Journal of Applied Mathematics and Physics (ZAMP) 0044-2275/86/003387-08 3.10/0 Vol. 37, May 1986 9 Birkh~iuser Verlag Basel, 1986 A note on quantization in rotating co-ordinates By Faheem Hussain*), International Centre for Theoretical Physics, Trieste, Italy and Asghar Qadir**), Mathematics Department, Quaid-i-Azam University, Islamabad, Pakistan 1. Introduction Letaw and Pfautsch [1] considered the problem of canonical quantisation of the free scalar field for the six classes of stationary coordinate systems in Minkowski space [2]. As an example of class-C coordinate systems based on a Killing vector field with components (1 + ~ x, z t - -c y, ~ x, o), ~ > ~, they stud- ied what they called "rotating coordinates" (t, r, ~0, z) which are related to the cylindrica] Minkow ski coordinates (t', r', ~0', z') through the transformations t=t', r=r', qo=qo'-fAt', z=z'. (1 ) They demonstrated that the vacuum defined in these "rotating" coordinates is just the Minkowski vacuum. Similar conclusions were reached in two earlier papers by the same authors [3] and by Denardo and Percacci [4]. We argue that the transformation given by Eq. (1 l) does not correspond to relativisitic uniform, circular motion. The correct rotating coordinates were given much earlier by Post [5] and we consider the problem of quantisation of the free scalar field in these coordinates and reach the conclusion that, here too, the vacuum is just the Minkowski vacuum. 2. The transformation to uniformly rotating coordinates There has been much work done on the problem of identifying the correct relativistic transformation to rotating coordinates for uniform circular motion and quantizing the scalar field in the background metric so obtained. Most authors including Letaw and Pfautsch [1, 2, 3], consider the transformation giv- en in Eq. (1) as the correct transformation from the cylindrical Minkowski coordinates (t', r', (p', z') to the "rotating" coordinates (t, r, ~o, z). As is obvious, *) Permanent address: Physics Department, Quaid-i-Azam University, islamabad, Pakistan. **) Also, Centre of Basic Sciences, UGC, Islamabad, Pakistan.
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Journal of Applied Mathematics and Physics (ZAMP) 0044 -227 5/86/0 0338 7-083.10/0Vol. 37, M ay 1986 9 Birkh~iuserVerlag B asel, 1986
A n o t e o n q u a n t i z a t i o n i n r o t a t i n g c o - o r d i n a t e s
B y F a h e e m H u s s a i n * ) , I n t e r n a t i o n a l C e n t r e f o r T h e o r e t i c a l P h y si cs , T ri es te ,
I ta ly a n d A s g h a r Q a d i r * * ), M a t h e m a t i c s D e p a r t m e n t , Q u a i d - i - A z a m
U n i v e r s i t y , I s l a m a b a d , P a k i s t a n
1 . In tro d u ct io n
L e t a w a n d P f a u t s c h [1] c o n s i d e r e d t h e p r o b l e m o f c a n o n i c a l q u a n t i s a t io n o f
t h e f re e s c a la r f i e ld f o r t h e s ix c la s s e s o f s t a t i o n a r y c o o r d i n a t e s y s t e m s i n
M i n k o w s k i s p a c e [ 2 ]. A s a n e x a m p l e o f c l a ss - C c o o r d i n a t e s y s t e m s b a s e d o n a
K i l l i n g v e c t o r f i e ld w i t h c o m p o n e n t s ( 1 + ~ x , z t - -c y , ~ x , o ), ~ > ~ , t h e y s t u d -
i e d w h a t t h e y c a l l e d " r o t a t i n g c o o r d i n a t e s " ( t, r , ~0, z ) w h i c h a r e r e l a t e d t o t h e
c y l i n d r i c a ] M i n k o w s k i c o o r d i n a t e s ( t', r ', ~ 0', z ') t h r o u g h t h e t r a n s f o r m a t i o n s
t = t ' , r = r ' , q o = q o ' - f A t ' , z = z ' . (1 )
T h e y d e m o n s t r a t e d t h a t t h e v a c u u m d e f i n e d i n t h es e " ro t a t in g " c o o r d i n a t e s i sj u s t t h e M i n k o w s k i v a c u u m . S i m i l ar c o n c l u s i o n s w e re r e a c h e d i n t w o e a r li e r
p a p e r s b y t h e s a m e a u t h o r s [ 3] a n d b y D e n a r d o a n d P e r c a c c i [4 ].
W e a r g u e t h a t t h e t r a n s f o r m a t i o n g i v e n b y E q . (1 l ) d o e s n o t c o r r e s p o n d t o
r e la t iv i s it ic u n i f o r m , c i r c u la r m o t i o n . T h e c o r r e c t r o t a t i n g c o o r d i n a t e s w e r e
g i v e n m u c h e a r li e r b y P o s t [5] a n d w e c o n s i d e r t h e p r o b l e m o f q u a n t i s a t i o n o f
t h e f re e s c a l a r fi e ld i n th e s e c o o r d i n a t e s a n d r e a c h t h e c o n c l u s i o n t h a t , h e r e t o o ,
t h e v a c u u m is j u s t t he M i n k o w s k i v a c u u m .
2 . T h e t r a n s f o r m a t i o n t o u n i f o r m l y r o t a t i n g c o o r d i n a t e s
T h e r e h a s b e e n m u c h w o r k d o n e o n t h e p r o b l e m o f i d e n ti fy i n g t h e c o rr e c t
r e la t iv i s ti c t r a n s f o r m a t i o n t o r o t a t i n g c o o r d i n a t e s f o r u n i f o r m c i r c u la r m o t i o n
a n d q u a n t i z i n g t h e sc a l a r f ie ld i n t h e b a c k g r o u n d m e t r i c s o o b t a i n e d . M o s t
a u t h o r s i n c l u d i n g L e t a w a n d P f a u t s c h [1 , 2 , 3 ], c o n s i d e r t h e t r a n s f o r m a t i o n g iv -
e n i n E q . (1 ) a s t h e c o r r e c t t r a n s f o r m a t i o n f r o m t h e c y l i n d ri c a l M i n k o w s k i
c o o r d i n a t e s ( t', r ', (p ', z ') t o t h e " r o t a t i n g " c o o r d i n a t e s ( t, r , ~o, z ). A s i s o b v i o u s ,
*) Perm anent address: Physics Dep artment, Quaid-i-Aza m University, islamabad, Pak istan.**) Also, Centre of Basic Sciences, UG C, Islamabad, Pakistan.
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3 88 F . H u s s a i n a n d A . Q a d i r Z A M P
t h is t r a n s f o r m a t i o n is th e c y l i n d r i c a l e q u i v a l e n t o f t h e G a l i l e a n c o o r d i n a t e
t r a n s f o r m a t i o n : x - - x ' - v t '. T h e y t h e n d e r i v e t h e m e t r i c in " r o t a t i n g " c o o r d i -
n a t e s a s
d s 2 = ( 1 - r 2 ~ 2 ) d t 2 - - 2 Q r 2 d t dq~ - d r 2 - r 2 d ( o 2 - d z 2 . (2 )
I t h a s b e e n a r g u e d b y A d l e r , B a z i n a n d S c h if fe r t h a t t h e m e t r i c g i v en in
E q . ( 2 ) g i v e s t h e t i m e d i l a t i o n f a c t o r
7 = ( 1 - - r 2 ~ r ~ 2 ) - 1 /2 (3 )
w h i c h i s e x p e c t e d t o a r i s e d u e t o r e l a t i v e m o t i o n b e t w e e n t w o o b s e r v e r s , o n e
l o c a t e d o n t h e a x is o f r o t a t i o n a n d t h e o t h e r r o t a t i n g a t a c o n s t a n t a n g u l a r
v e l o c i t y , f2 , a t a d i s t a n c e r f r o m t h e a x i s. T h e y c o n s i d e r t h e p r o p e r t i m e
d r = ~ (4)
a n d p u t d r = d (p = d z = 0 , g i v i n g
dz -- (1 - r 2 ~ r - ~ 2 ) 1 / 2 d t = 7 - 1 d t (5 )
w h i c h t h e y a r g u e i s t h e r e q u i r e d t i m e d i l a t i o n f a c to r .
T h i s a r g u m e n t i s fa ls e fo r t h e f o l l o w i n g r e a s o n s :
I ) T h e l in e a r G a l i l e a n t r a n s f o r m a t i o n x = x ' - v t , t = t ' , does n o t l e av e d s 2
i n v a r i a n t a n d is , t h e r e fo r e , a n i n a d m i s s i b l e t r a n s f o r m a t i o n . S i m i l a r ly th e
t r a n s f o r m a t i o n g i v e n b y E q . (1), b e i n g a g e n e r a l i s a t i o n o f t h e G a l i l e a n t r a n s -
f o r m a t i o n , is n o t a d m i s s i b le f r o m a p u r e l y s p e c i al r e l at iv i s ti c p o i n t o f v ie w .L o c a l l y , s u c h a t r a n s f o r m a t i o n c a n n o t d e s c r ib e a u n i f o r m l y r o t a t in g c o o r d i n a t e
f r a m e .
2 ) T h e a r g u m e n t g i v e n a b o v e c a n b e tr i v ia l ly a p p l ie d t o t h e u s u a l l in e a r G a l l -
l e a n t r a n s f o r m a t i o n , f o r i n s u c h a c a s e
d s 2 - - ( 1 - - / ) 2 ) d t 2 _ 2 v d t d x - d x 2 - d y 2 - d z 2 ( 6 )
a n d , p u t t i n g d x = d y - - d z = 0 , w e w o u l d g e t a s e a si ly
dz = ~ 5 = (1 - v2) 1/2 d t (7)
a n d c l a i m , in t h e m a n n e r o f A d l e r , B a z i n & S c h i f fe r , t h a t E q . (7 ) g i v es t h e t i m e
d i l a t i o n f a c t o r . B u t t h i s a r g u m e n t i s o b v i o u s l y f als e. T h e t i m e d i l a t i o n f a c t o r is
n o t o b t a i n e d b y m a n u f a c t u r i n g a (1 - v 2) f a c t o r m u l t i p l y i n g t h e d t a t e r m i n d s 2 .
I n f a ct , th e c o r r ec t L o r e n t z t r a n s f o r m a t i o n t o a u n i f o r m l y m o v i n g f r a m e
t = ~ ( t ' - v x ' ) , x = ~ ( x ' - v t ' ) ,
l e a d s t o t h e m e t r i c
d s 2 = d t 2 - - d x 2 - d y 2 - d z 2
y = y ' , z = z ' , (8 )
(9 )
w h i c h d o e s n o t h a v e t h e 7 - 2 f a c t o r p r e s e n t i n (6). N o 7 f a c t o r s a p p e a r i n th e
L o r e n t z m e t r i c b u t o n l y i n th e c o o r d i n a t e t r a n s f o r m a t i o n .
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Vol. 37, 1986 A note on quant ization in rotating co-ordinates 389
T r iv i a l ly , w e c a n s ee t h a t t h e G a l i l e a n t r a n s f o r m a t i o n a n d t h e t r a n s f o r m a -
t i o n g i v e n b y E q . (1 ) i m p l y t h a t t h e r e i s n o t i m e d i l a t i o n b e c a u s e , s i n c e t = t ' i n
b o t h c a s es , w e m u s t h a v e A t - - A t ', w h e r e a s o n e c a n s e e t h a t A t = 7 A t ' f o r t h e
L o r e n t z t r a n s f o r m a t i o n g i v e n b y E q . (8). T i m e d i l a t i o n a r i s e s b e c a u s e , i n t h e
L o r e n t z t r a n s f o r m a t i o n , w e h a v e a n e x p l ic i t f a c t o r in th e t r a n s f o r m a t i o n o f t h e
t i m e c o o r d i n a t e .
I n a n s w e r t o t h e o b j e c t i o n t h a t E q . (1 ) i s a G a l i l e a n t r a n s f o r m a t i o n , i t
m a y b e a r g u e d t h a t g e n e r a l r e l a t i v i t y a l l o w s a ll c o o r d i n a t e t r a n s f o r m a t i o n s
a n d t h a t c o o r d i n a t e a r e m e r e l y a m e a n s t o p r o v i d e c o n v e n i e n t n u m b e r s t o
l a b e l s p a c e - t im e e v e n ts . T h e r e is n o d o u b t t h a t a l l c o o r d i n a t e t r a n s f o r m a t i o n s
a r e a l l o w e d b y g e n e r a l r e l a ti v i t y b u t n o t e v e r y t r a n s f o r m a t i o n c o r r e s p o n d s
t o , f o r e x a m p l e , u n i f o r m l in e a r m o t i o n . T h u s , t h e G a l i l e a n t r a n s f o r m a t i o n ,
w h i l e p e r f e c t l y v a l i d a s a " t r a n s f o r m a t i o n " , d o e s n o t a p p l y t o r el a ti v is t i c ,
u n i f o r m , l i n e a r m o t i o n . S i m i l a rl y , E q . (1 ) d o e s n o t a p p l y t o u n i f o r m c i r c u l a r
m o t i o n .
A m u c h e a r li e r p a p e r b y P o s t [5] a l r e a d y p r o v i d e s t h e c o r r e c t t r a n s f o r m a -
t i o n t o a r o t a t i n g f r a m e
t = 7 - 1 t ' , r = r ' , ( p = q o ' - f 2 t ' , z = z ' ( 1 0 )
b y s i m p l y a p p l y in g t h e c o m p l e t e L o r e n t z t r a n s f o r m a t i o n s
t ' = 7 ( t + v . r )
i ( ( v . r ) t ( 1 1 )r ' = r - - v l - y ) v~ 7
t o u n i f o r m c i r c u l a r m o t i o n , r e m e m b e r i n g t h a t f o r c i r c u l a r m o t i o n v 9 r = 0 ,
a t e a c h p o i n t . T h e t i m e d i l a t i o n i s s e e n r e a d i l y f r o m E q s . (1 0) . T h i s e l e g a n t
p a p e r b y P o s t h a s g o n e la r g e ly u n n o t i c e d . P o s t h a d a l s o n o t e d t h a t t h e
t r a n s f o r m a t i o n (1 ) d o e s n o t g iv e t h e re q u i r e d t i m e d i l a t i o n . T h e a b o v e t r a n s -
f o r m a t i o n i s o n l y l o c a l ly d e f in e d a s w o u l d b e e x p e c t e d f o r a c c e l e r a t e d o b s e r v e r s .
U s i n g E q . ( i 1), t h e l i n e e l e m e n t f o r a r o t a t i n g o b s e r v e r b e c o m e s
d s 2 = dt 2 A- 2 r 0 2 ? 2 t d t d r - 2 r 2 Q 7 d t d o
- (1 - r 2 4 7 4t2 ) d r 2 - 2 r 3 ~23 7 3 d r d o - r 2 d c p 2 - d z 2 . ( 1 2 )
T h e d e t e r m i n a n t o f t h is m e t r i c t e n s o r i s - r 2 7 2. W e c a n s ee , i m m e d i a t e l y , t h a t
t h i s m e t r i c h a s a c o o r d i n a t e s i n g u l a r i t y a t r f2 = 1 , i n c o n t r a s t t o t h e m e t r i c
g i v e n b y E q . (2 ), w h i c h h a s a d e t e r m i n a n t - r 2. T h e s u r f a c e r = f 2 - 1 i s a s i n g u l a r
s p a c e - l i k e s u r f a c e f o r t h e o b s e r v e r a n d is n o t a h o r i z o n .
T h i s m e t r i c a d m i t s , a m o n g o t h e r s , t h e t i m e - l ik e a n d s p a c e - l i k e K i l l i n g v e c -
t o r f ie ld s , i n c o v a r i a n t a n d c o n t r a v a r i a n t f o r m :
K , = (7 , r ~ 2 r t , 0 , 0 ) , K " = (7 - ~ , 0 , - - O , 0 ) , )( 1 3 )
M , = ( r 2 ~ 7 , r 2 ~ 3 7 3 t , 0 , 0 ) , M " = ( 0 , 0 , - 1 , 0 ) .
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390 F. Hussa in and A. Q ad ir Z A M P
w h i c h h a v e m a g n i t u d e s 1 a n d - r 2 r e s p e c t i v e l y . T h e l i n e a r c o m b i n a t i o n
R u = K s - ~ ? M s , w h i c h h a s t h e c o m p o n e n t s 7 - 1 (1 , 0 , 0 , 0 ), is a l s o a K i l l in g
v e c t o r w h i c h i s t i m e - l i k e o n l y f o r r < ~ 2 - t , s i n c e R " R u = 1 - - ~22 r 2.
M s i s t h e g e n e r a t o r o f r o t a t i o n s a r o u n d t h e z - a xi s a n d i s e v e r y w h e r e s p a c e -
l i k e ; R s i s t h e t i m e t r a n s l a t i o n K i l l i n g v e c t o r i n t h e r o t a t i n g f r a m e ( it i s t h e
t a n g e n t v e c t o r t o t h e t r a j e c t o r i e s o f t h e p o i n t s w i t h f i x ed z, r , (p) a n d i s t i m e - l i k e
o n l y f o r r < ~ 2 - 1 . F i n a l l y , K " is t h e t i m e t r a n s l a t i o n K i l l i n g v e c t o r in t h e n o n -
r o t a t i n g ( M i n k o w s k i ) f r a m e ( i t is t h e t a n g e n t v e c t o r t o t h e t r a j e c t o r ie s w i t h fi xe d
z ' , r ' , q / ) ; i t i s g l o b a l l y t i m e - l i k e a n d o r t h o g o n a l t o t h e h y p e r s u r f a c e s
7 t = t ' = co ns tan t ; i n f ac t , K u i s t h e u s u a l t i m e t r a n s l a t i o n K i l l i n g v e c t o r o f
M i n k o w s k i s p a c e w r i t t e n i n u n u s u a l c o o r d i n a t e s .
T h e m e t r i c g i v e n i n E q . (1 2) i s n e i t h e r s t a t i o n a r y ( gu y t i m e i n d e p e n d e n t ) n o r
s t a t i c ( g o i - - 0 , i = 1 , 2 , 3 ). T h e r e a r e n o s t a t i c f o r m s o f t h i s m e t r i c b e c a u s e t h e r e
i s n o s u r f a c e e v e r y w h e r e o r t h o g o n a l t o t h e K i l l i n g v e c t o r f i e l d R s . Since th i s
c o o r d i n a t e s y s t e m i s n o t s t a t i o n a r y i t d o e s n o t f a ll , i n t o a n y o f t h e c a t e g o r i e s
e n u m e r a t e d b y L e t a w a n d P f a u t s c h [ 2] . T h e i r a n a l ys i s n e ed s t o b e l o o k e d a t
m o r e c a r e f u l l y a s w e l l . T h e y w o r k o u t t h e " m o v i n g t e t r a d " f o r a c c e l e r a t e d
m o t i o n i n M i n k o w s k i s p a c e a n d c la s si fy i t a c c o r d i n g t o v a r i o u s p a r a m e t e r s o f
t h e K i l l in g v e c t o r s . I n t h e i r a n a l y s i s t h e y s h o w t h a t , f o r u n i f o r m c i r c u la r m o t i o n ,
t h e y h a v e a u n i t t i m e - l ik e K i l li n g v e c t o r f ie ld w h i c h i s h y p e r s u r f a c e o r t h o g o n a l
i n c o n t r a d i c t i o n t o o u r c l a im h e r e . T h e r e a s o n f o r th i s d i ff e re n c e is t h a t t h e y a r e
a n a l y s i n g m o t i o n a l o n g a c i r c l e a n d t h e y t a k e r t o b e c o n s t a n t . T h u s , f o r
c o n s t a n t g2, t h e y g e t a c o n s t a n t 7 , a n d h e n c e t h e K i l l in g v e c t o r h a s c o n s t a n tm a g n i t u d e . B y r e s c a li n g t h e m a g n i t u d e o f t h e K i l l in g v e c t o r i t c a n b e c h o s e n t o
b e u n i ty . H o w e v e r , t h i s w o r k s o n l y f o r m o t i o n i n a g i ve n c ir cl e. I f t w o o b s e r v e r s
a t d i f f e re n t v a l u e s o f r a r e r o t a t i n g w i t h t h e s a m e a n g u l a r s p e e d , ~2, a n d c o m m u -
n i c at in g , t h e y c a n n o t a g r e e th a t b o t h h a v e u n i t t im e - l i k e K i ll i n g v e c to r s . T o d o
f i el d t h e o r y w e n e e d t o c o n s i d e r v a r i a t i o n o f r . I f r is a v a r i a b l e , 7 i s n o t a c o n s t a n t
a n d h e n c e t h e K i l li n g v e c t o r f ie ld d o e s n o t h a v e c o n s t a n t m a g n i t u d e . T h u s t h e
c l a s s i f i c a t i o n o f L e t a w [2 ] w h i l e c o r r e c t f o r m o t i o n i n a g i v e n c i r c le is n o t
c o m p l e t e l y r e l e v a n t f o r d o i n g f i e l d t h e o r y .
F o r t h e r e a s o n s s t a t e d e a r li e r , E q . (1 2 ) g i v e s t h e c o r r e c t m e t r i c f o r a re l a t iv -i s t i c , u n i f o r m l y r o t a t i n g s y s t e m . I n o t h e r w o r d s , a r o t a t i n g o b s e r v e r c a n n o t b e
r e p r e s e n t e d b y a s t a t i o n a r y m e t r ic . T h e c o o r d i n a t e s u r f ac e s o f t h e s e re l a ti v is t ic
r o t a t i n g c o o r d i n a t e s a r e d i s p l a y e d i n F ig . 1 . S u r f a c e s o f c o n s t a n t r a re i d e n t ic a l
t o c o o r d i n a t e s u r f a c e s i n c y l in d r i c a l M i n k o w s k i c o o r d i n a t e s . S u r f a c es o f c o n -
s t a n t t a r e g i v e n b y
tt ' = (14)
( 1 - r 2 ~ ' ~ 2 ) 1 / 2
i n M i n k o w s k i c o o r d i n a t e s . T h e t = 0 s u r fa c e is a p l a n e w h e r e a s t h e t @ 0 su r -
f a c e s a r e s u r f a c e s g o i n g o f f t o o o a t r = ~ 2 - 1. L i n e s o f c o n s t a n t q~ a r e d i s p l a c e d
r e l a t i v e t o l i n es o f c o n s t a n t ~o' a t a c o n s t a n t r a t e t 2 ( r e la t i v e t o t h e c o o r d i n a t e
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Vol. 37, 1986 A no te on quantization in rotating co-ordinates
Figure 1Ro tating coordinate surfaces displayedin rectangular Minkowski coordinates.Constant t surfaces are give n by t ' =
t
in Minkowski coordi-(1 - - r 2 ~ ' ~ 2 ) 1 / 2n a r e s . T h e t = 0 s u r f a c e i s a p l a n e
whereas the t + 0 surfaces going off toat r = ~ - 1. Constant r surfaces are
circular cylinders identical to surfacesin cylindrical Minkowski coordinates.Con stant ~p surfaces resemble an Ar-chimedean screw.
t = C o n s t a n t
d
391
x
- - k = C o n s t a n t
" ~ d = C o n s t a n t
y
t i m e t ' f o r t h e M i n k o w s k i s y s t e m o n l y ) t h u s f o r m i n g h e li ce s r a t h e r t h a n v e r t ic a l
s t r a i g h t l i n e s , a s
q~' = r + y g 2 t = q~ + ~2 t' . ( 15 )
A s a r e s u l t o f t h is d i s p l a c e m e n t s u r f a c e s o f c o n s t a n t ~p a r e r i g h t c o n o i d s r e s e m -
b l in g t h e s c r e w o f A r c h i m e d e s .
T h e g e o d e s i c e q u a t i o n s f o r t h e c o r r e c t r o t a t i n g m e t r i c a r e q u i te c o m p l i -
c a t e d . H o w e v e r , i f w e o n l y c o n s i d e r i n s t a n t a n e o u s r a d i a l m o t i o n , ~b = 0 , a n d
r e s t r ic t o u r s e l v e s t o t = 0 , w e f i n d t h a t w e o b t a i n t h e u s u a l c e n t r i f u g a l a n d
C o r i o l i s f o r c e s
i" = r Q 2 , r ~ + 2 O f = 0 . (1 6)
3 . Q u a n t i z a t i o n o f th e s c a l a r fi e ld
I n r o t a t i n g c o o r d i n a te s t h e K l e i n - G o r d o n e q u a t i o n is
I ( 8 2
82 1 ~ 82 82
+ 2 r ( 2 2 7 2 t Or O t r ~r Or 2 2 ~ 7 - 1 - -8 t8~o
1 ~ 2 ~ 2 ]
- - r2 7 2 ~ q ~ 2 ~ ~ z ~ = - M 2 ~ . (1 7)
A t f i rs t s i g h t th i s e q u a t i o n is n o t e a s i ly s e p a r a b l e . H o w e v e r , o n e c a n e a s i l y c h e c k
t h a t t h i s e q u a t i o n h a s t h e s i m p l e s o l u t i o n s
~kq ,, ,;~o = N e - i~ t (~ - mr) e im~ e ik~ j , , , (q r) (18 )
w i t h i n t e g e r m , q > 0 , 0) 2 = k 2 -4- q 2 -4- M 2 a n d N a n o r m a l i s a t i o n f a c t o r t o b e
o b t a i n e d b e l o w .
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3 9 2 F . H u s s a i n a n d A . Q a d i r Z A M P
T h e n o r m a l i s a t i o n f a c t o r N is o b t a i n e d b y n o r m a l i s in g t h e m o d e f u n c t io n s
o v e r a t = 0 = t ' h y p e r s u r f ac e Z :
i I ~ b k * m ; o , ~ U ~ k k a m ; 'o d X u = b m m ' 6 ( k - - k ' ) l 6 ( q - - q ' ) 'q(19)
w h e r e
y , = I g l l / 2 g , ~ ~ X v ~ X v{ g ] l / 2 guy. (20)
T h i s p r o c e d u r e y i e l d s t h e n o r m a l i s a t i o n c o n s t a n t
1N = (21)
2~12co11/2
s o t h a t t h e n o r m a l i s e d m o d e f u n c t i o n s a r e
1e - i r t ~ o, - m Q ) e i m ~ e ik ~ J m ( q r ) (22)
~ k k q , . ; o , ( t , x ) - - 2n12r
L e t u s n o w c o n s i d e r t h e d i s t i n c t i o n b e t w e e n p o s i t i v e a n d n e g a t i v e e n e r g y
m o d e s . T h e L i e d e r i v a t iv e w i t h r e s p e c t t o t h e t i m e - l i k e K i l li n g v e c t o r R u , f o r t h e
r o t a t i n g c o o r d i n a t e s , i s
L ~~ - - R " V . = 7 - 1 8O t (23)
w e , t h e r e f o r e , g e t
5 e s O k q m ; o , = - - i ( 0 9 - - m r 2 ) ~ kk q ,. ; ,o (24)
* * ( 2 5 )s O k q ~ ; = i ( c o - - m s ~ bk ~ =; ~ o .
I t is ev ide n t t ha t t he m o de 0kqm;o , (0*q,. ;,~) i s no t ge ne ra l ly o f pos i t i ve (nega t ive)
e n e r g y . H e r e o u r r e s u lt s c o i n c i d e w i t h t h o s e o f L e t a w a n d P f a u t s c h [1 , 3] a l-
t h o u g h t h e y s t a r te d w i t h a w r o n g t r a n s f o r m a t i o n . T h e r o t a t i n g m o d e s ~ a r e
i d e n ti c a l to t h e c y l in d r ic a l M i n k o w s k i m o d e s
1e i ,, ,t ' e i m r e i k z ' (26)
_ 27z 12co11/2 J , , ( q r ' )
w h e n b o t h m o d e s a r e e x p r e s se d in t h e s a m e c o o r d i n a t e s y st e m . N o t e t h a t ~ is
n o r m a l i s e d o v e r t h e s a m e h y p e r s u r f a c e a s ~ .
T h e f ie ld o p e r a t o r 4~ c a n n o w b e w r i t t e n i n t e r m s o f t h e p o s i t iv e e n e r g y
r o t a t i n g m o d e s a s
= ~ I ( a k q ,~ ~ k k o m + ak + qm ~ '~ q m ) q d q d k . (27)m = ~
t o - m O > O
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V o l . 3 7 , 1 98 6 A n o t e o n q u a n t i z a t i o n i n r o t a t i n g c o - o r d i n a t e s 3 93
S i n c e w e r e q u i r e co - m ~2 > 0 f o r p o s i t i v e e n e r g y , t h e v a c u u m w i l l n o t b e t h e
l o w e s t e n e r g y s t a t e .
T h e B o g o l i u b o v t r a n s f o r m a t i o n b e t w e e n t h e tw o s e ts o f m o d e s i s
- - + + ( 2 8 )k q m ~ C L kqm , a k q m ~ - -- a k q m
w h e r e g ( a) r e fe r t o M i n k o w s k i ( r o t at i n g ) c o o r d i n a t e s . T h u s , a s fo r L e t a w a n d
P f a u t s c h [1 , 3], t h e v a c u u m d e f i n e d i n r o t a t i n g c o o r d i n a t e s i s j u s t t h e M i n k o w s k i
v a c u u m . O u r f i n al re s u l t a g r e e s w i t h L e t a w a n d P f a u t s c h [ 1, 3 ] i n t h a t t h e f ie l d
t h e o r i e s i n r o t a t i n g a n d c y l i n d r ic a l c o o r d i n a t e s d if fe r o n l y b y a r e d e fi n i ti o n o f
t h e e n e r g y o f a m o d e . H o w e v e r , w e b e li e v e t h a t t h e e a r l ie r d e r i v a t i o n s o f t h is
r e s u lt b y L e t a w a n d P f a u t s c h [ 1, 3 ] a n d b y D e n a r d o a n d P e r c a c c i [4] w e r e f l a w e d
b y t h e u s e o f a w r o n g c o o r d i n a t e s y s te m .
4 . C o n c l u s i o n s
W e h a v e s t u d i e d t h e q u a n t i z a t i o n o f a s c a l a r f ie ld in t h e m e t r i c o f a r e la t iv -
i st ic r o t a t i n g o b s e r v e r . T h e v a c u u m i n th i s m e t r i c d o e s n o t d i ff er f r o m t h e
M i n k o w s k i v a c u u m , a re s u lt id e n ti c a l to t h a t o b t a i n e d b y q u a n t is i n g i n a
G a l i l e a n r o t a t i n g f r am e . I t a p p e a r s t h a t t h e m e r e e x i s t e n c e o f a s u r f ac e o n w h i c h
t h e m e t r i c i s s i n g u l a r i s n o t s u f f i c i e n t t o p r o d u c e a v a c u u m w h i c h i s d i f f e r e n t
f r o m t h e M i n k o w s k i v a c c u m . T o p r o d u c e a d if fe r en t v a c u u m i t s e e m s th a t i t is
n e c e s s a r y t o h a v e a n e v e n h o r i z o n [1 , 7 ].
A c k n o w l e d g e m e n t
T h e a u t h o r s w o u l d l ik e to t h a n k P r o f e s s o r G . D e n a r d o f o r u s e fu l d i s cu s -
s i o n s a n d P r o f . A b d u s S a l a m , t h e I n t e r n a t i o n a l A t o m i c E n e r g y A g e n c y a n d
U N E S C O f o r h o s p i t a li t y a t t h e I n t e r n a t i o n a l C e n t r e f or T h e o r e t i c a l P h y s ic s ,
T r i e s t e w h e r e p a r t o f t h is w o r k w a s d o n e . W e a r e a l s o g r a t e fu l t o J . R . L e t a w a n d
J . D . P f a u t s c h f o r t he ir c o m m e n t s o n a n e a rl ie r v e r s io n o f t hi s w o r k w h i c h
a l l o w e d u s to c la ri fy o u r o w n c o n c e p t s a n d t o s h a r p e n o u r a r g u m e n t s . F i n a ll y ,w e w o u l d l i k e to a c k n o w l e d g e t h a t o u r F i g . 1 is a m o d i f i e d v e r s i o n o f F i g . 2 o f
t h e p a p e r b y L e t a w a n d P f a u t s c h [ 2 ] .
References
[ 1 ] J . R . L e t aw an d J . D . P f au t s ch , Ph y s . Rev . D 2 4 , 1491 (1981).[2 ] J. R . L e t aw a n d J . D . P f au t s ch , J . Ma t h . Ph y s . 2 3 , 425 (1982).
[ 3 ] J . R . L e t aw an d J . D . P f au t s ch , Ph y s . Rev . D 2 2 , 1345 (1980).
[ 4 ] G . D e n a r d o a n d R . P e r c a c c i , N u o v o C i m e n t o 4 8 , 81 (1967).
[ 5 ] E . J . Po s t , Rev . Mo d . Ph y s . 3 9 , 475 (1967).
[6 ] R . A d l e r , M. B as i n an d M . Sch i ff e r , I n t r o d u c t i o n t o G e n e r a l R e l a t i v i t y . M c G r a w H i ll , N e w Y o r k1975.
[7 ] B. S . DeWi t t , Phys . Rep . 1 9 c , 295 (1975).
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394 F. Hussain and A. Qadir ZAMP
Abstract
The quantization of scalar fields in a uniformly rotating frame is reconsidered. It is pointedout that the coordinates usually used do not represent a frame describing unifo rm circular motionand the metric in the correct coordinates has a singularity. The result obtained earlier with the wrong
coordinates, that there is no radiation, nevertheless holds.
Zusammenfassung
Die Quantisierung skalarer Felder in einem gleichf6rmig rotierenden Koordinatensystem wirderneut betrachtet. Dabei wird hervorgehoben, dab die gew6hnlich verwendeten Koordinaten diegleiehf6rmige Kreisbewegung nicht beschreiben und dab die Metrik in den richtigen Koord inateneine Singularitfit aufweist. Trotzdem bleibt das fri iher mit den falschen Koordinaten abgeleitete
Resultat erhalten, dab nfimlich ein strahlungsfreier Zustand existiert.
(Eingegangen: December 12, 1985)
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