Pram~na, Vol. 14, No. 4, April 1980, pp. 253-262. (g) Printed in India.
Some solutions of the Einstein-Maxwell equations in general relativity
P V B H A T T a n d L K P A T E L *
Department of Mathematics, Saurashtra University, Rajkot 360 005, India
*Department of Mathematics, Gujarat University, Ahmedabad 380 009, India MS received 19 November 1979; revised 3 March 1980
Abstract. A solution of the Einstein-Maxwell equations corresponding to source-free electromagnetic field plus pure radiation is obtained. The solution is algebraically special. A particular case of the solution is considered which encompasses many known solutions. Among them is a radiating Ruban metric.
Keywords. General relativity; algebraically special solution; Einstein-Maxwell equa- tions; electromagnetic field.
1. Introduction
There are two types o f solutions o f the Einstein-Maxwell equations in general rela- tivity, n a m e l y algebraically general solution and algebraically special solution. In spite o f the fact that an exact gravitational solution radiating f r o m a finite source must be algebraically general (Sachs 1961) the problem o f obtaining algebraically special solutions has received much attention due to several reasons. One r e a s o n is t h a t the Schwarzschild solution, the K e r r solution ( K e r r 1963) and the N U T solution ( N e w m a n e t a l 1963) are familiar members o f this class. In the present paper, using the complex vectorial formalism as formulated by Cahen e t a l (1967) we obtain some algebraically special solutions o f the Einstein's equations plus pure radiation which are equivalent to (Bkatt and Patel 1978)
E , , ~ = - - 2 F , , F ~ 8 8
~
(1)d F + = O. (2)
In (1) and (2), Er~ is the hermitian tensor corresponding to the trace-free part o f the Ricci tensor a n d / ~ is the density o f flowing radiation. F + -~ F p Z p is the self-dual part o f the electromagnetic field tensor, Z p forming a basis for the complex 3-space o f self-dual 2-forms. T h e bar denotes the complex conjugation. A detailed a c c o u n t o f this formalism is given by Israel (1970). We shall use his notations. In particular, the G r e e k and the first h a i l o f the Latin indices will range from 1 to 4 and second h a l f o f the L a t i n indices will range from 1 to 3.
253
254 P V :Bhatt and L K P a t e l 2. The metric and the M a x w e l l equations
We consider the metric (Vaidya et al 1976)
d s g - - 2 ( d u + g sin a dfl) d x - - 2 L ( d u + g sin a d[3)9--M 9 ( d a 2 + s i n 2 adfl2). (3) H e r e L a n d M are functions o f u, a a n d x and g is a function o f ~ only. W e use u, a,/3 a n d x as the coordinates. Introducing the basic 1-forms
01 : du + g sin ~ dfl, %/509 = M ( d ~ + i sin adfl),
04 : d x - - LO !, 03 : 02, (4)
we c a n express (3) as
ds 9 = 2(0104 -- 0903). (5)
Using (4) we c a n o b t a i n dO", which by using the defining expressions Z 1 - - 0 3 A O ~, Z 2 = 0 1 A 0 2 , Z 3 = ( 1 / 2 ) ( 0 1 A 0 4 - 0 2 A 0 3 ) ,
f o r Zp, will give us dZ~'. Using these expressions f o r d Z p, C a r t a n ' s first equations o f structure
dZp = ( 1 / 2 ) • "pmn ¢~m A Z n
will then determine the connection 1-forms %, E p"*" being the Levi-Civita's p e r m u t a - tion symbol. The detailed calculations o f d Z r a n d % are given in Bhatt a n d Patel (1978) a n d we shall reproduce the expressions for % for r e a d y reference:
~1 - - 2 [ - - ( M x / M ) + i (f/M2)]O 2,
~9 : - - V / 2 [ ( L ~ / M ) + ig(Lu/M)] 01 + 2 [ ( 1 / M ) ( M u + L g x )
- - i(Lf/Mg)] 0 z, (6)
~r 3 : : - - 2[L:, + i(Lf/Mg)]o 1 - - X / 2 [ F + i g ( m J M ) ] O 9 +
~ / 2 [ F - - i g ( M , / M ) ] 03 -k 2i(f/M2)O ~.
H e r e 2 f = g , ~ + g cot a and M e F = M,~ + M cot a, and suffixes denote partial derivatives, viz., M,~ = OM/~a, etc.
T h e absence o f terms involving 08 and 0a in 0.1 indicates t h a t the congruence k"
o f null tangents is geodesic as well as sb.ear-free.
One c a n n o w use % given by (6) a n d Cartart's second e q u a t i o n o f structure Z' e = d % - - (1/2) %,,, ,r" A ~"
Einstein-Maxwell equations in general relativity
255 t o obtain the curvature 2-forms Z' o. The expressions f o r Z' o are recorded in B h a t t and Patel (1978) a n d are not repeated here. These expressions f o r Z'p along with the identity~,
= c , . z o - (1/6) R ~,,~ z , + E,~ Z . .will then determine the h e r m i t i a n tensor
Ep;~,
the curvature scalar R and the complex- valued trace-free symmetric tensor C~q which correspond to the Weyl tensor. Ep?and R are given by El" ~ = E t ~ = E I ~ = E27 :
+
Ea~ - - +
E s ~ =
+
R : --Ea~ + 4L E~7 + 4Lxx +
(2/M ~)[4MMx~ + 4MMx Lx
+ (8Lf2/M2)].
(7b)In the above equations, the variable y replaces the variable a, the defining relation (2/M) [M,., - -
(p/MS)I,
E~T = 0,
E3 T = ~/-2 (g/M) { (Ux/U) r -- (f/U2)~ + i ((Mx/M)~ + (f/U2)y~],
L 2 E t 7 + (1/M 2)[gZ (L~ +Lyr) + 2fLy + 2L~ MM~
4L M M ~ -- 2L~ MM~ + 2MM~],
E~] = ~ 2 LEa- i -- i~¢/2 (g/M) [Lx + ( M J M ) + i (2f L/m2)]
~v/2 (g/M) [Lx + (Mr~M) + i (2f L/M~)],.,
(7a) ( 2 / M 2){g~ [(mdm)~ + (MJe)d - - 2M~M~ - - 2L M~
(2f My/M) + (6L fZ/M2)} + 2Lxx.
(8) EI~ =
--2F~
F2 that either We take F 1 = 0 and assume beingg d~ = dy.
Since E I ~ = 0, it follows from the field equation (i) F l = 0 or (ii) f 2 = 0 or (iii) F 1 = 0 and F2~---0.
the self-dual 2-form F ÷ as
F ÷ = C z ~ + ~ z 3, (9)
where 4' and ~b are complex-valued functions o f u, a and x. Since F 1 = 0, it follows f r o m the field equations (1) that E 17 = 0 and E I ~ = 0. These equations involve only one u n k n o w n f u n c t i o n M. They can be solved to get
( l o ) M ~ _-- ( f / Y ) (X ~ + Y~),
where X, = - - Y , , X , --- Yu, Xx = - - 1 , Y,~ ----0. (11)
256 P V jBhatt and L K Patel
With M given by (10) and (11) and Z' 1 given in Bhatt and Patel (1978) we have verified that Cxl ---- 0, C13 = 0. Therefore, the space-time given by the metric (3) is algebraically special.
Using F + given by (9), M 2 given by (I0) and (11) and dZP listed in Bhatt and Patel (1978) we have verified that the Maxwell equations (2) imply the following equations for ~ and ~b:
~bx - - 2~b [ - - ( n x / M ) + i (f/M~)] = O, (12)
~bu q- i ~b, ---- 0, (13)
2 V 2 (¢M)x -- g (i ~b. -1- ¢,) = 0, (14)
V 2 ( g / M ) (--i ¢. ~-by) + "V/2 ¢ [ F - - i ( M u / n ) ] - - ¢ . --L~b,,
- - 2~b[(MJM) - - L ( - - (M,,/M) q- i (f]M2))] : O. (15) We can use (10) to solve (12) for ~b. The function ~b is given by
~b -= K ( X - - i l O - ' , (16)
where K is a complex function of u and y. With ¢ given by (16), equation (13) implies that
e. -= hy, ey -= --h., K = e+ih. (17)
Using ~b given by (16) and (17), equation (14) gives us the following form for the function ¢:
¢ -: (ig/V'2M) [K](X--i Y)],. (18)
Finally, using all the relevant results of this section, equation (15) implies that
K -= H Y , (19)
where H is a complex function o f y only.
where
3. The remaining Einstein-Maxwell equations
We set R = 0 and use M 2 given by (10) and (11) to determine the following form of function 2L:
2L = - - ( Y J Y ) X + 2G + ( X 2 + r~) -1 ( 2 F X + 2E Y), (20) 2G -~ ( r / f ) [(1/2)g 2 V 2 log (fly) q-f, --1 - - 3 f ( rj,/r)],
V 2 -~ O~/Ou ~ + 02/Oy 2. (21)
E i n s t e i n - M a x w e l l e q u a t i o n s in g e n e r a l r e l a t i v i t y 257 E and F a r e undetermined functions o f u and y. With the f o r m (20) o f 2L and with M given by (10) and (11), the field equation Ea~ : --2Faff 3 then implies that
E : - - 2 G Y - - Y Y y - - ( 1 / 4 Y ) KY,. (22)
It then follows f r o m E a ff : --2FaF~ that
F . = - - [ E q - ( K K / 4 Y)],, F, = [ E q - ( K K / 4 Y)].. (23) Using E~2 = - - 2 F ~ F 2 - - I ~ , a straightforward but lengthy calculation gives the radia- tion density/z. The expression for/~ is recorded in the appendix.
The corresponding electromagnetic field tensor Fa~ can easily be obtained:
Fa~ = g ( x ¢ l q - y ¢ 2 ) , , Fla = - - g sin a ( x ¢ l + y ¢ 2 ) ,
F 1 4 = - ¢1~ F23 = g2 sin a(x~bl+y¢2)y - - M 2 sin a ~b2, (24)
F g 4 ~ - 0 , F3a = g sin a ('1, ¢ = ¢1+i¢2 •
Here we have n a m e d the coordinates
X 1 ~--- U~ X 2 = o,~ X 3 = fl~ X 4 = X.
We have so far worked with the general line element (3). A case f = Y has been treated in B h a t t a n d PateI (1978). It has been shown that the r a d i a t i n g D e b n e y - K e r r Sehild (1969) metric and the radiating Brill (1964) metric can be incorporated in (3) with M 2 == X ~ + y2 and 2L given by (20). In the next section we shall consider one more case which seems to be o f physical interest.
4. The case f ¢ Y, Y = Y(y)
We consider a case in which Y = Y ( y ) . I t then follows f r o m (11) that
Y = - - a y + b, X = au - - x , (25)
a and b being constants o f integration. N o additional constant is added in X because such a constant can always be incorporated in x coordinate.
Since Yis a function o f y ordy, equations (21) and (19) imply that G and K a t e also so. Consequently (17) would imply that K is a complex constant and E given by (22) will be a function o f y only. It then follows from (23) that
E - } - ( K K / 4 Y ) : - k y q- I, F = - - k u -- m , (26) k , / , m being constants o f integration.
We now introduce a variable and a function p as follows:
( f ] y ) l / ~ d a = dO, ( f ] y ) z / s sin a = p(O). (27)
258 P V 2~hatt and L K P a t e l Then we find f r o m (21) that
2 G = (P oo/P) q-- 2a.
This expression for 2G along with equations (22) a n d (26), then implies that
P oo q- [a --b (1/Y) ( k y -b l)] p ---- 0. (28)
We n o w consider a ease in which a -b (1/Y) ( k y ÷ l) is a constant, say ~, where
= 1, O, - - 1. It then follows that
k -~ a(a - - •), 1 = - - b(a - - O. (29)
E q u a t i o n (28) can then be integrated. We get
I: ° I
p ~ when • = 0 (30)
t, sinh 0, eosh 0, e 0 - - 1,
The expressions for 2G, E -~ ( K K / 4 Y ) and F then become
2 G - - - - 2 a - - • , E %- (K/~4 Y) = - - (a - - ~) Y, F = - - a ( a - - O u - - m .
(31) The foregoing expressions for 2G, E and F determine 2L as
EL ---- • - - (X ~ + y~)-i ( 2 [(a - - E)x + m ] X q- (1/2) (e ~ + h2)~ -, (32) The density o f flowing radiation/~ and the electromagnetic field F~B in this case are given by
t.~ ~- - - 2a(a - - e ) / ( X 2 -b Y~).
F12 = ag~b 2, F14 = ~1' F~4 = O,
where ~b --~ ~b 1 q- i¢2 and ~b is given by (16).
Fin = ag sin a~b l,
F2z ---- - - sin ~(ag ~ + M~)4,e, Fa4 = g sin ~ @1,
(33)
(34)
In view o f the relations (8), (25) and (27), we have the following relations between Y and g sin e:
2p Y -~ (g sin ~)0' where p is given by (30).
differential equation:
P YO = - - ag sin ~,
These relations together then give
(35) the following
YOO -k (pO/p) Yo -k 2a Y = O. (36)
Einstein-Maxwell equations in general relativity 259 With p given by (30), (36) is a differential equation for Y. One can solve it and use Y in (35) to get g sin ~. M ~ and 2L are given by (10), (11) and (32). Thus all the metric potentials are determined. The case ~ = 1 and a ¢ 0 has been discussed earlier by us (Bhatt and Patel 1978). In the next section we shall give some o t h e r particular
cases.
5. Particular cases
We cite several particular cases here. In all the cases we shall give the forms o f the line elements along with the corresponding values o f the density o f flowing radiation /~ and the electromagnetic field tensor F F
5.1. Case (i): a : 0 The metric in this case is
ds ~ =- 2 ( du q- g sin adfl) d x - - ( x 2 q- b 2) (dO ~ q_ p2 ((9) dfl ~)
-~- ~ - - (X ~ -~ bZ) -1 [2mx q- 2~b~--(1/2) (e 2 -k- hZ)]) " (du q - g sin adfl) 2, (37) where p (0) is given by (30) and g sin a is given by
g sin ~ = bO ~ when E =
2b cos hO, 2b sin hO, 2be o
(38)
/~ = O. (39)
F14 ~- ~bl, Fza = - - (x ~ + b 2) p( O) ~bz, F ~ = g sin a ~b 1.
(40)
where g sin ~ and ~b =~b 1-1-i ~b z are given by (38) and (16), respectively. The metric (37) along with (38), (39) and (40) is the same as that obtained by R u b a n (1972) with ?t = 0 there.
5.2. Case (ii): a = E
The subcase ~ = 0 is discussed in case (i). Therefore, we shall give the results for
= 1 and E = --1. The metric in this case is
ds 2 = 2 (du-[-g sba ~ tiff) d x - - ( X2 + Y~) (dO2-~ p~( O) tlff ~)
_ { E _ ( X S _ k y~)-x [ 2 m X + ( l / 2 ) (e2-}-h2)] } ( d u + g s i n ~dfl) 2, (41) where X = ~ u - - x and p (0) is given by (30).
by
Y = ~ k cos 0
k cos hO, k sin hO, ke o
The functions Y a n d g sin a are given I 1
when e = , (42)
-1
260
and
P V Bhatt and L K Patel
g sin a = ~k
s i n ~,0
( lc
sin h ~ 0, k cos h 2 O, ke ~o ---0,Fl~. = e g ~'2 ,
F14 =~1 F24 = 0
when e = I - - i '
Fin = ¢ g sin a $1,
F~3 = - - sila c~ (e g2 + X 2 + yz) $2, F ~ =-g sin a $1,
(43)
(44)(45)
F24 = 0 , F ~ = - - ( l / a ) e-0 YO 1/11.
5.4. Case (iv): e = 0 , p(O)--O, a # O .
The space-time geometry of this case is described by the line element ds'~=2 ( d u - - ( l / a ) 0 Yo d~) a x - ( x ~ + Y~) (dO~+O ~ d~ ~)
(X2 + y2)-I [2 ( a x + m ) X + (1/2) (e2+he)] (du--(1/a) 0 I1o d~) 2,
(50)
where g sin ~ and $ = ~ b 1 + i ~z are given by (43) and (16), respectively. Here it should be noted that a and 0 are related byda/sin ~ = dO/p (0).
The metric (41) is an electromagnetic generalisation o f Kinnersley's (1969) vacuum metrics (cases IIA, liB, IIC, IID with l = 0 there).
5.3. Case(iii): e : - - - l , p = e °, a # O . The metric in this case is
ds ~ = 2 (du--(1/2) eOy o d[3) d x - - ( X ~ + Y~) (dO2-t-e 2o d]3 2)
+ {1 + (X z + yz)-i [2rex -I- (1/2) (e ~ -5- h2)]~ - (du - - (l/a) eOg o d[3) ~, (46) where Y = e - 0 1 2 [Ae aO/~ + ,!~e-aOIZ] , q = (1 - - 8 a ) 1/2
X = au - - x, (47)
= ( a + 1) x + m ,
t ~ ~--, --2a ( a + l ) / ( X z + y2), (48)
r ~ = - - I7o ~2, F13 = --eO I1o 4'1,
/714 = ~ 1 ' F23 = - - e 0
[(l/a) ( Y ~ + X 2 + Y~)I ¢2, (49)
Einstein-Maxwell equations in general relativity 261
where Y = b J o ( a/2aO) -~ c Yo ( V~2aO), (51)
J0 and Yo being Bessel functions o f zero order o f the first and second kind, respectively. The radiation density ~ and the electromagnetic field tensor F~F in this case are given by
= _ 2a2/(X ~ + y2), F ~ = - r'0~ ~,
F ~ = ~ , F2t = 0,
where ~b -- ~b 1 q- i~b 2 is given by (16).
(52)
F13 = - 0 Y 0 ~ ,
F23 = -- O[(1/a)Y~ q- X ~ q- Y2]@2, (53) r a = - ( 1 / a ) O Yoq'~,
When c -- 0, it can be verified that as a --> 0, (1/a)OY o ~ - - bO ~, and so the metric (50) will go over to the metric (37) with E = 0.
Since Y is singular for a = 0, it follows that the metric (50) is also singular for a -- 0. Thus, the metric (50) is a radiating Ruban metric for the case when the cosmological constant ~ vanishes.
Acknowledgement
The authors would like to thank Professor P C Vaidya for stimulating discussions and the referee for his valuable comments.
Appendix
The expression for density o f flowing radiation t~ is given as follows:
- - t* = (ge/M2){ V 2G -- ( f/M2)v~(KK/4 Y ) - - 2(M:,/M)r (K~2/4 Y),, q- 2 ( M ~ / M ) , (KK/4 Y)r q- ( X z q- y2)-1 [(e~ -t- h~) -I- 2KJ~((1/2) (X ~ + y~)-i (X~ q- g2) _. (log V ) t "2 -~- y2), (log v e 2 + h2).
÷ (tan -1 (h/e). (tan -1 (Y/X)).)]} ÷
(X~ q_ irz)-x [ _ y2( y . / Y ) u --2Fu -I- 3F( Y J Y) -I- 2 YGj, -- 3 Y~
+ 2 YY.~] q- (X 2 -t- y2)-2 [ _ 2 Y2(KK]4 Y), q- 2 X Y ( K I ~ 4 r ) . -- (K/~4 Y) ( X Y . - - YX.)I.
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