\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,epsfig}

\textwidth 18cm
\textheight 23.5cm
\topmargin -1.5cm \hoffset -2.5cm
%\pdfoutput=1
%\usepackage{xcolor}
%\usepackage{jheppub}
%\usepackage{showkeys}
\usepackage{mathtools}
\DeclareMathOperator{\im}{im}
\usepackage{epstopdf}
\usepackage{graphicx}
\usepackage{epsfig}
\usepackage{dcolumn}  
\usepackage{bm}    
\usepackage{amssymb} 
\usepackage{amsmath,bm}
\usepackage{amsfonts}    
\usepackage{slashed}  
%\usepackage{youngtab}
 %\usepackage{relsize}
\usepackage[mathscr]{euscript}
\usepackage{epsfig}
\usepackage{verbatim}
%\usepackage[position=t,singlelinecheck=off]{subfig}
%\usepackage{caption}
\hyphenation{ALPGEN}
\hyphenation{EVTGEN}
\hyphenation{PYTHIA}
%\usepackage{framed}
%\usepackage{tcolorbox}


%\newdimen\tableauside\tableauside=1.0ex
%\newdimen\tableaurule\tableaurule=0.4pt
%\newdimen\tableaustep
%\def\phantomhrule#1{\hbox{\vbox to0pt{\hrule height\tableaurule width#1\vss}}}
%\def\phantomvrule#1{\vbox{\hbox to0pt{\vrule width\tableaurule height#1\hss}}}
%\def\sqr{\vbox{%
%		\phantomhrule\tableaustep
%		\hbox{\phantomvrule\tableaustep\kern\tableaustep\phantomvrule\tableaustep}%
%		\hbox{\vbox{\phantomhrule\tableauside}\kern-\tableaurule}}}
%\def\squares#1{\hbox{\count0=#1\noindent\loop\sqr
	%	\advance\count0 by-1 \ifnum\count0>0\repeat}}
%\def\tableau#1{\vcenter{\offinterlineskip
%		\tableaustep=\tableauside\advance\tableaustep by-\tableaurule
%		\kern\normallineskip\hbox
%		{\kern\normallineskip\vbox
%%			\kern\normallineskip\kern\tableaurule}%
	%	\kern\normallineskip\kern\tableaurule}}
%	\def\gettableau#1 {\ifnum#1=0\let\next=\null\else
	%\squares{#1}\let\next=\gettableau\fi\next}

%\tableauside=1.0ex
%\tableaurule=0.4pt






\newcommand{\ysm}{\Yboxdim{8.5pt}\Ylinethick{.9pt}}
\def\tyng{\ysm\yng}







\newcommand{\be}{ \begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}[1]{\begin{eqnarray}\label{#1} }
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\eq}[1]{(\ref{#1})}
\def\ZZZ{{\hskip-3pt\hbox{ Z\kern-1.6mm Z}}}
\def\zzz{{\hskip-3pt\hbox{ z\kern-1mm z}}}
\newcommand{\rrho}{r}
\newcommand{\RRR}{{\rm R \hskip -7pt R}}
\newcommand{\bX}{\bar X}
\newcommand{\te}{\tilde e}
\newcommand{\bx}{\bar x}
\newcommand{\bw}{\bar w}
\newcommand{\tx}{\wt x}
\newcommand{\bF}{\bar F}
\newcommand{\bbF}{{\bf F}}
\newcommand{\bN}{{\bf N}}
\newcommand{\bbb}{{\bar b}}
\newcommand{\gam}{\tau}
\newcommand{\eps}{\epsilon}
\newcommand{\ra}{\rangle}
\newcommand{\la}{\langle}
\newcommand{\T}{\chi_{T}(k)}
\newcommand{\Tm}{\chi_{T}(k')}
\newcommand{\Cn}{{\cal C}_n}
\newcommand{\vp}{\varphi}
\newcommand{\ve}{\varepsilon}
\newcommand{\vt}{\vartheta}
\newcommand{\gl}{\mu}
\newcommand{\dt}{(\vec \nabla T)^2}
\newcommand{\hp}{{\wh\Phi}}
\newcommand{\hq}{{\wh Q_B}}
\newcommand{\he}{{\wh\eta_0}}
\newcommand{\ha}{{\wh{A}}}
\newcommand{\lllb}{\Bigl\langle\Bigl\langle}
\newcommand{\rrrb}{\Bigr\rangle\Bigr\rangle}
\newcommand{\tf}{\wt f}
\newcommand{\sss}{{\cal L}_{av}}
\newcommand{\GP}{\Psi}
\newcommand{\gp}{\psi}
\newcommand{\gt}{\theta}
\newcommand{\gf}{\phi}
\newcommand{\gk}{\chi}
\newcommand{\GO}{\Omega}
\newcommand{\GG}{\Gamma}
\newcommand{\hf}{{1\over 2}}
\newcommand{\thf}{{3\over 2}}
\newcommand{\ga}{\alpha}
\newcommand{\gb}{\beta}
\newcommand{\p}{\partial}
\newcommand{\vv} {\bar v}
\newcommand{\uu} {\bar u}
\newcommand{\K}{{\rm K_1}}
\newcommand{\Kt}{{\rm \widetilde K_1}}
\newcommand{\vptl}{\tilde\varphi}
\newcommand{\B}{b'}
\newcommand{\C}{c'}
\newcommand{\bB}{\bar b'}
\newcommand{\Bu}{B_{\vec u}}
\newcommand{\VV}{{\cal V}}
\newcommand{\BB}{{\cal B}}
\newcommand{\II}{{\cal I}}
\newcommand{\AAA}{{\cal A}}
%\newcommand{\GG}{{\cal G}}
\newcommand{\KK}{{\cal K}}
\def\Z{{\cal Z}}
\def\s{\sigma}
\def\bal#1\eal{\begin{align}#1\end{align}}
\def\lm{\lambda}
\def\V#1#2{ {V^{(#1)}_{#2}} }

\newcommand{\fff}{{\bf f}}
\newcommand{\ccc}{{\bf c}}
\newcommand{\FF}{{\cal F}}
\newcommand{\HH}{{\cal H}}
\newcommand{\MM}{{\cal M}}
\newcommand{\CC}{{\cal C}}
\newcommand{\DD}{{\cal D}}
\newcommand{\bC}{{\bf C}}
\newcommand{\OO}{{\cal O}}
\newcommand{\QQ}{{\cal Q}}
\newcommand{\PP}{{\cal P}}
\newcommand{\EE}{{\cal E}}
\newcommand{\ZZ}{{\cal Z}}
\newcommand{\LL}{{\cal L}}
\newcommand{\W}{{\cal W}}
\newcommand{\f}{{\rm f}}


%\newcommand{\rrr}{\rangle\rangle}
%\newcommand{\square}{\Box}
\newcommand{\half}{{1\over 2}}
\newcommand{\wt}{\widetilde}
\newcommand{\wh}{\widehat}
\newcommand{\wc}{\check}
\newcommand{\wb}{\bar}
\newcommand{\bd}{\bar{\rm D}}
\newcommand{\RR}{{\cal R}}
\newcommand{\nn }{{\cal N}}
\newcommand{\TT}{{\cal T}}
\newcommand{\bg}{\bar g}
\newcommand{\bb}{\bar b}
\newcommand{\bT}{\bar \Theta}
\newcommand{\SSS}{{\cal S}}
\newcommand{\tlx}{\left(\tilde \mu ; X^0(0) \right)}
%\newcommand{\al}{\alpha}

\newcommand{\omk}{\omega_n(\vec k)}
\newcommand{\onk}{\omega^{(N)}_{\vec k_\perp}}
\newcommand{\sldel}{\slash{\!\!\!\! \nabla}}

\newcommand{\refb}[1]{(\ref{#1})}
\newcommand{\pa}{\partial}
\newcommand{\sectiono}[1]{\section{#1}\setcounter{equation}{0}}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
%\renewcommand{\theequation}{\arabic{equation}}
\def\one{{\hbox{ 1\kern-.8mm l}}}
\def\zero{{\hbox{ 0\kern-1.5mm 0}}}

%\def\thefootnote{\fnsymbol{footnote}}
\def\Tr{\,{\rm Tr}\, }
\def\Trqy{\,{\rm Tr}_M^{q,y}\, }
\def\t{\tau}
\def\x{\xi}
\def\c{\chi}
\def\k{\kappa}
\def\L{\Lambda}
\def\e{\,{\rm e}}

\def\hn{\hat{n}}
\def\tg{\tilde{g}}



\newcommand{\hs}[1]{\mbox{hs$[#1]$}}
\newcommand{\w}[1]{\mbox{$\mathcal{W}_\infty[#1]$}}
\def\th{\theta}






\begin{document}

\title{Supertranslation and superrotation from  soldering transformations}

\author{{Srijit Bhattacharjee}${}^{a}$ and {Arpan Bhattacharyya}${}^{b}$\\
\it $^a$ Indian Institute of Information Technology, Allahabad,\\ 
Devghat, Jhalwa, Uttar Pradesh-211015, India\\
\it  $^b$ Department of Physics and Center for Field Theory and Particle Physics\\, Fudan University,
220 Handan Road, 200433 Shanghai, P. R. China}
\maketitle 
%\emailAdd{
%srijitb@iiita.ac.in,\quad bhattacharyya.arpan@yahoo.com}





\abstract{When two spacetimes are stitched together across a null shell placed at the horizon of a black hole, BMS-supertranslation like soldering freedom arises if one demands that, the induced metric on the shell should remain invariant under the translations generated by the null generators of the shell. We revisit this phenomenon for null shells and demonstrate that not only BMS-supertranslation but also {\it BMS-superrotation like symmetry} can emerge if we solder two metrics across a null hypersurface for which the induced metric is invariant up to a conformal rescaling. We demonstrate the appearance of this new type of soldering symmetry for the case of BTZ and Schwarzschild black holes. We compute the expression for the conserved charge (Energy) corresponding to this symmetry for BTZ black hole and relate that with the existing results.
}
\tableofcontents
%\preprint{}

%\baselineskip 3.5ex

%\maketitle



%make math in all titles bold
%\%makeatletter
%\g@addto@macro\bfseries{\boldmath}
%\makeatother
%end code
 %{\color{red} Tasks:\begin{itemize} %\item Comment about the singular structure of the superrotation 
% \item Add few sentences strongly emphasizing the difference between our superrotaion like transformation and the usual superrotation , the charges differ etc.  \end{itemize}}
  
  
\newpage  
\section{Introduction}

Symmetry consideration has been proven to be a very powerful approach to study many physical systems, and General Relativity is not an exception. Recently there has been a growing interest in determining the structure of asymptotic symmetries in gravity. Many years ago Bondi, van der Burg, Metzner, and Sachs (BMS) studied the diffeomorphisms that preserve the asymptotic structure of any asymptotically flat spacetime at future null infinity $I^{+}$ and to their great surprise, the asymptotic symmetry group turned out to be infinite dimensional, a semi direct product of Lorentz group and supertranslations (angle dependent translations) \cite{BBM, Sachs,Newman1}. These supertranslations constitute an infinite dimensional abelian subgroup of the BMS group and they map one asymptotically flat solution of Einstein's equation to another. Not long ago, BMS group has got extended and new symmetries like superrotations have emerged at the null infinities (both future and past) of asymptotically flat spacetimes \cite{Barnich:2006av,Barnich:2009se,Barnich:2011ct,Barnich}.\footnote{Recently this has been extended to Conformal BMS group in \cite{Haco}.} In simple terms this new symmetry rotates around each generator of $I^{\pm}$ separately. Due to superrotations the Lorentz part of the BMS group gets enlarged to full Virasoro group. Superrotation like symmetries have also emerged from diffeomorphisms preserving the near horizon geometry of black holes \cite{Pinoetal,Pinoetal1,Shi1}. On the other hand it has been shown that, there is a deep connection between the infrared structure of gravity (and also few gauge theories) and the asymptotic symmetries of it. Certain subgroup of BMS$^+ \times $BMS$^-$ has emerged as an exact symmetry of quantum gravitational S matrix \cite{Strominger:2013lka,Campiglia:2015qka,Campiglia:2014yka, Strominger:2013jfa, He:2014laa,Cachazo,He1,Campiglia1,Kapec1,Dumitrescu:2015fej,Lysov2, Kapec3}. For a more recent and comprehensive review on this subject, interested readers are referred to  \cite{Strominger} and also encouraged to consult various references therein. These interconnections have generated the intriguing possibility of resolving the black hole information paradox. The idea is that black holes are imparted by infinite number of soft hairs corresponding to  diffeomorphisms that act non-trivially on the phase space \cite{Hawking:2016msc,Hawking,Hawking1,Carney}.\footnote{Recently there are some works which suggest that the soft charges do not play any role in resolving the information paradox. Interested readers are referred to \cite{Mirbabayi,Bousso,Donnelly}.}

The supertranslation like transformation also arises in another way when one tries to stitch two spacetimes across a thin null shell assumed to be situated at the event horizon of a black hole \cite{Blau}. It has been shown that there exists considerable amount of freedom to solder two metrics across a null shell for which the induced metric remains invariant under the translations along the null generators. In fact the group of soldering transformations turns out to be infinite dimensional and have identical structure- like supertranslations in BMS group. 

In \cite{Blau}, a detailed analysis has been presented to find the soldering group of Schwarzschild spacetime. In this note, as a warm up exercise, we extend the results in \cite{Blau} for the case of rotating spacetimes. Particularly we have analysed null shells placed at the horizon of a slowly rotating Kerr and a rotating BTZ blackhole and demonstrate that BMS like symmetry (supertranslation) again emerges. The intrinsic properties of the shell are also studied. 

Next, the more important issue that we have addressed here and which is also the main focus of the paper is: Whether there is any way to obtain a superrotation type symmetry while joining two spacetimes across a null shell. Although people have already recovered this symmetry by relaxing some of the fall-off conditions either of asymptotic metric structure or of the near horizon metrics in 3 and 4 dimensional spacetimes \cite{Barnich:2009se,Barnich:2011ct, Barnich, Pinoetal,Pinoetal1,Koga} but here we want to reconsider those results from the point of view of soldering transformations. We want to see if we demand that the induced metric on the null shell across which we stitch the metrics is invariant up to conformal rescaling (thereby relaxing the condition on the induced metric of the shell to allow conformal isometries instead of only isometries) do we get something like superrotation type symmetries on the black hole horizon? Our analyses suggest the answer is affirmative but some pathological structure in the matter content residing on the shell. In the case of a BTZ black hole we have found that we can patch two spacetimes across the horizon shell allowing the following freedom in the angular coordinate 
\be \phi \rightarrow \phi + T(\phi)\ee
if we consider that the induced metric on the shell allows conformal isometries. This is nothing but superrotation freedom that emerged in the analysis of asymptotic symmetries near $I^{\pm}$ in recent past. We also show in a similar way superrotation arises in asymptotically flat spacetimes (Schwarzschild). The construction depicted here may appear closely related to Newman-Unti group of transformations that allow conformal isometries of the celestial spheres at null infinity \cite{Newman:1962cia, Barnich:2011ty}. Our approach has also similarities with the Nutku-Penrose construction where two local vacuum solutions of Einstein equations are glued together across a null shell \cite{Nutku}. In this case conformal transformations ensure the continuity of the induced metric across the shell. We don't know exactly how these constructions are related to our analyses but certainly there exist enough evidences for the relevance of conformal isometries in the context of horizon shells.

Role of conformal invariance in the near horizon physics of black hole has been explored quite extensively in recent past \cite{Brown-Hennaux,Strominger:1997eq, Carlip:1998wz,Carlip:1999cy}. Much of the recent activities related to BMS algebra are reconsideration of asymptotic symmetries in relation to flat space holography \cite{Arcioni:2003xx, Arcioni:2003td,Barnich:2010eb} ; that is, extending the AdS/CFT  correspondence for asymptotically flat spacetimes. The emergence of superrotation as a new symmetry of the near horizon physics is an important outcome of that \cite{Pinoetal}. It is thus not quite unexpected thing to discover superrotation symmetry from the perspective of soldering freedom in a slightly modified version of Israel junction condition \cite{Barrabes}. In the case of null shells allowing supertranslation like freedom to solder two metrics across it, it was shown  in \cite{Blau} that, the conserved charge associated with that symmetry has strikingly similar form as that of the BMS-supertranslation charge at $I^{+}(I^-)$ or the near horizon counterpart of it. We also expect that superrotation like freedom will also bear such similarity with the corresponding charges at $I^{\pm}$. In this paper we have shown that, when a {\it superrotated} and a non-superrotated spacetime (or the seed spacetime where the horizon shell is actually stationed) are soldered we get some conserved quantities that depend upon the conformal factor (the parameter for superrotation), although the properties intrinsic to the null hypersurface become nontrivial as stress tensor corresponding to the matter layer at the horizon shell possesses singularities. Setting aside these issues one can still find quite compelling relationship between BMS-supertranslation and superrotation symmetries and horizon shell dynamics. In fact we have shown in the case of weakly superrotated transformation, one may get well defined physical quantities corresponding to the horizon shell.  


In section 2 we review the Israel junction condition briefly and in the next section emergence of BMS like transformations from soldering transformations is reviewed. In Section 4 we deduce BMS like transformations for slowly rotating Kerr and rotating BTZ spacetimes and obtain conserved quantities.  Section 5 is devoted to explain how one can include conformal isometries in soldering freedom. In section 6 the stress energy tensor for the shell is evaluated. Finally we conclude with discussions on our results and propose few scopes of further investigations.


\section{Brief review of Israel junction condition} \label{first}
In this section we start by briefly reviewing all the essential features of Israel junction 
\cite{Barrabes,Barrabes1,Poisson} conditions.  
Most commonly, in general relativity, the problem is to find the surface dynamics of a thin shell, where the surface is embedded in a spacetime ($\mathcal{M}$) of the form $\mathcal{M}=\mathcal{M}_{+}\cup \mathcal{M}_{-}.$ $\mathcal{M}_{-}$ and $\mathcal{M}_{+}$
 denote respectively the manifolds inside and outside of the shell together with the corresponding intrinsic metrics $g^{-}_{\mu\nu}(x_{-}^{\mu})$ and $g^{+}_{\mu\nu}(x_{+}^{\mu}).$  Both the manifolds have boundaries, with $\mathcal{M}_{+}$ and  $\mathcal{M}_-$ are defined to the future and past of null hypersurfaces $\Sigma_+$ and $\Sigma_-$ respectively. Now let us suppose we can define a set of intrinsic coordinates $\zeta^{a}$ on the surface of the shell $\Sigma$ , across which the two manifolds will be joined. Now we can project both $g^{+}_{\mu\nu}$ and $g^{-}_{\mu\nu}$ on the surface from  both sides. Then the junction condition is, 
 \be \label{junc}
 g_{ab}= g^{+}_{\mu\nu}e^{+\mu}_{a}e^{+\nu}_{b}|_{\Sigma_+}=g^{-}_{\mu\nu}e^{-\mu}_{a}e^{-\nu}_{b}|_{\Sigma_-}.
 \ee
 $e^{\pm \mu}_{a}=\frac{\partial x_{\pm \mu}}{\partial \zeta^a}$ are the tangent vectors to the surface. We will use the Greek alphabets for the spacetime indices and Latin ones for the hypersurface indices. The junction condition simply says that, the hypersurfaces are isometric, i.e $\Sigma_{+}=\Sigma_{-}=\Sigma.$ Then the condition (\ref{junc}) determines the  functional dependence of the coordinates (although not uniquely)  $x^{ \mu}_{\pm}$ on $\zeta^a.$ In the literature this junction condition is often written in the following form, 
 \be
 [g_{\alpha\beta}]=g^{+}_{\alpha\beta }|_{\Sigma_{+}}- g^{-}_{\alpha\beta}|_{\Sigma_{-}}=0.
\ee
We introduced here the box notation which means for any tensor $A^{\mu}$,
\be
[A^{\mu}]= A^{\mu}|_{\Sigma_{+}}-A^{\mu}|_{\Sigma_{-}}.
\ee
In the rest of the paper we will be only considering null shell.  Next we define the normal vectors $n_{\alpha}^{\pm}=\chi\partial_{\alpha}(\Sigma_{\pm})$  for both $\Sigma_{\pm}$ which are also generators for the null congruences orthogonal to both the hypersurfaces. We always work with future directing normal vectors for each side of the shell. $\chi$ is an arbitrary normalization. To complete the basis we also have to define the auxiliary vector $N^{\pm}$ such that , $N.N|_{\pm}=0,n. N=-1|_{\pm}.$ So together with $e^{\pm\mu}_a$ they define a complete basis. From the continuity of the null congruence it follows,
\be \label{junc1}
[n^{\mu}]=0.
\ee
The normal vectors must satisfy $n.e_{a}|_{\pm}=0.$ Also if the induced metric is unique, 

\be \label{junc2}
[e^{\mu}_{a}]=0.
\ee
Now we can use the distributional tensor calculus to derive the form of the stress tensor for thin shell such that the Einstein equations will be satisfied.\footnote{We have only quoted the important results. For detail derivations interested readers are referred to \cite{Barrabes,Barrabes1,Poisson}.}  Suppose we define a coordinate chart ($x^{\mu} $) such that it covers both sides of the shell and on the hypersurface coincide with $\zeta^{a},$ then we express the metric covering both sides as a distribution valued tensor ,
\be \label{junc3}
g_{\mu\nu}=g^{+}_{\mu\nu}\mathcal{\theta}(\Sigma)+g^{-}_{\mu\nu}\mathcal{\theta}(-\Sigma),
\ee
where, both $g^{+}$ and $g^{-}$ has been expressed in terms of the single coordinate chart $x^{\mu}.$ Now if we compute the derivative of (\ref{junc3}),
\be \label{junc4}
\partial_{\alpha}g_{\mu\nu}=\partial_{\alpha}g^{+}_{\mu\nu}\mathcal{\theta}(\Sigma)+\partial_{\alpha} g^{-}_{\mu\nu}\mathcal{\theta}(-\Sigma)+[g_{\mu\nu}](\partial_{\alpha}\Sigma)\delta(\Sigma).
\ee
 Using (\ref{junc}) and (\ref{junc2})  the last term in (\ref{junc4}) becomes zero. So we end up with the following form for the Christoffel symbol,
\be
\Gamma^{\mu}_{\alpha\beta}=\Gamma^{+\mu}_{\alpha\beta}\mathcal{\theta}(\Sigma)+\Gamma^{-\mu}_{\alpha\beta}\mathcal{\theta}(\Sigma).
\ee
and the Riemann tensor takes the following form,
\be \label{junc5}
R^{\alpha}{}_{\beta\gamma\delta}= R^{+\alpha}{}_{\beta\gamma\delta}\mathcal{\theta}(\Sigma)+R^{-\alpha}{}_{\beta\gamma\delta}\mathcal{\theta}(-\Sigma)+\delta(\Sigma) Q^{\alpha}{}_{\beta\gamma\delta},
\ee
where, $Q^{\alpha}{}_{\beta\gamma\delta}=-\Big([\Gamma^{\alpha}{}_{\beta\delta}]n_{\gamma}-[\Gamma^{\alpha}{}_{\beta\gamma}]n_{\delta}\Big).$
To satisfy the Einstein equation we start with the following form for the stress tensor, 
\be \label{junc6}
T_{\alpha\beta}=T^{+}_{\alpha\beta}\mathcal{\theta}(\Sigma)+T^{-}_{\alpha\beta}\mathcal{\theta}(\Sigma)+S_{\alpha\beta}\delta (\Sigma),
\ee
where, 
\be \label{junc6b}
8\pi S_{\alpha\beta}=Q_{\alpha\beta}-\frac{1}{2}Q g_{\alpha\beta}.
\ee
(\ref{junc6b}) follows from satisfying the $\delta(\Sigma)$ part of the Einstein equations. 
Then the stress tensor of the shell ($S_{ab}$) can be found by projecting (\ref{junc6b})  to the surface,
\be
S_{ab}=S_{\alpha\beta}e^{\alpha}_{a}e^{\beta}_{b}.
\ee
%The stress tensor contains only delta function singularity.
We further observe that because of the junction conditions there is no discontinuity in the tangential derivatives of the metric, only there is a jump in the normal direction.
\be \label{junc7}
[\partial_{\alpha} g_{\mu\nu}]=-\gamma_{\mu\nu}n_{\alpha}.
\ee

Projecting $\gamma_{\mu\nu}$ to the surface gives us a unique induced metric on the surface of the shell. So using this we can finally recast $S_{\alpha\beta}$ after some manipulation \cite{Barrabes,Barrabes1,Poisson} in the following form,
%{\bf AB: Correct this equation}
\be
S^{\alpha\beta}=\mu n^{\alpha}n^{\beta}+j^{A}(n^{\alpha}e^{\beta}_{A}+e^{\alpha}_{A}n^{\beta})+p  \sigma^{AB}e^{\alpha}_{A}e^{\beta}_{B}, \label{ST}\ee
where, $\sigma_{AB}$ is the non degenerate metric of the spatial slice of the surface of the null shell. Capital indices $A,B$ denote the spatial indices of the null surface. $\mu, J^{A}$ and $p$ can be interpreted as the surface energy density, current and pressure of shell respectively. These quantities are defined as follows,
\begin{align}
\begin{split} \label{junc8}
\mu=-\frac{1}{16 \pi}(\sigma^{AB})\gamma_{AB}, J^{A}=\frac{1}{16 \pi}(\sigma^{AB}
\gamma_{B}),p=-\frac{1}{16 \pi}\gamma_{\alpha\beta}n^{\alpha}n^{\beta},\end{split}
\end{align}
where, $\gamma_{AB}=\gamma_{\alpha\beta}e^{\alpha}_{A}e^{\beta}_{B}, \gamma_{B}=\gamma_{\alpha\beta}e^{\alpha}_{B} n^{\beta}$.

\section{Soldering freedom and BMS like transformations}  \label{third}
It is a well known fact that, there exist considerable freedom in the choice of intrinsic coordinate on $\Sigma$  in the  null direction. This is often termed as ``soldering freedom."  Recently in \cite{Blau}, this fact has been utilized to show that it generates BMS (supertranslation) type transformations on the null shell horizon. In this section we will briefly review all the essential points of this development. To make things concrete, throughout the paper we will work in Kruskal coordinates. On the surface  $\Sigma$ of the null shell we define a coordinate chart such that $\zeta^{a}=\{V,x^{A}\}.$ $V$ is the parameter along the hypersurface generating null congruences.  In this coordinate system the normal vector takes the following form, 
\be n=\partial_{V}.\ee
On $\Sigma$ the metric will take the form $g_{ab}d\zeta^ad\zeta^b= g_{AB} dx^Adx^B.$ One then adopts a single chart $x=\{ U, x^{a}\}$ such that $x^{a}|_{\Sigma}=\zeta^{a},$ such that not only the conditions (\ref{junc}), (\ref{junc1}) and (\ref{junc2}) hold but all the components of the metric $g_{\mu\nu}$ become continuous across the junction. Equipped with this coordinate system, one can now ask, what are the possible allowed coordinate transformations on either side of the shell preserving the junction condition (\ref{junc}). In our adapted coordinate system this boils down to solve for the Killing vectors ($Z^a$)  of $g_{ab},$ the metric on $\Sigma$ i.e 
\be \label{killing}
\mathcal{L}_{Z} g_{ab}=0.
\ee
From (\ref{killing}) we get,
\be \label{killing1}
Z^{c}\partial_{c}g_{ab}+(\partial_{a}Z^{c})g_{cb}+(\partial_{b} Z^{c})g_{ca}=0.
\ee
In \cite{Blau} the authors have worked with the metric where $g_{aV}=0.$ From the $a\,V$ components of (\ref{killing1}) we can easily conclude ,
\be
\partial_{V} Z^{A}=0.
\ee
 This implies that the isometry transformations along the spatial directions of null surface is independent of $V$.  Next if we consider the spatial components $ A\, B$  of (\ref{killing1}) we get,
\be \label{killing2}
Z^{V}\partial_{V} g_{AB}+ Z^{C}\partial_{C}g_{AB}+(\partial_{A}Z^{C})g_{CB}+(\partial_{B}
Z^{C}) g_{C A}=0.
\ee
Now if the metric is generically dependent on $ V$  then setting $Z^{V}=0$  in (\ref{killing2}) we get the isometry generators of the spatial slice.\footnote{For an exceptional case interested readers are referred to \cite{Nutku}.} But most importantly when the metric is independent of $V$  we are free to choose $Z^{V}$ as an arbitrary function of $(V,x^{A}).$
\be
Z^{V}= F (V, X^{A}).
\ee
Next one can investigate the effect of this isometry transformation on the normal vector $ n^{a}.$ Following \cite{Blau} $\mathcal{L}_{A}n^{a} \sim n^a.$ If one consider only those transformations such that $\mathcal{L}_{Z} n^a=0$, then it gives,
\be \label{killing3}
\partial_{V} Z^{V}=0.
\ee
(\ref{killing3}) implies,
\be\label{killing4}
Z^{V}= F(x^A).
\ee
 (\ref{killing4}) implies that it generates the following transformation,
\be
V\rightarrow  V+ F (X^A).
\ee
This generates supertranslation like soldering transformation.

 The emergence of BMS like symmetries while soldering two manifolds across a null surface can also be viewed in the following way: We always have a seed metric (like Schwarzschild metric) and the horizon shell sort of divides the spacetime into two manifolds. Since there exist a lot of freedom to stich together two spacetimes across a null hypersurface, the upshot of the construction depicted in \cite{Blau} is, one can perform supertranslation like coordinate transformation in one side (lets $+$ side of the hypersurface) of the horizon and stitch that with the other side ($-$) without altering the metric induced on the hypersurface.  The generators of those supertranslations also preserve the induced metric on the hypersurface, which in turn let the Israel's junction conditions unaltered. To appreciate this fact let us consider the BTZ metric in (\ref{rotBTZ}) in $+$ side. For simplicity we set the rotation parameter to zero so that $\tilde r_{h}=0.$ Now we do an active supertranlsation transformation,
\be
V\rightarrow V+T(\phi),
\ee
and the drop the $+$ subscript. This gives the metric on the $-$ side. So we get the the following metric $g_{\mu\nu}^{-}$,
\be
ds_{-}^2= \frac{1}{(1+U (V+T(\phi)))^2}\Big[-4 l^2 dU dV -4 l^2 T'(\phi)dU d\phi +r_{h}^2(1- U(V+T(\phi)))^2d\phi^2\Big].
\ee
This amounts to the fact, that  we solder a BTZ metric with a supertranslated BTZ metric.  Now we can easily check that induced metric from the side on $\Sigma$ is same and the usual juncntion condition is maintained. We still have,
\be
[g_{\alpha\beta}]=0.
\ee
Equipped we this we can compute the conserved quantities  from the jump of the extrinsic curvature in the usual way \cite{Barrabes,Barrabes1, Poisson}. We compute the following,
\be
[\mathcal{K}_{ab}]=[e^{\alpha}_{a}e^{\beta}_{b}\nabla_{\alpha}n_{\beta}].
\ee
Only $[\mathcal{K}_{\phi\phi}]$ is non-vanishing.  Then we get,
\be
\mu=-\frac{1}{8\pi}\sigma^{AB}[\mathcal{K}_{AB}]=-\frac{1}{8\pi l^2}\Big(\frac{l^2T''(\phi)-r_h^2T(\phi) }{r_h^2 }\Big)
\ee
The total energy once again is obtained by integrating $\mu$ on the spatial slice of the horizon.
\be
E=\frac{1}{8\pi }\int d\phi \Big(\kappa T(\phi)\Big),
\ee
This reproduces the correct charge corresponding to the supertranslation on the horizon \cite{Pinoetal}. This can be extended to the Schwarzschild case also. 





\section{Rotating spacetime and soldering transformation}
In this section we first generalize the case studied in \cite{Blau} for the metrics where $g_{a V} \neq 0.$  We take two specific examples : Kerr spacetime in slow rotation limit and rotating BTZ. As before, we will use the coordinates of $\mathcal{M}_{-}$ as the intrinsic coordinates covering both sides, $x^{\alpha}_-=x^\alpha.$ Also $ x_{-}^{\alpha}|_{\Sigma}= \zeta^a.$ The horizon in the Kruskal coordinates is identified by setting $U=0$ for both $\mathcal{M}_{\pm}.$  The two spacetimes will be isometrically soldered at $U=U_+=0.$ The components $g_{aV}$ is proportional to $U$ such that at $U=0,$ $g_{aV}$ will go to zero. So it can be easily checked that (\ref{killing1}), (\ref{killing2}) and (\ref{killing3}) remains the same.  On the horizon we have $Z^{V}=F(V,X^B),$ which corresponds to the soldering freedom along the null direction. This will again give the supertranslation like transformations. We then extend this construction off the shell inside a small neighborhood of $U=0.$ There will be corrections to the finite transformations due to the presence of rotation compared to the non-rotating case considered in \cite{Blau}. Then we compute the stress energy tensor of the shell and comment on the physical constraints needed to be imposed to obtain supertranslation like soldering transformation. 
%So  let us consider now two examples and write this down explicitly. 
\newpage

\subsection *{\begin{itemize} \item Kerr spacetime: In Slow rotation limit \end{itemize}}

We consider first, the  Kerr metric in slow rotation limit \footnote{Virtue of the slow rotation limit is that, we will have more analytical control. It will be an interesting future problem to repeat this analysis without taking the slow rotation limit.}. In Kruskal coordinates the metric takes the following form, 
\begin{align}\begin{split} \label{kerr1}
ds^2=&r^2(d\theta^2+\sin(\theta)^2d\tilde \phi^2)-\Big(\frac{32m^3}{r}\Big)e^{-r/2m}dUdV\\&+\frac{2a}{r}\sin(\theta)^2e^{-r/2m}(r^2+2 m r+4 m^2) d\tilde \phi (U dV-V dU).
\end{split}
\end{align}$a$ is the rotation parameter and we expand all the quantities in small $a$ and keep only term linear in $a$. 
 On  horizon ($\Sigma$) the induced metric takes the following form,
\be
ds|_{\mathcal{N}}^2=r|_{\mathcal{N}}^2(d\theta^2+\sin(\theta)^2d\tilde \phi^2)
\ee
As mentioned before we take the $\{U_{-}, V_{-},\theta_{-},\tilde \phi_{-}\}$ as the intrinsic coordinates,  $\{U, V,\theta, \tilde \phi\}.$ On the horizon because of the  soldering freedom once again we get, from the killing equations (\ref{killing1}), $Z^{V}= F(V,\theta,\tilde \phi).$ The angular components of the killing equations give the usual isometry transformations on the 2-sphere. Then we extend this soldering transformations off the shell such that, the entire metric across the junction remains continuous i.e $[g_{\alpha\beta}]=0.$ So  following \cite{Blau} we take the ansatz for the off shell soldering transformation,
\be
Z_{+}=Z^{V}\partial_{V}+U z^{\alpha}\partial_{\alpha}.
\ee
On the  horizon,
\be
Z_{+}|_{\mathcal{\Sigma}}=Z^{V}\partial_{V},\quad \partial_{a} Z|_{\mathcal{\Sigma}}=\partial_{a} Z^{V}\partial_{V}.
\ee
Since on the horizon ,
\be
L_{Z_{+}}g_{ab}|_{\mathcal{\Sigma}}=0,
\ee
from (\ref{killing1}), We need to just impose the following for extending it off the shell,
\be
L_{Z_{+}}g_{U\alpha}|_{\mathcal{N}}=0.
\ee
Below we write down all the components explicitly. 
\begin{align}
\begin{split} \label{kerr2}
&Z^{V}\partial_{V}g_{UU}+2 z^{\tilde\phi}g_{\tilde \phi U}+ 2 z^{V}g_{VU}=0,\\&Z^{V}\partial_{V}g_{UV}+\Big(z^{U}+(\partial_{V}Z^{V})\Big)g_{UV}=0,\\&Z^{V}\partial_{V}g_{UA}+z^{\alpha}g_{\alpha A}+(\partial_{A} Z^{V})g_{UV}=0.
\end{split}
\end{align}
Solving (\ref{kerr1}) upto $\mathcal{O}(a)$,
\begin{align}
\begin{split}
&z^{V}=-z^{\tilde\phi}\frac{g_{\tilde \phi U}}{g_{VU}}=-z^{\tilde \phi}\frac{3 a V }{2 m}\sin(\theta)^2=-(\partial_{\tilde \phi} Z^{V})\frac{3 a V}{e m},\\&z^{U}=-\partial_{V} Z^{V},\\&z^{\tilde\phi}=\Big(Z^{V} -(\partial_{V} Z^{V})V\Big)\frac{3 a}{e m}+(\partial_{\tilde \phi} Z^{V})\frac{2}{e \sin(\theta)^2},\\&z^{\theta}=\frac{2}{e}(\partial_{\theta} Z^{V}).
\end{split}
\end{align}
Then using $Z^{V}= V\omega(V,\theta,\tilde \phi)$ we get the required Killing vectors for off-shell transformation,
\begin{align}
\begin{split} \label{kerr3}
Z_{+}&=\omega (V \partial_{V}-U \partial_{U}) +U V \Big(\frac{2}{e \sin(\theta)^2} (\partial_{\tilde\phi} \omega)\partial_{\tilde\phi}+\frac{2}{e}(\partial_{\theta}\omega)\partial_{\theta} -(\partial_{V}\omega)\partial_{U}\Big)\\&-\frac{ 3 a \,U V^2}{e m} \Big(( \partial_{\tilde \phi} \omega)\partial_{V} +  (\partial_{V}\omega)  \partial_{\tilde\phi} \Big).
\end{split}
\end{align}
We now consider the following ansatz to obtain the finite counterparts of the infinitesimal transformations given in (\ref{kerr3}),
\begin{align}
\begin{split} \label{kerr4}
V_{+}&=F(V,\theta, \tilde \phi)+U A(V,\theta,\tilde \phi),U_{+}=U C(V,\theta,\tilde \phi),\\& \theta_{+}=\theta+U B^{\theta}(V,\theta,\tilde \phi),\tilde \phi_{+}=\tilde \phi+U B^{\tilde \phi} (V,\theta,\tilde \phi).
\end{split}
\end{align}


Across the junction, demanding that the full spacetime metric is continuous at the leading order in $U$ upto $\mathcal{O}(a)$ we get,
 \begin{align}
\begin{split} \label{kerr5}
&C=\frac{1}{\partial_V F}\,\\&
B^{\theta}=\frac{2}{e} \frac{\partial_{\theta}F}{\partial_V F}\,\\&
B^{\tilde \Phi}= \frac{2}{e}\frac{1}{\sin(\theta)^2}\frac{\partial_{\tilde \phi}F}{\partial_{V} F}+\frac{3a}{2e\,m }(\frac{F}{\partial_{V} F}- V)\,\\&
A=\frac{e}{4}\partial_{V} F \Big((\frac{2}{e} \frac{\partial_{\theta}F}{\partial_V F})^2+\sin(\theta)^2(\frac{2}{e}\frac{1}{\sin(\theta)^2}\frac{\partial_{\tilde \phi}F}{\partial_{V} F})^2\Big)-\frac{3 a\, \partial_{\tilde \phi}F \, V}{ 2e m }.
\end{split}
\end{align}
%+\frac{a\, e  \,\sin(\theta)^2}{4}\partial_{V} F\Big(2(\frac{2}{e}\frac{1}{\sin(\theta)^2}\frac{\partial_{\tilde \phi} F}{\partial_{V} F})(\frac{3}{2e\,m }(\frac{F}{\partial_{V} F}- V))\\&
Now we want to evaluate $\gamma_{ab}$ and the conserved quantities are as defined in (\ref{junc7}) and (\ref{junc8}).  On $\mathcal{M}_{-}$ we have,
\be
r(UV)^2=4 m^2- \frac{8 m^2}{e} U V+\cdots.
\ee
and on $\mathcal{M}_{+}$ we have,
\be
r(U_{+} V_{+})^2=4 m^2- \frac{8 m^2}{e} U_{+} V_{+}+\cdots= 4m^2-\frac{8 m^2}{e} \frac{U F}{\partial_{V} F }.
\ee
Also
\be
\sin(\theta_{+})^2=\sin(\theta)^2+2 U B^{\theta}\sin(\theta)\cos(\theta)+\cdots
\ee

We now expand the tangential components of the metric off the shell towards both the side upto linear order in $a$ and $U.$ For $\mathcal{M}_{-}$ 

\begin{align}
\begin{split}
g^{-}_{ab}dx^{a} dx^{b}=g^{0}_{AB} dx^{A} dx^{A}- U \frac{8 m^2}{e} \Big( -\frac{3\, a}{2 m} \sin(\theta)^2 dV d\tilde \phi+V d\theta^2+ V \sin(\theta)^2 d \tilde \phi^2\Big)
\end{split}
\end{align}
For $\mathcal{M}_{+}$ we have, 
\begin{align}
\begin{split}
g^{+}_{ab}dx^{a} dx^{b}=g^{0}_{AB} dx^{A} dx^{B}+8 m^2 U &\Big(-\frac{2}{e}dC dF+\sigma_{AB} dx^A(dB^B-(F/eF_V) dx^{B})+\\& \sin(\theta)\cos(\theta)\,\Theta\,\, d\tilde\phi^2+\frac{6 \, a \,\sin(\theta)^2}{4 e m } d\tilde \phi(CdF-F dC) \Big)
\end{split}
\end{align}
Now,
\be
\gamma_{ab}=N^{\alpha}[\partial_{\alpha} g_{ab}]= N^{U}[\partial_{U} g_{ab}]+N^{\phi}[\partial_{\tilde \phi} g_{ab}]
\ee

and 
\be
N^{U}=\frac{e}{8 m^2}\,, N^{\tilde \phi}=\frac{3 a V}{16  m^3}.
\ee
Using (\ref{kerr5}) we get, 
on $\mathcal{M}_{+}$ , 
\begin{align}
\begin{split}
g^{+}_{ab}dx^{a} dx^{b}=g^{0}_{AB} dx^{A} dx^{B}+8 m^2 U &\Big(\frac{2}{e}\Big( \frac{\partial_{V}\partial_{a} F}{\partial_{V} F }dV dx^a\Big)+\frac{2}{e}\frac{\partial_{A}\partial_{B} F}{\partial_{V} F}dx^{A} dx^{B}-\frac{4}{e}\cot(\theta)\frac{\partial_{\tilde \phi} F}{\partial_{V} F} d\theta d \tilde \phi\\&-\sigma_{AB} dx^A(F/eF_V) dx^{B}+ \frac{2}{e}\sin(\theta)\cos(\theta)\,\frac{\partial_{\theta} F}{\partial_{V} F}  d\tilde \phi^2\\&+\frac{3a}{2em}\sin(\theta)^2 d\tilde \phi\Big(2\frac{\partial_{B} F}{\partial_{V} F}dx^{B} +dV \Big)\Big)\end{split}
\end{align}
For $\mathcal{M}_{-}$ we have,
\begin{align}
\begin{split}
g^{-}_{ab}dx^{a} dx^{b}=g^{0}_{AB} dx^{A} dx^{B}- U \frac{8 m^2}{e} \Big( -\frac{3\, a}{2 m} \sin(\theta)^2 dV d\tilde \phi+V d\theta^2+ V \sin(\theta)^2 d \tilde \phi^2\Big)
\end{split}
\end{align}
So we will have upto $\mathcal{O}(a)$,
\begin{align}
\begin{split} \label{kerr6}
&\gamma_{V a }=2\frac{\partial_{V}\partial_{a} F}{\partial_{V} F },\\&\gamma_{\theta \theta }= 2\Big(\frac{\nabla_{\theta}^{(2)} \partial_{\theta} F}{\partial_{V} F }-\frac{1}{2}\Big(\frac{F}{\partial_{V} F}-V\Big)\Big),\\&\gamma_{\theta \tilde \phi }= 2\Big(\frac{\nabla_{\theta}^{(2)} \partial_{\tilde \phi} F}{\partial_{V} F  }+\frac{3 a\sin(\theta)^2}{2 m}\frac{\partial_{\theta} F}{\partial_{V} F}\Big),\\&\gamma_{\tilde \phi \tilde \phi }= 2\Big(\frac{\nabla_{\tilde\phi}^{(2)} \partial_{\tilde \phi} F}{\partial_{V} F }-\frac{1}{2}\sin(\theta)^2\Big(\frac{F}{\partial_{V} F}-V\Big)+\frac{3 a \sin(\theta)^2 }{ 2m }\frac{\partial_{\tilde \phi} F}{\partial_{V} F}\Big).\end{split}
\end{align}

Using (\ref{kerr6}) we can write down the intrinsic energy momentum tensor of the shell. From that we get, 
\begin{align}
\begin{split} \label{kerr7}
&p=-\frac{1}{16 \pi} \gamma_{VV}=-\frac{1}{8\pi}\frac{\partial_{V}^2 F}{\partial_{V} F},\\&j^{A}=\frac{1}{32 m^2 \pi}\sigma^{AB}\frac{\partial_{B}\partial_{V}F}{\partial_{V} F},\\&\mu==-\frac{1}{32 m^2 \pi \partial_{V} F}\Big(\nabla^{(2)} F-F +V\partial_{V}F  +\frac{3 a}{2m}\partial_{\phi} F\Big).
\end{split}
\end{align}
Now when, $p=0,$ $\partial_{V}^2 F =0,$ which gives us, 
\be
F= V + T(\theta,\tilde \phi),
\ee
where $T(\theta,\tilde \phi)$ is an arbitrary function of $\theta$ and $\phi.$
Also when $J^{A}=0$  we have, $\partial_{B}\partial_{V} F=0$, which will imply,
\be
F=a V+B(\theta,\tilde \phi),
\ee
where, $a$ is a scale factor and $B(\theta,\tilde \phi)$ is an arbitrary function of angular coordinates. 
So in general upto a rescaling factor we have,
\be
F(V,\theta,\tilde \phi)= V+ T(\theta,\tilde \phi).
\ee
The corresponding energy density of this kind of the shell is ,
\be
\mu=-\frac{1}{32 m^2 \pi} \Big(\nabla^{(2)} T-T+\frac{3 a}{2m}\partial_{\tilde \phi} T \Big),
\ee
which will imply,
\be
\partial_{V} \mu=0.
\ee
The total energy is given by integrating $\mu$ over the spacelike cross-section of the horizon.
\be \label{kerr7a}
E= \frac{1}{32m^2 \pi}\int d\theta d\tilde \phi \sqrt{g}\,T,\ee
where, $\sqrt{g}$ is the determinant of the induced metric of the spacelike cross-section of the horizon of the null shell. Here we have thrown away the $\nabla^{(2)}T$ and $\partial_{\tilde \phi} T$ terms by doing an integration by parts. So at least upto $\mathcal{O}(a)$ we do not get any corrections coming from the rotation parameter to (\ref{kerr7a}). Next we consider another example of rotating metric when the ambient spacetime is BTZ.\\


\subsection*{\begin{itemize} \item Rotating BTZ \end{itemize}}

The matching conditions are insensitive to the asymptotic structure of spacetime. In view of this, here we study three dimensional BTZ black hole which is not asymptotically flat but the supertranslation like soldering freedom still appears. Unlike the Kerr metric we do not have to consider slow rotation limit for this case. We get an exact analytic answer for arbitrary rotation. So it will be interesting to see whether we get any qualitative changes because of the presence of rotation. The metric takes the following form, 
\be
ds^2= - f(r) dt^2+ \frac{dr^2}{f(r)}+r^2 \Big[N^{\phi}dt +d\phi\Big]^2,
\ee
where, 
$f(r)=-M+(\frac{r}{l})^2+\frac{J^2}{4 r^2}$ and $N^{\phi}(r)=\frac{J}{2}\frac{r^2-r_{h}^2}{r^2 r_{h}^2}.$
Also,
\be
r_{h}^2=\frac{1}{2}\Big(Ml^2+\sqrt{(Ml^2)^2-J^2 l^2}\Big), \tilde r_{h}=\frac{1}{2}\Big(Ml^2-\sqrt{(Ml^2)^2-J^2 l^2}\Big).
\ee
We express all the quantities in terms of $r_{h}$ and $\tilde r_h.$
\begin{align}
\begin{split}
M=\frac{r_{h}^2+\tilde r_{h}^2}{l^2}, \quad J=\frac{2 r_{h} \tilde r_{h}}{l}.
\end{split}
\end{align}
Then we have,
\begin{align}
\begin{split}
N^{\phi}(r)=\frac{\tilde r_{h}}{r_{h}}\frac{r^2-r_{h}^2}{l r^2}\,\quad f(r)=\frac{(r^2-r_{h}^2)(r^2-\tilde r_{h}^2)}{l^2 r^2}.
\end{split}\end{align}
Then we change to Kruskal coordinates \cite{Ross}. 
\be
U=-e^{-\kappa u},\quad V=e^{\kappa v},
\ee
where, $u, v= t\pm r_{*}.$ Also,
\be
r^{*}=\frac{1}{2\kappa}\log\Big(\frac{\sqrt{r^2-\tilde r_{h}^2}-\sqrt{r_{h}^2-\tilde r_{h}^2}}{\sqrt{r^2-\tilde r_{h}^2}+\sqrt{r_{h}^2-\tilde r_{h}^2}}\Big)
\ee
phi)and 
\be
\kappa=\frac{r_{h}^2-\tilde r_{h}^2}{l^2 r_{h}}.
\ee
So finally we get \cite{Ross},
\be \label{rotBTZ}
ds^2=\frac{1}{(1+ UV)^2}\Big(-4 l^2 dUdV- 4 l \tilde r_{h}( U dV-V dU)d\phi+[(1- UV)^2r_{h}^2+ 4 U V \tilde r_{h}^2]d\phi^2\Big).
\ee
Like the Kerr metric we will again get $Z^{V}=F(V,\phi)$ on the horizon. So there is again an arbitrary soldering freedom in the $V$ direction. Next extending this off the shell and  proceeding as before, we find the generators for the off-shell transformation,
\begin{align}
\begin{split} \label{kerr8}
Z_{+}&=\omega (V \partial_{V}-U \partial_{U})+U V \Big(-(\partial_{V} \omega) \partial_{U}+\frac{4 l^2}{r_h^2}[(\partial_{\phi}\omega)+\frac{\tilde r_{h}V }{l} (\partial_{V}\omega)]\partial_{\phi}\Big)\\&+U V^2 \Big(\frac{4 l \tilde r_{h}}{r_{h}^2}[(\partial_{\phi}\omega)+ \frac{\tilde r_{h} V}{l}(\partial_{V}\omega)]\partial_{V}\Big),
\end{split}
\end{align}
where $ F(V,\phi)=V \omega(V,\phi).$
The finite transformations will be generated by the following,
\begin{align}
\begin{split}
V_{+}= F(V,\phi)+ U A(V,\phi), \,\, \phi_{+}=\phi+ U B^{\phi}(V,\phi),\,\,U_{+}= U C.
\end{split}
\end{align}
Then demanding that the entire metric is continuous across the shell we get,
\begin{align}
\begin{split} \label{kerr9}
&C=\frac{1}{\partial_{V} F},\,\,B^{\phi}=\frac{2 (l^2 \partial_{\phi} F-l \tilde r_h F)}{ r_{h}^2\partial_{V}F}+\frac{2 l \tilde r_{h} V}{r_{h}^2},\\& A=\frac{ (l\partial_{\phi}F-\tilde r_{h} F +\tilde r_h V \partial_{V} F)(l \partial_{\phi}F +\tilde r_h F +\tilde r_h V \partial_{V}F)}{\partial_{V}F r_h^2}.
\end{split}
\end{align}
Using the following expressions,  
\be
N^{U}=\frac{1}{2l^2}, N^{V}=-\frac{V^2 \tilde r_{h}^2}{r_{h}^2 l^2}, N^{\phi}=-
\frac{ V \tilde r_{h}}{ r_{h}^2 l}.
\ee
we compute $\gamma_{ab}$ across the junction defined in (\ref{junc7}) as,  

\begin{align}
\begin{split} \label{kerr10}
&\gamma_{VV}=\frac{\partial_{V}^2 F}{\partial_{V} F },\,\,\gamma_{V\phi}=\frac{2 \partial_{V}\partial_{\phi}F}{\partial_{V}F},
\\&
\gamma_{\phi\phi}=\Big(\frac{l^2\partial_{\phi}^2 F-2 l\tilde r_h\partial_{\phi}F-(r_h^2-\tilde r_h^2)(F-V\partial_{v}F) }{\partial_V F l^2}\Big),
\end{split}
\end{align}
Using (\ref{kerr10}), the expression for the pressure and the current will be given by,
\begin{align}
\begin{split}
p=-\frac{1}{8\pi}\frac{\partial_{V}^2 F}{\partial_{V} F},J^{\phi}=\frac{1}{4\pi} \frac{ \partial_{V}\partial_{\phi}F}{r_{h}^2\partial_{V}F}, \mu=-\frac{1}{8\pi}\Big(\frac{l^2\partial_{\phi}^2 F-2 l\tilde r_h\partial_{\phi}F-(r_h^2-\tilde r_h^2)(F-V\partial_{v}F) }{r_h^2 l^2 \partial_V F}\Big).
\end{split}
\end{align}
Again setting $p=0$ we get,
\be
F(V,\phi)= V + T(\phi),
\ee
where $T(\phi)$ is an arbitrary function of angular coordinate. 
Also, if we want to set $J^{\phi}=0$ we will get, 
\be
F(V,\phi)= a V+ B(\phi). 
\ee
$B(\phi)$ is an arbitrary function of angular coordinate.  

Also for $p=0$  (or $J^\phi=0)$ case we again retrieve the fact that, $\partial_{V}\mu=0,$
where,
\be
\mu=-\frac{1}{8\pi l^2}\Big(\frac{l^2T''(\phi)-2 l\tilde r_h T'(\phi)-(r_h^2-\tilde r_h^2)T(\phi) }{r_h^2 }\Big).
\ee
The total energy once again is obtained by integrating $\mu$ on the spatial slice of the horizon.
\be
E=\frac{1}{8\pi }\int d\phi \Big(\kappa T(\phi)\Big),
\ee
where we have thrown away both $\partial_{\phi}^2 $ and $\partial_{\phi} T $ term as we have performed an integration by part. This matches with the supertranslation charge derived in the literature, for example readers are referred to (\cite{Pinoetal,Barnich:2009se}).\\


\section{Conformal isometries and modified soldering conditions} \label{sec5}

\subsection{Conformal Isometry }
So far we have considered the case of isometries of the induced metric on the horizon. This is of course nothing new, if one solder two spacetimes across a null shell, it will preserve the fundamental junction condition and the resulting geometry of the hypersurface will have some non triivial properties. The particular case of interest is the one where one gets the pressure of the stress-tensor zero, the coordinate freedom that corresponds to this kind of soldering turns out to be angle dependent translations or supertranslations. Now, can one generalize the freedom of soldering to allow conformal isometry (of the induced metric) to this picture? This is the question we are going to address now.   
We will adopt the same process like the previous section. We take a metric on hypersurface then allow conformal isometries by solving the conformal Killing equation, and then obtain the soldering freedom. After this we take a transformed metric obtained via this new kind of soldering freedom (on one side of hypersurface) and stitch it to the other side of the metric just as done earlier. Although this is not directly related to the superrotation kind of symmetry that is recovered on the celestial sphere at asymptotic infinity but this ikind of analysis we can expect would produce the superrotation type symmetry from the soldering freedom perspective.  

 Let us consider the situation when one makes conformal transformation to the boundary manifold (spacetime) of one side ($\cal{M_-}$) and join with the other side $\cal{M_+}$ which remains as the background seed spacetime,
\be \label{conf2}
g^{+}_{cd}|_{\Sigma_{+}}= \Omega(\zeta)^2\, g^{-}_{cd}|_{\Sigma_{-}}\ee
 This basically means, that the induced metric from both sides of the shell will be equal to each other upto a conformal factor. 
%This particular situation may not be very common but still this type of deviations are perfectly allowed in the Israel's junction condition formalism \cite{Barrabes}. In fact any jump in the first junction condition may be taken care of by switching to a new coordinate system and having a jump in the first derivatives so that continuity ($C^1$) of the induced metric is maintained \cite{Mansouri:1996ps}. 

As a consequence of (\ref{conf}), any infinitesimal coordinate transformation of the form, $x_{\pm}^{a}+ Z_{\pm}^{a},$ generated by $Z_{\pm}^{a}$ preserving (\ref{conf2}), obeys,
\be \label{conf3}
Z_{\pm}^{ c}\partial_{c}g_{ab}+(\partial_a Z_{\pm}^{c})g_{cb}+(\partial_b Z_{\pm}^{ c})g_{ac}=(1- \Omega(\zeta)^2) g_{ab}.
\ee
(\ref{conf3}) is nothing but the conformal Killing equation. Next we explore the solutions of the Conformal Killing vectors on the surface of the null shell. 

%We define a common coordinate system $x$ such that, we can express the both sides of the shell using this same coordinate system. Then we again use the distributional method to analyze the Einstein equation as defined in section (\ref{first}). 
%\be
%g_{\alpha\beta}(x)= g^{+}_{\alpha\beta}(x)\Theta(\Sigma)+g^{-}_{\alpha\beta}(x)\Theta(-\Sigma).
%\ee
%From this we get,
%\be \label{conf7a}
%\partial_{\mu}g_{\alpha\beta}(x)= \partial_{\mu} g^{+}_{\alpha\beta}(x)\Theta(\Sigma)+\partial_{\mu}g^{-}_{\alpha\beta}(x)\Theta(-\Sigma)+[g^{+}_{\alpha\beta}|_{\Sigma_{+}}-g^{-}_{\alpha\beta}|_{\Sigma_{-}}]n_{\mu}\delta(\Sigma).
%\ee
%The third term again vanishes because of (\ref{conf2}). Armed with this result we can safely conclude that the the distributional method can be applied as before and there will be only jump in the normal direction. Consequently we can apply the results of the section (\ref{first}) for computing the conserved quantities.  Then the question of soldering freedom in this case boils down to the fact that,  what are possible coordinate transformations one can do say on $\mathcal{M}_{+}$ (or $\mathcal{M}_{-}$) such that the condition (\ref{conf}) will hold. This says that we need to solve the conformal killing equation (\ref{conf2}) on the horizon. At this point let us take some concrete examples .
%\vspace{-1cm}
\subsection*{\begin{itemize}\item Rotating BTZ \end{itemize}} \label{second}
First we consider rotating BTZ metric in Kruskal coordinates \cite{Ross} as shown in (\ref{rotBTZ}).
%\be \label{conf1}
%ds_{\pm}^2=\frac{1}{(1+ U_{\pm}V_{\pm})^2}\Big(-4 l^2 dU_{\pm}dV_{\pm}- 4 l \tilde r_{h}( U_{\pm} dV_{\pm}-V_{\pm} dU_{\pm})d\phi_{\pm}+[(1- U_{\pm}V_{\pm})^2r_{h}^2+ 4 U_{\pm} V_{\pm} \tilde r_{h}^2]d\phi_{\pm}^2\Big).
%\ee

%$\pm$ denote the two spacetimes inside and outside the shell. $U_{\pm}=0$ denote the horizon ($\Sigma_{\pm}$) for these two metrics. Now we want to identify these two surfaces in a way such that, (\ref{conf}) holds. For this case, it amounts to the fact that, on the surface of the shell from both the side we will have,
%\begin{align}
%\begin{split}\label{conf6a}
%&r_{h}^2 d\phi_{+}^2=r_{h}^2 \Omega(\phi)^2 d\phi^2,\\&r_{h}^2 d\phi_{-}^2=r_{h}^2 \Omega(\phi)^2 d\phi^2
%\end{split}
%\end{align}

%This will gives us $\phi_{\pm}$ as function $\phi.$ So we have,
%\be
%\phi_{+}=\Omega(\phi)+c_1, \phi_{-}=\Omega(\phi)+c_2.
%\ee
%So $\phi_{+}$ and $\phi_{-}$  differ upto a constant factor, so that $d\phi_{+}=d\phi_{-}.$
On $\Sigma$ defined by $U_{\pm}=0,$  the coordinates are $\{V, \phi\},$ which we can set as the intrinsic coordinates of the shell. The killing vector takes the following form,
\begin{align}
\begin{split}
Z|_{\mathcal{N}}=\Big(Z^{V}\partial_{V}+Z^{\phi}\partial_{\phi}\Big)|_{\mathcal{N}}. 
\end{split}
\end{align}
First we check $\phi\,V$ component of (\ref{conf3}). This gives,
\begin{align}
\begin{split}  \label{conf4}
\partial_{V} Z^{\phi}=0.
\end{split}
\end{align}
This implies, that there cannot be any $V$ dependent transformation for the spatial coordinate. 
Then lets look at the $\phi\,\phi$ component of the Killing equations.
\begin{align}
\begin{split}
Z^{\lambda}\partial_{\lambda} g_{\phi\phi}+(\partial_{\phi} Z^{\lambda})g_{\lambda \phi}+(\partial_{\phi} Z^{\lambda}) g_{\lambda\phi}=(1-\Omega(V,\phi)^2) g_{\phi\phi}.
\end{split}
\end{align}

Further simplifying we get, 
\begin{align}
\begin{split} \label{conf5}
2(\partial_{\phi} Z^{\phi})=(1-\Omega(V,\phi)^2).
\end{split}
\end{align}
Using (\ref{conf4}) we get, $\partial_{V} \Omega(V,\phi)=0.$ This will imply, $\Omega(V,\phi)=\Omega(\phi).$ So we get, 
\begin{align}
\begin{split} \label{conf6}
&Z^{V}=F(V,\phi),\\&
Z^{\phi}= \frac{1}{2}\int_{1}^{\phi} (1- \Omega(\tilde \phi)^2) d\tilde \phi+C_1= \mathcal{T}(\phi)+C_1,
\end{split}
\end{align}
both are arbitrary functions. %Second of (\ref{conf6}) actually consistent with the condition (\ref{conf6a}). 
 Let us now look at the normal vectors generating the null congruence orthogonal to the hypersurface. As $\Sigma$ is identified by both $U_{+}=U_{-}=U=0,$ so $[n^{\alpha}]=0.$ This implies the continuity of the  null congruence on both sides. Now if we start from the following,
\be
g_{ab}n^{b}=0,
\ee
we get again like in the section~(\ref{third}), 

\be
\mathcal{L}_{Z} n^{a}\sim n^a.
\ee
Then demanding  that it exactly preserves the normal vector, we get,
\be
F(V,\phi)= F(\phi).
\ee
Therefore on the horizon we get the following two transformations,
\begin{align}
\begin{split} \label{tt3}
V\rightarrow V+ F(\phi),
\phi\rightarrow \mathcal{T}(\phi)+ C_1,
\end{split}
\end{align}
which are nothing but the supertranslation and superrotation like transformations. 
%We would like to make one more observation, if we are working with the usual Israel junction condition, then we can choose either of the coordinates of $\mathcal{M}_{\pm}$ as the intrinsic coordinates for $\Sigma$ without loss of any generality. It seems that for our new condition (\ref{conf}) it is not possible. If we want to do that, it will mean the following,
%\be
%\Sigma_{+}(x_{+})=\Omega(x_{-})^2\,\Sigma_{-}(x_{-})=\Sigma(x_{-}).
%\ee
%For this case of BTZ we will have the following,
%\be \label{conf7}
 %r_{h}^2 d\phi_{+}^2= \Omega (\phi_{-})^2r_{h}^2 d\phi_{-}^2.
%\ee
 %We can set, $\phi_{-}=\phi.$ It will give us the required coordinate transformation. $\phi_{+}=\Omega(\phi)+ c_1.$ To achieve this condition consistently (by projecting the bulk metric to the surface) we need to rescale at least the $\phi\,\phi$ component of the metric of $\mathcal{M}_{-}$ by the conformal factor $\Omega(\phi).$ Then again we can infer that, the third term in (\ref{conf7a}) goes to zero. But there will be a jump in the tangent vector as,
%\be
%e^{+\phi_{+}}_{\phi}=\Omega(\phi), e^{-\phi_{-}}_{\phi}=1.
%\ee
%So there will be a discontinuity in the tangential derivative of the metric. We can avoid this discontinuity by using the condition as shown in (\ref{conf2}).

%Henceforth we will work with the condition (\ref{conf2}). But we emphasize one more time here, utilizing the rescaling freedom we can indeed obtain both the supertranslation and superrotation like transformations by using the appropriate soldering conditions. 
%\vspace{-1cm}

\subsection*{\begin{itemize} \item Schwarzschild \end{itemize}}
We know look at the Schwarzschild case. We follow the same strategy as in previous section and proceed to solve the conformal Killing equations on the horizon shell. 

First we write down the Schwarzschild metric in double null coordinate,
 \begin{align}
 \begin{split}
 ds^2=-2 G(r) dU dV + r^2(d\theta^2 +\sin(\theta)^2 d\phi^2).
 \end{split}
 \end{align}
 We do the following coordinate transformations,
 \be
 z=e^{i \phi} \cot (\frac{\theta}{2}),\bar z=e^{-i \phi} \cot (\frac{\theta}{2}).
 \ee
 Then in terms of these complex variables we get,
 \be
ds^2=-2 G(r) dU dV + r^2\frac{4\, dz\, d \bar z}{(1+z\bar z)^2} . 
\ee
 
 On the horizon the Killing vectors takes the following form,
 \be
 \mathcal{Z}|_{\mathcal{N}}=\Big(Z^{V}\partial_{V}+Z^{z}\partial_{z}+Z^{\bar z}\partial_{\bar z}\Big)
 \ee
The we proceed to solve the conformal killing vector equations,
\be
\nabla_{a} Z_b+\nabla_b Z_a-(\nabla_a Z^a)g_{ab}=0.
\ee
From $V\,A$ components of these equations we again obtain,
\begin{align}
\begin{split}
\partial_{V} Z^{C}=0.
\end{split}
\end{align}

Remaining components give, 
\begin{align}\label{conf8}
\begin{split}
\partial_{z} Z^{\bar z}&=0,\\
 \partial_{\bar z} Z^{z}&=0.
 %Z^C\partial_{C} g_{z\bar z}+ \partial_{z}Z^{ z} g_{z\bar z}+\partial_{\bar z} Z^{\bar z} g_{z\bar z}&=(1- \Omega(z,\bar z)^2)  g_{z\bar z}.  
\end{split}
\end{align}
(\ref{conf8}) gives,
\be
Z^{\bar z}= \mathcal{  \bar T}(\bar z)+c_2, Z^{z}=\mathcal{T}(z)+c_3,
\ee
Where, $T(\bar z) $ and $T(z)$ are arbitrary functions of only $z$ (or $\bar z$). 
%Third equation of (\ref{conf8}) after rearranging gives,
%\be
%-\frac{\bar z }{1+ z \bar z} Z^{z}-\frac{ z }{1+ z \bar z} Z^{\bar z}+\partial_{z}Z^{ z} +\partial_{\bar z} Z^{\bar z} =(1- \Omega(z,\bar z)^2)
%\ee
%Then we can find $\Omega(z,\bar z)$ as,
%\be \label{conf9}
%1-\Omega(z,\bar z)^2=-\frac{\bar z}{1+z\bar z}T(z)-\frac{z}{1+z\bar z}\bar T (\bar z)+T'(z)+\bar T'(\bar z).
%\ee

So any transformations of the following is permitted 
%as long as (\ref{conf9}) is satisfied,
\be
z\rightarrow  \mathcal{T}(z)+c_2,\,\, \bar z\rightarrow  \mathcal{T} (\bar z)+c_3.
\ee
A point to be noted here, $T(z)$ and $\bar T (\bar z)$ can be in general singular at several points on the 2-sphere. If we require these transformations to be globally well defined on the 2-sphere, then it will only pick out the appropriate spherical harmonics. 
%as in those cases the conformal factors (written in terms of spherical coordinates ) will be either $\cos(\theta)$ or $\sin (\phi) \sin(\theta)$ or $\cos(\phi)\sin(\theta).
%\footnote{Details of this calculation of conformal killing vectors on 2-sphere in spherical coordinate are demonstrated in the Appendix~ (\ref{append}).}
 Also we will have the following transformation, 
\be
V \rightarrow V + B (V, z,\bar z).
\ee
By demanding again that it preserves the normal vector, we can set $B(V,z,\bar z)= B(z,\bar z).$ So we will have both supertranslation and superrotation type soldering on the horizon shell.   Now these transformations can also be extended offshell in the  analogous way as presented in \cite{Blau}. We compute the offshell killing vectors below.

\subsection*{\begin{itemize} \item Rotating BTZ: Off-shell Killing vector \end{itemize}}

We now start with the following form of the killing vectors, 
\be
Z=Z^{V}\partial_{V}+Z^{\phi}\partial_{\phi}+ U z^{\alpha}\partial_{\alpha}.
\ee

On the junction,
\be
Z|_{\mathcal{N}}=Z^{V}\partial_{V}+ Z^{\phi}\partial_{\phi},\quad \partial_{a} Z|_{\mathcal{N}}=\partial_{a} Z^{V}\partial_{V}+\partial_{a} Z^{\phi}\partial_{\phi}.
\ee

Since $U=0$ on the horizon, $Z_{+}$ satisfy the conformal Killing vector equations with respect to $g_{ab},$
 we just need to impose that the $Z_{+}$ satisfy the conformal killing equations  with respect to the $g_{U\alpha}.$ 
This will give, 
\begin{align}
\begin{split} \label{offshell}
&z^{\phi}g_{ U\phi}+z^{V}g_{UV}=0,\\&z^{U}+(\partial_{V} Z^{V})=(1-\Omega(\phi)^2),\\& Z^{V}\partial_{V}g_{U\phi}+ z^{U}g_{U \phi}+z^{\phi}g_{\phi\phi}+ (\partial_{\phi} Z^{V}) g_{UV}=0.
\end{split}
\end{align}

Here $g_{\phi\phi}, g_{U\phi}$ and $g_{UV}$ are evaluated on the horizon. 
(\ref{offshell}) gives the following solutions,
\begin{align}
\begin{split}\label{offshell1}
&z^{V}=-z^{\phi}\frac{g_{\phi U}}{g_{VU}}=z^{\phi}\frac{\tilde r_{h} V}{l},\\&z^{U}=(1- \Omega (\phi)^2)-(\partial_{V} Z^{V}),\\&z^{\phi}=\frac{4 l^2}{r_{h}^2 }(\partial_{\phi} Z^{V})+\frac{ 4 l \tilde r_{h}}{r_{h}^2}(V(\partial_{V} Z^{V})-Z^{V}).
\end{split}
\end{align} 

Setting $ Z^{V}=V \omega (V,\phi)$ we get,
\begin{align}
\begin{split} \label{t1}
&Z=\omega (V \partial_{V}-U \partial_{U})+U  \Big((1-\Omega(\phi)^2-V(\partial_{V} \omega)) \partial_{U}+\frac{4 l^2 V}{r_h^2}[(\partial_{\phi}\omega)+\frac{\tilde r_{h}V }{l} (\partial_{V}\omega)]\partial_{\phi}\Big)\\&+U V^2 \Big(\frac{4 l \tilde r_{h}}{r_{h}^2}[(\partial_{\phi}\omega)+ \frac{\tilde r_{h} V}{l}(\partial_{V}\omega)]\partial_{V}\Big)+Z^{\phi}\partial_{\phi}.
\end{split}
\end{align}


\subsection*{\begin{itemize} \item Schwarzschild: Off-shell Killing vector \end{itemize}}

For this case, proceeding similarly as before, we get the following equations,
\begin{align}
\begin{split}
& z^{V}g_{VU}=0,\\&z^{U}+(\partial_{V}Z^{V})=(1-\Omega(z,\bar z)^2,\\&z_{A}+(\partial_{A} Z^{V})g_{UV}=0.
\end{split}
\end{align}
This gives the following solutions,
\begin{align}
\begin{split}
&z^{V}=0,\\& z^{U}=(1-\Omega(z,\bar z)^2)-(\partial_{V} Z^{V}),\\&
z^{A}=4 l^2 g^{AB}(\partial_{B}  Z^{V}).
\end{split}
\end{align}
\begin{align}
\begin{split}\label{physical7}
Z_{+}&=\omega \Big(V  \partial_{V}- U\partial_{U}\Big)+ U  \Big((1-\Omega(z,\bar z)^2-V(\partial_{V}\omega))\partial_{U}+4 l^2 V g^{AB}(\partial_{B}\omega)\partial_{A}\Big)+Z^{A}\partial_{A}.
\end{split}
\end{align}
 In principle we can exponentiate this generators to find the corresponding conserved charges. But in general that is very difficult tasks to do.  Instead we will proceed to compute the bulk stress energy tensor much like what one do for the Israel junction condition but keeping in mind that now we have modified it. 

\section{Stress energy tensor for  modified junction condition}

Following the discussion in the begining of the previous section we will consider solder of two spacetime which are being superrotated relative to each other.   We first consider the example of the rotating BTZ space time which will demonstrate all the essential points. Like the the case of the supertranslation we will solder two metric : one of which is the non-rotating BTZ metric on $\mathcal{M}_{+}$ and other one is superrotated i.e we have performed an active transformation of the form $\phi \rightarrow \mathcal{T}(\phi)+c_1$ following from (\ref{tt3}) on $\mathcal{M}_{-}$. Then as before we drop the subscript $\pm$ and solder the BTZ metric with the,
\be
ds_{-}^2=\frac{1}{(1+ UV)^2}\Big(-4 l^2 dUdV %- 4 l \tilde r_{h}( U dV-V dU)\mathcal{T}'(\phi)d\phi
+(1- UV)^2r_{h}^2%+ 4 U V \tilde r_{h}^2
\mathcal{T}'(\phi)^2d\phi^2\Big).
\ee
 So the total metric (on $\mathcal{M}_{+} \cup \mathcal{M}_{-}$) is ,
 \begin{align}
 \begin{split}
& g_{UV}=-\frac{2 l^2}{(1+U V)^2}, g_{U\phi}= \frac{2\, l\, \tilde r_{h} V (-\Theta (\Sigma) \mathcal{T}' (\phi )+\Theta (\Sigma)+\mathcal{T}' (\phi ))}{(1+U V)^2},\\&g_{V\phi}=\frac{2\, l\, \tilde r_{h} U (\Theta (\Sigma) (\mathcal{T}' (\phi )-1)-\mathcal{T}(\phi )')}{(1+U V)^2},g_{\phi\phi}=\frac{\left(r_{h}^2 (U V-1)^2+4 \tilde r_{h}^2 U V\right) \left(\Theta (\Sigma)-(\Theta (\Sigma)-1) \mathcal{T}' (\phi )^2\right)}{(1+U V)^2}
 \end{split}
 \end{align}
Now we proceed to compute the Einstein tensor and identify the singular part of it. The singular part will be related to the shell's stress energy tensor. One thing we should note that unlike the usual junction condition in this case we have $$[e^{\mu}_{a}]=(\mathcal{T}'(\phi)-1)\delta^{\mu}_{a}.$$ Due to this tangential discontinuity we will have more stronger singularity than simple $\delta(\Sigma)$ in the Einstein equation.  The full Einstein tensor is quite lengthy and cumbersome. We present the full equation in a simplified picture,  considering the $\mathcal{T}(\phi)$ very close to one.
\be
\mathcal{T}(\phi)=1+\epsilon \tilde T(\phi).
\ee
We expand all the quantities in the $\epsilon$ and keep only leading order term. 
Below we present only the leading order part of the stress tensor.
\begin{align}
\begin{split} \label{sing}
&
8\pi S^{VV}=-\frac{V\,\epsilon }{2} \tilde T'(\phi )\delta(\Sigma)+\frac{ \epsilon\, (1+ U V)^4 }{4}\tilde T' (\phi )\delta'(\Sigma)+\mathcal{O}(\epsilon^2).
%8\pi S^{V\phi}= \frac{\tilde r_{h} V^2 \epsilon  \left(\left(2 r_{h}^2-11 \tilde r_{h}^2\right)\tilde  T(\phi )+2 \tilde r_{h} \tilde T'(\phi )\right)}{54 r_{h}^4}\delta(\Sigma)-\frac{\tilde r_{h} V \epsilon  T(\phi )}{54 r_{h}^2}\delta'(\Sigma)+\mathcal{O}(\epsilon^2),\\&
%8\pi S^{\phi\phi}=-\frac{3 \tilde r_{h} \epsilon  \tilde T(\phi )}{2 r_{h}^4}\delta(\Sigma)+\mathcal{O}(\epsilon^2).
\end{split}
\end{align}
Upon using the identity $f(x)\delta'(x)=-f'(x)\delta(x)$ we can write the above expression as

\begin{align}
\begin{split} \label{sing}
&
8\pi S^{VV}=-\frac{3\,V\,\epsilon }{2} \tilde T'(\phi )\delta(\Sigma)+\mathcal{O}(\epsilon^2).
%8\pi S^{V\phi}= \frac{\tilde r_{h} V^2 \epsilon  \left(\left(2 r_{h}^2-11 \tilde r_{h}^2\right)\tilde  T(\phi )+2 \tilde r_{h} \tilde T'(\phi )\right)}{54 r_{h}^4}\delta(\Sigma)-\frac{\tilde r_{h} V \epsilon  T(\phi )}{54 r_{h}^2}\delta'(\Sigma)+\mathcal{O}(\epsilon^2),\\&
%8\pi S^{\phi\phi}=-\frac{3 \tilde r_{h} \epsilon  \tilde T(\phi )}{2 r_{h}^4}\delta(\Sigma)+\mathcal{O}(\epsilon^2).
\end{split}
\end{align}

In the process of writing this we have used $\theta (\sigma)\delta(\Sigma)=\frac{1}{2}\delta(\Sigma)\,\,$\footnote{This is a standard result in distributional calculus \cite{Balasin-Steinbauer}.} and $\Theta(\Sigma)^2=\Theta(\Sigma).$ 
Then from  (\ref{conf6}) we have $ \mathcal{T}'(\phi)=1-\Omega(\phi)^2$ and assume the conformal factor to be infinitesimal , i.e $\Omega(\phi)=1+\epsilon\, \tilde \Omega(\phi)$ we have,
\begin{align}
\begin{split} \label{sing1}
&
8\pi S^{VV}=3\,V\,\epsilon\, \tilde \Omega(\phi)\, \delta(\Sigma)+\mathcal{O}(\epsilon^2).
\end{split}
\end{align}
%{\bf AB: Write some sentences here}

  Following (\ref{ST}) one may now identify the parts that will contribute to the energy density, current and pressure terms. However, we should be careful in reading off the shell intrinsic quantities from (\ref{ST}) as there will be extra contributions in the structure of $S^{\alpha \beta}$ due to the nonzero jump in the metric. From the expressions of the components of stress tensor, to $\epsilon$ order, the $VV$ %and $\phi-\phi$ 
 component is showing that the shell intrinsic quantities depend on $\tilde \Omega(\phi)$, the component of  superrotation-like generator. We should mention that there are higher order singular terms $\delta^2$ present in the full expression of the stress tensor which we can not avoid. The appearance of terms like $\delta^2$ in stress tensor of null horizon shell indicates that the matter residing on the shell is exotic in nature. The physical interpretation of this kind of a situation is not simple and may need more delicate investigations. In electrodynamics, we usually deal with ``smooth'' surface charge distributions across a thin surface which ensures that the tangential components of the electric field remain continuous across the surface (although normal component suffers from a discontinuity). The analogous situation in Gravity arises in the case of supertranslational kind of soldering, where the metric and it's tangential derivatives remain continuous. However the Riemann (and Ricci) tensor of resulting spacetime contains delta function like singularities that can be thought to be produced by some material surface layer on the horizon shell.  The situation under hand is a departure from this usual scenario, since to allow conformal isometries on the null shell, we have got discontinuities in the tangential derivative of the metric, the corresponding  material layer of the shell has now developed more stronger singularities in the form of $\delta^2$. Therefore the distributional algorithm of joining two spacetimes suffers from interpretational difficulties. This situation may also arise when one deals with charge distributions of singular nature across a surface in electrodynamics.\footnote{ In \cite{Zhiboedov} authors have proposed a physical interpreation for the finite superrotation trsnaformation. They argued that these trsnformations act on the asymptotically flat spacetime with isolated singular points due the cosmic string defects. There might be connection between their construction and the presence of the stronger singularity like $\delta^2$ that we have found. We leave this for future study. }

The $\delta^2$ type singular terms arising in Junction conditions are sometimes dealt with care using sophisticated distributional calculus. One replaces the Delta function by a limit of a sequence and regulates the terms. But one has to ensure that finally  the expression should be independent of the regulator \cite{Balasin-Steinbauer}. There also exists theory of nonlinear generalized functions developed by Colombeau and others \cite{Columbeau}, which can cure these kind of singular terms. We are not in a position to apply those sophisticated techniques here.
 
If we restrict ourselves to the case where $\Omega$ is small, then to leading order in $\epsilon$ we only get terms proportional to $\delta$ in the stress tensor. This indicates the distributional algorithm holds quite well in this situation when the discontinuity of the tangential derivative of the metric is mild. In this weak transformation limit one may extend the analysis and go on to define a charge corresponding to superrotation like soldering transformation (it will be proportional to $\mu$) but it would not be equal to the superrotaion charge calculated at the celestial sphere of asymptotic boundary. In general there is no a priori reason to expect that (although the supertranslation part indeed match) the charge calculated in this way will match with the expression exists in literature. Even the conserved charge computed in \cite{Pinoetal,Barnich}, used Noether procedure from the gravitational action, but for our case we are just computing the intrinsic quantities of the shell which comes from the stress energy tensor required for satisfying the Einstein equations. 

\par Let us briefly touch upon the results when the above procedure is employed to non-rotating BTZ case for the sake of completeness.
\be
ds_{-}^2=\frac{1}{(1+ UV)^2}\Big(-4 l^2 dUdV - 4 l \tilde r_{h}( U dV-V dU)\mathcal{T}'(\phi)d\phi
+\Big[(1- UV)^2r_{h}^2+ 4 U V \tilde r_{h}^2
\mathcal{T}'(\phi)^2\Big]d\phi^2\Big).
\ee
We write down the singular part of these stress energy tensor below. 
\begin{align}
\begin{split}
&8\pi S^{VV}= \frac{V \epsilon  \left(\left(r_{h}^2-4 \tilde r_{h}^2\right) \Omega (\phi )+\tilde r_{h} \Omega '(\phi )\right)}{r_{h}^2}\delta(\Sigma)-\Big(\frac{\epsilon  (U V+1)^4 \Omega (\phi ) \left(r_{h}^2 (U V-1)^2+4 \tilde r_{h}^2 U V\right)}{2 r_{h}^2 (U V-1)^2}\Big)\delta'(\Sigma)+\mathcal{O}(\epsilon^2),\\&
8\pi S^{V\phi}=-\frac{3 \tilde r_{h}\, \epsilon \,\Omega (\phi )}{2 r_{h}^2}\delta(\Sigma)+\mathcal{O}(\epsilon^2).
\end{split}
\end{align}
Then using the identity $f(x)\delta'(x)=-f'(x)\delta(x)$ we can rewrite the above expression for $S^{VV}$ as,
\begin{align}
\begin{split}
&8\pi S^{VV}= \frac{V \epsilon  \left(\left(3r_{h}^2-2 \tilde r_{h}^2\right) \Omega (\phi )+\tilde r_{h} \Omega '(\phi )\right)}{r_{h}^2}\delta(\Sigma)+\mathcal{O}(\epsilon^2).
\end{split}
\end{align}
Like the non rotating case we get $\delta, \delta'$ terms in $VV$ component. Also in the rotating case we have non zero term proportional to the $\delta$ in the $V\phi$ component. We have checked that for arbitrary conformal factor there is no stronger singular term like $\delta ^2$ in the $V\phi$ component. Only we get those stronger singular terms in the $VV$ component.  We can conclude that energy density  (coefficient of the $S^{VV}$ ) of the shell is singular but the current density of the shell is non-singular. To conclude, the upshot of our exercise is: if one tries to patch a superrotated spacetime with the one which is unchanged counterpart of the former (to allow conformal isometries on the horizon shell), one has to deal with exotic matter on the shell. However, if we restrict our attention to the case when the transformation parameter is small then we find well defined quantities on the shell. In this section we have only considered the BTZ spacetime, one can easily extend this exercise  exercise for Scwarzschild metric. Apart from stationary cases, one may be interested to know what happens to these soldering transformations for non-stationary spacetimes. In the appendix A, we discuss this issue briefly.   

\section{Summary and Discussion}
 We have revisited the dynamics of thin null shell situated at the horizon of black holes in general relativity and explored the consequences when the induced metric on the shell is invariant up to a conformal rescaling. In the process we have to consider discontinuous tangential derivative in the distributional algorithm of junction conditions, to include the effect of conformal isometries. We have demonstrated that conformal isometries induce {\it superrotation type} soldering transformation  like pure isometries induce supertranslations. The nullity of the horizon shells play crucial role to generate these freedom to solder two spacetimes. For shells in BTZ black holes, superrotation being generated by an arbitrary conformal factor. We have also computed the shell's intrinsic stress energy tensor and found the energy density, current density and pressure, all depending on arbitrary conformal factor. The stress tensor of the shell however is not globally well defined as singular terms like $\delta^2$ arise that do not have well defined meaning. This indicates that while patching a conformally transformed (superrotated) spacetime with another unaltered seed spacetime we bound to get some exotic feature on the material layer existing on the shell. One may have to apply nonlinear distributional calculus like Colombeau calculus to rectify these pathologies.  We leave this for a future study to understand better the implications of these results. However, if we consider matching of weakly superrotated spacetime with the one without being changed, then the singularities in the stress tensor do not appear to the leading order. In this case we could get well defined quantities- like the charge corresponding to the superrotation like soldering transformation. We should also mention that the matching conditions that we have used to generate superrotation type symmetries are locally well defined but we don't know whether there can be any global inconsistency arising from the effects of these conformal isometries on horizon topology. This needs further investigations. So the summary of our exercise is: if we allow conformal isometries on the null shell and look for the soldering freedom, the soldering transformation looks like a superrotation but the  material surface layer of the horizon shell develops some irregularities. If we deal with weakly superrotated spacetmes, we get enough control over the irregularities and make connections with the conserved charges of superrotation type soldering transformations.     
 
 \par

We anticipate that the results obtained here can have far reaching consequences specially in the context of black hole information paradox.  It will be interesting to investigate the soldering symmetries for collapsing shells. At late time we expect the emerging symmetry will be close to the BMS like soldering transformations \cite{Tanabe}. We hope to get back to this issue in near future. Also it would be interesting to study any possible connection between black hole membrane paradigm and supertranslation-superrotation transformations and how the BMS like soldering transformations affect the properties of the horizon fluid \cite{Penna:2015gza,Eling}. 
 
 \section*{Acknowledgements}
Authors would like to thank Alok Laddha, Amitabh Virmani, Avirup Ghosh, Swastik Bhattacharya, and Sudipta Sarkar for their valuable comments. AB would like to thank Prof. Ling-Yan Hung for discussions and constant encouragements and  acknowledges the support from the Thousand Young Talents Program, and Fudan University. SB acknowledges the support from IIIT Allahabad.

\appendix
\section{Vaidya Spacetime}
We briefly apply our construction for non stationary spacetimes. From (\ref{killing2}) it is evident that if we consider Vaidya type spacetime we will get, $Z^{V}=0.$ But now we will consider our modified junction condition where we are patching two spacetimes by allowing a conformal rescaling freedom. In the following  we look at both three and four dimensional Vaidya spacetime. For Vaidya spacetime,
\be
ds^2=- f(U,V) dU dV+ r(U,V)^2 d\Omega_{D-2}^2,
\ee
where the $U=0$ is the horizon of the collapsing shell. $d\Omega_{D-2}^2$ is the metric of the $D-2$ sphere expressed in the standard spherical polar coordinate. The functions $f(U,V)$  and $r(U,V)$ are such that even at $U=0$ they are nontrivial functions of $V.$ So for this case we will have  $\partial_{V}g_{AB} \neq 0$ even at $U=0.$ Next we consider briefly two specific examples.
\vspace{-1cm}
\subsection*{\begin{itemize} \item BTZ-Vaidya spacetime \end{itemize}}
Form (\ref{conf3}) it follows,
\begin{align}
\begin{split} \label{vaidya}
\partial_{V} Z^{\phi}=0.
\end{split}
\end{align}
and 
\begin{align}
\begin{split} \label{vaidya1}
Z^{\lambda}\partial_{\lambda} g_{\phi\phi}+(\partial_{\phi} Z^{\lambda})g_{\lambda \phi}+(\partial_{\phi} Z^{\lambda}) g_{\lambda\phi}=(1-\Omega(V,\phi)^2) g_{\phi\phi}.
\end{split}
\end{align}
Now we will have ,
\begin{align}
\begin{split}\label{vaidya2}
Z^{V}\partial_{V} g_{\phi\phi}+2(\partial_{\phi} Z^{\phi})g_{\phi \phi}=(1-\Omega(V,\phi)^2) g_{\phi\phi}.
\end{split}
\end{align}
%So if  $\Omega(V,\phi)=\Omega(\phi)$ , then, $ Z^{V}=0$ and using (\ref{vaidya1}) we conclude that,
%\be
%\partial_{\phi} Z^{\phi}= (1-\Omega (\phi)^2).
%\ee
%This again gives,
%\be
%Z^{\phi}=\frac{1}{2}\int_{1}^{\phi}(1- \Omega(\tilde\phi)^2) d\tilde \phi+ C_2.
%\ee
If $\Omega(V,\phi)$ is an arbitrary function of $ V$ and $\phi$ we get,
\be 
Z^{\phi}=C_3
\ee
and 
\be Z^{V}= \frac{g_{\phi\phi}}{\partial_{V} g_{\phi\phi} }(1-\Omega (V,\theta)^2) .
\ee
So  we generate a supertranslation type soldering transformation.
%\newpage
\vspace{-1cm}
\subsection*{\begin{itemize} \item Schwarzschild-Vaidya spacetime \end{itemize}}
For this case we will have ,
\be \label{vaidya3}
\partial_{V} Z^{A}=0.
\ee
Other equations are, 
\begin{align}
\begin{split} 
&\frac{ 2 \partial_{V} r}{r}Z^{V}+2(\partial_{\theta} Z^{\theta}) =
(1-\Omega(V,\theta,\phi)^2),\\&\frac{ 2 \partial_{V} r}{r}Z^{V}+2(\partial_{\phi} Z^{\phi})+2 Z^{\theta}\cot(\theta) =(1-\Omega(V,\theta,\phi)^2 ),\\&
(\partial_{\theta} Z^{\phi})\sin(\theta)^2+(\partial_{\phi} Z^{\theta})=0.
\end{split}
\end{align}
%Again there are two case . When $\Omega(V,\theta,\phi)= \Omega(\theta,\phi)$ we get 
%\be Z^{V}=0.
%\ee
%Also we get the conformal killing vectors for 2-sphere.  
Again, $\Omega(V,\theta,\phi)$ is an arbitrary function of $V$, $\theta$ and $\phi,$ we get the Killing vectors on 2-sphere satisfying the following equations,
\begin{align}\begin{split}&(\partial_{\theta} Z^{\theta})=0 ,\\&(\partial_{\phi} Z^{\phi})+ Z^{\theta}\cot(\theta) =0,\\&
(\partial_{\theta} Z^{\phi})\sin(\theta)^2+(\partial_{\phi} Z^{\theta})=0.
\end{split}
\end{align}
Apart from that we get,
\be
\frac{ 2 \partial_{V} r}{r}Z^{V}=(1-\Omega(V,\theta,\phi)^2),\quad Z^{V}= \frac{r}{2\partial_{V} r}(1- \Omega(V,\theta,\phi)^2). 
\ee
We get supertranslation type transformation. 
So we can conclude this section by observing that we can get  supertranslation type soldering transformation on the horizon even for the nonstationary spacetime. Next logical step would be to compute the intrinsic stress energy tensor of the shell, which we leave for future study. 





 \begin{thebibliography}{99}
 
\bibitem{BBM} Bondi, M. van der Burg, and A. Metzner, "Gravitational waves in general
relativity. 7. Waves from axisymmetric isolated systems," Proc.Roy.Soc.Lond. A269
(1962) 21-52.

\bibitem{Sachs} R. Sachs, "Gravitational waves in general relativity. 8. Waves in asymptotically 
flat space-times," Proc.Roy.Soc.Lond. A270 (1962) 103-126.

\bibitem{Newman1}
  E.~T.~Newman and R.~Penrose,
 ``Note on the Bondi-Metzner-Sachs group,''
  J.\ Math.\ Phys.\  {\bf 7} (1966) 863.
  doi:10.1063/1.1931221

  

\bibitem{Barnich:2006av} 
  G.~Barnich and G.~Compere,
  ``Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions,''
  Class.\ Quant.\ Grav.\  {\bf 24}, F15 (2007)
  doi:10.1088/0264-9381/24/5/F01, 10.1088/0264-9381/24/11/C01
  [gr-qc/0610130].
  

\bibitem{Barnich:2009se} 
  G.~Barnich and C.~Troessaert,
  ``Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited,''
  Phys.\ Rev.\ Lett.\  {\bf 105}, 111103 (2010)
  doi:10.1103/PhysRevLett.105.111103
  [arXiv:0909.2617 [gr-qc]].
  

\bibitem{Barnich:2011ct} 
  G.~Barnich and C.~Troessaert,
  ``Supertranslations call for superrotations,''
  PoS, 010 (2010)
  [Ann.\ U.\ Craiova Phys.\  {\bf 21}, S11 (2011)]
  [arXiv:1102.4632 [gr-qc]].
  

  
  \bibitem{Barnich}
  G.~Barnich and C.~Troessaert,
  ``BMS charge algebra,''
  JHEP {\bf 1112} (2011) 105
  doi:10.1007/JHEP12(2011)105
  [arXiv:1106.0213 [hep-th]].
  
    \bibitem{Haco}
  S.~J.~Haco, S.~W.~Hawking, M.~J.~Perry and J.~L.~Bourjaily,
 ``The Conformal BMS Group,''
  arXiv:1701.08110 [hep-th].

\bibitem{Pinoetal}
  L.~Donnay, G.~Giribet, H.~A.~Gonzalez and M.~Pino,
 ``Supertranslations and Superrotations at the Black Hole Horizon,''
  Phys.\ Rev.\ Lett.\  {\bf 116} (2016) no.9,  091101
  doi:10.1103/PhysRevLett.116.091101
  [arXiv:1511.08687 [hep-th]].
 
 \bibitem{Pinoetal1}
  L.~Donnay, G.~Giribet, H.~A.~Gonzalez and M.~Pino,
 ``Extended Symmetries at the Black Hole Horizon,''
  JHEP {\bf 1609} (2016) 100
  doi:10.1007/JHEP09(2016)100
  [arXiv:1607.05703 [hep-th]].
  
  \bibitem{Shi1}
  C.~Shi and J.~Mei,
 ``Extended Symmetries at Black Hole Horizons in Generic Dimensions,''
  Phys.\ Rev.\ D {\bf 95} (2017) no.10,  104053
  doi:10.1103/PhysRevD.95.104053
  [arXiv:1611.09491 [gr-qc]].
  
  
  \bibitem{Strominger:2013lka} 
  A.~Strominger,
  ``Asymptotic Symmetries of Yang-Mills Theory,''
  JHEP {\bf 1407}, 151 (2014)
  doi:10.1007/JHEP07(2014)151
  [arXiv:1308.0589 [hep-th]].
  
\bibitem{Strominger:2013jfa} 
  A.~Strominger,
  ``On BMS Invariance of Gravitational Scattering,''
  JHEP {\bf 1407}, 152 (2014)
  doi:10.1007/JHEP07(2014)152
  [arXiv:1312.2229 [hep-th]].

\bibitem{He:2014laa} 
  T.~He, V.~Lysov, P.~Mitra and A.~Strominger,
  ``BMS supertranslations and Weinberg’s soft graviton theorem,''
  JHEP {\bf 1505}, 151 (2015)
  doi:10.1007/JHEP05(2015)151
  [arXiv:1401.7026 [hep-th]].
  
\bibitem{Cachazo}
  F.~Cachazo and A.~Strominger,
``Evidence for a New Soft Graviton Theorem,''
  arXiv:1404.4091 [hep-th].


\bibitem{He1}
  T.~He, P.~Mitra, A.~P.~Porfyriadis and A.~Strominger,
 ``New Symmetries of Massless QED,''
  JHEP {\bf 1410} (2014) 112
  doi:10.1007/JHEP10(2014)112
  [arXiv:1407.3789 [hep-th]].
  
  \bibitem{Lysov2}
  V.~Lysov, S.~Pasterski and A.~Strominger,
  ``Low’s Subleading Soft Theorem as a Symmetry of QED,''
  Phys.\ Rev.\ Lett.\  {\bf 113} (2014) no.11,  111601
  doi:10.1103/PhysRevLett.113.111601
  [arXiv:1407.3814 [hep-th]].

\bibitem{Campiglia:2014yka} 
  M.~Campiglia and A.~Laddha,
  ``Asymptotic symmetries and subleading soft graviton theorem,''
  Phys.\ Rev.\ D {\bf 90}, no. 12, 124028 (2014)
  doi:10.1103/PhysRevD.90.124028
  [arXiv:1408.2228 [hep-th]].
  
\bibitem{Campiglia1}
  M.~Campiglia and A.~Laddha,
 ``New symmetries for the Gravitational S-matrix,''
  JHEP {\bf 1504} (2015) 076
  doi:10.1007/JHEP04(2015)076
  [arXiv:1502.02318 [hep-th]].

\bibitem{Campiglia:2015qka} 
  M.~Campiglia and A.~Laddha,
  ``Asymptotic symmetries of QED and Weinberg’s soft photon theorem,''
  JHEP {\bf 1507}, 115 (2015)
  doi:10.1007/JHEP07(2015)115
  [arXiv:1505.05346 [hep-th]].
  
  \bibitem{Kapec1}
  D.~Kapec, M.~Pate and A.~Strominger,
 ``New Symmetries of QED,''
  arXiv:1506.02906 [hep-th].
  
  \bibitem{Dumitrescu:2015fej}
  T.~T.~Dumitrescu, T.~He, P.~Mitra and A.~Strominger,
``Infinite-Dimensional Fermionic Symmetry in Supersymmetric Gauge Theories,''
  arXiv:1511.07429 [hep-th].
  
  \bibitem{Kapec3}
  D.~Kapec, M.~Perry, A.~M.~Raclariu and A.~Strominger,
 ``Infrared Divergences in QED, Revisited,''
  arXiv:1705.04311 [hep-th].
 
  \bibitem{Strominger}
  A.~Strominger,
 ``Lectures on the Infrared Structure of Gravity and Gauge Theory,''
  arXiv:1703.05448 [hep-th].
  

  \bibitem{Hawking:2016msc} 
  S.~W.~Hawking, M.~J.~Perry and A.~Strominger,
  ``Soft Hair on Black Holes,''
  Phys.\ Rev.\ Lett.\  {\bf 116}, no. 23, 231301 (2016)
  doi:10.1103/PhysRevLett.116.231301
  [arXiv:1601.00921 [hep-th]].
  

 \bibitem{Hawking}
  S.~W.~Hawking,
  ``The Information Paradox for Black Holes,''
  arXiv:1509.01147 [hep-th].
 
  \bibitem{Hawking1}
  S.~W.~Hawking, M.~J.~Perry and A.~Strominger,
 ``Superrotation Charge and Supertranslation Hair on Black Holes,''
  arXiv:1611.09175 [hep-th].
  
 \bibitem{Carney}
  D.~Carney, L.~Chaurette, D.~Neuenfeld and G.~W.~Semenoff,
 `Infrared quantum information,''
  arXiv:1706.03782 [hep-th].
 
\bibitem{Mirbabayi}
  M.~Mirbabayi and M.~Porrati,
 ``Dressed Hard States and Black Hole Soft Hair,''
  Phys.\ Rev.\ Lett.\  {\bf 117} (2016) no.21,  211301
  doi:10.1103/PhysRevLett.117.211301
  [arXiv:1607.03120 [hep-th]]. 
 
 \bibitem{Bousso}
  R.~Bousso and M.~Porrati,
 ``Soft Hair as a Soft Wig,''
  arXiv:1706.00436 [hep-th].\\
 R.~Bousso and M.~Porrati,
 `Observable Supertranslations,''
  arXiv:1706.09280 [hep-th].
 
 \bibitem{Donnelly}
 W.~Donnelly and S.~B.~Giddings,
 ``How is quantum information localized in gravity?,''
  arXiv:1706.03104 [hep-th].
  
 
 
 \bibitem{Blau}
  M.~Blau and M.~O'Loughlin,
 ``Horizon Shells and BMS-like Soldering Transformations,''
  JHEP {\bf 1603} (2016) 029
  doi:10.1007/JHEP03(2016)029
  [arXiv:1512.02858 [hep-th]].\\
  M.~Blau and M.~O'Loughlin,
 ``Horizon Shells: Classical Structure at the Horizon of a Black Hole,''
  Int.\ J.\ Mod.\ Phys.\ D {\bf 25} (2016) no.12,  1644010
  doi:10.1142/S0218271816440107
  [arXiv:1604.01181 [hep-th]].

  \bibitem{Koga}
  J.~i.~Koga,
 ``Asymptotic symmetries on Killing horizons,''
  Phys.\ Rev.\ D {\bf 64} (2001) 124012
  doi:10.1103/PhysRevD.64.124012
  [gr-qc/0107096].
 
  \bibitem{Newman:1962cia} 
  E.~T.~Newman and T.~W.~J.~Unti,
  ``Behavior of Asymptotically Flat Empty Spaces,''
  J.\ Math.\ Phys.\  {\bf 3}, no. 5, 891 (1962).
  doi:10.1063/1.1724303
  17
  
  \bibitem{Barnich:2011ty} 
  G.~Barnich and P.~H.~Lambert,
  ``A note on the Newman-Unti group,''
  Adv.\ Math.\ Phys.\  {\bf 2012}, 197385 (2012)
  [arXiv:1102.0589 [gr-qc]].
  

 
\bibitem{Nutku} Nutku and R. Penrose, On Impulsive Gravitational Waves, Twistor Newsletter 34 (1992)
9 (http://people.maths.ox.ac.uk/lmason/Tn/).


\bibitem{Brown-Hennaux} D. Brown and M. Henneaux, “Central Charges in the
Canonical Realization of Asymptotic Symmetries : An
Example from Three-Dimensional Gravity,”
Commun.Math.Phys. 104 (1986) 207–226.

\bibitem{Strominger:1997eq} 
  A.~Strominger,
  ``Black hole entropy from near horizon microstates,''
  JHEP {\bf 9802}, 009 (1998)
  doi:10.1088/1126-6708/1998/02/009
  [hep-th/9712251].
  
\bibitem{Carlip:1998wz} 
  S.~Carlip,
  ``Black hole entropy from conformal field theory in any dimension,''
  Phys.\ Rev.\ Lett.\  {\bf 82}, 2828 (1999)
  doi:10.1103/PhysRevLett.82.2828
  [hep-th/9812013].
  
  
\bibitem{Carlip:1999cy} 
  S.~Carlip,
  ``Entropy from conformal field theory at Killing horizons,''
  Class.\ Quant.\ Grav.\  {\bf 16}, 3327 (1999)
  doi:10.1088/0264-9381/16/10/322
  [gr-qc/9906126].
  
  
\bibitem{Arcioni:2003xx} 
  G.~Arcioni and C.~Dappiaggi,
  ``Exploring the holographic principle in asymptotically flat space-times via the BMS group,''
  Nucl.\ Phys.\ B {\bf 674}, 553 (2003)
 doi:10.1016/j.nuclphysb.2003.09.051
  [hep-th/0306142].
  

  \bibitem{Arcioni:2003td} 
  G.~Arcioni and C.~Dappiaggi,
  ``Holography in asymptotically flat space-times and the BMS group,''
  Class.\ Quant.\ Grav.\  {\bf 21}, 5655 (2004)
  doi:10.1088/0264-9381/21/23/022
  [hep-th/0312186].
  

\bibitem{Barnich:2010eb} 
  G.~Barnich and C.~Troessaert,
  ``Aspects of the BMS/CFT correspondence,''
  JHEP {\bf 1005}, 062 (2010)
  doi:10.1007/JHEP05(2010)062
  [arXiv:1001.1541 [hep-th]].
 

\bibitem{Barrabes}
  C.~Barrabes and W.~Israel,
 ``Thin shells in general relativity and cosmology: The Lightlike limit,''
  Phys.\ Rev.\ D {\bf 43} (1991) 1129.
  doi:10.1103/PhysRevD.43.1129
  
   \bibitem{Barrabes1}
  C. Barrabes, P. Hogan. 2003. World Scientific. Singular Null Hypersurfaces in General Relativity.

  
  \bibitem{Poisson}
  E.~Poisson,
 ``A Reformulation of the Barrabes-Israel null shell formalism,''
  gr-qc/0207101.

\bibitem{Ross}
  A.~P.~Reynolds and S.~F.~Ross,
 ``Butterflies with rotation and charge,''
  Class.\ Quant.\ Grav.\  {\bf 33} (2016) no.21,  215008
  doi:10.1088/0264-9381/33/21/215008
  [arXiv:1604.04099 [hep-th]].


  \bibitem{Mansouri:1996ps} 
  R.~Mansouri and M.~Khorrami,
``Equivalence of Darmois-Israel and distributional methods for thin shells in general relativity,''
  J.\ Math.\ Phys.\  {\bf 37}, 5672 (1996)
  doi:10.1063/1.531740
  [gr-qc/9608029].
 
  \bibitem{Zhiboedov}
  A.~Strominger and A.~Zhiboedov,
 ``Superrotations and Black Hole Pair Creation,''
  Class.\ Quant.\ Grav.\  {\bf 34} (2017) no.6,  064002
  doi:10.1088/1361-6382/aa5b5f
  [arXiv:1610.00639 [hep-th]].
  
  \bibitem{Loran}
  F.~Loran and M.~M.~Sheikh-Jabbari,
  ``O-BTZ: Orientifolded BTZ Black Hole,''
  Phys.\ Lett.\ B {\bf 693} (2010) 184
  doi:10.1016/j.physletb.2010.08.022
  [arXiv:1003.4089 [hep-th]].\\
   F.~Loran and M.~M.~Sheikh-Jabbari,
  ``Orientifolded Locally $AdS_3$ Geometries,''
  Class.\ Quant.\ Grav.\  {\bf 28} (2011) 025013
  doi:10.1088/0264-9381/28/2/025013
  [arXiv:1008.0462 [hep-th]].
 
 
\bibitem{Compere}
  G.~Compère and J.~Long,
  ``Vacua of the gravitational field,''
  JHEP {\bf 1607} (2016) 137
  doi:10.1007/JHEP07(2016)137
  [arXiv:1601.04958 [hep-th]].


\bibitem{Tanabe}
  K.~Tanabe, T.~Shiromizu and S.~Kinoshita,
 ``Late-time symmetry near black hole horizons,''
  Phys.\ Rev.\ D {\bf 85} (2012) 024048
  doi:10.1103/PhysRevD.85.024048
  [arXiv:1112.0408 [gr-qc]].
  
  
  \bibitem{Penna:2015gza} 
  R.~F.~Penna,
 ``BMS invariance and the membrane paradigm,''
  JHEP {\bf 1603}, 023 (2016)
  doi:10.1007/JHEP03(2016)023
  [arXiv:1508.06577 [hep-th]].\\
  R.~F.~Penna,
 ``Horizon symmetries as fluid symmetries,''
  arXiv:1703.07382 [hep-th].
 
 
 \bibitem{Eling}
  C.~Eling and Y.~Oz,
 ``On the Membrane Paradigm and Spontaneous Breaking of Horizon BMS Symmetries,''
  JHEP {\bf 1607} (2016) 065
  doi:10.1007/JHEP07(2016)065
  [arXiv:1605.00183 [hep-th]].\\
  C.~Eling,
 ``Spontaneously Broken Asymptotic Symmetries and an Effective Action for Horizon Dynamics,''
  JHEP {\bf 1702} (2017) 052
  doi:10.1007/JHEP02(2017)052
  [arXiv:1611.10214 [hep-th]].
  
   
  \bibitem{Balasin-Steinbauer}H. Balasin, ‘‘Geodesics for impulsive gravitational waves and the multiplication of distributions,’’ Class. Quantum Grav.
14, 455–462 ~1997!. R. Steinbauer, ‘‘The ultrarelativistic Reissner-Nordstro ” m field in the Colombeau algebra,’’ J. Math. Phys. 38, 1614–1622
~1997!.
  
  \bibitem{Columbeau}J. F. Colombeau, New Generalized Functions and Multiplication of Distributions ~North Holland, Amsterdam, 1984!. J. F. Colombeau, Multiplication of Distributions ~Springer, Berlin, 1992!.
 
 
 \end{thebibliography}

\bibliographystyle{JHEP}



\end{document}









\section{Conformal Killing vectors on 2-sphere}\label{append}
We solved the conformal Killing vector on 2-sphere in polar coordinate. Although it is well know we provide here a detailed derivation of the conformal killing vectors on 2-sphere for the sake of completeness to complement our discussions in the section ~(\ref{sec5}). We start with the killing vector equations,
\begin{align}
\begin{split}
\nabla_{A}Z_{B}+\nabla_{B} Z_{A}=(\nabla^{C} Z_{C})g_{AB}.
\end{split}
\end{align}
We denote $\nabla^{C} Z_{C}=F.$ Then we have,
\begin{align}
\begin{split}
&\nabla^{A}\nabla_{A} Z_{B}+\nabla^{A}\nabla_{B} Z_{A}=\nabla_{B} F,\\&
\nabla^{B}\nabla^{A}\nabla_{A}Z_{B}+\nabla^{B}\nabla^{A}\nabla_{B}Z_{A}=\nabla^{2}F.
\end{split}
\end{align}
 After few manipulations we get,
 \begin{align}
 \begin{split}
& \nabla^2 F+ F=\frac{1}{2}\nabla^2 F\,\\&
\nabla^2 F+ 2 F =0.
 \end{split}
 \end{align}
 The general solution for this equation is,
 \begin{align}
 \begin{split}
F(\theta,\phi)=\sum_{c_m}e^{i m\phi} P^{m}_{l}(\theta),
 \end{split}
 \end{align}
where, $l=1$ and $m$ takes values $-1,0,1.$$c_m$ are arbitrary constants. $P^{m}_{l}(\theta)$ is the associated Legendre polynomial.  
For $m=0,$ $F(\theta,\phi)=\cos(\theta)$ we have the following solutions,
\begin{align}
\begin{split}
&J_1=\frac{1}{2}\sin(\theta)\partial_{\theta},\\&
J_2=\partial_{\phi},\\&
J_3=-\cos(\phi)\partial_{\theta}+\sin(\phi)\cot(\theta)\partial_{\phi},\\&
M_3=-\sin(\phi)\partial_{\theta}-\cos(\phi)\cot(\theta)\partial_{\theta}.
\end{split}
\end{align}
When $F(\theta,\phi)=\sin(\theta)\cos(\phi),$
\be
M_1=-\frac{1}{2}\cos(\theta)\cos(\phi)\partial_{\theta}+\frac{1}{2}\frac{\sin(\phi)}{\sin(\theta)}\partial_{\phi}.
\ee
When $F(\theta,\phi)=\sin(\theta)\sin(\phi),$
\be
M_2=-\frac{1}{2}\cos(\theta)\sin(\phi)\partial_{\theta}-\frac{1}{2}\frac{\cos(\phi)}{\sin(\theta)}\partial_{\phi}.
\ee
So we have our 6 conformal Killing vectors on 2-sphere. We now take the following linear combination,
\begin{align}
\begin{split}
P^1=i (J_1+M_1),\,P^2=i (J_2+M_2),\,D=i M_3,\,J^{12}=i J_3,\,K^1=i(J_1-M_1),\,K^2=i (J_2-M_2).
\end{split}
\end{align}
With this identification we can readily show that,the conformal algebra is satisfied. 
\begin{align}
\begin{split}
&[P^a,P^b]=0,[P^a,D]=P^a,[J^12,D]=0,[J^12,P^1]=-i P^2,[J^12,P^2]=i P^1,\\&[D,K^2]=-i K^a,[K^1,P^2]=-2 i J^{12},[K^1,P^1]=2i D,[K^2,P^1]=2i J^{12},[K^2,P^2]=\frac{2 i}{\sin(\theta)^2}D,
\end{split}
\end{align}
where $a=1,2.$










  We now want to extend the above conformal soldering transformations off the shell such that the junction condition remains same.
 %We will do it for both the superrotation and supertranslation type transformations. 
 The infinitesimal superrotation generators which act on the celestial sphere at null infinity are singular and it is not very clear whether there exist finite version of these transformations\footnote {This issue has been addressed in \cite{Zhiboedov} and a physical interpretation is given for finite superrotation.}. But in our context (at least for the BTZ) there seems to be no obstacle to generalize the infinitesimal transformations to finite ones. 
\vspace{-1cm}
  \subsection*{\begin{itemize} \item Rotating BTZ\end{itemize}}

 We start with the seed rotating BTZ metric for both the side \footnote{It will be interesting to extend our construction for orientifolded BTZ  spacetimes \cite{Loran} which involve orbifolding of by $Z_2$ transformation. We leave this as a future problem.}. Then we introduce a coordinate system such that all the components of the metric remains continuous across the junction. At the surface the intrinsic coordinates are $\zeta^a$ as before. Also we introduce a set of common coordinates $x^{\alpha}$ such that $x^{\alpha}|_{\Sigma}=\zeta^a.$ We can extend the soldering transformations towards both $\mathcal{M}_{+}$ and $\mathcal{M}_{-}$ direction and it will be of the same form for both the cases. So without loss of any generality we extend this in the direction of $\mathcal{M}_{+}.$ 
Similar expression can easily be obtained for $\mathcal{M}_{-}.$ 

The finite transformations generated by $Z_{+}$ can be achieved  by the following set of  transformations,
\be
V_{+}= F(V,\phi)+ U A (V,\phi), \phi_{+}=\int_{1}^{\phi}\Omega (\phi)d\phi +U B^{\phi} (V,\phi), U_{+}=U C (V,\phi).
\ee
Similarly for the offshell extension towards $\mathcal{M}_{-}.$
\be
V_{-}= \tilde F(V,\phi)+ U \tilde A (V,\phi), \phi_{-}=\int_{1}^{\phi}\Omega (\phi)d\phi +U \tilde B^{\phi} (V,\phi), U_{+}=U \tilde C (V,\phi).
\ee
Then the conserved quantities of the shell will be related to the difference between  $F(V,\phi)$ and $\tilde F(V,\phi)$ and there derivatives. The intrinsic coordinates covering both sides of the shell can be denoted by, $x^{\alpha}=\{U,V,\zeta^A\}$.
At the surface , 
\be \label{physical}
g^{+}_{\alpha\beta}|_{\Sigma_{+}}=g^{-}_{\alpha\beta}|_{\Sigma_{-}}=\Omega^2(\zeta) g_{ab}(\zeta)|_{\Sigma}.
\ee
We also demand that in our new coordinate chart $x$ both the metrics of $\mathcal{M}_{+}$ and $\mathcal{M}_{-}$ will be of the following form,
\be
ds^2_{+}(x)=g^{+}_{\alpha\beta}(x)dx^\alpha dx^\beta \equiv \Omega(\phi)^2 g_{\alpha\beta }(x)dx^\alpha dx^\beta,
\ee
and 
\be
ds^2_{-}(x)=g^{-}_{\alpha\beta}(x)dx^\alpha dx^\beta \equiv\Omega(\phi)^2 g_{\alpha \beta }(x)dx^\alpha dx^\beta.
\ee
We get the following solutions,
\begin{align}
\begin{split}
&C=\frac{\Omega(\phi)^2}{\partial_{V} F},\tilde C=\frac{\Omega(\phi)^2}{\partial_{V} \tilde F},\\& B^{\phi}=\frac{2 \Omega(\phi) (l^2\partial_{\phi} F- l\tilde r_h F \Omega(\phi)+l\tilde r_h V \partial_V F)}{ r_{h}^2\partial_{V}F},\tilde B^{\phi}=\frac{2 \Omega(\phi)(l^2 \partial_{\phi} \tilde F-l \tilde r_h \tilde F \Omega(\phi)+l\tilde r_g V \partial_V \tilde F)}{ r_{h}^2\partial_{V}\tilde F},\\& A=\frac{ l^2 ((\partial_{\phi}  F)^2-\tilde r_h^2 \Omega^2 F^2  +2\,l \tilde r_h V \partial_{\phi} F \partial_V  F +\tilde r_h^2 V^2 (\partial_V F)^2  }{\partial_{V} F r_h^2},\\& \tilde A=\frac{ l^2 ((\partial_{\phi} \tilde F)^2-\tilde r_h^2 \Omega^2\tilde F^2  +2\,l \tilde r_h V \partial_{\phi}\tilde F \partial_V \tilde F +\tilde r_h^2 V^2 (\partial_V \tilde F)^2  }{\partial_{V}\tilde F r_h^2}.
\end{split}\end{align}
Now the virtue of the condition (\ref{physical}) is that, we can use the usual distributional calculus as (\ref{junc7}) still holds. Hence the expressions for conserved quantities of the shell still takes the form as shown in (\ref{junc8}).
So following the prescription outlined in section (\ref{third}) we get,
\begin{align}
\begin{split} \label{physical2}
\gamma_{VV}&=2\Big(\frac{\partial^2_{V} F}{\partial_{V} F}-\frac{\partial^2_{V} \tilde F}{\partial_{V} \tilde F}\Big),\\\gamma_{V\phi}&=4\Big(\frac{\partial_{V}\partial_{\phi} F}{\partial_{V} F}-\frac{\partial_{V}\partial_{\phi} \tilde F}{\partial_{V} \tilde F}\Big),\\\gamma_{\phi\phi}&=2\Big(\frac{\partial_{\phi}^2 F}{\partial_{V} F}-\frac{\partial_{\phi}^2 \tilde F}{\partial_{V} \tilde F}-\frac{\partial_{\phi}F \Omega'(\phi)}{\partial_{V} F \Omega(\phi)}+\frac{\partial_{\phi}\tilde F \Omega'(\phi)}{\partial_{V} \tilde F \Omega(\phi)}-\frac{F \Omega(\phi)^2 r_{h}^2}{\partial_{V}F l^2}+\frac{\tilde F \Omega(\phi)^2 r_{h}^2}{\partial_{V}\tilde F l^2}\\&+\frac{ F \Omega(\phi)^2 \tilde r_h^2}{\partial_V F l^2}-\frac{ \tilde F \Omega(\phi)^2 \tilde r_h^2}{\partial_V \tilde F l^2}+\frac{2 \partial_{\phi} \tilde F \Omega(\phi) \tilde r_h}{\partial_V \tilde F l}-\frac{2 \partial_{\phi} F\Omega(\phi) \tilde r_h}{ \partial_V F l}\Big).
\end{split}
\end{align}


From this we get,
\begin{align}
\begin{split} \label{physical3}
p&=-\frac{1}{8 \pi}\Big(\frac{\partial^2_{V} F}{\partial_{V} F}-\frac{\partial^2_{V} \tilde F}{\partial_{V} \tilde F}\Big),\\
J^{\phi}&=\frac{1}{4 \pi \Omega(\phi)^2 r_{h}^2}\Big(\frac{\partial_{V}\partial_{\phi} F}{\partial_{V} F}-\frac{\partial_{V}\partial_{\phi} \tilde F}{\partial_{V} \tilde F}\Big),\\\mu&=-\frac{1}{8\pi \Omega(\phi)^2 r_{h}^2}\Big(\frac{\partial_{\phi}^2 F}{\partial_{V} F}-\frac{\partial_{\phi}^2 \tilde F}{\partial_{V} \tilde F}-\frac{\partial_{\phi}F \Omega'(\phi)}{\partial_{V} F \Omega(\phi)}+\frac{\partial_{\phi}\tilde F \Omega'(\phi)}{\partial_{V} \tilde F \Omega(\phi)}-\frac{F \Omega(\phi)^2 r_{h}^2}{\partial_{V}F l^2}+\frac{\tilde F \Omega(\phi)^2 r_{h}^2}{\partial_{V}\tilde F l^2}\\&+\frac{ F \Omega(\phi)^2 \tilde r_h^2}{\partial_V F l^2}-\frac{ \tilde F \Omega(\phi)^2 \tilde r_h^2}{\partial_V \tilde F l^2}+\frac{2 \partial_{\phi} \tilde F \Omega(\phi) \tilde r_h}{\partial_V \tilde F l}-\frac{2 \partial_{\phi} F\Omega(\phi) \tilde r_h}{ \partial_V F l}\Big).
\end{split}
\end{align}
Now if we try to set $p=0$ as before, that will leave us with several possibilities.
One possibility is to simply set $F=\tilde F .$ This will not only make $p=0$ but all also make $\mu=0$ and $J^{\phi}=0.$ This is expected as we are soldering two exactly same spacetime which are obtained by the same amount of supertranslation from the seed BTZ metric.\par
Second possibility is to set both $\partial_{V} F=\partial_{V}\tilde F =0$ individually, but still $F(V,\phi) \neq \tilde F (V,\phi).$ This will give us, $F=V+T(\phi)$ and $\tilde F = V +\tilde T (\phi).$
We can set $J^{\phi}=0$ and this will give, upto a factor, $F=a V +T(\phi)$ and $\tilde F= b V+\tilde T (\phi).$ As $V$ is the intrinsic coordinate covering both sides of the shell we set $a=b,$ so that we can rescale $V$ in the same way everywhere.
Both $p=0$ and $J^{\phi}=0$ will give, upto a overall factor, supertranslation like transformation which is already expected from the work \cite{Blau}.
 We now analyze the form of the energy density $\mu$. $$\mu= -\frac{1}{8\pi \Omega(\phi)^2 r_{h}^2}\Big(\mathcal{T}''(\phi) -\frac{\Omega'(\phi)}{ \Omega(\phi)}\mathcal{T}'(\phi)-\frac{\Omega(\phi)^2 (r_{h}^2-\tilde r_h^2)}{l^2}\mathcal{T}(\phi)-\frac{2\Omega(\phi) \tilde r_h}{l}\mathcal{T}'(\phi)\Big).$$ Where we have defined $\mathcal{T}(\phi)= T(\phi)-\tilde T (\phi).$ From this it is evident that it satisfy $\partial_{V} \mu=0.$ Using this we can define the energy of the shell,
\be \label{physical4}
E=\frac{1}{8 \pi r_{h}} \int d\phi \Big(-\partial_{\phi}\Big(\frac{T'(\phi)}{ \Omega(\phi)}\Big)+\frac{\Omega(\phi) (r_{h}^2-\tilde r_h^2)}{l^2}\mathcal{T}(\phi)+\frac{2  \tilde r_h}{l}\mathcal{T}'(\phi)\Big).
\ee
Dropping the first and last term by doing an integration by parts we get,
\be \label{physical4a}
E=\frac{1}{8 \pi} \int d\phi \Big(\kappa \Omega(\phi) \mathcal{T}(\phi)\Big).
\ee
 When $\Omega(\phi)=1,$ (\ref{physical4}) recovers the usual result \cite{Pinoetal}. Next we set, $\Omega(\phi)=1+ \epsilon\, \tilde \Omega (\phi),$ where $\epsilon$ is a small parameter. We then expand (\ref{physical4a}) in $\epsilon$ and keep only term upto $\mathcal{O}(\epsilon).$

\begin{align}
\begin{split} \label{physical5}
E&=\frac{1}{8\pi}\int d\phi\Big( \kappa \mathcal{T}(\phi)+\epsilon\,  \kappa \, \tilde \Omega (\phi)\mathcal{T}(\phi)\Big).
\end{split}
\end{align}

(\ref{physical5}) shows that the total energy gets contribution from both the supertranslation  and superrotation like soldering transformations on the horizon. We compare the expression for the energy (\ref{physical5}) with the conserved charge calculated in \cite{Pinoetal}. The first term matches with that of the supertranslation contribution to the conserved charge computed in \cite{Pinoetal}. At the leading order in $\epsilon$ there is no contribution from superrotation type soldering transformation. It appears only at the $\mathcal{O}(\epsilon)$ order. Although it contains the superrotation factor but this  $\mathcal{O}(\epsilon)$ term is not quite similar to the superrotation charge as computed in \cite{Pinoetal,Barnich}. In general there is no a priori reason to expect that (although the supertranslation part indeed match). The conserved charge computed in \cite{Pinoetal,Barnich} has been done using Noether procedure from the gravitational action, but for our case we are just computing the intrinsic quantities of the shell which comes from the stress energy tensor required for satisfying the Einstein equations. Although we have recovered an infinite dimensional symmetry which closely resembles superrotation but from the perspective of charge algebra our findings do not exactly match with the existing results  \cite{Pinoetal,Pinoetal1,Barnich:2009se,Barnich:2011ct,Barnich}. At this point we leave this issue for a detail study in a future publication.
\vspace{-1cm}


 \subsection*{\begin{itemize}\item Schwarzschild\end{itemize}}
 The corresponding finite transformations may again be generated analogously as what we have seen for BTZ case before. But one has to check this  more carefully as the finite superrotation is in general not globally well defined and hence the interpretation for the corresponding conserved charge is subtle \cite{Compere}. We leave this as a future problem.

%In this section we will briefly consider non stationary spacetime. 





























One can extend this soldering transformation off the shell surface $\Sigma.$ 
%We introduce first a coordinate chart such that the all the components of the metric remain continuous across the shell.
We  follow then the convention and notations of \cite{Blau}. 
One starts with, 
\be
Z=Z^{V}\partial_{V}+ U z^{\alpha}\partial_{\alpha}.
\ee
Demanding the $\mathcal{L}_{Z}(g_{U\alpha})=0$ we can arrive at the following form
\be \label{killing5a}
Z= \omega (V \partial_{V}-U \partial_{U})+ U (z^{A}\partial_{A}-V (\partial_{V}\omega\partial_{U}).
\ee
Here we have reparametrized $Z^V$ as $ Z^{V}= V \omega(V,X^{A})$ and $ z^{A}
=-g_{UV}g^{AB}\partial_{B} Z^{V}.$ $g_{UV}$ is evaluated on $\Sigma.$
Here we have written the soldering transformations as local killing transformations. Now one can demonstrate if one start with the following form of the coordinate transformations,
\be \label{killing5}
V_{+}=F (x^a)+ U A(x^a), x_{+}^A= X^{A}+ U B^A(x^a), U_{+}=U C(x^a),
\ee
it will give us the finite transformation corresponding to (\ref{killing5a}). Here we have assumed without loss of any generality that, we can choose the coordinates $( x_{-}^{\alpha}) $ of $\mathcal{M}_{-}$ as the intrinsic coordinates on the shell and which can be used to cover both sides of the shell $(x_{-}^{\alpha}=x^{\alpha})$. The undetermined functions in (\ref{killing5}) can be determined  by demanding the continuity of the metric $g_{U\alpha}$ across the shell. One important consequence of this off-shell extension is that we can now read of the intrinsic quantities of the shell using the transformations defined in (\ref{killing5}). Before that, we remind ourselves, that the $\gamma_{\mu\nu}$ defined in (\ref{junc7}) behaves as three tensor and after suitably projecting it on the surface $\Sigma$, $\gamma_{ab}=\gamma_{\mu\nu}e^{\mu}
_{a}\e^{\nu}_b$ gives all the informations about the intrinsic properties of the shell as defined in  (\ref{junc8}). Now,
\be  \label{killing6}
\gamma_{ab}=N^{\alpha}[\partial_{\alpha} g_{ab}],
\ee
where we have used $n.N=-1$. To  determine this $\gamma_{ab}$ we  will extend the tangential components of the of the metric on $\Sigma$ off the shell, in both  direction of $\mathcal{M}_{\pm}$ but using the same coordinate chart. 
%$Basically we use the transformations defined in (\ref{killing5}) and assume that we can use the coordinate of $\mathcal{M}_{-}$ as the common coordinate for both side. 
Then we end up with,
\begin{align}
\begin{split}
g^{\pm}_{ab}=(g^{(0)}_{ab}+ U g^{\pm (1)}_{ab}) dx^adx^b
\end{split}
\end{align}
From this ,
\be
\gamma_{ab}=\frac{1}{\chi}\Big(g^{+(1)}-g^{-(1)}\Big)|_{\Sigma},
\ee
where, $\chi=g_{UV}|_{\Sigma}.$ Also, $n=\partial_{U}$ and $N=\frac{1}{\chi}\partial_{V}$ in terms of Kruskal coordinates. Then the quantities defined in (\ref{junc8}) will be functions of $F (V,X^A)$ and its derivatives. It has been demonstrated in \cite{Blau} that, to get the supertranslation like transformation coming from the soldering freedom, one needs to put certain conditions on the intrinsic properties of the shell, for example, for the Schwarzschild case, the pressure ($p$) or current ($ J^{A}$) or both of them has to be zero. At this point we would like to emphasize that, we only get supertranslation like transformations following this construction. But the full BMS group (in three and four dimensions) consists of both supertranslation and superrotation (apart from the usual Poincare transformations) \cite{Barnich:2009se}. The main goal of this paper as explained before to explore whether it is possible to generate superrotation like transformations on the horizon shell by appropriately modifying the junction condition. Before that, we do a simpler exercise, we extend the above construction to rotating spacetime.