NOTE BREVI
Interacting Electromagnetic Shock Waves in General
Relativity.
M. HALILSOu
Nuclear Engineering Department, King Abdulaziz University P.O. Box 9027, Geddah 21413, Saudi Arabia
(ricevuto il 12 Marzo 1987)
Summary. - - We study the interacting electromagnetic shock waves with nonconstant profiles in general relativity. It is shown that by modi- fying the metric functions of the Bell-Szekeres solution, such solutions can be obtained.
PACS. 04.20. - General relativity.
1 . - I n t r o d u c t i o n .
As a m a n i f e s t a t i o n of t h e nonlinear feature of general relativity, t w o light waves scatter each other to develop a new region of space-time k n o w n as the interaction region. A l t h o u g h this problem, in a n a l o g y with p h o t o n - p h o t o n scattering of q u a n t u m electrodynamics is an i m p o r t a n t one, little has been achieved t o w a r d a complete u n d e r s t a n d i n g of it. The first a n d a c t u a l l y the only available solution so far was given b y BELL a n d SZEKE~ES (1) (hence- forth, BS), describing t h e i n t e r a c t i o n of two constant-profile shock electro- m a g n e t i c (e.m.) waves. A m i n o r c o n t r i b u t i o n t o t h e BS solution was given later b y showing t h a t t h e n u m b e r of i n c o m i n g shock waves can be arbitrarily in- creased (2). I n this case the i n t e r a c t i o n region emerges as a region of m a n y BS cells whose e x a c t n u m b e r is d e t e r m i n e d b y t h e n u m b e r of i n c o m i n g shocks. The m a i n features of t h e BS solution however, such as g r a v i t a t i o n a l impulse waves (1) P. BELL and P. SZ~K~RES: Gen. l~el. Gray., 5, 275 (1974).
(3) M. GURSES and M. HALInSOY: Lett. Nuovo Cimento, 34, 588 (1982).
9 6 1K. HALILSOY
arising a t t h e null b o u n d a r i e s , c o n f o r m a l p r o p e r t i e s a n d the r e m o v a b l e sin- gularities r e m a i n u n c h a n g e d . T h e i n t e r a c t i o n region of t h e BS solution is iso- m e t r i c to t h e B e r t o t t i - l ~ o b i n s o n solution (a,4), which is k n o w n to be t h e u n i q u e c o n f o r m a l l y flat solution to Einstein-Maxwell (EM) equations for n o n n u l l e.m. field.
I n t h i s p a p e r we consider t h e i n t e r a c t i o n of n o n c o n s t a n t - p r o f i l e e.m. shock w a v e s a n d show t h a t ~he solutions can be o b t a i n e d b y m o d i f y i n g t h e a r g u m e n t s of t h e m e t r i c functions in t h e BS solution. T h e conclusion is t h a t locally we r e c o v e r t h e BS line e l e m e n t w i t h u o n c o n s t a n t e.m. field strength, w h e r e a s iu t h e null co-ordinates t h e t w o solutions differ. W e s t a t e our result d e p e n d i n g on t w o functions r e s t r i c t e d b y t w o c o n s t r a i n t conditions a n d t h e BS choice of null co-ordinates h a p p e n s t o b e t h e simplest f o r m satisfying those conditions.
2 . - I n t e r a c t i o n o f e l e c t r o m a g n e t i c w a v e s .
T h e generic f o r m of t h e 1Lae e l e m e n t describing t h e i n t e r a c t i o n region of t h e collinearly polarized (~) e.m. waves is g i v e n b y (~)
(1) ds~ = 2 e x p [ - M ] d u d v - - e x p [ - - U ] ( e x p [ V ] d x ~ + e x p [ - - V ] d y 2) ,
w h e r e t h e m e t r i c functions d e p e n d on t h e null co-ordinates u a n d v. T h e in- c o m i n g s t a t e s (regions I I a n d I I I ) are c h a r a c t e r i z e d b y nonflat m e t r i c s as- sociated w i t h incoming o.m. waves. Region I contains no e.m. w a v e s a n d m u s t n a t u r a l l y be fiat. F o r t h e details of t h e s p a c e - t i m e p i c t u r e we refer t o (5). T h e Maxwell a n d Einstein-Maxwell (EM) field equations as d e r i v e d in BS a r e given as follows (note t h a t we a d o p t t h e s a m e n o t a t i o n s of BS):
(2) 2q~,~ = u ~ : - - V.~o, (3) 2~o,,. = u,,q~o- v , , ~ , (4) ~1 = 0 , (5)
u.~= u~u~,
(6)
2 u . ~ - - ~ + 2 V . i ~ V~ + 4kl~l ~ , (7) 2 u ~ - ~ + 2 u~ ~rs~ = v~ + ~k[~o[,, (s)2M.v+ u.u~= v.v~,
(k = const = G ) ,(a) B. BERTOTTI: Phys. Rev., 116, 1331 (1959).
where t h e e.m. field a m p l i t u d e s (i.e. t h e s p i n o r c o m p o n e n t s ~o a n d ~ ) a r e t h e (~ scale i n v a r i a n t )> ones as defined in (5). T h e s o l u t i o n t o this s e t of e q u a t i o n s as o b t a i n e d b y B S is
(~o)
a b
M ~ 0 , ~2 : ~2 - - ~/~. - - c o n s t , ~0 ~-- ~0 ~-- V/~ - - c o n s t ,
cos (au -- by)
e x p [ - - U]---- cos (au -- by) cos (au ~- bv) , e x p IV] - -
cos (au -~ bv) T h e s o l u t i o n g i v e n in ref. (~), o n t h e o t h e r h a n d , is o b t a i n e d f r o m B S b y m o d i f y i n g t h e a r g u m e n t s of cosine t e r m s , n a m e l y b y m a k i n g t h e s u b s t i t u t i o n s
(:11)
auO(u) -~ ~ a~(u
--u,)O(u
--us),
n$
bvO(v) --> ~_, b~(v --v~)O(v -- vi)
(ai, bi c o n s t a n t s , r e l a t e d t o t h e fluxes of t h e s h o c k s ) .
T h e t h o r n f u n c t i o n s in t h e a r g u m e n t s g u a r a n t e e t h e c o n s i s t e n t m a t c h i n g of t h e disjoint s p a c e - t i m e regions, a l b e i t i t r e s t r i c t s t h e m e t r i c t o b e of class C ~ a n d piecewise C 1. T h e f a c t t h a t t h i s s u b s t i t u t i o n y i e l d s a n o n t r i v i a l r e s u l t is seen b y c o m p u t i n g t h e c o m p o n e n t s of t h e R i e m a u n t e n s o r .
3. - Interacting e l e c t r o m a g n e t i c w a v e s w i t h n o n c o n s t a n t field strength.
W e p r o v i d e n o w solutions to t h e a b o v e set of e q u a t i o n s (2)-(9) f o r t h e case w h e n ~o a n d ~, a r e n o t c o n s t a n t s (with still ~1--~ 0). F o r t h i s p u r p o s e we s u b s t i t u t e u --~ ](u) a n d v -> g(v), w h o r e ] a n d g a r e f u n c t i o n s t o b e d e t e r m i n e d below. T h e a n s a t z s o l u t i o n s e e k e d is e x p r e s s e d in t h e R o s e n f o r m b y
(12)
ds ~ ~ 2 ] ' g' d u dv -- cos ~ [a/(u) O(u) -- bg(v) 0(v)[ 4x ~ - -- - cos ~ [a/(u) O(u) ~- bg(v) 0(v)[ d y ~ , w h e r e ]'-~ d//du, g ' = dg/dv a n d t h e e.m. s p i n o r c o m p o n e n t s a r e c h o s e n as
a b
(.13) ~ 2 = ~ / ' O ( u ) , % = ~ g ' O ( v ) .
9 8 5& HALILSOY
s i s t e n t l y yield t h e following conditions:
(J4)
/(u)~(u)
=o,
(h
(~5)
~
O(u) ~ o
a n d similar conditions f o r t h e f u n c t i o n g. I t is r e a d i l y o b s e r v e d t h a t t h e simplest possible ](u) satisfying t h e s e c o n s t r a i n t s is t h e one corresponding to t h e BS solu- tion, n a m e l y , ](u) -~ u ( a n d g(v) -~ v). O t h e r i n t e r e s t i n g values which s~tisfy t h e a b o v e c o n s t r a i n t s w i t h o u t giving rise to d e g e n e r a c y in the s p a c e - t i m e m e t r i c a r e t h e following:
](u) = sinu, s i n h u , t g h u , u e x p [u], u c o s u , - ~ - ~ , ... ~ cos
(~6)
/
v }
g(v) = sin v, sinh v, t g h v, v e x p [v], v cos v, - - ~ , . . . . co8
I t is o b s e r v e d t h a t t h e n o n c o n s t a n c y of e.m. spinor c o m p o n e n t s serve t o gen- e r a t e t h e m e t r i c f t m c t i o n M~ which vanishes in t h e case of BS. I n t h e null t e t r a d lug-- e x p [ - - M/2]6~, n u = e x p [ - - M/216~,
(1
7) - - %/2mu : e x p[--
U/'!](exp [V/'216~ + i e x p [V/216J), t h e scale-invariant, n o n v a n i s h i n g W e y l c o m p o n e n t s a r e given b y(18)
I T ~ = - - a ( ~ u ) $ ( u ) t g b g ( ~ ) O ( v ) ,I
~ g o = - - b(~
~ ~(v) t g a ] ( u ) 0 ( u ) . L e t us n o t e t h a t t h e c o n s t r a i n t (15) is j u s t t h e r e q u i r e m e n t to p r o v i d e a non- flat m e t r i c .I ~ order to i d e n t i f y t h e i n c o m i n g waves in t h e B r i n k m a n n (e) c o - o r d i n a t e s y s t e m , which is a h a r m o n i c s y s t e m , we express o u r solution in t h e BS f o r m
(19) ds 2 ---- 2 d] d g - - cos 2 (a/-- bg) dx ~ -- cos ~ (a] + bg) dy ~ ,
t h a t reduces in t h e region I I ( g - = 0) to
(20)
ds ~ ~ 2 d/dg -- cos ~ a](dx ~ + dy2).This line element is obtained from t h e B r i n k m a n n metric
(21) ds ~ : 2 d U d V - d X ~ - d Y ~ - O ( U ) ( X 2 + y 2 ) d U s
b y the following co-ordinate t r a n s f o r m a t i o n (which is equivalent to a null rotation) :
/ U = / ,
X = x F ,
Y = y E ,
(22)
/
V ---- g -{- 89 (x * -{- y : ) F F , ,
where E ~ cos a]. I t is observed t h a t ] a n d g correspond to the null co-ordinates of regions I I a n d I I I , respectively, when t h e waves are expressed in har- monic co-ordinates.
4. - Electromagnetic-potential approach.
The basic El~r equations (5), (8) a n d (9) are obtained b y t h e variational principle of t h e Lagrangian
(23) L - - e x p [ - - U ] ( M ~ U , + M~U~-[- U ~ U v - - V ~ V ~ ) - - 2 k e x p [ - - V ] A u A v ,
where the e.m. 4-potential is given b y A , = A ~ . The e.m. spinor components are defined b y
('2~t
T h e variational equation, ~ L / ~ A : 0, yields the equation satisfied b y t h e potential function A, which is equivalent to the Maxwell equations given in eqs. (2) a n d (3). This is given b y
(25) (exp [-- V ] A , ) v + (exp [-- V I A l ) ~ : 0 ,
which admits t h e solution
We introduce now the following co-ordinates:
1 0 0 M. HALILSOY
so t h a t t h e space-time describing i n t e r a c t i n g e.m. waves is expressed b y
(~8)
1 ( d~ ~ d(~ ~ _ _ ( l _ _ ~ 2 ) d x ~ ( l _ ( ~ 2 ) d y 2d s 2 = 2 a b ~ - - T 2 1 - - a 2]
a n d t h e o.m. p o t e n t i a l becomes A ~--~/(2/k)a.
A n o t h e r useful co-ordinate s y s t e m is p r o v i d e d b y t h e choice
(29) -~ a] + bg , ~ ---- a ] - - bg.
This co-ordinate system p r o v e s to be useful in s t u d y i n g geodesics m o t i o n a n d t h e Dirac equation in t h e i n t e r a c t i o n region (7). T h e space-time m e t r i c reads (we scaled x and y for obvious reason in t h e sequel)
(30) ds 2 = ~ [d~] 2 - d~ 2 - - cos2~ dy '2 - - cos 2 ~dx '*] .
This is easily t r a n s f o r m e d into B o r t o t t i - R o b i n s o n solution b y t h e following t r a n s f o r m a t i o n s :
(3:1) sin ~ r ' t x' ~ , 2y' In (r 2 - t 2) , ~ ~ ~ - - O, 2ab ] e 2
5 . - D i s c u s s i o n .
T h e p r o p e r solution for t h e problems of i n t e r a c t i n g (colliding) waves in general r e l a t i v i t y should go from region I (flat space), t h r o u g h regions IX a n d I I I (incoming regions) into region I V (the i n t e r a c t i o n region). T h e reverse order, n a m e l y from region I V to regions I I and I I I , a l t h o u g h h a p p e n s to be t h e simpler route, results m o s t l y in nonphysical incoming states. T h e choice of realistic wave forms yield u n f o r t u n a t e l y sot of coupled systems of p a r t i a l differential equations whose e x a c t solutions b e c o m e almost impossible.
F o r e.m. case, we h a v e shown t h a t changing t h e incoming waves o n l y serves to m o d i f y t h e metric functions in t h e BS metric. No m a t t e r how t h e profile of t h e incoming waves is chosen f r o m a set t h a t satisfies certain constraints, t h e e.m. waves continue into t h e i n t e r a c t i o n region unchanged. This m a y be a general f e a t u r e of i n t e r a c t i n g e.m. waves, a n d for this reason we p r e f e r to n a m e t h e problem interaction, r a t h e r t h a n collision. Locally, in t h e co-ordinates (v, o) or (~, 7) all solutions are expressed in BS f o r m , b u t in the null co-ordinates t h e details of incoming e.m. waves m o d i f y t h e a r g u m e n t s .
