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{
\usetheme[]{Warsaw}}
\usepackage[english]{babel}
\title{Internal Structure of AdS Black Holes}
\author{\Brown{Srijit Bhattacharjee}\\
%\small{Astroparticle Physics and Cosmology Division}\\
{{\tiny{IIIT Allahabad}}}}
\institute[PAMU, Indian Statistical Institute, Kolkata]
{
\Red{29$^{th}$ IAGRG Meeting\\ IIT Guwahati\\}
\vspace{1cm}
S. Bhattacharjee, S. Sarkar, and A. Virmani; Phys.Rev. D93 (2016) no.12, 124029; arXiv:1604.03730.
}

%\author{\Violet{Srijit Bhattacharjee}
%\vspace{0.1cm}
%\footnotesize{\Brown{High Energy Theory Group}}\\
%{{\footnotesize{\Brown{Institute Of Physics\\ Bhubaneswar, Odisha}}}}}
%\institute[PAMU, Indian Statistical Institute, Kolkata]
%{
%\Red{IIIT Allahabad, Uttar Pradesh\\}
%}
%\Blue{13th MARCEL GROSSMANN MEETING \\ Stockholm, Sweden\\
%}
%}
%\AtBeginSection[]

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\begin{document}

\begin{frame}
\titlepage

\end{frame}


\begin{frame}{\small{Internal structure of Schwarzschild black hole}}
\begin{small}
\begin{figure}[t]
 \centering
  % \resizebox{30mm}{!}{\includegraphics{Fig_Sch.eps}}
   \includegraphics[width=0.6\textwidth]{Fig_Sch}
     %\caption{Penrose diagram of Schwarzschild spacetime}
    %\label{figure1}
\end{figure}
\begin{itemize}
 \item Schwarzschild spacetime is geodesically incomplete! Spacetime is inextendible as $C^0$ metric at $r=0$
 
 \item An observer will face violent tidal forces as (s)he approaches the surface $r=0\rightarrow$a spacelike singularity. The incompleteness also {\bf stable} under perturbation of initial data. {\tiny{Penrose's incompleteness theorem (1965)}}
\end{itemize}
\end{small}
\end{frame}

\begin{frame}
 \begin{figure}[t]
 \centering
  % \resizebox{30mm}{!}{\includegraphics{Fig_Sch.eps}}
   \includegraphics[width=1\textwidth]{beyond}
     %\caption{Penrose diagram of Schwarzschild spacetime}
    %\label{figure1}
\end{figure}
\end{frame}



\begin{frame}
\begin{itemize}
 \item {\bf Weak Cosmic Censorship}(Penrose 1969): The future null infinity ($\scriptstyle{I^{+}}$) is complete for all {\it generic} AF initial data! 
 \item {\bf Event horizon} is the boundary of the closure of the causal past of $\scriptstyle{I^{+}}$. 
 \item Far away observers do not ``see'' the incompleteness. Singularity is hidden behind the event horizon.
 \end{itemize}
\end{frame}

\begin{frame}{\small{Internal structure of charged black hole}}
\begin{figure}[t]
 \centering
    \includegraphics[width=0.6\textwidth]{Fig_RN1.pdf}
     % \caption{Penrose diagram of Reissner-Nordstr\"{o}m spacetime}
    %\label{figure1}
\end{figure}
\begin{itemize}
 \item There are causal curves which don't intersect $\Sigma$ originating beyond $r=r_- \sim${\bf Cauchy Horizon}
 
 \item Part of the spacetime smoothly extendible beyond the null boundary CH but the spacetime is severely nonunique! 
 
 \item A new type of incompleteness!
\end{itemize}
\end{frame}


 
\begin{frame}{Penrose's resolution}
\begin{figure}[t]
    \begin{center}
      \includegraphics[width=0.5\textwidth]{blueshift}
     %\caption{Plot of the Effective Potential as a function of $\frac{\langle{\rho}\rangle}{M}$.}
     % \label{test}
    \end{center}
  \end{figure} 
  
 \begin{small}
 \begin{itemize}
  \item $O_1$ emits a photon every year, it emits infinite no of photons infinite time. $O_2$ receives all of those photons in finite time!!
  \item Cauchy Horizon becomes a singular surface due to infinite blue shift. Hope was that the blue shift effect will revert the CH into a spacelike singularity! {\tiny{Penrose (1968)}}\\
  
  This motivated the following conjecture-
   \end{itemize}
\end{small} 
    \end{frame}
  
  \begin{frame}
    \begin{itemize}
  \item {\bf Strong Cosmic Censorship} (Penrose 1972): For {\it generic} AF initial data maximal Cauchy development is future inextendible as a {\it suitably regular} Lorentzian manifold.\\ {\bf Future is uniquely determined by past!}
  
    \item  {\it Generic} $\approx$ allow wave-like dynamic dof (gravity/photon) $\Rightarrow$ you are solving full coupled Einstein-matter system.
    
    \item CH becomes unstable but geodesics may cross smoothly the CH to a sptm which is not globally hyperbolic. So SCC seems to be invalid for charged or Kerr BHs.
   
   \item But we can reasonably accept SCC for generic Kerr-Newman family for suitable Einstein-matter system. {\tiny{Christodoulou, Dafermos}(2006)}
      \end{itemize}
     \end{frame}

     \begin{frame}{Mass Inflation}
     \begin{small}
      \begin{itemize}
       \item Precise nature of the singularity at CH was unknown! Analyses by Hiscock, Chandrasekhar-Hartle, Poisson-Israel, Ori determined the nature of the CH singularity for AF black holes.
       
       \item {\bf Poisson-Israel MI model (1989,1990)}: Spherically symmetric charged star undergoing gravitational collapse. The blue shifted perturbations correspond to the back-scattering of the emission carrying away non-spherical inhomogeneities from the surface of the star. 
       
       \item The decay rate of  such inhomogeneities is governed by the {\bf Price's law}(1972). The in-falling flux decays as a power law in the advanced time $\sim v^{-n}$.
       
       \item {\bf Influx}: The ingoing perturbation to RN background is modeled as stream of massless particles. The sptm becomes charged Vaidya geometry.
       
       \item {\bf Outflux}: Outflux represents back-scattering of the ingoing particles once they are inside the outer event horizon. 
       
       \item The effective mass parameter of the internal spacetime diverges as the inner horizon is approached. However the sptm metric is regular at CH but curvature scalars diverge. $\Rightarrow$ sptm $C^2$ inextendible $\rightarrow$ SCC. 
        \end{itemize}
     \end{small}
     \end{frame}

     \begin{frame}{Mass inflation in AdS sptm}
     \begin{small}
      \begin{itemize}
       \item Can MI model be generalized to asymptotically AdS spacetime?
       
       \item How MI influences the Physics behind the horizon of AdS BHs? Can the inner horizon instability be probed with the aid of AdS/CFT? 
       
       \item {\bf Obstacle:} How the perturbation decay AdS? Unlike the asymptotically flat case, for AdS BHs there are no power law tails for decaying perturbation. {\tiny{Horowitz,Hubeny (2000); Wang, Lin, Molina (2004); Berti, Cardoso, Pani(2012)}} 
       
     \item One expects that the late time decay is governed by the lowest lying quasinormal mode. Decay is exponential $\sim e^{-\omega_{I} v}$.
       
      \item This simple guess may not be true. The decay of generic scalar perturbation in Kerr-AdS spacetimes is actually logarithmic $\sim (log|V|)^{-n}$. {\tiny{Holzegel and Smulevici}}
       
       %\item MI occurs at the inner horizon of AdS-RN BH! {\tiny{SB, SS, AV (2016)}}
      \end{itemize}
      \end{small}
     \end{frame}

     
  \begin{frame}{Ori model and mass inflation in AdS}
  \begin{small}
  {\bf Ori, PRL (1991)}: Only presence of outgoing flux is important. Ori modeled the outgoing flux as a delta-function shell and got MI by matching two charged Vaidya solutions across the shell. \end{small}
 \begin{figure}[t]
 \centering
  \includegraphics[width=0.4\textwidth]{PenroseDiagram}
     %\caption{Penrose diagram of the spacetime formed by matching two charged Vaidya spacetimes along the thin null shell $S$.}
    \label{figure1}
\end{figure}
%\end{small}
\end{frame}

  \begin{frame}{Divergence of mass functions}
   \begin{itemize}
   \item  MI for exponential fall-off:
    \be
     m_2\propto |-V|^{-\fr{ \k_- - \a \kappa_+}{\k_-}}
\ee
\hfill{{\tiny{SB, SS, AV (2016)}}}

\item MI for log fall-off:
 \be
     m_2\approx |V|^{-1}|(-\log|V|)|^{-1} |(\log{| \log V|})|^{-3}\ee
\hfill{{\tiny{SB, SS, AV (2016)}}}

    
   \item {\bf Weak null singularity}: In a regular coordinate system the tidal force near the inner horizon diverges. However the tidal distortion (integrating geodesic deviation equation twice) experienced by an object crossing the inner horizon is finite!   
   
   \item For exp. fall-off the tidal forces grows as $\propto |-V|^{-2}|-V|^{\frac{\a\k_+}{\k_-}}.$ \\ {\tiny{SB, SS, AV (2016)}}
    \end{itemize}
     \end{frame}
     
     \begin{frame}{Summary and Outlook}
     \begin{small}
      \begin{itemize}
       \item The inner horizon instability of an AF BH is a weak null curvature singularity. The metric is regular, curvature components diverge but tidal distortions for an observer approaching the inner horizon remains finite.
       
       \item This phenomenon supports the SCC for AF black holes in a refined form.
       
       \item Inner horizon of a charged spherical AdS black hole seems also be unstable. A key element of the analysis -- required as an input -- is the fall-off behaviour of the mass-function in the external Vaidya spacetime. For exponential as well as logarithmic fall-off functions the weak null singularity of CH has also been observed.
       
       \item For a rotating BTZ BH, similar analysis has been done and we get a MI singularity there also. {\tiny{SB,PR,AV in progress..}}
       
       \item As the singularity is weak, classical continuation of sptm beyond the singularity is not excluded.
       
       \item Fate of the outgoing section of inner horizon?
       
       \item Probing the MI singularity in AdS/CFT setting will be an interesting study! {\tiny{Balasubramanian, Levi}}
       
      \end{itemize}

     \end{small}
     
     \end{frame}


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 \begin{center}
 \Purple{ {\cal {\srijit Thank You}}}
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