CC BY
15
DOI: 10.18721/JPM.10410 UDC 530.12:531.51
ON THE SOLUTIONS OF FIELD EQUATIONS DUE TO ROTATING BODIES IN GENERAL RELATIVITY V. Kumar1, L. Kaur2
1DAV College Pundri Haryana, India;
2Jaypee Institute of Information Technology, Noida, India.
A metric, describing the field due to bodies in stationary rotation about their axes and compatible with a stationary electromagnetic field, has been studied in the present paper. Using the Lie symmetry reduction approach, we have herein examined, under continuous groups of transformations, the invariance of field equations due to rotation in General Relativity, which are expressed in terms of a coupled system of partial differential equations. We have exploited the symmetries of these equations to derive some ansatz leading to the reduction of variables, where the analytic solutions are easier to obtain by considering the optimal system of conjugacy inequivalent subgroups. Furthermore, some solutions are considered by using numerical methods due to complexity of reduced ordinary differential equations.
Key words: General Relativity, electromagnetic field, Lie symmetry method, exact solution
Citation: V. Kumar, L. Kaur, On the solutions of field equations due to rotating bodies in General Relativity, St. Petersburg Polytechnical State University Journal. Physics and Mathematics. 10 (4) (2017) 124-133. DOI: 10.18721/JPM.10410
О РЕШЕНИИ УРАВНЕНИЙ ГРАВИТАЦИОННОГО ПОЛЯ, СОЗДАННОГО ВРАЩАЮЩИМИСЯ ТЕЛАМИ, В ОБЩЕЙ ТЕОРИИ ОТНОСИТЕЛьНОСТИ В. Кумар1, Л. Каур2
1Колледж Дайянанд англо-ведической (DAV) системы образования, г. Пандри (Харьяна), г. Пандри, Индия;
2Институт информационных технологий Джепи, г. Нойда, Индия
Изучена метрика, которая описывает гравитационное поле, создаваемое телами при стационарном вращении вокруг своих осей, совместимая со стационарным электромагнитным полем. При этом, используя метод понижения симметрии групп Ли, мы изучили инвариантность (относительно непрерывных групп преобразований) уравнений поля, вызванного вращением, в общей теории относительности. Преобразования приводят к связанной системе дифференциальных уравнений с частными производными. Мы воспользовались симметрией этих уравнений, для того чтобы достичь некоторых упрощений, приводящих к сокращению числа переменных, когда аналитические решения легче получить, если рассмотреть оптимальную систему сопряженных неэквивалентных подгрупп. Кроме того, часть решений получена с применением численных методов, в связи со сложностью обыкновенных дифференциальных уравнений даже сниженного порядка.
Ключевые слова: общая теория относительности; электромагнитное поле; метод понижения симметрии групп Ли; точное решение
Ссылка при цитировании: Кумар В., Каур Л. О решении уравнений гравитационного поля, созданного вращающимися телами, в общей теории относительности // Научно-технические ведомости СПбГПУ. Физико-математические науки. 2017. Т. 10. № 4. С. 124-133. DOI: 10.18721/JPM.10410
Introduction
General Relativity describes phenomena on all scales in the Universe, from compact objects such as black holes, neutron stars and supernovae to large-scale structure formations such as those involved in creating the distribution of clusters of galaxies. For many years, physicists, astrophysicists and mathematicians have striven to develop techniques for unlocking the secrets contained in Einstein's theory of gravity. More recently, solutions of Einstein field equations have added their expertise to the endeavor. Those who study these objects face a daunting challenge that the equations are among the most complicated in mathematical physics. Together, they form a set of coupled, nonlinear, hyperbolic-elliptic partial differential equations that contain many thousands of terms.
The gravitational field due to a rotating body was firstly attempted by Thirring who used Einstein field equations in the linear approximation and showed that a rotating thin spherical shell produces near its center forces similar to the Coriolis and centrifugal forces of classical machines. Later on, this work has been revised by Bass and Pirani [1] who supplemented the energy tensor of incoherent material by a term representing the elastic interaction between the particles of the shell. Bach considered the field due to a slowly rotating sphere by successive approximations taking the Schwarzschild solution as his zeroth approximation. Special cases of stationary fields have been considered by Lanczos [2] who applied the results to cosmo-logical problems.
Lewis [3] found the field due to a rotating infinite cylinder and thus obtained two different methods of successive approximations for constructing solutions of a more general type which behave in an assigned manner at infinity and on a surface of revolution enclosing the rotating matter to which the field is due. Clark tried to solve the empty gravitational field equations, using successive approximations, with forms of appropriate to the gravitational field of ^ rotating body. This introduction provides a sample of the idea that these equations have been a subject of extensive and intensive study both by mathematicians and physicists. For a de-
tailed study of exact solutions of Einstein field equations, the reader may refer to Stephani et al. [4]. Recent years have been dedicated to studying the field equations of General Relativity for their solutions [5 — 16], these solutions are important in the sense that they represent the physical models in an analytical manner.
In the present paper, we have considered a metric [3] which is supposed to describe the field due to bodies in stationary rotation. Further in this case we furnished a consistent set of partial differential equations for determining
in the empty space-time. It is shown that by using a selective form of g^v, the problem of solving four equations in three unknowns has been reduced to a system of two partial differential equations in two unknowns and then Lie group analysis is applied to generate various symmetries of this coupled system of partial differential equations, which are then used to identify the associated basic vector fields of the optimal system for systematic study of the group invariant solutions admitted by the system.
Nature of field equations
The following metric described the field due to bodies in stationary rotation about their
axes
ds2 = - exp(2A,)(dp2 + dz2) - Cdф2 + + Ddt2 + 2Ed фdt,
(1)
where 1, C, D and E are functions of p and £ only.
Following Lewis [3], we have made use of canonical coordinates in the sense of Weyl. The choice of these coordinates is possible only in matter free space as it can be easily verified by a procedure similar to that of Synge. Consequently, in domains occupied by matter, the canonical coordinates cannot be used. In canonical coordinates we have
CD + E2 =p2,
(2)
and therefore the expressions for the Einstein tensor are given by:
G G \ CiDi + Ei - C2D2 - E2 (3)
G11 = -G22 =----~2-,(3)
p 4p2
22
G33 = exp
(-2b)
-2C (Ьц + ^22) + Сц +
Cl р
3C
2р2
+ C22 —^ + ^Cr(Ci^i + E2 + C2D2 + El)
G44 = exp
(-2b)
2D(bn + ^22) - Du -
(3)
DL
р
3D
2р2
- D22 - 32 (C1D1 + E12 + C2D2 + E22)
G34 = exp
(-2b)
-2E(bn + b22) - E11 -
El р
- E22 + ^-^CDi + E12 + C2D2 + E22) 2р
G = b2 C1D2 + 2E1E2 + C2D1 g12 = 7"2
р 4р2
where the lower suffixes 1 and 2 after the unknown functions imply partial differentiation with respect to p and £ respectively.
Now we have considered the determination equation
\G -sg 1 = 0. (4)
UV Ouv
We found that two of the eigenvalues of G with respect to g are given by
1
si = exp(-2X)(G222 + G?2)2, i = 1, 2, (5)
and the other two are given by following equation:
s2 + Rs - -1 = (G33G44 - G324) = 0 (6)
where R is the curvature scalar.
It is clear from Eqs. (5) and (6) that, in general, two eigenvalues of G are equal and opposite while the other two are different. Therefore, metric (1) in canonical coordinates cannot represent a perfect fluid distribution. But if we do not consider the canonical coordinates then all the eigenvalues of the Einstein tensor are different in general. Thus, in this case metric (1) can be utilized to describe the space time in the domains occupied by matter.
In case of an electromagnetic field, we have R = 0, therefore, Eq. (6) gives
Sj =± -^(gg - G34), j = 3,4. (7) p
Thus, in this case the other two eigenvalues are equal and opposite. In fact, the eigenvalues are k, —k, m, —m, where
k = exp(-2b^(G222 + Gl22) ,
m = (G33G44 - g34 ), р
(8)
and if we further assume that k = m, the eigenvalues become k, k, —k, —k, which characterize an electromagnetic field.
The field equations for empty space-time. The field equations, in terms of a coupled system of partial differential equations, for empty space-time, corresponding to Eq. (1) are given by
. b 1 = Ci Di + E12 Л11 + л22 _
b11 + b22 +
1
р 2р2
b 1 = C2 D2 + E2
2р2
(9) (10)
b2 = (CD + CD + E2 + E22), (11) 4p
C11 + C22 1 + ~ x
P P2 (12)
X (C1D1 + C2D2 + E2 + E22) = 0,
D11 + D22 - D + D x 11 22 P p2 (13)
X (C1D1 + C2D2 + E2 + E22) = 0,
E E E1 E
E11 + E22 + ~ X
P P2 (14)
X (C1D1 + C2D2 + E2 + E22) = 0.
From Eqs. (9) and (10), the condition of integrability can be easily verified for above system of partial differential equations. Also from Eqs. (9) and (10), we obtain
b + b C D1 + C2 d2 + E2 + E22)
b11 + b22 =-2-, (15)
P2
= (-C1D1 + C2D2 - E2 + Ef) . 1 4p '
Also Eq. (15) is consistent with Eqs. (11)
P
and (16). Thus, the problem of solving Eqs. (2), (9) — (14) is reduced to determining C, D and E from (2), (12), (13) and (14) and then 1 will be given by Eqs. (11) and (16).
We made the substitutions as follows:
C = p exp(-|) cos 9, D = p exp(|) cos 9, E = p sin 9
(17)
where ^ and 9 are functions of p and z.
Consequently, Eqs. (12), (13) and (14) are reduced to
cos 9(|n + |22 + — - 2 tan 9(^191 + |292) + P
(18)
9
+ sin 9(911 + 922 + — -P
-2 sin 9 cos 9(|2 + |2) = 0,
cos 9(|11 + |22 + 11 - 2 tan 9(|191 + |292) -P
- sin 9(911 + 922 + ^ - (19)
P
- 2 sin 9 cos 9(|2 +|2) = 0,
9
911 + 922 + - sin 9 cos 9(|2 + |2) = 0.(20) P
From Eqs. (18) and (19) we obtain, in view of Eq. (20), the single equation:
l11 + l22 +11 - tan9(|191 + |292) = 0, (21) P
and Eqs. (11) and (16) lead to the following equations:
= --1- - - (91 + 92 - cos2 9(|1 - |2)), (22) 4p 4 1 1
^ =2(cos2 9|1|2 -9192). (23)
Thus, the problem of solving four equations in three unknowns has been reduced to a system of partial differential equations consisting of two Eqs. (20) and (21) in two unknowns 9 and Also we can determine C, D and E by using the expressions of C, D and E in Eq. (17) and then 1 can be determined from Eqs. (22) and (23).
Solutions of field equations
It is well-known that the Lie symmetries,
originally advocated by the Norwegian mathematician Sophus Lie in the beginning of the 19th century, are widely applied to investigate nonlinear differential equations for constructing their exact and explicit solutions. Considering the tangent structural equations under one or several parameter transformation groups is the basic idea of Lie symmetry analysis. It has been shown that Lie symmetry analysis has been effectively used to seek exact and explicit solutions to both ordinary differential equations (ODEs) and partial differential equations (PDEs). There are a lot of papers and many excellent books [15 — 24] devoted to such applications.
In the present section, we have performed Lie group classification of Eqs. (20) and (21). That is, we have furnished all the possible forms of Lie point symmetries, admitted by Eqs. (20) and (21), and then constructed symmetry reductions and group-invariant solutions using the optimal system of subalgebras of the Lie algebras of the equations.
The classical Lie method [18] has been applied to Eqs. (20) and (21) by considering the one-parameter Lie group of infinitesimal transformations in p, z, 9, ^(p, £), £,2(p, £), n1(P, z) and n2(p, Z). This transformation leaves invariant the following set:
S - {9(p, z), |(p, z):Ai(9, |) = 0,
A2(9, |) = 0}, of the solutions of Eqs. (20) and (21), where
A1 = 911 + 922 + — - sin 9 cos 9(|i2 + i^), P 1
A2 = I11 + I22 +11 - tan 9(i191 + I292). P
(24)
(25)
The associated Lie algebra of infinitesimal symmetries is the set of vector fields of the form
t—, (.1 д r2 д 1 д 2 д
Г = г,1 — + £,2— + П1 — + n2 — дp дz д9 д|
(26)
The set is invariant under the one-
A
parameter transformations provided that
Pr(2)(r) |a=o = 0,
where Pr(2)(r) is the second prolongation of the vector field r, which is explicitly given in terms
of Ç1, E,2, n1 and n2.
After determining the infinitesimals of Eqs.
(20) and (21), the similarity variables are derived by solving invariant surface conditions
$1 - -n1 = 0,
P (27)
$2 - ^9p + ^ - n2 = 0.
The symmetries under which Eqs. (20) and
(21) are invariant can be spanned by the following three linearly independent infinitesimal generators:
д д д д 4= p— + z—,Г2=—,Г3 = —
дp дz дz д9
(28)
It is easy to verify that r1, r2 and r3 are closed under the Lie bracket. So we can see that the generator of the invariant group r of Eqs. (20) and (21) constructs three-dimensional Lie algebra, which is spanned by the basis r1, r2 and r3. Thus, we have the corresponding one-parameter group of symmetries of Eqs. (20) and (21):
G : (p, Z, 9, |) ^ (exp(e)p, (exp(e)z, 9, |), g2 : (p, Z, 9, |) ^ (p, exp(e)z, 9, |), (29)
g3 : (p, z, 9, |) ^ (p, z, exp(e)p, (exp(e)9, |).
We can see that G1 is a space translation, G2 is a time translation and G3 is a scaling transformation. We have used the subalgebraic structure of symmetries (28) to construct an optimal system [19] of one-dimensional subgroups. The optimal system yields only the following symmetry combinations:
(i )r1 + pr3, (ii )r2 + ar3, (iii )r3, (30)
where a and p are arbitrary constants.
Symmetry reductions. In this subsection, we have derived symmetry reductions of Eqs. (20) and (21) associated with the vector fields in the optimal system (28) by using similarity variables and further attempted to furnish exact solutions
(i )r +pr3.
Corresponding to this vector field, the form of the similarity variable and similarity solution are as follows:
Z = -, 9(p, Z) = f(Z), |(P, Z) = p log Z + g(Z).
Z
(31)
Substituting the expressions of similarity variable and similarity solution into Eqs. (20) and (21) yields the following system of reduced ODEs:
Z3F" + F' + Z + 2F'Z2 - ZP2 sin F cos F -
- 2 sin F cos FZ2p G' - sin F cos FZ3G"2 -- sin F cos F ZG' 2 = 0,
ZG" + G' + pZ- G"Z2 - 2Z2G' -- 2ZF'G' tan F + 2Z2pF ' tan F -- 2Z3F'G' tan F = 0.
In this case because of the complexity of reduced system (31), the following two particular cases have been worked out.
Case I. By considering F (Z) = 0, we found that metric (1) is reduced to static axially symmetric metric of Weyl in canonical coordinates and system (31) becomes
Z G" + G' + pZ - Z2G" - 2Z2G' = 0. (32)
Solving Eq. (32), we obtained the solutions as follows:
G(Z) = pln(-2Z) - p ln(-2Z + 2) +
+ J2ln(Z- 1) + qEi(l,2Z) -- c1 exp(-2)Ei(1,2Z - 2) + c2,
(33)
where c1 and c2 are arbitrary constants and Ei is the exponential integral.
Now, we have obtained solution of Eqs.
(20) and (21) for static axially symmetric metric and further by back substitution to original variables, the exact solution of Eqs. (20) and
(21) is given by:
|(p, z) = 4lni-^
4 I z
-в ln
-2p
+ 2 1 +
+ в in fP-1 1 + Ci
^Ei fl, 2
(34)
-q(exp(-2)Ei ^1, ^ - 2J + c2.
In Fig. 1, the picture representation of the solution (34) is given for p = 1, c1 = c2 = 1.
Case II. By putting G(Z) = 0, metric (1) is reduced to
Fig. 1. Picture representation of Coriolis and centrifugal force solution (34), produced by rotating spherical shell in General Relativity, to field Eqs. (20) and (21) with p = 1, c1 = c2 = 1
ds2 = - exp(2b)(d p2 + dz2) -- p cos 0(dф2 - dt2) + 2p sin 0dфdt,
(35)
and then solving Eq. (31) and reverting back to the original variables. Thus, we have obtained the following exact solution of Eqs. (20) and (21):
0(p, z) = C3 +
(36)
where c3 and c4 are arbitrary constants.
In Fig. 2, the picture representation of the solution (36) is given for c3 = c4 = 1.
Now, we attempt to furnish exact solutions
(ii )r2 + ar3.
For this vector field, the forms of the similarity variable and the similarity solution are as follows:
(37)
Z = p, 0(p, £) = F (Z), U(p, £) = G (Z) + yz.
Using these forms in Eqs. (20) and (21), we obtain the system of reduced ODEs:
ZF" + F' + Z sin F cos F(a2 + G'2) = 0,
ZF" + G'- 2Z tan FG = 0,
where the prime denotes the differentiation with respect to the variable Z.
Now under this vector field, we are unable to obtain the nontrivial exact solutions. So we have used the well-developed numerical technique to solve the reduced problem. For this purpose, we have obtained the following four first-order equations:
dyx dz dy1 dz
+ za2y42 sin y1 cos yjy2 • z^1,
= У2,
= [-y2 + a z sin y1 cos y1 +
(38)
dy1 = dz 3'
dy4 dz
-y4 + 2 y2 y4 z tan y1
with
y1(45) = 1.2, y2(45) = 0, y3(45) = 0, y5(45) = 0.1.
(39)
-15
Fig. 2. Coriolis and centrifugal force solution (36), produced by rotating spherical shell in General Relativity, to field Eqs. (20) and (21) with
C3 = C4 = 1
a)
c)
48 49 50 51 52
Similarity variable (z)
48 49 50 51 52
Similarity variable (z)
48 49 50 51 52
Similarity variable (z)
48 49 50 51 52
Similarity variable (z)
Fig. 3. Numerical solutions to field Eqs. (20) and (21) with respect to the reduced IVP (38), (39) when a2 = 5, with the initial value z = 45 at h = 0.01; the profiles of y{ (a), (b), ^ (c), y4 (d) are presented
The numerical solutions to the initial value problem (IVP) (38), (39) are depicted above.
In Fig. 3, numerical solutions of field Eqs. (20) and (21) are obtained with respect to the reduced IVP (38), (39). Now the profiles of and y2 show that the solution is periodic, and the profiles of y3 and y4 show that the solution is unbounded and damped oscillatory respectively.
Now, if we attempt to furnish exact solutions
(iii)
corresponding to this vector field, no such invariant solution exists.
Discussion and concluding remarks
In the present investigation, we have successfully implemented Lie symmetry
reduction to obtain the Lie symmetries admitted by field equations extracted from a metric which is supposed to describe the fields due to bodies in stationary rotation about their axes. The infinitesimal generators in optimal system of sub-algebras of the full Lie algebra of the coupled system of nonlinear partial differential equations of second order of field equations are considered. We completely solved the determining equations for the infinitesimal generators of Lie groups. Further, the group classification from the point of view of the optimal system of non-conjugate sub-algebras of the symmetry algebra of the nonlinear system has been performed under the adjoint action of the symmetry group. The various fields in the optimal system have been then exploited to get the reductions of PDEs into ODEs.
Due to the complexity of reduced ODEs, it is impossible to obtain the nontrivial exact solutions, so under the vector field (i )AX + ßA2, particular exact solutions are obtained for field Eqs. (20) and (21). Graphical representation of solutions (34) and (36) to field Eqs. (20) and (21) described the center forces, similar to the Coriolis and centrifugal forces of classical machines, produced by a rotating spherical shell in general relativity as shown in Figs. 1, 2. Now under the vector field (ii )A2 +aA3, it is again impossible to obtain the nontrivial
exact solutions with respect to the reduced ODEs (37). So, under this vector field, IVP is posed for numerical solution. As a result, a numerical solution is found which is periodic, unbounded and damped oscillatory as shown in Fig. 3.
Acknowledgments
The authors thank the referees for their careful reading of the manuscript and for valuable suggestions that were helpful in improving this paper.
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THE AUTHORS
KUMAR Vikas
DAV College Pundri Haryana Kaithal, Haryana, 132026, India [email protected]
KAUR Lakhveer
Jaypee Institute of Information Technology
A-10, Sector-62, Noida, Uttar Pradesh, 201309, India
СПИСОК ЛИТЕРАТУРЫ
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4. Stephani H., Kramer D., MacCallum M., Hoenselaers C., Herlt E. Exact solutions of Einstein's field equations. Cambridge: Cambridge University Press, 2003.
5. Misra M. A non-singular electromagnetic field // Proc. Camb. Phil. Soc. 1962. Vol. 58. No. 4. Pp. 711-712.
6. Bardeen J.M. A variational principle for rotating stars in General Relativity // Astrophys. J. 1970. Vol. 162. Pp. 71-95.
7. Bonnor W.B. A rotating dust cloud in General Relativity // J. Phys. A: Math. Gen. 1977. Vol. 10. No. 10. Pp. 1673-1678.
8. Islam J.N. Rotating fields in General Relativity. Cambridge: Cambridge University Press, 1985.
9. Grosso R., Stephani H. Bianchi types of Lie point symmetries arising for differentially rotating stationary axisymmetric fluids // Astron. Nachr. 1990. Vol. 311. No. 5. Pp. 283-286.
10. Marchildon L. Lie symmetries of Einstein's vacuum equations in N dimensions // J. Nonlinear Math. Phys. 1998. Vol. 5. No. 1. Pp. 68-81.
11. Barnes A. On Stephani's rotating dust solutions // Class. Quantum Grav. 1999. Vol. 16. No. 3. Pp. 919-926.
12. Klepac P. Some cosmological solutions of Einstein equations // J. Phys.: Conf. Ser. 2007. Vol. 82. No. 1. Pp. 012004.
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Статья поступила в редакцию 10.02.2016, принята к публикации 10.10.2017.
СВЕДЕНИЯ ОБ АВТОРАХ
КУМАР Викас — сотрудник кафедры математики Колледжа Дайянанд англо-ведической (DAV) системы образования, г. Пандри (Харьяна), г. Пандри провинции Харьяна, Индия. Kaithal, Haryana, 132026, India [email protected]
КАУР Лакхвир — сотрудник кафедры математики Института информационных технологий Дже-пи (Джепи — группа с широким диапазоном деловых интересов), г. Нойда, Индия. A-10, Sector-62, Noida, Uttar Pradesh, 201309, India [email protected]
© Санкт-Петербургский политехнический университет Петра Великого, 2017
