Information Storage in Black Holes
M. D. Maia
70919-970
Bras\'{\i}lia
DF
The information loss paradox for Schwarzschild black holes is examined, using the ADS/CFT correspondence extended to the $M_6 (4,2)$ bulk. It is found that the only option compatible with the preservation of the quantum unitarity is when a regular remnant region of the black hole survives to the black hole evaporation process, where information can be stored and eventually retrieved.
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\begin{document}
\title{Information Storage in Black Holes}
  
\author{M. D. Maia\\
The Institute of Physics, University of Brasilia\\ 70919-970,
Bras\'{\i}lia, DF,  maia@unb.br}
\maketitle
\begin{abstract}
The information loss paradox for  Schwarzschild black holes   is examined,  using  the  ADS/CFT correspondence  extended to  the  $M_6 (4,2)$ bulk. It is found that  the only option compatible with the preservation of  the quantum unitarity is when   a  regular  remnant region  of the black hole survives to the black hole  evaporation process, where  information  can be  stored and eventually  retrieved.  
\end{abstract}

\begin{center}
{\large The  ADS/CFT  Correspondence}
\end{center}


 Conformal  covariance  in physics  was   firstly  introduced in electrodynamics back in  1909, but its  applications to fundamental physics 
 are  still  lagging \cite{Fulton}.
  Only very recently   the  ADS/CFT correspondence,   between   conformal fields in the Minkowski space  $M_4$    and gauge field theory  on the five-dimensional  deSitter gravity $AdS_5$,  started to make an impact on some relevant  issues related to  the  quantum  field  theory  on curved  space-times	\cite{Maldacena}. 
  
  
  The  ADS/CFT  correspondence  can be  easily
   understood  when it is  seen  from   the point of  view   of   the  theory of  sub-manifolds.  Indeed, the four-dimensional anti-deSitter space-time  $AdS_4$ is   a hypersurface of negative constant  curvature embedded in the flat space $M_5 (3,2)$ \cite{Rosen}. By simply adding   one  extra   space-like   dimension, we  obtain    the five-dimensional negative constant curvature  surface   $AdS_5$ 
 as  a  hypersurface embedded    in the flat space $M_6 (4,2)$.    As it happens,  the 15-parameter conformal group   $C_o (3,1)$ acting on Minkowski's space-time  is  isomorphic  to the  group  $SO(4,2)$.   
  Since  $AdS_5$ is  a maximally symmetric  sub-space  of $M_6 (4,2)$,  it follows that any  (pseudo)  rotation  on  that  sub-space  corresponds  to  a conformal transformation  in $M_4$.   Thus,  for  a given   conformally invariant  field defined in Minkowski's space-time,  in  principle there  should be   a  corresponding   isometrically invariant field  in   the  $AdS_5$  space. 
    
  The  construction of the correspondence itself  may be not so  simple  as it depends on the nature and properties of the fields in  question.
  For example, we know how difficult it is to exactly solve  the non-Abelian Yang-Mills equations  in $M_4$. However,   in the  five-dimensional $AdS_5$  space   there exists  a particular   solution  for  the super-symmetric gauge-field, associated with the internal five-sphere $S^5$.  This solution appears in the derivation of the    heterotic  $E_8 \times  E_8$ string theory in the  10-dimensional  space $AdS_5 \times  S^5$ from  M-theory, through  a $Z_2$-symmetric orbifold compactification of the  $11^{th}$  dimension \cite{HW}. Therefore,  we obtain  a  super-conformal Yang-Mills field in  $M_4$ corresponding to  a  super-symmetric Yang-Mills field  in  $AdS_5$. Reciprocally,   the ADS/CFT  correspondence  can be used to explain how  the  unitarity   of  the quantum super-conformal Yang-Mills  field theory in  $M_4$, also  holds  in the five-dimensional $AdS_5$ gravitational environment \cite{Hawking2004}.
    
Four-dimensional gravity
can be recovered from this scheme as  a  brane-world of the 
 $AdS_5$ bulk. Accordingly, the  equations of  motion of the brane-world result from Einstein's  equations for the bulk geometry, together with   the four-dimensional  confinement  of  gauge fields,  and with the  exclusive probing of the extra dimensions by  gravitons. Although the full degrees of freedom required by the dynamical evolution of the brane-world may require more than five dimensions  \cite{ADD}, a  particular (albeit  very popular) model uses the  $AdS_5$  bulk by imposing  the same  $Z_2$   symmetry  across   fixed   boundary brane-worlds \cite{RS}. 
 
 However, the probing of the extra dimensions by gravitons should be  consistent   with  the classical  limit of the  future theory of quantum  branes,   generating 
   small deformations (or perturbations)  of the classical brane-world geometry. Coincidently, the theory of  sub-manifolds 
  developed  by   J. Nash describes such perturbations of   geometry,  requiring only that    the embedding functions  should  remain differentiable and regular \cite{Nash,Greene}. The regularity is  relevant to the application of the   inverse functions theorem, allowing us   to  keep a local  correspondence between  the original Riemannian  and the embedded sub-manifold. The problem appears  when  we try to apply Nash's  theorem  to   perturbations of a $Z_2$-symmetric brane-worlds  embedded in the   $AdS_5$ bulk. We find that  the  result is  not always compatible   with   the required regularity,
essentially   because each perturbation of a given manifold would have  a  mirror image  on the opposite side,
leading to an ambiguous  definition of its tangent vectors \cite{MaiaGDE}.  

 Another  difficulty facing a  five-dimensional  bulk with constant curvature
 is that  it  poses severe restrictions  to the  embedded brane-world.  For  example, in  the particular case of  a sub-manifold  with a
 spherically symmetric  metric, the  extrinsic  curvature turns  out to be  proportional to the metric: $k_{\mu\nu}= \alpha_0\; g_{\mu\nu}$, where  $\alpha_0$ denotes an integration  constant, characterizing umbilical points. With this, the  line element of   a  spherically  symmetric four-dimensional vacuum brane-world in a  five-dimensional constant curvature bulk  is like
 \begin{eqnarray*}
ds^2\!\! =\!\! (1\!\! -\! \frac{2m}{r}\!+\beta_0^2 r^2)^{-1}dr^2  +r^2 d\theta^2 
+ r^{2}\sin^{2}\theta d\phi^2
-(1\!\!-\!\!\frac{2m}{r} + \beta_0^2 r^{2})dt^2 
\end{eqnarray*}
where   $\beta_0^2 =( 3\alpha_{0}^2 -\Lambda_* )$  and where the    cosmological constant $\Lambda_*$  results from
 the  bulk's  constant curvature. Even considering 
$\Lambda_* =0$,  the constant $\alpha_0$  cannot  be  zero, under the penalty of  producing  just a trivial  solution $k_{\mu\nu}=0$ (which means a plane!).  The consequence is  that  the  good old  vacuum  Schwarzschild's  solution does not appear  in  the  $AdS_5$  bulk, but we may  have  the  Schwarzschild-Anti-deSitter solution.

\begin{center}
 { \large Extending the ADS/CFT  Correspondence}
\end{center}

 Since  the  $AdS_5$ and   $M_6 (4,2)$ have the same 15-parameter group of isometries,  all  arguments  based  on the  symmetries  of  the   $AdS_5$,
 including the  correspondence with conformal group $C_o (3,1)$, can be  extended to the whole space $M_6 (4,2)$. The  advantage of such extension is that $M_6 (4,2)$  contains  embedded sub-manifolds, including the Schwarzschild solution, which are not   present  in the  $AdS_5$  space.  Furthermore, at the points of  $M_6 (4,2)$ where  the embedding is regular, at least some of the  properties of  conformal fields in  $M_4$ can be  extended to these sub-manifolds,   and not just to the   $AdS_5$ space.
 
   As an example, twistors  are  usually defined  as conformally covariant spinors  on  $M_4$, but   they   can   also be  seen  as    spinors  defined by representations of the Clifford algebra in  $M_6 (4,2)$ \cite{Penrose,Murai}.
Therefore, we may extend some of the properties of twistors to the curved space  $AdS_5$, and  also to any   regular   vicinity of  the  Schwarzschild solution  embedded in $M_6 (4,2)$.
 
 This   extended ADS/CFT correspondence can also be used to   ensure that the   unitarity of the  quantum super-conformal  Yang-Mills theory in  $M_4$ can be extended to    points of  $M_6 (4,2)$ located
near the Schwarzschild's  black hole, by means of the regular embedding functions.
 Using such extension,  we may apply  the  same  arguments  of the 2004  version  of the 
information loss  theorem  to the standard  Schwarzschild black hole, as it appears in the 1975 theorem, instead of  depending only on the presence of extremal black holes in the 
$AdS_5$ bulk. Therefore, admitting that the   geometry  of the  brane-world   remains classical, we look again at the  three   possible  outcomes described at the conclusion of the original theorem  \cite{Hawking1975}: 
  
(a)  \emph{After the evaporation  everything disappears,  leaving  empty space  with no  singularity  at the origin}.  
 
 As in the    1975 paper, all information  is lost during the black hole thermal evaporation process, and  after  the evaporation anything that remains is  an  empty subspace of  the bulk.  On the other hand,   admitting that the unitarity is preserved under the  extended ADS/CFT  correspondence, some information should be retrievable. However,  the complete  evaporation  would imply  also in a   change from the multiply connected  to the simply connected topological configuration of  the sub-manifold geometry,  conflicting  with Geroch's theorem on the maintenance of  topology in a  classical geometry \cite{Geroch}.  
 
  
 (b) \emph{ After the evaporation   a   naked  singularity  remains at $r=0$}. 

 The presence of the  singularity at  $r=0$, means  that after the evaporation the  Schwarzschild  space-time  has been  transformed into  its maximal  analytic  extension \cite{Fronsdal}, implying that 
   the   signature of  the bulk  change  from  $M_6 (4,2)$  to  $M_6 (5,1)$ \cite{Kruskal}. Consequently,   the  bulk symmetry  $SO(4,2)$ changes  to $SO(5,1)$,  and both   the standard and the extended ADS/CFT  correspondences  break  down. Therefore, the  unitarity of the conformal  quantum field near the  naked singularity can  no longer  guaranteed, at least  by use of such correspondences.
    

(c)\emph{ After the evaporation there is  a  remnant of the  black hole in a region  with a  finite diameter defined by  the fundamental  energy scale}.  

 In this scenario the topology  of the brane-world and the signature of the bulk are not affected by the evaporation, so that   the  extended ADS/CFR  correspondence and the  unitarity are preserved. Furthermore, admitting that  the  remnant   is  a regular  manifold  embedded in the  $M_6 (4,2)$  bulk,
 it may act as  an information  storage space. That is, the information that passes through  the  horizon and falls into the 
  remnant,  may eventually be retrieved by use of the  local   inverse  of the embedding map.
  This  regular remnant  is essentially  the same region  envisaged by Hawking in the  1975 paper, with the difference that  here it  is described as a regular sub-manifold embedded in   $M_6 (4,2)$, whose radius of curvature is  defined by the  fundamental energy scale.
\vspace{3mm}
  
  Although this extension of  the  ADS/CFT correspondence  may be  useful  to  understand the behavior of conformal  fields  near  curved  space-times,   it also implies  in   the emergence of   the 11-dimensional  space  $M_6 (4,2) \times  S^5$,  whose meaning, perhaps  in the context of  M-theory,  still remains to  be understood.  
   
     
\begin{thebibliography}{10}


\bibitem{Fulton} T. Fulton, F. Rohrlich
\&  L. Witten, Rev. Mod. Phys. \textbf{ 34}, 442  (1962). 

\bibitem{Maldacena} J. Maldacena,
Adv. Theor. Math. Phys. \textbf{2}, 231 (1998).

\bibitem{Rosen}  
J. Rosen, Rev. Mod. Phys. {\bf {37}, 204, (1965)} (also  C.  Misner,  K.S. Thorn \&  J.A. Wheeler,  Gravitation,  W.H. Freeman, $1^{st}$ ed. page  745 (1973).
\bibitem{HW}  P. Horava  \&  E. Witten,
Nuc. Phys. \textbf{B475}, 94  (1996)
\bibitem{Hawking2004} S. Hawking,  talk  at  the  GR17,  Dublin, Ireland,  July  (2004).
 (www.math.ucr.edu/home/baez/week207).

\bibitem{ADD}
N. Arkani-Hamed et al,  Phys. Lett. {\bf{B429}}, 263
(1998), Phys. Rev. Lett. {84}, 586, (2000).

\bibitem{RS} 
L. Randall \& R. Sundrum,  Phys. Rev. Lett. {\bf{83}}, 3370 (1999);  Ibid  Phys. Rev. Lett.  {\bf 83}, 4690 (1999).
\bibitem{Nash} 
J. Nash, Ann.  Maths. {\bf{63}}, 20 (1956).
\bibitem{Greene} 
R. Greene, Mem. Amer. Math. Soc.  
 \textbf{97}, 1 (1970).

\bibitem{MaiaGDE}   M. D. Maia  et al, to appear in Class. Quant. Grav. (2005),  astro-ph/0403072  (Appendix  A).

\bibitem{Penrose}  R.  Penrose, J. Math. Phys.  \textbf{8}, 345 (1967).
\bibitem{Murai}Y. Murai, Prog. Theor. Phys.  \textbf{9}, 147 (1952),
ibid  \textbf{11}, 441  (1954).
\bibitem{Hawking1975}  S. Hawking, Commun. Math. Phy.\textbf{ 43}, 199 (1975).


\bibitem{Geroch} R. Geroch,  Jour. Math. Phys.  \textbf{8},  782  (1967).


\bibitem{Fronsdal} C. Fronsdal, Phys.  Rev.  \textbf{116}, 778  (1959). 
 \bibitem{Kruskal} M. Kruskal,  Phys. Rev.  \textbf{119}, 1743 (1960).





 \end{thebibliography}

 

\end{document}
\bye

\bibitem{Hellaby}  C. Hellaby,  A. Sumeruk \& G. F. R. Ellis, Int.Jour. Mod.Phys  \textbf{D6}, 211  (1997),
gr-qc/9907042   
 In order  to  keep   the   evaporation theorem consistent with the associated  topological  change, it becomes necessary   to  enlarge  the bulk  in such a way that  its  isometries allows for a   mixed  rotation from the  subspace  $M_6 (4,2)$  to  $M_6 (5,1)$.  
 
 
 
 This can be obtained with the  12-dimensional  space  $M_6 (4,2) \times I\!\!R^6$,  where  we considered the  5-sphere  $S^5$  embedded in the  $I\!\!R^6$ space. The  group of  isometries  of  such space  $SO(10,2)$  is  large enough  to  contain   the  $SO(4,2)$
 the $dS_5$,  the  $AdS_5$  and  the  $SO(5,1)$  subgroups.  
 
 
In such  12-dimensional bulk an  observer  sitting in the  Schwarzschild  subspace, sends  a particle towards the  horizon 
 but  he  will never see  what happens with the in falling particle   because  all geodesics  end  there. But  of  course, he  is  also  sitting  in a  subset of 
   Kruskal's space-time,  whose geodesics
   go all  the way to  $r=0$.  However,
 He will be able  to see the inside the horizon only  this perception  can  occur  only  with   a  change of  signature 
 of one of the extra dimensions  by action of the  big group  $SO(10,2)$. 
 In doing  so,  the  symmetry  $SO(4,2)$  is no longer accesible to  him as it brekes to   $SO(5,1)$.  In this process, 
 his  correspondence with  the  conformal  symmetry also disappears. 
 
 
  \emph{We  conclude that the  topological  change  associated with  the black hole  evaporation breaks the   AdS/CFT  correspondence. However, this  can be repaired  when  we look at the  isometric  subspaces of  $M12 (10,2)$}.


It  is  interesting to notice that  
in the  extended  correspondence  to   $M_6 (4,2) \times  S^5$ both versions of  Hawking's  information theorems  hold
 true,  depending on   what  observer we are talking  with. For   a   four-dimensional  observer  in the subspace  $M_6 (4,2)$   the   interior of the black  hole is  unreachable  and information is  lost according with  the  1975 version. However, 
 in that   same  subspace   the  five-dimensional  observer, but now  sitting in the   $Ads_5$  hypersurface   can retrieve the  information, although mingled  somehow,  according  with the 2004 version.  
 
 On  the other hand, in the same  12-dimensional bulk,  another 
 observer sitting  on  a  subspace  of  $M_6 (5,1)$   sees the particle's geodesics
 as being   complete, going all the  way to  the center of the black-hole.  But  then,   he  will not acknowledge a
   correspondence    with the   super conformal Yang-Mills theory,  while   the second  observer  recovers the  mingled information,   while still keeping  the   conformal field  correspondence. 
       
  
 % that  we cannot  have the  two   de Sitter  spaces embedded in the  same  five dimensional space.  However,  both spaces would  fit     in the six-dimensional space  $M_6 (4,2)$,  if  we  allow  
 % for the  $SO(5,1)$ and   of  $SO(4,2)$ subgroups of  the  conformal group.
 











\emph{ 
 The $D$ isometric   embedding  functions $\cal Z^A$ 
 and its   derivative map  ${\cal Z}^*$  
 with  components   ${\cal Z}^A_{,\mu}$  are the carriers  of the bulk-brane correspondence,  according with the embedding equations
  \begin{equation}
\phantom{x}\hspace{-0.5cm}\!{\cal Z}^{A}_{,\alpha}{\cal Z}^{B}_{,\beta}{\cal G}_{AB}\!\!
=\!\!g_{\alpha\beta},\;
{\cal Z}^{A}_{,\alpha}{\eta}^{B}_{b}{\cal G}_{AB}\!\!=\!\!y^{c}A_{\alpha cb},\;
{\eta}^{A}_{a}{\eta}^{B}_{b}{\cal G}_{ab}\!\!=\!\!g_{ab}
\!\!\!\! \label{eq:Z}
\end{equation}
where  $\eta^{a}$  are  the $N$ normal  vectors generating the  extra-dimensional (with  co dimension  $N= D-4$) subspace.
 The first equation tells  that  the  
 metric  geometry of the brane  is  induced  by the  geometry of  the bulk. }
 
  %The advantage over the   pure  Riemannian  description of  
 %the space-time  is  that it  provides a more  detailed   description
 %of the local   shape  through the extrinsic  curvature $k_a$ and  through  the  torsion (vector) $\tau_ab$  in the cases of  co-dimension greater  that two.  
 %The  above equations  tell that  $\{  \cal Z, \eta\} $  are  the  information carriers as  long  as the  $\cal Z$  is  regular.
The regularity of  $\cal Z$  is also  relevant   for the  
 perturbative description of  the   brane-worlds  as dynamically evolving objects in the bulk  \cite{Nash}.
 % which incidently  may  have  an impact on the perturbative analysis of  quantum  brane-worlds. 
%  But the latter  symmetrymetry group is  the same  as  that of the bulk.  But the latter  symmetry
 %is  shared  by all  properly  embedded  geometries  (better say,  
 %the  isometry  of the  $Ads_5$  is   the same  of the bulk  which is  transferred or  shared  to any other  regular embedded  geometry
 %which  i regularly embedded through  \rf{ Z}.     
 
One very common    mechanism   to  obtain   four-dimensional branes  as  hypersurfaces  of  $AdS_5$ is  the use of the same   orbifold  compactification   $S^1/Z_2$ of the Horava-Witten M-theory, where  the  $Z_2$  symmetry is  applied across the  brane-world.
 \cite{RS}. However,  as   it  has  been  pointed out in several occasions, this  impose  severe  constraints 
 on the  embedding  conditions of the brane-world. Essentially,  the    $Z_2$ symmetry  over the brane-world  implies that  the embedding 
 functions  become   non-regular, so  that a  differentiable  embedded structure  cannot be guaranteed  }.
 However, it  includes  all relevant  sub-manifolds  used in
 our present analysis: The Schwarzschild, Kruskal, deSitter and  anti-deSitter   space-times. 
     
One very common    mechanism   to  obtain   four-dimensional branes  as  hypersurfaces  of  $AdS_5$ is  the use of the same   orbifold  compactification   $S^1/Z_2$ of the Horava-Witten M-theory, where  the  $Z_2$  symmetry is  applied across the  brane-world.
 \cite{RS}. However,  as   it  has  been  pointed out in several occasions, this  impose  severe  constraints 
 on the  embedding  conditions of the brane-world. Essentially,  the    $Z_2$ symmetry  over the brane-world  implies that  the embedding 
 functions  become   non-regular, so  that a  differentiable  embedded structure  cannot be guaranteed  }.






for  any  observer  sitting in the  four-dimensional  brane-world 
there is  a  corresponding super conformal Yang-Mills   Field  theory  in  the Minkowski space-time. 
   
   
  
I
 
 
 
%\emph{Therefore,  the  topology  change  associated  with the evaporation of  black-holes   imply that the  arguments for  the  2004  information loss theorem,  based on the  
%AdS/CFT  correspondence through the symmetry $SO(4,2)$  cannot apply}.
An  alternative  way to  explain  the breakdown  of the conformal symmetry is  through  the  use of  an  M-Like  theory. Instead  of the  10-dimensional  space  $Ads_5 \times  S^5$   we  may  use   the  11-dimensional  $M_6  (4,2) \times  S^5$   space.
The group     $SO(4,2)$   is  the isometry  group of the   entire flat  space   $M_6 (4,2)$,  so  that   we may keep  the  same    correspondence  with the 
 conformal field  in  $M_4$.   The  advantage of  such  generalization is  that   we may   chose  different four-dimensional   observers,  sitting   in  different  subspaces of  $M_6 (4,2)$. Besides the  
 $AdS_4$   as  a  circle in  $AdS_5$   we may also include  the  Schwarzschild  space-times.



 %

%$Z_2$   symmetry.  Another    problem is   that  as  a  consequence of  Gauss'  equations, the  $AdS_5$  bulk    lends  a   cosmological term  to  most of  its  submanifolds.  Thus, for example, if  one forces the embedding  of  Schwarzschild's  or  Kerr  space times,  we end  up  with  a  Schwarzschild-Anti-deSitter and  a  Kerr-Anti-de Sitter.  Thus  we lose the   usual   black-hole  horizon at  $r=2m$.
   
%On the other hand,  considering that the    which  is  also the isometric embedding  space of the  Schwarzschild  space-time,  we may  consider the  correspondence  
%$$   
%C_{M_{4}} \sim   SO(4,2)  \stackrel{\cal Z}{\rightarrow} Schw %\subset  M_6 (4,2)  \times  S^5
%$$
%Here  an  observer   sitting on a  Schwarzschild  brane-world  
%can  see  the  same  horizon  and   therefore  he  may experience the same loss of  information  as in the  1975   theorem.

%However,  in this  case we   have an   11-dimensional  space, more likely  to represent a  M-theory  solution of the  information theorem. Instead of  depending   on the   orbifold  compactification here  we just  take the  embedding  theorems  as  a  mechanism  of    dimensional reduction.

%\section{Now  you see,  now  you d'ont see}
A  further  generalization  is possible  by
use of the  embeding of   $S^5$  in  a  six-dimensional  Euclidean  space  $I	\!\!  R^6$,  which  can  be  merged  with the other  six  dimensional space into a  12-dimensional  space  $M_{12}(10,2)$,  which can embed   a  much larger  class of  space-times,  
including the  deSitter     and  Kruskal  space-times  in  addition to the   previous mentioned ones. Some  interesting  situations, not  included  in the  
$AdS_5  \times  S^5$ case    may  occur.  In fact,  
% This  is  so because the  isometry group of  such space $SO(10,2)$  may  rotate  space-like and  time-like  extra dimensions  so as  to  make it  possible to.
%However in doing so  the   conformal  symmetry is  broken. 
   
     submanifold  embedded in a given space, it can  be  deformed   (that is, have its  geometry  perturbatively  modified) along  the  extra dimensions,  and  then,  ask  if the deformed manifold is  again  a  sub-manifold of the  same space, using only  the  differentiable structure, .  As  we know, the answer is  a yes,  in some extended sense besides being differentiable .
     
     
     
   
 Of  course, if  all physics   would remain  just at a fixed boundary,  taken  as   rigidly embedded,
then  these mathematical restriction can be  lifted but so  also the physics  which would be  associated with the perturbations.   In particular,  this affects some issues  related to  the loss of  information problem.
  
  
First we need to comment on  the following point: Since Nash's theorem applies  to any  differentiable and  regular manifold, then  it  can  also  be applied to  the  bulk. Accordingly, we may always  apply a sequence of  embeddings until  reaching the simplest structure  which that of  a flat  bulk,  at the cost of  an  increased number of  dimensions.    


\footnotemark[2]\footnotetext[2]{ 
 It  should be  also  noted that  in the  cosmological applications the $AdS_5$ bulk
 with the  $Z_2$ symetry across the  brane-world has   met  with  
 inflationary  constraints  and  that  with the  removal of  that symmetry  the  
 cosmological   solution is  compatible with the   observations
  \cite{MaiaGDE}.}
  
  
   Could  we  have  taken the wrong road  back in  1905?
 The   use of the   ADS/CFT   correspondence   in the   2004  theorem  may  be   an  indication that  we  are attempting a course  correction.  That is,  supposing that   we  could go back in time  and tell   Einstein,  Minkowski and  Lorentz  to  wait  a little  further,  until 1909,  where  the conformal invariance  of  electromagnetism  would be   discovered  \cite{Fulton},  then  we  would  have  a different  physics in 2005,  in which  Minkowski  space-time   would be  six-dimensional,  with  two  times.
 Dirac  spinors  would be replaced by  massless twistors  \cite{Penrose},  the  Higgs  would  intervene  to  produce masses, unitary quantum gravity   would  be around in the form  of  quantum  brane-worlds, exchanging information through   worm-holes. 
  
  
  
 (a) After the evaporation  everything disappears,  leaving  empty space  with no  singularity  at the origin.  
 
 Since  there  was  a  horizon  before  the evaporation (or  a true  singularity  at  $r=0$)  and  nothing is  left after, the    space-time   undergoes a classical topological change  from   multiply connected  to simply  connected after the evaporation, which is  forbiden by 
 
 
 
 
 ******
 
 
 the inverse  arguments of the classic  Geroch's theorem \cite{Geroch}. On the other  hand, 
 under the  extended  ADS/CFT,  unitarity is preserved through the bulk  $M_6 (4,2)$,
  so that  this is  also forbidden by quantum theory.
 
  
  
 (b)  The geometry  remains  classical
but  after the evaporation   a   naked  singularity remains at $r=0$. 

 The presence of the  singularity at  $r=0$, means  that after the evaporation the  Schwarzschild  space-time    transforms into  its maximal  analytic  extension  (Kruskal's brane-world) \cite{Kruskal}. In turn, this  implies that the signature of  the bulk must change  from  $M_6 (4,2)$  to  $M_6 (5,1)$  \cite{Fronsdal}. Correspondingly, this change in signature  means  that  the  bulk symmetry  $SO(4,2)$ changes  to $SO(5,1)$, so  that  the  extended (and the usual) ADS/CFT  correspondence  breaks  down.  Therefore  the   unitarity of the  conformal fields  cannot be applied  in this case. Unless we remove the point singularity to   preserve  the regularity of the embedding, the paradox remains.


(c) The  geometry  reacts back  to the   quantum  fluctuations of the source, so that after the evaporation there is  a quantum remnant of the  black hole in a region  with curvature radius corresponding to  the fundamental gravitional energy scale (usually in the  Tev  scale).  
 In this case  the topology  of the brane-world is maintained and the geodesics  are incomplete so that the signature of the bulk remain. 
Consequently,   the  extended
ADS/CFR  correspondence apply and  unitarity is preserved.   Nonetheless,   the information   is  lost inside  the  remnant  region.
 
This  paradox is only apparent because in the context of regular brane-worlds  the
possibility envisaged by  Hawking in  1975 
applies  with  all its  right:   that is, since unitarity is  preserved, the  information is not really lost but simply transferred to  a   regular  region  between the  horizon  $r=2m$ and the quantum remnant,  acting like  a   storage room for  the  would be lost  information. Due to the regularity  
of the semi extended  Schwarzschild   manifold embedded  in this  region,  the  stored  information  can be  retrieved  by the  embedding derivative map. Of course, 
some of this information may get  into the quantum remnant.  But then  we  cannot add
anything  before we may define  a  quantum theory of brane-worlds. 


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