Dasgupta, Arundhati

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Indian J. Phys. 73B  (6), 827-834 (1999)U P  B• an in te rn atio n al jo u rn a lBlack hole therm odynam icsArundhati DasguptaThe Institute of Mathematical Sciences, C. I. T. Campus, Chennai-600113A b s t r a c t .  We review latest results on the string theoretic description of the laws of black hole thermodynamics.K eyw ords : Black holes, thermodynamics, quantum aspects PA C S N os. : 97.60.Lf, 04.70.DyB lack  h o le s  are know n as classica l g ra v ita tin g  o b jec ts  which trap  all o b je c ts  in c lu d in g  lig h t . A s  lig h t ca n n o t escap e th em , th ey  co n stitu te  litera lly  ‘black  h o le s’ in sp a c e -tim e . In 1974, S .W . H aw king, changed th is defin ition  o f  black  h oles, w h en  he sh o w ed  th a t  th ey  are n o t really ’b lack’ but h o t rad iatin g  b od ies  a t a  certa in  te m p era tu re . U n d erstan d in g  th ese  new co n cep ts  o f  tem p era tu re , black h o le  ra d ia tio n  and o th er  th erm od yn am ic q u a n tities  like en tropy  w hich  ca m e to  b e  a sso c ia te d  w ith  th e  black hole has been an o u tsta n d in g  prob lem  o f  th e  la s t  few  d eca d es .B lack  h o le s  ar ise  a s so lu tio n s  o f  E in ste in ’s  th eory  o f  G eneral R e la tiv ity . T h o u g h  it  is  very  d ifficu lt to  ob serve black holes, to d a y  th ere are a  d ozen  o f  black h o le  c a n d id a te s  in our universe, d etected  by using ind irect ev id en ces. E in ste in ’s  th eo r y  o f  g ra v ity  re la tes th e  curvature o f  sp a ce-tim e to  m a tter . T h ro u g h  th e  u se  o f  th is , th e  natu re o f  m etric and curvature o f  sp a ce-tim e  for certa in  m a tte r  co n figu ration s, e .g ., a  s ta tic  poin t m ass h as so m e very in­te restin g  p ro p er tie s  w hich  m akes it a  black hole. It can be sh ow n  th a t  th e  cu rvatu re  is  s in g u la r  a t  th e  origin o f  sp a ce-tim e, and th e  m etric is such th a t  c o o r d in a te s  w h ich  are tim e-lik e  b ecom e space-like after a  certa in  cr itica l d is­ta n ce  aw a y  from  th e  s in gu larity . T h is  g ives rise to  tw o  cau sa lly  d isco n n ected  reg ion s w h ich  c a n n o t co m m u n ica te  w ith  each o th er . T h e  cr itica l d ista n c e  is  called  th e  h o r izo n , an d  d eterm in es th e  size  o f  th e  black hole. For a  s ta tio n a ry  m a ss, i t  is  g iv en  by 2 G M / c 2, w here G  =  N e w to n ’s  co n sta n t, M  m a ss o f  th e  p artic le , an d  c  th e  v e lo c ity  o f  ligh t. T h e  boundary o f  th ese  tw o  regions is a  null su rface  lo c a te d  a t  th e  cr itica l radius, called  th e  ev e n t horizon . S in ce a  p article828 Arundhati DasguptaFigure 1. Time- Slice of a Schwarzschild Black Hole: A Hole in Spacewhich falls inside the  horizon canno t com m unicate with an observer s tand ing  outside, it is lost to  the  world. By laws of gravity, the  particle ends up a t the singularity , w here physical laws fail to  describe it anym ore.T his was th e  understand ing  of black holes, when it was found th a t there exists a  special relation betw een the  change in m ass of the  black hole, i t ’s area  of horizon, angu lar m om entum  and electrom agnetic charge. These laws, known as the  laws of black hole m echanics were derived from simple considerations of black hole processes. T he set o f laws were soon supplem ented by the area increasing theorem , which s ta te s  th a t  in any physical process, the  area of a black hole horizon always increases. Taken together these laws can be w ritten  as:(IMbH = T“7^ A b h +UBHdJ + &bh<IQ (1)oTTCidAsH  > 0 (2)W here Mbh, J, Q are m ass, angu lar m om entum , charge of black hole, Abh, a rea  of horizon, Qbh and 4>jg// are angular velocity and electrom agnetic po­ten tia l a t  th e  horizon; k is surface gravity. T he laws resem ble the  first and the  second laws of therm odynam ics. It is easy to  see th a t  Mbh is like the  in ternal energy of th e  black hole, and th e  term s t lB n d J  +  $BH dQ  a re  the  analogs o f P d V  te rm  in th e  first law of therm odynam ics. Hence the  correspondence is exact, if we identify th e  A rea of th e  black hole A b h  as the  entropy, and the  surface gravity, k  w ith th e  tem p era tu re  of the  black hole [1]. T he surface grav­ity o f a  black hole is a  geom etric quan tity , which can be defined as th e  force exerted by an  observer s tand ing  far away from the  black hole to  keep a  unitBlack hole thermodynamics829Figure 2. A Black Hole Space-timem ass tes t partic le  s ta tio n a ry  a t the  horizon. It is rem arkable th a t  this force is co n stan t over th e  horizon, and is determ ined uniquely in term s of black hole param eters . T he  identification of a  black hole with a  therm odynam ic system  however could no t be achieved based on these evidences. T he im p o rtan t point to be clarified was th a t  if there  exists a  hot body a t a  certain tem pera tu re , then it should rad ia te  too . T he black hole a s | t  was known till then, could only absorb , w ith  th e  event horizon serving as a  due way m em brane.It w as in 1974 th a t  Stephen Hawking {2] cam e with the idea of using qu an tu m  m echanics near a  black hole. It was well known th a t, quantum  gravity  becomes im p o rta n t for g rav ita tin g  objects whose Com pton wavelength is of th e  o rder o f i t ’s Schwarzschild radius. This quan tity  called P lanck’s m ass is very sm all, ~  10~ 5#m . For black holes g rea ter than  th is m ass, gravity  can be taken  as a  fixed classical background, while o ther fields can be trea ted  as q u an tu m . T h e  s tu d y  of quan tum  fields on a  fixed curved background is called th e  sem i-classical approxim ation  for a  theory  with gravity. Using the  sem i-classical approx im ation  for scalar fields, Hawking showed th a t  the  scalar vacuum  a t  p a s t infinity evolves in to  a  therm al s ta te  in th e  fu tu re . T he obvious conclusion is th a t  th e  black hole rad ia tes particles in a  therm al spectrum . T he830 Arundhati DasguptaFigure 3. An evaporating black hole space-timem ost interesting p a rt of this result is th a t the tem perature of the distribution is k/ 27t, exactly w hat was required to complete the therm al description of the black hole.Though the tem pera tu re  for a  solar mass black hole ~  10~8A", there can be black holes formed in the  early universe whose Hawking tem perature  is much greater than  the am bient tem peratu re  of the Universe. The questions which were raised by Hawking’s remarkable discovery were, th a t-  Is the black hole a  therm odynam ic system  with properties similar to  or­dinary therm odynam ic system s like a  black body? If so, then, is there a heat bath  associated with the black hole? Is there a  sta tistical descrip­tion in term s of m icro-states, which would determ ine entropy in term s of microscopic degeneracy of sta tes?— For asym ptotically flat black holes, the  system  is unstable against ra ­diation, and hence dissipative in nature, e.g. for a  Schwarzschild black hole tem pera tu re  T h  oc 1/M , which m eans th a t  a  slight radiation in­creases the  tem peratu re , and hence the radiation rate . If the  black hole completely evaporates due to  radiation, then it implies a  non-unitary evo­lution as initial pure s ta te s  evolve to  a  therm al s ta te . For a  burning pieceBlack hole thermodynamics 831of coal, which m acroscopically has the sam e behavior, we know th a t  there  is an  underly ing  microscopic unitary  process which involves excitations of m olecules due  to  ex ternal heat. In quantum  mechanics, un itary  evolution is a  p rim ary  requirem ent, and hence does the presence of black hole imply a  con trad ic tion  to  th e  laws of quantum  mechanics?It is these tw o basic questions which we try  to address subsequently. In the  beginning, H aw king’s resu lts were probed for their validity w ithin th e  semi- classical fram ew ork. It was investigated w hether the  in teractions are basically unitary, and subsequen t averaging leads to  therm alization. This involved con­sidering back reaction  of m a tte r  particles on the gravitational field of th e  black hole. L a te r w ith  th e  adven t of string  theoretical black holes, it was observed th a t  black holes can be trea ted  as ordinary therm odynam ical system s, w ith a s ta tis tica l in te rp re ta tio n  in term s of string  m icro-states. W hether the  above succeed in im plying a un itary  evolution is a  question still being investigated. In the  following, we describe the  work done using the string  approaches.A m ajo r b reak th rough  cam e in the work of [3], where certain string  the ­oretic black holes were described as comprised of string solitonic s ta tes . The focus then  shifted to  understanding  string  black holes and a new physics emerged.S tring  theo ry  is a  cand idate  for the  theory of everything, including quan tum  gravity. In s tr in g  theo ry  the  fundam ental ob jects are not point particles, bu t strings, w ith a  given fundam enta l length ls, associated with them . T he various vibrational m odes of th e  string  appear as particle excitations. One of the  m odes of v ib ra tions yields a field to  which the string  does not couple. T he objects which couple to  these  fields are called D  -  Branes (for an in troduction see[4]). T he  D -p-branes or D irichlet branes are so called as they are described in p e rtu rb a tiv e  s tr in g  theory  as p dim ensional hypersurfaces w ith open strings stuck on them . T h e  strings thus have to  satisfy Dirichlet boundary conditions a t the  position of th e  D -branes. W hen these extended objects are wrapped on com pact spaces, in th e  super-gravity  lim$t, they give rise to  black hole solutions. I t is these  black holes which we shall call string  black holes, and try  to  understand  th e ir  therm odynam ics.In [3], th e  s tr in g  black hole considered was m ade by w rapping D -l-b ranes  and D-5 b ranes of type  IIB  S tring  Theory on a  com pact manifold K 3 x  5 1 ( K 3 s tan d s  for kahler m anifold). It was shown th a t  under certain  restrictions of the  num ber of charges, th e  m etric can be m apped to  an extrem al five dim ensional R eissner-N ordstorm  black hole, a  charged black hole solution of ordinary  gen­eral relativ ity . T h e  system  of D -l/D -5  branes a t  low energies is described by a  conform al field theo ry  w ith a  central charge. In [3] it was shown th a t  for large no. of D -l and  D-5 branes, the  log of num ber of s ta te s  of th is  C F T  is equal to  A Bh /4G, and  hence th e  en tropy  of the  black hole. T he result had rem arkable im plications in th e  in th e  a tte m p ts  for a  sta tis tica l description of black holes. T h e  o th e r  a spec t of th e  D -brane description of black holes, is of course832 Arundhati DasguptaPARTICLE)Figure 4. String theorythe  study  of Haw king rad ia tion , and hence the  un ita rity  puzzle. It was soon shown in [5, 6] th a t  th e  decay ra te  of the  collection of D -Branes into scalars, have th e  sam e rad iation  ra te  as the  corresponding black hole. We exam ined [7] the  case for ferm ions, for a  particu la r four dim ensional black hole obtained by com pactifying M theory  on T 6 x S 3. T he brane configuration consists of th ree M-5 branes in tersecting  along a com m on line . T he com putation  of the  black hole decay ra te  was somehow involved, as th e  ferm ion propagation on these black hole backgrounds is not exactly  determ ined . Under certain  approxim a­tions, we showed th a t  the  ra te  had a s tru c tu re  in which, there was a product of two therm al d istribu tions, a t  two different characteristic  tem pera tu res  T l and T r , such th a t  4tt/ k =  1 /7 / ,+  1 /7 r . One of the  therm al d istribu tions was ferm ionic and the  o th e r was bosonic. T he in te rp re ta tion  of th is in term s of a microscopic theory  had to  be done by looking a t the  branes. We took help of evidences of the  fact th a t  the  excitations along the  com m on intersection lines of th e  brane con stitu ted  a  1+1 conform al field theory. We used an in teraction between the  C F T  m odes and th e  bulk grav itino , which corresponds to  two open strings colliding to  give a closed string  in the  bulk for L)-Branes. The grav itino  w ith vector polarization along the  com pact d irections gives rise to  a fermion in the  four non-com pact d irections, and our calculations reproduced th e  black hole emission rates. T he two characteristic  tem pera tu res  were the effective left and right tem p era tu res  o f the  1+1 dim ensional C F T .As described above, th e  s trin g  black holes, have a  microscopic description in term s of a  C onform al field theory, constitu ted  by the  excitations of the  D- B rane6 o r M -B rane configurations. T his fac t has su p p o rt from th e  evidence th a t  th e  degeneracy of s ta te s  of th is  C F T  and th e  decay ra te  in to  scalars, ferm ions, e tc  m atch the  black hole rad iation  rates. B ut, the  question is, is this sufficient evidence for a  proof? O ne of the  m ajo r shortcom ings of th e  above approach  is th a t  th e  descrip tion of th e  B ranes exists when the  background m etric  is fla t. W hen th e  g rav ita tiona l coupling and the  in teraction  between the branes becam e large, th e  b ranes collapses to  give a  black hole, and it rem ains to  be answ ered w hether under th is collapse process, the  degeneracy and  theBlack hole thermodynamics 83Srad ia tion  ra te  do no t undergo renorm alization. Due to  this problem  ano ther crucial question  rem ains unansw ered, and th a t  is of the  location of degrees of freedom  of th is  microscopic theory in the  black hole background. In the  m icroscopic descrip tion  of any therm odynam ic system , e.g. a  gas, we identify th e  m olecules of the  gas as the  microscopic degrees of freedom. However, in presence of th e  black hole where concepts erf horizon arise, the brane degrees of freedom  becom e obscure. In try ing to  investigate this question it was found in [8], th a t  th e  near horizon geom etries of alt these black holes, have the  sam e isom etries as a n ti-s itte r  space, and the product of a  com pact manifold. T h is  led to  a  con jec tu re  th a t  string  theory on an ti-de S itte r space is dual to  a  conform al field theo ry  which lives on i t ’s boundary [8 , 9].T he near horizon geom etries of some 5 and 4 dim ensional black holes are found to  be 3 dim ensional anti-de S itte r space with some global identifications. T h is space-tim e has two horizons, and is a  black hole solution called the BTZ black hole [101. T h e  m ost in teresting aspect abou t the BTZ black hole is th a t  i t ’s en tropy  can  be reproduced from a C F T  [11], and has the  exact value as th e  higher dim ensional black hole. We showed th a t  the  Hawking emi«Hon ra tes  for m assless ferm ions which are non-minimally coupled to  gauge e s h i834 Arundhati Dasguptahigher dimension have a radiation rate in the BTZ space-time, which is same as the higher dimensional black hole [12]. These results confirm and enrich the conjecture that the microscopic degrees of freedom of the particular black holes considered are the degrees of freedom of a Conformal Field Theory, and they lie on the boundary of the near horizon geometry, obeying a ‘holographic’ principle.It is undoubtedly true tha t the description of certain supergravity black holes in terms of microscopic theory is a major step in understanding black hole thermodynamics. However, this result excludes a class of black holes, including the Schwarzschild black hole. The black holes which are expected to be found in nature belong precisely to the above. Hence much work needs to be done in this direction. Also, though we understand the thermodynamics of the string black holes, the crucial issue of non-unitary evolution in Hawking radiation is not understood very well. It still remains to be seen whether we can think of a black hole just like a burning piece of charcoal.R eferences1. J M Bardeen, R Carter and S W Hawking Comm. Math. Phys. 31 161 (1973)2. S W Hawking, Comm. Math. Phys. 43 199 (1975)3. A Strominger and C Vafa Phys. Lett. B 379 99 (1998)4. J Polchinski, S Chaudhuri and C V Jhonson Notes on D-Branes (hep-th/9602052)5. C Callan and J Maidacena Nucl. Phys. B472 591 (1996)6. S R  Das and S D Mathur Phys. Lett. B375 103 (1996)7. S Das, A Dasgupta, P Majumdar and T Sarkar (hep-th/9707124)8. J MalcLcena Adv. Theor. Math. Phys. 2 231 (1998)9. E Witten Adv. Theor. Math. Phys. 2 253 (1998)10. M Banados, C Teitelboim and J Zanelli Phys. Rev. Lett. 69 1849 (1992)11. S Carlip Class. Quant. Grav 15 3609 (1998)12. A Dasgupta Phys. Lett. B 445 279 (1999)

Black Hole Thermodynamics

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Black Hole Thermodynamics

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