# Extension property of continuous functions in a Riemannain manifold with a
pole

Absos Ali Shaikh1∗ and Chandan Kumar Mondal2

###### Abstract.

The Brouwer fixed point theorem says that any continuous function from disc to
itself has a fixed point. By using simple geometrical technique we have
generalized the result in manifold and proved that any continuous function on
the boundary of a bounded convex domain of a $2$-dimensional Riemannian
manifold with a pole having at least one fixed point can be extended to the
convex domain without any interior fixed point.

††footnotetext: ∗ Corresponding author
$\mathbf{2010}$ Mathematics Subject Classification: 53C22, 58C30.
Key words and phrases: Convex set, geodesic convex set, fixed point theorem,
Riemannian manifold.

##  1\. Introduction

The well-known classical Brouwer fixed point theorem states that any
continuous function from a closed disc $D$ to itself has a fixed point. An
amazing application of this theorem is the stirring of coffee. If someone
stairs the coffee, then there must be some points on the top of the coffee
where the points are not moving at all. But what will happen if the boundary
points are fixed? One can easily stair the coffee so that the boundary points
are not moving as well as there are some fixed inner points also. Naturally
the question arises is there any process of stirring so that boundary points
are moving but not any inner point. If the face of the coffee mug is circle,
then there is a positive solution, see [1]. But in case where the face is not
circle? For any arbitrary shape the answer is still unknown. In this paper we
have given a positive answer of this question in the case of bounded geodesic
convex subset of a $2$-dimensional Riemannian manifold.

A subset $C$ in the Euclidean plane is said to be convex if any two points can
be joined by a straight line which lies in $C$. The boundary of disc $D$ and
$C$ will be denoted by $S$ and $\partial C$ respectively. A subset $A$ of the
Riemannian manifold $M$ is said to be geodesic convex [5] if for any two
points, the minimal geodesic connecting them lies in $A$. A point $o$ of the
Riemannian manifold is called a pole [3] if the map $exp_{o}$ is a
diffeomorphism from $T_{o}M$ to $M$. Riemannian manifold with a pole will be
denoted by $(M,o)$. In $(M,o)$ any geodesic starting from $o$ is minimal [4].
A Riemannian manifold with a pole is automatically complete.

The mathematical formulation of the above question can be stated as follows:
Is it possible to extend any continuous function on the boundary of a convex
domain with some fixed points in a continuous function of the convex domain
without any interior fixed point? By using the technique of complex analysis
Brown and Greene [1] proved the following:

######  Theorem 1.1.

[1] Given a continuous function $f:S\rightarrow S$ with at least one fixed
point, there exists a map $G:D\rightarrow S$ such that $G|_{S}=f$ and in
particular $G$ has no fixed point on $Int(D)$.

We have given an alternative geometrical proof of the result by using rotation
and translation of circle. The technique developed in this proof will be used
in proving the main result of the paper. By using the alternative technique of
the proof of Theorem 1.1, we have given a purely geometrical proof of the
following:

######  Theorem 1.2.

Let $C$ be a convex subset of the Euclidean plane. Then any continuous
function $f:\partial C\rightarrow\partial C$ with at least one fixed point can
be extended to a continuous function $\psi:C\rightarrow\partial C$.

######  Corollary 1.2.1.

Let $T$ be a starshaped subset of the Euclidean plane. Then any continuous
function $f:\partial T\rightarrow\partial T$ with at least one fixed point can
be extended to a continuous function $\xi:T\rightarrow\partial T$.

The main result of this paper is the following:

######  Theorem 1.3.

Suppose $(M,o)$ is a $2$-dimensional Riemannian manifold with a pole $o$ and
$V$ is a bounded closed convex subset of $M$ with boundary $\partial V$ such
that $o\in V$. Any continuous function $f:\partial V\rightarrow\partial V$
with a fixed point can be extended to a continuous function from $V$ to $V$
without any interior fixed point.

##  2\. Proof of the results

Proof of Theorem 1.1: Suppose that $p\in S$ is the fixed point of $f$, see
Figure 1. The disc $D$ can be covered by the concentric circles $S_{t}$ of
radius $t$ for $t\in[0,1]$, i.e., $D=\cup_{t\in[0,1]}S_{t}$. Now consider a
continuous

bijection $i:[0,1]\rightarrow[0,2\pi]$ such that $i(0)=0$ and $i(1)=2\pi$. Let
$q_{t}\in D$. Then there exists $t\in[0,1]$ such that $q_{t}\in S_{t}$.
Suppose $\overline{oq_{t}}$ makes an angle $\theta$ with $\overline{op}$. Now
rotate the circle $S_{t}$ clockwise by an angle $i(t)$. The action is denoted
by $\theta_{t}$. Therefore the image of $q$ is
$q^{\prime}_{t}=\theta_{t}(q_{t})$ and the angle $\angle q^{\prime}_{t}op$ is
$i(t)+\theta$. Now extend the line $oq^{\prime}_{t}$ so that it intersects $S$
at the point $q^{\prime\prime}_{t}$. Now define the function $G:D\rightarrow
D$ by $G(q_{t})=f(q^{\prime\prime}_{t})\ \text{ for all }q_{t}\in D.$ It can
easily be seen that $G(o)=p$ and for any boundary point $q\in S$, $G(q)=f(q)$.
Since the function $G$ is constructed by using the composition of rotation,
translation and $f$ function, $G$ is continuous. Observe that image of $G$ is
the boundary set $S$. Therefore, $G$ can not have any interior fixed point.
Hence, the result is proved. $\Box$

Proof of Theorem 1.2: Since $C$ is a convex subset, there exists a unique
circumscribed circle $S$ of $C$, see [6]. Now fix a point $O$ inside $C$ such
that $O\in Int(C)$, see Figure 2. Take a point $q\in C$, then extend the
straight line from $O$ to $q$ so that it intersects $S$ at the point
$q^{\prime}$. Therefore, we get a continuous function $h:C\rightarrow S$,
defined by $h(q)=q^{\prime}$ for all $q\in C$. Again, if we restrict $h$ on
$\partial C$, then we get a bijection from $\partial C$ to $S$, which will be
denoted by $\partial h$.

Now the function $f^{\prime}:S\rightarrow S$ defined by
$f^{\prime}(q)=\partial h\circ f\circ\partial h^{-1}(q)\ \text{ for all }q\in
S.$ This map is continuous being the composition of continuous maps. Suppose
$p\in\partial C$ is a fixed point of $f$ and $p^{\prime}=\partial h(p)\in S$.
Now $f^{\prime}(p^{\prime})=\partial h\circ f\circ\partial
h^{-1}(p^{\prime})=\partial h\circ f(p)=\partial h(p)=p^{\prime}$. This
implies that $f^{\prime}$ has a fixed point. Now by the Theorem 1.1,
$f^{\prime}$ can be extended to a continuous map $G^{\prime}:D\rightarrow S$
such that $G^{\prime}|_{S}=f^{\prime}$. Consider the function
$\psi:C\rightarrow\partial C$ defined by $\psi(q)=\partial h^{-1}\circ
G^{\prime}\circ h(q)$. If we take $q\in\partial C$, then $\psi(q)=\partial
h^{-1}\circ G^{\prime}\circ h(q)=\partial h^{-1}\circ
G^{\prime}(q^{\prime})=\partial h^{-1}(f^{\prime}(q^{\prime}))=\partial
h^{-1}\circ\partial h\circ f\circ\partial h^{-1}(q^{\prime})=f(q).$ Hence, we
get a continuous function $\psi:C\rightarrow\partial C$ such that
$\psi|_{\partial C}=f$.$\Box$

Proof of Corollary 1.2.1: To prove it just choose the focus of $T$ as the
point $O$ and run through the arguments of the Theorem 1.2 and Theorem 1.1.
Hence the corollary is proved.$\Box$

Proof of Theorem 1.3: Since $V$ is a convex subset of $M$ containing $o$, the
set $V_{0}=\\{exp^{-1}_{o}q:q\in V\\}\subset T_{o}M$ is starshpaed with focus
$o$. Now consider the map $g:\partial V_{0}\rightarrow\partial V_{0}$, defined
by $g(q)=exp^{-1}_{o}\circ f\circ exp_{o}(q)$ for all $q\in\partial V_{0}$.
Suppose $p\in\partial V$ is a fixed point of $f$ and take
$p^{\prime}=exp^{-1}_{o}p$. Now $g(p^{\prime})=exp^{-1}_{o}\circ f\circ
exp_{o}(q^{\prime})=exp^{-1}_{o}\circ f(p)=exp^{-1}_{p}=p^{\prime}.$ So $g$
has a fixed point on $\partial V_{o}$. Hence in view of Corollary 1.2.1, we
get a function $g^{\prime}:V_{o}\rightarrow\partial V_{o}$ such that
$g^{\prime}|_{\partial V_{o}}=g$.

Now the function $H:V\rightarrow V$ defined by $H(q)=exp_{o}\circ
g^{\prime}\circ exp^{-1}_{o}(q)$ for $q\in V$ is the extension of $f$ on $V$
and $H$ does not have any interior fixed point. Hence, the theorem is proved.
$\Box$

## References

  * [1] Brown, R. E. and Greene, R. E., An interior fixed point property of the disc, Amer. Math. Monthly, 101(1) (1994), 39–47.
  * [2] Carmo, M. P., Riemannian geometry, Birkhäuser, 1992.
  * [3] Gromoll, D., Meyer, W., On complete open manifolds of positive curvature, Ann. Math., 90 (1969), No. 1, 75–90.
  * [4] Innami, N., Shiohama, K. and Soga, T., The cut loci, conjugate loci and poles in a complete Riemannian manifold, Geom. Funct. Anal., 22 (2012), 1400–1406.
  * [5] Udrişte, C., Convex functions and optimization methods on Riemannian manifolds, Kluwer Academic Publisher, 1994.
  * [6] Zamfirescu, T., Inscribed and circumscribed circles to convex curves, Proc. Amer. Math. Soc., 80 (1980), No. 3, 455–457.

1The University of Burdwan,

Department of Mathematics,

Golapbag, Burdwan-713104,

West Bengal, India.

<EMAIL_ADDRESS><EMAIL_ADDRESS>

2Netaji Subhas Open University,

School of Sciences,

Durgapur- 713214, West Bengal,

<EMAIL_ADDRESS>


# Gravitational-wave tails of memory at 4PN order

David Trestinia,b,∗ and Luc Blancheta
a)${\mathcal{G}}{\mathbb{R}}\varepsilon{\mathbb{C}}{\mathcal{O}}$, Institut
d’Astrophysique de Paris,
UMR 7095, CNRS, Sorbonne Université, 98bis boulevard Arago, 75014 Paris,
France.
b) Laboratoire Univers et Théories, Observatoire de Paris,
Université PSL, Université Paris Cité, CNRS, F-92190 Meudon, France.

###### Abstract

We study a novel cubic nonlinear effect, the tails-of-memory, which consist of
a combination of the tail effect (backscattering of linear gravitational waves
against the curvature of spacetime generated by the source) and the memory
effect (due to reradiation of gravitational waves by linear gravitational
waves themselves). Our final result is consistent with a straightforward
direct computation of the memory effect, but also involves many non-trivial
tail-like terms.

##  1 Introduction

The recent completion of the flux and phase evolution of gravitational waves
generated by compact binary systems at fourth and a half post-Newtonian order
(4.5PN), [1, 2, 3] necessitated the control of the nonlinear propagation
effects of gravitational waves through spacetime from the source to the
observer. These effects are encoded in the relation between the radiative
moments, which parametrize the multipolar structure of the waveform as
measured by an asymptotic far-away observer, and the so-called canonical
moments, which parametrize the _linearized_ vacuum metric outside the source.
The canonical moments are in turn connected _via_ a known procedure to the
parameters of the physical matter source. The relations between radiative and
canonical moments generically contain instantaneous terms, that only depend on
the canonical moments at retarded time $u=t-r/c$, but also _hereditary_ terms,
which entail a dependence of the radiative moments on the whole past history
of the source.

Some well-known hereditary effects are: the tail terms, which arise at 1.5PN
order beyond quadrupole radiation, due to an interaction between the ADM mass
of the system $\mathrm{M}$ and the canonical quadrupole $\mathrm{M}_{ij}$; the
memory effect at 2.5PN, due to the $\mathrm{M}_{ij}\times\mathrm{M}_{ij}$
interaction; and the tail-of-tail
$\mathrm{M}\times\mathrm{M}\times\mathrm{M}_{ij}$ at 3PN. In this
contribution, we are interested in the
$\mathrm{M}\times\mathrm{M}_{ij}\times\mathrm{M}_{ij}$ interaction which
enters at 4PN order, dubbed _tails-of-memory_ , which can be represented by
the following three Feynman diagrams.

(a)

(b)

(c)

Figure 1: Feynman diagrams associated to the tails-of-memory.

##  2 Multipolar post-Minkowkian setup

In previous applications, the vacuum metric exterior to an isolated source was
investigated using a multipolar post-Minkowskian (MPM) construction in
_harmonic_ coordinates [10], parametrized by two sets of canonical moments
$\mathrm{M}_{L}$ and $\mathrm{S}_{L}$. The asymptotic behavior of the metric
for $r\rightarrow\infty$ (and $u=$ const) then exhibited logarithms, which
could ultimately be removed by a well-chosen coordinate transformation. Here,
we present the _radiative_ MPM construction introduced in Ref. [9], see Ref.
[5] for a detailed comparison. This construction has the great advantage of
automatically avoiding the appearance of far-zone logarithms. The starting
point is to recast the vacuum field equations of general relativity in the
form of a wavelike equation by defining the “gothic” metric deviation, namely
$h^{\mu\nu}\equiv\sqrt{-g}g^{\mu\nu}-\eta^{\mu\nu}$ with standard notation. In
a general coordinate system, the equations read

$\Box h^{\mu\nu}-\partial H^{\mu\nu}=\Lambda^{\mu\nu}\bigl{[}h,\partial
h,\partial^{2}h\bigr{]}\,,$ (1)

where $\Box$ is the flat d’Alembertian operator, the divergence of the metric
is $H^{\mu}\equiv\partial_{\nu}h^{\mu\nu}$ and we introduce the useful
shorthand notation $\partial\theta^{\mu\nu}\equiv
2\partial^{(\mu}\theta^{\nu)}-\eta^{\mu\nu}\partial_{\rho}\theta^{\rho}$ for
any $\theta^{\mu}$. With the MPM construction the radiative metric is formally
decomposed to any PM order as
$h^{\mu\nu}=\sum_{n=1}^{+\infty}G^{n}h_{n}^{\mu\nu}$, where each of the
coefficients is a functional of two sets of STF multipole moments
$\overline{\mathrm{M}}_{L}$ and $\overline{\mathrm{S}}_{L}$ (different from
$\mathrm{M}_{L}$ and $\mathrm{S}_{L}$ in harmonic coordinates) which
parametrize the linear approximation $h_{1}^{\mu\nu}$ defined as

$h^{\mu\nu}_{1}\equiv
h_{\text{can}\,1}^{\mu\nu}[\overline{\mathrm{M}}_{L},\overline{\mathrm{S}}_{L}]+\partial\xi_{1}^{\mu\nu}\,,$
(2)

where the “canonical” linear metric reads explicitly 111We use the symmetric-
trace-free (STF) formalism, with $L=i_{1}\cdots i_{\ell}$ a multi-index with
$\ell$ spatial indices, and $\hat{n}_{L}=\text{STF}[n_{i_{1}}\cdots
n_{i_{\ell}}]$ the usual STF harmonics of order $\ell$. Similarly we shall
denote
$\hat{\partial}_{L}=\text{STF}[\partial_{i_{1}}\cdots\partial_{i_{\ell}}]$.
[11, 10]

$\displaystyle
h_{\text{can}\,1}^{00}[\overline{\mathrm{M}}_{L},\overline{\mathrm{S}}_{L}]$
$\displaystyle=-\frac{4}{c^{2}}\sum_{\ell\geqslant
0}\frac{(-)^{\ell}}{\ell!}\partial_{L}\left[\frac{1}{r}\,\overline{\mathrm{M}}_{L}\left(t-\frac{r}{c}\right)\right]\,,$
(3a) $\displaystyle
h_{\text{can}\,1}^{0i}[\overline{\mathrm{M}}_{L},\overline{\mathrm{S}}_{L}]$
$\displaystyle=\frac{4}{c^{3}}\sum_{\ell\geqslant
1}\frac{(-)^{\ell}}{\ell!}\left\\{\partial_{L-1}\left[\frac{1}{r}\,\overline{\mathrm{M}}_{iL-1}^{(1)}\left(t-\frac{r}{c}\right)\right]+\frac{\ell}{\ell+1}\epsilon_{iab}\partial_{aL-1}\left[\frac{1}{r}\,\overline{\mathrm{S}}_{bL-1}\left(t-\frac{r}{c}\right)\right]\right\\}\,,$
(3b) $\displaystyle
h_{\text{can}\,1}^{ij}[\overline{\mathrm{M}}_{L},\overline{\mathrm{S}}_{L}]$
$\displaystyle=-\frac{4}{c^{4}}\sum_{\ell\geqslant
2}\frac{(-)^{\ell}}{\ell!}\left\\{\partial_{L-2}\left[\frac{1}{r}\,\overline{\mathrm{M}}_{ijL-2}^{(2)}\left(t-\frac{r}{c}\right)\right]+\frac{2\ell}{\ell+1}\partial_{aL-2}\left[\frac{1}{r}\,\epsilon_{ab(i}\overline{\mathrm{S}}_{j)bL-2}^{(1)}\left(t-\frac{r}{c}\right)\right]\right\\}\,.$
(3c)

It satisfies the harmonic gauge condition,
$\partial_{\nu}h_{\text{can}\,1}^{\mu\nu}=0$. The gauge vector in (2) is

$\xi_{1}^{\mu}=\frac{2\mathrm{M}}{c^{2}}\eta^{0\mu}\ln\left(\frac{r}{b_{0}}\right)\,,$
(4)

with $b_{0}$ an arbitrary constant length scale. The role of this gauge
transformation is to ensure that the retarded time $u=t-r/c$ is asymptotically
a null coordinate when $r\to\infty$.

Once the linearized solution is defined, the nonlinear corrections are readily
obtained by injecting the PM expansion into the field equations and solving
these equations iteratively. At any order $n$, the general equation to solve
is

$\Box h_{n}^{\mu\nu}-\partial H^{\mu\nu}_{n}=\Lambda_{n}^{\mu\nu}\,,$ (5)

where $\Lambda_{n}^{\mu\nu}\equiv\Lambda_{n}^{\mu\nu}[h_{1},\cdots,h_{n-1}]$
is built out from previous iterations. The construction of the solution is
done in three steps. We first construct a particular retarded solution to the
wave equation, satisfying $\Box u_{n}^{\mu\nu}=\Lambda_{n}^{\mu\nu}$, as

$u_{n}^{\mu\nu}=\mathop{\mathrm{FP}}_{B=0}\Box^{-1}_{\text{ret}}\biggl{[}\left(\frac{r}{r_{0}}\right)^{B}\Lambda_{n}^{\mu\nu}\biggr{]}\,,$
(6)

where $\Box^{-1}_{\text{ret}}$ is the standard retarded integral operator,
$r_{0}$ is an arbitrary length scale and taking the finite part
($\mathop{\mathrm{FP}}_{B=0}$) when the complex parameter $B\to 0$ takes care
of the divergence of the multipole expansion when $r\to 0$. We then construct
$v_{n}^{\mu\nu}$ such that it satisfies $\Box v_{n}^{\mu\nu}=0$ and
$\partial_{\nu}v_{n}^{\mu\nu}=-\partial_{\nu}u_{n}^{\mu\nu}$. Finally, one can
prove [9] that
$\Lambda_{n}^{\mu\nu}=r^{-2}{k^{\mu}k^{\nu}}\sigma_{n}(u,\mathbf{n})\,+\,{\mathcal{O}}\left({r^{-3}}\right)$
when $r\to\infty$, $u=$ const, where $k^{\mu}=(1,\mathbf{n})$ is the outgoing
Minkowskian vector. This allows us to define the gauge vector

$\xi_{n}^{\mu\nu}=\mathop{\mathrm{FP}}_{B=0}\Box^{-1}_{\text{ret}}\biggl{[}\left(\frac{r}{r_{0}}\right)^{B}\frac{k^{\mu}}{2r^{2}}\int_{0}^{+\infty}\mathrm{d}\tau\,\sigma_{n}(u-\tau,\mathbf{n})\biggr{]}\,.$
(7)

The solution of the radiative MPM construction is then given by

$h_{n}^{\mu\nu}\equiv
u_{n}^{\mu\nu}+v_{n}^{\mu\nu}+\partial\xi_{n}^{\mu\nu}\,,$ (8)

which satisfies (5) by construction. The gauge transformation
$\partial\xi_{n}^{\mu\nu}$ ensures that $u$ is a null coordinate to
$n^{\text{th}}$ order and removes the far-zone logarithms at that order.

Note that the usual _harmonic_ MPM construction is exactly the same as the
previous _radiative_ one, except that at each step $n\geqslant 1$, one sets
$\xi_{n}^{\mu}=0$, and our “seed” linearized metric (3) is parametrized by
_harmonic_ canonical moments $\mathrm{M}_{L}$ and $\mathrm{S}_{L}$ (note that
$\mathrm{M}=\overline{\mathrm{M}}$ for the ADM mass). Finally one can prove
that the metric in the _radiative_ MPM construction is free of far-zone
logarithms at all post-Minkowskian orders [9, 5], such that the leading order
contribution in $1/r$ in a transverse-traceless (TT) gauge has the multipolar
structure

$h_{ij}^{\text{TT}}=-\frac{4G}{c^{2}r}\perp_{ij,ab}^{\text{TT}}\sum_{\ell=2}^{+\infty}\frac{1}{c^{\ell}\ell!}\left(n_{L-2}\,\mathcal{U}_{abL-2}(u)-\frac{2\ell}{c(\ell+1)}n_{cL-2}\,\epsilon_{cd(a}\,\mathcal{V}_{b)dL-2}(u)\right)+\mathcal{O}\left(\frac{1}{r^{2}}\right)\,,$
(9)

where $\perp_{ij,ab}^{\text{TT}}$ is the usual TT projection operator and
$\mathcal{U}_{L}$ and $\mathcal{V}_{L}$ are the radiative moments. The
radiative moments differ from the canonical moments by nonlinear terms,

$\mathcal{U}_{L}=\mathrm{M}_{L}^{(\ell)}+{\mathcal{O}}(G)\,,\qquad\mathcal{V}_{L}=\mathrm{S}_{L}^{(\ell)}+{\mathcal{O}}(G)\,,$
(10)

##  3 The memory effect

The “memory” in the waveform of any radiative source can be expressed as the
following contributions to the _mass-type_ radiative moments
$\mathcal{U}_{L}^{\mathrm{mem}}$ (but $\mathcal{V}_{L}^{\mathrm{mem}}=0$), see
_e.g._ [8, 7, 6]:

$\mathcal{U}_{L}^{\mathrm{mem}}=\frac{2c^{\ell-2}(2\ell+1)!!}{(\ell+1)(\ell+2)}\int\mathrm{d}\Omega\,\hat{n}_{L}\int_{0}^{+\infty}\mathrm{d}\tau\,\frac{\mathrm{d}E_{\mathrm{gw}}}{\mathrm{d}t\mathrm{d}\Omega}(u-\tau)\,,$
(11)

where the effective power density of gravitational waves is given by

$\frac{\mathrm{d}E_{\mathrm{gw}}}{\mathrm{d}u\mathrm{d}\Omega}=\lim_{r\rightarrow+\infty}\frac{r^{2}c^{3}}{32\pi
G}\,\dot{h}_{ij}^{\text{TT}}\,\dot{h}_{ij}^{\text{TT}}\,.$ (12)

For our purpose, we are interested in the dominant memory contribution due to
the $\mathrm{M}_{ij}\times\mathrm{M}_{ij}$ interaction 222Strictly speaking
the memory is the non-oscillatory part (DC) of the waveform. followed by the
subdominant one $\mathrm{M}\times\mathrm{M}_{ij}\times\mathrm{M}_{ij}$ arising
at the 1.5PN order beyond leading order. For this, we need the expression of
the radiative quadrupole at 1.5PN order [7],

$\mathcal{U}_{ij}=\mathrm{M}_{ij}^{(2)}+\frac{2G\mathrm{M}}{c^{3}}\int_{0}^{+\infty}\mathrm{d}\tau\,\mathrm{M}_{ij}^{(4)}(u-\tau)\left[\ln\left(\frac{c\tau}{2b_{0}}\right)+\frac{11}{12}\right]+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,.$
(13)

Injecting (13) into the TT waveform (9) and then in (12)–(11), and restricting
attention to the $\mathrm{M}_{ij}\times\mathrm{M}_{ij}$ and
$\mathrm{M}\times\mathrm{M}_{ij}\times\mathrm{M}_{ij}$ interactions, a little
tensor algebra leads to the result

$\displaystyle\mathcal{U}_{ij}^{\text{mem}}=$
$\displaystyle-\frac{2G}{7c^{5}}\int_{0}^{+\infty}\\!\\!\mathrm{d}\tau\,\mathrm{M}_{a\langle
i}^{(3)}(u-\tau)\,\mathrm{M}_{j\rangle a}^{(3)}(u-\tau)$
$\displaystyle-\frac{8G^{2}\mathrm{M}}{7c^{8}}\int_{0}^{+\infty}\\!\\!\mathrm{d}\rho\,\mathrm{M}_{a\langle
i}^{(3)}(u-\rho)\int_{0}^{+\infty}\\!\\!\mathrm{d}\tau\ln\left(\frac{c\tau}{2b_{0}}\right)\mathrm{M}_{j\rangle
a}^{(5)}(u-\rho-\tau)\,,$ (14)

where we have neglected the term associated with the constant $\frac{11}{12}$
in (13), which is instantaneous, and does not contribute to the memory effect.
Moreover, we can integrate the cubic term in (3) by parts — the surface term
is actually a tail term, and can be discarded as it does not contribute to the
memory effect — and find

$\displaystyle\mathcal{U}_{ij}^{\text{mem}}=$
$\displaystyle-\frac{2G}{7c^{5}}\int_{0}^{+\infty}\\!\\!\mathrm{d}\tau\,\mathrm{M}_{a\langle
i}^{(3)}(u-\tau)\,\mathrm{M}_{j\rangle a}^{(3)}(u-\tau)$
$\displaystyle+\frac{8G^{2}\mathrm{M}}{7c^{8}}\int_{0}^{+\infty}\\!\\!\mathrm{d}\rho\,\mathrm{M}_{a\langle
i}^{(4)}(u-\rho)\int_{0}^{+\infty}\\!\\!\mathrm{d}\tau\ln\left(\frac{c\tau}{2b_{0}}\right)\mathrm{M}_{j\rangle
a}^{(4)}(u-\rho-\tau)\,.$ (15)

##  4 Obtaining the full cubic expression for the tails-of-memory

In the radiative MPM iteration, the wave equation we need to solve can always
be reduced to a wave equation whose source term admits a definite
multipolarity $\ell$, say

$\Box\Psi_{L}=\hat{n}_{L}\,S(r,t-r/c)\,,$ (16)

where $S(r,u)$ is an arbitrary function of $r=|\mathbf{x}|$ and $u=t-r/c$,
that verifies straightforward smoothness properties, and tends sufficiently
rapidly to zero when $r\to 0$. 333Namely $S(r,u)=\mathcal{O}(r^{\ell+5})$ when
$r\rightarrow 0$ with $t-r/c$ kept fixed, see Theorem 6.1 of [10]. If this
condition is not satisfied, we can multiply the source by the regularization
factor $(r/r_{0})^{B}$ with $B\in\mathbb{C}$, which ensures convergence when
$\Re(B)$ is large enough, and take the finite part in the $B\rightarrow 0$
expansion in the end. We define

$R_{\alpha}(\rho,s)\equiv\rho^{\ell}\int_{\alpha}^{\rho}\mathrm{d}\lambda\,\frac{(\rho-\lambda)^{\ell}}{\ell!}\left(\frac{2}{\lambda}\right)^{\ell-1}S(\lambda,s)\,,$
(17)

where $\alpha$ is an arbitrary constant. Then the solution of (16) can be
written as

$\Psi_{L}=c\int_{-\infty}^{t-r}\mathrm{d}s\,\hat{\partial}_{L}\left[\frac{R_{\alpha}\left(\frac{t-r/c-s}{2},s\right)-R_{\alpha}\left(\frac{t+r/c-s}{2},s\right)}{r}\right]\,,$
(18)

see Eq. (6.4) in Ref. [10]. Thanks to this equation, and after an involved
computation detailed in [4], we obtain the radiative quadrupole
$\mathcal{U}_{ij}$ in term of $\overline{\mathrm{M}}_{ij}$ and $\mathrm{M}$.
We then go back to the usual harmonic-type moments, that are necessary to
relate our result to previous ones, using [5]

$\displaystyle\overline{\mathrm{M}}_{ij}=\mathrm{M}_{ij}$ $\displaystyle\
+\frac{G\,\mathrm{M}}{c^{3}}\,\mathrm{M}_{ij}^{(1)}\left[-\frac{26}{15}+2\ln\left(\frac{r_{0}}{b_{0}}\right)\right]+\frac{G^{2}\mathrm{M}^{2}}{c^{6}}\,\mathrm{M}_{ij}^{(2)}\left[\frac{124}{45}-\frac{52}{15}\ln\left(\frac{r_{0}}{b_{0}}\right)+2\ln^{2}\left(\frac{r_{0}}{b_{0}}\right)\right]$
$\displaystyle\
+\frac{G^{2}\mathrm{M}}{c^{8}}\bigg{[}-\frac{8}{21}\,\mathrm{M}_{a\langle
i}\mathrm{M}_{j\rangle a}^{(4)}-\frac{8}{7}\,\mathrm{M}^{(1)}_{a\langle
i}\mathrm{M}_{j\rangle a}^{(3)}-\frac{8}{9}\,\mathrm{M}^{(3)}_{a\langle
i}\mathrm{S}_{j\rangle|a}\bigg{]}+\mathcal{O}\left(\frac{1}{c^{9}}\right)\,.$
(19)

One must then take into account all the lower-order contributions to the
radiative quadrupole in order to correctly implement this redefinition.
Finally, we find that the tails-of-memory read

$\displaystyle\mathcal{U}_{ij}^{\mathrm{M}\times\mathrm{M}_{ij}\times\mathrm{M}_{ij}}$
$\displaystyle=\frac{8G^{2}\mathrm{M}}{7c^{8}}\Bigg{\\{}\int_{0}^{+\infty}\\!\mathrm{d}\rho\,\mathrm{M}_{a\langle
i}^{(4)}(u-\rho)\int_{0}^{+\infty}\\!\mathrm{d}\tau\,\mathrm{M}_{j\rangle
a}^{(4)}(u-\rho-\tau)\left[\ln\left(\frac{c\tau}{2r_{0}}\right)-\frac{1613}{270}\right]$
$\displaystyle\quad-\frac{5}{2}\int_{0}^{+\infty}\\!\mathrm{d}\tau\,(\mathrm{M}^{(3)}_{a\langle
i}\mathrm{M}^{(4)}_{j\rangle
a})(u-\tau)\left[\ln\left(\frac{c\tau}{2r_{0}}\right)+\frac{3}{2}\ln\left(\frac{c\tau}{2b_{0}}\right)\right]$
$\displaystyle\quad-3\int_{0}^{+\infty}\\!\mathrm{d}\tau\,(\mathrm{M}^{(2)}_{a\langle
i}\mathrm{M}^{(5)}_{j\rangle
a})(u-\tau)\left[\ln\left(\frac{c\tau}{2r_{0}}\right)+\frac{11}{12}\ln\left(\frac{c\tau}{2b_{0}}\right)\right]$
$\displaystyle\quad-\frac{5}{2}\int_{0}^{+\infty}\\!\mathrm{d}\tau\,(\mathrm{M}^{(1)}_{a\langle
i}\mathrm{M}^{(6)}_{j\rangle
a})(u-\tau)\left[\ln\left(\frac{c\tau}{2r_{0}}\right)+\frac{3}{10}\ln\left(\frac{c\tau}{2b_{0}}\right)\right]$
$\displaystyle\quad-\int_{0}^{+\infty}\\!\mathrm{d}\tau\,(\mathrm{M}_{a\langle
i}\mathrm{M}^{(7)}_{j\rangle
a})(u-\tau)\left[\ln\left(\frac{c\tau}{2r_{0}}\right)-\frac{1}{4}\ln\left(\frac{c\tau}{2b_{0}}\right)\right]$
$\displaystyle\quad-2\mathrm{M}^{(2)}_{a\langle
i}\int_{0}^{+\infty}\\!\mathrm{d}\tau\,\mathrm{M}^{(5)}_{j\rangle
a}(u-\tau)\left[\ln\left(\frac{c\tau}{2r_{0}}\right)+\frac{27521}{5040}\right]$
$\displaystyle\quad-\frac{5}{2}\,\mathrm{M}^{(1)}_{a\langle
i}\int_{0}^{+\infty}\\!\mathrm{d}\tau\,\mathrm{M}^{(6)}_{j\rangle
a}(u-\tau)\left[\ln\left(\frac{c\tau}{2r_{0}}\right)+\frac{15511}{3150}\right]$
$\displaystyle\quad+\frac{1}{2}\,\mathrm{M}_{a\langle
i}\int_{0}^{+\infty}\\!\mathrm{d}\tau\,\mathrm{M}^{(7)}_{j\rangle
a}(u-\tau)\left[\ln\left(\frac{c\tau}{2r_{0}}\right)-\frac{6113}{756}\right]\,\Bigg{\\}}\,.$
(20)

In particular, we recover in the first line the memory term of (3) but we also
obtain many non-trivial tail-like terms. This result was crucial for the
completion of the 4PN waveform and 4.5PN flux for compact binary system on
quasicircular orbits [2, 3]. Finally note that the two constants $r_{0}$ and
$b_{0}$ nicely vanish in the final results [2, 3].

## Acknowledgments

The authors thank Laura Bernard, Guillaume Faye, Quentin Henry, François
Larrouturou and Stavros Mougiakakos for interesting discussions.

## References

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