fields are always reducible to static electrovac fields when the rnetric elements ... tic fields are investigated. It is found .... Roy. Soc. (London) A244, 524 (1958). 3.
Revista Brasileira de Fsica, Vol. 8, N? 2, 1978
Cylindrically Symmetric Stationary Fields in General Relativity M.M. SOM and N.O. SANTOS Departamento de Frsica Matemática*, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ
Recebido em 21 de Novembro de 1977
It is demonstrated that cylindrically symmetric stationary electrovac fields are always reducible to static electrovac fields when the rnetric a n d g are linearly connected. elements CJ tt' gt4 44 Demonstra-se que campos eletrovac estacionários com simetria cilindrica ca são redutiveis a campos eletrovac estáticos sempre que os elementos da métrica g tt, gt4 e g4@ forem relacionados linearmente.
1. INTRODUCTION In an earlier work, ~ewis' first examined the stationary cylindrically symnetric vacuum field and obtained a special class of solutions of Einstein's equations, which are linear combinations of the static cylindrical ly symmetric solutions2 wi th constants coeff icients. Later Som e t aZ.3 extended his work to a stationary axially symmetric case and presented a class of solutions which are linear combinations of the static stuaxial ly symnetric curzon4 f ields. Recentl y ~rehland', Som et.aZ. died the general solution of Einstein's equations for stationary cylindrically syrnmetric vacuum field and found that these solutionsarealways reducible to the static f ield by a sui table coordinate transformation. In fact they obtained the coordinate transformations which diagonalize the stationary cylindrically symmetric line element. It seerns worthwhile to the present author to study whether this property ---
*
-
Postal address: Ilha do Fundão, Cidade Universitária, Rio de Janeiro
RJ, Brasil.
,
i s a general f e a t u r e o f t h e c y l i n d r i c a l symmetry w i t h respect stationary fields.
to
other
I n t h e present work some s p e c i a l s o l u t i o n s correspon-
d i n g t o t h e source- free s t a t i o n a r y c y l i n d r i c a l l y symmetric electromagnet i c f i e l d s are i n v e s t i g a t e d .
I t i s found t h a t i f g tt>g4$ and glat are 1i-
n e a r l y r e l a t e d , t h e s t a t i o n a r y s o l u t i o n s can be reduced t o t h e known sol u t i o n s corresponding t o t h e s t a t i c e l e c t r o v a c s .
2. FIELD EQUATIONS For a s t a t i o n a r y c y l i n d r i c a l l y symmetric m e t r i c ,
t h e most general l i n e
elernent may be w r i t t e n as:
where f, Ji, R and m a r e f u n c t i o n s o f r alone. We number t h e co- ordinates t, r, z, @ as 0,1,2,3,
and take t h e u n i t s i n which G = c = 1 .
The Eins-
tein-Maxwell f i e l d equations i n an otherwise empty space are
where
f
bei ng t h e electromagnetic f i e l d tensor whose o n l y non- vanishi ng con-
t r a v a r i a n t components f o r a s t a t i o n a r y c y l i n d r i c a l l y symmetric f i e l d a r e ~
1 =3 - ~ 3 1 and
F1Q
-p'O1.
For such a f i e l d one has
so t h a t from Eq. ( 2 ) one obtains
I n view o f ( 7 ) , one can introduce W e y l - l i k e canonical co- ordinates
such
that3
f~ + rn2 = r 2
.
( 8)
From Eq. (4) one has
and
W i t h the h e l p of Eqs. ( I ) , be w r i t t e n from Eq. ( 2 ) as
( 3 ) , (8) , (9) and (10) the f i e l d equations can
I f one w r i tes f / R
and Eqs.
(13)-(16)
= u and
m/R = v , then Eq. (8) reduces t o
take the simple forms
i f one now assumes t h a t t h e r e i s a l i n e a r r e l a t i o n between u and v
v = a u + b ,
(2 1 )
then f o r non-nu1 l r e a l f i e l d s one must have 1 o f s o l u t i o n s o f Eqs. ( 1 1 ) - ( 1 2 ) and (18)-(20)
+ 4ab
=
u2 >
isobtained
7
as
O.
One c l a s s
where k , c and r
o
a r e constants and
Another. c l a s s o f s o l u t i o n s i s 8
[ R ~- u 2 r 2 R - 7 ,
(28)
R = y2
E 2 r 2- w
(29)
rn = y2w
E 2 r 2-
f =
Y2
R'
where c i s a constant o f i n t e g r a t i o n and O
2- c R = rc + br Eqs.
,
~ R ~ ,
1,
