## LATEX

### AdS2 black holes and dilaton gravity

It is well known that in two dimensions all negatively curved spacetimes are locally  ${\bf AdS}^2$ and thus stable black hole solutions in two dimensions do not exist in a naive way.  This is similarly to the fact that in three dimensions all negavtively curved spacetimes are locally an ${\bf AdS}^3$ and thus stable black hole solutions in three dimensions do not  also exist in a naive way. Yet, in three dimensions the celebrated  BTZ black hole \cite{Banados:1992wn} is a stable black hole solution which differs from ${\bf AdS}^3$ by global identification and in two dimensions the SS black hole \cite{Spradlin:1999bn} is also a stable black solution which differs from ${\bf AdS}^2$ by global identification (we choose the Killing time $t$ at infinity such that the region $-\infty\lt t\lt +\infty$ does not cover all of the boundary of ${\bf AdS}^2$). These black holes are therefore locally identical with the corresponding anti-de Sitter spacetimes and differ from them only topologically.

As we will see in the following dilaton gravity in two dimensions provides another way of obtaining stable ${\bf AdS}^2$ black holes which are locally identicall to ${\bf AdS}^2$ spacetime but differ from it only globally precisely through the value of the dilaton field.

${\bf AdS}^2\times {\bf S}^2$ as a near-horizon geometry of extremal black holes

The single most important fact (in our opinion) about ${\bf AdS}^2$ geometry  is its appearance as a near-horizon geometry  of extremal black holes in both general relativity and string theory.  The typical example is Einstein gravity coupled to Maxwell electromagnetism and its celebrated four-dimensional Reissner-Nordstrom black hole given by the metric \cite{RN}
\begin{eqnarray}
ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_2^2~,~f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}.
\end{eqnarray}
This black hole is characterized by a mass $M$ and a charge $Q$ where $M\geq Q$ (otherwise if $M\lt Q$ a naked singularity appears which is forbidden by cosmic censorship \cite{Penrose:1969pc}).  In the Reissner-Nordstrom black hole solution the electric field (which we are not writing explicitly) plays a fundamental role by supporting the whole geometry.

The near-horizon geometry of this solution is approximately  a Rindler spacetime (recall the Schwarzschild solution) which does not solve Einstein equations. However, for extremal black holes (those with mass $M=Q$ or equivalently zero temperature $T=0$) the nera-horizon geometry is anti-de Sitter spacetime ${\bf AdS}^2$ (times a sphere ${\bf S}^2$ because of rotational invariance) which is actually an exact solution of Einstein equations. Thus, a quantum black hole with mass $M\gt Q$ will evaporate until it reaches the extremal mass $M=Q$ where the temperature vanishes and the evaporation stops , i.e. the extremal quantum black hole acts as a stable ground state in the case of a charged black hole \cite{Hawking:1974sw}.

In the extremal limit $M=Q$ (or $T=0$) the inner and outer horizons  $r_-$ and $r_+$ respectively coincide $r_+=r_-=Q$ and the horizon becomes a double zero since $f(r)=(1-Q/r)^2$. We define
\begin{eqnarray}
r=Q(1+\frac{\lambda}{z})~,~t=\frac{QT}{\lambda}.
\end{eqnarray}
The near-horizon geometry of the extremal solution is obtained by letting $\lambda\longrightarrow 0$. By substituting these definitions in the metric and taking the limit $\lambda\longrightarrow 0$ we obtain
\begin{eqnarray}
ds^2=\frac{Q^2}{z^2}(-dT^2+dz^2)+Q^2d\Omega_2^2.
\end{eqnarray}
This is the metric of ${\bf AdS}^2\times{\bf S}^2$ where the charge $Q$ appears as the radius of both factors ${\bf AdS}^2$ and ${\bf S}^2$ \cite{carter}.

${\bf AdS}^2$ black holes in dilaton gravity

\begin{eqnarray}
S=\int d^4x \sqrt{-{\rm det}g^{(4)}} e^{-2\phi}(R^{(4)}-F_{\mu\nu}F^{\mu\nu}).
\end{eqnarray}
The closely related low-energy effective actions  of string theory with similar black holes physics are found in \cite{Garfinkle:1990qj,Giddings:1992kn}).

A spherically symmetric non-singular black hole solution of the equations of motion stemming from this action is given by the monopole hedgehog configuration, the black hole spacetime metric and the dilaton field \cite{Cadoni:1994uf}
\begin{eqnarray}
F_{ij}=\frac{Q_M}{r^2}\epsilon_{ijk}n_k.
\end{eqnarray}
\begin{eqnarray}
ds^2=-(1-\frac{r_+}{r})dt^2+\frac{dr^2}{(1-\frac{r_+}{r})(1-\frac{r_-}{r})}+r^2d\Omega_2^2.\label{4dBH}
\end{eqnarray}
\begin{eqnarray}
e^{2(\phi-\phi_0)}=\frac{1}{\sqrt{1-\frac{r_-}{r}}}.
\end{eqnarray}
The inner radius $r_-$ and the outer radius $r_+$ (with $r_+\geq r_-$) are given in terms of the mass $M$ and the charge $Q_M$ of the black hole by the relations \cite{Cadoni:1994uf}
\begin{eqnarray}
2M=r_+~,~ Q_M^2=\frac{3}{4}r_+r_-.
\end{eqnarray}
The temperature and the entropy of the black hole are given on the other hand by the relations \cite{Cadoni:1994uf}
\begin{eqnarray}
T=\frac{1}{4\pi r_+}\sqrt{1-\frac{r_-}{r_+}}~,~S=\pi r_+^2
\end{eqnarray}
The extremal limit $T\longrightarrow 0$ of this black hole configuration is then given by $r_+=r_-=Q=2Q_M/\sqrt{3}$ or equivalently $M=Q_M/\sqrt{3}$.

The spatial sections of this black hole solution coincide  with those of the Reissner-Nordstrom black hole. However, this solution corresponds to a non-singular black hole where the spacetime manifold is cut at $r=r_-$ while it is asymptotically flat. Indeed, the maximal extension of this metric yields a Penrose diagram identical to that of the Schwarzschild solution except that the singularity $r=0$ is replaced by the boundary of the manifold at $r=r_-$ \cite{Cadoni:1994uf}.

For the extremal solution $r_+=r_-=Q$ we introduce the coordinates
\begin{eqnarray}
r=Q(1+\frac{4\lambda^2}{z^2})~,~t=\frac{QT}{\lambda}.
\end{eqnarray}
The metric and the dilaton in the near-horizon limit $\lambda\longrightarrow 0$ take then the form
\begin{eqnarray}
ds^2=\frac{4Q^2}{z^2}(-dT^2+dz^2)+Q^2d\Omega_2^2.
\end{eqnarray}
\begin{eqnarray}
e^{2(\phi-\phi_0)}=\frac{z}{2\lambda}.
\end{eqnarray}
This shows explicitly that the near-horizon geometry of the extremal black hole is indeed ${\bf AdS}^2\times{\bf S}^2$.

We can perform a spherical reduction of this solution by decomposing the metric as follows
\begin{eqnarray}
ds^2&=&g_{\mu\nu}^{(4)}dx^{\mu}dx^{\nu}\nonumber\\
\end{eqnarray}
The scalar field $\Phi$ is a dilaton field due to the spherical reduction. We compute then (see \cite{Grumiller:2001ea} and references therein)
\begin{eqnarray}
&&\sqrt{-{\rm det}g^{(4)}}=\Phi^2 \sqrt{-{\rm det}g^{(2)}}\sqrt{{\rm det}\gamma }\nonumber\\
&&R^{(4)}=R^{(2)}-\frac{2}{\Phi^2}(-1+\partial_a\Phi\partial^a\Phi)-\frac{4}{\Phi}\Delta\Phi.
\end{eqnarray}
And hence

\begin{eqnarray}
\int d^4x \sqrt{-{\rm det}g^{(4)}} R^{(4)}&=&4\pi \int d^2x \sqrt{-{\rm det}g^{(2)}} (\Phi^2 R^{(2)}+2\partial_a \Phi\partial^a\Phi+2).
\end{eqnarray}
Hence the action reduces to

\begin{eqnarray}

S&=&4\pi \int d^2x \sqrt{-{\rm det}g^{(2)}} e^{-2\phi}(\Phi^2 R^{(2)}+2\partial_a \Phi\partial^a\Phi+2-\Phi^2 F^2).
\end{eqnarray}
For Schwarzschild-like coordinates the dilaton field $\Phi$ is given by $\Phi=r$. However, in the current case the spherical reduction is performed on a sphere of constant radius $r= Q=2Q_M/\sqrt{3}$, i.e. $\Phi=Q$. We get then the action (with $\Lambda=1/2Q$)
\begin{eqnarray}
S&=&4\pi Q^2 \int d^2x \sqrt{-{\rm det}g^{(2)}} e^{-2\phi}(R^{(2)}+2\Lambda^2).
\end{eqnarray}
This is called the Jackiw-Teitelboim action \cite{JT} which is one of the most important dilatonic gravity models in two dimensions. The most general solution (see \cite{Cadoni:1993rn} and references therein) of the equations of motion stemming from the Jackiw-Teitelboim action  is given by  the metric field (in the so-called Schwarzschild coordinates)
\begin{eqnarray}
ds^2=-(\Lambda^2r^2-a^2)dt^2+\frac{dr^2}{\Lambda^2r^2-a^2}.
\end{eqnarray}
And the dilaton field (with $\Phi=\exp(-2\phi)$)
\begin{eqnarray}
e^{2(\phi-\phi_0)}=\frac{1}{\Lambda r}\iff \Phi=e^{-2\phi}=\Phi_0\Lambda r.
\end{eqnarray}
The parameter $a^2$ in the metric is an integration constant related to the mass $M$ of the solution by the relation
\begin{eqnarray}
M=\frac{\Lambda}{2}a^2\Phi_0.
\end{eqnarray}
The above metric corresponds, for all values of $a^2$, to a two-dimensional spacetime with a constant negative curvature $R=-2\Lambda^2$, i.e. an anti-de Sitter spacetime ${\bf AdS}^2$. Furthermore, it was shown in \cite{Cadoni:1994uf} that this metric in Schwarzschild coordinates describes the two-dimensional sections of the extremal four-dimensional black hole (\ref{4dBH}).

The solution for $a^2=0$ is exactly ${\bf AdS}^2$ spacetime and it plays the role of the ground state of the theory (analogous to Minkowski spacetime). For example, this solution has mass $M=0$ and the mass of the other solutions is computed with respect to this one.

The solution $a^2>0$ is our ${\bf AdS}^2$ black hole with a horizon at $r_H=a/\Lambda$ which  can not be distinguished locally from the actual ${\bf AdS}^2$ spacetime with $a^2=0$ (as we will see this is the analogue of Rindler spacetime). Indeed, by means of an appropriate coordinates transformation we can bring the solution $a^2>0$ into the form of the solution $a^2=0$. The difference between the two cases is strictly topological in character originating from the global properties of the solution encoded in the behavior of the dilaton field. To see this crucial point more explicitly we consider the coordinates transformation
\begin{eqnarray}
r^{\prime}=a\Lambda t r~,~2a\Lambda t^{\prime}=\ln\big(\Lambda^2t^2-\frac{1}{\Lambda^2 r^2}\big).
\end{eqnarray}
We can then check immediately that
\begin{eqnarray}
-(\Lambda^2r^{\prime 2}-a^2)dt^{\prime 2}+\frac{dr^{\prime 2}}{\Lambda^2r^{\prime 2}-a^2}=-\Lambda^2r^2 dt^2+\frac{dr^2}{\Lambda^2r^2}.
\end{eqnarray}
However, the dilaton field changes in a non-trivial way under the above coordinates transformation, viz
\begin{eqnarray}
\Phi_0\sqrt{\frac{\Lambda^2 r^{\prime 2}}{a^2}-1}e^{-a\Lambda t^{\prime}}=\Phi_0\Lambda r.
\end{eqnarray}
Thus, although the solution with $a^2=0$ (${\bf AdS}^2$ spacetime) is locally equivalent to the solution with $a^2 \gt 0$ (${\bf AdS}^2$ black hole) these two solutions are globally different due to the behavior of the dilaton field which effectively sets the boundary conditions on the spacetime.

Furthermore, the solution with $a^2\gt 0$ can be seen to represent really an ${\bf AdS}^2$ black hole from the fact that it must be cutoff at $r=0$ otherwise the dilaton field $\Phi=\exp(-2\phi)$ will become negative when we maximally extend the corresponding metric beyond $r=0$ which in turn will translate in four dimensions (recall that the two-dimensional theory is obtained by spherical reduction from four dimensions) into a negative value for the area of the transverse sphere which is physically unacceptable. Therefore $r=0$ is a boundary for the   ${\bf AdS}^2$ black hole with $a^2>0$ corresponding to the boundary $r=r_-$ of the extremal four-dimensional regular black hole (\ref{4dBH}).

The temperature and the entropy of this ${\bf AdS}^2$ black hole can be computed in the usual way and one finds \cite{Cadoni:1994uf}
\begin{eqnarray}
T=\frac{a\Lambda}{2\pi}~,~S=4\pi \sqrt{\frac{\Phi_0M}{2\Lambda}}.
\end{eqnarray}
The solution with the value $a^2\lt 0$ corresponds to a negative mass and although this makes sense in two dimensions (it corresponds to no naked singularities) it will translate in four dimensions into a naked singularity which is unacceptable by cosmic censorship. Hence the solution with $a^2\lt 0$ is unphysical (from the four-dimensional point of view) and should be discarded.

In summary, our ${\bf AdS}^2$ black hole (the solution with $a^2\gt 0$) is characterized by a horizon at $r_H=a/\Lambda$ and a boundary at $r=0$. For the semi-classical process of Hawking radiation the boundary at $r=0$ is not required and therefore one can work in a system of coordinates where the boundary is not accessible. We introduce then the light-cone coordinates
\begin{eqnarray}
\sigma^{\pm}=t\pm r_*
\end{eqnarray}
where $r_*$ is the tortoise coordinate defined as usual by the requirement
\begin{eqnarray}
(\Lambda^2 r^2-a^2)dr_*^2=\frac{dr^2}{\Lambda^2 r^2-a^2}\iff r_*=-\frac{1}{a\Lambda}{\rm arctanh}(\frac{a}{\Lambda r}).
\end{eqnarray}
Equivalently, we can work in the light-like coordinates $x^{\pm}$ defined by
\begin{eqnarray}
x^{\pm}=\frac{2}{a\Lambda}\tanh \frac{a\Lambda}{2}\sigma^{\pm}.\label{coc}
\end{eqnarray}
The metric and the dilaton fields in the light-like coordinates take the form (conformal gauge)
\begin{eqnarray}
ds^2&=&-\frac{a^2}{\sinh^2\frac{a\Lambda}{2}(\sigma^--\sigma^+)}d\sigma^-d\sigma^+\nonumber\\
&=&-\frac{4}{\Lambda^2}\frac{1}{(x^--x^+)^2}dx^-dx^+.\label{meme}
\end{eqnarray}
\begin{eqnarray}
e^{2(\phi-\phi_0)}&=&\frac{1}{a}\tanh\frac{a\Lambda}{2}(\sigma^--\sigma^+)\nonumber\\
&=&\frac{\Lambda}{2}\frac{x^--x^+}{1-\frac{a^2\Lambda^2}{4}x^-x^+}.
\end{eqnarray}
The ${\bf AdS}^2$ spacetime (in the conformal gauge) corresponds to setting $a^2=0$ (or equivalently $x^{\pm}=\sigma^{\pm}$) in these expressions. In other words, the coordinates $x^{\pm}$ can be thought of as describing ${\bf AdS}^2$ spacetime even for $a^2\ne 0$ since they can be easily extended to the whole of spacetime. We also observe that the boundary of ${\bf AdS}^2$ spacetime is located at $x^-=x^+$ and that we must have $x^-\geq x^+$ (corresponding to $r\geq 0$ in the Schwarzschild coordinates) in order for the dilaton field $\exp(2(\phi-\phi_0))$ to remain positive. Furthermore, it is clear that the coordinates $\sigma^{\pm}$ for $a^2\gt 0$ cover only the region $-2/a\Lambda\lt x^{\pm}\lt +2/a\Lambda$ of the  ${\bf AdS}^2$ spacetime (corresponding to the solution $a^2=0$ in the conformal gauge). This region corresponds to the region $r\gt r_H$ in the Schwarzschild coordinates whereas the boundary at $r=0$ in the Schwarzschild coordinates corresponds now to the line $1-\frac{a^2\Lambda^2}{4}x^-x^+=0$.

Another  interesting system of coordinates consists of  the Poincare coordinates $\hat{t}$ and $z$ defined for our ${\bf AdS}^2$ black hole by the change of coordinates
\begin{eqnarray}
&&\hat{t}=\frac{1}{a\Lambda}e^{a\Lambda t}\cosh a\Lambda r_*\longrightarrow t+\frac{1}{a\Lambda}~,~a\longrightarrow 0\nonumber\\
&&z=-\frac{1}{a\Lambda}e^{a\Lambda t}\sinh a\Lambda r_*\longrightarrow -r_*~,~a\longrightarrow 0
\end{eqnarray}
The metric in the Poincare patch is given by the usual form
\begin{eqnarray}
ds^2=\frac{1}{\Lambda^2 z^2}(-d\hat{t}^2+dz^2).
\end{eqnarray}
For $a=0$ (the ${\bf AdS}^2$ spacetime) the boundary is located at $z=0$  or equivalently $x^--x^+=0$ and the Poincare patch covers $z\gt 0$ or equivalently $x^--x^+\gt 0$. This result shows also that our ${\bf AdS}^2$ black hole is indeed locally equivalent to a pure ${\bf AdS}^2$ spacetime. In fact the difference between them is fully encoded in the value of the dilaton field which reflects the boundary conditions imposed on the spacetime and its consequent  topological features.

Hawking process

The relationship between the ${\bf AdS}^2$ spacetime corresponding to the solution $a^2=0$ (denoted from now on by ${\bf ADS}_0$) and the ${\bf AdS}^2$ black hole corresponding to the solution $a^2\gt 0$ (denoted from now on by ${\bf ADS}_+$) is identical to the relationship between the the two-dimensional Minkowski spacetime with metric
\begin{eqnarray}
ds^2=-dt^2+dx^2
\end{eqnarray}
and the  Rindler wedge with metric (with $-\infty<\tau,\sigma<+\infty$)
\begin{eqnarray}
ds^2=\exp(2\alpha \sigma)(-d\tau^2+d\sigma^2)
\end{eqnarray}
confined to the quadrant $x\gt |t|$. The change of coordinates $(t,x)\longrightarrow (\tau,\sigma)$ is given by

\begin{eqnarray}
t=\frac{1}{\alpha}\exp(\alpha\sigma)\sinh \alpha\tau~,~x=\frac{1}{\alpha}\exp(\alpha\sigma)\cosh \alpha\tau~, ~x>|t|.\label{cha}
\end{eqnarray}
Indeed, the parameter $a^2$ (which is proportional to the mass of the black hole) in our case  is the analogue of the acceleration $\alpha$ with which the Rindler observer is uniformly accelerating in Minkowski spacetime creating thus a horizon at $x=|t|$ separating the Rindler wedge from the rest of Minkowski spacetime.

Similarly here, a non-zero value of the parameter $a^2$ is associated with the existence of a horizon at $r=r_H=a/\Lambda$ separating the exterior of the black hole ${\bf ADS}_+$ (described by the light-like coordinates $\sigma^{\pm}$ which indeed covers only the region $r\gt r_H$) from its interior $0\lt r\lt r_H$.

The asymptotic behavior of this ${\bf AdS}^2$ black hole ${\bf ADS}_+$ is given by the ${\bf AdS}^2$ spacetime ${\bf ADS}_0$ which can be fully covered by the light-like coordinates $x^{\pm}$. The change of coordinates (\ref{coc}) which relates the two sets of coordinates $x^{\pm}$ and $\sigma^{\pm}$  (although in our case it does not correspond to any motion of physical observers and therefore is connecting two different manifolds) play exactly the role of the boost which connects the Minkwoski coordinates $(t,x)$ to the Rindler coordinates $(\eta,\sigma)$.

Therefore, quantization of fields and semi-classical considerations of Hawking radiation in the ${\bf ADS}_+$ background with ${\bf ADS}_0$ taken as a ground state should then proceed along the same steps as the analogous calculation performed in the Rindler wedge with the Minkowski spacetime taken as a ground state. As we will explain, there will be two vacuum states $|0_+\rangle$ and $|0_0\rangle$  corresponding to the two different spacetimes (observers) ${\bf ADS}_+$ and  ${\bf ADS}_0$ and as a consequence a thermal radiation will be observed in each vacuum state by the other observer associated with the other spacetime which is precisely Hawking radiation.

However, we should also recall that anti-de Sitter spacetime is not a globally hyperbolic space and thus one should be careful with the boundary conditions at infinity (transparent boundary conditions) as discussed for example in \cite{Cadoni:1993rn}.

We will use in the following a slightly different parameterization of ${\bf ADS}_+$ and ${\bf ADS}_0$ made possible by the $SL(2,R)$ symmetry of the metric (\ref{meme}) given by the transformations
\begin{eqnarray}
\end{eqnarray}
From the first line of (\ref{meme}) the metric on the ${\bf AdS}^2$ black hole ${\bf ADS}_+$ is given by (with the change of notation $t\longrightarrow \tau$ and $r_*\longrightarrow \sigma$)
\begin{eqnarray}
ds^2=\frac{a^2}{\sinh^2 a\Lambda\sigma}(-d\tau^2+d\sigma^2).
\end{eqnarray}
The black hole coordinates $\tau$ and $\sigma$ are defined in the range $-\infty\lt \tau\lt +\infty$ and $0\lt\sigma\lt \infty$ with corresponding light-cone coordinates defined by $\sigma^{\pm}=\tau\mp\sigma$.

The metric on the ${\bf AdS}^2$ spacetime ${\bf ADS}_0$ is assumed to be of the Poincare form, viz
\begin{eqnarray}
ds^2=\frac{1}{\Lambda^2 x^2}(-dt^2+dx^2).
\end{eqnarray}
The AdS coordinates $t$ and $x$ are defined in the range $-\infty\lt t\lt +\infty$ and $0\lt x\lt \infty$ with corresponding light-cone coordinates defined by $x^{\pm}=t\mp x$.

The change of coordinates $(t,x)\longrightarrow (\tau,\sigma)$ is given explicitly by
\begin{eqnarray}
t=\frac{1}{a\Lambda}e^{a\Lambda \tau}\cosh a\Lambda\sigma~,~x=\frac{1}{a\Lambda}e^{a\Lambda \tau}\sinh a\Lambda\sigma.
\end{eqnarray}
By comparing with (\ref{cha}) we can see that $a\Lambda$ in our anti-de Sitter black hole plays the role of the acceleration $\alpha$ in Rindler spacetime (the mathematics is identical although the underlying physics is quite different).

We also compute
\begin{eqnarray}
x^{\pm}=\frac{1}{a\Lambda}e^{a\Lambda\sigma^{\pm}}.
\end{eqnarray}
These coordinates define region or quadrant I of ${\bf ADS}_0$ (the exterior of our black hole) in which the timelike Killing vector field (which generates boosts in the $x-$direction) is $\partial_{\tau}$. This Killing vector field is future-directed.  Thus, the Killing horizons lie at $x=\pm t$.

The region or quadrant IV of ${\bf ADS}_0$ in which the timelike Killing vector field (given here by $\partial_{-\tau}=-\partial_{\tau}$) is past-directed is given by the coordinates
\begin{eqnarray}
x^{\pm}=-\frac{1}{a\Lambda}e^{a\Lambda\sigma^{\pm}}.
\end{eqnarray}
The change of coordinates $(t,x)\longrightarrow (\tau,\sigma)$ in this region is given by
\begin{eqnarray}
t=-\frac{1}{a\Lambda}e^{a\Lambda \tau}\cosh a\Lambda\sigma~,~x=-\frac{1}{a\Lambda}e^{a\Lambda \tau}\sinh a\Lambda\sigma.
\end{eqnarray}
Hence ${\bf ADS}^+$ covers the region of ${\bf ADS}_0$ given by the union of the two quadrants I and IV which is specified by the condition $x^+x^-\geq 0$ with the black horizon defined by the condition $x^+x^-=0$.

The equation of motion is the Klein-Gordon equation in the ${\bf AdS}^2$ black hole background ${\bf ADS}_+$ which is locally equivalent to the ${\bf ADS}^2$ spacetime ${\bf ADS}_0$, i.e. the equation of motion is effectively the Klein-Gordon equation in anti-de Sitter spacetime ${\bf AdS}^2$. Furthermore, the inner product  between two solutions $\phi_1$ and $\phi_2$ of the equation of motion is defined in the usual way by ($\Sigma$ is the spacelike surface $\tau=0$ and $n^{\mu}$ is the timelike unit vector normal to it)
\begin{eqnarray}
(\phi_1,\phi_2)&=&-i\int_{\Sigma} \big(\phi_1\partial_{\mu}\phi_2^*-\partial_{\mu}\phi_1.\phi_2^*\big) d\Sigma n^{\mu}\nonumber\\
&=&-i\int \big(\phi_1\partial_{\tau}\phi_2^*-\partial_{\tau}\phi_1.\phi_2^*\big) d\sigma.
\end{eqnarray}

A positive-frequency normalized plane wave solution of this equation of motion in region I ($x\gt 0$) is given by (with $\omega=|k|$)
\begin{eqnarray}
&&g_k^{(1)}=\frac{1}{\sqrt{4\pi \omega}}\exp(-i\omega \tau+ik\sigma)~,~{\rm I}\nonumber\\
&&g_k^{(1)}=0~,~{\rm IV}.\label{pos}
\end{eqnarray}
This is positive-frequency since $\partial_{\tau}g_k^{(1)}=-i\omega g_k^{(1)}$.

A positive-frequency normalized plane wave solution in region IV is instead given by
\begin{eqnarray}
&&g_k^{(2)}=0~,~{\rm I}\nonumber\\
&&g_k^{(2)}=\frac{1}{\sqrt{4\pi \omega}}\exp(i\omega \tau+ik\sigma)~,~{\rm IV}.
\end{eqnarray}
Since $\partial_{-\tau}g_k^{(2)}=-i\omega g_k^{(2)}$.

A general solution of the Klein-Gordon equation takes then the form
\begin{eqnarray}
\phi=\int_k \big(\hat{b}_k^{(1)}g_k^{(1)}+\hat{b}_k^{(2)}g_k^{(2)}+{\rm h.c}\big).
\end{eqnarray}
This should be contrasted with the expansion of the same solution in terms of the anti-de Sitter spacetime modes $f_k\propto \exp(-i(\omega t-kx))$ with $\omega=|k|$ which we will write as
\begin{eqnarray}
\phi=\int_k \big(\hat{a}_k^{}f_k^{}+{\rm h.c}\big).
\end{eqnarray}
The ${\bf ADS}_0$ vacuum $|0_0\rangle$ and the ${\bf ADS}_+$ vacuum $|0_+\rangle$ are defined obviously by
\begin{eqnarray}
\hat{a}_k|0_0\rangle=0.
\end{eqnarray}
\begin{eqnarray}
\hat{b}_k^{(1)}|0_+\rangle=\hat{b}_k^{(2)}|0_+\rangle=0.
\end{eqnarray}
In order to compute the corresponding Bogolubov coefficients we extend the positive-frequency modes $g_k^{(1)}$ and $g_k^{(2)}$ to the entire spacetime ${\bf ADS}_0$ thus replacing the corresponding annihilation operators $\hat{b}_k^{(1)}$ and $\hat{b}_k^{(2)}$ by new annihilation operators $\hat{c}_k^{(1)}$ and $\hat{c}_k^{(2)}$ which annihilate the anti-de Sitter  spacetime vacuum $|0_0>$ \cite{Unruh:1976db}.

Clearly, for $k>0$ we have in region I the behavior
\begin{eqnarray}
\sqrt{4\pi\omega}g_k^{(1)}&=&\exp(-i\omega\sigma^+)\nonumber\\
&=&(a\Lambda)^{-i\frac{\omega}{a\Lambda}}(x^+)^{-i\frac{\omega}{a\Lambda}}.
\end{eqnarray}
In region IV ($x<0$) we should instead consider
\begin{eqnarray}
\sqrt{4\pi\omega}g_{-k}^{(2)*}&=&\exp(-i\omega \sigma^+)\nonumber\\
&=&(-a\Lambda)^{-i\frac{\omega}{a\Lambda}}(x^+)^{-i\frac{\omega}{a\Lambda}}\nonumber\\
&=&e^{\frac{\pi\omega}{a\Lambda}}(a\Lambda)^{-i\frac{\omega}{a\Lambda}}e^{\frac{\pi \omega}{a\Lambda}}(x^+)^{-i\frac{\omega}{a\Lambda}}.
\end{eqnarray}
Thus for all $x$, i.e. along the surface $t=0$, we should consider for $k>0$ the combination
\begin{eqnarray}
\sqrt{4\pi\omega}\big(g_k^{(1)}+e^{-\frac{\pi\omega}{a\Lambda}}g_{-k}^{(2)*}\big)
&=&(a\Lambda)^{-i\frac{\omega}{a\Lambda}}(x^+)^{-i\frac{\omega}{a\Lambda}}.
\end{eqnarray}
A normalized analytic extension to the entire spacetime of the positive-frequency modes $g_k^{(1)}$ is given by the modes
\begin{eqnarray}
h_k^{(1)}&=&\frac{1}{\sqrt{2\sinh \frac{\pi\omega}{a\Lambda}}}\big(e^{\frac{\pi\omega}{2a\Lambda}} g_k^{(1)}+e^{-\frac{\pi\omega}{2a\Lambda}}g_{-k}^{(2)*}\big).\label{h1}
\end{eqnarray}
Similarly, a normalized analytic extension to the entire spacetime of the positive-frequency modes $g_k^{(2)}$ is given by the modes
\begin{eqnarray}
h_k^{(2)}&=&\frac{1}{\sqrt{2\sinh \frac{\pi\omega}{a\Lambda}}}\big(e^{\frac{\pi\omega}{2a\Lambda}} g_k^{(2)}+e^{-\frac{\pi\omega}{2a\Lambda}}g_{-k}^{(1)*}\big).\label{h2}
\end{eqnarray}
The field operator can then be expanded in these modes as
\begin{eqnarray}
\phi=\int_k \big(\hat{c}_k^{(1)}h_k^{(1)}+\hat{c}_k^{(2)}h_k^{(2)}+{\rm h.c}\big).
\end{eqnarray}
Obviously, the modes $h_k^{(1)}$ and $h_k^{(2)}$ share with $f_k$ the same anti-de Sitter spacetime vacuum $|0_0\rangle$, viz
\begin{eqnarray}
\hat{c}_k^{(1)}|0_0\rangle=\hat{c}_k^{(2)}|0_0\rangle =0.
\end{eqnarray}
The ${\bf ADS}_+$ number operator in region I is defined by
\begin{eqnarray}
\hat{N}_R^{(1)}(k)=\hat{b}_k^{(1)+}\hat{b}_k^{(1)}.
\end{eqnarray}
We can now immediately compute the expectation value of this number operator in region I in the anti-de Sitter vacuum $|0_0\rangle$ to find
\begin{eqnarray}
\langle 0_0|\hat{N}_R^{(1)}(k)|0_0\rangle&=&\langle 0_0|\hat{b}_k^{(1)+}\hat{b}_k^{(1)}|0_0\rangle\nonumber\\
&=&\frac{e^{-\frac{\pi\omega}{a\Lambda}}}{2\sinh\frac{\pi\omega}{2}}\langle 0_0|\hat{c}_{-k}^{(2)}\hat{c}_{-k}^{(2)+}|0_0\rangle\nonumber\\
&=&\frac{1}{e^{\frac{2\pi\omega}{a\Lambda}}-1}\delta(0).
\end{eqnarray}
This is a blackbody Planck spectrum corresponding to the temperature
\begin{eqnarray}
T=\frac{a\Lambda}{2\pi}.
\end{eqnarray}

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Vacuum States for AdS 2 Black Holes

The AdS/CFT Correspondence in Two Dimensions

NON-SINGULAR FOUR-DIMENSIONAL BLACK HOLES AND THE JACKIW-TEITELBOIM THEORY

Geometrodynamical Formulation of Two-Dimensional
Dilaton Gravity

ASYMPTOTIC SYMMETRIES OF AdS 2 AND
CONFORMAL GROUP IN d=1

AdS 2 Gravity as Conformally Invariant Mechanical
System

Open strings, 2D gravity and AdS/CFT correspondence

The Holographic Entanglement
Entropy of Schwarzschild Black Holes

Entanglement entropy of two-dimensional anti-de Sitter black holes

Near Extremal Black Hole Entropy
as Entanglement Entropy via AdS 2 /CFT 1