Kalam, Mehedi

Rahaman, Farook

IACS Institutional Repository

Indian J. Pkys. 75B (6), 505-507 (2001)U P  B— an international journalCharged AdS black holes and test particlesF a r o o k  R a h a m a n *  a n d  M e h e d i  K a la m  Department of Mathematics, Jadavpur University, Kolkata-700 032, India E-mail ’ jumath@cal.vsnl.nct.inReceived I December 2000, accepted 18 July 2001Abitract : We have presented a detailed analysis of the motion of the test particle around Rcissncr NordstrOm-AdS black holes in five dimensional space time. The study of the trajectories of the particles have been done using Hamilton Jacobi (H-J) formalism We have considered test particles with various masses and electric charges and examine its behaviour both in static and non-static casesKeywords * Charged AdS black hole, test particles, Hamilton-Jacobi formalismPACSNos. 97.60.Lf, 04 20,Jb. 04 70.-s1. IntroductionR e c e n t ly ,  a  n e w  f a m i l y  o f  e x a c t  b l a c k  h o le  s o lu t io n s  h a s  b e e n  d i s c o v e r e d  b o t h  in  f o u r  a n d  h ig h e r  d i m e n s io n a l  s p a c e  t im e s  [ 1 ] .  It  i s  in t e r e s t in g  to  s t u d y  b l a c k  h o le s  in  a n t i-d e  S i t t e r  ( A d S )  s p a c e  b e c a u s e  th e  A d S  c o n f o r m a l  f i e ld  t h e o r y  ( C F T )  d u a l i t y  r e la t e s  th e  p r o p e r t ie s  o f  th e s e  b la c k  h o le s  to  th e r m a l p r o p e r t ie s  o f  a  d u a l  c o n fo r m a l  f i e ld  t h e o r y  r e s id i n g  o n  th e  b o u n d a r y  o f  A d S  s p a c e  [ 2 ] .  A  f i v e  d i m e n s io n a l  A d S  a n a lo g u e  o f  R e i s s n e r - N o r d s t r d m  A d S  ( R N A d S )  s o lu t io n  o f  t y p e  I I B  s u p e r  g r a v i t y  w a s  d e r iv e d  b y  C h a m b l in  et al [ 2 ] .  T h e  a c t io n  f o r  th is  m o d e lis [2]/  = -I1 6 ; r G] (1)S o lu t io n s  o f  t y p e  I I B  s u p e r  g r a v i t y  d e s c r i b i n g  R e i s s n e r  N o r d s t r o m  A d S  b l a c k  h o le s  w it h  a n  in t e r n a l ^  is  [ 2 ]ds'^ =  - ^ r )  +  - ^ d r ^  + r ^ d n ]  (2)w ith  g (r) = ] - ^ + ^  + r^, ( 3 )w h e r e  M  a n d  Q  m e a s u r e  th e  b l a c k  h o l e ’ s  m a s s  a n d  c h a r g e  r e s p e c t iv e ly .  T h e  e le c t r o - m a g n c t ic  p o te n t ia l  in a  g a u g e  r e g u la r  o n  th e  o u te r  h o r iz o n  is-4, = [ ^ r ^ ) - H r ) ] d t ,w h e r e  ^ ( r )  =^ Q _ (4 )H e r e , R  is  th e  c u r v a t u r e  s c a la r  in  5 - d im e n s io n a l  s p a c e ,  is  th e  g a u g e  f i e ld ,  Fat ^ FahF°^) s ta n d s  f o r  M a x w e l l  f i e ld  s t r e n g t h . T h e  p a r a m e t e r  / m e a s u r e s  th e  s iz e  o f  th e  T h e  A d S s  5 *  g a u g e d  s u p e r  g r a v i t y  in  f i v e  d im e n s io n  h a s  an  S 0 ( 6 )  g a u g e  s y m m e t r y ,  a s s o c ia t e d  w ith  th e  g r o u p  o f  i s o m e t r ie s  o f  5 ® .H e r e  r+  is  th e  la r g e s t  r e a l  p o s i t i v e  r o o t  o f  g ( r )  =  0 . T h e  a im  o f  th e  p r e s e n t  w o r k  is  to  in v e s t ig a t e  th e  m o t io n  o f  th e  te s t  p a r t ic le s  in  th e  g r a v it a t io n a l  f i e ld  o f  a  f i v e  d im e n s io n a l  R N A d S  b l a c k  h o le  u s in g  H a m i l t o n - J a c o b i  ( H - J )  f o r m a l i s m  a n d  e x a m in e  b o th  s ta t ic  a n d  n o n - s t a t ic  c a s e s  a n d  a l s o  fo r  c h a r g e d  a n d  u n c h a r g e d  te s t  p a r t ic le s .2 .  T h e  H a m i l t o n  J a c o b i  f o r m a l i s mL e t  u s  c o n s id e r  a  te s t  p a r t ic le  h a v in g  m a s s  m a n d  c h a r g e  e m o v in g  in  th e  g r a v it a t io n a l  f i e ld  o f  a  f i v e  d im e n s io n a l'Corresponding AuthorPermanent address * Khodar Bazar. Baruipur, 24-Parganas (South), West Bengal-743 302, India©  2 0 0 1  I A C S506 Farook Rahaman and Mehedi KalamR N A d S  b l a c k  h o le .  S o  th e  H - J  e q u a t io n  f o r  th e  te s t  p a r t ic le  is  [ 3 ]d S+  eAi  ^ \ +  m^ = 0 , (5)w h e r e  a n d  Af, a r e  th e  c l a s s i c a l  b a c k g r o u n d  f i e ld s  ( 2 )  a n d  ( 4 )  r e s p e c t i v e l y  a n d  S  is  th e  s ta n d a r d  H a m i lt o n ’s  c h a r a c t e r is t ic  f u n c t io n .  F o r  m e t r i c  ( 2 ) ,  th e  e x p l i c i t  fo r m  o f  th e  H - J  e q .  ( 5 )  i s  [ 4 ]d S  J  1 1P r )H e r e ,  x\, X2 , xy a r e  th e  c o o r d in a t e s  o n  th e  s u r f a c e  o f  th e  3 - s p h e r e .  A s  t h e r e  is  n o  e x p l i c i t  d e p e n d e n c e  o f  t a n d  c o o r d in a t e s  o n  th e  3 - s p h e r e ,  a  n a tu r a l fo r m  f o r  th e  H - J  f u n c t io n  S {i,  r ,  jc i ,  xi, x^) w i l l  b eS  = - E - t p ] X \  P 2 X2 + P 3 -^ 3 -  ( 7 )H e r e ,  th e  c o n s t a n t  E  i s  id e n t i f i e d  a s  th e  e n e r g y  o f  th e  p a r t ic le  a n d  p\, p 2 , P 3 a r e  th e  m o m e n t u m  o f  th e  p a r t ic le  a lo n gd i f f e r e n t  a x e s  o n  th e  3 - s p h e r e  w ith p  p = {p\ a sth e  r e s u l t in g  m o m e n t u m  o f  th e  p a r t ic le .I f  w e  s u b s t i t u t e  th e  a n s a t z  ( 7 )  f o r  S  in  th e  H - J  e q u a t io n , th e n  w e  g e t  th e  f o l l o w i n g  e x p r e s s i o n  f o r  a s  [4 ]--- ------ dr. ( g )rg gH e r e ,  e= ± h  th e  s ig n  c h a n g e s  w h e n e v e r  r  p a s s e s  th r o u g h  a  z e r o  o f  th e  in t e g r a n d  in  ( 8 ).N o w ,  f o r  th e  t r a j e c t o r y  o f  th e  p a r t ic le  f o l l o w i n g  th e  H - J  m e t h o d , w e  c o n s id e r ,— T =  c o n s t a n t ,  —  =  c o n s t a n t ,^ E  a p \d S  d S   ^ ^d p 2  d p i ,   ^ JS o  th e  p a th  in  p a r a m e t r ic  fo r m  g i v e s  ( c h o o s i n g  c o n s ta n ts  to  b e  z e r o  w it h o u t  a n y  lo s s  o f  g e n e r a l i t y )  [4 ]P lLsr^g g (10)H e n c e ,  th e  r a d i a l  v e l o c i t y  o f  th e  p a r t ic le  i sdtE - e Q 1 1(11)g^F o r  th e  t u r n in g  p o in t s  o f  th e  p a th  o f  th e  p a r t ic le ,  w e  h a v e  dr =  0  a n d  c o n s e q u e n t ly  th e  p o t e n t ia l  c u r v e s  a r e  [4 ]pQm r l ^2 ^x | l -r m(12)d r ,  (9)S i n c e  th is  la r g e  e x p r e s s i o n  c o n t a in s  s o  m a n y  a r b it r a r y  p a r a m e t e r s  n a m e ly ,  e, Q, m, p, M, w e  c a n  n o t  d e f i n e  p h y s ic a l  c h a r a c t e r s  o f  th e  p o t e n t ia l  c u r v e .  B u t  o n e  c a n  f in d  a n a ly t ic a l l y  th e  e x t r e m e  p o in t s  o f  th e  p o t e n t ia l  c u r v e .I n  a  s t a t io n a r y  s y s t e m , V(r) m u s t  h a v e  a n  e x t r e m a l  v a lu e  g i v e n  b y(13)H e n c e ,  w e  g e t  th e  f o l l o w i n g  e q u a t io n( H IW e  s e e  th a t  r  =  r+ is  a  s o lu t io n  o f  ( 1 4 )  p r o v id e d  r *  + M rl -2Q^=0  [ s in c e  r. i s  a  r o o t  o f  g ( r )  =  0 ] T h e  a b o v e  e q u a t io n  im p l ie sr , ^ = S + r[ w h e r e  + a n d  T = ^Q^ ^H e n c e ,  t r a je c t o r y  o f  th e  te s t  p a r t ic le  is  b o u n d e d .Charged AdS black holes and test particles 5013. Test particle in static equilibriumIn s t a t ic  e q u i l ib r iu m ,  th e  m o m e n t u m  p  m u s t  b e  z e r o .  S o  f r o m  ( 1 4 ) ,  th e  v a l u e  o f  r  f o r  w h ic h  p o t e n t ia l  w i l l  b e  a n  e x t r e m a l ,  i s  g i v e n  b y( B )F r o m  th is ,  w e  g e t,1 6 ,I f  —  =  2 , w e  s e e  th a t it is  p o s s ib le  to  h a v e  b o u n d  o rb it  fo r  mth e te s t  p a r t ic le  p r o v id e d  M  < J 2 Q  s in c e  th is  e q u a t io n  h a s  at le a s t  t w o  r e a l  r o o t s  w ith  o n e  p o s i t iv e .W e  a l s o  s e e  th a t th is  is  a n  a l g e b r a i c  e q u a t io n  o f  e v e nd e g r e e  w ith  n e g a t iv e  la s t  te r m  p r o v id e d  —  >  2 .mT h i s  e q u a t io n  h a s  at le a s t  t w o  r e a l  r o o t s  w ith  o n e  is  p o s it iv e ,  s o  p a r t ic le  m u s t  b e  t r a p p e d  b y  R N A d S  b la c k  h o le .4. Test particle in non-equilibrium stateCase I  . Uncharged test particle (e ^ 0)N o w  th e  e x p r e s s i o n  ( 1 4 )  s i m p l i f i e s  to2/7^m( 2 M  -  'l[^75---- ^  + (17)T h u s ,  w e  g e t  th e  f o l l o w i n g  a l g e b r a i c  e q u a t io n+ { m ^ M -2 p ^ y * -A Q ~ m ^ r ~  =0. (18)T h i s  e v e n - d e g r e e  a l g e b r a i c  e q u a t io n  h a s  at le a s t  t w o  r e a l  ro o ts  s in c e  its  la s t  t e r m  is  n e g a t iv e  (w it h  o n e  r o o t  p o s i t iv e ) .S o  it i s  p o s s ib le  to  h a v e  a  b o u n d  o r b it  fo r  th e  te s t  p a r t ic le .Case 1 1 : Test partic le  with charge (i e. e ^  0)IT o m  e q . ( 1 4 ) ,  w e  h a v e  th e  a l g e b r a i c  e q u a t io n  4 ^ 4 ^ 1 6  + 4 m ^ (2 M n C  -^2Mp^ -2 p ^ )r^ ^- {4m ^p^Q^ + 4 e -Q ^ p ^  -2 M m ^  -2 M p ^  -^2p^)r^  + |2 ( 2 M /m  ^ - y lM p ^  - 2 p ^ ) ( 2 p ^ M - 4 m ^ Q ^ )+ 4e^Q^ m -  M  ^  4e^Q^ p ^ ] r ^-{\2p'^ Q'^ {2Mm^- +2Mp^ - 2 / j 7 - 4 c 2 ^ V A / } r ’+ {\2p^Q‘ [^r,r-Q- - 2 p -M )- 4e^ -Q p^^ -} r ‘+ 36/?2g2=0 (19)W h a t e v e r  r e s t r ic t io n s  w c  im p o s e  o n  th e  p h y s ic a l  p a r a m e t e r s ,  o n e  c a n  s e e  th a t  th e  a b o v e  a l g e b r a i c  e q u a t io n  ( o f  d e g r e e  1 6 )  c o n t a in s  e v e n  n u m b e r s  o f  v a r ia t io n  o f  s ig n s .  S o  b y  D e s c a r t e s  r u le  o f  s ig n ,  th is  e q u a t io n  h a s  e i t h e r  s o m e  r e a l  r o o t s  o r  n o  r e a l  ro o ts . S o  u n le s s  w e  g e t  e x a c t  n u m e r ic a l  v a l u e s  o f  th e p h y s ic a l  p a r a m e t e r s ,  w c  c a n  n o t  s a y  w h e th e r  th e  c h a r g e d  te s t  p a r t ic le  c a n  b e  t r a p p e d  b y  b la c k  h o le  o r  n o t .5. Concluding remarksIn  th is  p a p e r ,  w e  h a v e  e x a m in e d  th e  b e h a v i o u r  o f  a  le s t  p a r t ic le  in  th e  g r a v i t a t io n a l  f i e ld  o f  a R N A d S  b l a c k  h o le  in f i v e  d im e n s io n , u s in g  th e  f o r m a l i s m  d u e  to  H a m ilt o n  a n d  J a c o b i .  T h e  te s t  p a r t ic le  is  c o n s id e r e d  to  b e  b o th  s ta t ic  a n d  n o n - s t a t ic  a s  w e l l  a s  c h a r g e d  o r  u n c h a r g e d .In  s ta t ic  c a s e ,  w c  h a v e  s e e n  th at th e  te s t  p a r t ic le  c a n  b e  t r a p p e d  d u e  to  s o m e  r e s t r ic t io n  o n  p h y s i c a l  p a r a m e t e r s  s a y^  >  2  ( o r  =  2  a n d  M  < y/2 Q ).F o r  n o n  e q u i l ib r iu m  te s t  p a r t ic le ,  w e  h a v e  e x a m in e d  th e p o s s ib i l i t y  f o r  b o u n d  o r b it .W e  s e e  th a t u n c h a r g e d  te s t  p a r t ic le  a l w a y s  b e  t r a p p e d  b y  R N A d s  b la c k  h o le .W e  c a n  n o t  s a y  e x a c t l y  w h e th e r  a  c h a r g e d  te s t  p a r t ic le  is t r a p p e d  o r  n o t  b y  R N A d S  b la c k  h o le .F o r  fu tu r e  w o r k ,  it w i l l  b e  in t e r e s t in g  to  in v e s t ig a t e  d e t a i l s  h o w  th e  p h y s i c a l  p a r a m e t e r s  a f f e c t  o n  th e  t r a j e c t o r y  o f  th e  te s t  p a r t ic le  in  th e  g r a v it a t io n a l  f i e ld  o f  d i f fe r e n t  b l a c k  h o le s .AcknowledgmentA u t h o r s  a r e  t h a n k fu l  to  th e  r e f e r e e  fo r  h is  v a l u a b le  c o m m e n t s  a n d  c o n s t r u c t iv e  s u g g e s t io n s .Kefercnces[II D Garfikle, Horowitz and A Slrominger Phyx Rev D43 3140 (1991), J Prcskill, P Schwarz. A Shapcrc, S Trivcdi and F Wilezck Mod Phys U t t  A6 2353 (1991). F Written Phys Rev D44 314 (1991); A Chamblin, R Fmparan, C V Johnson and R C Myers Phys Rev D59 10640 10 (1999). S Sethi and L Suskind Phys Lett B400 265 (1997), G W Gibbons and K Meada Nucl Phys B298 741 (1998). M M Caldareiliad and D Klcmm AW/, Phys B545 434 (1999)[2] S W Hawking and H S Reall Phys Rev. D61 0240 14 (1999), A Chamblin, R Kmparam, C V Johnson and R C Myers Phys Rev D60 0640 18 (1999)[3] L D L^ andau The Ctassicai Theory  ^ of Fields (New York Pergamon) (1973)[4] S Chakraborty Gen Rel Grav 28 1115 (1996)

Charged AdS Black Holes and Test Particles

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Charged AdS Black Holes and Test Particles

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