credit to the
investigations in [27]. It was argued there that in advance of solving the
constraints, one could determine each of the ADM charges, the mass, center of
mass, and the linear and angular momentum of initial datasets relevant for
binary black hole systems. This could be done as the values of the ADM charges
are not affected by the particular functional form of the constrained
variables. However, we made two critical assumptions in [27]. First, even
though the explicit form of the constrained variables was irrelevant, their
falloff behavior—making them compatible with the strong asymptotic flatness of
the yielded complete data $(h_{ab},K_{ab})$—was assumed in deriving the ADM
charges. Second, it was also presumed that the original form of the parabolic-
hyperbolic form of the constraints holds. Given these points, it is
transparent that the results covered by the present paper provide a
significant verification of the assumptions made in [27], thus, also to the
determination of the ADM charges of initial datasets of binary black hole
systems.

At this point, we remind the reader that although the main concern of the
present paper is the near-Kerr initial data of a single black hole, there are
parallel studies that verify that the evolutionary formulation of the
constraints is capable of producing binary black hole initial data [13, 15,
17, 18]. We believe that the methods presented in this paper are very useful
for generating initial data for binary black holes with appropriate
asymptotics. However, the verification of the latter claim is left to future
studies.

Finally, it would be good to know that the novel method introduced in this
paper will find its way to be adopted in analytic investigations. It may
provide stimulating munition to prove the global existence of solutions to the
evolutionary forms of the constraints or derive exact decay rates for various
asymptotically flat near-Kerr initial datasets. Applying our new proposal to
other more challenging asymptotically flat configurations would also deserve
further investigation.

##  5 Acknowledgments

The authors would like to thank Florian Beyer and Joshua Ritchie for helpful
comments. This work was supported in part by NKFIH Grant No. K-115434. K. C.
also acknowledges support by NSF CAREER Award No. PHY-2047382.

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