A Class of Transcendent Solution to Ernst System

LETT:E:R~E AL 2qL'OVO CI~E~,-TO VOL. 44, X. 2 16 Settembre 1985

A Class of Transcendent Solutions to Ernst System.

~Tuclear .Engineering Department, King Abdulaziz University t2.0. Box 9027, Jeddah, Saudi Arabia

(ricevuto il 12 Aprile 1985) PACS. 04.20. - General relativity.

Summary. - We present a method t h a t generates a class of transcendent solutions to E r n s t system in Einstein-Maxwell theory of general relativity.

I n an earlier paper (~) we presented the general similarity integral to the E r n s t equations in Einstein-Maxwell theory if), where the similarity variables were arbitrary harmonic functions in cylindrical co-ordinates. I t has been shown later (3) t h a t b y em- ploying a particular independent variable the basic differential equation transforms into the type of Painleve's fifth transceudent, resulting therefore in a transcendental solution to Einstein-Maxwell equations. I n this paper we shall give criteria about how this class of transcendental solutions can be increased in number.

The powerful technique t h a t aided in complete similarity integral was based on the harmonic mappings, ~ : M -~ M', where

(1)

(2)

M : d s 2 = g~zdxzdx~= d ~ 2 + d z 2 + ~2d~v2 ( / ~ , v = 1 , 2 ) , M' : d s ' 2 = g,tB(~) dq)~ d~b B (A, B ~ 1, 2, 3, 4), ds'2 = ($~ + V~-- 1)-2{d~d$( 1 -- ~ ) + dv d~(1 -- ~$) + ~ / d ~ d$ + ~75 d~e d~/}.

Upon variation of the energy functional ( ~ the action)

f ~q~ c~q~B

(3) E [ r = gAR(q~) ~X~ ~X~ g'~]gl~d2x'

M

(1) 1VI. ttALILSOu Lett. Nuovo Cimento, 37, 231 (1983). (2) F . J . ERNST: P h y s . Rev., 168, 1415 (1968).

(3) ]3. LEAUTE a n d G. MARCIL~ACY: LetL Nuovo Cimenlo, 40, 102 (198~:). 88

A CLASS OF T R A N S C E N D E N T SOLUTION- TO E R N S T SYSTEM ~ 9

one obtains the pair of E r n s t equations (Ernst system)

(4)

{

I n ref. (~) we had integrated these equations under the assumption t h a t ~ is only a function of v which satisfied V~v = 0, i.e. v is harmonic. Employing nonharmonie variables transforms the problem n a t u r a l l y into a much complicated one (a). We show now t h a t when ~ depends on two independent harmonic functions the problem can be stated as a

Theorem. L e t @a be parametrized b y two harmonic functions v a n d ~ such t h a t

On(v, ~) is geodesic with respect to each of them independently. The field equations from the action principle then are satisfied, provided the constraint condition Vv- 9 V~ = 0 holds.

Proo]. The variational principle ~E[O] = 0 yields the covariant equation

(V/~ g~ 08xx~v

)

~ + ~,~ a -- O.

Using the fact now t h a t # a = #a(v, ~), one obtains

+ ~ j Wv + ~ - W~ + g,~(v,~,~ + v,~,~) ( ~ + ~ ~ - ~ ] 0 .

Harmonicity a n d the geodesic condition makes all foregoing expressions vanish, whereas the last term is made vanish b y virtue of the constraint

(5) Vv. V~ = 0 .

Let us note that Kerr solution can be reformulated in terms of two harmonic func- tions in conform to the above theorem. Indeed, in prelate spheroidal co-ordinates x, y, eq. (5) reads

(x ~ - 1 ) v x ~ + (1 -- y 2 ) v ~ = O,

a n d is solved b y the harmonic pair of functions v = t g h - l x a n d ~ = t g h - l y . Although the expression for OA(V, 3) can be more general in terms of v a n d ~, we shall proceed b y making a particular, separable dependence, namely, On(v, ~ ) =

= ]a(v)exp [ic~]. I n this choice c is an arbitrary, real constant a n d the results of

ref. (1) will be obtained in the limit e = 0. I n order to formulate our objective i n a eovariant language, we consider the harmonic map, Ca. M0_+ M', between the mani- folds, M 0 : ds~ = exp [2v] dv ~ + d~ 2 + exp [2v] d~ 2, a n d M ' , given b y expression (2).

90 M. HALILSOY

Our f u r t h e r p a r a m e t r i z a t i o n will be

{ ~ ; = ~ = y c o s T e x p [i(~ + c~)~ , (6) r V = Y sin W e x p

[i(fl

w h i c h casts t h e action f u n c t i o n a l into

(~2~ ~

q)4= ~ ,

(7)

,

]

+ y2(cos2 T . s + sin 2 T . fl,2) _ 4 y4 sin e 2 T ( a ' --

fl,)2 + c 2 y2 e2, ,

N o t e t h a t ~-dependence w a s h e s o u t f r o m t h e v a r i a t i o n a l principle, its overall effect being to b r i n g i n t o t h e action such a c - d e p e n d e n t t e r m . I t is r e a d i l y o b s e r v e d t h a t Y~, a n d fl e q u a t i o n s r e m a i n i n v a r i a n t a n d correspond to eqs. (10), (12) a n d (13) of ref. (1), r e s p e c t i v e l y . T h e y e q u a t i o n modifies due to t h e c-term, a n d reads as

[

k~] ' c2y(1 + y 2 )

(8)

y,,_ 2yy~ '2

_ + (y2_

y(a + b) ~--

-,-

y 2 _ 1 y 2

1

'

where, t h e c o n s t a n t / ~ , a a n d b arc t h e s a m e c o n s t a n t s b y i n t e g r a t i o n s a d o p t e d in ref. (1). C h a n g i n g v a r i a b l e s b y v = lncr a n d y 2 = y , t r a n s f o r m s this e q u a t i o n i n t o

2

[

k~]

y ' ~ + ~ Y ( ~ - ] ) ~ (a+b) ~-~-~ +

F +

=

w h i c h is identified as a p a r t i c u l a r P a i n l e v e ' s fifth t r a n s c e n d e n t (5). Once Y (and t h e r e f o r e y) is k n o w n , all t h e r e m a i n i n g W, ~ a n d fl e q u a t i o n s can b e r e d u c e d to q u a d r a t u r e s , as f u n c t i o n s of y. F r o m t h e o n e - t o - o n e c o r r e s p o n d e n c e b e t w e e n t h e E r n s t s y s t e m a n d E i n s t e i n - M a x w e l l ' s t h e o r y , t h e foregoing solution p r o v i d e s a t r a n s c e n d e n t a l solution to t h e l a t t e r .

T h e c o n s t r a i n t c o n d i t i o n Vv. V~ = 0 ( = ve~ e + v ~ ) a d m i t s t h e simplest solution as v = log e, ~ = z, w h i c h corresponds to t h e solution of ref. (a). A n e w solution is g i v e n b y t h e choice of h a r m o n i c functions,

(10)

v = (0 ~ + z 2 ) - 8 9

~ = l o g

~ +

1 + 0 ~ ] j "

W e r e m a r k t h a t it is n o t clear a b o u t h o w large t h i s set of h a r m o n i c f u n c t i o n s satisfying t h e a b o v e c o n s t r a i n t is. I n t h e c o m p l e x p l a n e it is w e l l - k n o w n t h a t this reduces to t h e class of h a r m o n i c a n d c o n j u g a t e h a r m o n i c functions. F u r t h e r , t h e K e r r s o l u t i o n does n o t a d m i t such a t r a n s c e n d e n t M extension, since its ~b-function is n o t in t h e s e p a r a b l e f o r m w h i c h we h a v e assumed.

I n conclusion, we n o t e t h a t t h e foregoing p r o c e d u r e to o b t a i n t r a n s c e n d e n t a l solu- tions can d i r e c t l y be a d o p t e d in a n y c o m p l e t e l y i n t e g r a b l e systems t h a t are expressible in t e r m s of h a r m o n i c maps. T h e self-duM S U~ g a u g e field p r o b l e m is one such e x a m p l e .