# Gravitational radiation from a binary system in odd-dimensional spacetime

M. Khlopunov<EMAIL_ADDRESS>Faculty of Physics, Lomonosov
Moscow State University, Moscow, 119899, Russia
Institute of Theoretical and Mathematical Physics, Lomonosov Moscow State
University, Moscow, 119991, Russia  D.V. Gal’tsov<EMAIL_ADDRESS>Faculty
of Physics, Lomonosov Moscow State University, Moscow, 119899, Russia

###### Abstract

We explore possible manifestations of an odd number of extra dimensions in
gravitational radiation, which are associated with violation of Huygens’
principle in flat odd-dimensional spacetime. Our setup can be regarded as the
limit of an infinite compactification radius in ADD model and is not viable as
realistic cosmology, but it still may be useful as a simple analytically
solvable model catching certain features of more realistic scenarios. The
model consists of two point masses moving inside a flat three-dimensional
brane, embedded in a five-dimensional Minkowski space and interacting only
through a massless scalar field localized on the same brane, while
gravitational radiation is emitted into the bulk. This setup avoids the
difficulties associated with taking into account the gravitational stresses
binding the system, which require the cubic terms in the perturbative
gravitational Lagrangian, and permits to limit ourselves to linearized theory.
We calculate radiation in a linearized five-dimensional gravity generalizing
the Rohrlich-Teitelboim approach to extract the emitted part of the retarded
gravitational field. The source term consists of a local contribution from
point particles and a non-local contribution from scalar field stresses, which
is calculated using the DIRE approach to post-Newtonian expansions. In the
nonrelativistic limit, we find an analog of the quadrupole formula containing
an integral over the history of the particles’ motion preceding the retarded
time. We also show that, for an observer on the brane, the radiation contains
a third polarization: the breathing mode.

###### pacs:

04.20.Jb, 04.50.+h, 04.65.+e

##  I Introduction

In the last twenty years, the interest has arisen in the theories of
gravitation with extra spacetime dimensions. Extra dimensions are
intrinsically present in the string theory Green:1987sp , holography
Arefeva:2014kyw ; Cardoso:2013vpa , and various models proposed to solve the
hierarchy problem Arkani-Hamed:1998jmv ; Randall:1999ee ; Randall:1999vf  and
puzzles of cosmology Dvali:2000hr ; Gabadadze:2007dv ; Gabadadze:2004dq ;
Deffayet:2000uy ; Deffayet:2001pu  (for review see, e.g., Rubakov:2001kp ;
Maartens:2010ar ; Cheng:2010pt ). New tools in astrophysics, such as
gravitational-wave astronomy and black hole shadows, are opening up new ways
to experimentally explore extra dimensions Visinelli:2017bny ; Andriot2017 ;
Pardo:2018ipy ; Andriot:2019hay ; Corman:2020pyr ; Andriot:2021gwv ;
Vagnozzi:2019apd ; Banerjee:2019nnj  as was anticipated about two decades ago
Barvinsky:2003jf . Presence of extra dimensions can manifest itself in a
number of ways in gravitational-wave observations (for review see, e.g.,
Yu:2019jlb ). One of the most common for different theories sign of extra
dimensions is the additional polarisations of the gravitational waves
Andriot2017 , the experimental observation of which will become available when
more gravitational-wave observatories will start operation. For compact extra
dimensions, gravitational-wave signals acquire, also, the tower of massive,
high-frequency Kaluza-Klein modes Andriot2017 ; Barvinsky:2003jf ;
Andriot:2019hay ; Andriot:2021gwv , which, however, are currently unavailable
for observation. In some theories, gravitational waves detected by the
observer on the brane are modified due to the additional contributions into
the source term in the effective Einstein equations for the induced metric on
the brane Shiromizu:1999wj ; Maeda:2003ar ; Garcia-Aspeitia:2013jea ;
Kinoshita:2005nx . Also, in some theories, the falloff condition for the
gravitational-wave signal is modified at the cosmological distances, compared
to that for the counterpart electromagnetic signal Deffayet:2007kf . Although,
this effect was not observed in the GW170817 event Pardo:2018ipy , it can be
observed in future by LISA Corman:2020pyr . Also, some theories predict the
time gap between the observation of the gravitational-wave and electromagnetic
signals from the binary neutron star merger due to the former taking the
shortcut through the extra dimensions during its propagation to the observer
Yu:2016tar . The first constraints on the parameters of such theories have
already been obtained from the GW170817 event data Visinelli:2017bny .

Extra dimensions, also, modify the tidal deformabilities of the black holes
and neutron stars Chakravarti:2018vlt ; Cardoso:2019vof ; Chakravarti:2019aup
. In particular, higher-dimensional black holes acquire non-vanishing tidal
Love numbers, in contrast with the four-dimensional ones. As well, extra
dimensions manifest themselves in the quasinormal modes of the black holes,
which determine the spectrum of the gravitational radiation of the binary
system at the final stage of its merger Chakraborty:2017qve . Particularly,
presence of extra dimensions significantly increases the damping time of the
quasinormal oscillations of the black holes, providing the rigorous
constraints on the parameters of corresponding theories Mishra:2021waw .
Another promising tool to probe extra dimensions is the photographs of the
black holes shadows. In some theories, presence of extra dimensions results in
the tidal charge of the black holes, which plays the role similar to the
square of the electric charge, but can take the negative values Dadhich:2000am
; Aliev:2005bi , significantly modifying the shadows of the black holes.
Parameters of the corresponding theories have already been constrained by
observation of shadow of the supermassive black hole M87* Vagnozzi:2019apd ;
Banerjee:2019nnj ; Neves:2020doc .

For large extra dimensions one can expect manifestation of the difference
between even and odd-dimensional spacetimes. Recall that in even dimensions
the Green’s function of the flat space d’Alembert equation is localized on the
light cone, implying applicability of the standard procedure to calculate
radiation. In odd dimensions Green’s function has support also inside the
light cone, leading to violation of the Huygens’s principle and certain
complications in computation of radiation. For this reason only the problems
of radiation in even dimensions were considered in most of the literature
Kosyakov1999 ; Cardoso:2002pa ; Mironov:2006wi ; Mironov:2007nk ;
Cardoso:2007uy ; Kosyakov:2008wa , while the odd dimensions were mainly
discussed in the context of the radiation reaction force Galtsov:2001iv ;
Kazinski:2002mp ; Kazinski:2005gx ; Yaremko2007 ; Shuryak:2011tt ; Dai:2013cwa
; Harte:2016fru  (see, also, Kosyakov:2007qc ; Kosyakov:2018wek ). Violation
of Huygens’s principle in odd-dimensional flat spacetime is known since the
works of Hadamard hadamard2014lectures , Courant and Hilbert
courant2008methods , Ivanenko and Sokolov Ivanenko_book . It implies that,
while in even dimensions the signal from the instant flash of the source,
having reached the observer in the interval of time required for it to
propagate with the speed of light, ends instantly, in odd dimensions the
endless tail signal decaying with time is observed after that. On the other
hand, free massless fields propagate exactly with the speed of light in all
dimensions. Therefore, while the total retarded gravitational field of
localized source propagates with all velocities up to that of light, its
gravitational radiation, being the free field far from it, should propagate
exactly with the speed of light. As a result, this mismatch makes the
extraction of the emitted part of retarded gravitational field in odd
dimensions a non-trivial task. In odd dimensions, there is the non-local tail
contribution into the emitted part of the field analogous to that found by
DeWitt and Brehme in the curved four-dimensional spacetime DeWitt:1960fc ;
Barack:2018yvs ; Galtsov:2007zz . However, in latter case, the tail is due to
the scattering of the waves on the curvature of spacetime and its evaluation
is rather complicated, while in odd dimensions the tail contribution is given
in closed analytic form. The tail term can be dealt with by use of the
effective field theory approach to the problems of radiation Porto:2016pyg ;
Cardoso:2008gn ; Birnholtz:2013ffa ; Birnholtz:2015hua . However, being based
on the calculations in the momentum space irrelevant to the dimensionality of
the spacetime, this method does not provide us with any information about the
structure of the retarded field in the wave zone and the role of the tail term
in the formation of radiation. As was shown recently, this obstacle can be
overcome in two ways: by Fourier transforming the retarded Green’s functions
over the temporal coordinate Chu:2021uea , or by modifying the radiation
definition Galtsov:2020hhn ; Galtsov:2021zpb . In this article, we follow the
second approach.

To implement these ideas for the two-body system one needs a simplified model
which would be tractable within the linearized gravity in Minkowski space.
However, gravitational radiation in pure gravity requires second order
expansion of the Ricci tensor to describe gravitational stresses binding the
system. To avoid this, we propose the combination of the scalar and
gravitational interactions to have the binding forces of nongravitational
nature. Then gravitational radiation can be computed within the linearized
gravity in Minkowski space. Our model also assumes that motion is restricted
to the four-dimensional subspace (brane), while gravity operates in the odd-
dimensional bulk (for simplicity we assume five dimensions). In contrary, the
scalar field is assumed to live on the brane too. Then, starting with initial
particles’ velocities lying on the brane, we will ensure motion remains on the
brane perpetually without imposing any confining mechanism. Of course, this
model cannot serve a realistic model of spacetime such as ADD or RS models,
and it has no purely four-dimensional interpretation. In fact, our model can
be regarded as the limit of the ADD model with an infinite compactification
length. This limiting theory clearly is not physically viable, but here we
sacrifice viability in favor of mathematical consistency and simplicity. Our
model still captures basic features of violation of Huygens’s principle in
odd-dimensions and demonstrate the emergence of tail and an extra polarisation
in gravitational radiation for an observer living on the brane.

We use the Rohrlich-Teitelboim radiation definition Rohrlich1961 ;
Teitelboim1970 ; rohrlich2007  (see, also, Kosyakov1992 ; Galtsov:2004uqu ;
Spirin2009 ; Galtsov:2010tny ), based on the Lorentz-invariant decomposition
of the on-shell energy-momentum tensor of the retarded field, to extract the
emitted part of the five-dimensional gravitational field. For consistency, we,
also, explicitly take into account the contribution from the stresses of the
field, through which particles forming binary system interact, into the
gravitational radiation of the system. The stresses contribution is dealt with
by use of the DIRE approach to the post-Newtonian expansions in the four-
dimensional GR Pati:2000vt . We find that the power of the five-dimensional
gravitational radiation from the binary system is given by the analog of the
famous quadrupole formula maggiore2008 , which, however, now depends on the
entire history of the system’s motion preceding the retarded moment of time.
In contrast, in four-dimensional theory power of the gravitational radiation
at some moment of time is completely determined by the state of the source at
the corresponding retarded time. The obtained result is in agreement with the
one found by use of the Fourier transformations of the retarded Green’s
functions Chu:2021uea .

This paper is organised as follows. In the second section, we discuss the
model under the consideration, polarisations of the gravitational radiation
generated by the system and briefly recall the Rohrlich-Teitelboim radiation
definition. Section III is devoted to the contribution of the point particles
into the gravitational radiation of the binary system. We illustrate the main
steps of its computation by a simple example of the five-dimensional scalar
radiation of the point particle. In Sec. IV, we find the contribution of the
scalar field’s stresses, through which point particles interact with each
other, into the gravitational radiation of the system by use of the DIRE
approach. In section V, we find the five-dimensional analog of the quadrupole
formula for the power of the gravitational radiation of the binary system and,
as for a simple example of its application, consider the gravitational
radiation of the binary system on the circular orbit and corresponding orbital
evolution. In Sec. VI we discuss the obtained results and their relation to
the other works.

##  II The Setup

As is known, the gravitational radiation of point particles interacting
gravitationally differs from the electromagnetic radiation of point charges in
the nonlocality of the source in the respective d’Alembert equation, which
must take into account the contribution of gravitational stresses. For this
reason, we cannot restrict ourselves to the linearized approximation of
general relativity, but must include, as part of the radiation source, the
quadratic part of the Ricci tensor as representing these stresses
Weinberg:1972kfs . To simplify this setup, we introduce a scalar field that
provides the forces needed to create acceleration, assuming that this force is
much greater than the force of their gravitational interaction, and the latter
can be neglected. In our model, it is possible to consistently describe
gravitational radiation using purely linear gravity, but still taking into
account the nonlocality of the source in the d’Alembert equation for the
gravitational field $h_{MN}$ due to the contribution of interaction field
stresses. The consistency of such a model will be ensured if the interacting
particles do not leave the four-dimensional subspace (brane); this is
guaranteed because the scalar field is restricted to live on the brane. We do
not specify the physical mechanism of such localization and do not introduce
brane as a physical object. Our goal is to reveal the features of
gravitational radiation in odd-dimensional space-time due to violation of the
Huygens principle, which induces the effect of nonlocality of a different
nature.

###  II.1 Brane and bulk

The model assumes that the system of point particles $m_{a}$ with initial
velocities, bounded by the four-dimensional subspace of the five-dimensional
spacetime, moves under mutual scalar forces, forever remaining in this
subspace, which we loosely call the three-brane. We choose the five-
dimensional coordinates $x^{M}=(x^{\mu},x^{4})$ with $M=\overline{0,4}$ and
$\mu=\overline{0,3}$, so that the brane is orthogonal to $x^{4}$ and is
parametrized by the same bulk coordinates.

We start with the generally covariant action

$S=S_{g}+S_{\varphi}+S_{m},$ (II.1)

which is the sum of the five-dimensional Einstein-Hilbert term

$S_{g}=\frac{1}{2\kappa_{5}}\int d^{5}x\sqrt{-g}\,R,$ (II.2)

where $\kappa_{5}$ is the five-dimensional gravitational constant, the four-
dimensional scalar field term

$S_{\varphi}=-\frac{1}{8\pi}\int
d^{4}\sigma\sqrt{-\gamma}\gamma^{\mu\nu}\partial_{\mu}\varphi(\sigma)\partial_{\nu}\varphi(\sigma),$
(II.3)

where the induced metric $\gamma_{\mu\nu}=g_{\mu\nu}$ coincides with the four-
dimensional part of the bulk metric $g_{MN}$, chosen in the gauge
$g_{4\mu}=0$, and the particle term describes their motion along affinely
parameterized world lines $z_{a}^{\mu}(\tau)$ (omitting an index $a$ in
$\tau_{a}$)

$S_{p}=-\sum_{a}\int\left(m_{a}+g_{a}\varphi(z_{a})\right)\left(-\dot{z}_{a}^{2}\right)^{1/2}d\tau,\qquad\dot{z}_{a}^{2}=\dot{z}_{a}^{\mu}\dot{z}_{a}^{\nu}\gamma_{\mu\nu},\qquad\dot{z}_{a}^{\mu}=dz_{a}^{\mu}/d\tau.$
(II.4)

The scalar charges $g_{a}$ with this definition are dimensionless.

The corresponding Einstein equations will read

$R_{MN}-\frac{1}{2}g_{MN}R=2\kappa_{5}T_{MN},\qquad
T_{MN}=\delta^{\mu}_{M}\delta^{\nu}_{N}\,T_{\mu\nu}\,\delta(x^{4}),$ (II.5)

where $T_{\mu\nu}$ is the (restricted to the brane) matter energy-momentum
tensor consisting of the particles and scalar field terms
$T_{\mu\nu}=T_{\mu\nu}^{\rm P}+T_{\mu\nu}^{\rm F}$:

$\displaystyle T_{\mu\nu}^{\rm P}=\sum_{a}\int
d\tau\left(m_{a}+g_{a}\varphi(z_{a})\right)\dot{z}_{a\mu}\dot{z}_{a\nu}\frac{\delta^{(4)}(x-z_{a})}{\sqrt{-\gamma}},$
(II.6) $\displaystyle T_{\mu\nu}^{\rm
F}=\frac{1}{4\pi}\left(\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}\gamma_{\mu\nu}(\partial\varphi)^{2}\right).$
(II.7)

In accordance with the localisation of the scalar field on the three-brane, it
contains the distributional singularity provided by the five-dimensional delta
function under the four-dimensional integral over the brane.

We also find the equations of motion for the matter localised on the three-
brane. So, scalar field equation of motion has the form

${{}^{(\gamma)}}\square\varphi=\frac{1}{\sqrt{-\gamma}}\partial_{\mu}\left({\sqrt{-\gamma}}\gamma^{\mu\nu}\partial_{\nu}\varphi\right)=4\pi\sum_{a}g_{a}\int
d\tau\frac{\delta^{(4)}(x-z_{a})}{\sqrt{-\gamma}},$ (II.8)

Equations of motion for the point particles can be presented in the form

$m_{a}(\tau)\frac{Dz^{\mu}_{a}}{d\tau}=g_{a}\left(\gamma^{\mu\nu}+z_{a}^{\mu}z_{a}^{\nu}\right)\partial_{\nu}\varphi,$
(II.9)

where the variable mass is introduced

$m_{a}(\tau)=m_{a}+g_{a}\varphi.$ (II.10)

In fact, the Einstein equations (II.5) with point particle sources only make
sense in the linearized approximation, so our generally covariant formulation
was only intended to emphasize the geometric meaning of the quantities under
consideration. Now we assume

$g_{MN}=\eta_{MN}+h_{MN},\quad\left|h_{MN}\right|\ll 1.$ (II.11)

with the Minkowski metric $\eta_{MN}\equiv{\rm diag}(-1,1,1,1,1)$, and in what
follows perform manipulations with indices using the flat metric. Introducing
the reduced field

$\bar{h}_{MN}\equiv h_{MN}-\frac{1}{2}\eta_{MN}h,\qquad h=\eta^{MN}h_{MN},$
(II.12)

and imposing the Lorentz gauge condition

$\partial^{M}\bar{h}_{MN}=0,$ (II.13)

we find the linearised Einstein’s equation as

${{}^{(5)}}\square\bar{h}_{MN}=-2\kappa_{5}T_{MN},$ (II.14)

where ${{}^{(5)}}\square=\eta^{MN}\partial_{M}\partial_{N}$ is the d’Alembert
operator on the five-dimensional Minkowski space. Note, for the Lorentz gauge
condition (II.13) to be valid, the matter energy-momentum tensor has to be
conserved

$\partial^{M}T_{MN}=0,$ (II.15)

what is easily verified using the above definitions, and the four-dimensional
conservation equation

$\partial^{\mu}T_{\mu\nu}=0,$ (II.16)

which holds by virtue of the equations of motion (II.8,II.9).

###  II.2 Gravitational radiation: polarizations

We want to calculate the 5D gravitational radiation of our system. To do this,
we introduce another division of coordinates:
$x^{M}=(x^{0}=t,x^{i}),\,i=\overline{1,4}$.

The energy-momentum flux carried by the 5D transverse-traceless wave
polarizations is given by an effective energy-momentum tensor analogous to
that in maggiore2008

$t_{MN}=\frac{1}{4\kappa_{5}}\left\langle\partial_{M}h_{ij}^{\rm
TT}\partial_{N}h_{ij}^{\rm TT}\right\rangle,$ (II.17)

where we assume that the motion of the source is periodic and the brackets
$\langle\ldots\rangle$ denote the averaging over the period. In the
transverse-traceless gauge we use here, the following conditions hold:

$h_{0M}^{\rm TT}=0,\quad h_{ii}^{\rm TT}=0,\quad\partial^{j}h_{ij}^{\rm
TT}=0,$ (II.18)

which ensure that we deal with five polarisations. These can be obtained
acting on the unconstrained potentials by the projector maggiore2008

$\displaystyle h_{ij}^{\rm
TT}(x)=\Lambda_{ij,kl}(\mathbf{n})h_{kl}(x),\quad\mathbf{n}=\frac{\mathbf{x}}{|\mathbf{x}|}$
(II.19) $\displaystyle\Lambda_{ij,kl}(\mathbf{n})\equiv
P_{ik}P_{jl}-\frac{1}{3}P_{ij}P_{kl},\quad
P_{ij}(\mathbf{n})\equiv\delta_{ij}-n_{i}n_{j},$ (II.20)

which has the vanishing traces

$\Lambda_{ii,kl}(\mathbf{n})=\Lambda_{ij,kk}(\mathbf{n})=0.$ (II.21)

Therefore, the transverse-traseless potentials can be equivalently obtained
from the trace-reversed metric deviations

$h_{ij}^{\rm TT}(x)=\Lambda_{ij,kl}(\mathbf{n})\bar{h}_{kl}(x).$ (II.22)

From five polarizations of the bulk modes, three are present in the brane
subspace. Choosing the propagation vector $\mathbf{n}$ to be aligned with
$x^{3}$-direction, we will have

$\bar{h}_{ij}^{\rm
TT}(x)=\begin{pmatrix}h_{+}-\displaystyle\frac{1}{2}h_{\displaystyle\circ}&h_{\times}&0&h_{14}\\\
h_{\times}&-h_{+}-\displaystyle\frac{1}{2}h_{\displaystyle\circ}&0&h_{24}\\\
0&0&0&0\\\ h_{14}&h_{24}&0&h_{\displaystyle\circ}\end{pmatrix},$ (II.23)

where the ”cross” and ”plus” polarizations are standard

$h_{+}=\frac{1}{2}\left(h_{11}-h_{22}\right),\quad h_{\times}=h_{12},$ (II.24)

and the $h_{\displaystyle\circ}$ is the ”breathing” polarization Andriot2017
given as

$h_{\displaystyle\circ}=\frac{2}{3}h_{44}-\frac{1}{3}\left(h_{11}+h_{22}\right).$
(II.25)

So the brane-living observer will see only three polarizations – cross, plus,
and the breathing mode Andriot2017 . From the linearized geodesic deviation
equation maggiore2008 ; Andriot2017 , it can be seen that under the action of
breathing mode the circle of brane-living test masses lying in the plane
orthogonal to the propagation direction will uniformly shrink and stretch.

Polarizations amplitudes for a geneneric symmetric tensor of second order
$A_{ij}$ in transverse-traceless gauge for an arbitrary propagation direction
$\mathbf{n}$ are given in appendix A.

###  II.3 Dynamics of the non-relativistic binary

In the linear approximation (II.14), the particles and the scalar field
propagate on the flat Minkowski background $\gamma_{\mu\nu}=\eta_{\mu\nu}$,
and the equations of motion (II.8) and (II.9) reduce to

$\displaystyle{{}^{(4)}}\square\varphi(x)=4\pi\sum_{a}g_{a}\int
d\tau\,\delta^{(4)}\left(x-z_{a}\right),$ (II.26)
$\displaystyle\left(m_{a}+g_{a}\varphi(z_{a})\right)\ddot{z}_{a}^{\mu}=-g_{a}\partial^{\mu}\varphi(z_{a})-g_{a}\partial_{\nu}\varphi(z_{a})\dot{z}_{a}^{\nu}\dot{z}_{a}^{\mu},$
(II.27)

where ${{}^{(4)}}\square=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$ is the
d’Alembert operator on the four-dimensional Minkowski space. The particles’
motion is completely governed by their mutual interaction through the scalar
field given by the retarded solution of Eq. (II.26).

In what follows, we are interested in the gravitational radiation from the
non-relativistic binary system, in which case Eq. (II.27) simplifies
significantly. Moreover, we assume accordingly:

  * •

the particles are non-relativistic $|\dot{\mathbf{z}}_{a}|\ll 1$;

  * •

spatially separated $|\mathbf{z}_{2}-\mathbf{z}_{1}|\gg-
g_{1}g_{2}/m_{a},\,\forall a$;

  * •

the scalar field just governs the motion of particles and is not radiated.

Then, we neglect the terms proportional to the square of particle’s velocity
$\sim\dot{z}_{a}\dot{z}_{a}$ and the self-interaction terms

$\varphi(z_{a})z_{a}=\varphi_{b}(z_{a})z_{a},\quad b\neq a$ (II.28)

in the Eq. (II.27), where $\varphi_{a}(x)$ is the scalar field produced by
$a$-th particle. So the scalar field of $a$-th particle is a Coulomb field
Jackson:1998nia :

$\varphi_{a}(x)=-\frac{g_{a}}{|\mathbf{x}-\mathbf{z}_{a}(t)|},$ (II.29)

where $t=x^{0}$, and we neglected the retardation. Thus, combining Eqs.
(II.28) and (II.29) with the condition of spatial separation, we neglect the
influence of the scalar field on particles’ masses

$(m_{a}+g_{a}\varphi_{b}(z_{a}))\simeq m_{a}.$ (II.30)

As a result, we obtain the non-relativistic particles equation of motion
simply as

$m_{a}\ddot{\mathbf{z}}_{a}=-g_{a}\nabla\varphi_{b}(\mathbf{z}_{a}),$ (II.31)

where $\varphi_{b}(x)$ is given by the Eq. (II.29). Note that these four-
dimensional equations of motion admit the stable elliptical orbits. The point
particles and scalar field energy-momentum tensors reduce to

$\displaystyle T_{MN}^{\rm
P}=\delta_{M}^{\mu}\delta_{N}^{\nu}\sum_{a}m_{a}\int
d\tau\,\dot{z}_{a\mu}\dot{z}_{a\nu}\,\delta^{(4)}(x-z_{a})\delta(x^{4}),$
(II.32) $\displaystyle T_{MN}^{\rm
F}=\frac{1}{4\pi}\delta_{M}^{\mu}\delta_{N}^{\nu}\left(\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}\eta_{\mu\nu}\partial^{\alpha}\varphi\partial_{\alpha}\varphi\right)\delta(x^{4}).$
(II.33)

Clearly, as we tacitly assumed the initial particles’ velocities to lie on the
brane, no forces normal to the brane will arise, so they perpetually remain
there. Summarising, we consider the binary system of non-relativistic,
spatially separated point particles moving inside the flat three-brane and
interacting with each other only through the scalar field localised on the
same brane. Dynamics of the binary is described by Eq. (II.31), while the
scalar fields produced by the particles is given by Eq. (II.29). Gravitational
waves will arise as the solution of the five-dimensional d’Alembert equation
Eq. (II.14) with the matter energy-momentum tensor given as the sum of Eqs.
(II.32) and (II.33) which both contain delta-functions with the support on the
brane. Since this sum is four-dimensionally divergenceless on-shell, it will
be automatically five-dimensionally divergenceless, since the particle
velocities have zero component normal to the brane, while the scalar field has
no dependence on the fifth coordinate. This condition is necessary for
consistency of the Eq. (II.14).

###  II.4 Gravitational waves generated by the system

Gravitational waves generated by the binary system are given by the retarded
solution of Eq. (II.14) constructed by the corresponding Green’s function
courant2008methods ; hadamard2014lectures ; Ivanenko_book ; Galtsov:2001iv ;
Kazinski:2002mp ; Cardoso:2002pa ; Dai:2013cwa ; Galtsov:2020hhn

$\displaystyle\bar{h}_{MN}(x)=-2\kappa_{5}\int
d^{5}x^{\prime}\,T_{MN}(x^{\prime})\,G_{\rm ret}^{4+1}(x-x^{\prime}),$ (II.34)
$\displaystyle G_{\rm
ret}^{4+1}(x)=\frac{\theta(t)}{2\pi^{2}}\left[\frac{1}{2}\frac{\theta(-x^{2})}{(-x^{2})^{3/2}}-\frac{\delta(-x^{2})}{(-x^{2})^{1/2}}\right].$
(II.35)

Note that the Green’s function (II.35) is localised not only on the light cone
$x^{2}=0$, but also inside it, leading to the propagation of the retarded
gravitational field in space with all velocities up to that of light. However,
further we demonstrate, using the Rohrlich-Teitelboim approach to radiation,
that the retarded field derivative contains the component propagating exactly
with the speed of light and representing the emitted part of the field.

Also, while each term in Eq. (II.35) is separately divergent on the light
cone, the resulting field is finite Galtsov:2020hhn . Let us briefly discuss
the regularisation procedure used to make the finiteness of the field
explicit. We consider each term in Eq. (II.35) as a product of two
distributions, e.g., their derivatives are given by the Leibniz rule

$\displaystyle\frac{d}{dy}\left(y^{n}\delta(y)\right)=ny^{n-1}\delta(y)+y^{n}\delta^{\prime}(y),$
(II.36)
$\displaystyle\frac{d}{dy}\left(y^{n}\theta(y)\right)=ny^{n-1}\theta(y)+y^{n}\delta(y).$
(II.37)

It is clear that if $n$ is negative, these distributions are singular at
$y=0$. We introduce the regularising parameter $\epsilon\to+0$ by shifting the
arguments of delta and Heaviside functions as $y\to y+\epsilon$. Then,
integrating by parts the Heaviside function term in (II.34) we can extract the
divergence contained in it and cancel it out with the divergent delta function
term coming to the finite result. The same procedure is valid when one
considers the derivative of retarded field, taking into account the
differentiation rules (II.36) and (II.37).

We consider the point particles and scalar field contributions into the
gravitational waves separately, decomposing the retarded gravitational field
(II.34) into two terms

$\bar{h}_{MN}(x)=\bar{h}_{MN}^{\rm P}(x)+\bar{h}_{MN}^{\rm F}(x).$ (II.38)

Point particles contribution is computed using the Rohrlich-Teitelboim
approach to radiation, while the scalar field term is calculated by use of the
DIRE approach to the post-Newtonian expansions (for review see, e.g.,
Pati:2000vt ).

###  II.5 ADD-model with the infinite compactification radius

Let us see how our model can be interpreted as the limiting case of the ADD-
model with the infinite compactification radius of an extra dimension. We
start with the five dimensional ADD-model with the compactification radius
$R$. The scalar retarded Green’s function of the d’Alembert equation in such a
theory is written as Giudice:1998ck

$G_{\rm ret}^{\rm ADD}(x;y)=-\frac{1}{2\pi
R}\sum_{n=-\infty}^{+\infty}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{e^{im_{n}y}e^{ik_{\alpha}x^{\alpha}}}{k_{\mu}^{2}+m_{n}^{2}},$
(II.39)

where $y$ is the coordinate along the extra dimension and $m_{n}=n/R$ are the
Kaluza-Klein masses. Integrating over $k^{0}$ in accordance with the
retardation condition we find it as

$G_{\rm ret}^{\rm ADD}(x;y)=-\frac{\theta(t)}{2\pi
R}\sum_{n=-\infty}^{+\infty}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\sin\sqrt{\mathbf{k}^{2}+m_{n}^{2}}t}{\sqrt{\mathbf{k}^{2}+m_{n}^{2}}}e^{i\mathbf{k}\mathbf{x}}e^{im_{n}y}.$
(II.40)

Performing integration over the angular variables we obtain

$G_{\rm ret}^{\rm
ADD}(x;y)=\frac{\theta(t)}{4\pi^{3}Rr}\frac{\partial}{\partial
r}\sum_{n=-\infty}^{+\infty}e^{im_{n}y}\int
dk\,\cos{kr}\frac{\sin\sqrt{k^{2}+m_{n}^{2}}t}{\sqrt{k^{2}+m_{n}^{2}}}.$
(II.41)

Evaluating the remaining integral Ivanenko_book ; zwillinger2014table

$\int
dk\,\cos{kr}\frac{\sin\sqrt{k^{2}+m_{n}^{2}}t}{\sqrt{k^{2}+m_{n}^{2}}}=\frac{\pi}{2}J_{0}(m_{n}\sqrt{t^{2}-r^{2}})\left[\theta(t-r)+\theta(t+r)-1\right]$
(II.42)

and differentiating over the spatial distance, we get

$\displaystyle G_{\rm ret}^{\rm ADD}(x;y)$
$\displaystyle=\frac{\theta(t)\theta(t-r)}{8\pi^{2}R}\sum_{n=-\infty}^{+\infty}e^{im_{n}y}\frac{m_{n}J_{1}(m_{n}\sqrt{t^{2}-r^{2}})}{\sqrt{t^{2}-r^{2}}}-$
$\displaystyle-\frac{\theta(t)\delta(t-r)}{8\pi^{2}Rr}\sum_{n=-\infty}^{+\infty}e^{im_{n}y}\times\times
J_{0}(m_{n}\sqrt{t^{2}-r^{2}}).$ (II.43)

Two series over the Kaluza-Klein masses can be combined using the following
relations for the products of Heaviside and delta functions

$\theta(t)\theta(t-r)=\theta(t)\theta(t^{2}-r^{2}),\quad\frac{\theta(t)\delta(t-r)}{r}=2\theta(t)\delta(t^{2}-r^{2}),$
(II.44)

arriving at the following expression for the ADD retarded Green’s function

$G_{\rm ret}^{\rm
ADD}(x;y)=-\frac{1}{\sqrt{t^{2}-r^{2}}}\frac{\partial}{\partial\sqrt{t^{2}-r^{2}}}\left[\frac{\theta(t)\theta(t^{2}-r^{2})}{8\pi^{2}R}\sum_{n=-\infty}^{+\infty}e^{im_{n}y}J_{0}(m_{n}\sqrt{t^{2}-r^{2}})\right].$
(II.45)

In the limit of infinite compactification radius $R\to\infty$, one can pass to
continuous limit $m_{n}\to m\in\mathbb{R}$ and integrate over $m$:

$G_{\rm
ret}^{R\to\infty}(x;y)=-\frac{1}{\sqrt{t^{2}-r^{2}}}\frac{\partial}{\partial\sqrt{t^{2}-r^{2}}}\left[\frac{\theta(t)\theta(t^{2}-r^{2})}{4\pi^{2}}\int_{0}^{+\infty}dm\,\cos(my)J_{0}(m\sqrt{t^{2}-r^{2}})\right],$
(II.46)

where we have taken into account the positive parity of $J_{0}(x)$. Evaluating
the resulting integral zwillinger2014table , we arrive at the following
expression for the Green’s function

$G_{\rm
ret}^{R\to\infty}(x;y)=-\frac{1}{\sqrt{t^{2}-r^{2}}}\frac{\partial}{\partial\sqrt{t^{2}-r^{2}}}\frac{\theta(t)\theta(t^{2}-r^{2}-y^{2})}{4\pi^{2}\sqrt{t^{2}-r^{2}-y^{2}}}.$
(II.47)

Finally, after the differentiation we come to the expression equivalent to the
retarded Green’s function of the d’Alembert equation in the five-dimensional
Minkowski space (II.35)

$G_{\rm
ret}^{R\to\infty}(x;y)=\frac{\theta(t)}{2\pi^{2}}\left[\frac{1}{2}\frac{\theta(t^{2}-r^{2}-y^{2})}{(t^{2}-r^{2}-y^{2})^{3/2}}-\frac{\delta(t^{2}-r^{2}-y^{2})}{\sqrt{t^{2}-r^{2}-y^{2}}}\right].$
(II.48)

###  II.6 Rohrlich-Teitelboim definition of radiation

In four dimensions, the retarded electromagnetic field of the point charge
consists of two terms: one proportional to $1/r^{2}$ and being the deformed
Coulomb-like part of the field, and another, which depends on the acceleration
of the charge and scales as $1/r$. The latter gives a non-zero flux of the
field energy-momentum through the distant sphere, and therefore represents
radiation. To argue that this is radiation indeed, Rohrlich Rohrlich1961 ;
rohrlich2007  and Teitelboim Teitelboim1970  (see also Kosyakov1992 ;
Galtsov:2004uqu ; Kosyakov:2007qc ; Galtsov:2010tny ; Kosyakov:2018wek ),
computed the most long-range part of the on-shell energy-momentum tensor,
demonstrating that it exhibits special properties, meaning that the
corresponding part of the field energy-momentum propagates at the speed of
light.

The Rohrlich-Teitelboim approach is based on the use of certain covariantly
defined quantities, so we briefly recall their definitions. Let us consider a
point particle moving along a worldline $z^{M}(\tau)$ with the $D$-velocity
$v^{M}=dz^{M}/d\tau$. The observation point coordinates are denoted as
$x^{M}$. Consider the observation point as a top of the light cone in the
past, and denote the intersection point of this light cone with the particle’s
worldline as $\hat{z}^{M}={z}^{M}(\hat{\tau})$. The quantity $\hat{\tau}$ is
called the retarded proper time and is defined by the equation

$\left(x^{M}-z^{M}(\hat{\tau})\right)^{2}=0,\quad x^{0}\geq
z^{0}(\hat{\tau}).$ (II.49)

If the signal propagating with the speed of light were emitted at the retarded
moment, it would reach the observation point at the corresponding moment of
time $x^{0}$. Further, all the hatted quantities correspond to the retarded
proper time $\hat{\tau}$. We introduce three spacetime vectors: a lightlike
vector $\hat{X}^{M}=x^{M}-\hat{z}^{M}$ directed from $\hat{z}^{M}$ to the
observation point; a spacelike unit vector $\hat{u}^{M}$, orthogonal to the
particle’s velocity $\hat{v}^{M}$; and a lightlike vector
$\hat{c}^{M}=\hat{v}^{M}+\hat{u}^{M}$ being the sum of the formers. These
vectors have the following properties:

$\displaystyle\hat{v}^{2}=-\hat{u}^{2}=-1;\quad\hat{c}^{2}=0;\quad(\hat{c}\hat{v})=-(\hat{c}\hat{u})=-1;\quad(\hat{v}\hat{u})=0,$
(II.50)
$\displaystyle\hat{X}^{M}=\hat{\rho}\hat{c}^{M};\quad\hat{\rho}=-(\hat{v}\hat{X}),\quad\hat{X}^{2}=0,$
(II.51)

where $(AB)=A^{M}B_{M}$. Note that $\hat{\rho}$ is a Lorentz-invariant
distance equal to the spatial distance in the Lorentz frame comoving with the
particle at the retarded moment $\hat{\tau}$. Note also that for a point
charge, moving along the worldline $z^{M}(\tau)$ for an infinite proper time
$-\infty<\tau<\infty$, certain care is needed to correctly define the
asymptotic conditions for acceleration, for details see Teitelboim1970 . Here
we will not discuss this subtlety, considering the simple case of periodic
motion. Far from the region of the particle’s motion $\hat{\rho}\sim r$, so
the Lorentz-invariant definition of the distance is equivalent to the naive
definition. Nevertheless, in order to obtain Lorentz-covariant expansions of
tensors, one has to use $1/\hat{\rho}$ as an expansion parameter.

Based on the definitions above, in four-dimensional electrodynamics Teitelboim
Teitelboim1970  obtained the following decomposition of the the on-shell
energy-momentum tensor:

$T^{\mu\nu}=T_{\rm Coul}^{\mu\nu}+T_{\rm mix}^{\mu\nu}+T_{\rm rad}^{\mu\nu},$
(II.52)

where the Coulomb part falls down as $\hat{\rho}^{-4}$, the mixed part – as
$\hat{\rho}^{-3}$, and the last part – as $\hat{\rho}^{-2}$. Teitelboim found
that the most long-range term of the on-shell energy-momentum tensor has the
following properties:

  * •

It is separately conserved $\partial_{\nu}T_{\rm rad}^{\mu\nu}=0$.

  * •

It is proportional to the direct product of two null vectors
$\hat{c}^{\mu}\hat{c}^{\nu}$ and, therefore, $\hat{c}_{\mu}T_{\rm
rad}^{\mu\nu}=0$.

  * •

It scales as $1/\hat{\rho}^{2}$ and gives positive definite energy-momentum
flux through the distant sphere.

It is clear that the corresponding part of the field’s energy-momentum
propagates with the speed of light. Therefore, the radiation power can be
computed as the flux of the energy, associated with $T_{\rm rad}^{\mu\nu}$.
Note that, due to the electromagnetic field’s energy-momentum tensor being the
bilinear functional of the field derivatives $T_{\mu\nu}\sim\partial
A\,\partial A$, one can define the emitted part of the latter
$\left(\partial_{\mu}A_{\nu}\right)^{\rm rad}$, which contributes into the
radiated part of the energy-momentum tensor $T_{\rm rad}^{\mu\nu}$.

Similar decomposition and reasoning holds in any spacetime dimensions
Kosyakov1999 ; Kosyakov:2008wa ; Spirin2009  and, also, for the scalar
Spirin2009 ; Galtsov:2020hhn  and gravitational fields. The only difference is
that in $D$ dimensions area of the distant sphere is proportional to
$r^{D-2}$, thus, the relevant asymptotic behaviour of the field derivative in
the wave zone is $1/r^{D/2-1}$.

Passing to our theory of the five-dimensional gravitational radiation, we
calculate the long-range part (with respect to the Lorentz-invariant distance
$\hat{\rho}$) of derivative of the retarded gravitational field (II.34) and
substitute it into the bilinear functional (II.17). All the listed properties
of the emitted part of the energy-momentum tensor $T_{\rm rad}^{MN}$ hold for
the obtained tensor. Note that the transformation into the transverse-
traceless gauge can be performed after the extraction of the emitted part of
the field derivative, because the derivative of the $\Lambda$-tensor is
proportional to the inverse distance
$\partial_{m}\Lambda_{ij,kl}(\mathbf{n})\sim 1/r$ and, therefore, does not
contribute into the long-range part of the field derivative

$\left(\partial_{M}h_{ij}^{\rm TT}\right)^{\rm
rad}=\Lambda_{ij,kl}(\mathbf{n})\left(\partial_{M}\bar{h}_{kl}\right)^{\rm
rad}.$ (II.53)

However, in contrast with the four-dimensional theory, in five dimensions
emitted part of the field derivative depends on the entire history of
particle’s motion preceding the retarded moment of proper time $\hat{\tau}$,
while in four dimensions it depends only on the particle’s kinematic
characteristics at this moment. Nevertheless, due to the above properties of
the on-shell energy-momentum tensor, the energy-momentum associated with the
long-range component of the field derivative propagate at the speed of light.

The flux of the radiated energy-momentum passing per unit time through the
$(2\nu-1)$-dimensional sphere of radius $r$ is given by the integral

$W_{2\nu+1}^{\mu}=\int\,T_{\rm rad}^{\mu
i}\;n^{i}\,r^{2\nu-1}\,d\Omega_{2\nu-1},\quad i=\overline{1,2\nu},$ (II.54)

where $d\Omega_{2\nu-1}$ is an angular element, and $\bf{n}$ is a unit
spacelike vector in the direction of observation.

Using the Rohrlich-Teitelboim approach, the scalar synchrotron radiation from
a circularly moving particle in $2+1$ and $4+1$ dimensions was calculated
Galtsov:2020hhn . The correctness of calculations was verified by the
computation of the total radiation power using Fourier spectral decomposition,
which are indifferent to the dimensionality of spacetime and are free from the
problems discussed above.

##  III Point particles contribution

The point particles contribution into the gravitational waves produced by the
binary system is written as

$\bar{h}_{MN}^{\rm P}(x)=-2\kappa_{5}\int d^{5}x^{\prime}\,T_{MN}^{\rm
P}(x^{\prime})\,G_{\rm ret}^{4+1}(x-x^{\prime}),$ (III.1)

where the particles energy-momentum tensor is given by the Eq. (II.32) and the
retarded Green’s function by the Eq. (II.35).

We are interested in the non-relativistic approximation of the emitted part of
the gravitational field’s (III.1) derivative. To clarify the main steps of
derivation, let us start with the problem of scalar radiation of non-
relativistic point charge as a simple but instructive example of five-
dimensional radiation of a non-relativistic source.

###  III.1 Scalar radiation of a non-relativistic charge

In this subsection, we deviate from our basic setup and consider the case of a
scalar field living in the bulk. In this case, the bulk scalar radiation (in
the lowest approximation) can be calculated within the framework of a
linearized theory, since there is no need to take into account the stresses
that bind the system. This calculation may be interesting in itself, but for
us it also provides an easier way to explain the characteristics of radiation
in an odd-dimensional flat space. In fact, this gives a simplified setup for
gravitational radiation, neglecting the contribution of stresses.

By analogy with (II.26), equation of motion of a five-dimensional massless
scalar field interacting with a point charge moving along an arbitrary
worldline $z^{M}(\tau)$ is written as

${{}^{(5)}}\square\varphi(x)=2\pi^{2}g\int d\tau\,\delta^{(5)}(x-z).$ (III.2)

Retarded solution of Eq. (III.2) is constructed by the corresponding five-
dimensional Green’s function (II.35) and is found as

$\varphi(x)=g\int
d\tau\,\theta(X^{0})\left[\frac{1}{2}\frac{\theta(-X^{2})}{(-X^{2})^{3/2}}-\frac{\delta(-X^{2})}{(-X^{2})^{1/2}}\right],$
(III.3)

where we introduced the vector $X^{M}(\tau)=x^{M}-z^{M}(\tau)$ and omitted its
dependence on proper time for brevity. Computing its derivative we obtain

$\partial_{M}\varphi(x)=g\int
d\tau\,\theta(X^{0})\left[\frac{3}{2}\frac{\theta(-X^{2})}{(-X^{2})^{5/2}}+2\frac{\delta^{\prime}(-X^{2})}{(-X^{2})^{1/2}}-2\frac{\delta(-X^{2})}{(-X^{2})^{3/2}}\right]X_{M}.$
(III.4)

We integrate by parts the term containing derivative of delta function by use
of the relation

$\frac{dX^{2}}{d\tau}=-2(vX),$ (III.5)

arriving at

$\partial_{M}\varphi(x)=g\int
d\tau\,\theta(X^{0})\left[\frac{3}{2}\frac{\theta(-X^{2})}{(-X^{2})^{5/2}}X_{M}-\frac{\delta(-X^{2})}{(-X^{2})^{3/2}}X_{M}+\right.\\\
\left.+\frac{\delta(-X^{2})}{(vX)^{2}(-X^{2})^{1/2}}\left[(aX)+1\right]X_{M}-\frac{\delta(-X^{2})}{(vX)(-X^{2})^{1/2}}v_{M}\right],$
(III.6)

where we introduced the acceleration vector $a^{M}=d^{2}z^{M}/d\tau^{2}$.

To extract the leading $\hat{\rho}$-asymptotic of the Eq. (III.6), we use
relation for the delta function of complex argument transforming its product
with the Heaviside function as

$\theta(X^{0})\delta(-X^{2})=\frac{\delta(\tau-\hat{\tau})}{2\hat{\rho}}.$
(III.7)

Also, we rewrite the vector $X^{M}$ in the following form

$X^{M}=Z^{M}+\hat{\rho}\hat{c}^{M},\quad Z^{M}=\hat{z}^{M}-z^{M}.$ (III.8)

Then, expanding Eq. (III.6) in the inverse powers of the Lorentz-invariant
distance $\hat{\rho}$ we find the emitted part of the scalar field derivative
as

$\left(\partial_{M}\varphi(x)\right)^{\rm
rad}=\frac{g\hat{c}_{M}}{2^{5/2}\hat{\rho}^{3/2}}\int_{-\infty}^{\hat{\tau}}d\tau\left[\frac{3}{2}\frac{1}{(-Z\hat{c})^{5/2}}-\frac{\delta(\tau-\hat{\tau})}{(-Z\hat{c})^{3/2}}+\frac{2(a\hat{c})\delta(\tau-\hat{\tau})}{(v\hat{c})^{2}(-Z\hat{c})^{1/2}}\right],$
(III.9)

where we have taken into account that up to the leading order $X^{2}\sim
2\hat{\rho}(Z\hat{c})$ and $(vX)\sim\hat{\rho}(v\hat{c})$.

Each integral in (III.9) diverges on the upper integration limit
$\tau\to\hat{\tau}$, where
$(Z\hat{c})\to(\hat{v}\hat{c})(\hat{\tau}-\tau)=-(\hat{\tau}-\tau)$. However,
by use of the regularization procedure discussed in section II.D one can show
that the last two terms in Eq. (III.9) do not contain physical information
concerning the properties of the field in wave zone and just subtract the
divergences from the first one. Indeed, we introduce the regularising
parameter $\epsilon\to+0$ into the argument of delta function, obtaining the
divergent result

$\int_{-\infty}^{\hat{\tau}}d\tau\,\frac{\delta(\tau-\hat{\tau}+\epsilon)}{(-Z\hat{c})^{3/2}}=\frac{1}{\epsilon^{3/2}},$
(III.10)

which is rewritten as

$\frac{1}{\epsilon^{3/2}}=\frac{3}{2}\int_{-\infty}^{\hat{\tau}-\epsilon}\frac{d\tau}{(\hat{\tau}-\tau)^{3/2}}.$
(III.11)

The second term with delta function is transformed in an analogous manner
providing us with the integral

$\int_{-\infty}^{\hat{\tau}}d\tau\frac{(a\hat{c})\delta(\tau-\hat{\tau})}{(v\hat{c})^{2}(-Z\hat{c})^{1/2}}=\frac{1}{2}\int_{-\infty}^{\hat{\tau}-\epsilon}d\tau\frac{(\hat{a}\hat{c})}{(\hat{\tau}-\tau)^{3/2}}.$
(III.12)

Then, emitted part of the retarded scalar field derivative takes the form

$\left(\partial_{M}\varphi(x)\right)^{\rm
rad}=\lim_{\epsilon\to+0}\frac{g\hat{c}_{M}}{2^{5/2}\hat{\rho}^{3/2}}\int_{-\infty}^{\hat{\tau}-\epsilon}d\tau\left[\frac{3}{2}\frac{1}{(-Z\hat{c})^{5/2}}-\frac{3}{2}\frac{1}{(\hat{\tau}-\tau)^{5/2}}+\frac{(\hat{a}\hat{c})}{(\hat{\tau}-\tau)^{3/2}}\right].$
(III.13)

Further, we omit the regularising parameter in the upper integration limit for
simplicity, taking in mind that to make the finiteness of the obtained
expression explicit one needs to perform some transformation of the first term
in the integrand: usually it is the integration by parts, which extract the
divergences from the first term and subtract them with the last two integrals.
The obtained structure of the emitted part of the field is the general feature
of the odd-dimensional fields Galtsov:2020hhn  – it consists of several
separately divergent integrals, first of which contains all the information
concerning the field in the wave zone, while the remaining ones are just the
counter-terms eliminating the divergence from the first one. Note, also, that
the emitted part of the retarded field depends on the entire history of the
charge’s motion preceding the retarded proper time $\hat{\tau}$.

Now let us find the non-relativistic approximation of the emitted part of the
scalar field derivative (III.13). We make the following assumptions about the
charge’s motion:

  * •

it is non-relativistic $|\mathbf{v}|\ll 1,\,\forall\tau$;

  * •

it moves inside compact region of space $|\mathbf{z}|\leq d,\forall\tau$ ($d$
is the characteristic size of this region);

  * •

the observation point is far from this region $d\ll r$.

Then, the covariant retarded quantities are expanded up to the leading order
in small parameters $|\mathbf{v}|$ and $|\mathbf{z}|/r$ as

$\displaystyle\hat{\tau}\simeq\bar{t}+\mathbf{n}\bar{\mathbf{z}},\quad\mathbf{n}=\mathbf{x}/r,$
(III.14) $\displaystyle\hat{\rho}\simeq
r\left(1-\mathbf{n}\bar{\mathbf{v}}-\frac{1}{r}\mathbf{n}\bar{\mathbf{z}}\right),$
(III.15)
$\displaystyle\hat{c}^{M}\simeq\left[1+\mathbf{n}\bar{\mathbf{v}};\,\mathbf{n}\left(1+\mathbf{n}\bar{\mathbf{v}}+\frac{1}{r}\mathbf{n}\bar{\mathbf{z}}\right)-\frac{1}{r}\bar{\mathbf{z}}\right],$
(III.16)

where $\bar{t}=t-r$ is the retarded time calculated up to the leading
contribution, and all the barred quantities correspond to this moment. Using
the Eqs. (III.14–III.16), we rewrite the first term of the integrand in Eq.
(III.13) as

$\frac{1}{(-Z\hat{c})^{5/2}}\simeq\frac{1}{(\bar{t}-t^{\prime})^{5/2}}\left[1+\mathbf{n}\bar{\mathbf{v}}-\mathbf{n}\mathbf{s}(t^{\prime})\left(1+\mathbf{n}\bar{\mathbf{v}}+\frac{1}{r}\mathbf{n}\bar{\mathbf{z}}\right)-\frac{1}{r}\mathbf{s}(t^{\prime})\bar{\mathbf{z}}\right]^{-5/2},$
(III.17)

where we introduced the spacelike vector $\mathbf{s}(t^{\prime})$ defined as

$\mathbf{s}(t^{\prime})=\frac{\bar{\mathbf{z}}-\mathbf{z}(t^{\prime})}{\bar{t}-t^{\prime}},$
(III.18)

and replaced the proper time with the coordinate one $\tau=t^{\prime}$, given
their equivalence in the non-relativistic limit. It is clear that at the
retarded time $\bar{t}$ vector $\mathbf{s}(t^{\prime})$ has a finite value
$\displaystyle\lim_{t^{\prime}\to\bar{t}}\mathbf{s}(t^{\prime})=\bar{\mathbf{v}}$.
Rewriting the particle’s worldline in terms of its velocity

$\mathbf{z}(t)=\int_{t_{\rm
in}}^{t}\mathbf{v}(t^{\prime})dt^{\prime}+\mathbf{z}(t_{\rm in}),$ (III.19)

where $t_{\rm in}$ is the initial moment of time in the remote past and using
the first mean value theorem, we find that the vector $\mathbf{s}(t^{\prime})$
is of order of particle’s velocity

$|\mathbf{s}(t^{\prime})|\sim|\mathbf{v}|,\,\forall t^{\prime}.$ (III.20)

So, it should be used as another expansion parameter. Thus, we calculate the
integrand in (III.13) up to the leading order in small parameters arriving at

$(-Z\hat{c})^{-5/2}=\frac{1}{(\bar{t}-t^{\prime})^{5/2}}\left(1+\frac{5}{2}\left(\mathbf{n}\mathbf{s}(t^{\prime})-\mathbf{n}\bar{\mathbf{v}}\right)\right),\quad\hat{c}_{M}=\bar{c}_{M}=[-1,\mathbf{n}],\quad\hat{a}\hat{c}=\mathbf{n}\bar{\mathbf{a}}.$
(III.21)

Substituting the obtained expansions into the Eq. (III.13), we find the non-
relativistic approximation of the emitted part of scalar field derivative as

$(\partial_{M}\varphi(x))^{\rm
rad}=-\frac{g\bar{c}_{M}}{2^{5/2}r^{3/2}}\int_{-\infty}^{\bar{t}}dt^{\prime}\left[\frac{15}{4}\frac{\mathbf{n}\mathbf{s}(t^{\prime})}{(\bar{t}-t^{\prime})^{5/2}}-\frac{15}{4}\frac{\mathbf{n}\bar{\mathbf{v}}}{(\bar{t}-t^{\prime})^{5/2}}+\frac{\mathbf{n}\bar{\mathbf{a}}}{(\bar{t}-t^{\prime})^{3/2}}\right].$
(III.22)

To make the finitness of the emitted part of the scalar field explicit, we
introduce the regularising parameter into the upper integration limit and
integrate the first term in (III.22) by parts three times arriving at

$\frac{15}{4}\lim_{\epsilon\to+0}\int_{-\infty}^{\bar{t}-\epsilon}dt^{\prime}\frac{\mathbf{n}\mathbf{s}(t^{\prime})}{(\bar{t}-t^{\prime})^{5/2}}=\frac{5}{2}\lim_{\epsilon\to+0}\frac{\mathbf{n}\bar{\mathbf{v}}}{\epsilon^{3/2}}-2\lim_{\epsilon\to+0}\frac{\mathbf{n}\bar{\mathbf{a}}}{\epsilon^{1/2}}+2\int_{-\infty}^{\bar{t}}dt^{\prime}\frac{\mathbf{n}\dot{\mathbf{a}}(t^{\prime})}{(\bar{t}-t^{\prime})^{1/2}},$
(III.23)

where $\dot{\mathbf{a}}=d^{3}\mathbf{z}/dt^{\prime 3}$. It is clear that the
first two divergent terms in Eq. (III.23) are cancelled out by the ”counter-
terms” in (III.22) yielding the simple expression for the emitted part of the
scalar field derivative in the non-relativistic limit

$\left(\partial_{M}\varphi(x)\right)^{\rm
rad}=-\frac{g\bar{c}_{M}}{2^{3/2}r^{3/2}}\int_{-\infty}^{\bar{t}}dt^{\prime}\frac{\mathbf{n}\dot{\mathbf{a}}(t^{\prime})}{(\bar{t}-t^{\prime})^{1/2}}.$
(III.24)

Note that in five-dimensions, up to the leading order in non-relativistic
expansion, the rectilinear uniformly accelerated scalar charge does not
radiate, in contrast with the four-dimensional theory. However, as will be
shown later, this is not the case for the gravitational radiation.

Substituting the Eq. (III.24) into the energy-momentum tensor of the five-
dimensional massless scalar field, analogous to that of Eq. (II.33),

$T_{MN}=\frac{1}{2\pi^{2}}\left(\partial_{M}\varphi\partial_{N}\varphi-\frac{1}{2}\eta_{MN}\partial^{A}\varphi\partial_{A}\varphi\right),$
(III.25)

we find the radiation energy-momentum tensor as

$T_{MN}^{\rm
rad}(x)=\frac{g^{2}\bar{c}_{M}\bar{c}_{N}}{16\pi^{2}r^{3}}\left[\int_{-\infty}^{\bar{t}}dt^{\prime}\frac{\mathbf{n}\dot{\mathbf{a}}(t^{\prime})}{(\bar{t}-t^{\prime})^{1/2}}\right]^{2},$
(III.26)

which has the tensor structure corresponding to the propagation of the given
part of energy-momentum exactly with the speed of light. One can show that in
the case of charge’s uniform circular motion Eq. (III.26) yields the finite
value of radiation power coinciding with the one obtained from the spectral
decompositions of the radiation power indifferent to the dimensionality of
spacetime found in Galtsov:2020hhn . Note that in the non-relativistic limit
the radiation power (III.26) depends on the entire history of charge’s motion
preceding to the retarded time, while in the ultra-relativistic limit studied
in Galtsov:2020hhn  this dependence effectively reduces to the small interval
of proper time near the retarded moment.

###  III.2 Emitted part of the gravitational field

Let us return to the point particles contribution into the gravitational
radiation (III.1). Due to the particles energy-momentum tensor (II.32) being
just the sum of energy-momentum tensors of individual particles, it is
convenient to calculate the gravitational field of one particle and then take
the sum of two analogous terms.

Retarded gravitational field of a point particle with mass $m$ moving along an
arbitrary worldline $z^{M}(\tau)$, in accordance with Eq. (III.1), is given as

$\bar{h}_{AB}^{\rm P}=\frac{m\kappa_{5}}{\pi^{2}}\int
d\tau\,v_{A}v_{B}\,\theta(X^{0})\left[\frac{\delta(-X^{2})}{(-X^{2})^{1/2}}-\frac{1}{2}\frac{\theta(-X^{2})}{(-X^{2})^{3/2}}\right].$
(III.27)

By analogy with the scalar field, its derivative has the form

$\partial_{M}\bar{h}_{AB}^{\rm P}=-\frac{2m\kappa_{5}}{\pi^{2}}\int
d\tau\,v_{A}v_{B}\,\theta(X^{0})\left[\frac{3}{4}\frac{\theta(-X^{2})}{(-X^{2})^{5/2}}-\frac{\delta(-X^{2})}{(-X^{2})^{3/2}}+\frac{\delta^{\prime}(-X^{2})}{(-X^{2})^{1/2}}\right]X_{M}.$
(III.28)

We transform the term containing derivative of delta function by the
integration by parts using the relation (III.5). We, also, use relation
(III.7) to transform the products of delta and Heaviside functions. Thus, the
gravitational field derivative takes the form

$\partial_{M}\bar{h}_{AB}^{\rm
P}=-\frac{m\kappa_{5}}{2\pi^{2}}\int_{-\infty}^{\hat{\tau}}d\tau\left[3\frac{v_{A}v_{B}X_{M}}{(-X^{2})^{5/2}}-\frac{v_{A}v_{B}X_{M}}{\hat{\rho}(-X^{2})^{3/2}}\delta(\tau-\hat{\tau})-\right.\\\
-\left.\frac{2a_{(A}v_{B)}X_{M}-v_{A}v_{B}v_{M}}{\hat{\rho}(vX)(-X^{2})^{1/2}}\delta(\tau-\hat{\tau})+\frac{v_{A}v_{B}X_{M}[(aX)+1]}{\hat{\rho}(vX)^{2}(-X^{2})^{1/2}}\delta(\tau-\hat{\tau})\right],$
(III.29)

where $a^{M}=d^{2}z^{M}/d\tau^{2}$ is the acceleration five-vector, and we
defined symmetrisation over two indices as
$A_{(M}B_{N)}=\frac{1}{2}(A_{M}B_{N}+A_{N}B_{M})$.

We extract the emitted part of the gravitational field derivative (III.29) by
use of the relations (III.8), expanding the integrand in the inverse powers of
the Lorentz-invariant distance $\hat{\rho}$. By analogy with the scalar field,
it is found as

$(\partial_{M}\bar{h}_{AB}^{\rm P})^{\rm
rad}=-\frac{m\kappa_{5}\hat{c}_{M}}{2^{3/2}\pi^{2}\hat{\rho}^{3/2}}\int_{-\infty}^{\hat{\tau}}d\tau\left[\frac{3}{4}\frac{v_{A}v_{B}}{(-Z\hat{c})^{5/2}}-\frac{1}{2}\frac{v_{A}v_{B}}{(-Z\hat{c})^{3/2}}\delta(\tau-\hat{\tau})-\right.\\\
-\left.\frac{2a_{(A}v_{B)}}{(v\hat{c})(-Z\hat{c})^{1/2}}\delta(\tau-\hat{\tau})+\frac{v_{A}v_{B}(a\hat{c})}{(v\hat{c})^{2}(-Z\hat{c})^{1/2}}\delta(\tau-\hat{\tau})\right].$
(III.30)

Terms containing delta functions are transformed by analogy with Eqs. (III.10
– III.12)

$\displaystyle\int_{-\infty}^{\hat{\tau}}d\tau\frac{1}{2}\frac{v_{A}v_{B}}{(-Z\hat{c})^{3/2}}\delta(\tau-\hat{\tau})=\int_{-\infty}^{\hat{\tau}}d\tau\frac{3}{4}\frac{\hat{v}_{A}\hat{v}_{B}}{(\hat{\tau}-\tau)^{5/2}},$
(III.31)
$\displaystyle\int_{-\infty}^{\hat{\tau}}d\tau\frac{2a_{(A}v_{B)}}{(v\hat{c})(-Z\hat{c})^{1/2}}\delta(\tau-\hat{\tau})=-\int_{-\infty}^{\hat{\tau}}d\tau\,\frac{\hat{a}_{(A}\hat{v}_{B)}}{(\hat{\tau}-\tau)^{3/2}},$
(III.32)
$\displaystyle\int_{-\infty}^{\hat{\tau}}d\tau\frac{v_{A}v_{B}(a\hat{c})}{(v\hat{c})^{2}(-Z\hat{c})^{1/2}}\delta(\tau-\hat{\tau})=\int_{-\infty}^{\hat{\tau}}d\tau\frac{1}{2}\frac{\hat{v}_{A}\hat{v}_{B}(\hat{a}\hat{c})}{(\hat{\tau}-\tau)^{3/2}}.$
(III.33)

Therefore, the emitted part of the retarded gravitational field takes the form
analogous to that of the scalar field (III.13)

$(\partial_{M}\bar{h}_{AB}^{\rm P})^{\rm
rad}=-\frac{m\kappa_{5}\hat{c}_{M}}{2^{5/2}\pi^{2}\hat{\rho}^{3/2}}\int_{-\infty}^{\hat{\tau}}d\tau\left[\frac{3}{2}\frac{v_{A}v_{B}}{(-Z\hat{c})^{5/2}}-\frac{3}{2}\frac{\hat{v}_{A}\hat{v}_{B}}{(\hat{\tau}-\tau)^{5/2}}+\frac{2\hat{a}_{(A}\hat{v}_{B)}+\hat{v}_{A}\hat{v}_{B}(\hat{a}\hat{c})}{(\hat{\tau}-\tau)^{3/2}}\right].$
(III.34)

Note that, by analogy with the scalar field (III.13), all the information
concerning the gravitational field in the wave zone is contained in the first
term of Eq. (III.34), while the remaining terms just subtract the divergences
present in the first one at the upper integration limit. Also, as in the case
of scalar field, the emitted part of five-dimensional gravitational field
depends on the history of the particle’s motion.

###  III.3 Gravitational radiation of non-relativistic particle

Now we turn to the computation of gravitational radiation from the non-
relativistic particle. By analogy with the case of scalar field, we assume
that: (i) particle is non-relativistic $|\mathbf{v}|\ll 1,\,\forall\tau$; (ii)
it moves inside the compact region of space $|\mathbf{z}|\leq d,\forall\tau$;
(iii) the observation point is far from this region $d\ll r$. We are
interested only in the spatial components of metric perturbations

$(\partial_{M}\bar{h}_{ij}^{\rm P})^{\rm
rad}=-\frac{m\kappa_{5}\hat{c}_{M}}{2^{5/2}\pi^{2}\hat{\rho}^{3/2}}\int_{-\infty}^{\hat{\tau}}d\tau\left[\frac{3}{2}\frac{v_{i}v_{j}}{(-Z\hat{c})^{5/2}}-\frac{3}{2}\frac{\hat{v}_{i}\hat{v}_{j}}{(\hat{\tau}-\tau)^{5/2}}+\frac{2\hat{a}_{(i}\hat{v}_{j)}+\hat{v}_{i}\hat{v}_{j}(\hat{a}\hat{c})}{(\hat{\tau}-\tau)^{3/2}}\right],$
(III.35)

because it is them, which contribute into the effective energy-momentum tensor
of gravitational field (II.17).

By analogy with the scalar field, using expansions of the retarded covariant
quantities Eqs. (III.14–III.16) and introducing the spacelike vector
$\mathbf{s}(t^{\prime})$ defined by Eq. (III.18) (here we replaced the proper
time with coordinate one), we obtain the non-relativistic approximation of the
emitted part of the gravitatonal field up to the leading order in small
parameters $|\mathbf{z}|/r,|\mathbf{v}|,|\mathbf{s}|\ll 1$ as

$(\partial_{M}\bar{h}_{ij}^{\rm P})^{\rm
rad}=-\frac{m\kappa_{5}\bar{c}_{M}}{2^{5/2}\pi^{2}r^{3/2}}\int_{-\infty}^{\bar{t}}dt^{\prime}\left[\frac{15}{4}v_{i}v_{j}\frac{\mathbf{n}\mathbf{s}-\mathbf{n}\bar{\mathbf{v}}}{(\bar{t}-t^{\prime})^{5/2}}+\frac{3}{2}\frac{v_{i}v_{j}}{(\bar{t}-t^{\prime})^{5/2}}-\frac{3}{2}\frac{\bar{v}_{i}\bar{v}_{j}}{(\bar{t}-t^{\prime})^{5/2}}+\right.\\\
+\left.\frac{2\bar{a}_{(i}\bar{v}_{j)}+\bar{v}_{i}\bar{v}_{j}\mathbf{n}\bar{\mathbf{a}}}{(\bar{t}-t^{\prime})^{3/2}}\right].$
(III.36)

Here, all the information concerning the gravitational field in the wave zone
is contained in the first two terms of the integrand, while the remaining two
terms just subtract the divergences present in them at the upper integration
limit.

To make the convergence of integral (III.36) explicit, by analogy with the
scalar field, one needs to integrate the first two terms of the integrand by
parts reducing the powers of denominators to $1/2$. Detailed discussion of
this integration is relegated to the Appendix B, while here we just present
the result remaining after the mutual cancellation of divergences

$(\partial_{M}\bar{h}_{ij}^{\rm P})^{\rm
rad}=-\frac{m\kappa_{5}\bar{c}_{M}}{2^{5/2}\pi^{2}r^{3/2}}\int_{-\infty}^{\bar{t}}dt^{\prime}\left[2v_{i}v_{j}\frac{\mathbf{n}\dot{\mathbf{a}}}{(\bar{t}-t^{\prime})^{1/2}}+12a_{(i}v_{j)}\frac{\mathbf{n}\mathbf{a}}{(\bar{t}-t^{\prime})^{1/2}}+\right.\\\
+\left.\left(\dot{a}_{i}v_{j}+2a_{i}a_{j}+v_{i}\dot{a}_{j}\right)\frac{6\mathbf{n}\mathbf{v}-5\mathbf{n}\bar{\mathbf{v}}+2}{(\bar{t}-t^{\prime})^{1/2}}-2\left(\ddot{a}_{i}v_{j}+3\dot{a}_{i}a_{j}+3a_{i}\dot{a}_{j}+v_{i}\ddot{a}_{j}\right)\frac{\mathbf{n}\bar{\mathbf{z}}-\mathbf{n}\mathbf{z}}{(\bar{t}-t^{\prime})^{1/2}}\right].$
(III.37)

However, the terms in the integrand of Eq. (III.37) does not contribute
equivalently into the non-relativistic approximation of the emitted part of
the gravitational field.

To perform power counting, let us consider the Fourier transform of the
particle’s world line

$z^{k}(t)=\int\frac{d\omega}{2\pi}\,\tilde{z}^{k}(\omega)e^{-i\omega
t},\quad\tilde{z}^{k}(\omega)=\int dt\,z^{k}(t)e^{i\omega t}.$ (III.38)

When particle performs a periodic motion inside the compact region of space
$|\mathbf{z}|\leq d$, the Fourier transform of its world line satisfies the
inequality

$\tilde{z}^{k}(\omega)\leq d\int dte^{i\omega t}=2\pi d\delta(\omega),$
(III.39)

where the right-hand side corresponds to the particle at rest at the border of
this region. Thus, the Fourier transform of periodically moving particle’s
world line has to be of form

$\tilde{z}^{k}(\omega)=2\pi df^{k}(\omega),$ (III.40)

where $f^{k}(\omega)$ are the functions localised around a characteristic
frequency of periodic motion $\omega_{\rm ch}$. Then, the $n$-th derivative of
the particle’s world line is found as

$\frac{d^{n}z^{k}}{dt^{n}}=(-i)^{n}\int
d\omega\,d\,\omega^{n}f^{k}(\omega)e^{-i\omega
t}\sim-i\frac{|\mathbf{v}|}{d}\frac{d^{n-1}z^{k}}{dt^{n-1}},$ (III.41)

where, due to the localisation of integration interval around $\omega_{\rm
ch}$, product $d\omega$ has to be of order of particle’s velocity
$|\mathbf{v}|$. The case of circular motion is the most transparent example

$\displaystyle z^{k}(t)=R_{0}\cos(\omega_{0}t+\phi),$ (III.42)
$\displaystyle\frac{d^{n}z^{k}}{dt^{n}}=\omega^{n-1}_{0}|\mathbf{v}|\cos\left(\omega_{0}t+\phi+\frac{\pi
n}{2}\right),$ (III.43)

where $\omega_{0}R_{0}$ is the particle’s velocity. Therefore, in the non-
relativistic limit, all the particle’s kinematic characteristics are of order
of its velocity $d^{n}z^{k}/dt^{n}\sim|\mathbf{v}|\ll 1$. Following the same
reasoning, one can show that
$z^{k}\,d^{n}z^{l}/dt^{n}\sim|\mathbf{v}|^{2},\,n\geq 2$.

Based on the power counting scheme above, we find that in the non-relativistic
limit only one term in the integrand in Eq. (III.37) contributes into the
emitted part of the gravitational field

$(\partial_{M}\bar{h}_{ij}^{\rm P})^{\rm
rad}=-\frac{m\kappa_{5}\bar{c}_{M}}{2^{3/2}\pi^{2}r^{3/2}}\int_{-\infty}^{\bar{t}}dt^{\prime}\frac{\dot{a}_{i}v_{j}+2a_{i}a_{j}+v_{i}\dot{a}_{j}}{(\bar{t}-t^{\prime})^{1/2}}.$
(III.44)

Note that, by analogy with the scalar field (III.24), in the non-relativistic
limit, emitted part of the gravitational field depends on the entire history
of particle’s motion preceding the retarded time.

###  III.4 Gravitational radiation of binary system

Having found the non-relativistic approximation of the gravitational field of
individual particle (III.44), we proceed to the case of non-relativistic
binary system. Without loss of generality, all the previous calculations could
be performed in the center-of-mass frame

$\mathbf{x}_{\rm
CM}=\frac{m_{1}\mathbf{z}_{1}+m_{2}\mathbf{z}_{2}}{M}=0,\forall t,$ (III.45)

where $M=m_{1}+m_{2}$ is the total mass of the system. Then, the particles’
coordinates are rewritten in terms of the relative coordinate
$\mathbf{z}=\mathbf{z}_{2}-\mathbf{z}_{1}$ in a standard manner

$\mathbf{z}_{1}=-\frac{m_{2}}{M}\mathbf{z},\quad\mathbf{z}_{2}=\frac{m_{1}}{M}\mathbf{z}.$
(III.46)

Therefore, in the center-of-mass frame, superposition of gravitational field’s
from individual particles forming the binary system is given by the integral
(III.44) calculated on the relative coordinate

$(\partial_{M}\bar{h}_{ij}^{\rm P})^{\rm
rad}=-\frac{\mu\kappa_{5}\bar{c}_{M}}{2^{3/2}\pi^{2}r^{3/2}}\int_{-\infty}^{\bar{t}}dt^{\prime}\frac{\dot{a}_{i}v_{j}+2a_{i}a_{j}+v_{i}\dot{a}_{j}}{(\bar{t}-t^{\prime})^{1/2}},$
(III.47)

where $\mu=m_{1}m_{2}/M$ is the reduced mass of the system. It is the Eq.
(III.47) that gives the point particles contribution into the gravitational
radiation of the binary system.

##  IV Scalar field contribution

Contribution of the scalar field into the gravitational radiation of the
binary system, by analogy with the one from the point particles (III.1), is
written as

$\bar{h}_{MN}^{\rm F}(x)=-2\kappa_{5}\int d^{5}x^{\prime}\,T_{MN}^{\rm
F}(x^{\prime})\,G_{\rm ret}^{4+1}(x-x^{\prime}),$ (IV.1)

where the scalar field energy-momentum tensor is given by Eq. (II.33) and the
retarded Green’s function is defined by Eq. (II.35).

Recall that to determine the energy-momentum flux of gravitational radiation
(II.17) one needs only the spatial components of metric perturbations in the
transverse-traceless gauge. Computing the contractions (II.22), we take into
account the vanishing traces of the projector (II.21) and that the second term
in the scalar field energy-momentum tensor $T_{ij}^{\rm F}$ is proportional to
the Kronecker delta. Therefore, the scalar field energy-momentum tensor
effectively reduces to

$T_{ij}^{\rm
F}=\frac{1}{4\pi}\partial_{i}\varphi\partial_{j}\varphi\,\delta(x^{4}).$
(IV.2)

Also, recall the condition for absence of the scalar radiation from the
system. Based on it, we omit the self-energy type terms in the scalar field
energy-momentum tensor

$T_{ij}^{\rm
F}=\frac{1}{4\pi}\partial_{i}\varphi_{1}\partial_{j}\varphi_{2}\,\delta(x^{4})+(i\leftrightarrow
j)$ (IV.3)

leaving only the interaction type ones. Therefore, we obtain the effective
scalar field contribution into the gravitational field as

$\displaystyle\bar{h}_{ij}^{\rm F}(x)=J_{ij}(x)+(i\leftrightarrow j),$ (IV.4)
$\displaystyle J_{ij}(x)=-\frac{\kappa_{5}}{2\pi}\int
d^{5}x^{\prime}\,\partial_{i}^{\,\prime}\varphi_{1}(x^{\prime})\,\partial_{j}^{\,\prime}\varphi_{2}(x^{\prime})\,G_{\rm
ret}^{5}(x-x^{\prime})\,\delta(x^{\prime 4}).$ (IV.5)

We compute the $J_{ij}(x)$ integral and symmetrise the obtained result over
the indices to find the gravitational radiation produced by the scalar field.

Figure 1: The spacetime splitting into the near and radiation zones.
Integration region for the tail integral ${\cal C}_{\rm in}$ and its
intersection with the near zone $\cal K$ (left figure). Integration region for
the cone integral ${\cal C}$ and its intersection with the near zone $\cal N$
(right figure).

Given the structure of retarded Green’s function (II.35), we split the
$J_{ij}(x)$ integral into two terms with different geometrical interpretations

$\displaystyle J_{ij}(x)=J_{ij}^{\rm T}(x)+J_{ij}^{\rm C}(x),$ (IV.6)
$\displaystyle J_{ij}^{\rm T}(x)=-\frac{\kappa_{5}}{8\pi^{3}}\int
d^{5}x^{\prime}\,T_{ij}(x^{\prime})\,\theta(X^{0})\,\theta(-X^{2})\,\delta(x^{\prime
4}),$ (IV.7) $\displaystyle J_{ij}^{\rm C}(x)=\frac{\kappa_{5}}{4\pi^{3}}\int
d^{5}x^{\prime}\,C_{ij}(x^{\prime})\,\theta(X^{0})\delta(-X^{2})\,\delta(x^{\prime
4}),$ (IV.8)

where we introduced vector $X^{M}=x^{M}-x^{\prime M}$. The tail integral
(IV.7) corresponds to the integration of tensor

$T_{ij}(x^{\prime})=\frac{\partial_{i}^{\,\prime}\varphi_{1}(x^{\prime})\,\partial_{j}^{\,\prime}\varphi_{2}(x^{\prime})}{(-X^{2})^{3/2}}$
(IV.9)

over the cross-section of the light cone in the past with top at the
observation point $x^{M}$ and of its interior by the hypersurface $x^{\prime
4}=0$. The corresponding integration region is denoted as ${\cal C}_{\rm in}$,
see Fig. (1). Analogously, the cone integral (IV.8) corresponds to the
integration of tensor

$C_{ij}(x^{\prime})=\frac{\partial_{i}^{\,\prime}\varphi_{1}(x^{\prime})\,\partial_{j}^{\,\prime}\varphi_{2}(x^{\prime})}{(-X^{2})^{1/2}}$
(IV.10)

over the cross-section of the light cone by the hypersurface $x^{\prime 4}=0$.
This integration region is denoted as $\cal C$, see Fig. (1). Further, we
demonstrate that, by analogy with the point particles contribution, the tail
integral contains all the information about the gravitational field in the
wave zone, while the cone one is just the counter-term subtracting the
divergences contained in the former on the light cone.

###  IV.1 Spacetime splitting – near and radiation zones

In our computations we follow the DIRE approach to the post-Newtonian
expansions in the four-dimensional GR Pati:2000vt  (see, also, Thorne:1980ru ;
Will:1996zj  for compendium of formulas), limiting ourselves to the leading
order contributions. In this approach, splitting the spacetime into zones
yielding different in orders of magnitude contributions into the gravitational
field seems to be essential. So, let us briefly recall their definitions.

In what follows, we work in the center-of-mass frame (III.45) and denote the
characteristic size of the binary system as ${\cal S}$. First, the source zone
${\cal T}$ is defined as the world tube covering the region of particles
motion

${\cal T}=\left\\{x^{M}|r<{\cal S};-\infty<x^{0}<+\infty\right\\}.$ (IV.11)

It is clear that outside the source zone $\cal T$ particles energy-momentum
tensor vanishes $T_{MN}^{\rm P}=0$. In the center-of-mass frame, the origin
lies inside $\cal T$.

We assume that the non-relativistic particles move with characteristic
velocity $v\ll 1$ determining the characteristic wavelength $\lambdabar_{\rm
GW}\sim{\cal S}/v\gg{\cal S}$ of the gravitational waves generated by the
system. Then, the near zone $\cal D$ is defined as the world tube of radius
${\cal R}={\cal S}/v$

${\cal D}=\left\\{x^{M}|r<{\cal R};-\infty<x^{0}<+\infty\right\\},$ (IV.12)

see Figs. (1). Inside $\cal D$, we consider the gravitational and scalar
fields as the instantaneous functions of the particles world lines, i.e., we
neglect the retardation of the fields inside the near zone. Finally, the
radiation zone is defined as the region exterior to the near zone, $r>{\cal
R}$. As we are interested in the gravitational radiation of the system, the
observation point is assumed to lie in the radiation zone,
$|\mathbf{x}|\gg{\cal R}$.

Based on the spacetime zones above, we split the integration regions of the
tail and cone integrals. We denote the intersection of the near zone with the
tail integral’s region as ${\cal K}={\cal C}_{\rm in}\cap{\cal D}$, see Fig.
(1). Analogously, intersection of the near zone with the cone integral’s
hypersurface is denoted as ${\cal N}={\cal C}\cap{\cal D}$, see Fig. (1).
Then, the Eqs. (IV.7) and (IV.8) split into the integrals over the near and
radiation zones

$\displaystyle J_{ij}^{\rm T}(x)=-\frac{\kappa_{5}}{8\pi^{3}}\int_{{\cal
K}}d^{5}x^{\prime}\,T_{ij}(x^{\prime})-\frac{\kappa_{5}}{8\pi^{3}}\int_{{\cal
C}_{\rm in}\setminus{\cal K}}d^{5}x^{\prime}\,T_{ij}(x^{\prime}),$ (IV.13)
$\displaystyle J_{ij}^{\rm C}(x)=\frac{\kappa_{5}}{4\pi^{3}}\int_{{\cal
N}}d^{5}x^{\prime}\,C_{ij}(x^{\prime})+\frac{\kappa_{5}}{4\pi^{3}}\int_{{\cal
C}\setminus{\cal N}}d^{5}x^{\prime}\,C_{ij}(x^{\prime}).$ (IV.14)

However, as we assume particles to be non-relativistic and we are interested
only in the leading contributions into the tail and cone integrals, we suppose
the radius of the near zone $\cal R$ to be large enough for the scalar field
energy-momentum tensor $T_{ij}^{\rm F}$ to be vanishingly small in the
radiation zone. Thus, the tail and cone integrals effectively reduce to

$J_{ij}^{\rm T}(x)=-\frac{\kappa_{5}}{8\pi^{3}}\int_{{\cal
K}}d^{5}x^{\prime}\,T_{ij}(x^{\prime}),\quad J_{ij}^{\rm
C}(x)=\frac{\kappa_{5}}{4\pi^{3}}\int_{{\cal
N}}d^{5}x^{\prime}\,C_{ij}(x^{\prime}).$ (IV.15)

Also, based on the reasoning above, we suppose that the boundary terms, which
would arise from the integrations by parts of the spatial integrals in
(IV.15), are defined by regions $|\mathbf{x}^{\prime}|\sim{\cal R}$ and, thus,
negligibly small. By analogy, we neglect the contributions into resulting
integrals proportional to ${\cal R}^{-n},n>0$.

###  IV.2 Tail integral

We start with computation of the tail integral (IV.15) and its contribution
into the emitted part of gravitational field. Restoring its explicit form we
arrive at

$J_{ij}^{\rm T}(x)=-\frac{\kappa_{5}}{8\pi^{3}}\int_{{\cal
K}}d^{4}x^{\prime}\,\partial_{i}^{\,\prime}\varphi_{1}(x^{\prime})\,\partial_{j}^{\,\prime}\varphi_{2}(x^{\prime})\,\theta(X^{0})\left.\frac{\theta(-X^{2})}{(-X^{2})^{3/2}}\right|_{x^{\prime
4}=0},$ (IV.16)

where the subscript $\cal K$ denotes that we limit integration region to the
near zone, and we performed integration over the $x^{\prime 4}$-coordinate by
use of the delta function.

Heaviside functions in Eq. (IV.16) determine the interval for the temporal
integral

$t^{\prime}\leq t-\sqrt{(\vec{x}-\vec{x}^{\prime})^{2}+(x^{4})^{2}}=t_{\rm
ret}(\vec{x}^{\prime}),$ (IV.17)

where $t=x^{0}$, and we denote coordinates on the brane as $\vec{x}$.
Separating the spatial and temporal integrals, we obtain the tail integral as

$J_{ij}^{\rm T}(x)=-\frac{\kappa_{5}}{8\pi^{3}}\int_{{\cal
K}}d^{3}x^{\prime}\int_{-\infty}^{t_{\rm
ret}}dt^{\prime}\frac{\partial_{i}^{\,\prime}\varphi_{1}(x^{\prime})\,\partial_{j}^{\,\prime}\varphi_{2}(x^{\prime})}{\left[(t-t^{\prime})^{2}-(\vec{x}-\vec{x}^{\prime})^{2}-(x^{4})^{2}\right]^{3/2}}.$
(IV.18)

Note that for the observation point in the radiation zone
$|\vec{x}^{\prime}|\leq{\cal R}\ll r$. Then, expanding the retarded time
(IV.17) and the denominator in Eq. (IV.18) up to the leading order in small
parameter $|\vec{x}^{\prime}|/r\ll 1$ as

$\displaystyle t_{\rm
ret}(\vec{x}^{\prime})\simeq\bar{t}+\vec{x}\vec{x}^{\prime}/r,$ (IV.19)
$\displaystyle(t-t^{\prime})^{2}-(\vec{x}-\vec{x}^{\prime})^{2}-(x^{4})^{2}\simeq(t-t^{\prime})^{2}-r^{2}+2\vec{x}\vec{x}^{\prime},$
(IV.20)

we come to the tail integral in form

$J_{ij}^{\rm T}(x)=-\frac{\kappa_{5}}{8\pi^{3}}\int_{{\cal
K}}d^{3}x^{\prime}\int_{-\infty}^{\bar{t}+\vec{x}\vec{x}^{\prime}/r}dt^{\prime}\frac{\partial_{i}^{\,\prime}\varphi_{1}(x^{\prime})\,\partial_{j}^{\,\prime}\varphi_{2}(x^{\prime})}{\left[(t-t^{\prime})^{2}-r^{2}+2\vec{x}\vec{x}^{\prime}\right]^{3/2}}.$
(IV.21)

As we are interested only in the leading contribution into the tail and cone
integrals, we expand the temporal integral in (IV.21) up to the leading order
in $|\vec{x}^{\prime}|/r$, obtaining

$J_{ij}^{\rm T}(x)=-\frac{\kappa_{5}}{8\pi^{3}}\int_{{\cal
K}}d^{3}x^{\prime}\left[\int_{-\infty}^{\bar{t}}dt^{\prime}\frac{\partial_{i}^{\,\prime}\varphi_{1}(x^{\prime})\,\partial_{j}^{\,\prime}\varphi_{2}(x^{\prime})}{[(t-t^{\prime})^{2}-r^{2}]^{3/2}}+{\cal
O}\left(\frac{\vec{x}^{\prime}}{r}\right)\right].$ (IV.22)

Therefore, in Eq. (IV.22) the spatial and temporal integrals decoupled and we
can calculate them independently.

To transform the Eq. (IV.22) into the form analogous to that of the point
particles contribution (III.47), we have to compute the spatial integral. We
integrate it by parts

$J_{ij}^{\rm T}(x)=\frac{\kappa_{5}}{8\pi^{3}}\int_{{\cal
K}}d^{3}x^{\prime}\int_{-\infty}^{\bar{t}}dt^{\prime}\frac{\varphi_{1}(x^{\prime})\partial_{i}^{\,\prime}\partial_{j}^{\,\prime}\varphi_{2}(x^{\prime})}{[(t-t^{\prime})^{2}-r^{2}]^{3/2}},$
(IV.23)

omitting, as discussed above, the boundary terms. Given that inside $\cal K$
retardation of the fields is negligible, the scalar fields are given by Eq.
(II.29), and the obtained integral is easily computed using the methods
developed in the DIRE approach. We relegate the detailed discussion of this
calculation to the Appendix C, while here we just present the result

$\int_{{\cal
K}}d^{3}x^{\prime}\,\varphi_{1}(x^{\prime})\partial_{i}^{\,\prime}\partial_{j}^{\,\prime}\varphi_{2}(x^{\prime})=-\pi\mu[a_{i}(t^{\prime})z_{j}(t^{\prime})+z_{i}(t^{\prime})a_{j}(t^{\prime})]$
(IV.24)

in terms of the relative coordinate of the binary system $z_{i}(t^{\prime})$,
where $\mu$ is its reduced mass. Thus, the tail integral takes the form

$J_{ij}^{\rm
T}(x)=-\frac{\mu\kappa_{5}}{8\pi^{2}}\int_{-\infty}^{\bar{t}}dt^{\prime}\frac{a_{i}z_{j}+z_{i}a_{j}}{[(t-t^{\prime})^{2}-r^{2}]^{3/2}}.$
(IV.25)

Also, we transform the denominator in the Eq. (IV.25), rewriting the
observation time $t$ in terms of the retarded time $t=\bar{t}+r$ and arriving
at

$J_{ij}^{\rm
T}(x)=-\frac{\mu\kappa_{5}}{8\pi^{2}}\int_{-\infty}^{\bar{t}}dt^{\prime}\frac{a_{i}z_{j}+z_{i}a_{j}}{(\bar{t}-t^{\prime})^{3/2}[\bar{t}-t^{\prime}+2r]^{3/2}}.$
(IV.26)

It is clear that the leading contribution into this integral comes from the
small region around the retarded time $\bar{t}-t^{\prime}\ll 2r$. Therefore,
the leading contribution into the tail integral has the form

$J_{ij}^{\rm
T}(x)=-\frac{\mu\kappa_{5}}{2^{9/2}\pi^{2}r^{3/2}}\int_{-\infty}^{\bar{t}}dt^{\prime}\,\frac{a_{i}z_{j}+z_{i}a_{j}}{(\bar{t}-t^{\prime})^{3/2}}$
(IV.27)

analogous to the contribution of the point particles into the gravitational
field of the system (III.47). Note that the obtained integral diverges on the
upper integration limit.

###  IV.3 Cone integral

Now we turn to the calculation of the cone integral (IV.15), which explicitly
is written as

$J_{ij}^{\rm C}(x)=\frac{\kappa_{5}}{4\pi^{3}}\int_{{\cal
N}}d^{4}x^{\prime}\,\partial_{i}^{\,\prime}\varphi_{1}(x^{\prime})\,\partial_{j}^{\,\prime}\varphi_{2}(x^{\prime})\,\theta(X^{0})\left.\frac{\delta(-X^{2})}{(-X^{2})^{1/2}}\right|_{x^{\prime
4}=0},$ (IV.28)

where we performed the integration over $x^{\prime 4}$-coordinate by use of
the delta function.

We transform the product of delta and Heaviside functions by use of the
formula for delta function of complex argument

$\left.\theta(X^{0})\delta(-X^{2})\right|_{x^{\prime
4}=0}=\frac{\delta(t^{\prime}-t_{\rm
ret})}{2\sqrt{(\vec{x}-\vec{x}^{\prime})^{2}+(x^{4})^{2}}},$ (IV.29)

where $t_{\rm ret}$ is the retarded time defined by Eq. (IV.17). Thus, the
cone integral takes the form

$J_{ij}^{\rm C}(x)=\frac{\kappa_{5}}{8\pi^{3}}\int_{{\cal
N}}d^{3}x^{\prime}\int
dt^{\prime}\,\frac{\partial_{i}^{\,\prime}\varphi_{1}(x^{\prime})\,\partial_{j}^{\,\prime}\varphi_{2}(x^{\prime})\,\delta(t^{\prime}-t_{\rm
ret})}{\sqrt{(\vec{x}-\vec{x}^{\prime})^{2}+(x^{4})^{2}}\sqrt{(t-t^{\prime})^{2}-(\vec{x}-\vec{x}^{\prime})^{2}-(x^{4})^{2}}}.$
(IV.30)

By analogy with the tail integral, we expand the Eq. (IV.30) with respect to
the small parameter $|\vec{x}^{\prime}|/r\ll 1$ keeping only the leading
contribution

$J_{ij}^{\rm C}(x)=\frac{\kappa_{5}}{8\pi^{3}r}\int_{{\cal
N}}d^{3}x^{\prime}\int
dt^{\prime}\,\frac{\partial_{i}^{\,\prime}\varphi_{1}(x^{\prime})\,\partial_{j}^{\,\prime}\varphi_{2}(x^{\prime})\,\delta(t^{\prime}-\bar{t})}{\sqrt{(t-t^{\prime})^{2}-r^{2}}},$
(IV.31)

where the spatial and temporal integrals decouple.

Spatial integral in (IV.31) is equivalent to the the one obtained in the
computation of the tail integral and is given by Eq. (IV.24). Thus, the cone
is written as

$J_{ij}^{\rm C}(x)=\frac{\mu\kappa_{5}}{8\pi^{2}r}\int
dt^{\prime}\,\frac{a_{i}z_{j}+z_{i}a_{j}}{\sqrt{(t-t^{\prime})^{2}-r^{2}}}\,\delta(t^{\prime}-\bar{t}),$
(IV.32)

where $z^{i}(t^{\prime})$ is the relative coordinate of the binary system.
Rewriting the coordinate time in terms of the retarded time $t=\bar{t}+r$ and
taking into account that the leading contribution into the integral (IV.32)
comes from the point $t^{\prime}=\bar{t}$, due to the delta function, we
transform denominator of the integrand by analogy with Eq. (IV.27), obtaining

$J_{ij}^{\rm C}(x)=\frac{\mu\kappa_{5}}{2^{7/2}\pi^{2}r^{3/2}}\int
dt^{\prime}\,\frac{a_{i}z_{j}+z_{i}a_{j}}{(\bar{t}-t^{\prime})^{1/2}}\,\delta(t^{\prime}-\bar{t}).$
(IV.33)

Transforming the Eq. (IV.33) by analogy with Eqs. (III.10) and (III.11), we
come the cone integral in the form

$J_{ij}^{\rm
C}(x)=\frac{\mu\kappa_{5}}{2^{9/2}\pi^{2}r^{3/2}}\int_{-\infty}^{\bar{t}}dt^{\prime}\,\frac{\bar{a}_{i}\bar{z}_{j}+\bar{z}_{i}\bar{a}_{j}}{(\bar{t}-t^{\prime})^{3/2}}.$
(IV.34)

It is clear that, by analogy with the point patricles contribution, the cone
integral (IV.33) is just the counter-term subtracting the divergence contained
in the tail integral (IV.27) at the upper integration limit.

###  IV.4 Emitted part of the gravitational field

Combining Eqs. (IV.27) and (IV.34) and taking into account Eq. (IV.4), we find
the scalar field’s contribution into the gravitational radiation of the system

$\bar{h}_{ij}^{\rm
F}(x)=-\frac{\mu\kappa_{5}}{2^{7/2}\pi^{2}r^{3/2}}\int_{-\infty}^{\bar{t}}dt^{\prime}\left[\frac{a_{i}z_{j}+z_{i}a_{j}}{(\bar{t}-t^{\prime})^{3/2}}-\frac{\bar{a}_{i}\bar{z}_{j}+\bar{z}_{i}\bar{a}_{j}}{(\bar{t}-t^{\prime})^{3/2}}\right].$
(IV.35)

In accordance with the Rohrlich-Teitelboim approach, one needs to find the
emitted part of the gravitational field derivative. Note that all the
observation point dependence of the Eq. (IV.35) is contained in the prefactor,
proportional to the inverse distance $1/r$, and in the retarded time
$\bar{t}$. Differentiation of the prefactor increases the decay power of the
field asymptotic and, thus, does not contribute to the emitted part. In
addition, differentiation of the upper limit of integration containing
retarded time leads to the sum of two diverging terms that cancel each other
out, and thus also do not contribute to gravitational radiation. Therefore,
the emitted part of the gravitational field produced by the scalar field
stresses has the form

$(\partial_{M}\bar{h}_{ij}^{\rm F})^{\rm
rad}=\frac{\mu\kappa_{5}(\partial_{M}\bar{t})^{\rm
rad}}{2^{7/2}\pi^{2}r^{3/2}}\int_{-\infty}^{\bar{t}}dt^{\prime}\left[\frac{3}{2}\frac{a_{i}z_{j}+z_{i}a_{j}}{(\bar{t}-t^{\prime})^{5/2}}-\frac{3}{2}\frac{\bar{a}_{i}\bar{z}_{j}+\bar{z}_{i}\bar{a}_{j}}{(\bar{t}-t^{\prime})^{5/2}}+\right.\\\
+\left.\frac{\bar{\dot{a}}_{i}\bar{z}_{j}+\bar{a}_{i}\bar{v}_{j}+\bar{v}_{i}\bar{a}_{j}+\bar{z}_{i}\bar{\dot{a}}_{j}}{(\bar{t}-t^{\prime})^{3/2}}\right].$
(IV.36)

By analogy with the contribution of point particles, the first term in the Eq.
(IV.36) contains all the physical information concerning the gravitational
field in the wave zone, while two remaining terms are just the counter-terms
subtracting the divergences contained in the first one at the upper
integration limit. By analogy with Eq. (III.23), to make the convergence of
the integral (IV.36) explicit, we introduce the regularising parameter
$\epsilon\to+0$ into the upper integration limit $\bar{t}\to\bar{t}-\epsilon$
and integrate the first term by parts twice, arriving at

$\lim_{\epsilon\to+0}\frac{3}{2}\int_{-\infty}^{\bar{t}-\epsilon}dt^{\prime}\frac{a_{i}z_{j}+z_{i}a_{j}}{(\bar{t}-t^{\prime})^{5/2}}=\lim_{\epsilon\to+0}\frac{\bar{a}_{i}\bar{z}_{j}+\bar{z}_{i}\bar{a}_{j}}{\epsilon^{3/2}}-2\lim_{\epsilon\to+0}\frac{\bar{\dot{a}}_{i}\bar{z}_{j}+\bar{a}_{i}\bar{v}_{j}+\bar{v}_{i}\bar{a}_{j}+\bar{z}_{i}\bar{\dot{a}}_{j}}{\epsilon^{1/2}}+\\\
+2\int_{-\infty}^{\bar{t}}dt^{\prime}\frac{\ddot{a}_{i}z_{j}+2\dot{a}_{i}v_{j}+2a_{i}a_{j}+2v_{i}\dot{a}_{j}+z_{i}\ddot{a}_{j}}{(\bar{t}-t^{\prime})^{1/2}}.$
(IV.37)

The first two divergent terms in Eq. (IV.37) are cancelled out by the last two
terms of the integral (IV.36) leaving us with the finite integral.

The derivative of the retarded time $\bar{t}$ is found as

$(\partial_{M}\bar{t})^{\rm rad}=\partial_{M}\bar{t}=-\bar{c}_{M}.$ (IV.38)

Therefore, the emitted part of the gravitational field due to the scalar field
contribution takes the form

$(\partial_{M}\bar{h}_{ij}^{\rm F})^{\rm
rad}=-\frac{\mu\kappa_{5}\bar{c}_{M}}{2^{5/2}\pi^{2}r^{3/2}}\int_{-\infty}^{\bar{t}}dt^{\prime}\frac{\ddot{a}_{i}z_{j}+2\dot{a}_{i}v_{j}+2a_{i}a_{j}+2v_{i}\dot{a}_{j}+z_{i}\ddot{a}_{j}}{(\bar{t}-t^{\prime})^{1/2}},$
(IV.39)

analogous to the emitted part of the gravitational field of point particles
(III.47).

##  V Five-dimensional quadrupole formula

Having found the point particles and scalar field contributions into the
emitted part of gravitational field, we turn to the calculation of total
gravitational radiation power from the binary system.

###  V.1 Emitted part of total gravitational field

Combining the Eqs. (III.47) and (IV.39) we come to the emitted part of the
total gravitational field of the binary system

$(\partial_{M}\bar{h}_{ij})^{\rm
rad}=-\frac{\mu\kappa_{5}\bar{c}_{M}}{2^{5/2}\pi^{2}r^{3/2}}\int_{-\infty}^{\bar{t}}dt^{\prime}\,\frac{\ddot{a}_{i}z_{j}+4\dot{a}_{i}v_{j}+6a_{i}a_{j}+4v_{i}\dot{a}_{j}+z_{i}\ddot{a}_{j}}{(\bar{t}-t^{\prime})^{1/2}}.$
(V.1)

Rewriting the denominator in Eq. (V.1) as

$\ddot{a}_{i}z_{j}+4\dot{a}_{i}v_{j}+6a_{i}a_{j}+4v_{i}\dot{a}_{j}+z_{i}\ddot{a}_{j}=\partial_{t^{\prime}}^{4}(z_{i}z_{j})$
(V.2)

we arrive at

$(\partial_{M}\bar{h}_{ij})^{\rm
rad}=-\frac{\mu\kappa_{5}\bar{c}_{M}}{2^{5/2}\pi^{2}r^{3/2}}\int_{-\infty}^{\bar{t}}dt^{\prime}\,\frac{\partial_{t^{\prime}}^{4}(z_{i}z_{j})}{(\bar{t}-t^{\prime})^{1/2}}.$
(V.3)

As we consider the non-relativistic particles, we define, by analogy with the
four-dimensional theory maggiore2008 , the second mass moment of the system as

$M^{ij}=\sum_{a}m_{a}\int
d^{4}x\,x^{i}x^{j}\,\delta^{(4)}(\mathbf{x}-\mathbf{z}_{a})=\sum_{a}m_{a}z_{a}^{i}z_{a}^{j}.$
(V.4)

Turning into the center-of-mass frame (III.45), the second mass moment takes
the form

$M_{ij}=\mu z_{i}z_{j},$ (V.5)

where $z^{i}$ is the relative coordinate of the system defined by Eq.
(III.46), and $\mu$ is its reduced mass. Thus, emitted part of the total
gravitational field is written as

$(\partial_{M}\bar{h}_{ij})^{\rm
rad}=-\frac{\kappa_{5}\bar{c}_{M}}{2^{5/2}\pi^{2}r^{3/2}}\int_{-\infty}^{\bar{t}}dt^{\prime}\,\frac{\ddddot{M}_{ij}}{(\bar{t}-t^{\prime})^{1/2}}.$
(V.6)

Therefore, by analogy with the gravitational radiation in four dimensions
maggiore2008 , the five-dimensional gravitational radiation of the non-
relativistic binary system is determined by its second mass moment.

Moreover, recall that we are interested in the emitted part of gravitational
field in the transverse-traceless gauge given by Eq. (II.53). Then, taking
into account the vanishing traces of the projector (II.21) one can replace the
second mass moment of the system with its quadrupole moment

$Q_{ij}=M_{ij}-\frac{1}{4}\delta_{ij}M_{kk},$ (V.7)

arriving at the emitted part of total gravitational field in the transverse-
traceless gauge in form

$\displaystyle\left(\partial_{M}h_{ij}^{\rm TT}\right)^{\rm
rad}=-\frac{\kappa_{5}\bar{c}_{M}}{2^{5/2}\pi^{2}r^{3/2}}{\cal A}_{ij}^{\rm
TT},$ (V.8) $\displaystyle{\cal A}_{ij}^{\rm
TT}=\int_{-\infty}^{\bar{t}}dt^{\prime}\frac{\ddddot{Q}_{ij}^{\rm
TT}}{(\bar{t}-t^{\prime})^{1/2}},$ (V.9)

where $Q_{ij}^{\rm TT}=\Lambda_{ij,kl}(\mathbf{n})Q_{kl}$. Obtained emitted
part of the total gravitational field is analogous to that of the four-
dimensional theory maggiore2008 . However, while in four dimensions
gravitational radiation at some moment of time is completely determined by the
state of the source at the corresponding retarded moment, in five dimensions
it depends on the entire history of the source’s motion preceding the retarded
time.

###  V.2 Quadrupole formula

Using the Eqs. (II.17) and (II.54) we find the five-dimensional analog of the
quadrupole formula maggiore2008  for the angular distribution of the
gravitational radiation power of the non-relativistic binary system

$\frac{dW_{4+1}}{d\Omega_{3}}=\frac{\kappa_{5}}{128\pi^{4}}\left\langle{\cal
A}_{ij}^{\rm TT}{\cal A}_{ij}^{\rm TT}\right\rangle,$ (V.10)

where ${\cal A}_{ij}^{\rm TT}(x)$ is given by the Eq. (V.9).

To rewrite Eq. (V.10) in terms of the polarisation amplitudes we follow the
derivation analogous to that in four dimensions maggiore2008 . We note that
the contraction of tensors ${\cal A}_{ij}^{\rm TT}$ is invariant under the
SO(4)-transformations. Thus, the contraction computed in the arbitrary
oriented $x$-frame is equal to the one calculated in the $x^{\prime}$-frame,
where the observation direction vector $\mathbf{n}^{\prime}$ is aligned with
the $x^{\prime 4}$ coordinate

${\cal A}_{ij}^{\rm TT}(x){\cal A}_{ij}^{\rm TT}(x)={\cal
A^{\prime}}_{ij}^{\rm TT}(x^{\prime}){\cal A^{\prime}}_{ij}^{\rm
TT}(x^{\prime}),$ (V.11)

see Eq. (A.1). Then, by analogy with the computation of the polarisation
amplitudes in Appendix A, we calculate it in the $x^{\prime}$-frame and
rewrite the components of tensor ${\cal A^{\prime}}_{ij}^{\rm TT}$ in terms of
${\cal A}_{ij}^{\rm TT}$ by use of the Eq. (A.8). As a result, using Eq. (A.2)
we find

${\cal A^{\prime}}_{ij}^{\rm TT}{\cal A^{\prime}}_{ij}^{\rm
TT}=\operatorname{tr}\left({\cal A^{\prime}}^{\rm TT}{\cal A^{\prime}}^{\rm
TT}\right)=2\left({\cal A}_{+}^{2}+{\cal A}_{\times}^{2}+\frac{3}{4}{\cal
A}_{\displaystyle\circ}^{2}+{\cal A}_{\oplus}^{2}+{\cal
A}_{\otimes}^{2}\right),$ (V.12)

where we have taken into account the definitions of the polarisation
amplitudes (A.9 – A.13) and the symmetry of ${\cal A^{\prime}}_{ij}^{\rm TT}$.

Therefore, the gravitational radiation power takes the following form

$\frac{dW_{4+1}}{d\Omega_{3}}=\frac{\kappa_{5}}{64\pi^{4}}\left\langle{\cal
A}_{+}^{2}+{\cal A}_{\times}^{2}+\frac{3}{4}{\cal
A}_{\displaystyle\circ}^{2}+{\cal A}_{\oplus}^{2}+{\cal
A}_{\otimes}^{2}\right\rangle.$ (V.13)

The obtained Eq. (V.13) determines the contributions of the independent
polarisations into the gravitational radiation of the system. In particular,
it is clear that the ”breathing” mode ${\cal A}_{\displaystyle\circ}$ carries
less energy, than the other polarisation modes of the gravitational field.

###  V.3 Binary system in a circular orbit

As a simple but instructive example, we consider the five-dimensional
gravitational radiation from the non-relativistic binary system in a circular
orbit. Relative coordinate of such a system is written as

$z^{i}=\\{R_{\rm s}\cos{\omega_{\rm s}t},R_{\rm s}\sin{\omega_{\rm
s}t},0,0\\},$ (V.14)

where $R_{\rm s}$ is the radius of the orbit, and $\omega_{\rm s}$ is the
frequency of the orbital motion.

As discussed above, calculating the gravitational radiation one can
equivalently use tensor ${\cal A}_{ij}(x)$ defined in terms of the second mass
moment (V.5) or quadrupole moment (V.7) of the system. In what follows, we use
the former for simplicity

${\cal
A}_{ij}(x)=\int_{-\infty}^{\bar{t}}dt^{\prime}\,\frac{\ddddot{M}_{ij}}{(\bar{t}-t^{\prime})^{1/2}}.$
(V.15)

The non-vanishing components of its derivative have the form

$\displaystyle\ddddot{M}_{11}=-\ddddot{M}_{22}=8\mu R_{\rm s}^{2}\omega_{\rm
s}^{4}\cos{2\omega_{\rm s}t^{\prime}},$ (V.16)
$\displaystyle\ddddot{M}_{12}=\ddddot{M}_{21}=8\mu R_{\rm s}^{2}\omega_{\rm
s}^{4}\sin{2\omega_{\rm s}t^{\prime}}$ (V.17)

From Eqs. (V.16) and (V.17) we find the non-zero components of ${\cal
A}_{ij}(x)$

$\displaystyle{\cal A}_{11}(x)=-{\cal A}_{22}(x)=\sqrt{32\pi}\mu R_{\rm
s}^{2}\omega_{\rm s}^{7/2}\sin\left(2\omega_{\rm
s}\bar{t}+\frac{\pi}{4}\right),$ (V.18) $\displaystyle{\cal A}_{12}(x)={\cal
A}_{21}(x)=\sqrt{32\pi}\mu R_{\rm s}^{2}\omega_{\rm
s}^{7/2}\sin\left(2\omega_{\rm s}\bar{t}-\frac{\pi}{4}\right),$ (V.19)

For detailed derivation of the Eqs. (V.18) and (V.19) see Appendix D.

In accordance with Eq. (V.13), to compute the power of gravitational radiation
of the system we find the polarisations amplitudes of the gravitational field.
Substituting into the Eqs. (A.9 \- A.13) the non-vanishing components of
${\cal A}_{ij}(x)$ tensor (V.18) and (V.19), we arrive at

$\displaystyle{\cal A}_{+}=-\sqrt{8\pi}\mu R_{\rm s}^{2}\omega_{\rm
s}^{7/2}(1+\cos^{2}\theta)\sin\left(2\omega_{\rm
s}\bar{t}+\frac{\pi}{4}-2\phi\right),$ (V.20) $\displaystyle{\cal
A}_{\times}=\sqrt{32\pi}\mu R_{\rm s}^{2}\omega_{\rm
s}^{7/2}\cos\theta\cos\left(2\omega_{\rm
s}\bar{t}+\frac{\pi}{4}-2\phi\right),$ (V.21) $\displaystyle{\cal
A}_{\displaystyle\circ}=\frac{\sqrt{32\pi}}{3}\mu R_{\rm s}^{2}\omega_{\rm
s}^{7/2}\left(2\sin^{2}\theta\cos^{2}\zeta-\cos^{2}\theta+1\right)\sin\left(2\omega_{\rm
s}\bar{t}+\frac{\pi}{4}-2\phi\right),$ (V.22) $\displaystyle{\cal
A}_{\oplus}=\sqrt{32\pi}\mu R_{\rm s}^{2}\omega_{\rm
s}^{7/2}\sin\theta\cos\zeta\cos\left(2\omega_{\rm
s}\bar{t}+\frac{\pi}{4}-2\phi\right),$ (V.23) $\displaystyle{\cal
A}_{\otimes}=\sqrt{8\pi}\mu R_{\rm s}^{2}\omega_{\rm
s}^{7/2}\sin{2\theta}\cos\zeta\sin\left(2\omega_{\rm
s}\bar{t}+\frac{\pi}{4}-2\phi\right).$ (V.24)

Note that for the observation point lying on the brane $\zeta=\pi/2$ the
”breathing” mode (V.22) is non-zero, while the ${\cal A}_{\oplus}$ and ${\cal
A}_{\otimes}$ polarisations vanish here and, therefore, observer on the brane
detect only three independent polarisations of the gravitational waves, as was
found earlier in Ref. Andriot2017 .

Averaging the squares of polarisation amplitudes in Eq. (V.13) over the period
of orbital motion

$\left\langle\sin^{2}\left(2\omega_{\rm
s}\bar{t}+\frac{\pi}{4}-2\phi\right)\right\rangle=\left\langle\cos^{2}\left(2\omega_{\rm
s}\bar{t}+\frac{\pi}{4}-2\phi\right)\right\rangle=\frac{1}{2},$ (V.25)

we come to the angular distribution of the gravitational radiation power of
the binary system in circular motion

$\frac{dW_{4+1}^{\rm circ}}{d\Omega_{3}}=\frac{\kappa_{5}\mu^{2}R_{\rm
s}^{4}\omega_{\rm
s}^{7}}{4\pi^{3}}\left[\left(\frac{1+\cos^{2}\theta}{2}\right)^{2}+\cos^{2}\theta+\sin^{4}\theta\frac{(1+2\cos^{2}\zeta)^{2}}{12}+\sin^{2}\theta\cos^{2}\zeta(1+\cos^{2}\theta)\right].$
(V.26)

Integrating the Eq. (V.26), we obtain the total gravitational radiation power
of the non-relativistic binary system on the circular orbit

$W_{4+1}^{\rm circ}=\frac{5}{9\pi}\kappa_{5}\mu^{2}R_{\rm s}^{4}\omega_{\rm
s}^{7}.$ (V.27)

###  V.4 Evolution of the orbit

As we emphasized from the very beginning, this calculation should be
considered as a mathematically consistent but physically unrealistic model, so
we refrain from making any quantitative physical predictions. In particular,
the evolution of the orbit, which we consider in this subsection as a logical
extension of our model, cannot be considered as an attempt to extract
experimentally available predictions and gives a result that clearly
contradicts the observational data of binary pulsars. However, we present this
calculation simply as an example of a toy model showing how different
frequency dependences give different predictions for the lifetime of a binary
system.

Consider the orbital evolution of the system due to gravitational radiation.
As in the non-relativistic limit the gravitational-wave energy loss over the
period of system’s motion is small compared to its total energy, we assume
that the system moves on a quasi-circular orbit. Indeed, by analogy with the
four-dimensional theory maggiore2008 , we consider the orbital radius $R_{\rm
s}(t)$ and the frequency of orbital motion $\omega_{\rm s}(t)$ as the slowly
varying functions of time satisfying condition

$-\dot{R}_{\rm s}\ll\omega_{\rm s}R_{\rm s},$ (V.28)

corresponding to the slow shrinking of orbit. The orbital evolution is
governed by the energy conservation law

$\frac{dE_{\rm tot}}{dt}=-W_{4+1}^{\rm circ},$ (V.29)

where $E_{\rm tot}$ is the total mechanical energy of the non-relativistic
binary system, and gravitational radiation power is given by the Eq. (V.27).

We rewrite the mechanical energy of the system and its radiation power in
terms of the frequency of orbital motion by use of its relation to the orbital
radius. Substituting into the Eq. (II.31) relative coordinate of the system
(V.14), up to the leading order contributions, we arrive at

$\omega_{\rm s}^{2}=\frac{g_{1}g_{2}}{\mu R_{\rm s}^{3}}.$ (V.30)

Using the Eq. (V.30) we rewrite the energy of the system and its gravitational
radiation power as

$\displaystyle E_{\rm
tot}=\frac{\mu\mathbf{v}^{2}}{2}-\frac{g_{1}g_{2}}{|\mathbf{z}|}=-\frac{1}{2}(\sqrt{\mu}g_{1}g_{2})^{2/3}\omega_{\rm
s}^{2/3},$ (V.31) $\displaystyle W_{4+1}^{\rm
circ}=\frac{5}{9\pi}\kappa_{5}(\sqrt{\mu}g_{1}g_{2})^{4/3}\omega_{\rm
s}^{13/3}.$ (V.32)

Therefore, substituting the Eqs. (V.31) and (V.32) into the energy
conservation law (V.29), we come to the equation for the evolution of the
orbital frequency

$\displaystyle\dot{\omega}_{\rm
s}=\frac{5\kappa_{5}}{3\pi}(\sqrt{\mu}g_{1}g_{2})^{2/3}\omega_{\rm s}^{14/3},$
(V.33) $\displaystyle\dot{\omega}_{\rm s}\ll\omega_{\rm s}^{2},$ (V.34)

where Eq. (V.34) is the condition for the quasi-circular motion (V.28) in
terms of the orbital frequency.

Solution to the Eq. (V.33) is found as

$\omega_{\rm
s}(t)=\left(\frac{9\pi}{55\kappa_{5}}\right)^{3/11}(\sqrt{\mu}g_{1}g_{2})^{-2/11}s^{-3/11},$
(V.35)

where $s=t_{\rm coal}-t$, and $t_{\rm coal}$ is the moment of coalescence of
the binary system. From the Eq. (V.30) we find the equation for the orbital
radius

$\dot{R}_{\rm s}=-\frac{2}{3}R_{\rm s}\frac{\dot{\omega}_{\rm s}}{\omega_{\rm
s}},$ (V.36)

and from the Eq. (V.35) we determine the evolution of the orbital radius as

$R_{\rm s}(t)=R_{0}\left(\frac{t_{\rm coal}-t}{t_{\rm
coal}-t_{0}}\right)^{2/11},$ (V.37)

where $R_{0}$ is the orbital radius at some moment of time $t_{0}$. Note that
the Eqs. (V.35) and (V.37) differ significantly from that in four dimensions
maggiore2008

$\omega_{\rm s,\,4D}(t)\sim(t_{\rm coal}-t)^{-3/8},\quad R_{\rm
s,\,4D}(t)\sim(t_{\rm coal}-t)^{1/4},$ (V.38)

as could be expected, given that in our work gravity is described by the five-
dimensional GR with the infinite-volume extra dimension being
phenomenologically non-viable.

##  VI Conclusion

In this article, we have considered (partially inspired by the DGP model) a
mixed environment with five-dimensional gravity and four-dimensional matter
consisting of point particles and a scalar field. The mechanism of dynamic
confinement was not imposed, and the localization of particle motion in the
brane was assumed to be purely kinematic due to the appropriate choice of
initial conditions. The physical viability of this setup as a cosmological
model is ruled out, but mathematically it is a consistent model that opens the
way for calculating bulk gravitational radiation in the framework of
linearized gravity. This model reflects a typical feature of radiation in odd
space-time dimensions, which is the appearance of tails due to violation of
the Huygens principle.

We have introduced techniques that can be useful in other problems of
radiation in odd-dimensional spacetime, namely, the extraction of the
radiative part of the retarded potential using a modification of the Rohrlich-
Teitelboim approach. It has also been shown that the contribution of field
stresses through which two point masses interact to gravitational radiation
can be explicitly taken into account using the DIRE approach to post-Newtonian
expansions in four dimensions. As a result, it turned out that in odd
dimensions the radiated part of the gravitational field depends on the entire
history of the system’s motion preceding the retarded time, in contrast to
even dimensions, where it depends only on the state of the system at the
retarded moment of proper time. Another interesting feature of odd dimensions
is that the retarded gravitational field is given by the sum of separately
divergent integrals. However, these divergences cancel each other out, and the
resulting retarded field and its radiated part are finite.

We considered the gravitational radiation from the binary system of two non-
relativistic point particles deriving the five-dimensional analog of the
quadrupole formula for gravitational radiation power (V.10). We found that in
contrast with the ultrarelativistic synchrotron radiation in odd dimensions
Galtsov:2020hhn , the dependence of the gravitational radiation of a
nonrelativistic source on its motion history is not localized near the
retarded time.

Another feature concerns polarization. In five dimensions free gravitational
waves have five polarizations. If motion is restricted to four-dimensional
branes, an observer living on the brane will see three of them – two standard
”cross” and ”plus” polarisations and an additional ”breathing” mode. This
feature was also provided from other Andriot2017  considerations and may be of
interest for more realistic models

Now let’s briefly discuss the relationship of our results with other work. Our
quadrupole formula agrees with the formula obtained by Chu Chu:2021uea  using
the Fourier transforms of the retarded Green’s functions. In addition, our
results fit into the general schemes for the quadrupole formula and the
radiation power of the system in a circular orbit obtained by Cardoso et al.
Cardoso:2002pa  for the arbitrary even dimensions. We also expect that the
resulting formula for the radiation power of the system at the circular orbit
can be verified using the results obtained by Cardoso et al. Cardoso:2008gn
in effective field theory approach framework.

## Acknowledgements

The work of M. Kh. was supported by the “BASIS” Foundation Grant No.
20-2-10-8-1. This research was also supported by the Russian Foundation for
Basic Research on the project 20-52-18012Bulg-a, and the Scientific and
Educational School of Moscow State University “Fundamental and Applied Space
Research”.

##  Appendix A Polarizations in an arbitrary observation direction

The following derivation is analogous to that in maggiore2008 . We calculate
the polarizations of some symmetric rank-2 tensor ${\cal A}_{ij}^{\rm TT}$
with respect to the appropriately chosen frame. Then, we rewrite its
components used in the polarizations in terms of the components defined with
respect to the arbitrarily oriented frame.

We begin with the tensor ${\cal A}_{ij}^{\prime}$ in the $x^{\prime}$-frame,
where the observation direction vector $\mathbf{n}^{\prime}$ is aligned with
the $x^{\prime 4}$-coordinate

${n}^{\prime i}=\\{0,0,0,1\\}.$ (A.1)

Using the Eq. (II.19) we find the ${\cal A}_{ij}^{\prime}$ tensor in the
transverse-traceless gauge as

${{\cal A}^{\prime}}_{ij}^{\rm TT}=\begin{pmatrix}{\cal
A}_{+}-\displaystyle\frac{1}{2}{\cal A}_{\displaystyle\circ}&{\cal
A}_{\times}&{\cal A}_{\oplus}&0\\\ {\cal A}_{\times}&-{\cal
A}_{+}-\displaystyle\frac{1}{2}{\cal A}_{\displaystyle\circ}&{\cal
A}_{\otimes}&0\\\ {\cal A}_{\oplus}&{\cal A}_{\otimes}&{\cal
A}_{\displaystyle\circ}&0\\\ 0&0&0&0\end{pmatrix},$ (A.2)

where the polarizations are defined by analogy with the Eqs. (II.24) and
(II.25)

$\displaystyle{\cal A}_{+}=\frac{1}{2}\left({\cal A}_{11}^{\prime}-{\cal
A}_{22}^{\prime}\right),\quad{\cal A}_{\times}={\cal
A}_{12}^{\prime},\quad{\cal A}_{\displaystyle\circ}=\frac{2}{3}{\cal
A}_{33}^{\prime}-\frac{1}{3}\left({\cal A}_{11}^{\prime}+{\cal
A}_{22}^{\prime}\right),$ (A.3) $\displaystyle{\cal A}_{\oplus}={\cal
A}_{13}^{\prime},\quad{\cal A}_{\otimes}={\cal A}_{23}^{\prime}.$ (A.4)

Now we rewrite the components of ${\cal A}_{ij}^{\prime}$ in terms of the
components of ${\cal A}_{ij}$ defined with respect to the arbitrary oriented
$x$-frame, where the observation direction vector $\mathbf{n}$ has the form

${n}^{i}=\\{\cos\phi\sin\theta\sin\zeta,\sin\phi\sin\theta\sin\zeta,\cos\theta\sin\zeta,\cos\zeta\\},$
(A.5)

and we use the hyperspherical coordinates Landim:2009arx . It is related to
$\mathbf{n}^{\prime}$ by the SO(4)-transformation

$n^{i}=R^{i}_{\,j}n^{\prime\,j},$ (A.6)

where the transformation matrix $R^{i}_{\,j}$ is written as

$R^{i}_{\,j}=\begin{pmatrix}\sin\phi&\cos\phi\cos\theta&\cos\phi\sin\theta\cos\zeta&\cos\phi\sin\theta\sin\zeta\\\
-\cos\phi&\sin\phi\cos\theta&\sin\phi\sin\theta\cos\zeta&\sin\phi\sin\theta\sin\zeta\\\
0&-\sin\theta&\cos\theta\cos\zeta&\cos\theta\sin\zeta\\\
0&0&-\sin\zeta&\cos\zeta\end{pmatrix}.$ (A.7)

Analogously, components of ${\cal A}_{ij}^{\prime}$ are written in terms of
the components of ${\cal A}_{ij}$ as

${\cal A}_{mn}^{\prime}=\left(R^{\rm T}{\cal A}R\right)_{mn},$ (A.8)

where $(PQ)_{ij}=P_{ik}Q_{kj}$ is the matrix product and $R^{\rm T}$ is the
transposed transformation matrix.

As a result, after some algebra we obtain the polarizations for the arbitrary
observation direction $\mathbf{n}$. First, the standard ”plus” and ”cross”
polarizations take the form

$\displaystyle{\cal A}_{+}$ $\displaystyle=\frac{1}{2}\left[{\cal
A}_{11}(\sin^{2}\phi-\cos^{2}\phi\cos^{2}\theta)+{\cal
A}_{22}(\cos^{2}\phi-\sin^{2}\phi\cos^{2}\theta)-A_{33}\sin^{2}\theta-\right.$
$\displaystyle-\left.{\cal A}_{12}(1+\cos^{2}\theta)\sin{2\phi}+({\cal
A}_{13}\cos\phi+{\cal A}_{23}\sin\phi)\sin{2\theta}\right],$ (A.9)
$\displaystyle{\cal A}_{\times}$ $\displaystyle=\frac{1}{2}({\cal
A}_{11}-{\cal A}_{22})\sin{2\phi}\cos\theta-{\cal
A}_{12}\cos{2\phi}\cos\theta-({\cal A}_{13}\sin\phi-{\cal
A}_{23}\cos\phi)\sin\theta.$ (A.10)

Note that the obtained polarisations (A.9) and (A.10) are that of the four-
dimensional theory maggiore2008 . ”Breathing” mode is found as

${\cal A}_{\displaystyle\circ}=\frac{1}{3}\left\\{{\cal
A}_{11}[(2\cos^{2}\zeta+1)\cos^{2}\phi\sin^{2}\theta-1]+{\cal
A}_{22}[(2\cos^{2}\zeta+1)\sin^{2}\phi\sin^{2}\theta-1]+{\cal
A}_{33}\right.\times\\\ \times[(2\cos^{2}\zeta+1)\cos^{2}\theta-1]+2{\cal
A}_{44}\sin^{2}\zeta+[{\cal A}_{12}\sin{2\phi}\sin^{2}\theta+({\cal
A}_{13}\cos\phi+{\cal A}_{23}\sin\phi)\times\\\
\times\left.\sin{2\theta}](2\cos^{2}\zeta+1)-2[({\cal A}_{14}\cos\phi+{\cal
A}_{24}\sin\phi)\sin\theta+{\cal A}_{34}\cos\theta]\sin{2\zeta}\right\\}.$
(A.11)

Note that it is non-vanishing even when the observation point and the source
are on the brane $\zeta=\pi/2,\,{\cal A}_{i4}=0$. Remaining polarizations
${\cal A}_{\oplus}$ and ${\cal A}_{\otimes}$ take the form

$\displaystyle{\cal A}_{\oplus}$ $\displaystyle=\frac{1}{2}\left({\cal
A}_{11}-{\cal A}_{22}\right)\sin{2\phi}\sin\theta\cos\zeta-{\cal
A}_{12}\cos{2\phi}\sin\theta\cos\zeta+\left({\cal A}_{13}\sin\phi-{\cal
A}_{23}\cos\phi\right)\times$
$\displaystyle\times\cos\theta\cos\zeta-\left({\cal A}_{14}\sin\phi-{\cal
A}_{24}\cos\phi\right)\sin\zeta,$ (A.12) $\displaystyle{\cal A}_{\otimes}$
$\displaystyle=\frac{1}{2}\left\\{\left({\cal A}_{11}\cos^{2}\phi+{\cal
A}_{22}\sin^{2}\phi-{\cal A}_{33}\right)\sin{2\theta}\cos\zeta+{\cal
A}_{12}\sin{2\phi}\sin{2\theta}\cos\zeta+2\left({\cal
A}_{13}\cos\phi+\right.\right.$ $\displaystyle+\left.\left.{\cal
A}_{23}\sin\phi\right)\cos{2\theta}\cos\zeta-2\left({\cal
A}_{14}\cos\phi+{\cal A}_{24}\sin\phi\right)\cos\theta\sin\zeta+{\cal
A}_{34}\sin\theta\sin\zeta\right\\}.$ (A.13)

In contrast with the ”breathing” mode, they vanish when the observation point
and the source are on the brane.

##  Appendix B Regularisation of the point particles term

Consider the first two terms in Eq. (III.36) and split it into three integrals

$\displaystyle
I_{ij}=\int_{-\infty}^{\bar{t}}dt^{\prime}\left[\frac{15}{4}v_{i}v_{j}\frac{\mathbf{n}\mathbf{s}(t^{\prime})-\mathbf{n}\bar{\mathbf{v}}}{(\bar{t}-t^{\prime})^{5/2}}+\frac{3}{2}\frac{v_{i}v_{j}}{(\bar{t}-t^{\prime})^{5/2}}\right]=I_{ij}^{1}+I_{ij}^{2}+I_{ij}^{3},$
(B.1) $\displaystyle
I_{ij}^{1}=\frac{15}{4}\int_{-\infty}^{\bar{t}}dt^{\prime}\,v_{i}v_{j}\frac{\mathbf{n}\mathbf{s}(t^{\prime})}{(\bar{t}-t^{\prime})^{5/2}},\quad
I_{ij}^{2}=-\frac{15}{4}\int_{-\infty}^{\bar{t}}dt^{\prime}\,v_{i}v_{j}\frac{\mathbf{n}\bar{\mathbf{v}}}{(\bar{t}-t^{\prime})^{5/2}},$
(B.2) $\displaystyle
I_{ij}^{3}=\frac{3}{2}\int_{-\infty}^{\bar{t}}dt^{\prime}\,\frac{v_{i}v_{j}}{(\bar{t}-t^{\prime})^{5/2}}.$
(B.3)

By analogy with the regularisation of scalar field (III.23), we introduce the
regularising parameter $\epsilon\to+0$ into the upper integration limit
$\bar{t}\to\bar{t}-\epsilon$ and integrate Eqs. (B.1–B.3) by parts reducing
the powers on denominators to $1/2$.

Using the Eq. (III.18) and integrating the $I_{ij}^{1}$ three times by parts
we extract its divergent contributions coming from the upper integration limit

$I_{ij}^{1}=\frac{5}{2}\lim_{\epsilon\to+0}\bar{v}_{i}\bar{v}_{j}\frac{\mathbf{n}\bar{\mathbf{v}}}{\epsilon^{3/2}}-2\lim_{\epsilon\to+0}\bar{v}_{i}\bar{v}_{j}\frac{\mathbf{n}\bar{\mathbf{a}}}{\epsilon^{1/2}}-10\lim_{\epsilon\to+0}\bar{a}_{(i}\bar{v}_{j)}\frac{\mathbf{n}\bar{\mathbf{v}}}{\epsilon^{1/2}}+2\int_{-\infty}^{\bar{t}}dt^{\prime}\,v_{i}v_{j}\frac{\mathbf{n}\dot{\mathbf{a}}}{(\bar{t}-t^{\prime})^{1/2}}+\\\
+12\int_{-\infty}^{\bar{t}}dt^{\prime}\,a_{(i}v_{j)}\frac{\mathbf{n}\mathbf{a}}{(\bar{t}-t^{\prime})^{1/2}}+6\int_{-\infty}^{\bar{t}}dt^{\prime}\left(\dot{a}_{i}v_{j}+2a_{i}a_{j}+v_{i}\dot{a}_{j}\right)\frac{\mathbf{n}\mathbf{v}}{(\bar{t}-t^{\prime})^{1/2}}-\\\
-2\int_{-\infty}^{\bar{t}}dt^{\prime}\left(\ddot{a}_{i}v_{j}+3\dot{a}_{i}a_{j}+3a_{i}\dot{a}_{j}+v_{i}\ddot{a}_{j}\right)\frac{\mathbf{n}\bar{\mathbf{z}}-\mathbf{n}\mathbf{z}}{(\bar{t}-t^{\prime})^{1/2}},$
(B.4)

where we have taken into account that
$\mathbf{s}(t)\to\bar{\mathbf{v}},\,t\to\bar{t}$. After analogous
transformations, the $I_{ij}^{2}$ integral is found as

$I_{ij}^{2}=-\frac{5}{2}\lim_{\epsilon\to+0}\bar{v}_{i}\bar{v}_{j}\frac{\mathbf{n}\bar{\mathbf{v}}}{\epsilon^{3/2}}+10\lim_{\epsilon\to+0}\bar{a}_{(i}\bar{v}_{j)}\frac{\mathbf{n}\bar{\mathbf{v}}}{\epsilon^{1/2}}-5\int_{-\infty}^{\bar{t}}dt^{\prime}\left(\dot{a}_{i}v_{j}+2a_{i}a_{j}+v_{i}\dot{a}_{j}\right)\frac{\mathbf{n}\bar{\mathbf{v}}}{(\bar{t}-t^{\prime})^{1/2}}.$
(B.5)

The first two terms of Eq. (B.5) cancel out the two divergent terms in the Eq.
(B.4). The last integral $I_{ij}^{3}$ is transformed to the following form

$I_{ij}^{3}=\lim_{\epsilon\to+0}\frac{\bar{v}_{i}\bar{v}_{j}}{\epsilon^{3/2}}-4\lim_{\epsilon\to+0}\frac{\bar{a}_{(i}\bar{v}_{j)}}{\epsilon^{1/2}}+2\int_{-\infty}^{\bar{t}}dt^{\prime}\frac{\dot{a}_{i}v_{j}+2a_{i}a_{j}+v_{i}\dot{a}_{j}}{(\bar{t}-t^{\prime})^{1/2}}.$
(B.6)

The remaining divergent terms in Eqs. (B.5) and (B.6) are cancelled out by the
”counter-terms” contained in the Eq. (III.36)

$\displaystyle-\frac{3}{2}\lim_{\epsilon\to+0}\int_{-\infty}^{\bar{t}-\epsilon}dt^{\prime}\frac{\bar{v}_{i}\bar{v}_{j}}{(\bar{t}-t^{\prime})^{5/2}}=-\lim_{\epsilon\to+0}\frac{\bar{v}_{i}\bar{v}_{j}}{\epsilon^{3/2}},$
(B.7)
$\displaystyle\lim_{\epsilon\to+0}\int_{-\infty}^{\bar{t}-\epsilon}dt^{\prime}\frac{2\bar{a}_{(i}\bar{v}_{j)}+\bar{v}_{i}\bar{v}_{j}\mathbf{n}\bar{\mathbf{a}}}{(\bar{t}-t^{\prime})^{3/2}}=2\lim_{\epsilon\to+0}\frac{2\bar{a}_{(i}\bar{v}_{j)}+\bar{v}_{i}\bar{v}_{j}\mathbf{n}\bar{\mathbf{a}}}{\epsilon^{1/2}},$
(B.8)

leaving us with the convergent integral (III.37) for the emitted part of
gravitational derivative.

##  Appendix C Spatial integral in scalar field contribution

Spatial integral arising in the Eqs. (IV.23) and (IV.31) has the form

$\int_{{\cal
R}}d^{3}x^{\prime}\,\varphi_{1}(x^{\prime})\partial_{i}^{\,\prime}\partial_{j}^{\,\prime}\varphi_{2}(x^{\prime}),$
(C.1)

where the subscript denotes the integration over the interior of the two-
dimensional sphere of radius $\cal R$ with center at the origin.

Given the non-relativistic motion of particles and neglecting the retardation
of the fields in the near zone, the scalar fields are given by the Eq.
(II.29). Rewriting the particles’ world lines in terms of the relative
coordinate (III.46) we obtain

$\int_{\cal
R}d^{3}x^{\prime}\frac{g_{1}}{|\vec{x}^{\prime}+s_{2}\vec{z}(t^{\prime})|}\partial_{i}^{\,\prime}\partial_{j}^{\,\prime}\frac{g_{2}}{|\vec{x}^{\prime}-s_{1}\vec{z}(t^{\prime})|},$
(C.2)

where $s_{1}=m_{1}/M$, and $s_{2}=m_{2}/M$. We shift the origin as

$\vec{y}=\vec{x}^{\prime}-s_{1}\vec{z},$ (C.3)

to get under derivatives the inverse distance from the origin. We, also,
change the integration region to the interior of the sphere centered at
$\vec{y}=0$. New integration region deviates from the original one just in the
regions $|\vec{y}|\sim{\cal R}$, where the energy-momentum density of the
scalar field is vanishingly small. Therefore, we neglect these deviations. As
a result, we arrive at the spatial integral in form

$3g_{1}g_{2}\delta_{i\mathbf{i}}\delta_{j\mathbf{j}}\int
dy\,d\Omega_{2}\frac{\hat{n}^{\mathbf{i}}\hat{n}^{\mathbf{j}}}{y|\vec{y}+\vec{z}|},\quad
y=|\vec{y}|,\quad\hat{n}^{\mathbf{i}}=\frac{y^{\mathbf{i}}}{y},$ (C.4)

where the bold indices denote the coordinates on the brane
$\mathbf{i},\mathbf{j}=\overline{1,3}$, we use the spherical coordinates, and
$\Omega_{2}$ is an angular element on the sphere.

To compute the integral over the sphere in Eq. (C.4) we expand the integrand
into the spherical harmonics (see, e.g., Pati:2000vt ; Will:1996zj ;
Thorne:1980ru ) as

$\frac{1}{|\vec{y}+\vec{z}|}=\sum_{l,m}\frac{4\pi(-1)^{l}}{2l+1}\frac{r_{<}^{l}}{r_{>}^{l+1}}Y_{lm}^{*}(n_{z}^{\mathbf{i}})Y_{lm}(\hat{n}^{\mathbf{i}}),$
(C.5)

where $r_{>(<)}$ is the greater (lesser) of $y$ and $z=|\vec{z}|$. Then, the
integral in Eq. (C.4) splits into two terms corresponding to the integration
over the interior of the sphere of radius $z$ and over the spherical shell
$y\in(z,{\cal R})$

$\int
dy\,d\Omega_{2}\frac{\hat{n}^{\mathbf{i}}\hat{n}^{\mathbf{j}}}{y|\vec{y}+\vec{z}|}=4\pi\sum_{l,m}\int_{0}^{z}dy\,\frac{y^{l-1}}{z^{l+1}}\int
d\Omega_{2}\,\frac{(-1)^{l}}{2l+1}\left(\hat{n}^{\mathbf{i}}\hat{n}^{\mathbf{j}}-\frac{1}{3}\delta^{\mathbf{i}\mathbf{j}}\right)Y_{lm}^{*}(n_{z}^{\mathbf{i}})Y_{lm}(\hat{n}^{\mathbf{i}})+\\\
+4\pi\sum_{l,m}\int_{z}^{\cal R}dy\,\frac{z^{l}}{y^{l+2}}\int
d\Omega_{2}\,\frac{(-1)^{l}}{2l+1}\left(\hat{n}^{\mathbf{i}}\hat{n}^{\mathbf{j}}-\frac{1}{3}\delta^{\mathbf{i}\mathbf{j}}\right)Y_{lm}^{*}(n_{z}^{\mathbf{i}})Y_{lm}(\hat{n}^{\mathbf{i}}),$
(C.6)

where we added the Kronecker deltas to the products of two unit vectors, given
that
$\delta_{i\mathbf{i}}\delta_{j\mathbf{j}}\delta^{\mathbf{i}\mathbf{j}}=\delta_{ij}$
and that in the transverse-traceless gauge these terms vanish.

The angular integral in the first term of Eq. (C.6) is computed by use of the
formulas of Ref. Pati:2000vt ; Thorne:1980ru ; Will:1996zj

$\sum_{m}\int
d\Omega_{2}\left(\hat{n}^{\mathbf{i}}\hat{n}^{\mathbf{j}}-\frac{1}{3}\delta^{\mathbf{i}\mathbf{j}}\right)Y_{lm}^{*}(n_{z}^{\mathbf{i}})Y_{lm}(\hat{n}^{\mathbf{i}})=\left(\hat{n}^{\mathbf{i}}_{z}\hat{n}^{\mathbf{j}}_{z}-\frac{1}{3}\delta^{\mathbf{i}\mathbf{j}}\right)\delta_{l2}.$
(C.7)

The remaining radial integral in the first term of Eq. (C.6) is found as

$\frac{4\pi}{5}\left(\hat{n}^{\mathbf{i}}_{z}\hat{n}^{\mathbf{j}}_{z}-\frac{1}{3}\delta^{\mathbf{i}\mathbf{j}}\right)\int_{0}^{z}dy\,\frac{y}{z^{3}}=\frac{2\pi}{5}\frac{z^{\mathbf{i}}z^{\mathbf{j}}}{z^{3}},$
(C.8)

where we omitted the Kronecker delta term in the right-hand side. Using the
Eq. (C.7), we compute the second term in Eq. (C.6) arriving at

$-\frac{4\pi}{15}\frac{z^{\mathbf{i}}z^{\mathbf{j}}}{{\cal
R}^{3}}+\frac{4\pi}{15}\frac{z^{\mathbf{i}}z^{\mathbf{j}}}{z^{3}}.$ (C.9)

Omitting, as discussed above, the term proportional to the inverse near zone
radius and combining the Eqs. (C.8) and (C.9) we obtain the spatial integral
(C.1) as

$\int_{{\cal
R}}d^{3}x^{\prime}\,\varphi_{1}(x^{\prime})\partial_{i}^{\,\prime}\partial_{j}^{\,\prime}\varphi_{2}(x^{\prime})=2\pi
g_{1}g_{2}\frac{z^{i}z^{j}}{z^{3}},$ (C.10)

Finally, using the particles equation of motion we rewrite it as

$\int_{{\cal
R}}d^{3}x^{\prime}\,\varphi_{1}(x^{\prime})\partial_{i}^{\,\prime}\partial_{j}^{\,\prime}\varphi_{2}(x^{\prime})=-\pi\mu[a^{i}z^{j}+z^{i}a^{j}].$
(C.11)

##  Appendix D Radiation of circular binary system

Here, we calculate only the ${\cal A}_{11}(x)$ component, and the remaining
ones are computed in analogous way. Substituting into the Eq. (V.15) the
derivative of the second mass moment (V.16) we arrive at

${\cal A}_{11}(x)=8\mu R_{\rm s}^{2}\omega_{\rm
s}^{4}\int_{-\infty}^{\bar{t}}dt^{\prime}\frac{\cos{2\omega_{\rm
s}t}}{(\bar{t}-t^{\prime})^{1/2}}.$ (D.1)

Introducing new integration variable $s=\bar{t}-t^{\prime}$ and expanding the
cosine of difference, we obtain

${\cal A}_{11}(x)=8\mu R_{\rm s}^{2}\omega_{\rm
s}^{4}\int_{0}^{+\infty}\frac{ds}{\sqrt{s}}\left(\cos{2\omega_{\rm
s}\bar{t}}\cos{2\omega_{\rm s}s}+\sin{2\omega_{\rm s}\bar{t}}\sin{2\omega_{\rm
s}s}\right).$ (D.2)

Two remaining integrals are just the Fresnel integrals zwillinger2014table

$\int_{0}^{+\infty}dx\frac{\cos(ax)}{\sqrt{x}}=\int_{0}^{+\infty}dx\frac{\sin(ax)}{\sqrt{x}}=\sqrt{\frac{\pi}{2a}},\,a>0.$
(D.3)

Therefore, using the formula for the sum of sine and cosine we find

${\cal A}_{11}(x)=\sqrt{32\pi}\mu R_{\rm s}^{2}\omega_{\rm
s}^{7/2}\sin\left(2\omega_{\rm s}\bar{t}+\frac{\pi}{4}\right).$ (D.4)

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