# Influence of gravitational waves on circular moving particles

Manfried Faber  Atominstitut, Technical University, Vienna, Austria  Martin
Suda  Atominstitut, Technical University, Vienna, Austria  AIT Austrian
Institute of Technology GmbH, Digital Safety & Security, Security &
Communication Technologies, Giefinggasse 4, 1210 Vienna, Austria
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate the influence of a gravitational wave background on particles
in circular motion. We are especially interested in waves leading to
stationary orbits. This consideration is limited to circular orbits
perpendicular to the incidence direction. As a main result of our calculation
we obtain in addition to the well-known alteration of the radial distance a
time dependent correction term for the phase modifying the circular motion of
the particle. A background of gravitational waves creates some kind of
uncertainty.

gravitational waves, circular orbits, linearized Einstein equations

###### pacs:

42.50.-p

##  I Introduction

One hundred years after Albert Einstein Einstein:1916vd  predicted the
existence of gravitational waves and after decades of searches, LIGO has
succeeded to detect gravitational waves 2016PhRvLb . Obviously, they
originated from a merger of two stellar-mass black holes. With sizes of the
interferometer arms of 4 km LIGO is most sensitive in the frequency band
100–300 Hz. This success of LIGO could support the idea of the existence of
gravitational waves in very different frequency ranges. In the Big Bang and in
the later history of the universe a background of gravitational waves
Allen:1997ad  could have been produced, still today influencing elementary
particles at the atomic scale, as one could speculate. The influence of such
waves on particles on circular orbits seem not to have been treated yet.
Especially interesting would be an influence on electrons in hydrogen. But
apart from this special example the formalism could be valid quite generally
for particles moving in a central force field.

The most important equation of physics of gravitational waves (GW) has been
derived from general relativity (GR) by A. Einstein himself Einstein:1916vd .
It is about radiation power or luminosity of GW and contains the quadrupole
tensor. The components of this tensor has been calculated for standard
problems by Misner, Thorne and Wheeler Misner:1974qy . The word-famous pulsar
(PSR B1913-16) was the first one to be detected by Taylor and Hulse
Hulse:1974eb ; Taylor:1979zz  where a GW-analysis has been executed. The two
neutron stars are approaching each other exactly according to the prediction
of GR caused by emission of GW. Important properties of this binary star
system can be found in Weisberg:2010zz ; Weisberg:2016jye ; Detweiler:1979wn .
The problem of spin precession has been developed further as well
Kramer:1998id . The binary pulsar system PSR J0737-3039A has been investigated
Kramer:2006nb ; Burgay:2003jj  confirming anew GR. Post-Newtonian
approximative numerical procedures are used in order to execute computerized
simulation of colliding black holes Hannam:2007ik ; Santamaria:2010yb ;
Baker:2005vv ; Bruegmann:2003aw ; Campanelli:2005dd . Ultimatively, in
September 14th 2015 the first GW (GW150914) from a merger of two black holes
have been observed from LIGO 2016PhRvLa . The direction where these GW have
arrived could be approximately estimated Connaughton:2016umz . A second signal
of GW (GW151226) has been observed too 2016PhRvLb . In the meanwhile a
numerical model of binary system of 2 stars between 40 and 100 solar masses
has been discussed Belczynski:2016obo . A binary system of a 3 solar mass
black hole with a X-ray star has been identified Altamirano:2011aa  as well as
spinning black holes James:2015yla .

Trajectory and precession of spinning particles in external gravitational and
electromagnetic fields have been treated analytically in Khriplovich:1997ni ;
Deriglazov:2015bqa ; Deriglazov:2015wde ; Deriglazov:2017jub .

Tiny ripples in spacetime curvature propagate as waves with the speed of light
and lead to periodic oscillations in the distance of test particles, as nicely
expounded in many books and review articles, e.g. in Carroll ; Cheng2005 ;
schroeder2007 ; sharan2009 ; Braccini2016 ; Rebhan10 . An overview of
detection of gravitational waves can be read in Roberto . Therein it is as
well described how gravitational waves arise from general relativity. In
Pereira  a foundational review about gravitational waves is given and a
critical review of the standard linear approach of the theory is depicted.

The aim of this paper is to investigate particles moving in a central force
field under the influence of gravitational waves. To achieve this goal we have
to modify the equation of geodesic deviation including additional force terms.
We are especially interested in uncertainties of the orbits and conditions for
stationary orbits.

The paper is organized as follows: In chapter II we repeat the results
obtained for distances of test particles in the presence of plane
gravitational waves. Afterwards, in chapter III, we discuss an extension of
the problem by considering the influence of such gravitational waves on
circular moving particles. In chapter IV results of the previous chapter are
graphically displayed and discussed. Finally in chapter V conclusions are
drawn.

##  II  Motion of test particles

In order to obtain a coordinate independent measure of the wave’s influence,
the relative motion of two nearby particles can be considered, see e.g. in
Carroll . It can be described by the geodesic equation in four-dimensional
space-time $x^{\mu}=(ct,x,y,z)$. The four-velocity is given by
$U^{\mu}=\frac{dx^{\mu}}{d\tau}$ and the distance vector $S^{\mu}$ is a
solution of the differential equation

$\displaystyle\frac{D^{2}}{d\tau^{2}}S^{\mu}=\gamma^{2}\partial_{0}^{2}S^{\mu}=S^{\sigma}U^{\nu}U^{\rho}{R^{\mu}}_{\nu\rho\sigma}$
(1)

with the Minkowski metric $\eta^{\mu\lambda}=\mathrm{diag}(-1,1,1,1)$ in the
flat background. The double differential $\frac{D^{2}}{d\tau^{2}}$ points to
the parallel transport in general relativity and the differential of the
proper time is given by $d\tau=\frac{1}{\gamma}dt$ with
$\gamma=\frac{1}{\sqrt{1-\beta^{2}}}$ and $\beta=\frac{v}{c}$.

In curved space-time the linearized Riemann-tensor $R_{\mu\nu\rho\sigma}$ is
given by

$\displaystyle\begin{aligned}
R_{\mu\nu\rho\sigma}=\frac{1}{2}(&\partial_{\rho}\partial_{\nu}h_{\mu\sigma}+\partial_{\sigma}\partial_{\mu}h_{\nu\rho}\\\
&-\partial_{\sigma}\partial_{\nu}h_{\mu\rho}-\partial_{\rho}\partial_{\mu}h_{\nu\sigma}).\end{aligned}$
(2)

Here the metric $g_{\mu\nu}$ has been approximated by
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with $|h_{\mu\nu}|\ll{1}$.

For a gravitational wave $h_{\mu\nu}$, propating in $z-$direction with the
velocity of light $c$, one obtains harmonic oscillations, which neglecting
additional phases may be written as (the real part of) plane waves Carroll ;
Cheng2005 ; schroeder2007 ; sharan2009 ; Braccini2016 ; Rebhan10

$\displaystyle h_{\mu\nu}=C_{\mu\nu}\cos(kz-\omega_{g}t)$ (3)

where $\omega_{g}=ck$ is the frequency of the gravitational wave ($k$ is the
wave number). The constant quantities $C_{\mu\nu}\ll 1$ form a symmetric
$(0,2)$ tensor with $C_{0\nu}=C_{3\nu}=0$ and $C_{22}=-C_{11}$ as well as
$C_{12}=C_{21}$. They generate a time-varying quadrupole deformation. The two
parameters $C_{11}$ and $C_{12}$ are sufficient to describe any quadrupole
deformation in the xy-plane.

Because the Riemann-tensor is first order, the corrections to $U^{\nu}$ may be
ignored, and, for slowly moving particles ($\tau=x^{0}=ct$), we have
$U^{\nu}=(1,0,0,0)$, setting $c=1$. As a consequence $R_{\mu
00\sigma}=\frac{1}{2}\partial_{0}^{2}h_{\mu\sigma}$ and the geodesic equation
(1) becomes

$\displaystyle\frac{\partial^{2}}{\partial
t^{2}}S^{\mu}\overset{\footnotesize(\ref{GG})}{=}S^{\sigma}\frac{\partial^{2}}{\partial
t^{2}}\frac{{h^{\mu}}_{\sigma}}{2}.$ (4)

$C_{11}$ and $C_{12}$ differ only by a $\pi/4$-rotation in the xy-plane.
Therefore we can choose without loss of generality $C_{12}=0$ resulting in

$\displaystyle\begin{aligned}
&\partial_{0}^{2}S^{1}\overset{\footnotesize(\ref{GE})}{=}S^{1}\partial_{0}^{2}\frac{h_{11}}{2},\quad&&&\partial_{0}^{2}S^{0}\overset{\footnotesize(\ref{GE})}{=}0,\\\
&\partial_{0}^{2}S^{2}\overset{\footnotesize(\ref{GE})}{=}-S^{2}\partial_{0}^{2}\frac{h_{11}}{2},\quad&&&\partial_{0}^{2}S^{3}\overset{\footnotesize(\ref{GE})}{=}0,\end{aligned}$
(5)

with $h_{11}$ given by Eq. (3). For $C_{11}\ll 1$ the solutions of these
equations are given in refs. Carroll ; Cheng2005 ; schroeder2007 ; sharan2009
; Braccini2016 ; Rebhan10  by

$\displaystyle\begin{aligned} &S^{1}\approx
S_{c}^{1}(t):=S_{c}^{1}(0)\left[1+\frac{h_{11}}{2}\right],\\\ &S^{2}\approx
S_{c}^{2}(t):=S_{c}^{2}(0)\left[1-\frac{h_{11}}{2}\right],\\\
&S^{0}:=S^{3}:=0.\end{aligned}$ (6)

$S_{c}^{1}(t)$ and $S_{c}^{2}(t)$ are the coordinates of the separation vector
in the xy-plane. The oscillations in x and y coordinates are $180^{\circ}$ out
of phase and lead therefore to linear oscillations around the vector
$(S_{c}^{1}(0),S_{c}^{2}(0))$, as shown in Fig. 1.

Figure 1: Linear oscillations of $(S_{c}^{1}(t),S_{c}^{2}(t))$ with time $t$
for various values of $S_{c}^{1}(0)$ and $S_{c}^{2}(0)$ with
$\sqrt{[S_{c}^{1}(0)]^{2}+[S_{c}^{2}(0)]^{2}}=1$ and $C_{11}=0.2$.

To increase the visibility we have chosen an unrealistic large value for
$C_{11}$. The functions in Eq. (LABEL:C11L) fulfil Eq. (5) up to order
$C_{11}^{2}$ only111With the approximation symbol “$\approx$” we indicate
accuracy up to order $C_{11}$.

$\displaystyle\begin{aligned}
S_{c}^{1}(0)&(-\omega_{g}^{2})\frac{h_{11}}{2}\overset{\footnotesize(\ref{C11})}{\approx}S_{c}^{1}(t)(-\omega_{g}^{2})\frac{h_{11}}{2},\\\
S_{c}^{2}(0)&\omega_{g}^{2}\,\frac{h_{11}}{2}\overset{\footnotesize(\ref{C11})}{\approx}-S_{c}^{2}(t)(-\omega_{g}^{2})\frac{h_{11}}{2},\end{aligned}$
(7)

##  III Circular motion with constant angular velocity

Let us consider a light particle rotating in the xy-plane on a circle with
radius $r$ with constant angular velocity $\omega$ around a heavy particle,
like in Bohr’s model for the hydrogen atom. We assume that the radius $r$ and
the azimuthal angle $\varphi=\omega t$ of this motion are slightly disturbed
by a gravitational wave with “+”-polarisation incident in 3-direction with
$h_{\mu\nu}$ given by Eq. 3 and the symmetric tensor

$\displaystyle C_{0\nu}=C_{12}=C_{3\nu}=0,\quad C_{22}=-C_{11}.$ (8)

In a generalised equation of motion we have to take into account the
gravitational force, the rhs. of Eq. (1), and the central force keeping the
light particle on a circle. For the assumed circular motion it can be
approximated by the expression for the centripetal force

$\displaystyle\partial_{0}^{2}S^{\mu}=S^{\sigma}\frac{U^{\nu}}{\gamma}\frac{U^{\rho}}{\gamma}{R^{\mu}}_{\nu\rho\sigma}+(0,\vec{a}_{r},0)^{\mu},$
(9)

where $\vec{a}_{r}$ is the radial acceleration in the xy-plane caused by the
external centripetal force.

The geometrical term $S^{\sigma}U^{\nu}U^{\rho}{R^{\mu}}_{\nu\rho\sigma}$
representing the curved space-time due to gravitational waves simplifies
drastically due to our special assumptions (3) and (8) — wave propagating in
z-direction and particle rotating in the xy-plane. $h_{\mu\nu}$ of Eq. (3)
depends only on $z$ and $t$. In the curvature tensor $R_{\mu\nu\rho\sigma}$ of
Eq. (2) the derivatives of $h_{\mu\nu}$ are always contracted either with
$S^{\mu}$ or $U^{\mu}$. Due to $S^{3}=U^{3}=0$ finally only time derivatives
of $h_{11}$ and $h_{22}=-h_{11}$ matter. Therefore, the contributions of the
geometrical term read

$\displaystyle\begin{aligned}
S^{\sigma}&U^{\nu}U^{\rho}{R^{0}}_{\nu\rho\sigma}\overset{\footnotesize(\ref{RT})}{=}\\\
&=-\partial_{0}^{2}\frac{h_{11}}{2}[S^{0}(U^{1}U^{1}-U^{2}U^{2})-S^{1}U^{0}U^{1}+S^{2}U^{0}U^{2}],\\\
S^{\sigma}&U^{\nu}U^{\rho}{R^{1}}_{\nu\rho\sigma}\overset{\footnotesize(\ref{RT})}{=}\\\
=&\partial_{0}^{2}\frac{h_{11}}{2}[-S^{0}U^{0}U^{1}+S^{1}U^{0}U^{0}],\\\
S^{\sigma}&U^{\nu}U^{\rho}{R^{2}}_{\nu\rho\sigma}\overset{\footnotesize(\ref{RT})}{=}\\\
&=\partial_{0}^{2}\frac{h_{11}}{2}[S^{0}U^{0}U^{2}-S^{2}U^{0}U^{0}],\end{aligned}$
(10)

As expected, due to the factor

$\displaystyle\partial_{0}^{2}\frac{h_{11}}{2}=-\frac{C_{11}}{2}\,\omega_{g}^{2}\,\cos(kz-\omega_{g}t).$
(11)

all these contributions are of first order in $C_{11}$.

In the spirit of the approximate solution (LABEL:C11L) we expect that the
distance vector of pure circular motion indicated by the subscript $0$

$\displaystyle
S_{0}^{\mu}=(0,r\cos\varphi,r\sin\varphi,0),\quad\varphi(t)=\omega t,$ (12)

is modified by terms linear in $C_{11}$. The modifications of the radial
distance according to Eq. (5) lead to Coriolis forces inducing variations of
the angular velocity. Therefore, we try the ansatz

$\displaystyle\;S^{0}:=S^{0}(t),$ (13) $\displaystyle\begin{aligned}
S^{1}:=&R_{1}(t)\cos\varphi_{1}(t)=\\\
:=&r\,[\,1+\frac{h_{11}(t)}{2}\,]\cos[\varphi(t)-\eta_{1}(t)],\end{aligned}$
(14) $\displaystyle\begin{aligned} S^{2}:=&R_{2}(t)\sin\varphi_{2}(t)=\\\
:=&r\,[\,1-\frac{h_{11}(t)}{2}\,]\sin[\varphi(t)+\eta_{2}(t)],\end{aligned}$
(15) $\displaystyle S^{3}:=0,$ (16)

where $S^{0}$, $\eta_{1}$ and $\eta_{2}$ are of first order in $C_{11}$. For
the velocity vector $U^{\mu}$ we get

$\displaystyle\frac{U^{0}}{\gamma}=1+\partial_{0}S^{0}(t),$ (17)
$\displaystyle\frac{U^{1}}{\gamma}=\partial_{0}S^{1}\overset{\footnotesize(\ref{ansatz1})}{=}\partial_{0}R_{1}\cos\varphi_{1}-R_{1}\sin\varphi_{1}\,\dot{\varphi}_{1},$
(18)
$\displaystyle\frac{U^{2}}{\gamma}=\partial_{0}S^{2}\overset{\footnotesize(\ref{ansatz2})}{=}\partial_{0}R_{2}\sin\varphi_{2}+R_{2}\cos\varphi_{2}\,\dot{\varphi}_{2},$
(19) $\displaystyle
U^{3}=\partial_{0}S^{3}\overset{\footnotesize(\ref{ansatz3})}{=}0$ (20)

using dots for time-derivatives $\partial_{0}$. With Eqs. (13)-(20) we are
ready to calculate the geometrical terms (10) up to order $C_{11}$

$\displaystyle\begin{aligned}
S^{\sigma}&\frac{U^{\nu}}{\gamma}\frac{U^{\rho}}{\gamma}{R^{0}}_{\nu\rho\sigma}\overset{\footnotesize(\ref{gT})}{\approx}\\\
&\approx-\partial_{0}^{2}\frac{h_{11}}{2}[-S_{0}^{1}\partial_{0}S_{0}^{1}+S_{0}^{2}\partial_{0}S_{0}^{2}]\overset{\footnotesize(\ref{S0})}{\approx}\\\
&\approx-\partial_{0}^{2}\frac{h_{11}}{2}\,r^{2}\omega\sin(2\varphi)\overset{\footnotesize(\ref{AblWel})}{=}\\\
&\overset{\footnotesize(\ref{S0})}{=}\frac{C_{11}}{2}\,\omega_{g}^{2}\,\cos(kz-\omega_{g}t)\,r^{2}\omega\sin(2\omega
t),\end{aligned}$ (21) $\displaystyle\begin{aligned}
S^{\sigma}&\frac{U^{\nu}}{\gamma}\frac{U^{\rho}}{\gamma}{R^{1}}_{\nu\rho\sigma}\overset{\footnotesize(\ref{gT})}{\approx}\partial_{0}^{2}\frac{h_{11}}{2}S^{1}\overset{\footnotesize(\ref{ansatz1})}{=}\\\
&=\partial_{0}^{2}R_{1}\cos\varphi_{1},\end{aligned}$ (22)
$\displaystyle\begin{aligned}
S^{\sigma}&\frac{U^{\nu}}{\gamma}\frac{U^{\rho}}{\gamma}{R^{2}}_{\nu\rho\sigma}\overset{\footnotesize(\ref{gT})}{\approx}-\partial_{0}^{2}\frac{h_{11}}{2}S^{2}\overset{\footnotesize(\ref{ansatz2})}{=}\\\
&=\partial_{0}^{2}R_{2}\sin\varphi_{2},\end{aligned}$ (23)

Thus, the equation of motion (9) for the time-component reads

$\displaystyle\begin{aligned}
\partial_{0}^{2}S^{0}&\overset{\footnotesize(\ref{gT0})}{\approx}\frac{C_{11}}{4}\,r^{2}\omega\,\omega_{g}^{2}\\\
&\left\\{\sin[(2\omega-\omega_{g})t+kz]+\sin[(2\omega+\omega_{g})t-kz]\right\\}\end{aligned}$
(24)

with a solution of order $C_{11}$

$\displaystyle\begin{aligned}
S^{0}&\overset{\footnotesize(\ref{BG0})}{\approx}-\frac{C_{11}}{4}\,r^{2}\omega\,\omega_{g}^{2}\\\
&\left\\{\frac{\sin[(2\omega-\omega_{g})t+kz]}{(2\omega-\omega_{g})^{2}}+\frac{\sin[(2\omega+\omega_{g})t-kz]}{(2\omega+\omega_{g})^{2}}\right\\}.\end{aligned}$
(25)

More terms we get for the x- and y-components

$\displaystyle\begin{aligned}
\partial_{0}^{2}S^{1}&\overset{\footnotesize(\ref{ansatz1})}{=}\partial_{0}^{2}R_{1}\cos\varphi_{1}-2\partial_{0}R_{1}\sin\varphi_{1}\dot{\varphi}_{1}-\\\
&-R_{1}\cos\varphi_{1}\dot{\varphi}_{1}^{2}-R_{1}\sin\varphi_{1}\ddot{\varphi}_{1},\end{aligned}$
(26) $\displaystyle\begin{aligned}
\partial_{0}^{2}S^{2}&\overset{\footnotesize(\ref{ansatz2})}{=}\partial_{0}^{2}R_{2}\sin\varphi_{2}+2\partial_{0}R_{2}\cos\varphi_{2}\dot{\varphi}_{2}-\\\
&-R_{2}\sin\varphi_{2}\dot{\varphi}_{2}^{2}+R_{2}\cos\varphi_{2}\ddot{\varphi}_{2}.\end{aligned}$
(27)

To simplify the notation for the four terms on the rhs we use two-dimensional
vectors in the xy-plane. Further on we denote them by
$\vec{a}_{1},\,\vec{a}_{2},\,\vec{a}_{3}$ and $\vec{a}_{4}$ and use dots for
time derivatives $\partial_{0}$. We realise that the first terms $\vec{a}_{1}$
in these two equations are the geometrical terms (22) and (23). The third
terms

$\displaystyle\vec{a}_{3}:=-\begin{pmatrix}R_{1}\cos\varphi_{1}\,\dot{\varphi}_{1}^{2}\\\
R_{2}\sin\varphi_{2}\,\dot{\varphi}_{2}^{2}\end{pmatrix}\overset{{}^{(\ref{ansatz1})}}{\underset{{}^{(\ref{ansatz2})}}{=}}-\begin{pmatrix}S_{1}\,\dot{\varphi}_{1}^{2}\\\
S_{2}\,\dot{\varphi}_{2}^{2}\end{pmatrix}$ (28)

include the centrifugal acceleration $\vec{a}_{r}$ acting in radial direction
$\vec{S}=\bigl{(}\begin{smallmatrix}S^{1}\\\ S^{2}\end{smallmatrix}\bigr{)}$,
see Fig. 2,

Figure 2: Decomposition of the acceleration $\vec{a}_{3}$ in components
parallel and perpendicular to the distance vector $\vec{S}$ in the xy-plane.

$\displaystyle\vec{a}_{r}:=\frac{\vec{a}_{3}\vec{S}}{{\vec{S}}^{2}}\,\vec{S}=-\frac{(S^{1})^{2}\dot{\varphi}_{1}^{2}+(S^{2})^{2}\dot{\varphi}_{2}^{2}}{(S^{1})^{2}+(S^{2})^{2}}\,\vec{S}.$
(29)

Inserting

$\displaystyle\begin{aligned}
\dot{\varphi}_{1}^{2}\overset{\footnotesize(\ref{ansatz1})}{=}(\dot{\varphi}-\dot{\eta}_{1})^{2}\overset{\footnotesize(\ref{S0})}{\approx}\omega^{2}-2\omega\dot{\eta}_{1},\\\
\dot{\varphi}_{2}^{2}\overset{\footnotesize(\ref{ansatz1})}{=}(\dot{\varphi}+\dot{\eta}_{2})^{2}\overset{\footnotesize(\ref{S0})}{\approx}\omega^{2}+2\omega\dot{\eta}_{2},\end{aligned}$
(30)

we get

$\displaystyle\begin{aligned}
\vec{a}_{r}\overset{\footnotesize(\ref{ar})}{=}&-\omega^{2}\vec{S}\left[1-\frac{2}{\omega}\frac{\dot{\eta}_{1}(S^{1})^{2}-\dot{\eta}_{2}(S^{2})^{2}}{{\vec{S}}^{2}}\right]=\\\
\overset{{}^{(\ref{ansatz1})}}{\underset{{}^{(\ref{ansatz2})}}{=}}&-\omega^{2}\vec{S}\left[1-\frac{2}{\omega}\left(\dot{\eta}_{1}\cos^{2}\varphi-\dot{\eta}_{2}\sin^{2}\varphi\right)\right].\end{aligned}$
(31)

The remainder of $\vec{a}_{3}$

$\displaystyle\begin{aligned}
\vec{a}_{\perp}:=&\vec{a}_{3}-\vec{a}_{r}\overset{{}^{(\ref{a3})}}{\underset{{}^{(\ref{azb})}}{=}}\\\
=&2\omega\begin{pmatrix}S^{1}[\dot{\eta}_{1}-\dot{\eta}_{1}\cos^{2}\varphi+\dot{\eta}_{2}\sin^{2}\varphi]\\\
S^{2}[-\dot{\eta}_{2}-\dot{\eta}_{1}\cos^{2}\varphi+\dot{\eta}_{2}\sin^{2}\varphi]\end{pmatrix}\\\
\overset{{}^{(\ref{ansatz1})}}{\underset{{}^{(\ref{ansatz2})}}{=}}&2\omega\begin{pmatrix}S^{2}(\dot{\eta}_{1}+\dot{\eta}_{2})\sin\varphi\cos\varphi\\\
-S^{1}(\dot{\eta}_{1}+\dot{\eta}_{2})\sin\varphi\cos\varphi\end{pmatrix}\end{aligned}$
(32)

contributes to the azimuthal accelerations together with the second and forth
term in Eqs. (26) and (27). Since there are no forces leading to azimuthal
accelerations we have to set their sum to zero

$\displaystyle\underbrace{\vec{a}_{3}-\vec{a}_{r}}_{\vec{a}_{\perp}}+\vec{a}_{2}+\vec{a}_{4}\overset{\footnotesize!}{=}0.$
(33)

This is a condition for the time dependence of the angular velocity. It leads
to

$\displaystyle\omega\sin(2\varphi)(\dot{\eta}_{1}+\dot{\eta}_{2})-\omega\,\partial_{0}h_{11}+\ddot{\eta}_{1}=0,$
(34)
$\displaystyle-\omega\sin(2\varphi)(\dot{\eta}_{1}+\dot{\eta}_{2})-\omega\,\partial_{0}h_{11}+\ddot{\eta}_{2}=0.$
(35)

The sum of these two equations

$\displaystyle\ddot{\eta}_{1}+\ddot{\eta}_{2}\overset{{}^{(\ref{aphi1})}}{\underset{{}^{(\ref{aphi2})}}{=}}2\omega\,\partial_{0}h_{11}$
(36)

can be solved by

$\displaystyle\dot{\eta}_{1}+\dot{\eta}_{2}\overset{\footnotesize(\ref{dd1+2})}{=}2\omega\,h_{11}$
(37)

and by

$\displaystyle\eta_{1}+\eta_{2}\overset{\footnotesize(\ref{GW})}{=}-2\,C_{11}\frac{\omega}{\omega_{g}}\sin(kz-\omega_{g}t).$
(38)

The difference of the two equations (34) and (35)

$\displaystyle\begin{aligned}
\ddot{\eta}_{2}-\ddot{\eta}_{1}\overset{{}^{(\ref{aphi1})}}{\underset{{}^{(\ref{aphi2})}}{=}}&2\omega\sin(2\varphi)(\dot{\eta}_{1}+\dot{\eta}_{2})\overset{\footnotesize(\ref{d1+2})}{=}4\omega^{2}\sin(2\varphi)\,h_{11}\overset{\footnotesize(\ref{GW})}{=}\\\
=&\,4\,C_{11}\,\omega^{2}\sin(2\varphi)\cos(kz-\omega_{g}t)\end{aligned}$ (39)

can be solved by

$\displaystyle\begin{aligned}
&\eta_{2}-\eta_{1}\overset{\footnotesize(\ref{dd2-1})}{=}2\,C_{11}\,\omega^{2}\\\
&\left\\{\frac{\sin[(2\omega-\omega_{g})t+kz]}{(2\omega-\omega_{g})^{2}}+\frac{\sin[(2\omega+\omega_{g})t-kz]}{(2\omega+\omega_{g})^{2}}\right\\}.\end{aligned}$
(40)

We get the results

$\displaystyle\eta_{1,2}\overset{{}^{(\ref{1+2})}}{\underset{{}^{(\ref{2-1})}}{=}}-C_{11}\,\left\\{\frac{\omega}{\omega_{g}}\sin(kz-\omega_{g}t)\right.$
(41)
$\displaystyle\left.\pm\omega^{2}\left[\frac{\sin[(2\omega-\omega_{g})t+kz]}{(2\omega-\omega_{g})^{2}}+\frac{\sin[(2\omega+\omega_{g})t-kz]}{(2\omega+\omega_{g})^{2}}\right]\right\\}=$
$\displaystyle\overset{\footnotesize(\ref{Sol0})}{=}\frac{\omega}{\omega_{g}}\left\\{-C_{11}\sin(kz-\omega_{g}t)\pm\frac{4S^{0}}{\omega_{g}r^{2}}\right\\},$

where according to Eq. (25) $S^{0}$ is of order $C_{11}$.
Result: We have obtained the time-dependent correction terms
$\eta_{1,2}=\eta_{1,2}(t)$ which have to be subtracted from (added to)
$\varphi(t)=\omega t$ in Eqs. (14) and (15) when circular moving particles are
influenced by gravitational waves, Eq. (3). We have found a solution $S^{\mu}$
according to the ansatz (13)-(16) fullfilling the equation of motion (9) up to
first order in $C_{11}$.

The distance vector $\vec{S}=(S^{1},S^{2})$ from the first to the second
particle, shown in Fig. 2, can be represented by the polar coordinates

$\displaystyle\begin{aligned}
R:=\sqrt{(S^{1})^{2}+(S^{2})^{2}},\quad\phi:=\arctan\frac{S^{2}}{S^{1}}\end{aligned}$
(42)

with $S^{1}$ and $S^{2}$ given in Eqs. (14) and (15).

The gravitational wave modifies the radius $R$ of the circular motion and in
consequence the angular velocity deviates from $\omega$.

##  IV Results and discussion

With the expressions (14) and (15) for $S^{\mu}$ we are now able to display
figures of the orbits, see Figs. 3–12. Since we want to compare the positions
of the two particles at equal times we put $S_{0}=0$ in Eq. (41). To allow for
better visibility we choose the amplitudes of the quadrupole oscillations in
most diagrams unrealistically large, $C_{11}/2=0.05$. Only in Fig. 6 and Fig.
10 we choose slightly smaller values. In Figs. 3-6 the orbits close after one
revolution $T$ due to the integer frequency ratios
$\frac{\omega_{g}}{\omega}$. With $\frac{\omega_{g}}{\omega}=1.25$ the path
does not close yet after 2 revolutions in Fig. 7, but it would after 4. In
Fig. 8, 9 and 10 we choose $\frac{\omega_{g}}{\omega}=2.1$ and $1.02$
respectively and need 10 and 50 revolutions to get a closed paths. In Figs. 9
and 10 we compare with different values for the amplitude $C_{11}$. In the
diagrams 11 and 12 we modify the synchronisation of oscillation and rotational
motion by choosing an additional phase, $kz=\pi/4$, leading to a rotation of
the diagram compared to the figures 3 and 4.

Figure 3: $\frac{C_{11}}{2}=0.05$, $kz=0$, $\frac{\omega_{g}}{\omega}=1$,
$t=T$. Figure 4: $\frac{C_{11}}{2}=0.05$, $kz=0$,
$\frac{\omega_{g}}{\omega}=2$, $t=T$. Figure 5: $\frac{C_{11}}{2}=0.05$,
$kz=0$, $\frac{\omega_{g}}{\omega}=3$, $t=T$. Figure 6:
$\frac{C_{11}}{2}=0.025$, $kz=0$, $\frac{\omega_{g}}{\omega}=20.$, $t=T$.
Figure 7: $\frac{C_{11}}{2}=0.05$, $kz=0.$, $\frac{\omega_{g}}{\omega}=1.25$,
$t=2~{}T$. Figure 8: $\frac{C_{11}}{2}=0.05$, $kz=0$,
$\frac{\omega_{g}}{\omega}=2.1$, $t=5~{}T$. Figure 9: $\frac{C_{11}}{2}=0.05$,
$kz=0.$, $\frac{\omega_{g}}{\omega}=1.02$, $t=25~{}T$. Figure 10:
$\frac{C_{11}}{2}=0.025$, $kz=0.$, $\frac{\omega_{g}}{\omega}=1.02$,
$t=50~{}T$. Figure 11: $\frac{C_{11}}{2}=0.05$, $kz=\pi/4$,
$\frac{\omega_{g}}{\omega}=1$, $t=T$. Figure 12: $\frac{C_{11}}{2}=0.05$,
$kz=\pi/4$ and $\frac{\omega_{g}}{\omega}=2$, $t=T$.

These examples clearly reveal the influence of gravitational waves on
circulating particles in the space-time structure of general relativity.

##  V Conclusion

In this work we attempted to investigate the influence of gravitational waves
on particles circulating in a central force field. We especially concentrated
on periodic orbits and their perturbations.

After the introduction we recapitulated in chapter II the very well known
approximate solution for slowly moving particles subjected to gravitational
waves with frequency $\omega_{g}$. In chapter III we expanded the problem to
particles circulating with frequency $\omega$ in a central force field. To get
a precision of the same order as for slowly moving test particles in refs.
Carroll ; Cheng2005 ; schroeder2007 ; sharan2009 ; Braccini2016 ; Rebhan10  we
had to take into account the centripetal force and to modify the time
dependence of the azimuthal angle $\varphi=\omega t$ to $\phi$ of Eq. (42).
Chapter IV is devoted to the graphical representation of the results. Besides
the amplitude of the gravitational wave the ratio $\omega_{g}/\omega$ plays an
important role. For integer values we get periodic orbits. For non-integer
values of $\omega_{g}/\omega$ the paths are disturbed. For the shape of these
perturbations we have to take into consideration that incident gravitational
waves will have different polarisations and therefore different positions of
the nodes. In any case a background of gravitational waves creates a kind of
uncertainty in rotating systems.

The idea to this work is inspired by the silicon oil drop experiment of Yves
Couder and his group. This is the only experiment, we know, which could give
some idea why we can describe nature perfectly by quantum mechanics. A silicon
droplet bouncing on a vibrating fluid bath creates waves interfering with the
background field Couder2006 . The resonant interaction of particle and field
creates a wave field guiding moving droplets.

Thinking about the nature of a subquantum medium which could guide elementary
particles we observe the importance of the Compton wave length which is
related to the mass of particles. A natural type of background waves which
could feel the mass of particles are gravitational waves. Waves which are not
in resonance would lead to disturbances and to uncertainties in the position
and momentum as we have seen in the investigation presented above. To get
closer to quantum mechanics it would be necessary that the size of these
displacements is related to the Compton wavelength. In the above linear
treatment we find closed orbits for any integer ratio $\omega_{g}/\omega$.
This does not agree with Bohr’s quantisation condition. It would be
interesting to take into account the non-linear terms of Einstein’s equation
and to investigate whether we get a relation to Bohr’s quantisation condition.

## Acknowledgments

We thank Alexei A. Deriglazov and Jorge Daniel Casaleiro Lopes for valuable
comments.

## Conflict of Interests

The authors declare that there are no conflicts of interest regarding the
publication of this paper.

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