Evaporation of two-dimensional black holes

We present a detailed analysis of results from a new study of the quantum evaporation of Callan-Giddings-Harvey-Strominger black holes within the mean-field approximation. This semiclassical theory incorporates backreaction. Our analytical and numerical calculations show that, while some of the assumptions underlying the standard evaporation paradigm are borne out, several are not. One of the anticipated properties we confirm is that the semiclassical space-time is asymptotically flat at right future null infinity ${\mathcal{I}}_{\mathrm{R}}^{+}$ yet incomplete in the sense that null observers reach a future Cauchy horizon in finite affine time. Unexpected behavior includes that the Bondi mass traditionally used in the literature can become negative even when the area of the horizon is macroscopic; an improved Bondi mass remains positive until the end of semiclassical evaporation, yet the final value can be arbitrarily large relative to the Planck mass; and the flux of the quantum radiation at ${\mathcal{I}}_{\mathrm{R}}^{+}$ is nonthermal even when the horizon area is large compared to the Planck scale. Furthermore, if the black hole is initially macroscopic, the evaporation process exhibits remarkable universal properties. Although the literature on Callan-Giddings-Harvey-Strominger black holes is quite rich, these features had escaped previous analyses, in part because of the lack of required numerical precision and in part due to misinterpretation of certain properties and symmetries of the model. Finally, our results provide support for the full quantum scenario recently developed by Ashtekar, Taveras, and Varadarajan and also offer a number of interesting problems to the mathematical relativity and geometric analysis communities.

Figure 1

Penrose diagram of the CGHS black hole formed by the gravitational collapse of a left-moving field $f_{+}$. The physical space-time is that part of the fiducial Minkowski space which is to the past of the spacelike singularity.

Figure 2

Penrose diagram of an evaporating CGHS black hole in the mean-field approximation. Because of quantum radiation the singularity now ends in the space-time interior and does not reach ${\mathcal{I}}_{\mathrm{L}}^{+}$ or ${\mathcal{I}}_{\mathrm{R}}^{+}$ (compare with Fig. 1.) Space-time admits a generalized dynamical horizon whose area steadily decreases. It meets the singularity at its (right) end point. The physical space-time in this approximation excludes a future portion of the fiducial Minkowski space (bounded by the singularity, the last ray, and the future part of the collapsing matter).

Figure 3

The Ricci scalar $R$ for $M^{\star }=8$. Left: 2D contour plot of $R^{1/5}$ showing the increase in $R$ as the singularity (dark vertical region near the middle) is approached and the asymptotically flat region ($R\to 0$) near ${\mathcal{I}}_R^{+}$ ($z^{+}\to \infty$). Right: $R^{1/5}$ as a function of $z^{+}$ on the lines $z^{-}-z_{\sin \mathrm{g}}^{-}=-{10}^{-5},-{10}^{-6},-{10}^{-8}$ (marked on the left panel as horizontal lines), showing a double peak, indicating the divergent behavior of ${\partial }_{+}{\partial }_{-}\Phi$ at the singularity. The fact that the peak is narrow rules out a strong thunderbolt singularity. Note that the dark color at the region of the singularity is due to the high density of contour lines and not directly due to negative values of $R$. While naive numerical calculation of $R$ very close to ${\mathcal{I}}_R^{+}$ does not yield reliable results due to catastrophic cancellation, it is already very small in the high $z^{+}$ values shown here, and the trend towards 0 is clear.

Figure 4

Left: Plot of ${log}_{10}(d{y}^{-}/d{z}^{-})$ vs ${log}_{10}\Delta$ for $M^{\star }=8$, $\overline{N}=1$, where $\Delta =(z_{\mathrm{sing}}^{-}-z^{-})$. Right: Slope of the curve on the left. If locally the function on the left behaves as $\sim (\kappa \Delta {)}^{-p}$, the curve on the right shows $-p$. In the distant past (rightmost region in both plots), $y^{-}$ tends to $z^{-}$. The intermediate region is similar to that in the classical solution where $(d{y}^{-}/d{z}^{-})\sim (\kappa \Delta {)}^{-1}$. As the last ray is further approached (leftmost region), we see that $(d{y}^{-}/d{z}^{-})$ increases much slower than $(\kappa \Delta {)}^{-1}$, leading to a finite value for $y^{-}$ at the last ray.

Figure 5

The ATV Bondi mass $M_{\mathrm{Bondi}}^{\mathrm{ATV}}$ (solid lines) and the traditional Bondi mass $M_{\mathrm{Bondi}}^{\mathrm{Trad}}$ (dashed lines) are plotted against $z^{-}-z_{\mathrm{sing}}^{-}$ (left) and the horizon area (right). This simulation corresponds to $M_{\mathrm{ADM}}=360$, $N=720$ (so $M^{\star }=12$). For high values of $N$, both formulas give a large nonzero Bondi mass at the last ray. $M_{\mathrm{Bondi}}^{\mathrm{Trad}}$ becomes negative when the area is still macroscopic. On the other hand, $M_{\mathrm{Bondi}}^{\mathrm{ATV}}$ remains strictly positive all the way to the last ray, where the GDH shrinks to zero area.

Figure 6

The value of $m^{\star }$ (i.e., $M_{\mathrm{Bondi}}^{\mathrm{ATV}}/\overline{N}$ at the last ray) is plotted against $M^{\star }$ (which equals $M_{\mathrm{ADM}}/\overline{N}$) for $M^{\star }\ge 1$. For macroscopic $M^{\star }$ (actually, already for $M^{\star }\gtrsim 4$) $m^{\star }$ has a universal value, approximately $0.864$.

Figure 7

Left: The affine parameter $y^{-}$ [defined in Eq. (2.12)] of the physical metric $g$ is plotted against the rescaled area ${\mathbf{a}}^{\star }≔({\mathbf{a}}_{\mathrm{GDH}}/\overline{N})$ of the generalized dynamical horizon [given by $(\Phi /\overline{N}-2)$] at the horizon for values of $M^{\star }$ from 4 to 14. Even though the curves are very similar in shape, they do not coincide. Right: Once the shifting freedom in $y^{-}$ is utilized, we see that a properly shifted version $y_{\mathrm{sh}}^{-}$ is universal with respect to ${\mathbf{a}}^{\star }$ for all macroscopic $M^{\star }$ values. $y_{\mathrm{sh}}^{-}$ can be used as a universal coordinate similar to the horizon area.

Figure 8

$F^{\star }=(F^{\mathrm{ATV}}/\overline{N})$ is plotted against the horizon area ${\mathbf{a}}^{\star }≔({\mathbf{a}}_{\mathrm{GDH}}/\overline{N})$ (left) and $y_{\mathrm{sh}}^{-}$ (right) for values of $M^{\star }$ from 4 to 14. For all $M^{\star }$ values, $F^{\star }$ starts at the value of 0 at the distant past ($\kappa y_{\mathrm{sh}}^{-}\ll -1$) and then joins a universal curve of $F^{\star }$. Note that once the GDH is formed (the rightmost beginning of each curve on the left plot), $F^{\star }$ is already slightly larger in magnitude than the Hawking or thermal value 0.5 and it increases steadily as one approaches the last ray (i.e., as ${\mathbf{a}}_{\mathrm{GDH}}$ and $y_{\mathrm{sh}}^{-}$ approach 0).

Figure 9

$M_{\mathrm{Bondi}}^{\star }=(M_{\mathrm{Bondi}}^{\mathrm{ATV}}/\overline{N})$ is plotted against the horizon area ${\mathbf{a}}^{\star }≔({\mathbf{a}}_{\mathrm{GDH}}/\overline{N})$ (left) and $y_{\mathrm{sh}}^{-}$ (right) for values of $M^{\star }$ from 4 to 14. For all these macroscopic $M^{\star }$, $M_{\mathrm{Bondi}}^{\star }$ starts at the value of $M_{\mathrm{ADM}}$ in the distant past ($\kappa y_{\mathrm{sh}}^{-}\ll -1$) and then joins a universal curve of $M_{\mathrm{Bondi}}^{\star }$. When the dynamical horizon first forms $M_{\mathrm{Bondi}}^{\star }$ is quite close to its initial value of $M^{\star }$. (This is difficult to see in the left plot where all the curves crowd.) This means that almost all of the evaporation occurs after the formation of the dynamical horizon.

Figure 10

The value of $m^{\star }$ (i.e., $M_{\mathrm{Bondi}}^{\mathrm{ATV}}/\overline{N}$ at the last ray) plotted against $M^{\star }$ (which equals $M_{\mathrm{ADM}}/\overline{N}$) for $M^{\star }\ge 1$. In addition to points corresponding to shell collapse ($w=0$) the plot now includes points with more general profiles with $w=0.25,0.5,1$. The universal value $m^{\star }\approx 0.864$ persists for $M^{\star }\ge 4$.

Figure 11

$F^{*}$ (left) and $M_{\mathrm{Bondi}}^{*}$ (right) plotted against $y_{\mathrm{sh}}^{-}$, for various incoming matter profiles ($w$ and $M_{\mathrm{ADM}}$ values), including several shell ($w=0$) cases. The time when the dynamical horizon first forms is marked on each flux curve (which is later for larger $w$). All the curves with the same $M_{\mathrm{ADM}}$ ($6$ in this example) are on top of each other and cannot be distinguished by the eye, showing that they have the same universal behavior throughout the evolution, including the early times. More generally all the asymptotic physical quantities depend only on the combination $M_{\mathrm{ADM}}$ of the profile parameters $M$ and $w$ as long as $\kappa w$ is small compared to the initial area of the GDH.

Figure 12

Plot of $d{y}^{-}/dz$ against the separation in $z^{-}$ from the singularity for various values of $M$ and $w$ with a fixed ADM mass $M^{\star }=6$. The functional dependence $y^{-}(z^{-})$ determines the physics of the outgoing quantum state completely [7, 8]. Coincidence of these curves in the mean-field theory suggests that the outgoing quantum state is likely to be universal within the class of initial data with the same ADM mass, so long as the collapse is prompt.