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\begin{document}
Any geometry that a null surface $\mathcal N$ inherits from the ambient spacetimes $\mathcal {M}^+$ and $\mathcal {M}^-$ can be classified into one of the following three types (in the order of increasing structure):


\begin{itemize}
 \item Type-I: The induced metrics match from both sides. 
 
 \item Type-II: type I with parallel transport of the normal $n$ along
the null generators matching.

\item Type-III: type II with parallel transport of any tangent vector
to $\mathcal N$  along the null generators matching.

\end{itemize}

In our case, when conformal isometries are allowed on the hypersurface (or one side, say $\mathcal M^+$, of $\N$), the geometry of hypersurface can be type-I only, otherwise the pressure of the shell wold be zero which is physically unacceptable. Also, if we want to match two geometries which are solutions of Einstein equation with matter, then the natural choice should be type-I geometry as conformal isometry will certainly render the shell to be of a nontrivial nature. Now there can be two types of soldering: i) Affine soldering where the surface pressure vanishes but surface density becomes time dependent. ii) Statically soldering: Here surface pressure and surface density are time independent. Now under conformal isometries, in spherically symmetric situation do the normal vectors and oblique normal vectors agree at the junction? It seems at least in spherisymmetric case a single parameter $r$ can be chosen which will play the role of soldering parameter. Further, under a conformal transformation a spacetime turns a vacuum to non-vacuum and that must be reflected to somewhere in the shell's intrinsic properties. Therefore it is better to take the shell's intrinsic geometry as $S^2\times R$ and choose one of the double null coordinates $u$ as the parameter across the generators and let the functional form of the metric functions in a generic geometry in which the shell is embedded completely arbitrary. The case ii) is more appropriate here. The normal and oblique normals are continuous across the shell. Remember, we are only making a conformal transformation on the intrinsic metric $\Sigma$. One then has to compute the jumps in transverse derivatives of metrics and their consequences. Following Blau, we can compute the $\gamma_{ab}$ and then construct the conserved quantities. We may have to redo the calculation without changing the fundamental junction condition. 

under conformal transformation,

$L_Z g_{ab}=\O^2g_{ab},\,L_Z n^a=\a n^a,\, L_Z N^a=-\a N^a $

Since null geodesics remain invariant under conformal isometries we can make the following constructions:

We perform a conformal transformation on one side $\Sigma^+$ and attach. The freedom of soldering can be obtained solving Conformal Killing equation of the induced metric. (Induced metrics are same from two sides) Then figure out the $\gamma$ from off-shell extension and then figure out the properties of the shell.

On a 3-manifold, the 4-geometry induces an intrinsic metric and an extrinsic curvature. These things should satisfy certain constraints. Our constraints are $q_{ab}l^b=0, (a,b=0,1,2), {\cal{L}}_lq_{ab}=0, D_al^b=\omega_al^b, {\mathcal{L}}_l\omega=0$


Essential points:

\begin{itemize}
 \item The kinematics and dynamics of a null hypersurface is tricky things to study. Unlike for the case of timelike or spacelike surfaces one cannot  decompose a vector normal to a null hypersurface into components that purely lie in the orthogonal and in the tangent to the hypersurface. Therefore to study the intrinsic and extrinsic properties of the null hypersurfaces one needs extra care. In Barrabes-Israel paper a first coordinate friendly approach was depicted by dealing kinematics(dnamics) of null surface as smooth limit of timelike or spacelike hypersurface. 
  
 Let us take $\mathcal {N}$ to be the null hypersurface. Let us call  $k$ and $l$ to be the null normal and auxiliary null normal respectively to it. One can map the metric of spacetime Manifold  $g$ to the one $h=\Phi^*g$ (first fundamental form)  at the hypersurface by pulling back through $\Phi$. The Weingarten map gives how the hypersurface is bent in the ambient spacetime. This map defines the second fundamental form : $\Theta_{\alpha\beta}=h^{\rho}_{\alpha}h^{\gamma}_{\beta}\nabla_{\rho} k_{\gamma}$, where $h^{\rho}_{\alpha}=\delta^{\rho}_{\alpha} + k^{\rho}_{\alpha} + l^{\rho}_{\alpha}$. Due to degeneracy of $\mathcal{N}$ , $\Theta$ is acting on the $\mathcal{S}_t$ i.e. the 
2-dimensional surface (spacelike) that foliates the $\mathcal{N}$. The second fundamental form $\Theta$ is also equal to the deformation rate of the 2-surface $\mathcal{S}$ with respect to the standard normal vector $k$ and given by : $\Theta_{cd}=\frac{1}{2}h^a_ch^b_d\mathcal{L}_k h_{ab} $ . Here $k$ is the tangent vector to the null geodesics generating the horizon. The null normals are defined only upto a rescaling of the form $l' =\alpha l $ and $k=\alpha^{-1} k$. However, the projector $h^{\alpha}_{\beta}$ is invariant under this rescaling. Also the second fundamental form becomes $\Theta'=\alpha \Theta$ and the expansion w.r.t. $k$ also has similar properties. The expansion w.r.t. transverse normal $l$ transforms according to $\Xi'=\alpha \Xi $. All these structures are independent of the fact that whether the metric $g$ of the ambient spacetime is a solution of Einstein equation or not. Now if one tries to take the second derivative or Lie derivative of the quantities then Einstein equation comes into the picture. 
 
 Since for null hypersurface, $k$ is simultaneously tangent and normal to the surface $\mathcal{N}$, one cannot differentiate between Gauss and Codazzi equation. In fact the trace of contracted Codazzi equation can also be seen to be as a component of null Gauss equation. These equations are to be satisfied or act as necessary and sufficient condition so that the null surface $\mathcal{N}$, endowed with a degenerate metric $h$, to be regarded as a submanifold of ambient spacetime $\mathcal{M}, g$.  
 Let us recall the way to get these equations, we start with the Ricci identity:
 
 \begin{equation}
 \nabla_{\mu}\nabla_{\nu}l^{\mu}-\nabla_{\nu}\nabla_{\mu}l^{\mu}=R_{\mu\nu}l^{\mu}h^{\nu}_{\alpha}
 \end{equation}
 
 Using $\nabla_{\mu}l^{\alpha}=\Theta^{\alpha}_{\mu}+l^{\alpha}\omega_{\mu}+l_{\mu}k^{\beta}\nabla_{\beta}l^{\alpha}$ and $\nabla.k=\kappa + \theta$ one gets,
 
 \begin{equation}
 R_{\alpha\mu}\Pi^{\mu}_{\nu}l^{\alpha}=-R_{\alpha\mu}l^{\alpha}l^{\mu}k_{\nu}+R_{\alpha\mu}h^{\mu}_{\nu}l^{\alpha}
 \end{equation}
 where $\bf{\Pi}=\bf{h}-l.k$.
 Now one contracts the Ricci identity along $l$ and onto the 2-d surface $\mathcal{S}_t$ and gets the null Raychaudhuri equation and Damour-Navier Stokes equation (6.15 of GG). 
 
 In the context of joining two spacetimes across any null hypersurface the surface possesses intrinsic properties that are determined by precisely the null Raychaudhuri and DNS equations. 
 
 To express the dynamical equations of null hyper surfaces, or horizon of a black hole we may also invoke 2+1 decomposition to the hypersurface. This will make the analysis more tractable. 
 
 \item Now if our induced metric falls in a conformal class, then the RE and DNS equations will change accordingly. {\bf One may start from Ricci identity and contract then use the identities in GG ch-10} . 
 If we allow a conformal transformation to a spatial metric $h$ as $\tilde{h}=\Omega^4 h$. We would like to find the effect of the above conformal transformation along with the conformal transformation of the extrinsic curvature- on constraint equations that need to be satisfied. For a spacelike hypersurface $\Sigma_t$ one starts with a conformal decomposition of extrinsic curvature 
 \begin{equation}
 K_{ab}=A_{ab}~+~1/3 K h_{ab}
 \end{equation}
 
 where, $A_{ab}$ is the trace free part of the tensor and $K$ is the trace of extrinsic curvature. The traceless part has a transformation $\tilde{A}_{ab}=\Omega^{-4} A_{ab}$. This changes the relation between time evolution of induced metric and extrinsic curvature : $\mathcal{L}_nh_{ab}=-2NK_{ab}$
 ($n$ is vector normal to the surface)
 \begin{equation}
 (\partial_t-\mathcal{L}_{\beta}) \tilde{h}_{ab}= -2 N \tilde{A}_{ab} - \frac{2}{3} \tilde{D}_k\beta^k h_{ab}
 \end{equation}
 $D$ is the covariant derivative operator w.r.t. $h$. 
 
 where $\beta$ is the shift vector $t^a=n^aN+\beta^a$. The momentum constraint is 
 
 \begin{equation}
 \tilde{D}_j\hat{A}^{ij}-\frac{2}{3}\Omega^6 \tilde{D}=8\pi \Omega^{10}p^i
 \end{equation}
 with $\hat{A}=\Omega^{10}A$.
 The Hamiltonian constraint 
 \begin{equation}
 \tilde{D}_i\tilde{D}^i\Omega - \frac{1}{8}\tilde{R}\Omega ~+~\frac{1}{8}\hat{A}^{ij}\hat{A}_{ij}\Omega^{-7}~+~\left(2\pi E-\frac{1}{12}K^2\right)\Omega^5=0
 \end{equation}
 
 GG2 6.105-6.112
 (Gour, Sec-10.4, Eq. 10.95, Eq. 10.98, 10.103, 10.114 )
 
 There are analogue equations for null surface? Using 2+1 decomposition one gets some equations for fields (expansion etc. ) on null hypersurface but constraints?
 
 The 2+1 version involves a spacelike normal vector $s$ to 2-d surface $\mathcal{S}_t$ intersection of $\mathcal{N}$ and spacelike surface $\Sigma_t$). This space like normal lies in $\Sigma_t$. If ${}^2D_A$ is denoted as the connection compatible with the induced metric $h_{ab}=g_{ab}~k_al_b~+l_ak_b$. The expansion in terms of 2+1 fields reads:
 
 \begin{eqnarray}
 \theta&=&N\Omega^{-2}\left(4\tl{s}^k\tl{D}_k\ln \Omega + \frac{2}{3} \tl{H}\right)-\frac{1}{2}\dot{\tl{h}}_{kl}\tl{s}^k\tl{s}^l\\ \nonumber &+&\frac{1}{3}\left[\left(N\Omega^{-2}-\tl{b}\right)\tl{H}~+~2\tl{s}^k\tl{D}_k\tl{b}-2V^A {}^{2}\tl{D}_A\ln \tl{M}~+~{}^{2}\tl{D}_AV^A~-~NK\right]
 \end{eqnarray}
 
 with $V^A \in \mathcal{T}_p(\mathcal{S}_t)$ and $b$ is defined as $\beta^a= b s^a-V^a$. $H_{ab}=h^{\mu}_{a}h^{\nu}_b\D_{\mu}s_{\nu}$. 
 
 $\Theta_{ab}=\Omega^4\{(N\Omega^{-2}-\tl{b})\tl{H}_{ab}~+~\frac{1}{2} ( {}^2D_AV_B~+~{}^2D_BV_A)+ \frac{1}{2}\dot{\tl{h}}_{kl}h^k_ah^l_b~+~[\frac{1}{3}(\tl{b}\tl{H}\tl{s}^k\tl{D}_k\tl{b}-NK-{}^2\tl{D}_AV^A-V^A{}^2\tl{D}_A\ln \tl{M})+2N \Omega^{-2}\tl{s}^k\tl{D}_k \ln \Omega]\tl{h}_{ab}\}$
 
 The conformal decomposition of transverse extrinsic curvature:
 
 Define $\Xi_{ab}=\mathcal{L}_l (g_{ab}+k_al_b+k_bl_a)$. 
 
 \begin{eqnarray}
 \Xi_{ab}=-\frac{\Omega^4}{2N^2}\{(N\Omega^{-2}~+~\tl{b})\tl{H}_{ab}-\frac{1}{2}\left[{}^2D_AV_B~+~{}^2D_BV_A~+~\dot{\tl{h}}_{kl}h^k_ah^l_b\right]~+~\\ \nonumber \left[2 \Omega^{-2}\tl{D}_{\tl{s}} \ln \omega +\frac{1}{3}(NK - \tl{b}\tl{h} -\tl{D}_{\tl{s}}\tl{b}+{}^2\tl{D}.V+{}^2\tl{D}_V \ln \tl{M})\right]h_{ab}\}
 \end{eqnarray}
 
 \item We can check for particular geometries like Schwarzschild/BTZ to verify these equations.)
\end{itemize}






\end{document}