LETTER]] AL NLIOV0 CIM]~!N'TO VOL. 44, ~-. 8 16 D i c e m b r e 1985
Scalar Plane Waves in General Relativity.
~r HALILSOYNuclear Engineering, King Abdulaziz University - P.O. Box
9027,Jeddah, Saudi Arabia
(rieevuto il 12 A p r i l e 1985)
P A C S . 04.20. - General r e l a t i v i t y .
Summary.
- A m o r e g e n e r a l class of scalar p l a n e w a v e s is p r e s e n t e d in g e n e r a l r e l a t i v i t y . I t is s h o w n t h a t no m a t t e r h o w t h e scalar field is modified, s p a c e - t i m e possesses an essential s i n g u l a r i t y .Scalar p l a n e w a v e s in g e n e r a l r e l a t i v i t y can be described e i t h e r b y t h e l o n g i t u d i n a l w a v e s of t h e B r i n k m a n n line e l e m e n t
(1)
d s 2 = 2 d u ' d v ' - - dx ' z - dy ' ~ - 2(x '2 ,a_ y , 2 ) h ( u ' ) d u '2 ,or b y t r a n s v e r s e w a v e s in t h e R o s e n f o r m
(2) ds 2 = 2 e x p [-- M(u)] du dv -- e x p [-- U(u)] (dx ~ + dy 2) .
W h i l e BRI*-KMANN f o r m is s u i t a b l e to h a n d l e s u p e r p o s i t i o n of p l a n e w a v e s , R o s e n f o r m is m o r e c o n v e n i e n t to s t u d y t h e collision of p l a n e waves. Such a s p a c e - t i m e s u p p o r t s s u i t a b l y a massless scalar field, w h i c h w i t h o u t loss of g e n e r a l i t y , we shall d e n o t e b y t h e r e a l f u n c t i o n q~(u).
P u r e g r a v i t a t i o n a l p l a n e w a v e s on t h e o t h e r h a n d are r e p r e s e n t e d b y t h e R o s e n line e l e m e n t
(3) d s 2 = 2 e x p [ - -
M ( u ) ] d u d v - - e x p [ - - U(u)](exp[V(u)]dx2+
e x p [ - -V(u)]dy~).
Collision b e t w e e n g r a v i t a t i o n a l p l a n e (1,2) a n d scalar p l a n e w a v e s (3) w e r e f o r m u l a t e d
(1) K . KHAN a n d R . PEN'ROSE: Nature, 229, 185 (1971). (3) 1 =~. SZEKERES" J. Malh. Phys. ( N . Y . ) , 13, 286 (1972). (8) ~V. Z. CHAO: J. Phys. A : Math., Nucl. Gen., 15, 2~29 (1982).
SCALAR PLAlffE WAVES I1r GENERAL RELATIVITY 545
before, so t h a t we shall make use of the field equations directly. The Einstein field equations for colliding gravitational waves are
(4)
u ~ = u=u.,
2 r ~ , - y ~ V ~ - y , r ~ =
o,
2 u . . + 2M. U.-- u~- r~= o,
9.co, + 2~u, ~ o - ~,~- v,~ = o .
2M,,~,+ U,,U~-- V~,V,= O,
whose general solution was presented by SZEK~ES (2). Field equations for colliding real scalar fields arc
(.5)
U ~ = YuU~,
2r
~qr
Z7,r
0,
2 v , , + 2M, Uo-- U,== ~,=,
The striking similarity between the sets of equations (4) a n d (5), prompts one to con- struck scalar field solutions from a given v a c u u m solution. I t can readily be observed that,if (U, V, M) is a know solution for set (4), then b y identifying, r = V, one ob- tains a solution for colliding scalar plane waves, n a m e l y for set (5). This observation was given in (2).
A more general class of scalar wave solution can be obtained b y the choice
(6)
~ = V + ~ U , O = U ,- g l = M + a V - t - 8 9
where, ~-= real, a r b i t r a r y constant. I t can be checked t h a t (r (7, ~ ) constitutes another solution of (5), whenever (U, V, M) is a k n o w n solution for (4). Note t h a t the class of solution (6) has no V -~ 0, limit, b u t it has, ~ = 0 limit which reduces in the real domain to the solution in (3). I n order to have no-scalar-wave case, we m u s t have b o t h V - = 0 a n d c~ = 0, simultaneously, a n d the resulting space-time then is fiat, as it should.
Using the solutions (U, V, M) of Szekeres, (6) reads explicitly,
(7)
exp [ ~] __ (w § P~kll2 (r + q~k'/~ t-2~ '
\ w -- p /
\ r -- q/
exp [-- g?] = t 2 ,546 where, t h e s t a n d a r d n o t a t i o n s are, as in (2) t 2 = w 2 _ p 2 = r 2 _ q ~ = 1 - - u * a - - v ~ * ~ , r z = l - - p 2 ,
p = u'~l~O(u),
b ~ v"~'~O(v),
k ~ - 8 ( 1 - - 1 / ,
X
n~l
M. HALILSOY w: = 1 -- q2 , i---- 1 , 2 .Also n o t e t h a t scaling of t h e n u l l co-ordinates, which is at o u r disposal, does n o t c h a n g e t h e f e a t u r e of t h e p r o b l e m .
E m p l o y i n g x as a (( screening )) p a r a m e t e r we p r o c e e d n o w to c o n s t r u c t r e g u l a r scalar fields, at least locally, since, as it s t a n d s in (7), tlle scalar field has u n d e s i r e d features.
W e consider two p a r t i c u l a r cases.
1) L e t t i n g e = -
kl/2 ( - kJ2),
yields in t h e i n c o m i n g regions w e l l - b e h a v e d scalar p l a n e waves. T h e n o n v a n i s h i n g c o m p o n e n t T u u = 092 ( T v v = q)2) of t h e energy- m o m e n t u m tensor, T~v = ~b~ q)v-- ~ gl~v q)a ~ is also finite. F o r t h e ~t = 2 case (impul-waves) one gets,
fdu T,~,=
s i r e 4.
T h e s p a c e - t i m e metric, h o w e v e r , possesses t h e
p(u) =
l(q(v) = 1) singularity. More- over, t h e r e g u l a r n a t u r e s of q) a n d Tar in t h e i n c o m i n g regions is n o t s h a r e d in t h e s c a t t e r i n g region.2) L e t , ~ ~ -- (k~ + k2)/2 (for simplicity, also l e t / c l = k2), one has
e x p [~5]
= (w + p)-~(r + q)-a,
w h i c h is r e g u l a r at t h e s p a c e - t i m e s i n g u l a r i t y p2 + q2 = 1 of t h e i n t e r a c t i o n region. U n f o r t u n a t e l y e n e r g y - m o m e n t u m t e n s o r of this scalar field d i v e r g e s for (p, q) -~ (1, 0) a n d (0, 1). M o r e o v e r , t h e i n c o m i n g w a v e l i m i t of this w a v e d i v e r g e s for p = 1 ( = q). T h e conclusion d r a w n o u t is t h a t using f r e e d o m s at our disposal it is impossible to ob- t a i n an e v e r y w h e r e r e g u l a r scalar field w i t h a p h y s i c a l e n e r g y - m o m e n t u m tensor. L o c a l l y this can be a t t a i n e d , b u t globally, for s c a t t e r i n g scalar w a v e s not. S p a c e - t i m e e m e r g e s i r r e s p e c t i v e of t h e chosen scalar field, a l w a y s singular. This reflects once m o r e t h e singular n a t u r e of colliding g r a v i t a t i o n a l w a v e s since, a f t e r all, we h a v e e m p l o y e d Szekeres singular solutions. T h i s r e s u l t is c o n f o r m w i t h t h e s i n g u l a r i t y t h e o r e m p r o v e d b e f o r e (4).
