We investigate a dilaton gravity model in
AdS
Article funded by SCOAP3
0$ prevents this singularity from reaching the boundary within a finite amount of time. ]]>
0$. Via the substitution \bea \label{BHx} X^{+}(u) \is \frac{1}{\sqrt{\kappa E}}\tanh\Bigl(\sqrt{\ketw}\, u\Bigr)\,,\qquad \ \ X^{-}(v) = \frac{1}{\sqrt{\kappa E}}\tanh\Bigl(\sqrt{\ketw}\, v\Bigr) \, , \eea we can write this black hole solition in a manifestly static form as \bea \label{BHmetr} \dd s^{2} \is -\frac{4\kappa E}{\sinh^2\bigl(\sqrt{\ketw}\spc (u-v)\bigr)}\dd u\, \dd v\,, \\[2mm] \Phi^2 \is 1 + a \sqrt{\kappa E} \coth\bigl(\sqrt{\ketw}\spc (u-v)\bigr) \, . \label{BHdilaton} \eea The future and past horizon are found at $u = \infty$ and $v=-\infty$, which in terms of the $X$ coordinates translates to $X^+ = 1/{\sqrt{\kappa E}}$ and $X^- = - 1/{\sqrt{\kappa E}}$. In the limit $E\to 0$, one recovers the standard AdS${}_2$ metric~(\ref{metricX}) in Poincar\'e coordinates, with dilaton profile $\Phi^2 = 1 + {a}/(X^+\!\! -\! X^-).$ As indicated in figure~\ref{fig:patches}, the black hole space-time outside the horizon can be embedded as a bounded triangular region within the Poincar\'e patch. Note that the size of the black hole patch shrinks with increasing $E$. The black hole metric and dilaton in equations~(\ref{BHmetr}) and~(\ref{BHdilaton}) are periodic in imaginary time with period $ \beta = {\pi}/{\sqrt{\kappa E}}$. The Hawking temperature of the black hole is thus identified as \bea \label{temperature} T = \frac 1 \pi {\sqrt{\kappa E}} \, . \eea \begin{figure}[t] \centering \includegraphics[width=0.2\textwidth]{patchesfig.pdf} ]]>
0$ \bea :T_{uu}(t): \is \frac{\ccc}{24\pi}\left\{\tau,t\right\} \, . \eea Hence we arrive at the somewhat tantalizing result that at times greater than $t=0$, the rate of change of the energy is proportional to the energy \begin{equation} -\frac{1}{2\kappa} \frac{\dd\ }{\dd t} \left\{\tau,t\right\} = \frac{\ccc}{24\pi}\left\{\tau,t\right\} \, . \end{equation} So the energy decays exponentially \begin{equation} {E(t) = E(t_0) \exp\left(- A (t-t_0) \right)} \, , \end{equation} with evaporation rate \bea A \is \frac{\kappa \spc c}{12\pi} \, = \, \frac{2G\spc c}{3a} \, . \eea So the characteristic timescale associated to this evaporation process is $t_{\rm evap} = 1/A$.\footnote{An exponentially decaying profile could have been anticipated since: \begin{equation} \frac{\dd M}{\dd t} \sim - \sigma_{SB} T^2 \sim -M \, , \end{equation} where the evaporation rate is given by Stefan-Boltzmann's law in 2d, and we used the fact that $T \sim \sqrt{M}$.} The exact solution for $\tau(t)$ is of the general form\footnote{Here $\alpha = \frac{24\pi}{c}\sqrt{\frac{E_0}{\kappa}}$. One can check that indeed $\tau(0)=0$, $\tau'(0)=1$, $\tau''(0)=0$ and $\tau'''(0)=-\frac{2\mu_0}{a\tau'(0)} = -2\kappa E_0$ as required to glue this solution to $\tau(t)=t$ for $t<0$ through an infalling pulse with magnitude $E_0$ using the boundary conditions for infalling pulses. These conditions fully fix all integration constants. } \bea \label{exsolu} \tau(t) \! \is \! \frac{1}{\alpha^2} \int_0^t\!\! {\dd x\, \frac{1}{\bigl(I_1\left(\alpha \right)K_0\bigl(\alpha e^{-\frac{Ax}{2}}\bigr) + K_1\left(\alpha \right) I_0\bigl(\alpha e^{-\frac{Ax}{2}}\bigr)\spc \bigr)^2}} \, . \eea This solution has the property that $\tau'(+\infty) = 0$, implying that $\tau(+\infty) < +\infty$, thus Poincar\'e time does not flow forever in this solution.\footnote{The integrand behaves for large $x$ as $\sim 1/x^2$ which is integrable as $x\to+\infty$.} The (quasi-static) Hawking temperature of the black hole as it evaporates equals \begin{equation} \label{qHawking} T(t) = \frac{1}{\pi}\sqrt{\kappa {E(t)}} = \frac{1}{\pi}\sqrt{\kappa{E}}\exp\left(-\frac{A}{2} t\right) \, , \end{equation} decaying at half the rate. We plot the exact solutions in figure~\ref{XevapA1}. One sees that as the evaporation rate increases, the profile approaches more and more the Poincar\'e profile. In the limiting case, the black hole evaporates instantaneously and the Poincar\'e time coordinate remains intact. For extremely low evaporation rates, one approaches the static black hole profile. Intermediate rates lead to postponing the Poincar\'e time $\tau$ at which this time coordinate stops flowing ($\partial_t \tau=0$). \begin{figure} \centering \setlength{\tabcolsep}{2pt} \begin{tabular}{ccc} \includegraphics[width=4.8cm]{A01B1v2.pdf} & \includegraphics[width=4.8cm]{A1B1v2.pdf} & \includegraphics[width=4.8cm]{A5B1v2.pdf} \\ (a) & (b) & (c) \\ \end{tabular} ]]>
1$): \begin{align} \tan y &= \sqrt{\frac{1}{1 - \kappa\omega}}\tan(\sqrt{1-\kappa \omega}\, t) , \ \quad \kappa\omega < 1 \, , \\ \tan y &= \sqrt{\frac{1}{\kappa\omega-1}}\tanh(\sqrt{\kappa \omega-1}\, t), \quad \kappa\omega > 1\, . \end{align} Clearly, in the former case, no periodicity in imaginary time is generated, and this does \emph{not} represent a black hole; the infalling pulse is too weak. In the latter case, the pulse generates a black hole, by identifying $\kappa\omega - 1 = \kappa E $ where $E$ is the black hole mass. This illustrates the fact that the global frame can be seen as being lower in energy by $-\frac{1}{\kappa}$ than the Poincar\'e patch, and this is easily confirmed by computing $\bigl\langle\, \hat{T}_{tt}\spc \bigr\rangle$. The energy of the different spacetimes we discussed so far is illustrated in figure~\ref{energyscheme}. As described in~\cite{Balasubramanian-ml-1999re} for the 3d case, this can be interpreted as the global frame having lower vacuum energy than the Poincar\'e patch. Furthermore, by starting in the global patch with $E = -\frac{a}{8\pi G}$ and letting the preferred frame evolve from there, one readily proves the consistency check: \begin{equation} \bigl\langle\, \hat{T}_{tt}\spc \bigr\rangle = -\frac{1}{2\kappa}\left\{\tan y,t\right\} = -\frac{1}{2\kappa}\{\tau,t\}\,. \end{equation} \begin{figure}[t] \centering \includegraphics[width=0.46\textwidth]{energy.pdf} ]]>