ЖЭТФ, 2022, том 162, вып. 3 (9), стр. 377-381
© 2022
CYLINDRICAL GRAVITATIONAL PULSE WAVEGUIDE
EXCITATIONS
J. J. Defoa,b*, V. K. Kuetchea,b,c**
a Department of Physics, Faculty of Science, University of Yaounde I
P. O. Box 812, Yaounde, Cameroon
b National Advanced School of Engineering, University of Yaounde I
P. O. Box 8390, Yaounde, Cameroon
c Department of Physics, Faculty of Science, University of Dschang
P. O. Box 67, Dschang, Cameroon
Received April 28, 2022,
revised version May 21, 2022
Accepted May 21, 2022
DOI: 10.31857/S0044451022090103
ating transformation methods [5]. Among these various
methods, we will make allegiance to the inverse scatter-
EDN: ELHFWH
The foundations and principles of the theory of
ing method (ISM) of Belinskii and Zahkarov [6] which
gravitation are based on the idea that space and time
rests on the integrability of the Einstein field equations
can be represented by a Riemannian (Lorentzian) vari-
in dimension two whose contruction gave rise to the
ety, which consists in imposing purely geometrical re-
“gravitational soliton.” This new concept from ISM al-
quirements. Taking into account these constraints, Ein-
lows to put in evidence the phenomenon of temporal
stein’s theory of relativity is recognized by relativists
shift [7] and many other phenomena mentioned with
as the ideal in gravitation theory [1]. Although this
success in this work [5]. We note that in this procedure
theory predicts the existence of gravitational waves,
emanating from the ISM, some solutions such as the
many doubts arise on this subject where Einstein de-
gravitational cylindrical soliton offer the possibility to
clares: “Together with a young collaborator I arrived
study the phenomenon of gravitational collapse [8]. In
at the interesting result that gravitational waves do
the same vein, this solution could serve as an interest-
not exist, though they had been assumed to be a cer-
ing element in the application of quantum information
tainty to the first approximation. This shows that non-
theory [9]. The integrability in dimension five of the
linear gravitational wave field equations tell us more
Einstein field equations allows the modification of the
or, rather, limit us more than we had believed up to
Belinskii and Zahkarov ISM [6] above, the improved
now” [2]. The doubt emitted by Einstein concerning
Pomeransky ISM [10] which is a fundamental tool in
the theory of the gravitational waves was to know if
the generation of black holes. This approach allows
the gravitational radiation has a real existence [3]. On
the construction of new gravitational solitons and their
this intriguing query, the numerical works of Piran and
direct applications [11-13]. It permits the clarification
Stark [4] confirm the real existence of the gravitational
of the studies on the applications of gravitational soli-
radiation. This significant advance leads us to ques-
tons [11, 14]. The real observation of the gravitational
tion about the exact solutions of the field equations
waves by the LIGO-Virgo science team [15] in a recent
and their physical interpretations concerning the rela-
investigation, showing the typical profile of the prop-
tion between field and matter [1, 5]. Concerning the
agating waves and therefore the existence of non zero
exact solutions of the field equations in relativity the-
energy densities of these structures, would actually fos-
ory, the 1970’s and late 1980’s are considered as the
ter the set-up of underlying analytical orientations to
legendary era with the appearance of the soliton gener-
unite, from the exact solutions of the field equations,
the points of convergences and divergences [3,4,16] con-
cerning the waves of impulses or waves of Einstein and
* E-mail: defo.jeanjules@yahoo.fr
** E-mail: vkuetche@yahoo.fr
Rosen (ER) [16]. For that, we account for the relevant
377
6*
J. J. Defo, V. K. Kuetche
ЖЭТФ, том 162, вып. 3 (9), 2022
remark of Alekseev [17] which shows that the soliton
and
can provide complete information during its propaga-
B+ = 2(ψ,t -ψ,ρ ).
(6)
tion in the spacetime of Kasner.
In our investigations, the quantity A+ represents the
In this paper, motivated by the above, we propose
amplitude of the explosion wave and B+ represents the
some underlying approach using the solution of two
amplitude of the implosion wave. We introduce the
cylindrical pole-conjugate solitons generated by the im-
wave vectors of explosions and implosions as defined
proved Pomeransky ISM [10] while associating the nu-
by Piran and Stark [4] in the following form:
merical method of Piran and Stark [4], to construct the
ER waves. In this procedure, we construct the explo-
1
u=
(t - ρ)
(7)
sion and implosion waves as described by Weber and
2
Wheeler [3]. Then, we show the existence of different
and
energy densities relative to the ER waves [14, 16]. In
1
v=
(t + ρ).
(8)
this context, we discuss the ER-metric [18] within the
2
viewpoint of investigation of the soliton dynamics.
Knowing the different expressions of vectors mentioned
The organization of the paper is set as follows: in
above, we simplify the expressions of the amplitudes of
Sec. 2, we present the ER-metric [18] as well as the
the explosive and implosive waves [13,14] in the follow-
field equations governing the behavior of the gravi-
ing form:
tational wave while introducing the soliton solutions
A+ = 2ψ,v
(9)
from Pomeransky’s ISM [10]. We introduce the ER-
and
metric [18] and the three Einstein field equations.
B+ = 2ψ,u .
(10)
Thus, we consider that a four-dimensional spacetime
has a symmetry, then giving the existence of two fields
We used these different expressions above to demon-
of navigating Killing vectors, an axisymmetric Killing
strate the decomposition of the cylindrical gravitational
vector ∂/∂φ and a spatially translational Killing vec-
pulse wave into explosion and implosion waves accord-
tor ∂/∂z, where the coordinate of the polar angle φ
ing to the radial ρ and temporal t coordinates, whose
and the coordinates z have the ranges 0 ≤ φ < 2π and
physical implications are represented in Figs. 1 and 3
-∞ < z < +∞. Validating these different hypothe-
followed with some descriptions in captions. Using the
ses of symmetry, we start from the general form of the
simplified expressions from the amplitudes of the explo-
Jordan and Ehlers metric [4,13,19]. We eliminate the
sion and implosion waves, we rewrite the field equations
nonlinear term ω = 0 and obtain the following expres-
in the following form:
sions:
A+ - B
+
A+,u =
(11)
2ρ
ds2 = e2(γ-ψ)(dρ2 - dt2) + ρ2e-2ψ dφ2 + e2ψ dz2, (1)
and
ψ,ρ
A+ - B+
ψ,tt -
- ψ,ρρ = 0,
(2)
B+,v =
(12)
ρ
2ρ
We introduce in the Einstein field equations for energy
γ,ρ = ρ(ψ,2t +ψ,2ρ ),
(3)
densities, the expression of the amplitudes of the ex-
γ,t = 2ρψ,t ψ,ρ .
(4)
plosion and implosion waves in the following form:
We note that (ρ, z, φ) represents the cylindrical coor-
ρ
γ,t=
(A2+ - B2+)
(13)
dinates and t the time. We specify that the different
8
functions ψ and γ depend on ρ and t. In this met-
and
ρ
ric including the Einstein field equations, ψ represents
γ,ρ=
(A2+ + B2+).
(14)
8
a dynamic degree of freedom of the gravitational field
We specify that the field equations related to the en-
and γ plays the role of the gravitational energy of the
system. It is also noted that the previous observables
ergy density γ,t represents the non-gravitational energy
written with comma as subscript denotes the partial
density of the wave and γ,ρ the gravitational energy
derivatives with the associated variables. We intro-
density. We note that the expressions of the differ-
duce, the solutions of Piran and Stark [4] relative to
ent energy densities as functions of the different ampli-
the field equations in the following form:
tudes A+ and B+ of the explosion and implosion waves
show that the propagation of the cylindrical gravita-
A+ = 2(ψ,t +ψ,ρ )
(5)
tional impulse wave is vector of energy, of which Figs. 2
378
ЖЭТФ, том 162, вып. 3 (9), 2022
Cylindrical gravitational pulse waveguide excitations
Fig. 1. We observe that the implosion wave B+ is focused near
Fig. 3. We show the different behaviors of explosive and
ρ = -t on the implosion course for negative values of t. While
implosive waves under the following conditions: (k, θ, q) =
we observe that the explosion wave A+ is concentrated near
= (2, nπ/4, 1)(n = 0) with ρ = 0, 1, 2
ρ = +t when it reexpands out from the axis of symmetry. We
use t = ±1, ±2, ±3 and (k, θ, q) = (2, nπ/4, 1)(n = 0)
Fig.
4. We have the behavior of the gravitational
Fig. 2. We have the representation of the energy density with
energy
density
concerning
the
following
values:
the following contions: (k, θ, q) = (4, nπ/4, 1)(n = 0) with
(k, θ, q)
= (4, nπ/4, 1)(n
= 0) with ρ
= 0, 1, 2. This
t = ±1,±2,±3. This shows that the gravitational waves are
shows the well-localized of the gravitational waves within the
well localized in the spacetime manifold
spacetime manifold
and 4 are tangible proof. All the equations obtained
and
with Piran and Stark [4] have a major and fundamen-
tal interest in the comprehension of the phenomena as
dρ2 =
mentioned in introduction. We introduce in the ex-
q2
(
)
=
x2(y2-1)2 dx2+y2(x2+1)2 dy2
+
pressions of the time t and the radial coordinates ρ,
(x2 + 1)(y2 - 1)
the cartesian coordinates in the following form [13]:
q2
(
)
+
2xy(x2 + 1)(y2 - 1) dx dy
(18)
(x2 + 1)(y2 - 1)
t = qxy,
(15)
√
From the different quantities calculated previously, we
ρ=q
(x2 + 1)(y2 - 1),
(16)
modify the ER-metric [18] into the form below:
where q represents a constant. We calculate the follow-
ds2 = e2ψ dz2 + ρ2e-2ψ dφ2 +
ing quantities:
(
2
)
dx2
dy
+q
2(x2 + y2)e2(γ-ψ) -
+
(19)
dt2 = q2(y2 dx2 + x2 dy2 + 2xy dx dy)
(17)
x2 + 1
y2 - 1
379
J. J. Defo, V. K. Kuetche
ЖЭТФ, том 162, вып. 3 (9), 2022
We determine the functions ψ and γ belonging to the
Expressions (24)-(27) are used to highlight the differ-
Einstein field equations as well as to the ER-metric [18].
ent behaviors of the gravitational pulse wave mentioned
For this, we use the solutions of the cylindrical soli-
above during its propagation in the ER-spacetime [18].
tons [10, 13] to construct the different amplitudes of
Next, in Sec. 3, we analyze the two soliton solu-
the explosion and implosion wave and we obtain the
tion obtained by calculating the amplitudes of the in-
following relations:
coming and outgoing waves. They are assimilated to
the explosions and implosions waves from Weber and
Y
e2ψ =
,
(20)
Wheeler [3], viewpoint taking into account the differ-
X
ent energy densities. Following the previous expres-
χ
e2(γ-ψ) =
,
(21)
sions, we observe how gravitational pulse waves would
4096q6(x2 + y2)6
propagate as exploding and imploding waves, as well
where
as different densities of energy in spacetime through
a multitude of viewpoints. For a clear understanding,
(
we introduce the different parameters which are the
χ = a4i(y-1)2(y+1)6+2a2i(y+1)2
a2r(y-1)2(y+1)4 +
(
(
)
(
)
))
modulus k and the angle θ of the complex parameter
+64q2
x4
y(9y-8)+1
+2x2
y(y+4)-3
y2+y6+y4
-
(
)(
)
k = |ar + iai| = |a|, θ = Arg(a). In the following
−512aiarq2x(y+1)2
x2 -(y-2)y
x2(2y-1)+y2
+
analysis, we only consider the case q = 1, because the
+ a4r(y - 1)2(y + 1)6 +
parameter q can be normalized by a scaling of the co-
(
(
)
ordinates. For the investigation of the given orienta-
+ 128a2rq2(y + 1)2
2x6 + x4
(8 - 3y)y - 1
+
(
)
)
tions, we define the following spacetime: -5 ≤ t ≤ 5,
+ 2x2y2
2(y - 2)y + 3
+y6 -y4
+
-5 ≤ ρ ≤ 5, and θ = nπ/4 with n = 0, . . ., 3. In the
+ 4096q2(x2 + y2)4
(22)
following, we aim at investigating detailed behavior of
waves propagating near the limits of spacetime, with a
and
particular focus on waves of explosions and implosions
(
as well as energy densities.
Y = a4i(y2 - 1)2 + 2a2i(y2 - 1)
a2r(y2 - 1)3 +
(
))
+ 64q2
x4(9y2 - 1) + 2x2(y2 + 1)y2 + y6 - y4
-
1. Timelike infinity. Next we consider the asymp-
− 1024aiarq2x(x2 + 1)y(y2 - 1)(x - y)(x + y) +
totic behaviors of the waves at late time t → ∞. At
(
t → ∞, the metric behaves as
+ a4r(y2 - 1)4 + 128a2rq2(y2 - 1)
2x6 + x4 +
)
+ (4x2 + 1)y4 - 3(x2 + 2)x2y2 + y6
+
(
)
(
)
a2r
a2r
+ 4096q4(x2 + y2)4.
(23)
ds2 =
1-
dz2 + ρ2 1+
dφ2 +
4t2
4t2
(
)
a2r
In the different expressions obtained, ai and ar are real-
+ 1+
(dρ2 - dt2).
(28)
4t2
valued variables. We introduce a relation between the
different cartesian and cylindrical variables, and we ob-
This metric allows us to highlight the representation
tain the following two relations [20].
obtained by Weber and Wheeler [3], by associating the
For ρ ≪ |t|, we obtain
two numerical solutions while paying particular inter-
ests to the energy density.
ρ2t
x=t-
+ Θ(ρ4)
(24)
2(1 + t2)
2. Spacelike infinity. Let us study the behavior of
and
these gravitational waves when ρ → ∞. At the limit of
2
ρ
ρ → ∞, we get a new expression of the metric in the
y=1+
+ Θ(ρ4).
(25)
2(1 + t2)
form
For ρ ≫ |t|, we obtain
(
)
4|a|2q
ds2 ≈
1-
dz2 +
2
(|a|2 + 64q2)ρ
1-t
y=ρ+
+ Θ(ρ-2)
(26)
(
)
4|a|2q
2ρ
+ρ2 1+
(
)
dφ2 +
|a|2 + 64q2
ρ
and
(|a|2 + 64q2)2
t
t(t2
− 1)
+
(dρ2 - dt2).
(29)
x=
+
+ Θ(ρ-4).
(27)
4096q2
ρ
2ρ3
380
ЖЭТФ, том 162, вып. 3 (9), 2022
Cylindrical gravitational pulse waveguide excitations
3. Axis. Now we look at the behavior of the waves on
4.
T. Piran, P. N. Sarfier, and R. F. Stark, Phys. Rev.
the axis of symmetry ρ = 0. Near the axis, the metric
D 32, 3101 (1985).
behaves as follows:
5.
V. A. Belinski and E. Verdaguer, Gravitational soli-
ton, Cambridge Univ. Press, Cambridge (2001).
4(q2 + t2)
ds2 ≈
dz2 +
4(q2 + t2) + (tar - qai)2
6.
V. Belinsky and V. Zakharov, Sov. Phys. JETP 48,
2
4(q2 + t2) + (tar - qai)
985 (1978).
+ρ2
dφ2 +
4(q2 + t2)
2
7.
A. D. Dagotto, R. J. Glleiser, and C. O. Nicasio,
4(q2 + t2) + (tar - qai)
+
(dρ2 - dt2).
(30)
Class. Quant. Grav. 8, 1185 (1991).
4(q2 + t2)
8.
R. Penrose, Gen. Relativ. Grav. 34, 7 (2002).
This metric allows us to show that in the vicinity of the
axis ρ = 0, the explosion and implosion waves during
9.
M. Lanzagorta, Quantum Information in Gravita-
their propagation in this region of space are merged. In
tional Fields, Morgan Claypool Publishers (2014).
this case, we study the representation of gravitational
waves as well as its gravitational density. We see that
10.
A. A. Pomeransky, Phys. Rev. D 73, 044004 (2006).
when ρ = 0, B+ = A+.
Finally, Sec. 4 is devoted to conclusion and perspec-
11.
S. Tomizawa and T. Mishima, Phys. Rev. D 90,
tives.
044036 (2014).
Acknowledgements. The authors would like to ex-
12.
T. Igata and S. Tomizawa, Phys. Rev. D 91, 124008
press their sincere thanks to the anonymous referees
(2015).
for their critical comments and appropriate sugges-
tions which made this paper more precise and read-
13.
S. Tomizawa and T. Mishima, Phys. Rev. D 91,
124058 (2015).
able. They also appreciate the contributions of Mr.
Joe Wabo and Ms. Zang Sedena Christine for the im-
14.
A. Tomimatsu, Gen. Relativ. Grav. 21, 613 (1989).
provement of the English language in the text.
15.
B. P. Abbott et al., LIGO-Virgo scientific collabora-
The full text of this paper is published in the English
tion, Phys. Rev. Lett. 116, 061102 (2016).
version of JETP.
16.
M. Halilsoy, Nuovo Cim. 102, 6 (1988).
17.
A. Alekseev, Phys. Rev. D 93, 061501(R) (2016).
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