Thermodynamics of dynamical black holes beyond perturbation theory

Abstract

The close similarities of the three laws of black hole mechanics, discovered by Bardeen, Carter and Hawking, with the laws of thermodynamics led to the identification of a multiple of the area of the event horizon with entropy. However, developments over the past two decades have shown that this paradigm has some important limitations, especially because of the teleological nature of event horizons. After a brief review of these limitations, we will show that they can be overcome using quasi-local horizons. Specifically, the new first law applies to black holes in general relativity that can be \emph{arbitrarily far from equilibrium} and refers to \emph{finite} changes that occur due to \emph{physical processes} at the horizon. The second law is now a \emph{quantitative} statement that relates the change in the area of a dynamical horizon segment due to fluxes of energy falling into the black hole. Together, they lead one to identify black hole entropy with the area of marginally trapped surfaces in quasi-local horizons, generalizing recent {perturbative} findings that it should be identified not with the area of the event horizon but with the area of a marginally trapped surface inside it.

Figures (4)

• Figure 1: Vaidya Space-time: Collapse of a null fluid, incident from $\mathscr{I}^{-}$. To the past of the null fluid, the space-time metric is Minkowskian. The EH $\mathfrak{eh}$ forms and grows already in this flat region although there is nothing happening there. To the future of the null fluid, the space-time metric is isometric to the Schwarzschild metric. The $r=2M(v)$ surface is the spherical dynamical horizon segment that forms in the fluid region, grows in area, and joins the EH once all of the null fluid has fallen in. None of it lies in the region where the metric is flat.
• Figure 2: Left Panel: A double Vaidya null fluid collapse. Again, the event horizon $\mathfrak{eh}$ forms and grows in the flat region of space-time. The QLH (in green) lies entirely in the curved region. It starts out as a (space-like) DHS that grows in area in response to infalling matter, then settles down to a (null) IHS in the space-time region $M_1$, only to become a DHS because of the second infall and finally settles to an IHS in $M_2$, the only segment that coincides with the EH. Our thermodynamic considerations apply to all segments of the QLH. Right Panel: A DHS. The portion $\Delta \mathcal{H}$ of a space-like DHS is bounded by two MTSs $S_1$ and $S_2$. $k^a,\,\underline{k}^a$ are the two future directed null normals to MTSs $S$, with $\theta_{(k)}=0$. $\hat{\tau}^a$ is the unit normal to the DHS and $\hat{r}^a$ is the unit normal to MTSs within the DHS.
• Figure 3: Projection maps: The upper-left part of the figure depicts a QLH which has a DHS (shown in blue), bounded by MTSs $S_1$ and $S_2$, and sandwiched between an IHS to the past of $S_1$ and another to the future of $S_2$. An equilibrium state $\mathring{\mathfrak{e}}$ corresponds to time-independent fields $(\mathring{q}_{ab}, \mathring\omega_{a})$ on an IHS $\mathfrak{ih}$, constructed from its geometry. The space ${\mathcal{E}_{\mathfrak{ih}}}$ of these $\mathfrak{ih}$-states --depicted in the lower-left part of the figure-- is infinite dimensional. A non-equilibrium state $\mathfrak{n}$ is specified by the values of an infinite number of fields on a MTS $S$ of a DHS $\mathfrak{Dh}$ constructed from the initial data $(q_{ab}, k_{ab})$ on $\mathfrak{Dh}$ and the null normals $(k^a,\, \underline{k}^a)$ to its MTSs. The space $\mathcal{N}$ of non-equilibrium states $\mathfrak{n}$ much larger than ${\mathcal{E}_{\mathfrak{ih}}}$ because more fields are needed to characterize $\mathfrak{n}$. From these fields, one can construct $(R_{\mathfrak{ih}},\, J_{\mathfrak{ih}})$ on $\mathfrak{ih}$ and $R_{{}_{\mathfrak{Dh}}} [S],\,J_{{}_{\mathfrak{Dh}}} [S]$ on $\mathfrak{Dh}$. Kerr Killing horizons $\mathfrak{Kh}$ constitute a 2-dimensional space $\mathcal{E}_{{\rm kerr}}$ of states $\mathfrak{e}_{\rm kerr}$, each representing a BH in global equilibrium. Each $\mathfrak{e}_{\rm kerr}$ is characterized by a pair of numbers $(R,\, J)$. The projection maps $\mathring{\Pi}$ and $\Pi$ (respectively) send any given $\mathring{\mathfrak{e}}$ and $\mathfrak{n}$ to a global equilibrium state $\mathfrak{e}_{\rm kerr}$ such that the 2-parameters $(R,\, J)$ labeling $\mathfrak{e}_{\rm kerr}$ equal $(R_{\mathfrak{ih}}, J_{\mathfrak{ih}})$ of $\mathring{\mathfrak{e}}$ and $(R_{{}_{\mathfrak{Dh}}} [S],\, J_{{}_{\mathfrak{Dh}}} [S])$ of $\mathfrak{n}$. Thus, $\mathring{\Pi}$ and $\Pi$ ignore all the rich information contained in $\mathring{\mathfrak{e}}$ and $\mathfrak{n}$ except for these two numbers. Via $\Pi$, the 1-parameter family of non-equilibrium states defined by $\mathfrak{Dh}$ project down to a trajectory in $\mathcal{E}_{{\rm kerr}}$ along which $R$ changes monotonically.
• Figure 4: Left Panel: Oppenheimer-Snyder collapse.$\mathfrak{Dh}$ is time-like if the collapsing star is homogeneous, or nearly homogenous. Right Panel: Semi-classical phase of a spherical, evaporating BH. To investigate the issue of information loss, the incoming state has to include the collapsing matter. Therefore, a stellar collapse is unsuitable but one can consider collapse of a massless scalar field that is in a coherent state on $\mathscr{I}^{-}$. Then, in the semi-classical approximation, the DHS formed during the collapse is space-like. At the end of the collapse it is instantaneously null and then becomes time-like in response to the infalling negative energy quantum flux. The portion of space-time that lies outside the domain of applicability of the semi-classical approximation has been excised.