## LATEX

### Dilaton gravity in two dimensions as a non-linear sigma model

The most general dilaton  gravity action  in two dimensions is given by (after an appropriate Weyl rescaling of the metric)
\begin{eqnarray}
S=\int d^2x\sqrt{-{\rm det} g}(\Phi R+V(\Phi)).\label{dg}
\end{eqnarray}
For example, the Jackiw-Teitelboim model coressponds to the potential $V(\Phi)=2\Lambda^2\Phi$.

There are two  physical degrees of freedom in this theory: one given by the dilaton field $\Phi$ and one contained in the metric $g_{\mu\nu}$. Indeed, in two dimensions the metric is of the generic form
\begin{eqnarray}
g_{\mu\nu}=\rho\left( \begin{array}{cc}
\alpha^2-\beta^2 &\beta \\
\beta & -1 \end{array}\right).
\end{eqnarray}
The "lapse function" $\alpha=\alpha(t,x)$ and the "shift vector" $\beta=\beta(t,x)$ are non-dynamical variables here whereas the scale factor $\rho=\rho(t,x)$ is the only metric dynamical variable in two dimensions.

The metric in two dimensions can also be rewritten in the form (with $u=(t+x)/2$ and $v=(t-x)/2$ being the conformal light-cone coordinates)
\begin{eqnarray}
ds^2=4\rho(u,v)dudv.\label{myu}
\end{eqnarray}
In other words, the metric (any metric in two dimensions) is locally conformally flat.

The above model (\ref{dg}) is completely integrable which means that it can be completely solved in terms of free fields \cite{Cavaglia:1998xj}.

We start from the equations of motion
\begin{eqnarray}
(\nabla_{(\mu}\nabla_{\nu)}-g_{\mu\nu}\nabla_{\sigma}\nabla^{\sigma})\Phi+\frac{1}{2}g_{\mu\nu}V=0. \label{em1}
\end{eqnarray}
\begin{eqnarray}
R+\frac{dV}{d\Phi}=0.\label{em2}
\end{eqnarray}
If
\begin{eqnarray}
H(g,\Phi)= \nabla_{\sigma}\Phi\nabla^{\sigma}\Phi\ne 0
\end{eqnarray}
then the equation of motion (\ref{em2}) is automatically satisfied if the equation of motion (\ref{em1}) is satisfied \cite{Cavaglia:1998xj}.

The so-called  Bäcklund transformation \cite{filippov} allows us to transform the interacting fields $\Phi$ and $g_{\mu\nu}$ into the free fields $M$ and $\psi$ as follows
\begin{eqnarray}
\nabla_{\mu}\psi=\frac{\nabla_{\mu}\Phi}{H(g,\Phi)}.
\end{eqnarray}
\begin{eqnarray}
\end{eqnarray}
Indeed, $\psi$ and $M$ are free fields since $\nabla_{\mu}\nabla^{\mu}\psi=0$ and $\nabla_{\mu}M=0$. These equations of motion are equivalent to the original equations of motion (\ref{em1}) and (\ref{em2}). In particular, the second equation $\nabla_{\mu}M=0$ means that the field $M$ is actually a locally conserved quantity (the ADM mass). In the conformal light-cone coordinates the solution of these free equations of motion is trivially given by
\begin{eqnarray}
\end{eqnarray}
We can explicitly determine the dependence of the original fields $\Phi$ and $\rho$ on the free fields $\psi$ and $M$ to be given by \cite{Cavaglia:1998xj}
\begin{eqnarray}
\frac{d\psi}{d\Phi}=\frac{1}{N(\Phi)-M}~,~\rho=(N(\Phi)-M)\partial_u\psi\partial_v\psi.
\end{eqnarray}
The metric (\ref{myu}) becomes then
\begin{eqnarray}
ds^2=4(N(\Phi)-M)dUdV.
\end{eqnarray}
In other words, $U$ and $V$ appear as conformal light-cone coordinates. Thus, we introduce together with $\psi=U+V$ a timelike coordinate $T$ by $T=U-V$. The metric becomes
\begin{eqnarray}
ds^2=-(N(\Phi)-M)dT^2+\frac{d\Phi^2}{(N(\Phi)-M)}.\label{mele}
\end{eqnarray}
The dilaton field $\Phi$ appears therefore as a radial coordinate and it is the only dynamical variable appearing in the above general solution. In some sense this result generalizes Birkhoff theorem (in Einstein gravity in four dimensions with spherical symmetry the only local constant of the motion is the Schwarzschild mass).

As an example, we consider the dimensional reduction of Einstein gravity in four dimensions on a sphere of radius $R^2=4\Phi$ (spherical reduction is consistent in the case of maximal rotational invariance) . The resulting action takes the form (\ref{dg}) with a potential $V(\Phi)=1/2\sqrt{\Phi}$ and as a consequence the metric element (\ref{mele}) reduces to the radial part of the Schwarzschild solution.

By taking the derivative of equation (\ref{ADM}) we obtain
\begin{eqnarray}
\nabla_{\mu}M=V.\nabla_{\mu}\Phi-\nabla_{\mu}H.
\end{eqnarray}
By employing this result we can express the potential $V$ in terms of $M$ and $\nabla_{\mu}\Phi$ as follows
\begin{eqnarray}
V=\frac{1}{H}\nabla^{\mu}\Phi\nabla_{\mu}M+\frac{1}{H}\nabla^{\mu}\Phi\nabla_{\mu}H.\label{piece1}
\end{eqnarray}
From the other hand, the Ricci scalar in two dimensions is locally given by a total divergence, viz
\begin{eqnarray}
R=2\nabla_{\mu}A^{\mu}~,~A^{\mu}=\frac{\nabla^{\mu}\nabla^{\nu}\chi.\nabla_{\nu}\chi-\nabla_{\nu}\nabla^{\nu}\chi.\nabla^{\mu}\chi}{\nabla^{\rho}\chi.\nabla_{\rho}\chi}.
\end{eqnarray}
In this equation $\chi$ is an arbitrary scalar field which we choose to be $\chi=\Phi$. We then compute
\begin{eqnarray}
\Phi.R&=&2\Phi.\nabla_{\mu}A^{\mu}\nonumber\\ &=&2\nabla_{\mu}(\Phi.A^{\mu})-2\nabla_{\mu}\Phi.A^{\mu}\nonumber\\
&=& 2\nabla_{\mu}(\Phi.A^{\mu})-\frac{2}{H}\nabla_{\mu}\Phi.(\frac{1}{2}\nabla^{\mu}H-\nabla_{\nu}\nabla^{\nu}\Phi.\nabla^{\mu}\Phi)\nonumber\\
&=&2\nabla_{\mu}(\Phi A^{\mu}+\nabla^{\mu}\Phi)-\frac{1}{H}\nabla_{\mu}\Phi.\nabla^{\mu}H.\label{piece2}
\end{eqnarray}
By putting (\ref{piece1}) and (\ref{piece2}) together  and using the fact that $H=N(\Phi)-M$ we can show that the original dilaton gravity action (\ref{dg}) takes the form (up to a surface term)
\begin{eqnarray}
S=\int d^2x\sqrt{-{\rm det} g}\frac{\nabla_{\mu}M\nabla^{\mu}\Phi}{N(\Phi)-M}.
\end{eqnarray}
This is a non-linear sigma model.

### References

%\cite{Cavaglia:1998xj}
\bibitem{Cavaglia:1998xj}
M.~Cavaglia,
Geometrodynamical formulation of two-dimensional dilaton gravity,''
Phys.\ Rev.\ D {\bf 59}, 084011 (1999)
doi:10.1103/PhysRevD.59.084011
[hep-th/9811059].
%%CITATION = doi:10.1103/PhysRevD.59.084011;%%
%24 citations counted in INSPIRE as of 12 Dec 2019

%\cite{filippov}
\bibitem{filippov}
A.~T. ~Filippov,
in: Problems in Theoretical Physics, Dubna, JINR, June 1996, p. 113;
Mod. Phys. Lett. A 11, 1691 (1996); Int. J. Mod. Phys. A 12, 13 (1997).