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\title{Near Horizon structures of Black Holes}
\author{\Brown{Srijit Bhattacharjee}\\
%\small{Astroparticle Physics and Cosmology Division}\\
{{\tiny{IIIT Allahabad}}}}
\institute[PAMU, Indian Statistical Institute, Kolkata]
{
\textcolor{blue}{CTP, Jamia Milia Islamia, Delhi}
}


%\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[]

 % \begin{frame}<beamer>{\Blue{Plan Of Talk}}
  %  \tableofcontents
  %\end{frame}
%}


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\begin{document}

\begin{frame}
\titlepage

\end{frame}

\begin{frame}{What is a Black Hole?}
 \begin{itemize}
  \item First proposal of a massive object that doesn't allow even light to escape from it's surface was made by \textcolor{red}{ John Michell} in 1784. Later in 1796 French Mathematician \textcolor{red} {Laplace} had also given a mathematical description of such objects.
  
  \vspace{1cm}
  \item In nineteenth century it became apparent that light is an electromagnetic radiation. People became skeptical about existence of objects that can gravitationally attract a radiation.
  
  \vspace{1cm}
  
  \item \textcolor{red}{Einstein} proposed \ textcolor{red}{gravity can influence matter as well as radiation} in 1915. Einstein's theory of gravity predicted that Black Holes exist.  
    \end{itemize}
   \end{frame}

\begin{frame}{BHs are exact solution of Einstein's gravity}
 \begin{itemize}
  \item \textcolor{red} {General Relativity} (GR) explains one of the fundamental interactions of spacetime: gravity.
  
  \item Einstein's proposal changed our perception about everyday phenomena like how things fall!
  
  \item \textcolor{red} {Gravity=Space-time curvature}
  
  \be
  R_{ab}-\frac{1}{2}g_{ab}R=8\pi G_N T_{ab}
  \ee
  \item \textcolor{red}{Black holes (BH) are exact solutions of GR}. Schwarzschild discovered a BH solution in 1916. 
 \end{itemize}

\end{frame}

\begin{frame}{How Black holes form?}
\begin{itemize}
 \item Chandrasekhar: Massive body $ < 1.44M_{\odot}$ becomes white dwarf (1930). If the limit exceeds it either forms neutron star or undergoes supernova explosion.
 \vspace{1cm}
 \item Tolman-Oppenheimer-Volkoff: Neutron stars of mass greater than $3 M_{\odot}$ undergo gravitational collapse and form black holes (1939). 
\end{itemize}
\end{frame}

\begin{frame}{How black holes are detected?}
\begin{small}
\begin{itemize}
\item Mass estimates from objects orbiting a black hole or spiraling into the core. For example, at the center of the Milky Way , we see an empty spot where all of the stars are circling around as if they were orbiting a really dense mass. That's where the black hole is situated. 

\item By observing the matter falling into the black hole. When material falls into a black hole from a companion star, it gets heated and settles in a disk around the black hole. The superheated materials emit radiations, for example X-rays, and we detect this X-rays through telescopes like Chandra X-ray Observatory. {\tiny{Cygnus X-1 is a strong X-ray source and is considered to be a good candidate for a black hole.}}
\end{itemize}
\end{small}
\begin{figure}[t]
 \centering
  % \resizebox{30mm}{!}{\includegraphics{Fig_Sch.eps}}
   \includegraphics[width=0.4\textwidth]{binaryblackHole}
     %\caption{Penrose diagram of Schwarzschild spacetime}
    %\label{figure1}
\end{figure}
\end{frame}

\begin{frame}{Relativist's black holes}
\begin{small}
 \textcolor{red}{Schwarzschild} solution: Vacuum Einstein's equation; Asymptotically flat, static, spherically symmetric.
 \be ds^2=-\left(1-{2M\over r}\right)dt^2+ \left(1-{2M\over r}\right)^{-1}dr^2 + r^2d\theta^2+r^2sin^2\theta d\phi^2\ee
 \begin{figure}[t]
 \centering
  % \resizebox{30mm}{!}{\includegraphics{Fig_Sch.eps}}
   \includegraphics[width=0.7\textwidth]{penrose-sch}
     %\caption{Penrose diagram of Schwarzschild spacetime}
    %\label{figure1}
\end{figure}
\end{small}
 \end{frame}
 
 \begin{frame}
 \begin{small}
 \vspace{0.2cm}
 \textcolor{red}{Reissner-Nordstorm} solution: \be ds^2=-\left(1-{2M\over r}+{Q^2\over r^2}\right)dt^2+ \left(1-{2M\over r}+{Q^2\over r^2}\right)^{-1}dr^2 + r^2d\theta^2+r^2sin^2\theta d\phi^2\ee
 
 \begin{figure}[t]
 \centering
  % \resizebox{30mm}{!}{\includegraphics{Fig_Sch.eps}}
   \includegraphics[width=0.28\textwidth]{RN}
     %\caption{Penrose diagram of Schwarzschild spacetime}
    %\label{figure1}
\end{figure}
\end{small}
\end{frame}

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

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

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

\begin{frame}{Physics of black holes: story so far}
\begin{small}
 \textcolor{blue} {Classical/Semiclassical picture:}
 \begin{itemize}
  \item Laws of Black hole mechanics {\tiny Bardeen, Carter, Hawking}
  \item Hawking radiation and black hole thermodynamics \hspace{0.2cm} {\tiny{Bekenstein, Hawking, Israel, Wald...}}
  \item Information puzzle {\tiny{Hawking}}
  \item Uniqueness theorems {\tiny{Israel}}
    \item No hair theorems {\tiny{Bekenstein}}
  \item Membrane paradigm {\tiny{Damour, Thorne, Price, D. T. Son, Minwalla, Rangmani...}}
  \item Singularities {\tiny {Penrose, Hawking}}
  \item Cosmic censorship conjecture {\tiny{Penrose}}
    \item etc.
   \end{itemize}
\textcolor{blue}{Quantum gravity:}
\begin{itemize}
 \item String theory, Quantum geometry (LQG), AdS/CFT or Holography
\end{itemize}
\end{small}
\end{frame}

\begin{frame}{Plan of Talk}
\begin{itemize}
 \item Introduction
 \vspace{0.2cm}
 \item Inner horizon instabilityingularity
 \vspace{0.2cm}
 \item Open issues and future plans
\end{itemize}
\end{frame}





\begin{frame}
 \Large{Inner horizon singularity}
\end{frame}


\begin{frame}{Singularity in BHs}
\begin{small}
\begin{figure}[t]
 \centering
  % \resizebox{30mm}{!}{\includegraphics{Fig_Sch.eps}}
   \includegraphics[width=.5\textwidth]{Fig_Sch}
     %\caption{Penrose diagram of Schwarzschild spacetime}
    %\label{figure1}
\end{figure}
\begin{itemize}
\item \textcolor{red}{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.  {\tiny{Penrose's incompleteness theorem (1965)}}
 \item \textcolor{red}{Weak Cosmic Censorship}(Penrose 1969): The future null infinity ($\scriptstyle{I^{+}}$) is complete for all {\it generic} AF initial data! 
 %\item \textcolor{red}{ 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{small}
\end{frame}

\begin{frame}{\small{Internal structure of charged black hole}}
\begin{figure}[t]
 \centering
    \includegraphics[width=0.7\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 $r=r_- \sim$\textcolor{red}{ 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)}}\\
  \end{itemize}
\end{small} 
    \end{frame}
  
  \begin{frame}
  \begin{small}
   This motivated the following conjecture-  
    \begin{itemize}
    
  \item \textcolor{red}{ Strong Cosmic Censorship} (Penrose 1972): For {\it generic} AF initial data maximal Cauchy development is future inextendible as a {\it suitably regular} Lorentzian manifold.\\ \textcolor{red}{ Future is uniquely determined by past!}
  
    \item  {\it Generic} $\approx$ allow wave-like dynamic dof (gravity/light).
    
    \item CH becomes unstable but geodesics may cross smoothly the CH to a sptm which is not globally hyperbolic. One has to carefully consider the word ``suitably regular'' in the statement of SCC.
   
  % \item But we can reasonably accept SCC for generic Kerr-Newman family for suitable Einstein-matter system. {\tiny{Poisson-Israel (1990), Christodoulou(1999), Dafermos(2006)}}
  \item Precise nature of the singularity at CH was unknown until the work of Poisson-Israel, Ori who determined the nature of the CH singularity for AF black holes.
      \end{itemize}
      \end{small}
     \end{frame}

     \begin{frame}{Mass Inflation}
     \begin{small}
      \begin{itemize}
       
       \item \textcolor{red}{ 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 amplitude of in-falling flux decays as a power law in the advanced time $\sim v^{-n}$ \textcolor{red}{ Price's law} (1972).
       
       \item \textcolor{red}{ Influx}: The ingoing perturbation to RN background is modeled as stream of massless particles. The sptm becomes charged Vaidya geometry.
       
       \item \textcolor{red}{ Outflux}: Outflux represents emission of radiation from the collapsing star. 
       
       \item The effective mass parameter of the internal spacetime diverges as the inner horizon is approached. 
        \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 \textcolor{red}{ 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 The decay of generic scalar perturbation in AdS-black holes is 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}{Mass inflation in AdS a la Ori}
  \begin{small}
  \textcolor{red}{ 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.\\
  CVS: $ds^2=-\left(1-2m(v)/r + q^2/r^2\right)dv^2 +2dvdr +r^2 d\O^2$. 
  \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 log fall-off:
 \be m_2(V)\approx |V|^{-1}|(-\log|V|)|^{-1} |(\log{| \log V|})|^{-3}\ee
\hfill{{\tiny{{\bf SB}, Sarkar, Virmani, PRD (2016)}}}

\item {\bf Weak null singularity}: In a regular coordinate system the tidal force (curvature components) 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 The sptm metric is regular at CH but curvature scalars diverge. $\Rightarrow$ Sptm $C^2$ inextendible. 
    \end{itemize}
     \end{frame}
     
     \begin{frame}{Future plans}
     \begin{small}
   \textcolor{blue}{Near horizon Physics (Outer)}
     \begin{itemize}
            \item Study of black hole mechanics in higher curvature theories (HCT) and well motivated alternative theories of GR. 
      
      \item Studying the second law for HCT without
assuming any symmetry. Vaidya sptm has spherical symmetry, one must check more generic cases by turning on more generic perturbations.

\item Second law beyond linear order: Whether one can establish an instantaneous increase law for all order in perturbation. (work in progress for GB gravity).

       \item Second law with matter (GSL): To see the second law if there is matter coupling to the higher curvature terms. In
that case it is not clear how to define the null energy condition. It seems natural that the matter couplings will get constrained by demanding the second law.
       
       \item No-hair theorems: For a generic scalar field (non-canonical Lagrangian) there is no proof of no-scalar hair theorem for AF black holes without assuming spherical symmetry. We would like to address this issue.  
       
       \item Late time behaviour of perturbations in black holes. 
                   
      \end{itemize}

     \end{small}
     
     \end{frame}
    
    
     \begin{frame}{Future plans}
     \begin{small}
   \textcolor{blue}{Near horizon Physics (Inner)}
     \begin{itemize}
     
\item Inner horizon instability of AdS black holes. Mass Inflation For rotating BTZ BH ({\tiny{{\bf SB}, PR, PP, BC, AV in progress..}}) . 
       
       %\item As the singularity is weak, classical continuation of sptm beyond the singularity is not excluded.
       
       \item Late time behaviour of perturbations inside the black holes.
       
       \item Fate of the outgoing section of inner horizon?
       
              
       \item Probing the MI singularity in AdS/CFT setting. To understand how the geodesic propagators, or an appropriate generalization thereof, probe the singularity. 
       \end{itemize}
     \end{small}
       \end{frame}

  \begin{frame}{Future plans}
     \begin{small}
   \textcolor{blue}{BMS-like symmetries and membrane paradigm of BHs:} 
     \begin{itemize}
      \item A detailed study of the asymptotic symmetries of black hole spacetimes. Analysing near horizon
counterparts of those symmetries in asymptotically flat, AdS, and other spacetimes. ({\tiny{{\bf SB}, AB arXiv:1707.01112 (2017))}}

\item A detailed exposition of near horizon symmetries in collapsing or dynamical black holes.

\item To establish the connection between \textcolor{red}{ membrane paradigm} and near horizon symmetries of black
holes. Particularly building a connection between fluid conservation laws with conservation laws
arising from BMS like symmetries at the horizon.

\item Setting up a statistical-mechanical description of horizon fluid. We want to develop a field
theory description of fluctuating horizon-fluid in flat spacetimes and compute various quantities (eg.
transport coefficients) with the aid of linear response theory. 

     \end{itemize}
     \end{small}
     \end{frame}
     
 \begin{frame}
  \begin{small}
  \begin{enumerate}
   \item  Doctoral Research:
   \begin{itemize}
   \item Gauge theories (gauge free description).
    \item Functional Integral methods in QFT.
    \item Effective potential and spontaneous symmetry breaking (Coleman-Weinberg mechanism).
       \end{itemize}
\item Post-doctoral and current research:
\begin{itemize}
 \item Black hole Thermodynamics.
 
  \item Internal structure of black holes.
 
 \item Late time perturbations of black holes.
 
 \item Asymptotic symmetries of black holes. 
 
 \item Membrane paradigm.
\end{itemize}

   \end{enumerate}
  \end{small}

 \end{frame}
  
     
     
     
\begin{frame}{Courses that I can offer}

Basic courses:
\begin{enumerate}
   \item Classical, Quantum \& Relativistic 
Mechanics 
 \item Introduction to Engineering 
Electromagnetics
  \item Modern Physics
   \item Quantum Physics.
   \item Mathematical Methods
   \item Relativistic Electrodynamics
   \item Statistical Physics
    
    \end{enumerate}
    Elective courses:
    \begin{enumerate}
    \item Advanced Quantum Mechanics
     \item Introduction to Quantum Field Theory 
     
     \item General Relativity $\&$ Cosmology
    \end{enumerate}

\end{frame}

\begin{frame}
 \begin{center}
 \textcolor{blue}{\cal {\srijit Thank You}}
 \end{center}
\end{frame}

\end{document}

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

Black holes are viewed as the most intriguing objects in the cosmos that offer many outstanding phenomena. Generic black holes possess two horizons, one is outer and the other is inner horizon. The outer or the ``event horizon`` of black holes acts as the boundary of causal communication and it has been the source of most of the outstanding issues related to black holes, such as: black hole thermodynamics, no-hair theorems of black holes etc.. On the other hand predictability of state of any dynamical field breaks down at the inner horizon of black holes. Many of these interesting phenomena are still far from being completely understood. In this talk I shall discuss few of these outstanding issues related to near horizon physics of black holes and highlight the progresses have been made in those areas so far.