2021

[1,2]Konstantin Osetrin

These authors contributed equally to this work.

These authors contributed equally to this work.

[1]Center for Mathematical and Computer Physics, Tomsk State Pedagogical
University, Kievskaya str. 60, Tomsk, 634061, Russia

2] National Research Tomsk State University, Lenina pr. 36, Tomsk, 634050,
Russia

# Gravitational wave of the Bianchi VII universe: particle trajectories,
geodesic deviation and tidal accelerations

<EMAIL_ADDRESS>Evgeny Osetrin<EMAIL_ADDRESS>Elena Osetrina
<EMAIL_ADDRESS>*  [

###### Abstract

For the gravitational wave model based on the type III Shapovalov wave space-
time, test particle trajectories and the exact solution of geodesic deviation
equations for the Bianchi type VII universe are obtained. Based on the found
4-vector of deviation, tidal accelerations in a gravitational wave are
calculated. For the obtained solution in a privileged coordinate system, an
explicit form of transformations into a synchronous reference system is found,
which allows time synchronization at any points of space-time with separation
of time and spatial coordinates. The synchronous reference system used is
associated with a freely falling observer on the base geodesic. In a
synchronous coordinate system, an explicit form of the gravitational wave
metric, a 4-vector of geodesic deviation, and a 4-vector of tidal
accelerations in a gravitational wave are obtained. The exact solution
describes a variant of the primordial gravitational wave. The results of the
work can be used to study the plasma radiation generated by tidal
accelerations of a gravitational wave.

###### keywords:

gravitation wave, cosmology, deviation of geodesics, tidal acceleration,
Hamilton-Jacobi equation, Bianchi universes, Stäckel spaces, Shapovalov spaces

##  1 Introduction

Gravitational waves are currently both an object of academic interest and an
object of experimental observation, which provides new data about the Universe
PhysRevLett.116.061102 ; PhysRevX.9.031040 ; PhysRevX.11.021053 . In addition,
from the point of view of theoretical cosmology, the discovery of primordial
gravitational waves can provide a weighty argument in support of the
inflationary model of the initial stage of the Universe. Obtaining exactly
integrable models in this direction, as the basic elements of the theory,
makes a significant contribution to the development of gravitational-wave
research.

This paper presents new results on primordial gravitational waves in Bianchi
type VII spaces, which are spatially homogeneous but not isotropic like the
standard Friedmann–Robertson–Walker model, inspired by observational data on
the anisotropy of the electromagnetic microwave background of the Universe
Bennett2013  and is of interest for studying the initial stages of the
dynamics of the Universe.

The main manifestation of the gravitational field from the point of view of
the motion of test particles is the deviation of the geodesics along which the
particles move. The deviation of geodesics leads to the appearance of tidal
accelerations in the gravitational wave, which can cause various physical
effects. For example, the primordial gravitational wave, acting on the primary
plasma, excites plasma density waves, which leads to the appearance of
additional electromagnetic fields that contributed to the characteristics of
the background microwave radiation, which is currently detected using
satellite telescopes Bennett2013 .

The calculation in such problems is quite complicated and is carried out
mainly by approximate and numerical methods. In this regard, finding exact
solutions to the equations of deviation of geodesics in a gravitational wave
seems to be an important basic element of the theory, which makes it possible
to exact calculate tidal accelerations and the resulting physical effects.
Exact solutions in any physical theory are the cornerstones that make it
possible to understand the qualitative behavior of physical objects in more
complex cases. In addition, exact solutions make it possible to effectively
debug approximate and numerical methods for more complex physical models, as
well as to train computer ”artificial intelligence” in recognizing
gravitational waves.

As an initial model for a gravitational wave, we use Shapovalov type III wave
spaces Osetrin2020Symmetry  in this paper, which allow the existence of
privileged coordinate systems, where the metric depends on only one wave
variable, along which the space-time interval vanishes. Shapovalov’s wave
spaces make it possible to construct a complete integral of the equation of
motion of test particles in the Hamilton-Jacobi formalism and, consequently,
to find the trajectories of motion of test particles in a gravitational wave
in quadratures. The presence of the complete integral of the Hamilton-Jacobi
equation also allows one to obtain the exact solution of the geodesic
deviation equations, and then calculate the tidal accelerations for particles
in a gravitational wave.

The privileged coordinate systems used to integrate the Hamilton-Jacobi
equation are based on the three commuting Killing vectors allowed by the
Shapovalov type III wave space, which determine the so-called "complete set"
of integrals of the motion of test particles and make it possible to carry out
analytical integration of the Hamilton-Jacobi equation by the method of
complete separation of variables Shapovalov1978I ; Shapovalov1978II ;
Shapovalov1979 .

An additional advantage of Shapovalov wave spaces is the ability to construct
an explicit coordinate transformation from a privileged coordinate system,
where the exact integration of the equations of motion of test particles in
the Hamilton-Jacobi formalism is allowed, into synchronous reference systems,
where time and spatial coordinates are separated, and an observer freely
falling in a gravitational field is at rest . Synchronous reference systems
make it possible to synchronize time at different points of space-time and
clarify the physical content of the obtained exact models.

Spaces that allow the existence of privileged coordinate systems, where the
Hamilton-Jacobi equation of test particles can be integrated by the method of
complete separation of variables, are widely used in the theory of gravity,
since they allow integrating field equations and obtaining exactly integrable
models. These are models with an electromagnetic field Obukhov1988 ;
Obukhov2022632 ; Obukhov2021695 , models with dust and pure radiation
OsetrinDust2016 ; OsetrinVaidya1996 ; OsetrinVaidya2009 ; OsetrinRadiation2017
, models with a scalar field OsetrinScalar2018 ; Obukhov2022142 ;
Obukhov2021134 ; Obukhov2021183 . Obtaining realistic models of primordial
gravitational waves of the Universe on the basis of Shapovalov’s wave spaces
is also possible for gravity models taking into account quantum and other
corrections (for example, f(R) - gravity, etc. Odintsov2007 ; Odintsov2011 ;
Capozziello2011 ; Odintsov2017 ).

Shapovalov’s wave spaces make it possible to obtain exact wave solutions in
the theory of gravity not only in Einstein’s general theory of relativity, but
also in modified theories of gravity OsetrinScalar312020 ; OsetrinScalar212020
; OsetrinSymmetry2021 , which makes it possible to compare analogous solutions
and select realistic theories and models.

Shapovalov’s wave spaces allow selection of classes of spatially homogeneous
models, which are interpreted as cosmological models of the Universe and,
accordingly, describe primordial gravitational waves. So, for example, type
III Shapovalov spaces allow IV, VI and VII types of Bianchi Universes (see for
example OsetrinHomog312002 ; OsetrinHomog2006 ), type II Shapovalov spaces
allow Bianchi universes of type III OsetrinHomog212020 .

In this paper, we will consider the Shapovalov wave spaces related to Bianchi
Universes of type VII. Trajectories of test particles will be found, exact
solutions of the geodesic deviation equations in a gravitational wave will be
found, and tidal accelerations of particles in a gravitational wave will be
calculated. The results are presented both in a privileged coordinate system,
where the equations for test particle trajectories are integrated, and in a
laboratory synchronous coordinate system associated with a freely falling
observer.

##  2  Shapovalov wave spacetimes

The Hamilton-Jacobi equation for a test particle of mass $m$ in a
gravitational field with the metric tensor $g_{{\alpha}{\beta}}(x^{\gamma})$
has the form (see LandauEng1 ):

$g^{{\alpha}{\beta}}\frac{\partial S}{\partial x^{\alpha}}\frac{\partial
S}{\partial x^{\beta}}=m^{2}c^{2},$ (1)
${\alpha},{\beta},{\gamma}...=0,...,(n-1),$

where the capital letter $S$ denotes the test particle action function, $n$ is
the space dimension. We will set the speed of light $c$ to be equal to unity.
In what follows, we will also use the lowercase letter $s$ to denote a space-
time interval.

######  Definition 1.

If the space allows the existence of a privileged coordinate system
$\\{x^{\alpha}\\}$, where the Hamilton-Jacobi equation (1) can be integrated
by the complete separation of variables method, when the complete integral $S$
for the test particle action function can be written in ”separated” form:

$S=\sum_{\alpha=0}^{n-1}\phi_{\alpha}(x^{\alpha},\lambda_{0},...,\lambda_{n-1}),$
(2)
$\lambda_{0},...,\lambda_{n-1}-\mbox{const},\qquad\det\Bigl{\lvert}\frac{\partial{}^{2}S}{\partial
x^{\alpha}\partial\lambda_{\beta}}\Bigr{\rvert}\neq 0,$

moreover, one of the non-ignored variables (on which the metric depends) is a
wave (null), i.e. the space-time interval along this variable vanishes, then
such a space is called the Shapovalov wave space.

Shapovalov spaces represent gravitational-wave models of space-time, which
allow integrating the equations of motion of test particles and radiation in
the Hamilton-Jacobi formalism in quadratures, i.e. make it possible to find an
explicit form of geodesic lines along which test particles move.

Shapovalov wave spaces are a subclass of wave models for the broader class of
Stäckel spaces (Paul Stäckel, see Stackel1891thesis ; Stackel1897145 ), which
allow separation of variables in the Hamilton-Jacobi equation. The theory of
Stäckel spaces was built by the efforts of a large number of researchers and
was basically completed by the construction of a classification and a detailed
description of these spaces in the works of Vladimir Shapovalov
Shapovalov1978I ; Shapovalov1978II ; Shapovalov1979 .

Shapovalov spaces admit a ”complete set” of Killing vectors and Killing
tensors of the second rank, which correspond to sets of integrals of motion of
test particles. Shapovalov wave spaces are classified according to the Abelian
group of motions they admit, which specifies the number of ignored variables
(on which the metric in the privileged coordinate system does not depend) and,
in the case of space-time, can admit from one to three commuting Killing
vectors in a ”complete set” and, accordingly, for the case of space-time there
are types I, II and III of Shapovalov wave spaces.

Shapovalov’s wave spaces allow explicit construction of the complete integral
of the Hamilton-Jacobi equation for test particles and make it possible to
obtain an analytical form of their motion trajectories in the Hamilton-Jacobi
formalism. The test particle coordinates $x^{\alpha}$ are functions of proper
time $\tau$ on the base geodesic line along which the test particle moves, and
is given in the Hamilton-Jacobi formalism by the particle trajectory equations
in the form:

$\frac{\partial
S(x^{\alpha},\lambda_{k})}{\partial\lambda_{j}}=\sigma_{j},\quad\tau=S(x^{\alpha},\lambda_{k})\rvert_{m=1},$
(3) ${i},{j},{k}=1,2,3\mathchar 24635\relax\;$

where $\lambda_{k}$, $\sigma_{k}$ are independent constant parameters
determined by the initial or boundary data for the motion of a test particle
along the base geodesic line, the variable $\tau$ is the proper time of the
particle.

Shapovalov type III wave space-time metric in a privileged coordinate system
depends on one wave variable and can be reduced to the following form (see
f.e. OsetrinHomog312002 ; OsetrinHomog2006 ; OsetrinHomog312020 ):

${ds}^{2}=2dx^{0}dx^{1}+g_{ab}(x^{0})\Bigl{(}dx^{a}+g^{a}(x^{0})\,dx^{1}\Bigr{)}$
$\Bigl{(}dx^{b}+g^{b}(x^{0})\,dx^{1}\Bigr{)},$ (4)

where indices a, b run through the values 2, 3. Thus, in the general case, the
metric includes five arbitrary functions of the wave (null) variable $x^{0}$.

Einstein’s equations with cosmological constant $\Lambda$ in vacuum

$R_{\alpha\beta}=\Lambda g_{\alpha\beta}$ (5)

for the metric (4) lead to the following necessary conditions:

$\Lambda=g^{a}=0.$ (6)

The space-time metric for a gravitational wave (4) has type N according to
Petrov’s classification.

##  3  Deviation of geodesics in Shapovalov wave spaces

The deviation of geodesics in the general theory of relativity and in modified
theories of gravity is the main manifestation of the gravitational field and
gravitational waves in particular. Geodesic deviation is used to detect the
gravitational field and provides information about the space-time curvature
tensor, and also allows you to calculate various effects of gravity on
physical objects, incl. tidal effects of the gravitational field.

The geodesic deviation equation has the following form:

$\frac{D^{2}\eta^{\alpha}}{{d\tau}^{2}}=R^{\alpha}{}_{{\beta}{\gamma}{\delta}}u^{\beta}u^{\gamma}\eta^{\delta}.$
(7)

Here $u^{\alpha}(\tau)$ is the four-velocity of the test particle on the base
geodesic line, $D$ is the covariant derivative, $\eta^{\alpha}(\tau)$ is the
geodesic deviation vector, $R^{\alpha}{}_{{\beta}{\gamma}{\delta}}$ is the
Riemann curvature tensor. This equation is considered along the base geodesic
line of a test particle with four-velocity $u^{\alpha}$ and, therefore, the
coordinates $x^{\alpha}$ are parametrized by one parameter (the proper time of
the test particle $\tau$).

For the four-velocity test particle, the normalization condition is satisfied

$g^{{\alpha}{\beta}}u_{\alpha}u_{\beta}=1.$ (8)

For four-velocity, assuming the mass of the test particle to be equal to
unity, we obtain the expression:

$u_{\alpha}=\frac{\partial S}{\partial x^{\alpha}}\,\biggl{\rvert}_{m=1},$ (9)

where $S$ is the test particle action function.

Thus, first finding the form of the metric from the field equations for the
corresponding gravitational model, then obtaining the trajectories of the test
particles from the Hamilton-Jacobi equations (1) and calculating the four-
velocities of the test particles from the relation (9), one can write the
deviation equations (7) explicitly and integrate the resulting system of
differential equations in a direct way.

On the other hand, if we are able to find the complete integral of the
Hamilton-Jacobi equation for a test particle $S(x^{\alpha},\lambda_{a})$, then
the deviation vector of geodesics $\eta^{\alpha}$ can be found also according
to the method proposed in the work Bazanski19891018 . Using the variational
method, a generalization of the Hamilton-Jacobi equations for the complete
integral of geodesics was found, which made it possible to obtain equations
for determining the deviation of geodesics through the complete integral of
the Hamilton-Jacobi equation. According to this method, the deviation vector
$\eta^{\alpha}$ can be found as a solution of a linear algebraic system of
equations on the trajectory of the ”base” test particle

$\eta^{\alpha}\,\frac{\partial
u_{\alpha}(x^{\beta},\lambda_{j})}{\partial\lambda_{i}}+\rho_{k}\frac{\partial^{2}S(x^{\beta},\lambda_{j})}{\partial\lambda_{i}\partial\lambda_{k}}=\vartheta_{i},$
(10) $u_{\alpha}(x^{\beta},\lambda_{i})\,\eta^{\alpha}=0,$ (11)
${i},{j},{k}=1\ldots 3\mathchar
24635\relax\;\qquad{\alpha},{\beta},{\gamma}=0\ldots 3,$

where $\lambda_{i}$, $\rho_{i}$, $\vartheta_{i}$ are independent constant
parameters. Here the constants $\lambda_{i}$ are determined by the initial
data or boundary conditions for the velocity (momentum) of the test particle
on the base geodesic line. The constants $\rho_{i}$ are determined by the
initial data for the deviation vector (that is, they are related to the
initial data adjacent geodetic line). The values ot the constants
$\vartheta_{i}$ are related to the definition of the origin of the variables
$x^{\alpha}$. The equation (11) expresses the condition that the deviation
vector $\eta^{\alpha}$ is orthogonal to the base geodesic line.The dependence
of the variables $x^{\alpha}$ on the proper time $\tau$ on the base geodesic
line along which the test particle moves is established in the Hamilton-Jacobi
formalism by the equations (3) for the particle trajectory.

Thus, the Shapovalov wave spaces make it possible to obtain exact solutions
for gravitational waves both of the equations of motion of test particles and
to find exact solutions of the geodesic deviation equations in a gravitational
wave.

##  4 Spatially homogeneous models of Shapovalov spaces

Earlier, in the study of spatially homogeneous symmetries of Shapovalov spaces
of type III it was shown that three types of spatially homogeneous models can
be built on the basis of these spaces: types IV, VI and VII according to
Bianchi (see f.e. OsetrinHomog312002 ; OsetrinHomog2006 ; OsetrinHomog312020
). Moreover, gravitational waves in Shapovalov spaces of type IV and VI types
of Bianchi are aperiodic in the wave variable.

Bianchi type VII Shapovalov wave space metric in a privileged coordinate
system, where the metric depends only on the wave (null) variable $x^{0}$
(along the variable $x^{0}$ the space-time interval vanishes), can be
represented as OsetrinHomog2006 :

${ds}^{2}=2dx^{0}dx^{1}$
$\mbox{}-\frac{{x^{0}}^{2\omega}}{\gamma\left({\delta}^{2}-1\right)}\,\biggl{[}\left(1-{\delta}\cos\left(\theta-2\log{x^{0}}\right)\right)\,\left({dx^{2}}\right)^{2}$
$\mbox{}-2\sin\left(\theta-2\log{x^{0}}\right)\,{dx^{2}}{dx^{3}}$
$\mbox{}+\left(1+{\delta}\cos\left(\theta-2\log{x^{0}}\right)\right)\,\left({dx^{3}}\right)^{2}\,\biggr{]},$
(12) ${g=\det
g_{ij}=}\frac{-{x^{0}}^{4\omega}}{\gamma^{2}\left({1-\delta}^{2}\right)},\qquad{\delta}^{2}<1,$
(13)

where $x^{0}$ is the wave variable, the constants ${\gamma}$, ${\delta}$,
$\theta$ and ${\omega}$ are independent parameters of the gravitational wave
model. The parameter ${\omega}$ is the parameter of the Bianchi type VII
spatial homogeneity group.

The space-time model under consideration can be interpreted as a model of a
propagating primordial gravitational wave against the background of an
expanding Universe.

The space with the metric (12) admits a covariantly constant vector $K$ and,
therefore, is a plane-wave space:

$\nabla_{\beta}K_{\alpha}=0\quad\to\quad
K_{\alpha}=\bigl{(}K_{0},0,0,0\bigl{)},$ (14)

where $K_{0}$ is a constant.

A space with a metric (12) admits a spatial homogeneity group with Killing
vectors $X_{(1)}$, $X_{(2)}$, and $X_{(3)}$, which can be chosen in the
privileged coordinate system in the form

$X^{\alpha}_{(1)}=\bigl{(}0,0,1,0\bigr{)},\qquad
X^{\alpha}_{(2)}=\bigl{(}0,0,0,1\bigr{)},$ (15)
$X^{\alpha}_{(3)}=\bigl{(}-x^{0},\,x^{1},\,\omega\,x^{2}-x^{3},\,x^{2}+\omega\,x^{3}\bigr{)}.$
(16)

The additional fourth Killing vector associated with the choice of the
privileged coordinate system commutes with the vectors $X_{(1)}$ and $X_{(2)}$
and has the form in this coordinate system

$X^{\alpha}_{(0)}=\bigl{(}0,1,0,0\bigr{)}.$ (17)

The Killing vector $X_{(0)}$ is a null vector, because
$g_{{\alpha}{\beta}}X^{\alpha}_{(0)}X^{\beta}_{(0)}=0$.

The commutation relations for the Killing vectors have the form:

$\left[X_{(0)},X_{(1)}\right]=0,\qquad\left[X_{(0)},X_{(2)}\right]=0,$ (18)
$\left[X_{(0)},X_{(3)}\right]=X_{(0)},$ (19) $\left[X_{(1)},X_{(2)}\right]=0,$
(20) $\left[X_{(1)},X_{(3)}\right]={\omega}X_{(1)}+X_{(2)},$ (21)
$\left[X_{(2)},X_{(3)}\right]=-X_{(1)}+{\omega}X_{(2)}.$ (22)

The Killing vectors $X_{(0)}$, $X_{(1)}$, $X_{(2)}$ and $X_{(3)}$ generate a
4-dimensional motion group, the $X_{(0)}$, $X_{(1)}$, $X_{(2)}$ generate a
3-dimensional abelian subgroup, vectors $X_{(1)}$, $X_{(2)}$ and $X_{(3)}$
generate a 3-dimensional subgroup of the spatial homogeneity of the type VII
Bianchi model.

##  5 Gravitational Wave for Type VII Bianchi Models in Einstein’s Theory of
Gravity

Consider the solution of the Einstein equations in vacuum (5) for the
gravitational wave metric (12). As a result, we obtain the following
exhaustive restrictions on the parameters of the gravitational wave:

${\delta}^{2}=\frac{\omega(\omega-1)}{\omega^{2}-\omega-1},$ (23)
$0\leq\omega\leq 1,\qquad 0\leq\delta^{2}\leq 1/5.$ (24)

Thus, three independent constant parameters of the gravitational wave remain
in the model under consideration: the Bianchi type VII homogeneity subgroup
parameter $\omega$, the $\gamma$ parameter related to the wave amplitude, and
the angular parameter $\theta$ related to the wave phase.

The $(+,-,-,-)$ metric signature imposes additional restrictions on the choice
of $\gamma$ sign:

$\gamma<0.$ (25)

To shorten the formulas, we will further use the notation ${\delta}$,
understanding it as an expression of the following form:

${\delta}(\omega)=\pm\sqrt{\frac{\omega(\omega-1)}{\omega^{2}-\omega-1}}.$
(26)

The nonzero components of the Riemann curvature tensor
$R_{\alpha\beta\gamma\delta}$ for the metric (12), subject to the constraints
(23) and (24) in the privileged coordinate system, take the following form:

${R}_{0202}=-\frac{\left(\omega^{2}-\omega-1\right){x^{0}}^{2\omega-2}}{\gamma}\Bigl{[}-2(\omega-1)\omega$
$\mbox{}+2\left(\omega^{2}-\omega-1\right){\delta}\cos\left(\theta-2\log{x^{0}}\right)$
$\mbox{}+(2\omega-1){\delta}\sin\left(\theta-2\log{x^{0}}\right)\Bigr{]},$
(27)
${R}_{0302}=\frac{\left(\omega^{2}-\omega-1\right){\delta}{x^{0}}^{2\omega-2}}{\gamma}\Bigl{[}$
$(2\omega-1)\cos\left(\theta-2\log{x^{0}}\right)$
$\mbox{}+2\left(-\omega^{2}+\omega+1\right)\sin\left(\theta-2\log{x^{0}}\right)\Bigr{]},$
(28)
${R}_{0303}=\frac{\left(\omega^{2}-\omega-1\right){x^{0}}^{2\omega-2}}{\gamma}\Bigl{[}2(\omega-1)\omega$
$\mbox{}+2\left(\omega^{2}-\omega-1\right){\delta}\cos\left(\theta-2\log{x^{0}}\right)$
$\mbox{}+(2\omega-1){\delta}\sin\left(\theta-2\log{x^{0}}\right)\Bigr{]}.$
(29)

Thus, for the value of the parameter $\omega=0,1$ the parameter $\delta$ and
the Riemann curvature tensor $R_{\alpha\beta\gamma\delta}$ vanish, the model
degenerates, and space-time becomes flat.

##  6 Integration of the Hamilton-Jacobi equation for test particles (case of
$\omega\neq 1/2$)

To obtain the trajectories of motion of test particles in a gravitational wave
(12), we consider the Hamilton-Jacobi equation (1) in the given space. In
accordance with the general properties of Shapovalov spaces, we will look for
the complete integral of the Hamilton-Jacobi equation in a separated form:

$S(x^{\alpha},\lambda_{k})=\phi_{0}(x^{0})+\sum_{k=1}^{3}\lambda_{k}x^{k},$
(30)

where the independent constant parameters $\lambda_{k}$ are determined by the
initial or boundary conditions for the motion of a test particle.

Then the Hamilton-Jacobi equation (1) gives for the function $\phi_{0}(x^{0})$
(${\lambda_{1}}\neq 0$):

$\phi_{0}{}^{\prime}=\frac{1}{2{\lambda_{1}}}+\frac{\gamma{x^{0}}^{-2\omega}}{2{\lambda_{1}}}\biggl{[}-{\lambda_{2}}^{2}-{\lambda_{3}}^{2}$
$\mbox{}-2{\delta}{\lambda_{2}}{\lambda_{3}}\sin\left(\theta-2\log{x^{0}}\right)$
$\mbox{}-{\delta}\left({\lambda_{2}}^{2}-{\lambda_{3}}^{2}\right)\cos\left(\theta-2\log{x^{0}}\right)\biggr{]}.$
(31)

The integration of the equation (31) has a singularity at $\omega=1/2$.
Therefore, we will further consider the case $\omega=1/2$ separately.

In this section, we will assume that $\omega\neq 1/2$. Then the equation (31)
gives for the function $\phi_{0}(x^{0})$:

$\phi_{0}(x^{0})=\frac{x^{0}}{2{\lambda_{1}}}+\frac{{x^{0}}^{1-2\omega}\gamma({\lambda_{2}}^{2}+{\lambda_{3}}^{2})}{2{\lambda_{1}}(2\omega-1)}$
$\mbox{}+\frac{{\gamma}{\delta}{x^{0}}^{1-2\omega}}{2{\lambda_{1}}\left(4\omega^{2}-4\omega+5\right)}\Bigl{[}$
$2\left({\lambda_{2}}^{2}+{\lambda_{2}}{\lambda_{3}}(2\omega-1)-{\lambda_{3}}^{2}\right)\sin\left(\theta-2\log{x^{0}}\right)$
$\mbox{}+\Bigl{(}(2\omega-1)\left({\lambda_{2}}^{2}+{\lambda_{3}}^{2}\right)-4{\lambda_{2}}{\lambda_{3}}\Bigr{)}\cos\left(\theta-2\log{x^{0}}\right)\Bigr{]}.$
(32)

Thus, we have found the complete integral of the Hamilton-Jacobi equation of
test particles $S(x^{\alpha},\lambda_{k})$.

Now we can write and solve particle trajectory equations (3) in the Hamilton-
Jacobi formalism.

Omitting obvious calculations, we present the result of solving the equations
(3) in the form of a test particle trajectory in a privileged coordinate
system:

$x^{0}(\tau)={\lambda_{1}}{\tau},$ (33)
$x^{1}(\tau)=\frac{({\lambda_{1}}{\tau})^{1-2\omega}}{2{\lambda_{1}}^{2}(2\omega-1)}\Bigl{(}(2\omega-1)({\lambda_{1}}{\tau})^{2\omega}+\gamma({\lambda_{2}}^{2}+{\lambda_{3}}^{2})\Bigr{)}$
$\mbox{}+\frac{\gamma{\delta}({\lambda_{1}}{\tau})^{1-2\omega}}{2{\lambda_{1}}^{2}(4\omega^{2}-4\omega+5)}\Bigl{[}2\Bigl{(}{\lambda_{2}}^{2}+{\lambda_{2}}{\lambda_{3}}(2\omega-1)-{\lambda_{3}}^{2}\Bigr{)}\sin\left(\theta-2\log\left({\lambda_{1}}{\tau}\right)\right)$
$\mbox{}+\Bigl{(}{\lambda_{2}}^{2}(2\omega-1)-4{\lambda_{2}}{\lambda_{3}}+{\lambda_{3}}^{2}(1-2\omega)\Bigr{)}\cos\left(\theta-2\log\left({\lambda_{1}}{\tau}\right)\right)\Bigr{]},$
(34)
$x^{2}(\tau)=-\frac{\gamma({\lambda_{1}}{\tau})^{1-2\omega}}{{\lambda_{1}}(2\omega-1)(4\omega^{2}-4\omega+5)}\Bigl{[}{\lambda_{2}}\left(4\omega^{2}-4\omega+5\right)$
$\mbox{}+(2\omega-1){\delta}(2{\lambda_{2}}+{\lambda_{3}}(2\omega-1))\sin\left(\theta-2\log\left({\lambda_{1}}{\tau}\right)\right)$
$+(2\omega-1){\delta}({\lambda_{2}}(2\omega-1)-2{\lambda_{3}})\cos\left(\theta-2\log\left({\lambda_{1}}{\tau}\right)\right)\Bigr{]},$
(35)
$x^{3}(\tau)=\frac{\gamma({\lambda_{1}}{\tau})^{1-2\omega}}{{\lambda_{1}}(2\omega-1)(4\omega^{2}-4\omega+5)}\Bigl{[}{\lambda_{3}}\left(-4\omega^{2}+4\omega-5\right)$
$\mbox{}-(2\omega-1){\delta}({\lambda_{2}}(2\omega-1)-2{\lambda_{3}})\sin\left(\theta-2\log\left({\lambda_{1}}{\tau}\right)\right)$
$\mbox{}+(2\omega-1){\delta}(2{\lambda_{2}}+{\lambda_{3}}(2\omega-1))\cos\left(\theta-2\log\left({\lambda_{1}}{\tau}\right)\right)\Bigr{]},$
(36)

where $\tau$ is the proper time of the test particle. In the process of
integrating the equations (3), we set the constants $\sigma_{k}$ equal to zero
by choosing the origin of the variables $x^{\alpha}$ and the proper time
$\tau$.

Obtaining an explicit form of test particle trajectories in a gravitational
wave here demonstrates the possibilities of exact integration of wave models
in privileged coordinate systems in Shapovalov spaces.

For the solutions obtained, we find the 4-velocity of the test particle in the
privileged coordinate system:

$u^{\alpha}(\tau)=\frac{Dx^{\alpha}}{d\tau},$ (37) $u^{0}={\lambda_{1}},\qquad
x^{0}={\lambda_{1}}\tau,$ (38)
$u^{1}=\frac{{x^{0}}^{-2\omega}}{2{\lambda_{1}}}\Bigl{[}{x^{0}}^{2\omega}-\gamma{\delta}\left({\lambda_{2}}^{2}-{\lambda_{3}}^{2}\right)\cos\left(\theta-2\log{x^{0}}\right)-2{\lambda_{2}}{\lambda_{3}}\gamma{\delta}\sin\left(\theta-2\log{x^{0}}\right)-\gamma({\lambda_{2}}^{2}+{\lambda_{3}}^{2})\Bigr{]},$
(39)
$u^{2}=\gamma{x^{0}}^{-2\omega}\left[{\lambda_{2}}{\delta}\cos\left(\theta-2\log{x^{0}}\right)+{\lambda_{3}}{\delta}\sin\left(\theta-2\log{x^{0}}\right)+{\lambda_{2}}\right],$
(40)
$u^{3}=\gamma{x^{0}}^{-2\omega}\left[{\lambda_{2}}{\delta}\sin\left(\theta-2\log{x^{0}}\right)-{\lambda_{3}}{\delta}\cos\left(\theta-2\log{x^{0}}\right)+{\lambda_{3}}\right],$
(41)

Having obtained complete the integral of the Hamilton-Jacobi equation
$S(x^{\alpha},\lambda_{k})$ and the 4-velocity of test particles, we can now
write and solve equations for the components of the geodesic deviation vector
(10)-(11).

Omitting the obvious calculations for solving the system of algebraic
equations (10)-(11), we present the exact solution for the geodesic deviation
vector $\eta^{\alpha}(\tau)$ in the privileged coordinate system ($\omega\neq
1/2$):

$\eta^{0}(\tau)={\rho_{1}}\tau-{\lambda_{1}}{\Omega},\qquad
x^{0}(\tau)=\lambda_{1}\tau,$ (42)
$\eta^{1}(\tau)={\vartheta_{1}}-\frac{\gamma{R_{2}}{x^{0}}^{1-2\omega}}{{\lambda_{1}}^{3}(2\omega-1)\left(4\omega^{2}-4\omega+5\right)}\Bigl{[}{\lambda_{2}}\left(4\omega^{2}-4\omega+5\right)$
$+(2\omega-1){\delta}(2{\lambda_{2}}+{\lambda_{3}}(2\omega-1))\sin\left(\theta-2\log{x^{0}}\right)$
$+(2\omega-1){\delta}({\lambda_{2}}(2\omega-1)-2{\lambda_{3}})\cos\left(\theta-2\log{x^{0}}\right)\Bigr{]}$
$+\frac{\gamma{R_{3}}{x^{0}}^{1-2\omega}}{{\lambda_{1}}^{3}(2\omega-1)\left(4\omega^{2}-4\omega+5\right)}\Bigl{[}{\lambda_{3}}\left(-4\omega^{2}+4\omega-5\right)$
$-(2\omega-1){\delta}({\lambda_{2}}(2\omega-1)-2{\lambda_{3}})\sin\left(\theta-2\log{x^{0}}\right)$
$+(2\omega-1){\delta}(2{\lambda_{2}}+{\lambda_{3}}(2\omega-1))\cos\left(\theta-2\log{x^{0}}\right)\Bigr{]}$
$+\frac{{x^{0}}^{-2\omega}}{2{\lambda_{1}}^{3}}\Bigl{[}\gamma{\delta}\left({\lambda_{2}}^{2}-{\lambda_{3}}^{2}\right)\left({\lambda_{1}}^{2}{\Omega}-{\rho_{1}}{x^{0}}\right)\cos\left(\theta-2\log{x^{0}}\right)$
$+2{\lambda_{2}}{\lambda_{3}}\gamma{\delta}\left({\lambda_{1}}^{2}{\Omega}-{\rho_{1}}{x^{0}}\right)\sin\left(\theta-2\log{x^{0}}\right)-{\rho_{1}}{x^{0}}^{2\omega+1}$
$\mbox{}-{\lambda_{1}}^{2}{\Omega}{x^{0}}^{2\omega}+({\lambda_{2}}^{2}+{\lambda_{3}}^{2})\gamma({\lambda_{1}}^{2}{\Omega}-{\rho_{1}}{x^{0}})\Bigr{]},$
(43)
$\eta^{2}(\tau)={\vartheta_{2}}-\frac{\gamma{x^{0}}^{-2\omega}\left({\lambda_{1}}^{2}{\Omega}-{\rho_{1}}{x^{0}}\right)}{{\lambda_{1}}^{2}}\Bigl{[}{\lambda_{2}}+{\lambda_{2}}{\delta}\cos\left(\theta-2\log{x^{0}}\right)$
$+{\lambda_{3}}{\delta}\sin\left(\theta-2\log{x^{0}}\right)\Bigr{]}$
$+\frac{\gamma{R_{2}}{x^{0}}^{1-2\omega}}{{\lambda_{1}}^{2}(2\omega-1)\left(4\omega^{2}-4\omega+5\right)}\Bigl{[}4\omega^{2}-4\omega+5$
$+2(2\omega-1){\delta}\sin\left(\theta-2\log{x^{0}}\right)+(1-2\omega)^{2}{\delta}\cos\left(\theta-2\log{x^{0}}\right)\Bigr{]}$
$-\frac{\gamma{R_{3}}{\delta}{x^{0}}^{1-2\omega}\left[(1-2\omega)\sin\left(\theta-2\log{x^{0}}\right)+2\cos\left(\theta-2\log{x^{0}}\right)\right]}{{\lambda_{1}}^{2}\left(4\omega^{2}-4\omega+5\right)},$
(44)
$\eta^{3}(\tau)={\vartheta_{3}}+\frac{\gamma{x^{0}}^{-2\omega}\left({\lambda_{1}}^{2}{\Omega}-{\rho_{1}}{x^{0}}\right)}{{\lambda_{1}}^{2}}\Bigl{[}-{\lambda_{2}}{\delta}\sin\left(\theta-2\log{x^{0}}\right)$
$+{\lambda_{3}}{\delta}\cos\left(\theta-2\log{x^{0}}\right)-{\lambda_{3}}\Bigr{]}$
$-\frac{\gamma{R_{2}}{\delta}{x^{0}}^{1-2\omega}\left[(1-2\omega)\sin\left(\theta-2\log{x^{0}}\right)+2\cos\left(\theta-2\log{x^{0}}\right)\right]}{{\lambda_{1}}^{2}\left(4\omega^{2}-4\omega+5\right)}$
$-\frac{\gamma{R_{3}}{x^{0}}^{1-2\omega}}{{\lambda_{1}}^{2}(2\omega-1)\left(4\omega^{2}-4\omega+5\right)}\Bigl{[}-4\omega^{2}+4\omega-5$
$+2(2\omega-1){\delta}\sin\left(\theta-2\log{x^{0}}\right)+(1-2\omega)^{2}{\delta}\cos\left(\theta-2\log{x^{0}}\right)\Bigr{]},$
(45)

where, for brevity, the auxiliary notation $R_{2}$, $R_{3}$ and $\Omega$ is
introduced:

$R_{2}={\lambda_{2}}{\rho_{1}}-{\lambda_{1}}{\rho_{2}},\qquad
R_{3}={\lambda_{3}}{\rho_{1}}-{\lambda_{1}}{\rho_{3}},$ (46)
$\Omega={\lambda_{1}}{\vartheta_{1}}+{\lambda_{2}}{\vartheta_{2}}+{\lambda_{3}}{\vartheta_{3}}.$
(47)

The exact solution obtained above for the deviation vector
$\eta^{\alpha}(\tau)$ includes a number of independent parameters. The
parameters $\omega$, $\gamma$ and $\theta$ determine the characteristics of
the gravitational wave. The parameters $\lambda_{k}$ define the motion of the
test particle on the base geodesic and are determined by the initial or
boundary conditions for the particle velocity on the base geodesic. The
parameters ${\rho_{k}}$ and ${\vartheta_{k}}$ are determined by the initial or
boundary conditions for the velocity and relative position of the particle on
the adjacent geodesic.

One of the important manifestations of geodesic deviation in a gravitational
field is tidal accelerations $A^{\alpha}=D^{2}\eta^{\alpha}/d^{2}\tau$, which
affect physical objects in a gravitational wave.

In a privileged coordinate system, the form of tidal accelerations in a
gravitational wave is quite cumbersome, so we will not present it here, we
only note that component $A^{0}=0$. Further, passing to the synchronous frame
of reference, we will find a compact form for tidal accelerations.

##  7 Gravitational wave in synchronous frame of reference (case of
$\omega\neq 1/2$)

Synchronous frames of reference are physically distinguished frames of
reference, since they allow one to synchronize time at different points in
space-time LandauEng1 . Also, in contrast to the privileged coordinate system,
which we used to integrate the Hamilton-Jacobi equations, in the synchronous
reference frame, time and spatial coordinates are separated, which is
important for the observer. Therefore, the possibility of passing to a
synchronous frame of reference is an advantage in the physical analysis of
solutions. Shapovalov wave spaces provide such an opportunity, since the
presence of the complete integral of the Hamilton-Jacobi equation and the
knowledge of trajectories for test particles allows one to find an explicit
analytical form of such a transition. Moreover, the proper time of an observer
freely falling along the base geodesic $\tau$ in a synchronous frame of
reference becomes a common time variable. According to the algorithm described
in LandauEng1 , the transformation from the privileged coordinate system
$x^{\alpha}$ to the synchronous frame $\tilde{x}{}^{\alpha}$ has the form:

$x^{\alpha}\to\tilde{x}{}^{\alpha}=\left(\tau,\lambda_{1},\lambda_{2},\lambda_{3}\right),$
(48) $x^{0}={\tilde{x}{}^{1}}{\tau},$ (49)
$x^{1}=\frac{\tau^{2}({\tilde{x}{}^{1}}{\tau})^{-2\omega-1}}{2(2\omega-1)(4\omega^{2}-4\omega+5)}\Bigl{[}\left(4\omega^{2}-4\omega+5\right)\Bigl{(}(2\omega-1)({\tilde{x}{}^{1}}{\tau})^{2\omega}+\gamma({\tilde{x}{}^{2}}^{2}+{\tilde{x}{}^{3}}^{2})\Bigr{)}$
$+2\gamma(2\omega-1){\delta}\Bigl{(}{\tilde{x}{}^{2}}^{2}+{\tilde{x}{}^{2}}{\tilde{x}{}^{3}}(2\omega-1)-{\tilde{x}{}^{3}}^{2}\Bigr{)}\sin\left(\theta-2\log\left({\tilde{x}{}^{1}}{\tau}\right)\right)$
$+\gamma(2\omega-1){\delta}\Bigl{(}{\tilde{x}{}^{2}}^{2}(2\omega-1)-4{\tilde{x}{}^{2}}{\tilde{x}{}^{3}}+{\tilde{x}{}^{3}}^{2}(1-2\omega)\Bigr{)}\cos\left(\theta-2\log\left({\tilde{x}{}^{1}}{\tau}\right)\right)\Bigr{]},$
(50)
$x^{2}=-\frac{\gamma{\tau}({\tilde{x}{}^{1}}{\tau})^{-2\omega}}{(2\omega-1)(4\omega^{2}-4\omega+5)}\Bigl{[}{\tilde{x}{}^{2}}\left(4\omega^{2}-4\omega+5\right)$
$+(2\omega-1){\delta}(2{\tilde{x}{}^{2}}+{\tilde{x}{}^{3}}(2\omega-1))\sin\left(\theta-2\log\left({\tilde{x}{}^{1}}{\tau}\right)\right)$
$+(2\omega-1){\delta}({\tilde{x}{}^{2}}(2\omega-1)-2{\tilde{x}{}^{3}})\cos\left(\theta-2\log\left({\tilde{x}{}^{1}}{\tau}\right)\right)\Bigr{]},$
(51)
$x^{3}=\frac{\gamma{\tau}({\tilde{x}{}^{1}}{\tau})^{-2\omega}}{(2\omega-1)(4\omega^{2}-4\omega+5)}\Bigl{[}{\tilde{x}{}^{3}}\left(-4\omega^{2}+4\omega-5\right)$
$-(2\omega-1){\delta}({\tilde{x}{}^{2}}(2\omega-1)-2{\tilde{x}{}^{3}})\sin\left(\theta-2\log\left({\tilde{x}{}^{1}}{\tau}\right)\right)$
$+(2\omega-1){\delta}(2{\tilde{x}{}^{2}}+{\tilde{x}{}^{3}}(2\omega-1))\cos\left(\theta-2\log\left({\tilde{x}{}^{1}}{\tau}\right)\right)\Bigr{]}.$
(52)

The chosen synchronous reference system is connected with the observer on the
base geodesic, which in the synchronous reference system will have constant
spatial coordinates. Four-velocity components of a freely falling observer on
a base geodesic (38)-(41) are converted to the following form:

$\tilde{u}^{\alpha}=\Bigl{\\{}1,0,0,0\Bigr{\\}}.$ (53)

Thus, the observer is at rest on the base geodesic relative to the chosen
synchronous frame of reference.

Now, by applying the transformations (49)-(52), we can write down the form of
the gravitational wave metric in the synchronous frame of reference:

$ds^{2}=d\tau^{2}-dl^{2}=d\tau^{2}+\tilde{g}{}_{{i}{j}}(\tau,{\tilde{x}{}^{k}})\,d{\tilde{x}{}^{i}}d{\tilde{x}{}^{j}},\qquad
i,j,k=1,2,3\mathchar 24635\relax\;$ (54)

where $\tau$ is the time variable (the observer’s proper time on the base
geodesic), $dl$ is the spatial distance element, ${\tilde{x}{}^{k}}$ are the
spatial coordinates.

The components of the gravitational wave metric (12) in the synchronous frame
take the following form ($\omega\neq 1/2$):

$\tilde{g}{}^{00}=1,\qquad\tilde{g}{}^{01}=\tilde{g}{}^{02}=\tilde{g}{}^{03}=0,$
(55)
$\tilde{g}{}^{1k}=-\frac{{\tilde{x}{}^{1}}{\tilde{x}{}^{k}}}{{\tau}^{2}},$
(56)
$\tilde{g}{}^{22}=\frac{(1-2\omega)^{2}\left(\omega^{2}-\omega-1\right)({\tau}{\tilde{x}{}^{1}})^{2\omega}}{5\gamma{\tau}^{2}}\biggl{[}-4\omega^{2}+4\omega-5$
$\mbox{}+(2\omega+1)(2\omega-3){\delta}\cos\Bigl{(}\theta-2\log({\tau}{\tilde{x}{}^{1}})\Bigr{)}$
$\mbox{}+4(2\omega-1){\delta}\sin\Bigl{(}\theta-2\log({\tau}{\tilde{x}{}^{1}})\Bigr{)}\biggr{]}-\frac{{\tilde{x}{}^{2}}^{2}}{{\tau}^{2}},$
(57)
$\tilde{g}{}^{23}=\frac{(1-2\omega)^{2}\left(\omega^{2}-\omega-1\right){\delta}({\tau}{\tilde{x}{}^{1}})^{2\omega}}{5\gamma{\tau}^{2}}\biggl{[}4(1-2\omega)\cos\Bigl{(}\theta-2\log({\tau}{\tilde{x}{}^{1}})\Bigr{)}$
$\mbox{}+(2\omega+1)(2\omega-3)\sin\Bigl{(}\theta-2\log({\tau}{\tilde{x}{}^{1}})\Bigr{)}\biggr{]}-\frac{{\tilde{x}{}^{2}}{\tilde{x}{}^{3}}}{{\tau}^{2}},$
(58)
$\tilde{g}{}^{33}=-\frac{(1-2\omega)^{2}\left(\omega^{2}-\omega-1\right)({\tau}{\tilde{x}{}^{1}})^{2\omega}}{5\gamma{\tau}^{2}}\biggl{[}4\omega^{2}-4\omega+5$
$\mbox{}+(2\omega+1)(2\omega-3){\delta}\cos\Bigl{(}\theta-2\log({\tau}{\tilde{x}{}^{1}})\Bigr{)}$
$\mbox{}+4(2\omega-1){\delta}\sin\Bigl{(}\theta-2\log({\tau}{\tilde{x}{}^{1}})\Bigr{)}\biggr{]}-\frac{{\tilde{x}{}^{3}}^{2}}{{\tau}^{2}},$
(59)

where $\omega$, $\gamma$ and $\theta$ are independent parameters of the
gravitational wave model, and the constant ${\delta}$ is given by the relation
(26).

Converting the components of the deviation vector $\eta{}^{\alpha}(\tau)$ from
the preferred coordinate system (42)-(45) to the synchronous reference system
gives the following result:

$\tilde{\eta}{}^{0}=0,$ (60)
$\tilde{\eta}{}^{1}(\tau)={\rho_{1}}-\frac{{\lambda_{1}}{\Omega}}{{\tau}},$
(61)
$\tilde{\eta}{}^{2}(\tau)=\frac{(1-2\omega)\left(1+\omega-\omega^{2}\right){\lambda_{1}}{\vartheta_{2}}({\lambda_{1}}{\tau})^{2\omega-1}}{5\gamma}\Bigl{[}2(1-2\omega){\delta}\sin\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}$
$\mbox{}-(1-2\omega)^{2}{\delta}\cos\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}+4\omega^{2}-4\omega+5\Bigr{]}$
$\mbox{}-\frac{(1-2\omega)^{2}\left(1+\omega-\omega^{2}\right){\delta}{\lambda_{1}}{\vartheta_{3}}({\lambda_{1}}{\tau})^{2\omega-1}}{5\gamma}\Bigl{[}2\cos\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}$
$\mbox{}+(1-2\omega)\sin\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}\Bigr{]}-\frac{{\lambda_{2}}{\Omega}}{{\tau}}+{\rho_{2}},$
(62)
$\tilde{\eta}{}^{3}(\tau)=-\frac{(1-2\omega)^{2}\left(1+\omega-\omega^{2}\right){\delta}{\lambda_{1}}{\vartheta_{2}}({\lambda_{1}}{\tau})^{2\omega-1}}{5\gamma}\Bigl{[}2\cos\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}$
$\mbox{}+(1-2\omega)\sin\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}\Bigr{]}$
$\mbox{}+\frac{(1-2\omega)\left(1+\omega-\omega^{2}\right){\lambda_{1}}{\vartheta_{3}}({\lambda_{1}}{\tau})^{2\omega-1}}{5\gamma}\Bigl{[}2(2\omega-1){\delta}\sin\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}$
$\mbox{}+(1-2\omega)^{2}{\delta}\cos\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}+4\omega^{2}-4\omega+5\Bigr{]}-\frac{{\lambda_{3}}{\Omega}}{{\tau}}+{\rho_{3}}.$
(63)

Here the parameters $\omega$, $\gamma$ and $\theta$ determine the
gravitational wave model, the constants $\lambda_{k}$, $\rho_{k}$ and
$\vartheta_{k}$ are given by the initial or boundary conditions for the
velocities and mutual positions of particles on neighboring geodesics, the
auxiliary constant $\Omega$ is determined by the relation (47), the constant
$\delta$ is determined by the relation (26).

Let us now find the form of tidal accelerations $\tilde{A}{}^{\alpha}$ of a
gravitational wave (12) in the synchronous reference system:

$\tilde{A}{}^{\alpha}=\frac{D^{2}\tilde{\eta}{}^{\alpha}}{d^{2}\tau},$ (64)
$\tilde{A}{}^{0}=\tilde{A}{}^{1}=0,$ (65)
$\tilde{A}{}^{2}=\frac{(2\omega-1)(\omega^{2}-\omega-1){\delta}{\vartheta_{2}}({\lambda_{1}}{\tau})^{2\omega}}{\gamma{\tau}^{3}}\Bigl{[}2\cos\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}$
$\mbox{}+(1-2\omega)\sin\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}\Bigr{]}$
$\mbox{}+\frac{(2\omega-1)(\omega^{2}-\omega-1){\delta}{\vartheta_{3}}({\lambda_{1}}{\tau})^{2\omega}}{\gamma{\tau}^{3}}\Bigl{[}2\sin\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}$
$\mbox{}+(2\omega-1)\cos\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}\Bigr{]}$
$\mbox{}+\frac{{R_{2}}\left(\omega^{2}-\omega-1\right){\delta}}{{\lambda_{1}}{\tau}^{2}}\Bigl{[}(1-2\omega)\sin\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}+2\cos\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}\Bigr{]}$
$\mbox{}+\frac{{R_{3}}}{{\lambda_{1}}{\tau}^{2}}\Bigl{[}2\left(\omega^{2}-\omega-1\right){\delta}\sin\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}$
$\mbox{}+(2\omega-1)(\omega^{2}-\omega-1){\delta}\cos\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}+\omega(2\omega-1)(1-\omega)\Bigr{]},$
(66)
$\tilde{A}{}^{3}=\frac{(2\omega-1)(\omega^{2}-\omega-1){\delta}{\vartheta_{2}}({\lambda_{1}}{\tau})^{2\omega}}{\gamma{\tau}^{3}}\Bigl{[}2\sin\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}$
$\mbox{}+(2\omega-1)\cos\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}\Bigr{]}$
$\mbox{}-\frac{(2\omega-1)(\omega^{2}-\omega-1){\delta}{\vartheta_{3}}({\lambda_{1}}{\tau})^{2\omega}}{\gamma{\tau}^{3}}\Bigl{[}2\cos\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}$
$\mbox{}+(1-2\omega)\sin\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}\Bigr{]}$
$\mbox{}+\frac{{R_{2}}}{{\lambda_{1}}{\tau}^{2}}\Bigl{[}\omega(2\omega-1)(\omega-1)+2\left(\omega^{2}-\omega-1\right){\delta}\sin\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}$
$\mbox{}+(2\omega-1)(\omega^{2}-\omega-1){\delta}\cos\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}\Bigr{]}$
$\mbox{}-\frac{{R_{3}}\left(\omega^{2}-\omega-1\right){\delta}}{{\lambda_{1}}{\tau}^{2}}\Bigl{[}2\cos\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}$
$\mbox{}+(1-2\omega)\sin\Bigl{(}\theta-2\log({\lambda_{1}}{\tau})\Bigr{)}\Bigr{]},$
(67)

where the parameters $\omega$, $\gamma$ and $\theta$ determine the
characteristics of the gravitational wave, the parameters $\lambda_{k}$ are
determined by the initial or boundary values of the particle velosity on the
base geodesic, the parameters ${\rho_{k}}$ and ${\vartheta_{k}}$ are
determined by the initial or boundary values of the velosity and relative
positions of the particle on the adjacent geodesic, auxiliary constants
$R_{2}$ and $R_{3}$ are defined by the relations (46).

In the synchronous reference frame used, the components of the gravitational
wave tidal acceleration are not equal to zero only in the plane of variables
${\tilde{x}{}^{2}}$, ${\tilde{x}{}^{3}}$. The gravitational wave propagates
along the space variable ${\tilde{x}{}^{1}}$.

##  8 A special case of a wave at $\omega=1/2$ in a privileged coordinate
system

When integrating the Hamilton-Jacobi equation of test particles for the
Shapovalov type III space in the Bianchi type VII cosmological model for the
metric (12), a special case arises when $\omega=1/2$. In this case, the wave
metric acquires the following form in the privileged coordinate system

${ds}^{2}=2dx^{0}dx^{1}-\frac{{x^{0}}}{\gamma^{2}-\alpha^{2}-\beta^{2}}\Bigl{[}$
$\left(\alpha\cos{\bigl{(}2\log{x^{0}}\bigr{)}}+\beta\sin{\bigl{(}2\log{x^{0}}\bigr{)}}-\gamma\right){dx^{2}}^{2}$
$+\left(\beta\cos{\bigl{(}2\log{x^{0}}\bigr{)}}-\alpha\sin{\bigl{(}2\log{x^{0}}\bigr{)}}\right){dx^{2}}{dx^{3}}$
$+\left(-\alpha\cos{\bigl{(}2\log{x^{0}}\bigr{)}}-\beta\sin{\bigl{(}2\log{x^{0}}\bigr{)}}-\gamma\right){dx^{3}}^{2}\Bigr{]},$
(68) ${g=\det g_{ij}=}-\frac{{x^{0}}^{2}}{\gamma^{2}-\alpha^{2}-\beta^{2}}.$
(69)

The Einstein vacuum equations (5) in this case lead to a condition on the
constants $\alpha$, $\beta$ and $\gamma$ of the form

$\gamma^{2}=5\left(\alpha^{2}+\beta^{2}\right),$ (70)

which allows us to introduce a two-parameter representation ($\gamma$,
$\theta$) using the constant angular parameter $\theta$ instead of the
parameters $\alpha$ and $\beta$:

$\alpha=\frac{\gamma\sin{\theta}}{\sqrt{5}},\qquad\beta=\frac{\gamma\cos{\theta}}{\sqrt{5}}.$
(71)

The metric signature $(+,-,-,-)$ leads to a negative value of the constant
$\gamma$:

$\gamma=-\sqrt{5\left(\alpha^{2}+\beta^{2}\right)}<0,$ (72)

Integrating the Hamilton-Jacobi equation of a test particle for the special
case $\omega=1/2$, we obtain the complete integral in the following form

$S=\sum_{k=1}^{3}{\lambda_{k}}{x^{k}}+\frac{1}{4{\lambda_{1}}}\biggl{[}2{x^{0}}-2\gamma\left({\lambda_{2}}^{2}+{\lambda_{3}}^{2}\right)\log{x^{0}}$
$\mbox{}+\Bigl{(}\beta\left({\lambda_{2}}^{2}-{\lambda_{3}}^{2}\right)-2\alpha{\lambda_{2}}{\lambda_{3}}\Bigr{)}\cos{\left(2\log{{x^{0}}}\right)}$
$\mbox{}+\Bigl{(}\alpha\left({\lambda_{3}}^{2}-{\lambda_{2}}^{2}\right)-2\beta{\lambda_{2}}{\lambda_{3}}\Bigr{)}\sin{\left(2\log{x^{0}}\right)}\biggr{]},$
(73)

where $\lambda_{k}$ are constant parameters, determined by the initial or
boundary values of the velocity of the test particle.

From the equations of motion in the Hamilton-Jacobi formalism, we find the
general form of the trajectories of motion of test particles in a
gravitational wave in a privileged coordinate system:

$x^{0}(\tau)={\lambda_{1}}{\tau},$ (74)
$x^{1}(\tau)=-\frac{1}{4{\lambda_{1}}^{2}}\Bigl{[}$
$\left(\alpha\left({\lambda_{2}}^{2}-{\lambda_{3}}^{2}\right)+2\beta{\lambda_{2}}{\lambda_{3}}\right)\sin{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}$
$+\left(2\alpha{\lambda_{2}}{\lambda_{3}}+\beta\left({\lambda_{3}}^{2}-{\lambda_{2}}^{2}\right)\right)\cos{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}$
$+2\gamma\left({\lambda_{2}}^{2}+{\lambda_{3}}^{2}\right)\log{({\lambda_{1}}{\tau})}-2{\lambda_{1}}{\tau}\Bigr{]},$
(75)
$x^{2}(\tau)=\frac{1}{2{\lambda_{1}}}\Bigl{[}(\alpha{\lambda_{2}}+\beta{\lambda_{3}})\sin{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}$
$+(\alpha{\lambda_{3}}-\beta{\lambda_{2}})\cos{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}+2{\lambda_{2}}\gamma\log{({\lambda_{1}}{\tau})}\Bigr{]},$
(76)
$x^{3}(\tau)=\frac{1}{2{\lambda_{1}}}\Bigl{[}(\beta{\lambda_{2}}-\alpha{\lambda_{3}})\sin{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}$
$+(\alpha{\lambda_{2}}+\beta{\lambda_{3}})\cos{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}+2{\lambda_{3}}\gamma\log{({\lambda_{1}}{\tau})}\Bigr{]},$
(77)

where $\tau$ is the proper time of the test particle.

The components of the 4-velocity vector $u^{\alpha}$ of test particles in a
gravitational wave take the form:

$u^{0}={\lambda_{1}},$ (78)
$u^{1}(\tau)=\frac{1}{2{\lambda_{1}}{x^{0}}}\Bigl{[}{x^{0}}-\gamma({\lambda_{2}}^{2}+{\lambda_{3}}^{2})$
$\mbox{}+\sin{\bigl{(}2\log{x^{0}}\bigr{)}}\left(2\alpha{\lambda_{2}}{\lambda_{3}}-\beta{\lambda_{2}}^{2}+\beta{\lambda_{3}}^{2}\right)$
$\mbox{}+\cos{\bigl{(}2\log{x^{0}}\bigr{)}}\left(\alpha\left({\lambda_{3}}^{2}-{\lambda_{2}}^{2}\right)-2\beta{\lambda_{2}}{\lambda_{3}}\right)\Bigr{]},$
(79)
$u^{2}(\tau)=\frac{1}{{x^{0}}}\Bigl{[}\sin{\bigl{(}2\log{x^{0}}\bigr{)}}(\beta{\lambda_{2}}-\alpha{\lambda_{3}})$
$\mbox{}+\cos{\bigl{(}2\log{x^{0}}\bigr{)}}(\alpha{\lambda_{2}}+\beta{\lambda_{3}})+{\lambda_{2}}\gamma\Bigr{]},$
(80)
$u^{3}(\tau)=\frac{1}{{x^{0}}}\Bigl{[}\cos{\bigl{(}2\log{x^{0}}\bigr{)}}(\beta{\lambda_{2}}-\alpha{\lambda_{3}})$
$\mbox{}-\sin{\bigl{(}2\log{x^{0}}\bigr{)}}(\alpha{\lambda_{2}}+\beta{\lambda_{3}})+{\lambda_{3}}\gamma\Bigr{]},$
(81)

where $x^{0}={\lambda_{1}}{\tau}$.

Solving the equations (10)-(11) for the deviation vector $\eta^{\alpha}(\tau)$
, we obtain its following form in the privileged coordinate system:

$\eta^{0}(\tau)={\rho_{1}}{\tau}-{\lambda_{1}}{\Omega},\qquad
x^{0}={\lambda_{1}}{\tau},$ (82)
$\eta^{1}(\tau)=\frac{{R_{2}}}{2{\lambda_{1}}^{3}}\Bigl{[}\sin{\bigl{(}2\log{x^{0}}\bigr{)}}(\alpha{\lambda_{2}}+\beta{\lambda_{3}})$
$+\cos\left(2\log{{x^{0}}}\right)(\alpha{\lambda_{3}}-\beta{\lambda_{2}})+2\gamma{\lambda_{2}}\log({x^{0}})\Bigr{]}$
$\mbox{}+\frac{{R_{3}}}{2{\lambda_{1}}^{3}}\Bigl{[}\sin{\bigl{(}2\log{x^{0}}\bigr{)}}(\beta{\lambda_{2}}-\alpha{\lambda_{3}})$
$+\cos{\bigl{(}2\log{x^{0}}\bigr{)}}(\alpha{\lambda_{2}}+\beta{\lambda_{3}})+2\gamma{\lambda_{3}}\log({x^{0}})\Bigr{]}$
$+\frac{1}{2{\lambda_{1}}^{3}{x^{0}}}\Bigl{[}\sin{\bigl{(}2\log{x^{0}}\bigr{)}}\left(-2\alpha{\lambda_{2}}{\lambda_{3}}+\beta{\lambda_{2}}^{2}-\beta{\lambda_{3}}^{2}\right)\left({\lambda_{1}}^{2}{\Omega}-{\rho_{1}}{x^{0}}\right)$
$+\cos{\bigl{(}2\log{x^{0}}\bigr{)}}\left(\alpha\left({\lambda_{2}}^{2}-{\lambda_{3}}^{2}\right)+2\beta{\lambda_{2}}{\lambda_{3}}\right)\left({\lambda_{1}}^{2}{\Omega}-{\rho_{1}}{x^{0}}\right)$
$\mbox{}-{\rho_{1}}{x^{0}}^{2}-{x^{0}}\left({\lambda_{1}}^{2}{\Omega}+\gamma{\rho_{1}}\left({\lambda_{2}}^{2}+{\lambda_{3}}^{2}\right)\right)+{\lambda_{1}}^{2}\gamma{\Omega}\left({\lambda_{2}}^{2}+{\lambda_{3}}^{2}\right)\Bigr{]}+{\vartheta_{1}},$
(83)

$\eta^{2}(\tau)=-\frac{\left({\lambda_{1}}^{2}{\Omega}-{\rho_{1}}{x^{0}}\right)}{{\lambda_{1}}^{2}{x^{0}}}\Bigl{[}\sin{\bigl{(}2\log{x^{0}}\bigr{)}}(\beta{\lambda_{2}}-\alpha{\lambda_{3}})$
$+\cos{\bigl{(}2\log{x^{0}}\bigr{)}}(\alpha{\lambda_{2}}+\beta{\lambda_{3}})+{\lambda_{2}}\gamma\Bigr{]}$
$+\frac{{R_{2}}}{2{\lambda_{1}}^{2}}\Bigl{[}-\alpha\sin{\bigl{(}2\log{x^{0}}\bigr{)}}+\beta\cos{\bigl{(}2\log{x^{0}}\bigr{)}}-2\gamma\log({x^{0}})\Bigr{]}$
$\mbox{}-\frac{{R_{3}}}{2{\lambda_{1}}^{2}}\Bigl{[}\alpha\cos{\bigl{(}2\log{x^{0}}\bigr{)}}+\beta\sin{\bigl{(}2\log{x^{0}}\bigr{)}}\Bigr{]}+{\vartheta_{2}},$
(84)

$\eta^{3}(\tau)=-\frac{\left({\lambda_{1}}^{2}{\Omega}-{\rho_{1}}{x^{0}}\right)}{{\lambda_{1}}^{2}{x^{0}}}\Bigl{[}-\sin{\bigl{(}2\log{x^{0}}\bigr{)}}(\alpha{\lambda_{2}}+\beta{\lambda_{3}})$
$+\cos{\bigl{(}2\log{x^{0}}\bigr{)}}(\beta{\lambda_{2}}-\alpha{\lambda_{3}})+{\lambda_{3}}\gamma\Bigr{]}$
$-\frac{{R_{3}}}{2{\lambda_{1}}^{2}}\Bigl{[}-\alpha\sin{\bigl{(}2\log{x^{0}}\bigr{)}}+\beta\cos{\bigl{(}2\log{x^{0}}\bigr{)}}+2\gamma\log({x^{0}})\Bigr{]}$
$\mbox{}-\frac{{R_{2}}}{2{\lambda_{1}}^{2}}\Bigl{[}\alpha\cos{\bigl{(}2\log{x^{0}}\bigr{)}}+\beta\sin{\bigl{(}2\log{x^{0}}\bigr{)}}\Bigr{]}+{\vartheta_{3}},$
(85)

where the constant parameters ${\rho_{k}}$ and ${\vartheta_{k}}$ are
determined by the initial or boundary values of the relative positions and
velocities of neighboring geodesics. Auxiliary constants ${R_{2}}$, ${R_{3}}$
and $\Omega$ are introduced for brevity and are determined by the relations
(46)-(47).

Having obtained the solution for the deviation vector, we can now write down
the form of tidal accelerations $A^{\alpha}(\tau)$ of neighboring geodesics.
Calculating the covariant derivative
$A^{\alpha}=D^{2}\eta^{\alpha}/{d\tau}^{2}$, we obtain the tidal acceleration
components in the preferred coordinate system in the form:

$A^{0}=0,\qquad x^{0}={\lambda_{1}}{\tau},$ (86)
$A^{1}(\tau)=\frac{{R_{2}}}{4{\lambda_{1}}{x^{0}}^{2}}\biggl{[}10{\lambda_{2}}\log({x^{0}})\left(\alpha\cos{\bigl{(}2\log{x^{0}}\bigr{)}}+\beta\sin{\bigl{(}2\log{x^{0}}\bigr{)}}\right)$
$-{\lambda_{3}}\left(10\alpha\log({x^{0}})\sin{\bigl{(}2\log{x^{0}}\bigr{)}}-10\beta\log({x^{0}})\cos{\bigl{(}2\log{x^{0}}\bigr{)}}+\gamma\right)\biggr{]}$
$+\frac{{R_{3}}}{4{\lambda_{1}}{x^{0}}^{2}}\biggl{[}{\lambda_{2}}\left(-10\alpha\log({x^{0}})\sin{\bigl{(}2\log{x^{0}}\bigr{)}}+10\beta\log({x^{0}})\cos{\bigl{(}2\log{x^{0}}\bigr{)}}+\gamma\right)$
$-10{\lambda_{3}}\log({x^{0}})\left(\alpha\cos{\bigl{(}2\log{x^{0}}\bigr{)}}+\beta\sin{\bigl{(}2\log{x^{0}}\bigr{)}}\right)\biggr{]}$
$-\frac{{5{\lambda_{1}}\vartheta_{2}}}{2\gamma{x^{0}}^{2}}\biggl{[}{\lambda_{2}}\left(\alpha\cos{\bigl{(}2\log{x^{0}}\bigr{)}}+\beta\sin{\bigl{(}2\log{x^{0}}\bigr{)}}\right)$
$+{\lambda_{3}}\left(\beta\cos{\bigl{(}2\log{x^{0}}\bigr{)}}-\alpha\sin{\bigl{(}2\log{x^{0}}\bigr{)}}\right)\biggr{]}$
$+\frac{{5{\lambda_{1}}\vartheta_{3}}}{2\gamma{x^{0}}^{2}}\biggl{[}{\lambda_{3}}\left(\alpha\cos{\bigl{(}2\log{x^{0}}\bigr{)}}+\beta\sin{\bigl{(}2\log{x^{0}}\bigr{)}}\right)$
$-{\lambda_{2}}\left(\beta\cos{\bigl{(}2\log{x^{0}}\bigr{)}}-\alpha\sin{\bigl{(}2\log{x^{0}}\bigr{)}}\right)\biggr{]},$
(87)
$A^{2}(\tau)=\frac{5{\lambda_{1}}^{2}{\vartheta_{2}}}{2\gamma{x^{0}}^{2}}\left(\alpha\cos{\bigl{(}2\log{x^{0}}\bigr{)}}+\beta\sin{\bigl{(}2\log{x^{0}}\bigr{)}}\right)$
$+\frac{5{\lambda_{1}}^{2}{\vartheta_{3}}}{2\gamma{x^{0}}^{2}}\left(\beta\cos{\bigl{(}2\log{x^{0}}\bigr{)}}-\alpha\sin{\bigl{(}2\log{x^{0}}\bigr{)}}\right)$
$-\frac{{R_{3}}}{4{x^{0}}^{2}}\left(-10\alpha\log({x^{0}})\sin{\bigl{(}2\log{x^{0}}\bigr{)}}+10\beta\log({x^{0}})\cos{\bigl{(}2\log{x^{0}}\bigr{)}}+\gamma\right)$
$-\frac{5{R_{2}}\log({x^{0}})}{2{x^{0}}^{2}}\left(\alpha\cos{\bigl{(}2\log{x^{0}}\bigr{)}}+\beta\sin{\bigl{(}2\log{x^{0}}\bigr{)}}\right),$
(88)
$A^{3}(\tau)=\frac{5{\lambda_{1}}^{2}{\vartheta_{2}}}{2\gamma{x^{0}}^{2}}\left(\beta\cos{\bigl{(}2\log{x^{0}}\bigr{)}}-\alpha\sin{\bigl{(}2\log{x^{0}}\bigr{)}}\right)$
$-\frac{5{\lambda_{1}}^{2}{\vartheta_{3}}}{2\gamma{x^{0}}^{2}}\left(\alpha\cos{\bigl{(}2\log{x^{0}}\bigr{)}}+\beta\sin{\bigl{(}2\log{x^{0}}\bigr{)}}\right)$
$+\frac{{R_{2}}}{4{x^{0}}^{2}}\left(10\alpha\log({x^{0}})\sin{\bigl{(}2\log{x^{0}}\bigr{)}}-10\beta\log({x^{0}})\cos{\bigl{(}2\log{x^{0}}\bigr{)}}+\gamma\right)$
$+\frac{5{R_{3}}\log({x^{0}})}{2{x^{0}}^{2}}\left(\alpha\cos{\bigl{(}2\log{x^{0}}\bigr{)}}+\beta\sin{\bigl{(}2\log{x^{0}}\bigr{)}}\right),$
(89)

where the constant parameters of the gravitational wave $\alpha$, $\beta$,
$\gamma$ are related by the relation (71), the constants $\lambda_{k}$,
$\rho_{k}$, ${\vartheta_{k}}$ are given by the initial conditions for
velocities and relative positions of particles on neighboring geodesics, the
constants ${R_{2}}$ and ${R_{3}}$ are determined by the relations (46).

The obtained exact solution for tidal acceleration in a gravitational wave can
be used to calculate various physical effects when a wave passes through a
material medium.

##  9 Special case of a wave for $\omega=1/2$ in synchronous reference system

To study the physical effects of a gravitational wave and the action of tidal
accelerations, it is convenient to switch to a synchronous reference frame,
where the observer is at rest on the base geodesic, time and spatial
coordinates are separated, and the observer’s proper time on the base geodesic
is a temporal variable, and time can be synchronized between different points
in space. time.

The transition to a similar synchronous reference frame $\tilde{x}{}^{\alpha}$
with a known complete integral of the Hamilton-Jacobi equation of test
particles is carried out according to the well-known LandauEng1  algorithm by
transformation to spatial variables, which are kept constant on the trajectory
of the basic test particle:

$x^{\alpha}\to\tilde{x}{}^{\alpha}=\left(\tau,\lambda_{1},\lambda_{2},\lambda_{3}\right),$
(90)

In our case $(\omega=1/2)$ the coordinate transformation takes the form:

$x^{0}={\tilde{x}{}^{1}}{\tau},$ (91)
$x^{1}=-\frac{1}{4{\tilde{x}{}^{1}}^{2}}\Bigl{[}$
$\left(\alpha\left({\tilde{x}{}^{2}}^{2}-{\tilde{x}{}^{3}}^{2}\right)+2\beta{\tilde{x}{}^{2}}{\tilde{x}{}^{3}}\right)\sin{\bigl{(}2\log{({\tilde{x}{}^{1}}{\tau})}\bigr{)}}$
$+\left(2\alpha{\tilde{x}{}^{2}}{\tilde{x}{}^{3}}+\beta\left({\tilde{x}{}^{3}}^{2}-{\tilde{x}{}^{2}}^{2}\right)\right)\cos{\bigl{(}2\log{({\tilde{x}{}^{1}}{\tau})}\bigr{)}}$
$+2\gamma\left({\tilde{x}{}^{2}}^{2}+{\tilde{x}{}^{3}}^{2}\right)\log{({\tilde{x}{}^{1}}{\tau})}-2{\tilde{x}{}^{1}}{\tau}\Bigr{]},$
(92)
$x^{2}=\frac{1}{2{\tilde{x}{}^{1}}}\Bigl{[}(\alpha{\tilde{x}{}^{2}}+\beta{\tilde{x}{}^{3}})\sin{\bigl{(}2\log{({\tilde{x}{}^{1}}{\tau})}\bigr{)}}$
$+(\alpha{\tilde{x}{}^{3}}-\beta{\tilde{x}{}^{2}})\cos{\bigl{(}2\log{({\tilde{x}{}^{1}}{\tau})}\bigr{)}}$
$+2{\tilde{x}{}^{2}}\gamma\log{({\tilde{x}{}^{1}}{\tau})}\Bigr{]},$ (93)
$x^{3}=\frac{1}{2{\tilde{x}{}^{1}}}\Bigl{[}(\beta{\tilde{x}{}^{2}}-\alpha{\tilde{x}{}^{3}})\sin{\bigl{(}2\log{({\tilde{x}{}^{1}}{\tau})}\bigr{)}}$
$+(\alpha{\tilde{x}{}^{2}}+\beta{\tilde{x}{}^{3}})\cos{\bigl{(}2\log{({\tilde{x}{}^{1}}{\tau})}\bigr{)}}$
$+2{\tilde{x}{}^{3}}\gamma\log{({\tilde{x}{}^{1}}{\tau})}\Bigr{]},$ (94)

Upon transition to the considered synchronous frame of reference, the
components of the gravitational wave metric will take the following form:

${ds}^{2}={d\tau}^{2}-{dl}^{2}={d\tau}^{2}+\tilde{g}{}_{ij}(\tau,\tilde{x}{}^{k})\,d\tilde{x}{}^{i}d\tilde{x}{}^{j},$
(95)

where ${dl}$ is the spatial distance element, which in the synchronous
reference frame is a function of time $\tau$ and spatial variables
$\tilde{x}{}^{k}$.

The four-velocity of the particle on the base geodesic in the chosen
synchronous frame of reference, as expected, takes the following form:

$\tilde{u}{}^{\alpha}=\Bigl{\\{}1,0,0,0\Bigr{\\}},$ (96)

those, in the selected synchronous frame of reference, the observer rests on
the base geodesic.

The metric of a gravitational wave in the Bianchi type VII space in the
considered synchronous reference frame in the case of $\omega=1/2$ will take
the following form:

$\tilde{g}{}^{00}=1,\qquad\tilde{g}{}^{01}=\tilde{g}{}^{02}=\tilde{g}{}^{03}=0,$
(97)
$\tilde{g}{}^{1k}=-\frac{{\tilde{x}{}^{1}}{\tilde{x}{}^{k}}}{{\tau}^{2}},$
(98) $\tilde{g}{}^{22}=-\frac{\left({\tilde{x}{}^{2}}\right)^{2}}{{\tau}^{2}}$
$\mbox{}+\frac{20\,{\tilde{x}{}^{1}}}{\gamma^{2}{\tau}\Bigl{(}1-20\log^{2}({\tilde{x}{}^{1}}{\tau})\Bigr{)}^{2}}\biggl{[}\gamma+20\gamma\log^{2}({\tilde{x}{}^{1}}{\tau})$
$+\sin{\bigl{(}2\log{({\tilde{x}{}^{1}}{\tau})}\bigr{)}}\left(20\beta\log^{2}({\tilde{x}{}^{1}}{\tau})-20\alpha\log{({\tilde{x}{}^{1}}{\tau})}-\beta\right)$
$+\cos{\bigl{(}2\log{({\tilde{x}{}^{1}}{\tau})}\bigr{)}}\left(20\alpha\log^{2}({\tilde{x}{}^{1}}{\tau})+20\beta\log{({\tilde{x}{}^{1}}{\tau})}-\alpha\right)\biggr{]},$
(99)
$\tilde{g}{}^{23}=-\frac{{\tilde{x}{}^{2}}{\tilde{x}{}^{3}}}{{\tau}^{2}}+\frac{20\,{\tilde{x}{}^{1}}}{\gamma^{2}{\tau}\Bigl{(}1-20\log^{2}({\tilde{x}{}^{1}}{\tau})\Bigr{)}^{2}}\biggl{[}$
$\Bigl{(}\alpha-20\log{({\tilde{x}{}^{1}}{\tau})}\left(\alpha\log{({\tilde{x}{}^{1}}{\tau})}+\beta\right)\Bigr{)}\sin{\bigl{(}2\log{({\tilde{x}{}^{1}}{\tau})}\bigr{)}}$
$+\left(20\beta\log^{2}({\tilde{x}{}^{1}}{\tau})-20\alpha\log{({\tilde{x}{}^{1}}{\tau})}-\beta\right)\cos{\bigl{(}2\log{({\tilde{x}{}^{1}}{\tau})}\bigr{)}}\biggr{]},$
(100)
$\tilde{g}{}^{33}=-\frac{\left({\tilde{x}{}^{3}}\right)^{2}}{{\tau}^{2}}$
$\mbox{}+\frac{20\,{\tilde{x}{}^{1}}}{\gamma^{2}{\tau}\Bigl{(}1-20\log^{2}({\tilde{x}{}^{1}}{\tau})\Bigr{)}^{2}}\biggl{[}\gamma\left(1+20\log^{2}({\tilde{x}{}^{1}}{\tau})\right)$
$\mbox{}+\cos{\bigl{(}2\log{({\tilde{x}{}^{1}}{\tau})}\bigr{)}}\left(\alpha-20\beta\log{({\tilde{x}{}^{1}}{\tau})}-20\alpha\log^{2}({\tilde{x}{}^{1}}{\tau})\right)$
$\mbox{}+\sin{\bigl{(}2\log{({\tilde{x}{}^{1}}{\tau})}\bigr{)}}\left(\beta+20\alpha\log{({\tilde{x}{}^{1}}{\tau})}-20\beta\log^{2}({\tilde{x}{}^{1}}{\tau})\right)\biggr{]},$
(101)

where the observer’s proper time on the base geodesic $\tau$ is the unified
time of the synchronous frame of reference, and the variables
${\tilde{x}{}^{k}}$ are the spatial coordinates.

The resulting coordinate transformation also allows calculating the components
of the geodesic deviation vector, and since the base geodesic in the used
synchronous reference system has become a time line on which the spatial
coordinates do not change, the deviation vector, up to a shift in the
reference point, becomes simply the spatial position (radius vector) of the
test particles on an adjacent geodesic.

The components of the 4-vector deviation $\tilde{\eta}{}^{\alpha}(\tau)$ in
the synchronous frame of reference in the case of $\omega=1/2$ take the form:

$\tilde{\eta}{}^{0}=0,$ (102)
$\tilde{\eta}{}^{1}(\tau)=-\frac{{\lambda_{1}}{\Omega}}{{\tau}}+{\rho_{1}},$
(103)
$\tilde{\eta}{}^{2}(\tau)=\frac{10{\lambda_{1}}{\vartheta_{2}}}{\gamma^{2}\left(20\log^{2}({\lambda_{1}}{\tau})-1\right)}\Bigl{[}-\alpha\sin{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}$
$+\beta\cos{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}+2\gamma\log({\lambda_{1}}{\tau})\Bigr{]}$
$\mbox{}-\frac{10{\lambda_{1}}{\vartheta_{3}}}{\gamma^{2}\left(20\log^{2}({\lambda_{1}}{\tau})-1\right)}\Bigl{[}\alpha\cos{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}$
$\mbox{}+\beta\sin{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}\Bigr{]}-\frac{{\lambda_{2}}{\Omega}}{{\tau}}+{\rho_{2}},$
(104)
$\tilde{\eta}{}^{3}(\tau)=\frac{10{\lambda_{1}}{\vartheta_{3}}}{\gamma^{2}\left(20\log^{2}({\lambda_{1}}{\tau})-1\right)}\Bigl{[}\alpha\sin{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}$
$-\beta\cos{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}+2\gamma\log{({\lambda_{1}}{\tau})}\Bigr{]}$
$\mbox{}-\frac{10{\lambda_{1}}{\vartheta_{2}}}{\gamma^{2}\left(20\log^{2}({\lambda_{1}}{\tau})-1\right)}\Bigl{[}\alpha\cos{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}$
$\mbox{}+\beta\sin{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}\Bigr{]}-\frac{{\lambda_{3}}{\Omega}}{{\tau}}+{\rho_{3}},$
(105)

where the quantities $\lambda_{k}$ now determine the constant spatial
coordinates of the observer on the base geodesic, $\tau$ is the time, the
constant parameters ${\rho_{k}}$ determine the asymptotic values of the
spatial components of the deviation vector for large values of $\tau$.

Let us find in the case of $\omega=1/2$ in the synchronous frame of reference
4-vector of tidal acceleration
$\tilde{A}{}^{\alpha}(\tau)=D^{2}\tilde{\eta}{}^{\alpha}/{d\tau}^{2}$ in a
gravitational wave:

$\tilde{A}^{0}=0,\qquad\tilde{A}^{1}=0,$ (106)
$\tilde{A}^{2}(\tau)=\frac{50{\lambda_{1}}{\vartheta_{2}}\log{({\lambda_{1}}{\tau})}\left(\alpha\cos{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}+\beta\sin{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}\right)}{\gamma^{2}{\tau}^{2}\left(20\log^{2}({\lambda_{1}}{\tau})-1\right)}$
$\mbox{}-\frac{5{\lambda_{1}}{\vartheta_{3}}\left(10\alpha\log({\lambda_{1}}{\tau})\sin{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}-10\beta\log{({\lambda_{1}}{\tau})}\cos{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}-\gamma\right)}{\gamma^{2}{\tau}^{2}\left(20\log^{2}({\lambda_{1}}{\tau})-1\right)}$
$\mbox{}-\frac{5{R_{3}}}{2{\lambda_{1}}\gamma{\tau}^{2}\left(20\log^{2}({\lambda_{1}}{\tau})-1\right)}\Bigl{[}-\alpha\left(20\log^{2}({\lambda_{1}}{\tau})+1\right)\sin{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}$
$\mbox{}+\beta\left(20\log^{2}({\lambda_{1}}{\tau})+1\right)\cos{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}+4\gamma\log{({\lambda_{1}}{\tau})}\Bigr{]}$
$\mbox{}-\frac{5{R_{2}}\left(20\log^{2}({\lambda_{1}}{\tau})+1\right)\left[\alpha\cos{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}+\beta\sin{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}\right]}{2{\lambda_{1}}\gamma{\tau}^{2}\left(20\log^{2}({\lambda_{1}}{\tau})-1\right)},$
(107)

$\tilde{A}^{3}(\tau)=-\frac{50{\lambda_{1}}{\vartheta_{3}}\log{({\lambda_{1}}{\tau})}\left[\alpha\cos{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}+\beta\sin{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}\right]}{\gamma^{2}{\tau}^{2}\left(20\log^{2}({\lambda_{1}}{\tau})-1\right)}$
$\mbox{}-\frac{5{\lambda_{1}}{\vartheta_{2}}\left[10\alpha\log{({\lambda_{1}}{\tau})}\sin{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}-10\beta\log({\lambda_{1}}{\tau})\cos{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}+\gamma\right]}{\gamma^{2}{\tau}^{2}\left(20\log^{2}({\lambda_{1}}{\tau})-1\right)}$
$\mbox{}+\frac{5{R_{2}}}{2{\lambda_{1}}\gamma{\tau}^{2}\left(20\log^{2}({\lambda_{1}}{\tau})-1\right)}\Bigl{[}\alpha\left(20\log^{2}({\lambda_{1}}{\tau})+1\right)\sin{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}$
$\mbox{}-\beta\left(20\log^{2}({\lambda_{1}}{\tau})+1\right)\cos{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}+4\gamma\log{({\lambda_{1}}{\tau})}\Bigr{]}$
$\mbox{}+\frac{5{R_{3}}\left(20\log^{2}({\lambda_{1}}{\tau})+1\right)\left[\alpha\cos{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}+\beta\sin{\bigl{(}2\log{({\lambda_{1}}{\tau})}\bigr{)}}\right]}{2{\lambda_{1}}\gamma{\tau}^{2}\left(20\log^{2}({\lambda_{1}}{\tau})-1\right)},$
(108)

where $\tau$ is the time in the synchronous frame of reference, the constants
$\lambda_{k}$ are the values of the spatial coordinates $\tilde{x}{}^{k}$ of
the observer on the base geodesic, the constants $\alpha$, $\beta$ and
$\gamma$ are the parameters of a gravitational wave in type VII Bianchi
spaces, related by an additional relation (70).

In the synchronous reference frame used, tidal accelerations appear only in
the plane of variables $\tilde{x}{}^{2}$ and $\tilde{x}{}^{3}$, while the wave
propagates along the coordinate $\tilde{x}{}^{1}$.

##  10 On retarded potentials generated by a charge in a gravitational wave

As an example of the application of the obtained exact solutions, consider the
problem of retarded potentials.

In the synchronous reference frame we use, the observer is at rest on the base
geodesic, and the motion of the test particle on the neighboring geodesic is
carried out with tidal acceleration, so if this particle has a charge, we can
formulate the problem of finding retarded potentials for a charge moving along
a given trajectory in a gravitational wave with respect to resting observer in
a synchronous frame of reference.

The exact solution of this problem requires the integration of Maxwell’s
equations in space-time with the metric (55)-(59), when the charge motion is
given by the deviation vector (60)-(63).

The solution of such a problem in flat space-time is given, as is known, by
the Lienard-Wiechert potentials $A^{\alpha}$ (in this section $A^{\alpha}$ is
used to denote electromagnetic potentials), which can be written as LandauEng1
:

$A^{\alpha}(\tau,\vec{r})=e\frac{u^{\alpha}}{R_{\beta}u^{\beta}},$ (109)

where $e$ is the charge of the particle, $u^{\alpha}$ is the 4-velocity of the
charge at time $\tau^{\prime}$,
$R^{\beta}=\\{c(\tau-\tau^{\prime}),\vec{r}-\vec{r}\,{}^{\prime}\\}$ is the
difference of the observer’s 4-vector at time $\tau$ and the 4-vector of the
charge at the time $\tau^{\prime}$, $c$ is the speed of light, $\vec{r}$ is
the spatial radius vector of the charge, and the time $\tau^{\prime}$ and the
components of the radius vector charges $\vec{r}\,{}^{\prime}$ are related to
$\tau$ and $\vec{r}$ by the following relation:

$R_{\alpha}R^{\alpha}=0.$ (110)

In the weak field approximation, when the space-time differs little from the
Minkowski space, we can apply the Lienard-Wiechert potential formula for an
approximate, qualitative study of the problem, but taking into account the
specifics of the exact solution we obtained for the deviation vector
$\eta^{\alpha}$.

In the weak field approximation, for the charge vector $x^{\alpha}(\tau)$ and
the vector $R^{\alpha}$ in the synchronous reference frame, one can take the
expressions

$\tilde{x}{}^{\alpha}(\tau)=\left\\{c\tau,\,\vec{\eta}(\tau)+\vec{\lambda}\,\right\\},$
(111)
$R^{\alpha}=\left\\{c(\tau-\tau^{\prime}),-\vec{\eta}(\tau^{\prime})\right\\},$
(112)
$\vec{\eta}=\left\\{\tilde{\eta}{}^{1},\tilde{\eta}{}^{2},\tilde{\eta}{}^{3}\,\right\\},\quad\vec{\lambda}=\left\\{{\lambda_{1}},{\lambda_{2}},{\lambda_{3}}\,\right\\}.$
(113)

Then the retarded potentials of the charge in the gravitational wave at the
observer’s point on the base geodesic can be represented as (109), where
$\tilde{\eta}{}^{\alpha}$ is the deviation vector (60)-(63), and in the
expression for the potentials $A^{\alpha}(\tau)$ the values on the right are
taken at the moment time ${\tau^{\prime}}$, which is related to the time
$\tau$ in the synchronous reference frame by the relation

$c^{2}(\tau-\tau^{\prime})^{2}+\tilde{g}{}^{kl}\,\tilde{\eta}{}_{k}\tilde{\eta}{}_{l}\,\Bigr{\rvert}_{x^{\alpha}=\\{c\tau^{\prime},\,\vec{\lambda}\\}}=0,$
(114)

where $k,l=1,2,3$ and the gravitational wave metric
$\tilde{g}{}^{\alpha\beta}$ in the synchronous reference frame is taken from
the relations (55)-(59).

##  11 Discussion

The exact solutions of the geodesic deviation equation obtained in this work
are the basis for calculating various physical effects in a gravitational
wave, including to calculate the radiation of charges moving with tidal
acceleration in a gravitational wave. These solutions make it possible to
estimate the influence of primordial gravitational waves on the parameters of
the microwave background of the Universe. Even a “naive” direct substitution
into the Lienard–Wiechert formulas of the solutions obtained in this work for
the deviation vector gives, in the weak field approximation, a qualitative
description of the radiation of charged plasma particles in the primordial
gravitational wave.

Obtaining exact mathematical models of geodesic deviation in primordial
gravitational waves with different types of space-time symmetries also makes
it possible to evaluate the influence of primordial gravitational waves on the
formation of the modern stage of homogeneous and isotropic space-time in order
to understand the process of "isotropization" in the early stages of the
Universe (including for describing violations of the homogeneity and isotropy
of the observed microwave electromagnetic background of the Universe), which
is important for constructing theoretical models of the early Universe.

Exact models of primordial gravitational waves for different types of space-
time symmetries will make it possible to reveal those parameters of the
electromagnetic microwave background that can characterize the presence of
certain types of spatial homogeneity symmetries in the initial stages of the
Universe, which is significant for constructing theoretical models of the
early Universe.

The approach proposed in this paper for obtaining solutions to geodesic
deviation equations in models of primordial gravitational waves for Universes
with Bianchi symmetries is applicable both in Einstein’s theory of gravity and
in modified theories of gravity. This allows, using exact solutions for the
deviation vector in various models and modified theories of gravitation, to
obtain different options for the influence of primary plasma radiation and to
find out the differences that may arise in the parameters of the microwave
electromagnetic background of the Universe at the present stage under the
influence of primordial gravitational waves in various models and modified
theories of gravity.

##  12 Conclusion

An exact solution of the geodesic deviation equations in a gravitational wave
is obtained for Bianchi type VII cosmological models in Einstein’s theory of
gravitation. The models of gravitational waves under consideration refer to
Shapovalov wave spaces of type III, which made it possible to obtain a
complete integral for the Hamilton-Jacobi equation of test particles in a
privileged coordinate system. The presence of the complete integral of the
Hamilton-Jacobi equation made it possible, in turn, to obtain both the
trajectories of motion of test particles and the solution of the geodesic
deviation equations in the models under consideration.

The obtained exact solutions of the geodesic deviation equation are presented
both in the privileged coordinate system and in the synchronous (laboratory)
coordinate system, relative to which the freely falling observer is at rest.
The transition to a synchronous coordinate system is based on the use of the
trajectories of motion of test particles obtained in the work. An explicit
form of metrics, geodesic deviation vectors, and an explicit form of tidal
force acceleration in a gravitational wave are obtained in this work both for
a privileged coordinate system and for a synchronous coordinate system.

The approach presented in the paper can be used both in the general theory of
relativity and in modified theories of gravity. The exact models obtained
describe the primordial gravitational waves of the Universe and can be used to
calculate the secondary physical effects that arise during the passage of a
wave, as well as to debug approximate and numerical methods of detection and
comparative analysis of the characteristics of gravitational waves in terms of
their effect on test particles and charges.

## Acknowledgments

The authors thank the administration of the Tomsk State Pedagogical University
for the technical support of the scientific project. The study was supported
by the Russian Science Foundation, grant No. 22-21-00265,
https://rscf.ru/project/22-21-00265/

## Statements and Declarations

### Data availability statement

All necessary data and references to external sources are contained in the
text of the manuscript. All information sources used in the work are publicly
available and refer to open publications in scientific journals and textbooks.

### Compliance with Ethical Standards

The authors declare no conflict of interest.

### Competing Interests and Funding

The study was supported by the Russian Science Foundation, grant No.
22-21-00265, https://rscf.ru/project/22-21-00265/

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