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Classes of Exact Solutions of the Einstein-Maxwell Equations.

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A N N A L E N D E R PHYSIK
7. Folge. Band 40.1983. Heft 4/5, S. 181-300
Classes of Exact Solutions of the Einstein- Maxwell Equations
By V. G. BAGROV
and V. V. OBUKHOV
Physikalische Fakultit der Moskauer Universitat Moskau, UdSSR
Abstract. All exact solutions of the Einstein-Maxwell equations with a cosmological term for
the case, when the Hamilton-Jacobi equations may be integrated by the method of separation of
variables by means of a complete isotropic set of integrals of motion are found.
Klassen exakter Lasungen der Einstein-Maxwellschen Gleichungen
Inhaltsubersicht. Es werden alle exakten Losungen der Einstein-Maxwellschen Gleichungen
mit einem kosmologischen Glied fur diejenigen Fiille angegeben, in denen die Hamilton-JaaobiGleichungen mittels eines vollstandigen isotropen Satzes von Bewegungsintegralen nach der Methode
der Separation der Variablen integrierbar sind.
Introduction
An article by CARTERpublished in 1968 [l]gave rise to systematic study of the general relativity fields allowing complete separation of variables in the Hamilton-Jacobi
equation for a massive particle in an external electromagnetic field. Later on, the problem of finding such fields was studied by others [2-51, but no complete solution was
obtained.
The problem is also interesting from physical point of view since the fields in which
the equations of motion for test particles can be integrated in closed form are most
convenient for investigation. However, it is very difficult to select such fields even from
a great number of the known exact solutions of the Einstein equations. Therefore we
set ourselves a target to find all exact solutions of a self-consistent system of the Einstein-Maxwell equations under the condition that the Hamilton-Jacobi equation for a
massive particle can be integrated by the method of complete separation of variables.
At the first stage we are interested in electro-vacuum solutions only. To find these solutions one needs to integrate a system of the Einstein-Maxwell vacuum equations (with
a cosmological term A).
Rii - GiiR/2
+ AGii = 4nxTii,
(1)
3
ZFii;
i=O
(2)
d = Q
where x is an arbitrary constant, Fij = Ai,i -
3
16nTii = GJi
2
FekFek
k,e=O
3
i4 2 pi@$,Gii and GJi are the covariant and contravariant components of the metk-0
ric tensor; the covariant and the partiel derivatives are denoted by semicolon and by
9
Tf = 0, then from ( 1 ) it follows that R = 42 = const.
comma, respectively. As
i=O
V. G . BAGROV
and V. V. OBUKHOV
182
Equation (1)can be reduced t o the following one:
+
Rii = 4nxTii lGii,
1= const.
(3)
Thus, our aim is to find all exact solutions of system (2), (3) subject to the condition
that the Hamilton-Jacobi equation for a massive particle in an external electromagnetic
field
3
2 Gii(S,i + Ai) (X,i + Ai) = m2
(4)
i,j=O
can be integrated by the method of complete separation of variables. I n the present
article all solutions for the case when the separation of variables can be accomplished
by the use of a complete isotropic set (defined in Q 1) are given.
1. Separation of Variables in Hamilton-Jacobi Equation
A theory of separation of variables is stated in detail in some original works beginning
with a classical work by Stackel [6]. More recently it was developed in papers [7-lo].
Therefore, we shall give only the most necessary information from theory, the proof of
which is presented in the works quoted.
Let us consider a particle of the mass m in a n external electromagnetic field defined
by the potential Ai in the pseudorieinannian space with the metric tensor G . .. The par??
ticle motion can be described by Eq. (4). If there is a coordinate system in which a
total integral of Eq. (4)can be written in the form
+
+
+
S = v0(x0) Fl("l) v2(z2) v3("3)1
(5)
the Eq. (4)is said to be solved by the method of separation of variables. The coordinate
system in which this is possible is called privileged. A necessary condition for complete
separation of variables is the existence in the space of a so-called complete set consisting
of threegeometric objects, i. e. of conmutating in paresKilling tensorsof a rank not higherthan 2. Let there are N Killing vectors Y k ( p = 1, . . . N ) in the set. Let us form a
3
2 GilYbY i . If
square iiiatrix
the rank of the matrix is lower than N the set is called
ij=O
isotropic. An isotropic set consisting of N Killing vectors is called a (N.l)-typeset, while
3
a nonisotropic one is called a (N.0)-type set (in this case rank
2
GiiYkYl = N ) .
i,j=O
In this paper we shall consider isotropic sets only. For the first time they were used in
a theory of separation of variables in Refs. [lo]. Einstein-Maxwell equations were not
investigated hitherto for the case of isotropic sets.
By the choice of the coordinate systeni one can get for the first N terms in Eq. (5)
to be linearly dependent of the first N coordinates. These coordinates are called ignorable. A privileged coordinate system defined by an isotropic set is such that coordinate hypersurfaces corresponding to one of the nonignorable coordinates coincide
with characteristic hypersurfaces of t h e Einstein equations. Such a coordinate is called
isotropic. Let us introduce the following notations. The independent variables of the
privileged coordinate systeiii are denoted by xi.The functions of one variable are denoted
by small letters with a subscript. The index meaning corresponds to the number of
variable which the function depends on. For example: vi = pi(xi). The constants are
denoted by small letters without any index. The rest of the functions are denoted by
capital letters.
The identity
1
4
Exact Solutions of the Einstein-Maxwell Equations
183
:3
takes place in a privileged coordinate system, where
Stackel matrix [6],
N-1
H,
-
=
(1 - 6:)CS,,)Z
+ s: z !eS,J,, +
p=O
-
2
2=
y;?@$
- is the
= Sf, 9:;’
N
N-1
3
2 hV”“,,S,,
p,q=o
+ 2 hgS,i+ it,,
i=O
h f = S ; h ; , v , , u = N , ..., 3 , z = 4 - N N , p , q , S s = 0 , ..., N - 1 ( O < N < 3 ) . Identity (6) defines the metric bensor Gii and the conditions imposed on the electromagnetic
potential :
All non-equivalent solutions of Eqs. ( 2 ) , ( 3 ) satisfying identity (6) are given in the
following paragraphs. Solutions are nonequivalent if they can not be reduced to each
other by coordinate transformation and gradient transformations of potential, or by
constant conformal transformation.
2. Type (1.1) Set
The coordinate xo is ignorable, 2, being isotropic. Identity (6) makes it possible t o
present the space linear element in the following form:
3
3
where, W , = t, - z, W , = z3 - z, W , = z, - z, A =
Therefore, one can present conditions (7), (8) in form :
2 (ovWv, A
v=
1
2
=
S,,W,,.
v=l
s
Ai
= A-l
2
j=O,v=l
GiiW,h$
(10)
3
2 [2h:hj! - h, - ( h : ) 2 ~ , , ] W=, ,0
V=l
There are only two non-equivalent solutions of system
(lo), (11):
3
I.
A , = 0,
A,
=
Wcl zh:W,.,
V=l
O, =
= h, = 0.
.
..
11.
~ 4=
0 a,,
A , = 0,
h, = 0 , h: = SiSfai.
One can easily make sure that
Y12 = T,, = T,, = 0.
As G,, = G,, = G,, = 0, it follows from (12) that
R,, = R,, = R2, = 0.
Eqs. (13) make it possible to bring the expression for the linear element to the form
[ll]:
as2 = - f l [ 2 d ~ o d ~-l AW-l d+
+ aF2Wdxg + U T ~ W ~ X ~ ] ,
(14)
where W = z, - t,,/1 = co2 co3. The formulae given allow to integrate system ( 2 ) ,
(3), (7), ( 8 ) completely. Results are given in Appendix I.
+
V. G. BAUROV
and V. V. OBWHOV
184
3. Type (2.1) Set
The coordinates xo, x1 are ignorable, x, being isotropic. Using identity (6) one can
represent the space linear element in the following form
+
ds2 = d G - ' ( f , d ~ ~ dx.,
+
-
dd~,),
- A(2dx0 dx, - o3dxg
+ +
where -G = ' 2 f 2 ~ 3 f . 3 ~Q,~ e3, A = f
Let's write down conditions (7), (8):
+
A,
= c,
i
=
p;: - 0,w3
=
G-'[2~,/3;
fJ1,
+
4 = -G-'(f28;
A,
A:
-
+ x3,
2 ~ 3
Ai
= 4 Z 2 ' x3),
83
- (72A
+
A3
A
=
6,
+
+ 6,.
d4)
(15)
= 0,
72)
AA'
+ (a2), + h, + h,].
0,
From Eqs. (17) it follows that
T,, = 0.
As G23 = 0, we have
R,, = 0.
19)
It appears that system ( 2 ) , ( 3 ) can be integrated completely without Ew. (17) but
using forniulae (15), (l(i),(18). The solutions obtained ensure complete separation of
variables in the Hamilton-Jacobi equation for a test neutral particle:
3
C GiiS,iS,j= 7n2.
a.i= 0
(20)
Provided that the solutions obey conditions (17) we can get solutions allowing complete
separation of variables in Eq. (4).I n this work solutions of both kinds are found. From
Eq. (19) it follows:
[ln(-GA-2)],3 = 0.
Thus, the functions G and A are connected by the correlation
G = -(bFrom here it follows t,hat the Eqs.
b,b;l
=
-2Id
(23)
2b3bj-'A,3A - (S2b2b3)' = 0
are the subsystem of system ( 2 , ) (3), (7),(8). This allows us t o establish that thesolutions of system (3) break up into nonequivalent classes defined by the conditions:
2A,,,
I
-A
-
+
A = X:
(&), 232 = (fzb,),
f kxa - 2%~'
U:
= b: =
(1/3)[x;
+
% ( d 2 ~ 3) ~ 3(82)4]
(the derivative with respect to the corresponding variable is dotted).
+
I1
A,, = f, = 0,
(a,), = p
24b3)2.
Eqs. ( 2 ) , (3), (18) can be now integrated. All the solutions are given in Appendix 11.
4. Type (3.1) Set
This set consists of three commuting in pars Killing vector fields consequently the
space admits a tree-parameter Abelian isometry group. The problem of integrating
Eqs. (2), (3) is trivial from mathematical point of view. Therefore, we confine ourselves
t o giving its solution.
Exsot Solutions of the Einstein-Maxwell Equations
186
The linear element of the space can be defined from the expression:
z, ax8 + exp (-t,) ax:
exp p3 - 2 dx,dx,
= -[exp
+ 2 11- exp 2 ( ~ , p),)
ax,
~ Z J
where
z3= E J [d- 2(i3
+ 6%- 21,) exp 2(e3 - p3)-
- (+3 exp %3 - e 3 exp %,I2 lI2 ax,,
exp 4v3- exp Ye3 9%)
A , = J exp (z34 e - 4)s) [k3I/exp 2(v3- e3)- 1
+ 6 1 2 ~ ~ x exp
- l (91,- t,) - i4] ax3,
Al=a3,
A,=A,=il=O,
E , E = fl.
1
+
Appendix I
I n this appendix all solutions of system (2), (3) are given for the case, when Eq. (4)
can be integrated by the method of separation of variables with the type (1.1)complete
set. The space linear element is defined by Eq. (14) and A, = A, = 0. The solutions
are defined by the functions fl, a,, a,, A,, A,, A, W, whose non-equivalent sets are given
below.
I.
a, = a, = 1,
A , = A , = A , = iz = 0.
1.
f, = 1, W = ch x2 - cos x,, W A , = ( k ex2) sh x, (n ex,) sin z,
+
A
= (rx,
COB
2.
+ q ) sh z, + (rx, + p ) sin z3+ x [ ( k + ex,)2 ch 5, + (n + ex3?
x,].
1, W = exp x,, A,
A = p sin x, qx, exp x,
f,
=
+
+ ( W 2 )exp - %,I.
3.
exp (-x,) sin (x,
+
+
A , = 5a($ - 4)
+ 4(kx, ex,)W-I,
+ 20a(kzX - ex!) t ( x t - xi) px, 95,
+
+
+
+
fl = W = 1, A , = 3a(d - xg) kz, ez,,
A = x [ 6 a z ( x ~+ x i ) 4a(kxg - ex;) (kz ea)xg
+
n(% - zg) 4-
5.
+ a) + bz,,
+ x[2kbxZcos (x, + a) + (bx3)2exp x,
=k
+ 4,
f, = 1, W = 4
A = 4x [&2(x; +)!x
+ 2k2 + 2e2].
4.
+ +
P
2
j,” = (4xaafi + 1Oef: + bf?),
A
= r(x$ - x i )
- e(x; + z!)
+ +
P%I
w = 4+ 4,
+ p x , + qx,.
- xi),
+ ex,,
6.
jf = x(ez + ka)fl + 2rff
11.
A = A , = A , = A , = 0, A, = ~ Wf, X
= 1. ~
W = x,, a; = 8A(X! + bg), a, = 1
W = x, - x,, a; = (-1y 8A($ + bz,2 + nx”), Y = 2 , 3 .
7.
8.
+ bf;, W = 1,
A = nxg - ( r + n)xg + px, + qx,.
A , = a(xg
A,
= kx,
,
186
V. G . BAGROV
and V. V. .OBUKHOV
Appendix I1
All solutions of Eqs. ( 2 ) ,( 3 )are given here for the case, when Eq. (4)can be integrated
by the method of separation of variables with the type (2.1) complete set. Moreover,
the solutions allowing complete separation of variables in the Hamilton-Jacobi equation
for a neutral particle (20) are given. The space linear element is defined by formula (15):
A , = a, f2A1, -4, = 0. The solutions are defined by the functions a,, b,, b,, w,, f,,
A , A , A,, A,. All non-equivalent sets of these functions are given below. By asterisk
we denote the solutions allowing complete separation of variables in Eq. (4).I n brackets
we point out how one should select the functions @$ 3/, h,, h,, y, to satisfy Eq. (17).
Not all functions necessary for the definition of the solution can be given for the formulae numbered with asterisk. Tn this case the missing function5 should be taken from
the formulae nunibered with the same numbers but without asterisk. The solutions
undenoted h p asterisk ensure separation of variables in Eq. (20).
+
48a; = 4%; = il[16x;
+ 24(f,b,~,)~- 3(f2b2)4]+ 48(kx,
+
6, x:, 46, =
f, = ~ c x , , b, = 1, a, = OJ, = A
sina] . [c(xg c2)]-l, A, = 0 ,
A
-
=
+
h, = -(2$
1 = 0,
+
C2)A1/9,,
6,
f2 = x;,
=
= 0,
A,
=
{@!;= 0 , 8,
p(2cx3 cos a
= Al(x;
h, = y, = O}.
I, U, = -2px2 sin a , A = (xg
+
+
2xp2),
+ (c2 - xg)
+ c2)bT2,
+ x?$bY2,
bT2, A , = 2p(x3 cos a x2 sin a ) (xg x;)-l, A , = 0,
{/3, = x;@ = 2 p x Z b ~cos~ a , h, = -4(px3bc1 cos a)2, h, = y 2 = 0).
il = 0, fl = 4(rfl ~ f , ) ,k = 0, p-2 = -2x, E = 0,1,
OJ,
=
+
+
6, = I$ = r & l , o, = 2spf,y2 sin a,
b, = f;',
A =fix$ + (%I2 + E ) X & w, = x$ + 2rx& A, = p(f2q2d)-1
x [2x3cp, cos a (9;- xg - 2~rf,d- ~ dsin
) a],
A, = pry7,A-l [291,x, cos a
(pig - x; - E A )sin a ] ,
+
E
+
{PI = 2px3(x; + r ) cos a , 8,
= 1,
=
2px3 cos a , h, = -4(px3 cos a)2,
h, = -(ro2)2, Y 2 = -ro2f2&
E = 0 , {PI = f2Ald2,
h3 = - (f2AA1)2, 8, = h, = y 2 = O } .
f2 =
b,
0,
=
A
1,
/ z2(rx2+ p2(22x + e)]b,(x)-*dx,
2
.
=
4x
0
/ bg2((x)dx[ 7b,4(t)dt[4x(rt2+ ep2 + 2ttp2)Z + 2p2(e+~ t ) ~ b i (+t )Q)] ,
z3
w3 = 4%
0
A,
=
0
2pxF1 sin a2
e
a,
A
=
-7~4b-2
3 3
/9!
=
-tP3,
a = nl2,
A,
= 2p sin a,
7 [(b$)a (-
1
0
X
+ t) + '
3
1dx.
3
a, = 2pe-1 sin ex,,
aB = ex,,
= a,
+ to,,
=
{/3!
h, = y, = O}.
0,
0, =
2px2 cos a,
,
013=
- t A , A , = 0, {/9 - 2pxiby2 sin a,
h, = - 2 ~ ~ ~ 1sin8 a, , h, = y, = 01
= A,
AA,, 8, = 2 p x ; b ~ ~ h,
, = -(2p~,by~)~,
2
+
Exact Solutions of the Einstein-Maxwell Equations
5.
f,
= 0,
b; = rb;
+
+ b,
(r,
187
= lc = p = A = 0,
~ ( 2 A r x : 2tAx3 - 3xe2), A , = a,,
+
A,
=
=
LC)~
-3(2A2~$)-~.
3(Az3)-l l/e2 - (u2bF1)2
a2 = (a/r)I/rb$
b, {@ = 3(Ax3)-l y e 2 - u 2 , 8, = h, = 0,
h, = -3a2(Ar2b;)-l(rb;
b ) , y, = 3aZ(Abg)-1}
U , = U X ~ , b, = x2, {@ = A,, /!I3 = 3a/A, h, = 0 , h, = -33a2%/A,
Yz
= 01
U, =
7.
+
0,
{/!I;
= A,,
8, = h,
= h, = y2 = O}.
(ez
b2 = 6, = 1, b, = x,,
+ b)dx
t
0
A,
=
(a3sin a2
+ g2)/)/-2ilx,
1
sin a
62
= )/-~(2xe2)-, sin
+ 2e1nx3 .
2ex,,
a,
=
2exz, g2e = t sin 2ex,.
cos a, a, = a, e = 0, g,
A = = - -t(AG)-',. CO, = AX,)^, A, = 0,
{/I3 = - ~ ~ ~ ( - 2 W x ) - ~ sin
/ ~ aa,, & = -t/?,A-l,
h, = y, = 0).
= 1/-22izx-'
X,
=
2tx2 cos a.
h,
= -(p3z3)2,
V. G. BAQROV
and V. V. OSWHOV
188
{@: = A ,
9.
h,
-@;x:,
h, = yz = O}.
L = a, = 0, p = 1, b,2 = (nb, + r)b& b, = d,, b, = x,,
A = z/xi, w, = t2xc2 + e x ( h x,), - rx$/4 b In q, A , = a,,
=
+
A,
9.*
10.
+ AAl, b3= a 3 ( 4 v - 2 h ) - ' ,
=
v e b , - a g In x ,
-
0, {& = veb, In x,, 8, = h, = h,
L = a, = 0, b, = 1, A = zx,, (0, = rxg nx,,
a2 = n = r =
+
= y, = O}.
A,
= a,,
lo.*
g, = --ta,,
{@: = @, = h, = 0 , y2 = a,6$/b& h, = -62a2b-2}.
2 2 2
In conclusiop we note that the solutions obtained are related to Wyman-Trollope
isotropic fields classes [ 1 2 ] with A = 0. These classes are known to be defined up to any
arbitrary solution of the second-order linear differential equation. I n particular, the
first-class solutions of Appendix I are planewave ones. As for the non zero A-term solutions we could not as yet to accertain their belonging to any of the well-known classes.
References
[l] CARTER,
B.: Commun. Math. Phys. 10 (1968) 280.
[a] IWATA,
G.: Natur. Sci. Rept. Ochanomizu Univ. 20 (1969) 2.
J.: J. Pliys. Lond. A: Math. Gen. 10 (1977) 5.
[3] COLLINSON,C. D.; FUGERE,
[4] DEYIANSKI,
M.; FRANCOVIGLLA,
M.: J. Phys. Lond. A: Math. Gen. 14 (1981) 173.
E.; MILLER,W.: J. Phys. Lond. A: Math. Gen. 14 (1981) 1675.
[5] BOYER,C.; KALNINS,
[GI STPCREL,P.: Math. Ann. 43 (1893) 637.
[7] SWPOVALOV,
V. N.: Sib. Mat. J. XX (1979) 6 , 1117.
[8] SHAPOVALOV,
V. N.: Izv. W Z Fiz. 1978, 9, 18.
M. S.: Prikl. Mat. Mekh. 27 (1963) 6.
[9] IAROV-IAROVOI,
[lo] BAGROV,
V. G.; MESHROV,
A. G.; SHAPOVALOV,
V. N.; SHAPOVALOV,
,4.V.: Izv. VUZ Fiz. 1973,
11, 66; 1973,12 45; 1974, 6, 74.
[I 11 OBUKEOV,
V. V. : Deponirivano VINITI, N 2640-77 Dep.
[12] WYMAN,M.; TROLLOPE,
R.: J. Math. Phys. 1965, 6, 12.
Bei der Redaktion eingegangen am 25. Februar 1981, revidiertes Manuskript am 21. Juli 1983.
Anschr. d. Verf.: Prof. Dr. M. BAGROV
and V. V. OBUKHOV
Institute of High Precision Electronics
Siberian Department of the Academy of Sciences of USSR
USSR - 634056 Tomsk - 55, Academic pr. 4
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