# Gravitational radiation with $\Lambda>0$

Béatrice Bonga<EMAIL_ADDRESS>Institute for Mathematics, Astrophysics
and Particle Physics, Radboud University, 6525 AJ Nijmegen, The Netherlands
Claudio Bunster<EMAIL_ADDRESS>Centro de Estudios Científicos (CECs), Av.
Arturo Prat 514, Valdivia, Chile
Universidad San Sebastián, Chile  Alfredo Pérez<EMAIL_ADDRESS>Centro
de Estudios Científicos (CECs), Av. Arturo Prat 514, Valdivia, Chile
Facultad de Ingeniería, Arquitectura y Diseño, Universidad San Sebastián, sede
Valdivia, General Lagos 1163, Valdivia 5110693, Chile

(August 28, 2024)

###### Abstract

We study gravitational radiation for a positive value of the cosmological
constant $\Lambda$. We rely on two battle-tested procedures: (i) We start from
the same null coordinate system used by Bondi and Sachs for $\Lambda=0$, but,
introduce boundary conditions adapted to allow radiation when $\Lambda>0$.
(ii) We determine the asymptotic symmetries by studying, à la Regge-
Teitelboim, the surface integrals generated in the action by these boundary
conditions. A crucial difference with the $\Lambda=0$ case is that the wave
field does not vanish at large distances, but is of the same order as de
Sitter space. This novel property causes no difficulty; on the contrary, it
makes quantities finite at every step, without any regularization. The
asymptotic symmetry algebra consists only of time translations and space
rotations. Thus, it is not only finite-dimensional, but smaller than de Sitter
algebra. We exhibit formulas for the energy and angular momentum and their
fluxes. In the limit of $\Lambda$ tending to zero, these formulas go over
continuously into those of Bondi, but the symmetry jumps to that of Bondi,
Metzner and Sachs. The expressions are applied to exact solutions, with and
without radiation present, and also to the linearized theory. All quantities
are finite at every step, no regularization is needed.

##  I Introduction

We study gravitational radiation for a positive value of the cosmological
constant $\Lambda$ as generated by compact sources such as stars and black
holes. We are guided by, and closely follow the steps of, Sachs’ classical
analysis of the concepts introduced by Bondi for $\Lambda=0$ (Bondi:1960jsa, ;
Sachs:1962wk, ; Sachs:1962zza, ). That analysis gave unambiguous expressions
for energy and energy flux, and also established the existence of an infinite-
dimensional symmetry algebra, now called the Bondi-Metzner-Sachs (BMS)
algebra.

Bondi and Sachs relied only on Einstein’s equations with boundary conditions
to study a large class of spacetimes, now known as “asymptotically flat”
solutions. They did not, and needed not to, invoke the action principle. They
required though, that the mass should decrease when radiation is emitted. This
demand was essential to arrive at unambiguous expressions for the mass and its
flux. We have not been able to implement the mass diminution requirement for
$\Lambda>0$, although we do recover this in the linearized case. In order to
arrive at the formulas for the mass and its flux, we have appealed instead to
the action principle, including in it appropriate surface integrals à la
Regge-Teitelboim.

The symmetry algebra consists only of time translations and space rotations,
even when gravitational radiation is present. It is not only finite-
dimensional but smaller than the de Sitter algebra. This result is crucially
linked to the $\mathbb{R}\times\mathbb{S}^{2}$ topology of the future
boundary, and as shown in (abk1, ), the symmetry group for asymptotically de
Sitter spacetimes depends crucially on the topology. In the limit
$\Lambda\rightarrow 0$, the mass and flux formulas coincide with those of
Bondi and Sachs. In contradistinction, the symmetry algebra has an enormous
jump: It becomes the BMS symmetry.

For waves of small amplitude, we recover the energy flux of the linear theory
on a de Sitter background. Interestingly, even waves with small amplitudes
reach infinity without decay. This is not the case for asymptotically flat
spacetimes, in which linear waves decay at least as $1/r$ at large distances.
For special solutions such as Kerr-de Sitter and Robinson-Trautman with
$\Lambda>0$, we recover the accepted expressions for the mass and angular
momentum. All of our conclusions follow from General Relativity, by bringing
into it boundary conditions which are the natural extension for $\Lambda>0$ of
those employed by Sachs for $\Lambda=0$.

Interestingly, all equations in this paper hold also for $\Lambda<0$. In that
case, the boundary conditions do not correspond to the typical reflecting ones
(see e.g. (Ashtekar:1984zz, ; Henneaux:1985tv, ; Ashtekar:1999jx, )).111This
is clear from the non-conformal flatness of the boundary metric. We find this
an appealing issue for exploration, but we will not address it here.

The key findings of this paper are summarized in Table 1.

Table 1:  This table summarizes the key results of this paper and compares those with the case $\Lambda=0$. | $\Lambda=0$ | $\Lambda>0$
---|---|---
Asymptotic region | Future null infinity | Future space-like infinity
Topology asymptotic region | $\mathbb{R}\times\mathbb{S}^{2}$ |  $\mathbb{R}\times\mathbb{S}^{2}$ (= $\mathbb{S}^{3}$ with two points removed)
|  | to describe radiation emitted by bounded sources
Conformal completion | Yes | Yes
of infinity possible? |  |
Coordinates | $ds^{2}=e^{2\beta}\frac{V}{r}du^{2}-2e^{2\beta}dudr$ | $ds^{2}=e^{2\beta}\frac{V}{r}du^{2}-2e^{2\beta}dudr$
|  $\qquad+r^{2}g_{AB}\left(dx^{A}-U^{A}du\right)\left(dx^{B}-U^{B}du\right)$, |  $\qquad+r^{2}g_{AB}\left(dx^{A}-U^{A}du\right)\left(dx^{B}-U^{B}du\right)$,
| $\det g_{AB}=\sin^{2}\theta$ | $\det g_{AB}=\sin^{2}\theta$
| (Bondi gauge with the Sachs condition) | (Bondi gauge with the Sachs condition)
Fall-off |  $\beta=O\left(r^{-2}\right)$, |  $\beta=O\left(r^{-2}\right)$,
| $U^{A}=-\frac{1}{2r^{2}}D_{B}C^{AB}$ | $U^{A}=U_{\left(0\right)}^{A}-\frac{1}{2r^{2}}D_{B}C^{AB}$
|  $\qquad-\frac{2}{3r^{3}}\left(N^{A}-\frac{1}{2}C^{AB}D^{C}C_{BC}\right)+\dots$, |  $\qquad-\frac{2}{3r^{3}}\left(N^{A}-\frac{1}{2}C^{AB}D^{C}C_{BC}\right)+\dots$,
|  $V=-r+2M+\dots$, |  $V=\frac{\Lambda r^{3}}{3}-D_{A}U_{\left(0\right)}^{A}r^{2}$$-\left(1+\frac{\Lambda}{16}C_{AB}C^{AB}\right)r+2M+\dots,$
| $g_{AB}=\gamma_{AB}+\frac{C_{AB}}{r}+\frac{\gamma_{AB}C_{CD}C^{CD}}{4r^{2}}+\frac{E_{AB}}{r^{3}}+\dots$ |  $g_{AB}=\gamma_{AB}+\frac{C_{AB}}{r}+\frac{\gamma_{AB}C_{CD}C^{CD}}{4r^{2}}+\frac{E_{AB}}{r^{3}}+\dots$,
|  | $D_{A}U_{\left(0\right)B}+D_{B}U_{\left(0\right)A}-\gamma_{AB}D_{C}U_{\left(0\right)}^{C}=\frac{\Lambda}{3}C_{AB}$
Radiation field vanishes | Yes | No
at infinity |  ($U^{A}_{(0)}=0$) |  ($U^{A}_{(0)}\neq 0$)
Imprint on the |  Symmetric traceless tensor $C_{AB}$ (“Bondi News”) |  Symmetric traceless tensor $C_{AB}$ (“Bondi News”)
metric of the most | arbitrary functions of the retarded time and | arbitrary functions of the retarded time and
general wave | the angles (generic graviton) | the angles (generic graviton)
Symmetry | Infinite-dimensional (“BMS”) Lie algebra: | Four-dimensional Lie algebra:
| $so\left(3,1\right)+\text{``supertranslations''}$ | $so\left(3\right)\oplus\mathbb{R}$
Energy (“Bondi mass”) | $E=\frac{1}{4\pi G}\oint d^{2}S\,M$ | $E=\frac{1}{4\pi G}\oint d^{2}S\,M$
Angular momentum | $\vec{J}=\frac{1}{8\pi G}\oint d^{2}S\,\hat{r}\epsilon^{AB}D_{A}N_{B}$ | $\vec{J}=\frac{1}{8\pi G}\oint d^{2}S\,\hat{r}\epsilon^{AB}D_{A}N_{B}$
Angular momentum | Yes | No
ambiguity | (angular momentum not invariant | (there are no supertranslations)
| under supertranslations) |
Energy flux | $\frac{dE}{du}=-\frac{1}{32\pi G}\oint d^{2}S\,N_{AB}N^{AB},$ | $\frac{dE}{du}=-\frac{1}{32\pi G}\oint d^{2}S\,\left[N_{AB}^{\left(\Lambda\right)}N^{\left(\Lambda\right)AB}+\frac{2\Lambda}{3}C^{AB}C_{AB}\right.$
|  with $N_{AB}:=\dot{C}_{AB}$ | $\qquad-\frac{\Lambda}{6}C^{AB}D^{2}C_{AB}+\frac{7\Lambda^{2}}{144}\left(C^{AB}C_{AB}\right)^{2}-\frac{\Lambda^{2}}{3}C^{AB}E_{AB}$
|  | $\qquad\left.+\left(4M+D_{A}D_{B}C^{AB}\right)\left(D_{C}U_{\left(0\right)}^{C}\right)\right],$
|  |  with $N_{AB}^{\left(\Lambda\right)}:=\dot{C}_{AB}+\mathcal{L}_{U_{\left(0\right)}}C_{AB}-\frac{1}{2}\left(D_{C}U_{\left(0\right)}^{C}\right)C_{AB}$
|  | $\qquad\qquad\qquad\qquad-\frac{\Lambda}{6}\gamma_{AB}C_{CD}C^{CD}$
Inputs to arrive at a | Equations of motion (asymptotic form of the | Equations of motion (asymptotic form of the
formula for the mass | solution should include the generic graviton) | solution should include the generic graviton)
and its variation |  |
(energy flux) | Mass should reduce to known expressions when | Mass should reduce to known expressions when
| there is no radiation | there is no radiation
| Energy flux should be negative or zero | Action principle should be well-defined

##  II Bondi revisited for $\Lambda>0$

Although the geometry is very different for $\Lambda>0$ and $\Lambda=0$, it
turns out that in the natural extension of the coordinate system used by Bondi
and Sachs the formulas for energy flux, energy, and the like turn out to be
remarkably simple, and furthermore reduce for $\Lambda=0$ to theirs. For this
reason, we will go right away into the analysis in that particular coordinate
system. In Appendix A, we show how these coordinates can be obtained from a
more geometric approach à la Penrose.

###  II.1 Asymptotic behavior of the metric

In the coordinate system $\left(u,r,\theta,\phi\right)$ originally introduced
by Bondi (Bondi:1960jsa, ) and generalized later to the non-axisymmetric case
by Sachs (Sachs:1962wk, ; Sachs:1962zza, ), the line element reads

$\displaystyle ds^{2}$
$\displaystyle=e^{2\beta}\frac{V}{r}\;du^{2}-2e^{2\beta}\;dudr$
$\displaystyle+r^{2}g_{AB}\left(dx^{A}-U^{A}\;du\right)\left(dx^{B}-U^{B}\;du\right)\;$
(1)

with $-\infty<u<\infty$ and $0<r<\infty$. The $x^{A}$ are coordinates on the
two-sphere, which we choose here to be the standard spherical one:
$x^{A}=\left(\theta,\phi\right)$ with $0\leq\theta\leq\pi$ and
$0\leq\phi<2\pi$.222Strictly speaking, one of course needs two charts to cover
the 2-sphere. The coordinate $u$ is null because when $du=0$ and $dx^{A}=0$,
one has $ds^{2}=0$. Radiation is “observed” as $r\rightarrow\infty$. In this
limit, one approaches the future boundary — often denoted by $\mathcal{I}$.
These coordinates nicely encode that the topology of $\mathcal{I}$ is
$\mathbb{R}\times\mathbb{S}^{2}$, which is the relevant setting for studying
gravitational radiation emitted by compact sources.

The functions $\beta$, $V$, $g_{AB}$ and $U^{A}$ depend on $x^{A}$, $u$, and
$r$. The procedure is to expand the metric components in powers of $r^{-1}$,
demand reasonable boundary conditions and impose Einstein’s equations order by
order in $r$. The latter step does not restrict the dependence on $u$ and
$x^{A}$, but leads to relationships between different coefficients in the
expansion. We will omit the details of this calculation and state the result
to the order needed for the determination of possible asymptotic “charges,”
and their fluxes.

One finds

$\displaystyle\beta$ $\displaystyle=-\frac{1}{32r^{2}}C^{AB}C_{AB}$
$\displaystyle\;\;+\frac{1}{128r^{4}}\left(\left(C^{AB}C_{AB}\right)^{2}-12C^{AB}E_{AB}\right)+\ldots,$
(2a) $\displaystyle V$ $\displaystyle=\frac{\Lambda
r^{3}}{3}-D_{A}U_{\left(0\right)}^{A}\;r^{2}-\left(1+\frac{\Lambda}{16}C^{AB}C_{AB}\right)r$
$\displaystyle\qquad+2M+\ldots$ (2b) $\displaystyle U^{A}$
$\displaystyle=U_{(0)}^{A}-\frac{1}{2r^{2}}D_{B}C^{AB}$
$\displaystyle\qquad-\frac{2}{3r^{3}}\left(N^{A}-\frac{1}{2}C^{AB}D^{C}C_{BC}\right)+\ldots,$
(2c) $\displaystyle g_{AB}$
$\displaystyle=\gamma_{AB}+\frac{C_{AB}}{r}+\frac{C^{CD}C_{CD}\gamma_{AB}}{4r^{2}}+\frac{E_{AB}}{r^{3}}+\ldots,$
(2d) $\displaystyle\det g_{AB}$ $\displaystyle=\sin^{2}\theta.$ (2e)

These expressions depend on $\Lambda$ explicitly in Eq. (2b) and through
$U_{(0)}^{A}\left(\Lambda\right)$, which vanishes for $\Lambda=0$, but depends
implicitly on it according to Eq. (4) below. When $\Lambda=0$, they reduce to
those of Sachs. Here $D_{A}$ is the covariant derivative with respect to the
metric of the unit two-sphere $\gamma_{AB}$. The indices $A,B$ are lowered and
raised with the metric $\gamma_{AB}$. The symmetric tensors $C_{AB}$ and
$E_{AB}$ are traceless: $\gamma^{AB}C_{AB}=\gamma^{AB}E_{AB}=0$.

Besides Eqs. (2), there are two further restrictions on the coefficients which
are of decisive importance in the analysis. They are the following: (i) The
zeroth order term in $g_{AB}$ is demanded to be the standard line element on
the unit two-sphere:

$\gamma_{AB}dx^{A}dx^{B}=d\theta^{2}+\sin^{2}\theta d\phi^{2}.$ (3)

This additional demand, imposed by Bondi, which does not follow from
Einstein’s equations and it is not a mere restriction on the coordinate
system, turns out to be of enormous consequence: It will guarantee later on
that no divergent quantities appear in the analysis of a problem that has no
physical singularities. In contradistinction, Eq. (2e) can be imposed to all
orders by a change of coordinates $r=f\left(r^{\prime},\theta,\phi\right)$.
(ii) Besides the relations between the coefficients in Eqs. (2), Einstein’s
equations imply

$2D_{(A}U_{B)}^{(0)}-\gamma_{AB}D_{C}U_{(0)}^{C}=\frac{\Lambda}{3}\;C_{AB}\;.$
(4)

Equation (4) exhibits the key difference in the imprint of the gravitational
wave on the metric for $\Lambda=0$ versus $\Lambda\neq 0$. In fact, as we will
see, the tensor $C_{AB}$ describes the field of the wave, and we see from (4)
that when $\Lambda=0$, the waves do not affect the metric to the lowest order.
However, when $\Lambda\neq 0$ the wave affects the metric even to the lowest
order through the shift vector $U_{\left(0\right)}^{A}$.

Note that the particular solution to Eq. (4) exclusively exhibits modes with
$\ell\geq 2$ that are inherited from the tensor $C_{AB}$. The information of
the gravitational wave is exclusively contained within these modes. On the
other hand, the solution of the homogeneous equation, specifically the
conformal Killing equation on the 2-sphere, only has $\ell=1$ modes and are
independent of the wave degrees of freedom. These latter modes represent the
freedom in selecting the frame at infinity and can be set to zero without loss
of generality.

_Remark._ The fact that no regularization is needed at any step in the present
work and that, in particular, all the charges are finite follows from allowing
a generic $U^{A}_{(0)}\neq 0$. Had we imposed $U^{A}_{(0)}=0$, we would have
been forced to led $\gamma_{AB}$ be a generic metric, but divergences would
appear.

###  II.2 Asymptotic symmetries for $\Lambda=0$ and $\Lambda>0$ compared and
contrasted

####  II.2.1 Mass for $\Lambda=0$

When the cosmological constant vanishes, Bondi proposed that the integral over
a two-sphere of the coefficient $M\left(u,\theta,\phi\right)$ appearing in Eq.
(2b)

$E=\frac{1}{4\pi G}\oint d^{2}S\;M,$ (5)

is the total energy of the system (with $d^{2}S=\sin\theta\,d\theta d\phi$).
To validate this guess, he observed first that for the static Schwarzschild
solution, $M$ was indeed the Schwarzschild mass. Then he moved on to
investigate dynamical cases with gravitational waves, when the integral of $M$
over a large sphere was expected to diminish as a function of $u$ due to an
energy flux emitted by a source within the sphere and going out to infinity
(the coordinate $u$ is a retarded coordinate because the sign of the $dudr$
term in the line element is negative). This crucial test was satisfied because
one can verify, from Einstein’s equations, that

$\frac{dE}{du}=-\frac{1}{32\pi G}\oint
d^{2}S\,N_{AB}N^{AB}<0\qquad(\Lambda=0).$ (6)

The mass expression in Eq. (5) has later also been derived using other methods
such as the Landau-Lifschitz approach based on a pseudo-tensor (see e.g.
Thorne:1980ru ) and covariant phase space methods (see e.g. Barnich:2011mi ;
Flanagan:2015pxa ).

####  II.2.2 Angular momentum for $\Lambda=0$

If one were to attempt guessing an expression for the angular momentum, one
would naturally focus on the shift $N_{A}$ because it carries the imprint of
being “stationary” (versus static). One would need a two-form to integrate
over the sphere constructed out of this shift. The simplest candidate is its
exterior derivative. So, one would write

$\displaystyle\vec{J}$ $\displaystyle=\frac{1}{8\pi G}\oint
d^{2}S\;\hat{r}\epsilon^{AB}D_{A}N_{B}.$ (7)

The first test would be to check if this formula gives the right value for the
angular momentum of the Kerr-de Sitter solution (which can be brought to
satisfy the boundary conditions in Eq. (2), see Sec. V.3). If one does so, one
finds that indeed the test is passed. One does not expect the angular momentum
flux to have a definite sign so that test is not available, but a complete
analysis of the asymptotically defined symmetries confirms its validity. The
vector $N^{A}$ is referred to as “angular momentum aspect”.333Beware,
conventions differ on the exact definition of the angular-momentum aspect:
some authors shift $N^{A}$ by terms proportional to $C_{AB}$ and its
derivatives, and/or multiply it by a numerical factor.

####  II.2.3 Symmetry for $\Lambda=0$

In order to prove that Eq. (5) and (7) are the energy and the angular
momentum, one needs to show that they generate time translations and spatial
rotations at infinity when acting on phase space. That proof, and much more,
was given by Sachs who, in a brilliant analysis did two things: (i) He
discovered, extending previous work of Bondi, Metzner and Van der Burg, that
the asymptotically defined symmetry is enormously larger than the expected
Poincaré group, and that the commutators of its Killing vectors form an
infinite-dimensional Lie algebra now called the Bondi-Metzner-Sachs algebra
(Sachs:1962wk, ). (ii) He _postulated_ a commutation rule for the two
independent components of the news $C_{AB}$ and showed that, with just that,
$M\left(\theta,\phi\right)$ and the Lorentz generators $J_{\mu\nu}$ that he
also constructed, generate the symmetry algebra (Sachs:1962zza, ). In
particular, the zero mode (5) generates time translations. By guessing the
commutation rule, Sachs did not need to use the action principle, but just the
equations of motion. Later developments have permitted to recover the
canonical generators of the Bondi-Metzner-Sachs algebra from the action
principle Barnich:2011mi ; Henneaux:2018cst ; Bunster:2018yjr .

####  II.2.4 Symmetry for $\Lambda>0$

For $\Lambda=0$, besides the energy and angular momentum, one has boosts
$\vec{K}$ and infinitely many supertranslation generators
$M\left(\theta,\phi\right)$, with spherical modes $\ell\geq 1$. The situation
is dramatically different for $\Lambda\neq 0$, in which case only $E$ and
$\vec{J}$ are present. The complete asymptotic symmetry algebra consists just
of time translations and spatial rotations, and _the expressions for the
generators are the same as for $\Lambda=0$_. This is why we have brought them
out especially above.

##  III Regge-Teitelboim analysis of the symmetries for $\Lambda\neq 0$

###  III.1 Preservation of the asymptotic behavior of the metric

Since $\Lambda$ does not appear explicitly in the asymptotic form (1) of the
metric, the form of the asymptotic Killing vectors for $\Lambda\neq 0$ is the
same as the one given by Sachs for $\Lambda=0$ (his equations III5-7 in
(Sachs:1962zza, )), that is

$\displaystyle\xi^{u}$ $\displaystyle=T\left(u,x^{A}\right),$ (8a)
$\displaystyle\xi^{r}$
$\displaystyle=-\frac{r}{2}\left(D_{A}X^{A}+D_{A}I^{A}-U^{A}D_{A}T\right),$
(8b) $\displaystyle\xi^{A}$
$\displaystyle=X^{A}\left(u,x^{A}\right)+I^{A}\left(u,r,x^{A}\right)$ (8c)
$\displaystyle\text{with}\;\;I^{A}=-\left(D_{B}T\right)\int_{r}^{\infty}dr^{\prime}\left(\frac{e^{2\beta}}{r^{2}}g^{AB}\right).$
(8d)

The preservation of Eq. (4) under the action of the asymptotic Killing vectors
implies that $X^{A}$ must obey the following differential equation

$2D_{(A}X_{B)}-\gamma_{AB}D_{C}X^{C}=2U^{(0)}_{(A}D_{B)}T-\gamma_{AB}U^{C}_{(0)}D_{C}T.$
(9)

The preservation of the fall-off of the metric also requires that the
parameters $T$ and $X^{A}$ obey the following first order differential
equations in time

$\displaystyle\dot{T}$
$\displaystyle=\frac{1}{2}D_{A}X^{A}-\frac{3}{2}U_{(0)}^{A}D_{A}T\,,$ (10)
$\displaystyle\dot{X}^{A}$
$\displaystyle=\dot{T}U^{A}_{(0)}-U^{A}_{(0)}U^{B}_{(0)}D_{B}T-\frac{\Lambda}{3}D^{A}T\,.$
(11)

In particular, Eq. (10) is obtained from the preservation of the decay of the
$g_{ur}$ component, and Eq. (11) from the $g_{uA}$ component. Eqs. (9)-(11)
constrain the algebra to three rotations and the time translation as we will
see in the next subsection.

###  III.2 Symmetry algebra

The symmetry algebra is determined from $T$ and $X^{A}$ satisfying Eqs.
(9)-(11). Eq. (9) constraints $T$ tremendously: from a generic function of
$u,\theta,\phi$ to a function of $u$ only. Eq. (10) then further requires that
$T$ is time-independent, so that $T$ can only be a constant. Using this, we
find that there are only three independent solutions for $X^{A}$ describing
exactly the three rotations on the sphere. We will now show in detail how this
comes about.

To analyze Eq. (9), it is useful to introduce $Y^{A}$

$\displaystyle Y^{A}$ $\displaystyle=X^{A}-U_{\left(0\right)}^{A}T\;,$ (12)

which explicitly separates a “frame rotation” at infinity. In which case, we
get

$2D_{(A}Y_{B)}-\gamma_{AB}D_{C}Y^{C}=-\frac{\Lambda}{3}TC_{AB}.$ (13)

This equation has the same form as the one obeyed by the zero order shift (Eq.
(4)), except for a negative sign — which is just a matter of convention in the
definition of $Y^{A}$ — and the appearance of the factor $T$ on the right-hand
side. Decomposing $Y^{A}$ into vector spherical harmonics, we see that the
left-hand side of Eq. (13) contains no $\ell=1$ modes as these are in the
kernel of the conformal Killing operator. Therefore, the right-hand side
cannot contain any $\ell=1$ modes. Decomposing $T$ and $C_{AB}$ into spin-
weighted spherical harmonics

$\displaystyle T$ $\displaystyle=\sum_{\ell,m}T_{\ell m}(u)\;{}_{0}Y_{\ell m}$
(14) $\displaystyle C_{AB}$ $\displaystyle=\sum_{\ell,m}C^{E}_{\ell
m}(u)\left({}_{-2}Y_{\ell m}m_{A}m_{B}+{}_{2}Y_{\ell
m}\bar{m}_{A}\bar{m}_{B}\right)$ $\displaystyle\qquad-iC^{B}_{\ell
m}(u)\left({}_{-2}Y_{\ell m}m_{A}m_{B}-{}_{2}Y_{\ell
m}\bar{m}_{A}\bar{m}_{B}\right)$ (15)

where $m_{A},\bar{m}_{A}$ are complex null vectors on the two-sphere
satisfying $m^{A}\bar{m}_{A}=1$, we find that if we project onto
$\bar{m}^{A}\bar{m}^{B}$, their product can be written as

$\displaystyle T\bar{m}^{A}\bar{m}^{B}C_{AB}=\sum_{\ell,m}\mathcal{C}_{\ell
m}\;{}_{-2}Y_{\ell m}\;.$ (16)

So we need to determine what the constraints on $T_{\ell m}$ are such that
$\mathcal{C}_{\ell m}$ does not contain any $\ell=1$ modes. We find that

$\displaystyle\mathcal{C}_{\ell m}$
$\displaystyle=\sum_{\ell^{\prime},m^{\prime},\ell^{\prime\prime},m^{\prime\prime}}T_{\ell^{\prime}m^{\prime}}\left(C^{E}_{\ell^{\prime\prime}m^{\prime\prime}}-iC^{B}_{\ell^{\prime\prime}m^{\prime\prime}}\right)\times$
$\displaystyle\qquad\qquad\qquad\int
d^{2}S\;{}_{0}Y_{l^{\prime}m^{\prime}}\;{}_{-2}Y_{\ell^{\prime\prime}m^{\prime\prime}}{}_{-2}\bar{Y}_{\ell
m}$ (17)
$\displaystyle=\sum_{\ell^{\prime},m^{\prime},\ell^{\prime\prime},m^{\prime\prime}}\sqrt{\frac{(2\ell+1)(2\ell^{\prime}+1)(2\ell^{\prime\prime}+1)}{4\pi}}\times$
$\displaystyle\qquad\qquad\qquad(-1)^{m}T_{\ell^{\prime}m^{\prime}}\left(C^{E}_{\ell^{\prime\prime}m^{\prime\prime}}-iC^{B}_{\ell^{\prime\prime}m^{\prime\prime}}\right)\times$
$\displaystyle\qquad\qquad\qquad\begin{pmatrix}\ell&\ell^{\prime}&\ell^{\prime\prime}\\\
m&m^{\prime}&m^{\prime\prime}\end{pmatrix}\begin{pmatrix}\ell&\ell^{\prime}&\ell^{\prime\prime}\\\
-2&0&2\end{pmatrix}.$ (18)

where in going from Eq. (17) to (18), we used that ${}_{s}\bar{Y}_{\ell
m}=(-1)^{m+s}{}_{-s}Y_{\ell m}$ and that the integral over three spin-weighted
spherical harmonics is given by the product of two 3$j$-symbols (NIST:DLMF, ,
Eq. (34.3.22)). Spin-weighted spherical harmonics ${}_{s}Y_{\ell m}$ are not
defined for $|s|>\ell$ so $C^{E}_{\ell m}/C^{B}_{\ell m}$ does not have any
modes with $\ell=0$ or $1$. Hence, $\mathcal{C}_{\ell m}$ contains no $\ell=1$
modes only if $T_{\ell m}$ is non-zero for $\ell=0$, because $C^{E}_{\ell
m}/C^{B}_{\ell m}$ is generically non-zero for $\ell\geq 2$ and the
3$j$-symbols are non-zero when
$|\ell^{\prime}+\ell^{\prime\prime}|\leq\ell\leq\ell^{\prime}+\ell^{\prime\prime}$.

So far, we have seen that $T(u,\theta,\phi)=T_{0}(u)$ and $X^{A}$ satisfies
the conformal Killing equation. Substituting this into Eq. (10), we obtain

$\dot{T}_{0}=\frac{1}{2}D_{A}X^{A}.$ (19)

The only consistent solution is if both sides of the equation vanish
independently. Hence, we find that $T_{0}$ is $u$-independent and $X^{A}$ is
also divergence-free. Finally, From Eq. (11), we obtain that $X^{A}$ is time-
independent. Therefore, we find that $T$ and $X^{A}$ are

$T=T_{0}\qquad\text{and}\qquad
X^{A}=\epsilon^{AB}D_{B}\left(\vec{\Omega}\cdot\hat{r}\right),$ (20)

for constant $T_{0}$ and $\vec{\Omega}$. Substituting this back into the form
of the asymptotic Killing vector fields, we obtain

$\displaystyle\xi^{u}$ $\displaystyle=T_{0},$ (21a) $\displaystyle\xi^{r}$
$\displaystyle=0,$ (21b) $\displaystyle\xi^{A}$
$\displaystyle=\epsilon^{AB}D_{B}\left(\vec{\Omega}\cdot\hat{r}\right).$ (21c)

One immediately recognizes this as the $\mathbb{R}\oplus so\left(3\right)$
algebra. This result generalizes the findings in (abk1, ), where it was shown
that the asymptotic symmetry group is exactly the 4-dimensional group of time
translations and rotations when $\mathcal{I}$ has
$\mathbb{R}\times\mathbb{S}^{2}$ topology _and_ the induced metric at
$\mathcal{I}$ is conformally flat. The requirement of conformal flatness,
which severely restricted the allowed gravitational radiation by essentially
cut the degrees of freedom of the gravitational field in half, can be lifted.

##  IV Revisiting Regge-Teitelboim for $\Lambda>0$ and radiation at future
space-like infinity

###  IV.1 Charges

The variation of the charge is obtained using the covariant approach of
Barnich and Brandt Barnich:2001jy , which — as they proved — is equivalent to
the standard Regge-Teitelboim analysis (Regge:1974zd, ).444Note that it is
also equivalent to the Wald-Zoupas method Lee:1990nz ; Wald:1999wa  for an
appropriate choice of boundary terms (see e.g. Compere:2018aar ), as well as
to the one of Abott, Deser and Tekin Abbott:1981ff ; Deser:2002rt ;
Deser:2002jk . In particular, if $h_{\mu\nu}:=\delta g_{\mu\nu}$ corresponds
to the functional variation of the spacetime metric, then the general
expression for the variation of the charge is given by

$\displaystyle\delta_{\xi}Q$ $\displaystyle=\frac{1}{16\pi
G}\oint_{\mathcal{I}}\left(d^{2}x\right)_{\mu\nu}\left[\xi^{\nu}\nabla^{\mu}h-\xi^{\nu}\nabla_{\sigma}h^{\mu\sigma}+\xi_{\sigma}\nabla^{\nu}h^{\mu\sigma}\right.$
$\displaystyle\left.\qquad+\frac{1}{2}h\nabla^{\nu}\xi^{\mu}+\frac{1}{2}h^{\nu\sigma}\left(\nabla^{\mu}\xi_{\sigma}-\nabla_{\sigma}\xi^{\mu}\right)-\left(\mu\leftrightarrow\nu\right)\right],$
(22)

where $\xi^{\mu}$ is the asymptotic Killing vector, $h:=g^{\mu\nu}h_{\mu\nu}$
and the volume form is

$\left(d^{2}x\right)_{\mu\nu}:=\frac{1}{4}\epsilon_{\mu\nu\lambda\sigma}dx^{\lambda}\wedge
dx^{\sigma}.$ (23)

Applying this to our set-up, we find

$\displaystyle\delta_{\xi}Q$ $\displaystyle=\frac{1}{16\pi
G}\oint_{\mathcal{I}}d^{2}S\,\left[T\delta\left(4M\right)+\frac{T}{2}N_{AB}^{\left(\Lambda\right)}\delta
C^{AB}\right.$
$\displaystyle\left.+X^{A}\delta\left(2N_{A}+\frac{1}{16}D_{A}\left(C_{BD}C^{BD}\right)\right)\right.$
$\displaystyle\left.-TU^{A}_{(0)}\delta\left(2N_{A}+\frac{1}{16}D_{A}\left(C_{BD}C^{BD}\right)\right)\right]\;,$
(24)

where the tensor $N_{AB}^{\left(\Lambda\right)}$ is defined by

$\displaystyle N_{AB}^{\left(\Lambda\right)}$
$\displaystyle:=\dot{C}_{AB}+\mathcal{L}_{U_{\left(0\right)}}C_{AB}-\frac{1}{2}\left(D_{C}U_{\left(0\right)}^{C}\right)C_{AB}$
$\displaystyle-\frac{\Lambda}{6}\gamma_{AB}C_{CD}C^{CD}.$ (25)

$N_{AB}^{(\Lambda)}$ generalizes the Bondi News tensor when the cosmological
constant is non-zero. This expression acquires a similar structure as the one
obtained in (Barnich:2011mi, ) for the asymptotically flat case with $M$
playing the role of the “Bondi mass aspect” and $N^{A}$ that of the “angular
momentum aspect”. However, there are some differences that come from the
presence of a non-zero cosmological constant. Apart from the correction coming
from $\Lambda$ in the “News tensor” $N_{AB}^{(\Lambda)}$ in Eq. (25), there is
an additional non-integrable term proportional to $U^{A}_{(0)}$ that vanishes
in the limit when $\Lambda\rightarrow 0$.

It is worth emphasizing that the variation of the charge is finite in the
limit when $r\rightarrow\infty$, _without the need of any ad-hoc
regularization procedure_. The only potentially divergent terms were those
proportional to $r$, which after some appropriate integration by parts on the
sphere acquire the form

$\displaystyle\delta_{\xi}Q_{\text{div}}$ $\displaystyle=-\frac{r}{32\pi
G}\oint d^{2}S\Big{(}2D_{(A}X_{B)}-\gamma_{AB}D_{C}X^{C}$
$\displaystyle-2U_{(A}^{(0)}D_{B)}T+\gamma_{AB}U^{C}_{(0)}D_{C}T$
$\displaystyle-T\left[2D_{(A}U^{(0)}_{B)}-\gamma_{AB}D_{C}U^{C}_{(0)}-\frac{\Lambda}{3}C_{AB}\right]\Big{)}\delta
C^{AB},$ (26)

which, by virtue of Eqs. (4) and (9), vanishes identically.

Thus, if one assumes that $\delta T_{0}=\delta\vec{\Omega}=0$, then the
integrable part (in the functional sense) of the variation of the charge,
takes the form

$\displaystyle Q^{\text{int}}[T_{0},\vec{\Omega}]$
$\displaystyle=T_{0}\;E+\vec{\Omega}\cdot\vec{J},$ (27)

where the energy $E$ and angular momentum $\vec{J}$ are

$\displaystyle E$ $\displaystyle=\frac{1}{4\pi G}\oint d^{2}S\;M,$ (28)
$\displaystyle\vec{J}$ $\displaystyle=\frac{1}{8\pi G}\oint
d^{2}S\;\hat{r}\epsilon^{AB}D_{A}N_{B}.$ (29)

Note that the term proportional to $T$ in the last line of Eq. (24) does not
contribute to the mass, because it does not contain any $\ell=0$ modes (this
can again be seen from an analysis of the $3j$-symbols and noting that
$U_{A}^{(0)}$ is only non-zero for $\ell\geq 2$). As we will show in Sec. V.3,
these expressions give the expected results for the mass and angular momentum
for the Kerr-de Sitter geometry, and allows to extend to notion of energy and
angular momentum to the case when gravitational waves are present.

###  IV.2 Fluxes

The fluxes of energy and angular momentum can be directly obtained by taking
the time derivative of Eqs. (28) and (29) in conjunction with Einstein’s
equation. In particular, Einstein’s equations yield the evolution of $M$ and
$N^{A}$, respectively. The resulting expressions are rather long but
manageable:

$\displaystyle\dot{M}$
$\displaystyle=\frac{1}{4}D_{A}D_{B}N_{(\Lambda)}^{AB}-\frac{1}{8}N_{AB}^{(\Lambda)}N_{(\Lambda)}^{AB}+\frac{\Lambda}{96}C^{AB}D^{2}C_{AB}$
$\displaystyle-\frac{\Lambda}{12}C^{AB}C_{AB}-\frac{\Lambda}{6}D_{A}N^{A}-\frac{\Lambda}{96}\left(D_{C}C_{AB}\right)\left(D^{C}C^{AB}\right)$
$\displaystyle+\frac{\Lambda^{2}}{24}C^{AB}E_{AB}-\frac{7\Lambda^{2}}{1152}\left(C^{AB}C_{AB}\right)^{2}$
$\displaystyle-
U_{(0)}^{A}D_{A}M-\frac{3}{2}MD_{A}U_{(0)}^{A}-\frac{1}{8}C^{AB}D_{A}D_{B}D_{C}U_{(0)}^{C}$
(30)

and

$\displaystyle\dot{N}^{A}$
$\displaystyle=D^{A}M+\frac{1}{4}D^{A}D^{B}D^{C}C_{BC}-\frac{1}{4}D_{B}D^{2}C^{AB}$
$\displaystyle+\frac{5}{16}C^{AB}D^{C}N_{BC}^{(\Lambda)}-\frac{3}{16}C_{BC}D^{B}N_{(\Lambda)}^{AC}-\frac{\Lambda}{2}D_{B}E^{AB}$
$\displaystyle-\frac{1}{2}N_{(\Lambda)}^{AB}D^{C}C_{BC}+\frac{1}{16}N_{(\Lambda)}^{BC}D^{A}C_{BC}+D_{B}C^{AB}$
$\displaystyle+\frac{5\Lambda}{32}C_{BD}C^{CD}D_{C}C^{AB}+\frac{7\Lambda}{48}C^{AB}C^{CD}D_{B}C_{CD}$
$\displaystyle-
U_{(0)}^{B}D_{B}N^{A}+N^{B}D_{B}U_{(0)}^{A}-2N^{A}D_{C}U_{(0)}^{C}$
$\displaystyle-\frac{1}{64}U_{(0)}^{A}\left(C_{BD}C^{BD}\right)-\frac{1}{64}\left(D^{2}U_{(0)}^{A}\right)C_{BD}C^{BD}$
$\displaystyle+\frac{1}{32}D^{A}\left(D_{C}U_{(0)}^{C}\right)\left(C_{BD}C^{BD}\right)\;.$
(31)

The energy flux is given by

$\displaystyle\frac{dE}{du}$ $\displaystyle=-\frac{1}{32\pi G}\oint
d^{2}S\left\\{N_{AB}^{(\Lambda)}N^{(\Lambda)AB}+\frac{2\Lambda}{3}C^{AB}C_{AB}\right.$
$\displaystyle-\frac{\Lambda}{6}C^{AB}D^{2}C_{AB}+\frac{7\Lambda^{2}}{144}\left(C^{AB}C_{AB}\right)^{2}-\frac{\Lambda^{2}}{3}C^{AB}E_{AB}$
$\displaystyle\left.+\left(4M+D_{A}D_{B}C^{AB}\right)\left(D_{C}U_{\left(0\right)}^{C}\right)\right\\}.$
(32)

The first term on the right-hand side has the same form as the one that
contributes to the loss of energy in the asymptotically flat case. However,
there are now also corrections coming from the presence of the cosmological
constant which are up to fourth order in the fields. These higher order terms
are characteristic of the full nonlinear theory and cannot be seen in the
linearized approximation. In Sec. V.4.1, we will show that when the higher
order terms are neglected, the total amount of energy radiated in a certain
interval of time precisely coincide with the one reported in (abk2, ;
Chrusciel:2020rlz, ; Kolanowski:2020wfg, ). An important difference with the
asymptotically flat case is that the flux of energy is not manifestly
negative. This was also observed for the case of homogeneous gravitational
perturbations on a de Sitter background in (abk2, ). Moreover, this can also
occur for Maxwell fields on a de Sitter background (abk2, ) and thus seems a
rather generic feature of spacetimes with $\Lambda>0$. This is likely due to
the fact that there is no global time-like Killing vector field in de Sitter
spacetime. However, as was pointed out in (abk3, ), and as we will show in
Sec. V.4, in the case of quadrupolar radiation in the linearized theory, the
flux of energy _is_ manifestly negative.

Analogously, the flux of angular momentum takes the form

$\frac{d\vec{J}}{du}=\frac{1}{8\pi G}\oint
d^{2}S\;\hat{r}\;\epsilon^{AB}D_{A}\dot{N}_{B},$ (33)

where $\dot{N}_{B}$ is given by Eq. (31). Due to the cosmological constant
there is no angular momentum ambiguity, because there are no abelian
supertranslations as is the case with $\Lambda=0$.

The flux of energy and angular momentum in Eqs. (32)-(33) can alternatively be
obtained from the non-integrable part of the variation of the charge in Eq.
(24) following the prescription in (Barnich:2011mi, ) (see also
(Bunster:2018yjr, ; Bunster:2019mup, )). We have verified this explicitly for
the energy flux.

##  V Application to special cases

In this section, we show explicitly that the fall off conditions in Eq. (2)
accommodate a wide range of physically interesting solutions to Einstein’s
equation. First, we discuss the de Sitter spacetime itself before moving on to
two black hole solutions in the presence of a positive cosmological constant:
the non-rotating Schwarzschild-de Sitter spacetime and the rotating Kerr-de
Sitter spacetime. Next, we discuss linearized solutions to Einstein’s
equations with $\Lambda>0$ representing gravitational radiation emitted by a
compact source. Finally, we describe a simple model of gravitational radiation
with a single degree of freedom known as the Robinson-Trautman spacetime.

###  V.1 de Sitter spacetime

The full de Sitter spacetime is not an example of the class of spacetimes we
have defined. This is not problematic, as the goal of this paper is to
describe radiation in the presence of $\Lambda$ in which case not the complete
de Sitter spacetime, but the Poincaré patch of de Sitter spacetime with an
additional point at $\mathcal{I}$ removed is relevant. The removal of this
additional point is natural as it represents the intersection of the future
boundary with the source generating radiation (see also (abk3, , Sec. II)). As
a result, the future boundary has topology $\mathbb{R}\times\mathbb{S}^{2}$
and is naturally coordinatized by $(t,r,\theta,\phi)$:

$ds^{2}=-\left(1-\frac{\Lambda
r^{2}}{3}\right)du^{2}-2dudr+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)\;.$ (34)

The time translation vector field $\frac{\partial}{\partial u}$ and the three
rotational Killing vector fields are not only asymptotic symmetries, but
symmetries of the entire spacetime. Translations and inverted translations,
which ae symmetries of the full de Sitter spacetime, do not leave $i^{0}$ and
$i^{+}$ invariant and are therefore not permissible (for a more extensive
discussion, see (abk1, )).

###  V.2 Schwarzschild-de Sitter spacetime

The simplest prototype for describing non-dynamical isolated gravitating
systems in the presence of a cosmological constant is undoubtedly the
Schwarzschild-de Sitter spacetime. This spacetime describes a non-rotating
black hole with $\Lambda>0$. We consider the metric in Eddington-Finkelstein
coordinates $(u,r,\theta,\phi)$:

$\displaystyle ds^{2}$
$\displaystyle=-\left(1-\frac{2m}{r}-\frac{r^{2}}{l^{2}}\right)du^{2}-2dudr$
$\displaystyle\;\;+r^{2}\left(d\theta^{2}+\sin^{2}\theta d\phi^{2}\right)\;,$
(35)

where $\Lambda=3l^{-2}$. The coordinate ranges are
$u\in(-\infty,\infty),r\in(0,\infty),\theta\in\left[0,\pi\right)$ and
$\phi\in\left[0,2\pi\right)$. While these coordinates do not provide a global
chart of the spacetime, they suffice to cover the asymptotic region near
$\mathcal{I}$. In terms of the asymptotic expansions of the metric in Eq. (2),
this metric has $M(u,\theta,\phi)=m$ and all other coefficients zero. In
particular, there is no gravitational radiation so that $C_{AB}$ and
$U_{(0)}^{A}$ are both zero.

This metric has four Killing vector fields: one Killing vector field generates
time translations and the other three describe the spherical symmetry of the
spacetime. A well-known property of (global) Killing vector fields is that
every Killing field of the physical spacetime admits an extension to the
boundary and is tangential to it. This is also the case for the above Killing
vector fields, which coincide exactly with the asymptotic symmetry vector
fields. All charges and fluxes vanish except for the mass, which is

$E_{{\rm Sch-dS}}=\frac{m}{G}\;.$ (36)

###  V.3 Kerr-de Sitter spacetime

Kerr-de Sitter spacetimes are stationary, vacuum solutions to Einstein’s
equations describing rotating black holes in the presence of a positive
cosmological constant. Let us consider the Kerr de-Sitter metric in standard
Boyer-Lindquist coordinates $(T,R,\Theta,\Phi)$ (Carter:1968ks, )

$\displaystyle ds^{2}$
$\displaystyle=-2a\sin^{2}\Theta\left(\frac{2mR}{a^{2}\cos^{2}\Theta+R^{2}}+\frac{a^{2}+R^{2}}{l^{2}}\right)dTd\Phi$
$\displaystyle-\left(1-\frac{2mR}{a^{2}\cos^{2}\Theta+R^{2}}-\frac{a^{2}\sin^{2}\Theta+R^{2}}{l^{2}}\right)dT{}^{2}$
$\displaystyle+\sin^{2}\Theta\left(\frac{2a^{2}mR\sin^{2}\Theta}{a^{2}\cos^{2}\Theta+R^{2}}+\left(a^{2}+R^{2}\right)\left(1+\frac{a^{2}}{l^{2}}\right)\right)d\Phi^{2}$
$\displaystyle+\left(a^{2}\cos^{2}\Theta+R^{2}\right)\left(\frac{dR^{2}}{R^{2}-\left(a^{2}+R^{2}\right)\frac{R^{2}}{l^{2}}-2mR+a^{2}}\right.$
$\displaystyle\left.+\frac{d\Theta^{2}}{1+\frac{a^{2}\cos^{2}\Theta}{l^{2}}}\right),$
(37)

where the parameter $a$ is related to the amount of rotation of this rotating
black hole. In the limit, $a\to 0$ one recovers the Schwarschild-de Sitter
metric in static coordinates. Note that these Boyer-Lindquist coordinates are
‘twisted’ at $\mathcal{I}$: for instance, surfaces of constant $T,R$ describe
deformed spheres (consequently, the range of $\Theta,\Phi$ is not the standard
range for coordinates on the sphere). Inspired by the coordinate
transformation used in (Henneaux:1985tv, ) to undo this twisting, we perform
the following asymptotic change of coordinates

$\displaystyle T$
$\displaystyle=u+l\,\text{arctanh}\left(\frac{r}{l}\right)-\frac{ml^{4}\left(1-\frac{a^{2}\sin^{2}\theta}{2l^{2}}\right)}{2\left(1+\frac{a^{2}\sin^{2}\theta}{l^{2}}\right)^{5/2}}\frac{1}{r^{4}}+\ldots$
$\displaystyle R$
$\displaystyle=r\sqrt{1+\frac{a^{2}\sin^{2}\theta}{l^{2}}}-\frac{\left(1+\frac{a^{2}}{l^{2}}\right)a^{2}\sin^{2}\theta}{2\left(1+\frac{a^{2}\sin^{2}\theta}{l^{2}}\right)^{3/2}}\frac{1}{r}$
$\displaystyle-\frac{ma^{2}\sin^{2}\theta}{2\left(1+\frac{a^{2}\sin^{2}\theta}{l^{2}}\right)^{2}}\frac{1}{r^{2}}+\ldots$
$\displaystyle\Theta$
$\displaystyle=\arccos\left(\frac{\cos\theta}{\sqrt{1+\frac{a^{2}\sin^{2}\theta}{l^{2}}}}\right)-\frac{a^{2}\sin\left(2\theta\right)\sqrt{1+\frac{a^{2}}{l^{2}}}}{4\left(1+\frac{a^{2}\sin^{2}\theta}{l^{2}}\right)^{2}}\frac{1}{r^{2}}$
$\displaystyle+\frac{3a^{4}\sin\left(2\theta\right)\left(1-2\sin^{2}\theta\left(1+\frac{a^{2}}{2l^{2}}\right)\right)\sqrt{1+\frac{a^{2}}{l^{2}}}}{16\left(1+\frac{a^{2}\sin^{2}\theta}{l^{2}}\right)^{4}}\frac{1}{r^{4}}\ldots$
$\displaystyle\Phi$
$\displaystyle=\frac{1}{1+\frac{a^{2}}{l^{2}}}\left(\phi+\frac{a\left(u+l\text{arctanh}\left(\frac{r}{l}\right)\right)}{l^{2}}\right)$
$\displaystyle+\frac{ma^{3}\sin^{2}\theta}{4\left(1+\frac{a^{2}}{l^{2}}\right)\left(1+\frac{a^{2}\sin^{2}\theta}{l^{2}}\right)^{5/2}}\frac{1}{r^{4}}+\ldots.$

The leading terms of the Kerr-de Sitter metric near $\mathcal{I}$ fit within
our asymptotic conditions. 555Note that the solution is not in the Bondi gauge
everywhere, but its asymptotic form to the orders needed is. Indeed
$g_{rr}=O\left(r^{-6}\right)$ and $g_{rA}=O\left(r^{-4}\right)$. The metric on
$\mathcal{I}$ with $u$=constant is the unit two-sphere with $\theta,\phi$
having their standard range, i.e.,
$\theta\in\left[0,\pi\right),\phi\in\left[0,2\pi\right)$ . Moreover,
$U_{\left(0\right)}^{A}$ and $C_{AB}$ are both equal to zero, which is
consistent with the fact that there is no gravitational radiation in this
spacetime. The mass and angular momentum aspect are given by:

$\displaystyle M$
$\displaystyle=m\frac{\left(1-\frac{a^{2}\sin^{2}\theta}{2l^{2}}\right)}{\left(1+\frac{a^{2}\sin^{2}\theta}{l^{2}}\right)^{5/2}}\;,$
(38) $\displaystyle N^{\theta}$ $\displaystyle=0\;,$ (39) $\displaystyle
N^{\phi}$
$\displaystyle=-\frac{3am}{\left(1+\frac{a^{2}\sin^{2}\theta}{l^{2}}\right)^{5/2}}.$
(40)

We also find that $E_{AB}$ is

$\displaystyle E_{\theta\theta}$
$\displaystyle=-\frac{ma^{2}\sin^{2}\theta}{\left(1+\frac{a^{2}\sin^{2}\theta}{l^{2}}\right)^{5/2}}\;,$
(41) $\displaystyle E_{\phi\phi}$
$\displaystyle=\frac{ma^{2}\sin^{4}\theta}{\left(1+\frac{a^{2}\sin^{2}\theta}{l^{2}}\right)^{5/2}}\;,$
(42) $\displaystyle E_{\theta\phi}$ $\displaystyle=0\;,$ (43)

so that $E_{AB}$ is traceless with respect to the unit two-sphere metric, as
it should.

The mass and the angular momentum can be directly computed from the
expressions for the charges in Eqs. (28) and (29) (which also define the
normalization of the Killing vectors here). They are given by

$E_{{\rm{Kerr-dS}}}=\frac{m}{G}\left(1+\frac{a^{2}}{l^{2}}\right)^{-2},\qquad
J_{z}=-a\;E_{{\rm{Kerr-dS}}}.$ (44)

These results coincide with the charges obtained using Hamiltonian methods by
Marolf and Kelly in (Kelly:2012zc, ).666These final expressions also agree
with the gravitational charges defined in terms of the electric part of the
Weyl tensor in (abk1, ) despite the fact that the mass and angular momentum
there refer to a differently normed Killing vector field. This is due to the
fact that in (abk1, ), the $\Theta,\Phi$ coordinates were assumed to have the
standard range on the two-sphere, which is not the case. If this is corrected,
the results here and in (abk1, ) differ exactly by the expected scaling with
the Killing vector field. Moreover, these expression also precisely coincide
with the ones obtained for Kerr-_anti_ -de Sitter spacetimes after replacing
$l\rightarrow il$ (Henneaux:1985tv, ). Since this spacetime is stationary, the
fluxes are trivially zero, which we verified by direct computation.

###  V.4 Linearized solutions in de Sitter spacetime

####  V.4.1 Linearized charges and fluxes

The expressions for the charges and fluxes simplify drastically in the
linearized context. Here, we will briefly comment on the linearized setting
and explicitly connect the resulting flux of energy radiated across
$\mathcal{I}$ to existing results in the literature.

Let us consider the linearized gravitational field in retarded null
coordinates $\left(u,r,x^{A}\right)$ around the de Sitter background metric

$d\bar{s}^{2}=-\left(1-\frac{\Lambda
r^{2}}{3}\right)du^{2}-2dudr+r^{2}\left(d\theta^{2}+\sin^{2}d\phi^{2}\right).$
(45)

The spacetime metric is then written as

$g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}\;.$ (46)

where $h_{\mu\nu}$ is kept only up to first order. Quantities that have
dimensions of length are compared with an external fixed length scale. The
linearization expression (46) is valid everywhere, not just asymptotically.
The fall-off of the metric in the linearized theory can be directly obtained
from our asymptotic conditions in Eq. (2) by neglecting the terms that are
quadratic in the fields. The asymptotic form of the metric then reads

$\displaystyle\beta$ $\displaystyle=\mathcal{O}\left(\frac{1}{r^{5}}\right),$
(47a) $\displaystyle V$ $\displaystyle=\frac{\Lambda
r^{3}}{3}+r+\left(-D_{A}U_{\left(0\right)}^{A}\;r^{2}+2M+\ldots\right),$ (47b)
$\displaystyle U^{A}$
$\displaystyle=\left(U_{(0)}^{A}-\frac{D_{B}C^{AB}}{2r^{2}}-\frac{2N^{A}}{3r^{3}}+\ldots\right),$
(47c) $\displaystyle g_{AB}$
$\displaystyle=\gamma_{AB}+\left(\frac{C_{AB}}{r}+\frac{E_{AB}}{r^{3}}+\ldots\right),$
(47d)

where $U_{(0)}^{A}$ obeys Eq. (4). Note that $U_{A}^{(0)}$, which is of order
zero in the asymptotic expansion, in which the reference length is $r$,
becomes of first order in the linearized theory, in which the reference length
is a fixed distance. So, both expansions do not coincide at large distances.
The time derivatives of $M$ and $N^{A}$ reduce to

$\displaystyle\dot{M}$
$\displaystyle=\frac{1}{2}D_{A}D_{B}N_{(\Lambda)}^{AB}-\frac{\Lambda}{6}D_{A}N^{A},$
(48) $\displaystyle\dot{N}^{A}$
$\displaystyle=D^{A}M+\frac{1}{4}D^{A}D^{B}D^{C}C_{BC}-\frac{1}{4}D_{B}D^{2}C^{AB}$
$\displaystyle\;\;-\frac{\Lambda}{2}D_{B}E^{AB}+D_{B}C^{AB},$ (49)

while the linearized version of the News tensor now takes the form

$N_{AB}^{(\Lambda)}=\dot{C}_{AB}.$ (50)

In the linearized limit, the symmetry algebra of the de Sitter background
metric can be naturally used (see Sec. V.1). The radiation rates in the linear
theory can be obtained from the corresponding expressions in the non-linear
theory by dropping the cubic and the quartic terms in the fields. In the case
of the energy, we have

$\displaystyle\frac{dE}{du}$ $\displaystyle=-\frac{1}{32\pi G}\oint
d^{2}S\left\\{N_{AB}^{(\Lambda)}N^{(\Lambda)AB}+\frac{2\Lambda}{3}C^{AB}C_{AB}\right.$
$\displaystyle-\frac{\Lambda}{6}C^{AB}D^{2}C_{AB}-\frac{\Lambda^{2}}{3}C^{AB}E_{AB}$
$\displaystyle\left.+\left(4M+D_{A}D_{B}C^{AB}\right)\left(D_{C}U_{\left(0\right)}^{C}\right)\right\\}.$
(51)

Remarkably, if one calculates the total flux of energy radiated in a finite
interval of time $\Delta u$ (assuming appropriate fall-off near the edges of
$\Delta u$), one obtains perfect agreement with the results obtained
independently by Chrúsciel, Hoque, and Smolka in (Chrusciel:2020rlz, ), and by
Kolanowski and Lewandowski in (Kolanowski:2020wfg, ). It also coincides with
the expression found by Ashtekar, Bonga and Kesavan in a different set of
coordinates (abk2, ). This can be seen as follows: If we re-express $C_{AB}$
in the terms with explicit $\Lambda$ dependence by using Eq. (4), we obtain

$\displaystyle\frac{dE}{du}$ $\displaystyle=-\frac{1}{32\pi G}\oint
d^{2}S\left(\dot{C}_{AB}\dot{C}^{AB}-2\Lambda
D^{B}U_{\left(0\right)}^{A}E_{AB}\right.$
$\displaystyle+4D^{B}U_{\left(0\right)}^{A}C_{AB}-D^{B}U_{\left(0\right)}^{A}D^{2}C_{AB}$
$\displaystyle\left.+\left(4M+D_{A}D_{B}C^{AB}\right)\left(D_{C}U_{\left(0\right)}^{C}\right)\right).$

After an integration by parts on the sphere, one obtains

$\displaystyle\frac{dE}{du}$ $\displaystyle=-\frac{1}{32\pi G}\oint
d^{2}S\left(\dot{C}_{AB}\dot{C}^{AB}+U_{\left(0\right)}^{A}\left[2\Lambda
D^{B}E_{AB}\right.\right.$
$\displaystyle\left.\left.-4D^{B}C_{AB}+D^{B}D^{2}C_{AB}-D_{A}\left(4M+D_{B}D_{C}C^{BC}\right)\right]\right).$

Using the linearized equation of motion for $N^{A}$ in (49), we can write this
compactly as

$\frac{dE}{du}=\frac{1}{8\pi G}\oint
d^{2}S\left(-\frac{1}{4}\dot{C}_{AB}\dot{C}^{AB}+U_{\left(0\right)}^{A}\dot{N}_{A}\right)\,.$

So that the total amount of energy radiated in the interval of time $\Delta u$
is given by

$\left.E\right|_{\Delta u}=\frac{1}{8\pi G}\int_{\Delta u}du\oint
d^{2}S\left(-\frac{1}{4}\dot{C}_{AB}\dot{C}^{AB}+U_{\left(0\right)}^{A}\dot{N}_{A}\right).$

Assuming that there is no flux of radiation outside this interval of time, we
can integrate by parts in the null time and discard the corresponding boundary
terms, so that we can write:

$\left.E\right|_{\Delta u}=-\frac{1}{8\pi G}\int_{\Delta u}du\oint
d^{2}S\left(\frac{1}{4}\dot{C}_{AB}\dot{C}^{AB}+\dot{U}_{\left(0\right)}^{A}N_{A}\right).$

Now, by virtue of our asymptotic conditions,

$h_{uA}=-\frac{1}{r^{2}}U_{A}.$

Therefore, if we consider an asymptotic expansion of the form

$h_{uA}=h_{uA}^{\left(2\right)}r^{2}+h_{uA}^{\left(1\right)}r+h_{uA}^{\left(0\right)}+\frac{h_{uA}^{\left(-1\right)}}{r}+\dots,$

we obtain

$h_{uA}^{\left(2\right)}=-U_{\left(0\right)A}\qquad,\qquad
h_{uA}^{\left(-1\right)}=\frac{2}{3}N_{A}.$

Thus, in terms of these variables, the total amount of energy radiated in the
interval of time $\Delta u$ acquires the following form

$\left.E\right|_{\Delta u}=-\frac{1}{32\pi G}\int_{\Delta u}du\oint
d^{2}S\left(\dot{C}_{AB}\dot{C}^{AB}-6\dot{h}_{uA}^{\left(2\right)}h_{u}^{\left(-1\right)A}\right).$

This expression precisely coincides with the ones found in (Chrusciel:2020rlz,
) and (Kolanowski:2020wfg, ).

####  V.4.2 Explicit solutions for quadrupolar modes

Using the set-up in the previous subsection, we will now study explicit
solutions to the linearized Einstein’s equation. The strategy is to solve for
the homogeneous solution of Einstein’s equation, which corresponds in a
partial wave expansion to the gravitational field away from the source
generating the gravitational waves (which is assumed to be bounded). In
particular, the homogeneous solutions corresponding to a fixed $\ell$ in the
spherical harmonic expansion, should necessarily be generated by a source with
multipole moment $\ell$. Even though the homogeneous solution is only valid
outside the source, in principle we could match this solution with the “inner”
solution using matched asymptotic expansions (Burke:1969zz, ). The matching
with an interior solution is beyond the scope of this paper, which focuses on
the solution far away from the source.

Since the background spacetime is spherically symmetric, it is convenient to
use similar techniques as Regge and Wheeler did when solving for linearized
perturbations off Schwarzschild (although we will not implement the Regge-
Wheeler gauge) (Regge:1957td, ). In particular, we will use separation of
variables and for the angular part of the perturbations, we introduce scalar,
vector and tensor spherical harmonics. Following the notation in (Martel-
Poisson, ), we note that the scalar harmonics are the usual spherical-harmonic
functions $Y_{\ell m}(x^{A})$ satisfying the eigenvalue equation $D^{2}Y_{\ell
m}=-\ell\left(\ell+1\right)Y_{\ell m}$. There are two types of vector
harmonics: even-parity $Y_{A}^{\ell m}$ (also known as electric) and odd-
parity $X_{A}^{\ell m}$ (also known as magnetic), which are related to the
scalar harmonics through the covariant derivative operator compatible with
$\gamma_{AB}$

$\displaystyle Y_{A}^{\ell m}$ $\displaystyle:=D_{A}Y_{\ell m}$ (52)
$\displaystyle X_{A}^{\ell m}$
$\displaystyle:=-\epsilon_{A}^{\;\;B}D_{B}Y_{\ell m}\;.$ (53)

The even- and odd-parity harmonics are orthogonal in the sense that $\int
d^{2}S\;\bar{Y}_{\ell m}^{A}X_{A}^{\ell^{\prime}m^{\prime}}=0$. The tensor
harmonics also come in the same two types:

$\displaystyle Y_{AB}^{\ell m}$ $\displaystyle:=D_{(A}Y_{B)}^{\ell
m}-\frac{1}{2}\gamma_{AB}D_{C}Y_{\ell m}^{C}$ (54) $\displaystyle X_{AB}^{\ell
m}$ $\displaystyle:=-\epsilon_{(A}^{\;\;\;\;C}D_{B)}Y_{C}^{\ell m}\;.$ (55)

These operators are traceless, i.e., $\gamma^{AB}Y_{AB}^{\ell
m}=0=\gamma^{AB}X_{AB}^{\ell m}$ and orthogonal in the same sense as the
vector harmonics are. The separation of variables takes the form

$\displaystyle h_{uu}$ $\displaystyle=\sum_{\ell,m}f_{\ell
m}\left(u,r\right)Y_{\ell m},$ (56a) $\displaystyle h_{ur}$
$\displaystyle=\sum_{\ell,m}h_{\ell m}\left(u,r\right)Y_{\ell m},$ (56b)
$\displaystyle h_{uA}$ $\displaystyle=\sum_{\ell,m}F_{1}^{\ell
m}\left(u,r\right)Y_{A}^{\ell m}+G_{1}^{\ell m}\left(u,r\right)X_{A}^{\ell
m},$ (56c) $\displaystyle h_{AB}$ $\displaystyle=\sum_{\ell,m}F_{2}^{\ell
m}\left(u,r\right)Y_{AB}^{\ell m}+G_{2}^{\ell m}\left(u,r\right)X_{AB}^{\ell
m},$ (56d)

where the sum here is restricted to $\ell\geq 2$ and $m$ ranges from $-\ell$
to $\ell$. We neglect the $\ell=0$ and $\ell=1$ multipoles, which are non-
radiative and require a special treatment. We have also set
$h_{rr}=h_{rA}=\gamma^{AB}h_{AB}=0$ to ensure that the linearized metric
satisfies the required fall-off in Eq.(47). This gauge choice can always be
made (in fact, there is some residual gauge freedom left that we will use in
the analysis below).

The even and odd-parity modes remain decoupled in the linearized Einstein’s
equation and so are all the $\ell,m$ modes in the spherical decomposition. We
will restrict ourselves in this section to the $\ell=2$ modes; the structure
of the solution is very similar for higher $\ell$ modes and we will briefly
comment on the form of the general solution at the end. Solving for the
simpler, odd-parity sector first, we find the following retarded solution:

$\displaystyle G_{1}^{\ell=2}\left(u,r\right)$
$\displaystyle=\sum_{m=-2}^{2}\left[\frac{1}{2}\dot{C}_{2}^{m}\frac{r^{2}}{l^{4}}-\left(l^{-2}\dot{C}_{1}^{m}-\dddot{C}_{1}^{m}\right)+\frac{2}{r}\ddot{C}_{1}^{m}\right.$
$\displaystyle\;\;\left.+\frac{3}{2r^{2}}\dot{C}_{1}^{m}\right],$ (57)
$\displaystyle G_{2}^{\ell=2}\left(u,r\right)$
$\displaystyle=\sum_{m=-2}^{2}\left[\left(C_{2}^{m}+C_{1}^{m}-\ddot{C}_{1}^{m}l^{2}\right)\frac{r^{2}}{l^{4}}\right.$
$\displaystyle\left.+r\left(l^{-2}\dot{C}_{1}^{m}-\dddot{C}_{1}^{m}\right)+\frac{1}{r}\dot{C}_{1}^{m}\right].$
(58)

with $C_{1}^{m},C_{2}^{m}$ dimensions of length squared. The term proportional
to $r^{2}$ in $G_{2}^{\ell=2,m}$ spoils the fall-off behavior of the angular
part of the metric in Eq. (2). This is, however, easily remedied by realizing
that the solution is not completely gauge fixed and with the residual gauge
freedom this part of the metric can be gauged away. With an appropriate gauge
choice, we set777By the Stewart-Walker lemma, the linearized Weyl tensor is
gauge-invariant and a straightforward computation shows that the linearized
Weyl tensor is independent of $C_{2}^{m}$. Therefore, the $C_{2}^{m}$ solution
is pure gauge and contains no physical degrees of freedom. This interpretation
of the solution is consistent with the gauge choice made in Eq. (59).

$C_{2}^{m}\equiv\ddot{C}_{1}^{m}l^{2}-C_{1}^{m}.$ (59)

This gauge choice is further preserved by the residual gauge freedom generated
by $\chi^{A}=\epsilon^{AB}D_{B}\left(\vec{\Omega}\cdot\hat{r}\right)$. With
this gauge choice, and introducing $B_{m}=\dot{C}_{1}^{m}$, we finally obtain
the following odd-parity solutions for the quadrupolar modes with $\ell=2$

$\displaystyle h_{uu}^{\text{odd}}$ $\displaystyle=0$ (60a) $\displaystyle
h_{ur}^{\text{odd}}$ $\displaystyle=0$ (60b) $\displaystyle
h_{uA}^{\text{odd}}$
$\displaystyle=\sum_{m=-2}^{2}\left[\frac{1}{2}\left(\ddot{B}_{m}-l^{-2}B_{m}\right)\frac{r^{2}}{l^{2}}+\left(\ddot{B}_{m}-l^{-2}B_{m}\right)\right.$
$\displaystyle\left.+\frac{2}{r}\dot{B}_{m}+\frac{3}{2r^{2}}B_{m}\right]X_{A}^{2m}$
(60c) $\displaystyle h_{AB}^{\text{odd}}$
$\displaystyle=\sum_{m=-2}^{2}\left[r\left(\ddot{B}_{m}-l^{-2}B_{m}\right)-\frac{1}{r}B_{m}\right]X_{AB}^{2m}$
(60d)

where $B_{m}=B_{m}\left(u\right)$ is an arbitrary function of the retarded
time $u$ with dimensions of length. Note that the leading order of the angular
metric is independent of the wave, but that $C_{AB}\neq 0$ and
$U_{(0)}^{A}\neq 0$ (with the constraint relating these metric coefficients in
Eq. (4) satisfied).

The solution for general $\ell$ is more complicated, but has these general
features:

$\displaystyle G_{1}^{\ell}(u,r)$
$\displaystyle=\sum_{m=-2}^{2}\left[\frac{1}{2}\dot{C}_{2}^{\ell
m}\frac{r^{2}}{l^{4}}+\sum_{i=0}^{\ell}a_{i}^{(\ell)}(r)\;\overset{(i)}{C_{i}^{\ell
m}}\right]$ (61) $\displaystyle G_{2}^{\ell}(u,r)$
$\displaystyle=\sum_{m=-2}^{2}\left[C_{2}^{\ell
m}\frac{r^{2}}{l^{4}}+\sum_{i=0}^{\ell}\left\\{\begin{array}[]{ll}b_{i}^{(\ell)}(r)\;\overset{(i)}{C_{i}^{\ell
m}}&\text{if }\ell\text{ even}\\\
b_{i}^{(\ell)}(r)\;\overset{(i+1)}{C_{i}^{\ell m}}&\text{if }\ell\text{
odd}\end{array}\right.\right]$ (64)

with the $C$-coefficients depending on $u$ only, the factor $(i)$ on top of
these coefficients indicate its $i$-th derivative with respect to $u$ and
$a_{i}^{(\ell)},b_{i}^{(\ell)}$ are polynomials in $r$ (and its inverse
powers) with the highest power being $r^{2}$. Note that the term proportional
to $C_{2}$ is in fact independent of $\ell$ and can always be gauged away. As
a result, even though generically $G_{2}$ contains terms proportional to
$r^{2}$ which could spoil the desired fall-off, these terms can always be set
to zero by a clever gauge choice for $C_{2}$ — similar to the case with
$\ell=2$. Hence, linearized solutions with odd-parity satisfy the desired
fall-off conditions for any $\ell\geq 2$.

The analysis for the even-parity sector mimicks that of the odd-parity sector,
but is more involved as more terms are non-zero. Nonetheless, also in this
case one can gauge fix the solution to obtain a linearized solution that
satisfies the fall-off conditions prescribed in Eq. (47). Specifically, the
retarded $\ell=2$ even-parity solutions for $h_{\mu\nu}$ takes the form

$\displaystyle h_{uu}^{\text{even}}$
$\displaystyle=\sum_{m=-2}^{2}\left[3\left(\ddot{A}_{m}-4l^{-2}A_{m}\right)\frac{r}{l^{2}}\right.$
$\displaystyle\left.+6\left(\ddot{A}_{m}-l^{-2}A_{m}\right)\frac{1}{r}+\frac{6}{r^{2}}\dot{A}_{m}+\frac{3}{r^{3}}A_{m}\right]Y_{2m},$
(65a) $\displaystyle h_{ur}^{\text{even}}$ $\displaystyle=0,$ (65b)
$\displaystyle h_{uA}^{\text{even}}$
$\displaystyle=\sum_{m=-2}^{2}\left[\left(2l^{-2}A_{m}-\frac{1}{2}\ddot{A}_{m}\right)\frac{r^{2}}{l^{2}}+\left(4l^{-2}A_{m}-\ddot{A}_{m}\right)\right.$
$\displaystyle\left.+\frac{2}{r}\dot{A}_{m}+\frac{3}{2r^{2}}A_{m}\right]Y_{A}^{2m},$
(65c) $\displaystyle h_{AB}^{\text{even}}$
$\displaystyle=\sum_{m=-2}^{2}\left[r\left(\ddot{A}_{m}-4l^{-2}A_{m}\right)+\frac{1}{r}A_{m}\right]Y_{AB}^{2m}$
(65d)

where $A_{m}=A_{m}(u)$ is an arbitrary function of the retarded time $u$ and
dimensions of length. Also, similar to the odd-parity sector, we have set
$h_{rr}=h_{rA}=\gamma^{AB}h_{AB}=0$. Note that there is backreaction on the
“background” metric as $r\to\infty$ through the leading term of $h_{uA}$, that
is, $U_{(0)}^{A}\neq 0$. The backreaction onto the leading order part is
unique to $\Lambda\neq 0$. In the limit $l\to\infty$, this backreaction
vanishes. This is immediately clear from the limit of the even- and odd-parity
solutions:

$\displaystyle\lim_{l\to\infty}h_{uu}$
$\displaystyle=\sum_{m=-2}^{2}\left[\frac{6\ddot{A}_{m}}{r}+\frac{6\dot{A}_{m}}{r^{2}}+\frac{3A_{m}}{r^{3}}\right]Y_{2m},$
(66a) $\displaystyle\lim_{l\to\infty}h_{ur}$ $\displaystyle=0,$ (66b)
$\displaystyle\lim_{l\to\infty}h_{uA}$
$\displaystyle=\sum_{m=-2}^{2}\left(\left[-\ddot{A}_{m}+\frac{2\dot{A}_{m}}{r}+\frac{3A_{m}}{2r^{2}}\right]Y_{A}^{2m}\right.$
$\displaystyle\left.+\left[\ddot{B}_{m}+\frac{2\dot{B}_{m}}{r}+\frac{3B_{m}}{2r^{2}}\right]X_{A}^{2m}\right),$
(66c) $\displaystyle\lim_{l\to\infty}h_{AB}$
$\displaystyle=\sum_{m=-2}^{2}\left(\left[\frac{\ddot{A}_{m}}{r}+\frac{A_{m}}{r^{3}}\right]r^{2}Y_{AB}^{2m}\right.$
$\displaystyle\left.+\left[\frac{\ddot{B}_{m}}{r}-\frac{B_{m}}{r^{3}}\right]r^{2}X_{AB}^{2m}\right)\;,$
(66d)

where $A_{m}$ and $B_{m}$ reduce to the standard quadrupole moments on flat
spacetime.

Connecting these results with the Bondi-Sachs expansions, we find that the
linear part of the metric coefficients is given by

$\displaystyle M$
$\displaystyle=\sum_{m=-2}^{2}3\left(\ddot{A}_{m}-l^{-2}A_{m}\right)Y^{2m}$
(67a) $\displaystyle U_{A}^{(0)}$
$\displaystyle=\frac{1}{l^{2}}\sum_{m=-2}^{2}\left(2l^{-2}A_{m}-\frac{1}{2}\ddot{A}_{m}\right)Y_{A}^{2m}$
$\displaystyle+\frac{1}{2}\left(\ddot{B}_{m}-l^{-2}B_{m}\right)X_{A}^{2m}$
(67b) $\displaystyle N_{A}$
$\displaystyle=\sum_{m=-2}^{2}-3\dot{A}_{m}\;Y_{A}^{2m}+3\dot{B}_{m}X_{A}^{2m}$
(67c) $\displaystyle C_{AB}$
$\displaystyle=\sum_{m=-2}^{2}\left(\ddot{A}_{m}-4l^{-2}A_{m}\right)Y_{AB}^{2m}+\left(\ddot{B}_{m}-l^{-2}B_{m}\right)X_{AB}^{2m}$
(67d) $\displaystyle E_{AB}$
$\displaystyle=\sum_{m=-2}^{2}A_{m}\;Y_{AB}^{2m}-B_{m}\;X_{AB}^{2m}$ (67e)

and all other coefficients vanishing or determined by lower order terms.

The radiation rate at the linearized level in Eq. (51) reduces after some
further simplifications to

$\displaystyle\frac{dE}{du}$ $\displaystyle=-\frac{3}{8\pi
G}\sum_{m=-2}^{2}\left[\left(\dddot{A}_{m}-4l^{-2}\dot{A}_{m}\right)^{2}\right.$
$\displaystyle\left.-3l^{-2}\ddot{A}_{m}\left(\ddot{A}_{m}-4l^{-2}A_{m}\right)+\left(\dddot{B}_{m}-l^{-2}\dot{B}_{m}\right)^{2}\right.$
$\displaystyle\left.+3l^{-2}\ddot{B}_{m}\left(\ddot{B}_{m}-l^{-2}B_{m}\right)\right]\;.$
(68)

If we consider the total energy radiated during some large time interval
$\Delta u$, where we assume that at far past and at far future the system will
not radiate so that we can remove the boundary terms in time, then

$\displaystyle E_{\Delta u}$ $\displaystyle=-\frac{3}{8\pi
G}\int_{-\infty}^{\infty}du\sum_{m=-2}^{2}\left[\left(\dddot{A}_{m}-\frac{\dot{A}_{m}}{l^{2}}\right)\right.$
(69)
$\displaystyle\left.\left(\dddot{A}_{m}-\frac{4\dot{A}_{m}}{l^{2}}\right)+\left(\dddot{B}_{m}-\frac{\dot{B}_{m}}{l^{2}}\right)\left(\dddot{B}_{m}-\frac{4\dot{B}_{m}}{l^{2}}\right)\right].$

In particular, after integration by parts, we have

$\displaystyle E_{\Delta u}$ $\displaystyle=-\frac{3}{8\pi
G}\int_{-\infty}^{\infty}du\sum_{m=-2}^{2}\left[\left(\dddot{A}_{m}\right)^{2}+\frac{5\ddot{A}_{m}^{2}}{l^{2}}+\frac{4\dot{A}_{m}^{2}}{l^{4}}\right.$
$\displaystyle\left.+\left(\dddot{B}_{m}\right)^{2}+\frac{5\ddot{B}_{m}^{2}}{l^{2}}+\frac{4\dot{B}_{m}^{2}}{l^{4}}\right].$
(70)

This flux is manifestly negative. Therefore, the total energy always decreases
for a source characterized by a quadrupole. The flat spacetime limit yields
the expected result for a quadrupolar source $A,B$

$\lim_{l\to\infty}E_{\Delta u}=-\frac{3}{8\pi
G}\int_{-\infty}^{\infty}du\sum_{m=-2}^{2}\left[\left(\dddot{A}_{m}\right)^{2}+\left(\dddot{B}_{m}\right)^{2}\right].$
(71)

Note that for $\Lambda<0$, i.e. $l\to il$, the energy flux is non-zero so that
the boundary is not reflective (as is typically imposed). In fact, the energy
flux can have an arbitrary sign depending on the values of $A$, its time
derivatives and $l$.

###  V.5 Robinson-Trautman spacetime

The Robinson-Trautman spacetime is an exact solution of Einstein equations
that describes the backreaction of a non-linear gravitational wave on a
Schwarzschild spacetime. The Robinson-Trautman solution is dynamical: it
models gravitational radiation expanding from a radiating object. Since
ultimately, we are interested in describing gravitational radiation emitted by
compact sources in the presence of a cosmological constant, this example is of
particular interest for our analysis. The original Robinson-Trautman solution
contained no cosmological constant (Robinson:1962zz, ), but it was soon
realized that the solution easily accommodates for a non-zero cosmological
constant. This class of spacetimes is the most general radiative vacuum
solution admitting a geodesic, shearfree and twistfree null congruence of
diverging rays. It has been shown that starting with arbitrary, smooth initial
data at some retarded time $u=u_{0}$ , the cosmological Robinson-Trautman
solutions converge exponentially fast to a Schwarzschild-de Sitter solution at
large retarded times ($u\to\infty$). Thus, these solutions also belong to the
class of solutions discussed in this paper. In this section, we will show this
explicitly by providing the form of this solution in Bondi-Sachs like
coordinates.

The line element of the Robinson-Trautman solution with a positive
cosmological constant is given by

$\displaystyle ds^{2}=$
$\displaystyle-2H\left(u,r,\theta,\phi\right)du^{2}-2dudr$
$\displaystyle+\frac{r^{2}}{P^{2}\left(u,\theta,\phi\right)}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (72)

with

$2H\left(u,r,\theta,\phi\right)=-\frac{r^{2}}{\ell^{2}}-\frac{2r\dot{P}}{P}+\frac{1}{2}\mathcal{R}_{h}-\frac{2m\left(u\right)}{r}.$

Here $P\left(u,\theta,\phi\right)$ is an arbitrary function of the retarded
time and the angles, and contains the information of the gravitational wave.
According to Einstein’s equations, the following equation governs the time
evolution of $m\left(u\right)$:

$\dot{m}=3m\frac{\dot{P}}{P}-\frac{1}{8}\Delta_{h}\mathcal{R}_{h}.$ (73)

The Laplacian $\Delta_{h}$ is defined with respect to the metric
$h_{AB}=P^{-2}\left(u,\theta,\phi\right)\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)$. In the particular case when $P=1$ (no radiation), the
Schwarzschild de Sitter solution is recovered.

The Robinson-Trautman solution, as written in Eq. (72), does not fit
immediately within the asymptotic conditions in Eqs. (2). The reason is the
presence of the function $P^{-2}$ appearing in front of the metric of the
2-sphere. In order to accommodate the solution one must perform an appropriate
change of coordinates. In general, the implementation of this change of
coordinates is technically a very hard task. However, a simplified analysis
can be achieved by considering the Robinson-Trautman metric with axial
symmetry. In addition, and for clarity to the reader, in this section we will
only consider a linearized version of the solution. The non-linear analysis
will be discussed in Appendix.

Assuming for simplicity axial symmetry, the linearized Robinson-Trautman
solution expanded around a Schwarzschild-de Sitter background is obtained by
expressing the function $P=P\left(u,\theta\right)$ as follows:

$P=1+\epsilon p\left(u,\theta\right).$

Here $\epsilon$ is a small parameter that controls the linearized expansion.
In this approximation, the leading order of eq. (73) becomes

$\displaystyle\dot{m}=$
$\displaystyle\frac{1}{4}\epsilon\left[12m\dot{p}+p^{(4)}-\cot^{2}\theta
p^{\prime\prime}+2\cot\theta p^{\prime\prime\prime}\right.$
$\displaystyle\left.+\cot\theta\left(\csc^{2}\theta+2\right)p^{\prime}\right]+O\left(\epsilon^{2}\right).$
(74)

Here the prime denotes derivatives with respect to $\theta$. In order to
accommodate the solution within our asymptotic conditions, one can implement
the following change of coordinates to linear order in $\epsilon$

$\displaystyle u$ $\displaystyle\rightarrow u-\epsilon
f\left(\mathit{u},\theta\right)+O\left(r^{-3}\right),$ $\displaystyle r$
$\displaystyle\rightarrow r\left(1+\epsilon
p\left(\mathit{u},\theta\right)\right)-\frac{1}{2}\epsilon
D^{2}f\left(\mathit{u},\theta\right)+O\left(r^{-2}\right),$
$\displaystyle\theta$
$\displaystyle\rightarrow\theta+\epsilon\frac{f^{\prime}\left(\mathit{u},\theta\right)}{r}+O\left(r^{-3}\right),$
$\displaystyle\phi$ $\displaystyle\rightarrow\phi,$

where

$p=\dot{f}.$ (75)

Thus, one finds

$\displaystyle M=m\left[1-\epsilon\left(3\dot{f}+f\dot{m}\right)\right],$
(76a) $\displaystyle
C_{\theta\theta}=-\csc^{2}\theta,\;\;C_{\phi\phi}=\epsilon\left(f^{\prime\prime}-\cot\theta
f^{\prime}\right),\;\;C_{\theta\phi}=0,$ (76b) $\displaystyle
U_{\left(0\right)}^{\theta}=\frac{\epsilon}{\ell^{2}}f^{\prime},\;\;\;\;U_{\left(0\right)}^{\phi}=0,$
(76c) $\displaystyle N^{\theta}=-3\epsilon mf^{\prime},\;\;\;\;N^{\phi}=0,$
(76d) $\displaystyle E_{AB}=0,$ (76e)

In addition, at linear order the News tensor in eq. (25) takes the simple form
$N_{AB}^{(\Lambda)}=\dot{C}_{AB}$.

To obtain the flux of energy one can replace (76) in equation (32), while
retaining only the terms up to order $\epsilon^{2}$. After some integrations
by parts one finally obtains

$\frac{dE}{du}=-\frac{\epsilon^{2}}{16\pi G}\oint
d^{2}S\left[\left(\dot{f}^{\prime\prime}-\cot\theta\dot{f}^{\prime}\right)^{2}+\frac{3m}{\ell^{2}}\partial_{u}\left(f^{\prime
2}\right)\right].$

##  VI Comparison with alternative approaches

Given the observational evidence for an accelerated expansion of our Universe
and the recent gravitational wave observations, the challenge of understanding
gravitational waves in the presence of a positive cosmological constant
$\Lambda$ has received considerable attention in recent years. At the
linearized level, most of the previous results in the literature agree with
each other. As we will see below, our results are also in agreement with them.

However, the situation is drastically different in the full non-linear theory.
Different methods and/or different boundary conditions are employed — some of
which even require regularization; the results in general do not agree. We
describe below some of these approaches without any pretense of being
exhaustive.

###  VI.1 Linearized gravity

An important starting point is a thorough understanding of weak gravitational
waves on a de Sitter background. There are two key issues predominantly
studied within this context: (1) a mathematically sound and physically
sensible notion of energy and its flux, and (2) finding explicit solutions for
gravitational waves generated by a compact source and their link to time-
changing quadrupole moments, thereby generalizing the well-known flat
result.888The gravitational memory effect in de Sitter spacetimes has been
investigated in (Bieri:2015jwa, ), however, this paper focused on the
cosmological horizon and is therefore more difficult to relate to the results
in this paper that exclusively apply near $\mathcal{I}$.

There are various notions of energy and its flux in the literature that mostly
distinguish themselves by the method used to derive it (as a consequence,
these notions typically are equivalent up to boundary terms), “where” in
spacetime the energy (flux) is evaluated (mostly on $\mathcal{I}$ or across
the cosmological horizon) and by the class of linearized solutions for which
the energy is defined. For instance, in (Kolanowski:2020wfg, ), the energy
flux across $\mathcal{I}$ is derived using the same symplectic methods as in
the earlier work in (abk2, ), but for a slightly larger class of linearized
solutions. In (Kolanowski:2021hwo, ), the authors use the Wald-Zoupas
prescription to define energy (and angular momentum). In all these cases, the
resulting energy flux is finite and the result gauge invariant.

On the other hand, the energy flux obtained in (Chrusciel:2020rlz, ) by direct
use of Noether currents is not finite (note also earlier work
(Chrusciel:2016oux, )). In that paper, they remedy this issue by isolating the
terms which would lead to infinite energy and introducing a “renormalised
canonical energy”. Their argument for the plausibility of this procedure is
based on the observation that the diverging terms have dynamics of their own,
which evolves independently from the remaining part of the canonical energy.

Yet another approach, which applies only for sources supporting the short
wavelength approximation, as it relies on the Isaacson effective stress-
tensor, matches the results in (abk3, ) if one identifies the transverse-
traceless gauge with a certain projection operation.999For linearized
solutions on Minkowski spacetime, this is a well-defined and consistent
procedure for the leading order components of the gravitational field. This is
shown explicitly in (ab, ), in which the first notion is referred to as ‘TT’
gauge and the second as ‘tt’ gauge. However, it is not clear that the two
notions are also equivalent for the leading order fields in de Sitter
spacetime.

In (Hoque:2018byx, ), the authors employ the symplectic current density of the
covariant phase space to show that the integrand in the energy flux expression
on the cosmological horizon is same as that on $\mathcal{I}$ . This result is
interesting as it suggests that at the linearized level propagation of energy
flux is along null rays in de Sitter spacetime, despite the fact that
gravitational waves themselves have tail terms due to back-scattering off the
background curvature.

The second key issue investigated is the link between time variation of some
compact source generating gravitational waves and the resulting gravitational
waves themselves. This was investigated in (abk3, ) by solving the linearized
Einstein’s equation on de Sitter background sourced by a (first order) stress-
energy tensor. To study the limit to $\mathcal{I}$, the authors introduce a
late time approximation in addition to the commonly used post-Newtonian
approximation. This allowed them to express the leading terms of the
gravitational waves in terms of the quadrupole moments of sources. Moreover,
the energy carried away by this gravitational waves was studied using
Hamiltonian methods on the covariant phase-space of the linearized solutions
introduced in their earlier paper (abk2, ). This showed that despite the fact
that in principle the energy for linearized perturbations on de Sitter
spacetime can be negative (note that this is not in contrast with the
finiteness discussed in the previous paragraph), the energy of gravitational
waves emitted by compact objects is always positive. This is also consistent
with our results in Sec. V.4. The quadrupolar solutions in (abk3, ) were also
reinterpreted in (He:2018ikd, ), by writing the solutions in Bondi-Sachs type
coordinates different from the ones introduced in this paper. The authors
showed that the quadrupolar solutions can be accommodated by a non-zero shear
for the leading order part of $g_{AB}$. This is different but not in
contradiction with the results in this paper, which show that the radiative
solution contributes to the sub-leading part of $g_{AB}$ _and_ to
$U^{A}_{(0)}$, while the leading order part of $g_{AB}$ is equal to
$\gamma_{AB}$. This is a gauge choice. Other papers relating the source
dynamics modeled by some compact stress-energy tensor to the gravitational
wave and the energy have relied on the short wave approximation (Date:2016uzr,
; Hoque:2017xop, ; Hoque:2018dcg, ). These results are consistent with the
results in (abk2, ).

###  VI.2 Full non-linear theory

Early investigations of the asymptotic structure of asymptotically de Sitter
spacetimes in full non-linear general relativity such as (Strominger:2001pn, ;
Anninos:2010zf, ; Anninos:2011jp, ) imposed too stringent boundary conditions
by demanding conformal flatness of the induced three-dimensional metric on
$\mathcal{I}$. As a result, these early investigations concluded that the
asymptotic symmetry group is the full de Sitter group. However, as was shown
in (abk1, ), imposing conformal flatness ruled out many physically relevant
spacetimes as they enforced a vanishing flux of radiation across $\mathcal{I}$
(and consequently all charges are strictly conserved, see also
(aneesh_conserved_2019, )).

The observation that demanding the asymptotic symmetry group to be the de
Sitter one ruled out gravitational waves sparked new interest in this
challenging problem. It lead the authors in (He:2015wfa, ) to consider Bondi-
Sachs type coordinates for asymptotically de Sitter spacetimes, which are not
conformally flat at $\mathcal{I}$. A nice property of their coordinates is
that the Weyl tensor has peeling behavior near $\mathcal{I}$ (Xie:2017uqa, ).
While these authors also rely on the Bondi framework, their fall-off
conditions on the metric coefficients are different from those considered
here. In particular, the authors did not fix $g_{AB}$ to be equal to the unit
two-sphere but instead allowed for a non-zero shear at leading order. Their
shear contains all the information about gravitational radiation. However,
based on our analysis, using their fall-off conditions the variation of the
charge is infinite. This makes these fall-off conditions not as attractive.
Moreover, the analysis in those papers was restricted to axi-symmetric
spacetimes and limited to the study of Einstein’s equations; these papers did
not study gravitational charges and fluxes.

Subsequently, various authors used the Newman-Penrose formalism to define and
study asymptotically de Sitter spacetimes (Saw:2016isu, ; Saw:2017hsf, ;
Mao:2019ahc, ). The two earlier papers by Saw used a special choice of null
foliation, thereby excluding the Robinson-Trautman spacetime with a positive
cosmological constant as part of their allowed class of spacetimes. The class
of null foliations was generalized in (Mao:2019ahc, ) by Mao to accommodate
for the Robinson-Trautman spacetime. A nice feature of the fall-off conditions
on the spin coefficients and Weyl scalars in those papers is that they have a
well-defined flat limit. However, Mao finds that the asymptotic symmetry
algebra consists of all diffeomorphisms on the two-sphere and translations in
the $u$-direction. In the limit $l\to\infty$, the asymptotic symmetry algebra
becomes the algebra of all diffeomorphisms on the two-sphere and
supertranslations known as the extended BMS algebra (Campiglia:2020qvc, )
instead of the BMS algebra. Gravitational charges and fluxes were not studied
in these papers.

Another set of recent papers on this topic uses similar techniques to those
used here (Compere:algebra, ; Compere:group, ). These authors find that the
asymptotic symmetries form a Lie algebroid instead of a Lie algebra, as they
used different fall-off conditions on the metric. Their asymptotic symmetries
consist of infinite-dimensional “non-abelian supertranslations” and
superrotations, and like in (Mao:2019ahc, ) reduces in the limit $l\to\infty$
to the extended BMS algebra. The resulting definitions for the Bondi mass and
angular momentum aspects is discontinuous in the flat limit. Moreover, the
authors also note that Kerr-de Sitter spacetime is ultimately not included in
the class of spacetimes considered in their work. These boundary conditions
were used in (erfani_bondi_2022, ) to define a Bondi news-like tensor using a
Newman-Penrose tetrad.

Inspired by the dictionary between Bondi and Fefferman-Graham gauges
(Poole:2018koa, ), the authors used earlier results in Fefferman-Graham gauges
to define a new class of asymptotically de Sitter spacetimes. In their follow-
up work (poole_charges_2022, ), in order to obtain finite charges and fluxes,
these authors introduce a holographic renormalization procedure while all
charges and fluxes are naturally finite in this paper and do not require any
_ad hoc_ regularization. The latter work also states more clearly that their
interest is in spacetimes with compact spatial slices, as opposed to this
work.

Other work has focused on studying the possible isometries of asymptotically
de Sitter spacetimes. One of the key results is that the asymptotic symmetry
algebra they find is maximally four-dimensional Kaminski:2022tum , which in
spirit agrees with our work.

Research in a different direction focused on the question of how to identify
the presence of gravitational radiation in the presence of $\Lambda$ using
geometric tools only and without referring to a specific coordinate system
(radiation-criterion, ). The criterion proposed is based on value of super-
Poynting vector at $\mathcal{I}$: if it vanishes, there is no gravitational
radiation across $\mathcal{I}$ while if it is non-zero, there is gravitational
radiation across $\mathcal{I}$. This criterion is straightforward to check as
the super-Poynting vector is the commutator of the leading order electric
$\mathcal{E}_{ab}$ and magnetic $\mathcal{B}_{ab}$ part of the Weyl tensor.
When the cosmological constant vanishes, this criterion is equivalent to the
standard ‘identification’ method of gravitational radiation at null infinity
through the means of the (non-)vanishing of the Bondi news tensor (radiation-
criterion-flat, ). When $\mathcal{B}_{ab}$ vanishes on $\mathcal{I}$, the
super-Poynting vector also vanishes and this criterion implies that there is
no radiation. However, the vanishing of $\mathcal{B}_{ab}$ is not a necessary
condition. In particular, certain Kerr-de Sitter generalized spacetimes have
non-vanishing $\mathcal{B}_{ab}$ on $\mathcal{I}$ yet their super-Poynting
vector vanishes. Here we find that there is gravitational radiation whenever
$C_{AB}$ (and hence $U_{(0)}^{A}$) is non-zero. When these are zero,
$\mathcal{B}_{ab}$ vanishes. Therefore, our criterion to establish the
presence of gravitational radiation seems to be stricter. In other words,
based on the super-Poynting vector criterion a spacetime may be labeled as
non-radiating, while based on $C_{AB}$ it would be considered radiating. It is
therefore not too surprising that the authors in follow-up work found that the
asymptotic symmetry algebra for the spacetimes they considered are infinite-
dimensional (Fernandez-Alvarez:2021yog, ; Fernandez-Alvarez:2021zmp, ;
Senovilla:2022pym, ).

###### Acknowledgements.

BB would like to thank Ahbay Ashtekar for many thought-provoking conversations
over the years, as well as CECs for its hospitality during her visits,
separated in time by force majeure, when this work was initiated and
completed. CB would like to thank Neil Turok for bringing the authors of this
paper together at the Path Integral for Gravity workshop held at the Perimeter
Institute in November 2017, when the discussions that led to this article
started. AP wishes to thank Jorge Noreña and Ricardo Troncoso for helpful
discussions. The research of AP is partially supported by Fondecyt grants No
1211226, 1220910 and 1230853.

##  Appendix A Link with geometric approach

The goal of this appendix is to show how the metric in Eq. (1) with the fall-
off conditions specified in Eq. (2) can be derived from a geometric starting
point.

We start with a geometric definition of asymptotically (Schwarzschild-)de
Sitter spacetimes (abk1, ), which is analogous to the geometric definition à
la Penrose and others for asymptotically flat spacetimes. Given this
definition, we construct Bondi-Sachs-like coordinates for the conformally
completed spacetime. This process naturally indicates the minimal fall-off
conditions for the metric coefficients. With these coordinates for the
conformally completed spacetime in hand, we perform a conformal transformation
to obtain coordinates for the physical spacetime. The result is consistent
with the fall-off conditions in Eq. (2).

###  A.1 Bondi-Sachs coordinates

In (abk1, ), the following definition for asymptotically de Sitter spacetimes
was introduced:

_Definition 1:_ A space-time $(M,g_{ab})$ is weakly asymptotically de Sitter
if there exists a manifold $\tilde{M}$ with boundary $\mathcal{I}$ equipped
with a metric $\tilde{g}_{ab}$ and a diffeomorphism from $M$ onto
$(\tilde{M}\,\setminus\,\mathcal{I})$ of $\tilde{M}$ (with which we identify
$M$ and ($\tilde{M}\,\setminus\,\mathcal{I}$)) such that:

  1. 1.

there exists a smooth function $\Omega$ on $\tilde{M}$ such that
$\tilde{g}_{ab}=\Omega^{2}g_{ab}$ on $M$; $\Omega=0$ on $\mathcal{I}$; and
$n_{a}:=\nabla_{a}\Omega$ is nowhere vanishing on $\mathcal{I}$;

  2. 2.

$g_{ab}$ satisfies Einstein’s equations with a positive cosmological constant
$\Lambda$, i.e., $R_{ab}-\frac{1}{2}Rg_{ab}+\Lambda g_{ab}=8\pi G\;T_{ab}$
where $\Omega^{-1}T_{ab}$ has a smooth limit to $\mathcal{I}$.

These weakly asymptotically de Sitter spacetimes contain three different sub-
classes which distinguish themselves by the topology of $\mathcal{I}$. We
restrict ourselves to the sub-class for which $\mathcal{I}$ has topology
$\mathbb{S}^{2}\times\mathbb{R}\simeq\mathbb{S}^{3}\setminus\\{i^{0},i^{+}\\}$,
which in (abk1, ) were coined asymptotically Schwarzschild-de Sitter
spacetimes.

From this definition, we will construct Bondi-Sachs type coordinates near
$\mathcal{I}$ for the conformally completed spacetime $\tilde{g}_{ab}$. We
start by introducing coordinates _on_ $\mathcal{I}$ for the induced metric.
Future null infinity $\mathcal{I}$ is a space-like surface with topology
$\mathbb{R}\times\mathbb{S}^{2}$, for which we choose any foliation of the
$\mathbb{R}$ direction and pick $\Omega$ such that when restricted to the base
$\mathbb{S}^{2}$, we have the unit round metric. This choice can _always_ be
made and is simply a convenient choice of the conformal factor. With this
choice, the divergence of $\tilde{\nabla}_{a}\Omega$ is non-zero. This is
different from the typical choice made in the asymptotically flat context,
where one often chooses $\Omega$ such that one is in a conformal divergence-
free frame because for those spacetimes this choice simplifies intermediate
results significantly.101010One can then show that one can _also_ choose
$\Omega$ to be such that  In this context, however, the more convenient choice
is to pick $\Omega$ such that the metric on $\mathbb{S}^{2}$ is the unit two-
sphere metric. Given that this is merely a convenient choice of the conformal
factor (a conformal gauge choice), it does not reduce the class of
asymptotically Schwarzschild-de Sitter spacetimes. In particular, when we
label the coordinates on the two-sphere with $x^{A}=(\theta,\varphi)$ and the
third coordinate as $u$ we find that:

$\displaystyle\tilde{g}_{\mu\nu}dx^{\mu}dx^{\nu}\mathrel{\mathop{\widehat{=}}}$
$\displaystyle\;UWdu^{2}$ (77)
$\displaystyle+\gamma_{AB}\left(dx^{A}-U^{A}du\right)\left(dx^{B}-U^{B}du\right),$

where $U,W,U^{A}$ are arbitrary functions of $(u,\theta,\varphi)$ only
constrained by the requirement that the pullback of this metric to
$\mathcal{I}$ is Riemannian. The hat on the equal sign indicates an equality
at $\mathcal{I}$. We choose the coordinate $u$ such that it is monotonically
increasing from $i^{+}$ to $i^{0}$. Away from $\mathcal{I}$, we have $\Omega$
as an additional coordinate. Consider null hypersurfaces transverse to
$\mathcal{I}$ that intersect $\mathcal{I}$ in the cross-sections $S_{u}$. In
fact, there are many such null surfaces in all directions. Fortunately, the
coordinate $u$ along $\mathcal{I}$ introduces a “special” direction. Using
this to select which transverse null hypersurfaces to select, in a
sufficiently small neighborhood of null infinity, these null hypersurfaces do
not intersect each other and thus generate a null foliation. We extend the
coordinate $u$ by demanding that it is constant along these null hypersurfaces
into the spacetime. Operationally, this means that we introduce a past-
directed null co-vector field $l_{\mu}$, which on $\mathcal{I}$ is given by
$l_{\mu}\hat{=}-\tilde{\nabla}_{\mu}u$ and extend it off of $\mathcal{I}$ by
demanding that it is geodesic ($l^{\mu}\tilde{\nabla}_{\mu}l^{\nu}=0$). One
can “normalize” $l_{\mu}$ so that $n^{\mu}l_{\mu}\hat{=}-1$, but we will keep
this arbitrary for now. We use this null vector field to extend the
coordinates $(u,\theta,\varphi)$ away from $\mathcal{I}$ by demanding that
$(\theta,\varphi)$ are parallel transported along $l^{\mu}$:
$l^{\mu}\tilde{\nabla}_{\mu}\theta=0$ and
$l^{\mu}\tilde{\nabla}_{\mu}\varphi=0$. This implies immediately that
$\tilde{g}^{uu}=0=\tilde{g}^{uA}$, which in turn implies that
$\tilde{g}_{\Omega\Omega}=0=\tilde{g}_{\Omega A}$. As a result, we can write

$\displaystyle\tilde{g}_{\mu\nu}dx^{\mu}dx^{\nu}$
$\displaystyle=e^{2\beta}W\;du^{2}+2e^{2\beta}\;dud\Omega$
$\displaystyle+q_{AB}\left(dx^{A}-U^{A}\;du\right)\left(dx^{B}-U^{B}\;du\right)\;,$
(78)

where we can expand all the metric coefficients in powers of $\Omega$ (since
$\tilde{g}_{\mu\nu}$ is smooth on $\mathcal{I}$ the leading order parts of the
metric start with arbitrary functions of $(u,\theta,\varphi)$):

$\displaystyle\beta$
$\displaystyle=\beta^{(0)}(u,\theta,\varphi)+\beta^{(1)}(u,\theta,\varphi)\Omega+\beta^{(2)}(u,\theta,\varphi)\Omega^{2}$
$\displaystyle+\beta^{(3)}(u,\theta,\varphi)\Omega^{3}+\ldots$ (79a)
$\displaystyle W$
$\displaystyle=W^{(0)}(u,\theta,\varphi)+W^{(1)}(u,\theta,\varphi)\Omega+W^{(2)}(u,\theta,\varphi)\Omega^{2}$
$\displaystyle+W^{(3)}(u,\theta,\varphi)\Omega^{3}+\ldots$ (79b)
$\displaystyle U^{A}$
$\displaystyle=U_{(0)}^{A}(u,\theta,\varphi)+U_{(1)}^{A}(u,\theta,\varphi)\Omega+U_{(2)}^{A}(u,\theta,\varphi)\Omega^{2}$
$\displaystyle+U_{(3)}^{A}(u,\theta,\varphi)\Omega^{3}+\ldots$ (79c)
$\displaystyle q_{AB}$
$\displaystyle=\gamma_{AB}+C_{AB}\Omega+d_{AB}\Omega^{2}+E_{AB}\Omega^{3}+\ldots$
(79d)

We will now set $l^{\mu}n_{\mu}\hat{=}-1$, which requires that
$\beta^{(0)}=0$. From the definition of asymptotically de Sitter spacetimes,
combined with Einstein’s equation, we know that
$\tilde{g}^{\mu\nu}\nabla_{\mu}\Omega\nabla_{\nu}\Omega\hat{=}-\frac{1}{\ell^{2}}$
(see e.g. (abk1, , Eq. (2.2))). As a result, we also know that

$W^{(0)}=\frac{1}{\ell^{2}}\;.$ (80)

Furthermore, imposing the Sachs condition, i.e., the determinant of $q_{AB}$
is the determinant of the unit two-sphere, we also find that

$\displaystyle\gamma^{AB}C_{AB}=0$ (81a) $\displaystyle
d_{AB}=\frac{1}{4}C^{CD}C_{CD}\gamma_{AB}$ (81b)
$\displaystyle\gamma^{AB}E_{AB}=0\;.$ (81c)

These conformal Bondi-Sachs coordinates for the unphysical spacetime
constructed above can be used to obtain asymptotic coordinates for the
physical metric. In the conformal Bondi-Sachs coordinates, the physical metric
is

$\displaystyle g_{\mu\nu}dx^{\mu}dx^{\nu}$
$\displaystyle=\Omega^{-2}\tilde{g}_{\mu\nu}dx^{\mu}dx^{\nu}$
$\displaystyle=\Omega^{-2}e^{2\beta}W\;du^{2}+2\Omega^{-2}e^{2\beta}\;dud\Omega$
$\displaystyle+\Omega^{-2}q_{AB}\left(dx^{A}-U^{A}\;du\right)\left(dx^{B}-U^{B}\;du\right)\;.$
(82)

Note that surfaces of constant $u$ are outgoing null surfaces for both the
unphysical and the physical spacetime, since conformal transformations do not
change the properties of null geodesics. To put this metric in a more familiar
form, we define a radial coordinate $r$ in the physical spacetime so that null
infinity is approached as the radial coordinate goes to infinity along the
null surfaces of constant $u$. The natural candidate for this radial
coordinate is $r=\Omega^{-1}$, in which case we exactly recover the metric
with fall-off conditions as in Eq. (2).

_Remark._ While in the definition of asymptotically de Sitter spacetimes and
at the level of Einstein’s equation it is relatively easy to include a non-
zero stress-energy tensor, to derive charges and fluxes one would also need to
specify a Lagrangian for the matter fields. In this paper, however, we have
restricted ourselves to a stress-energy tensor with compact support so that
effectively we are considering vacuum solutions only.

###  A.2 The electric and magnetic part of the Weyl tensor

In (abk1, ), also _strongly asymptotically de Sitter_ spacetimes were defined.
These classes of spacetimes satisfy the conditions in the definition for
weakly asymptotically de Sitter spactimes, but in addition have a conformally
flat metric at $\mathcal{I}$. It is clear from Eq. (77) that the class of
spacetimes considered in this paper do not belong to strongly asymptotically
de Sitter spacetimes. This can also been seen by studying the Weyl tensor. In
particular, conformal flatness of the metric at $\mathcal{I}$ is equivalent to
the vanishing of the next-to-leading order magnetic part of the Weyl tensor at
$\mathcal{I}$. Given the above expression for the metric, we can compute the
Weyl tensor explicitly. We find that – as expected – the leading order part of
the Weyl tensor vanishes on $\mathcal{I}$, that is,
$C_{abcd}\mathrel{\mathop{\widehat{=}}}0$ (abk1, ). The next-to-leading order
part is non-zero and we will decompose it into its electric and magnetic part:

$\displaystyle\mathcal{E}_{ab}$
$\displaystyle:=l^{2}\Omega^{-1}C_{acbd}n^{c}n^{d}$ (83)
$\displaystyle\mathcal{B}_{ab}$
$\displaystyle:=l^{2}\Omega^{-1}{}^{*}C_{acbd}n^{c}n^{d}=\frac{l^{2}}{2}\epsilon_{ac}^{\;\;\;mn}\;C_{mnbd}n^{c}n^{d}\;.$
(84)

where both $\mathcal{E}_{ab}$ and $\mathcal{B}_{ab}$ are symmetric, traceless
and orthogonal to $n^{a}$, which here is equal to
$n^{a}\partial_{a}=\partial/\partial u$. The resulting expressions are rather
long, so here we only show the part linear in the metric coefficients:

$\displaystyle\left.\mathcal{E}_{ab}dx^{a}dx^{b}\right|_{{\rm lin}}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}-\frac{1}{l^{2}}V^{(3)}\;\left(du+l^{2}d\Omega\right)^{2}$
$\displaystyle+\left(\frac{3}{2l^{2}}U_{A}^{(3)}+\frac{l^{2}}{2}\partial_{u}\left[D^{2}+1\right]U_{A}^{(0)}\right)\left(du\right.$
$\displaystyle+\left.\ell^{2}d\Omega\right)dx^{A}-\frac{l^{2}}{2}\left(\frac{3}{l^{4}}q_{AB}^{(3)}-\frac{1}{l^{2}}V^{(3)}\gamma_{AB}\right.$
$\displaystyle+\left[2l^{2}\partial_{u}^{2}+D^{2}-4\right]\left[D_{(A}U_{B)}^{(0)}-\frac{1}{2}\gamma_{AB}D_{C}U_{(0)}^{C}\right]$
$\displaystyle\left.-\left[D_{(A}D_{B)}-\frac{1}{2}\gamma_{AB}D^{2}\right]D_{C}U_{(0)}^{C}\right)dx^{A}dx^{B}$
(85)

and

$\displaystyle\left.\mathcal{B}_{ab}dx^{a}dx^{b}\right|_{{\rm lin}}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}\frac{1}{2}\left[D^{2}+2\right]\epsilon^{AB}D_{A}U_{B}^{(0)}\left(du+l^{2}d\Omega\right)^{2}$
$\displaystyle\frac{l^{2}}{2}\partial_{u}\left[D^{2}+1\right]\epsilon_{A}^{\;\;B}U_{B}^{(0)}(du+l^{2}d\Omega)dx^{A}$
$\displaystyle-\frac{l^{2}}{2}\left(\frac{1}{2}\left[D^{2}+2\right](\epsilon^{EF}D_{E}U_{F}^{(0)})\;\gamma_{AB}\right.$
$\displaystyle-2l^{2}\partial_{u}^{2}\left[D_{(A}(\epsilon_{B)}^{\;\;\;\;C}U_{C}^{(0)})\right.$
$\displaystyle\left.-\frac{1}{2}\gamma_{AB}D_{C}(\epsilon^{CE}U_{E}^{(0)})\right]$
$\displaystyle\left.-\left[D_{(A}D_{B)}-\frac{1}{2}\gamma_{AB}D^{2}\right]\epsilon^{EF}(D_{E}U_{F}^{(0)})\right)\times$
$\displaystyle\times dx^{A}dx^{B}\;.$ (86)

It is evident that $\mathcal{B}_{ab}$ is non-zero and consequently the class
of spacetimes considered in this paper are not strongly symptotically de
Sitter; as desired, because strongly asymptotically spacetimes remove half the
permissible data and have no fluxes of energy across $\mathcal{I}$.

Using the explicit linearized solutions in Sec. V.4, we find that for
quadrupolar gravitational waves, the electric and magnetic part of the Weyl
tensor are

$\displaystyle\mathcal{E}_{ab}dx^{a}dx^{b}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}\sum_{m=-2}^{2}\left\\{6l^{-2}\left(l^{-2}A_{m}-\ddot{A}_{m}\right)Y_{2m}\;du^{2}\right.$
$\displaystyle+\left(\partial_{u}\left(l^{-2}A_{m}-\ddot{A}_{m}\right)Y_{A}^{2m}\right.$
$\displaystyle\left.-\partial_{u}\left(4l^{-2}B_{m}-\ddot{B}_{m}\right)X_{A}^{2m}\right)dudx^{A}$
$\displaystyle+\left(\left[-\frac{3}{2}\left(l^{-2}A_{m}-\ddot{A}_{m}\right)\right.\right.$
$\displaystyle\left.+\frac{1}{2}\partial_{u}^{2}\left(l^{-2}A_{m}-\ddot{A}_{m}\right)\right]Y_{AB}^{2m}$
$\displaystyle-3\left(l^{-2}A_{m}-\ddot{A}_{m}\right)Y_{2m}\;S_{AB}$
$\displaystyle\left.\left.-\frac{1}{2}\partial_{u}^{2}\left(4B_{m}-l^{2}\ddot{B}_{m}\right)X_{AB}^{2m}\right)dx^{A}dx^{B}\right\\}$
(87)

and

$\displaystyle\mathcal{B}_{ab}dx^{a}dx^{b}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}\sum_{m=-2}^{2}\left\\{6l^{-2}\left(l^{-2}B_{m}-\ddot{B}_{m}\right)Y_{2m}\;du^{2}\right.$
$\displaystyle-\left(\partial_{u}\left(4l^{-2}A_{m}-\ddot{A}_{m}\right)Y_{A}^{2m}\right.$
$\displaystyle\left.-\partial_{u}\left(l^{-2}B_{m}-\ddot{B}_{m}\right)X_{A}^{2m}\right)dudx^{A}$
$\displaystyle+\left(\left[-\frac{3}{2}\left(l^{-2}B_{m}-\ddot{B}_{m}\right)\right.\right.$
$\displaystyle\left.+\frac{1}{2}\partial_{u}^{2}\left(l^{-2}B_{m}-\ddot{B}_{m}\right)\right]Y_{AB}^{2m}$
$\displaystyle\left.-3\left(l^{-2}B_{m}-\ddot{B}_{m}\right)Y_{2m}\;S_{AB}\right.$
$\displaystyle\left.\left.+\frac{1}{2}\partial_{u}^{2}\left(4A_{m}-l^{2}\ddot{A}_{m}\right)X_{AB}^{2m}\right)dx^{A}dx^{B}\right\\}$
(88)

We have explicitly verified that these expressions are symmetric, transverse
$\mathcal{E}_{ab}n^{b}\mathrel{\mathop{\widehat{=}}}0\mathrel{\mathop{\widehat{=}}}\mathcal{B}_{ab}n^{b}$
and traceless
$q^{ab}\mathcal{E}_{ab}\mathrel{\mathop{\widehat{=}}}0\mathrel{\mathop{\widehat{=}}}q^{ab}\mathcal{B}_{ab}$.
In taking the limit $l\to\infty$, one needs to be careful to rescale
$\mathcal{E}_{ab}$ and $\mathcal{B}_{ab}$; otherwise, due to the overall
factor of $l^{2}$ in the definition in Eqs. (83)-(84), this limit trivially
diverges. The flat limit is

$\displaystyle\lim_{l\to\infty}l^{-2}\mathcal{E}_{ab}dx^{a}dx^{b}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}\sum_{m=-2}^{2}-2\partial_{u}^{4}B_{m}X_{AB}^{2m}dx^{A}dx^{B}$
(89) $\displaystyle\lim_{l\to\infty}l^{-2}\mathcal{B}_{ab}dx^{a}dx^{b}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}\sum_{m=-2}^{2}2\partial_{u}^{4}A_{m}X_{AB}^{2m}dx^{A}dx^{B}\;.$
(90)

Note that the parity-even solution, which is also called an electric solution,
contributes to the magnetic part of the Weyl tensor and vice versa. There is
no contradiction here, as the names electric and magnetic refer to very
different notions.

In this linearized limit one can also explicitly show that the $\ell=1$ modes
in $U_{(0)}^{A}$ do not contribute to $\mathcal{E}_{ab}$ nor to
$\mathcal{B}_{ab}$.111111To show this, first decompose $U_{A}^{(0)}$ into an
‘electric’ and ‘magnetic’ part:
$U_{A}^{(0)}=D_{A}f+\epsilon_{A}^{\;\;B}D_{B}g$ and use that if $f$ and $g$
are $\ell=1$ modes $D_{A}D_{B}f=-\gamma_{AB}f$ and similarly for $g$. This
further supports the interpretation of those modes as non-radiative.

##  Appendix B Change of coordinates for the non-linear Robinson-Trautman
solution

In this appendix we show that the non-linear Robinson-Trautman solution can be
accommodated within the asymptotic conditions in Eq. (2). Starting from the
solution in Eq. (72) one can perform the following asymptotic change of
coordinates

$\displaystyle u\rightarrow$ $\displaystyle
f\left(u,\theta\right)+\frac{\ell^{2}\csc\theta\left(\sin\theta
h^{\prime}-\sin h\right)}{P}\frac{1}{r}+\frac{U_{2}}{r^{2}}+\dots,$
$\displaystyle r\rightarrow$ $\displaystyle\sin\theta\csc
hP\,r+R_{0}+\frac{R_{1}}{r}+\dots,$ $\displaystyle\theta\rightarrow$
$\displaystyle
h\left(u,\theta\right)-\frac{Pf^{\prime}}{r}-\frac{1}{2}\ell^{2}\left(h^{\prime\prime}+\cot\theta
h^{\prime}\right.$ $\displaystyle\left.-\frac{1}{2}\csc^{2}\theta\sin
2h\right)\frac{1}{r^{2}}+\dots,$ $\displaystyle\phi\rightarrow$
$\displaystyle\phi,$

with

$\displaystyle U_{2}$ $\displaystyle=\frac{\ell^{4}\left(\csc\theta\sin
h-h^{\prime}\right)}{4f^{\prime}P^{3}}\left[P\left(-2h^{\prime\prime}-2\cot\theta
h^{\prime}+\csc^{2}\theta\sin(2h)\right)\right.$
$\displaystyle\left.-2\left(\csc\theta\sin
h-h^{\prime}\right)\left(f^{\prime}\left(\partial_{f}P\right)+\left(\partial_{h}P\right)\left(h^{\prime}+\csc\theta\sin
h\right)\right)\right],$ $\displaystyle R_{0}$
$\displaystyle=\frac{1}{2}\ell\sin\theta\csc
h\left(\frac{\ell\left(-h^{\prime\prime}-\cot\theta
h^{\prime}+\frac{1}{2}\csc^{2}\theta\sin
2h\right)}{f^{\prime}}-\frac{f^{\prime}P\left(\partial_{h}P\right)}{\ell}+\frac{2\ell\left(\partial_{f}P\right)\left(h^{\prime}-\csc\theta\sin
h\right)}{P}\right),$ $\displaystyle R_{1}$
$\displaystyle=\frac{\ell^{4}\csc^{3}\theta\sin^{3}h}{16f^{\prime
2}P^{5}}\left\\{-2P^{2}\left(\partial_{h}P\right)\left(h^{\prime}\sin\theta\csc
h-1\right)^{2}\left(h^{\prime}\sin\theta\csc
h+1\right)\left(8f^{\prime}\sin\theta\csc
h\left(\partial_{f}P\right)\right.\right.$
$\displaystyle\left.+3\left(\partial_{h}P\right)\left(h^{\prime}\sin\theta\csc
h+1\right)\right)+4P^{3}\left(\sin^{2}\theta h^{\prime
2}\csc^{2}h-1\right)\left[4\sin\theta f^{\prime}\csc
h\left(\partial_{f}\partial_{h}P\right)\left(h^{\prime}\sin\theta\csc
h-1\right)\right.$ $\displaystyle\left.+2\partial_{h}^{2}P\left(h^{\prime
2}\sin^{2}\theta\csc^{2}h-1\right)+\left(\partial_{h}P\right)\left(\sin\theta\csc^{2}h\left(\sin\theta
h^{\prime\prime}+\cos\theta h^{\prime}\right)-\cot h\right)\right]$
$\displaystyle-8\ell^{2}P\left(\partial_{f}^{2}P\right)\left(h^{\prime}\sin\theta\csc
h-1\right)^{3}\left(h^{\prime}\sin\theta\csc
h+1\right)+8\ell^{2}\left(\partial_{f}P\right)^{2}\left(h^{\prime}\sin\theta\csc
h-1\right)^{3}\left(h^{\prime}\sin\theta\csc h+1\right)$
$\displaystyle+\csc^{2}hP^{4}\left[2h^{\prime\prime
2}\sin^{4}\theta\csc^{2}h+4h^{\prime\prime}\sin^{2}\theta\left(h^{\prime}\sin\theta\csc^{2}h\left(\cos\theta-2h^{\prime}\sin\theta\cot
h\right)+\cot h\right)\right.$
$\displaystyle+h^{\prime}\left(h^{\prime}\sin^{2}\theta\csc^{2}h\left(8h^{\prime}\sin\theta\left(h^{\prime}\sin\theta\csc^{2}h-\cos\theta\cot
h\right)+4\cos(2h)+\cos(2\theta)-11\right)+2\sin(2\theta)\cot h\right)$
$\displaystyle\left.\left.-3\cos(2h)+5\right]\right\\}$

and $P=P\left(f\left(u,\theta\right),h\left(u,\theta\right)\right)$. The
functions $f$ and $h$ are required to satisfy the following conditions

$h^{\prime 2}+\frac{P^{2}f^{\prime 2}}{\ell^{2}}=\csc^{2}\theta\sin^{2}h,$
(91)
$P\left(\dot{f}h^{\prime}-f^{\prime}\dot{h}\right)=\csc^{2}\theta\sin^{2}h.$
(92)

Note that in the linear approximation, Eq (91) indicates that $h=\theta$,
while Eq. (92) implies Eq. (75), i.e., $\dot{f}=-\omega$. In the flat limit
($\ell\rightarrow\infty$), Eq. (91) implies that $h=\theta$, while according
to Eq. (92) one has $\dot{f}=P^{-1}$. This condition is the same as that
obtained in vonderGonna:1997sh .

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