Geometric inequalities for black holes

Geometric inequalities have a long and interesting history. Perhaps the best-known example is the “isoperimetric inequality”. It states that the area $A$ of any domain in the Euclidean plane is related to its circumference $C$ by

\begin{equation}
\sqrt{4\pi A} \le C \ .
\end{equation}

Equality holds precisely if the domain is a disk. A similar inequality relating the volume of a domain $\Omega$ in Euclidean space $\mathbb{R}^n$ to its surface (hyper)area holds in higher dimensions, and the higher dimensional domains where the inequality is saturated are in this case balls. The 3-dimensional case already features in the famous story of Dido and the oxhide, an account of which is  given in the “Anaeid” by the Roman poet Virgil.

A geometric inequality of a somewhat similar nature, which applies to convex domains $\Omega$ in $\mathbb{R}^n$ with smooth boundary $\partial \Omega = \Sigma$, is the “Minkowski-inequality”. It states that

 

 

 

 

 

 

\begin{equation}
\int_\Sigma H \ge (n-1) \omega_{n-1} \left( \frac{|\Sigma|}{\omega_{n-1}} \right)^{\frac{n-2}{n-1}},
\end{equation}

where $H$ is the extrinsic curvature of the surface $\Sigma$, where $\omega_{n-1}$ is the hyperarea of the $(n-1)$-dimensional sphere, and where $|\Sigma|$ is the hyperarea (volume) of $\Sigma$. Also the Minkowski inequality has the property that it is saturated for a ball, i.e., the most symmetric among all convex bodies.

In the case of  black holes (in 3+1 dimensions), Penrose has conjectured that there holds an inequality relating the the ADM mass $m$ to the area of any outermost, marginally trapped surface $B$. A marginally (outer) trapped surface is intuitively a cross section of a null surface $\mathscr{H}$ whose “outward” directed null-rays tangent to $n$ have zero expansion, and whose inward directed null rays tangent to $l$ have negative expansion, i.e. contract, see the following figure. The concept of a marginally trapped surface is therefore closely related to that of an event horizon, but it is more general, and by contrast to the latter, is locally defined.

\begin{tikzpicture}[scale=.5, transform shape]

[+preamble]

\usetikzlibrary{decorations.pathreplacing}
\usetikzlibrary{calc}
\usetikzlibrary{shapes.geometric}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{decorations.pathmorphing}
[/preamble]
\node (rectnw) at (-6,2.3) {};
\node (rectne) at (8,2.3) {};
\draw[thick] (rectnw.center) — (rectne.center);
\node (coneupperleft) at (-3.5,8) {};
\node (conemiddleleft) at (-3,3) {};
\node (ellipsel1) at (-2, 1.5){};
\node (coneintersL) at (-.8,.7) {};
\node (conebase) at (0,-.2) {};
\node (coneintersR) at (.8,.7) {};
\node (ellipser1) at (2, 1.5){};
\node (conemiddleright) at (3,3) {};
\node (coneupperright) at (3.5,8) {};
\draw[opacity=0]
(coneupperleft.center) [rounded corners=15pt] — node[pos=.15](n3end){} node[pos=.4](ellipsel3){} node[pos=.65](n2end){} node[pos=.9](ellipsel2){}
(conemiddleleft.center) [rounded corners=15pt]  — (ellipsel1.south)[rounded corners=15pt] —
(coneintersL.center)[rounded corners=1pt]  —   (conebase.center) [rounded corners=15pt] — (coneintersR.center)[rounded corners=15pt]  — (ellipser1.south)[rounded corners=15pt]
—   node[pos=.5](HArrowtip){}
(conemiddleright.center)[rounded corners=15pt]  — node[pos=.1](ellipser2){} node[pos=.6](ellipser3){}
(coneupperright.center);
\shadedraw[shading=axis, shading angle=90, fill opacity=.8] (ellipsel1)[rounded corners=15pt] — (coneintersL.center)[rounded corners=1pt] — (conebase.center)[rounded corners=15pt] — (coneintersR.center) — (ellipser1);

\shadedraw[left color=blue, right color=white, shading=axis, shading angle=90, fill opacity=.7] let \p1 = (coneintersL), \n1={veclen(\x1,0)} in (rectne.center) — (5.5,\y1) node (rectse){} — (-8.5,\y1) node (rectsw){} — (rectnw.center) ;

\shadedraw[shading=axis, shading angle=90, fill opacity=.8] let
\p1 = (ellipsel1),
\n1 = {veclen(\x1,0)}
in  (coneupperleft.center)[rounded corners=15pt]   —
(conemiddleleft.center)[rounded corners=0pt]    — (\x1,\y1) arc [x radius=\n1, y radius=.4cm, start angle=180, end angle=360][rounded corners=15pt]  —  (conemiddleright.center)[rounded corners=15pt] — (coneupperright.center);

\draw[fill=white] (0,8) ellipse [x radius = 3.5, y radius= .5];

\draw[draw opacity = .4] (ellipsel1.center) to [ controls=++(90:.5) and ++(90:.5)]  (ellipser1.center);
\draw (ellipsel1.center) to [ controls=++(270:.5) and ++(270:.5)]  (ellipser1.center);

\draw[draw opacity = .4] (ellipsel2.center) to [ controls=++(90:.5) and ++(90:.5)]  (ellipser2.center);
\draw (ellipsel2.center) to [ controls=++(270:.5) and ++(270:.5)]  (ellipser2.center);

\draw[draw opacity = .4] (ellipsel3.center) to [ controls=++(90:.5) and ++(90:.5)]  (ellipser3.center);
\draw (ellipsel3.center) to [ controls=++(270:.5) and ++(270:.5)]  node[pos=.7, below](arrowBtip){} (ellipser3.center);
\node (cone1) at (-7.3,2) {};
\draw (cone1.center) — ++(.5,-.5) node (cone1b){} — ++(.5,.5);
\draw (cone1b) [yshift=.5cm] ellipse [ x radius= .5, y radius=.1];
\draw[->=latex,very thick] (cone1b.center) — ++(0,1.5) node[right]{time} ;
\node (cone2) at (-7.3,4.5) {};
\draw (cone2.center) — ++(.5,-.5) node (cone2b){} — ++(.5,.5);
\draw (cone2b) [yshift=.5cm] ellipse [ x radius= .5, y radius=.1];
\draw[->=latex,very thick]  (cone2b.center) — ++(0,1.5) node[right]{time} ;
\node (cone3) at (-7.3,7) {};
\draw (cone3.center) — ++(.5,-.5) node (cone3b){} — ++(.5,.5);
\draw (cone3b) [yshift=.5cm] ellipse [ x radius= .5, y radius=.1];
\draw[->=latex,very thick]  (cone3b.center) — ++(0,1.5) node[right]{time} ;
\node (cone4) at (-5,2) {};
\draw[rotate=355] (cone4.center) — ++(.5,-.5) node (cone4b){} — ++(.5,.5);
\draw[rotate=355] (cone4b) [yshift=.5cm] ellipse [ x radius= .5, y radius=.1];
\draw[rotate=355,->=latex,very thick] (cone4b.center) — ++(0,1.5);
\node (cone5) at (-5,4.5) {};
\draw[rotate=350] (cone5.center) — ++(.5,-.5) node (cone5b){} — ++(.5,.5);
\draw[rotate=350] (cone5b) [yshift=.5cm] ellipse [ x radius= .5, y radius=.1];
\draw[rotate=350,->=latex,very thick] (cone5b.center) — ++(0,1.5);
\node (cone6) at (-5,7) {};
\draw[rotate=350] (cone6.center) — ++(.5,-.5) node (cone6b){} — ++(.5,.5);
\draw[rotate=350] (cone6b) [yshift=.5cm] ellipse [ x radius= .5, y radius=.1];
\draw[rotate=350,->=latex,very thick] (cone6b.center) — ++(0,1.5);
\draw[->=latex, very thick] (ellipsel1.center) — node[pos=.7] (n1){} ++(130:1cm) node[above] {$n$};
\draw[rotate=350] (n1) arc [x radius=.6cm, y radius=.1cm, start angle=180, end angle=310] ;
\draw[rotate=350] (n1) arc [x radius=.55cm, y radius=.1cm, start angle=160, end angle=10] ;
\draw[->=latex, very thick] (ellipsel1.center) — ++(30:1cm) node[right]{$-l$};
\draw[->=latex,very thick] (ellipsel1.center) —  ++(77:1.2cm);
\draw[->=latex, very thick] (ellipsel2.center) — node[pos=.6] (n2){} (n2end) node[left] {$n$};
\draw[->=latex, very thick] (ellipsel2.center) —  ++(5:1cm) node[right]{$-l$};
\draw[rotate=325] (n2) arc [x radius=.6cm, y radius=.1cm, start angle=180, end angle=300] ;
\draw[rotate=320] (n2) arc [x radius=.5cm, y radius=.1cm, start angle=180, end angle=0] ;
\draw[->=latex,very thick] (ellipsel2.center) —  ++(50:1.2cm);
\draw[->=latex, very thick] (ellipsel3.center) — node[pos=.6] (n3){} (n3end) node[left] {$n$};
\draw[->=latex, very thick] (ellipsel3.center) — ++(3:1cm) node[right]{$-l$};
\draw[rotate=325] (n3) arc [x radius=.6cm, y radius=.1cm, start angle=180, end angle=300] ;
\draw[rotate=320] (n3) arc [x radius=.5cm, y radius=.1cm, start angle=180, end angle=0] ;
\draw[->=latex,very thick] (ellipsel3.center) —  ++(50:1.2cm);
\node[below, scale=1.5] (sigma) at (rectse) {$\Sigma$};
\draw[latex-, very thick] (arrowBtip) — ++(2, -.2) node[right]{$B=\{r=0, u=0\}$};
\draw[latex-, very thick] (HArrowtip.center) — ++(1.5,1) node[right]{$\mathscr{H}=\{r=0\}$};
\node[below, scale=1.5] at (sigma.south) {$n=\frac{\partial}{\partial u}, l=\frac{\partial}{\partial r}$};
\end{tikzpicture}

The Penrose inequality is

\begin{equation}
\sqrt{\frac{|B|}{16 \pi}} \le m \ ,
\end{equation}

and is saturated for the Schwarzschild black hole spacetime, i.e. the most symmetric among all 4-dimensional black holes. It is physically important due to its relationship with the famous “cosmic censorship” hypothesis. The idea is to consider the time evolution of a black hole spacetime that settles down to a Schwarzschild solution, or more generally a Kerr solution. The (Bondi-) mass of such an evolving spacetime is shown to decrease with time, whereas the area of horizon cross section is  shown to increase with time. For a Schwarzschild- or Kerr metric, the Penrose inequality is manifestly true, so it therefore better had to be true in the past. Thus, since marginally trapped surfaces are in the interior of black holes, it must hold for $B$. This is of course not a proof, since the underlying hypothesis that the spacetime actually settles down to Schwarzschild or Kerr is not proved yet. In fact, a proof of this “cosmic censorship” hypothesis is probably be even harder than that of the Penrose inequality. For this reason, the latter is generally seen as an important test of cosmic censorship. By contrast with the latter, it can presumably be proven by just considering the initial data of the spacetime on a Cauchy surface intersecting $\mathscr{H}$ in $B$. Actually, it has already been shown to hold for the restricted class of “time-symmetric” initial data. (For initial data that are essentially embedded hypersurfaces in $\mathbb{R}^4$, the Penrose inequality is in fact related to the Minkowski inequality.)

A complementary inequality, which was shown to hold under the assumption of axi-symmetry, has been established by Dain. It states that if $J$ is the angular momentum (which can be defined “quasi-locally” on $B$), then the area $|B|$ of $B$ has the upper bound

\begin{equation}
|B| \ge 8\pi |J| \ ,
\end{equation}

It is called an “area-angular momentum (A-J) inequality”. The inequality is interesting because, like (presumably) the Penrose inequality, it holds for dynamical (time dependent) spacetimes, for which few statements of such a general nature are known. Also, like in the case of the isoperimetric- and Penrose inequality, the case of equality sign $=$ is again shown to correspond to a very special kind of geometry, which in this case is the so-called “extremal Kerr-throat”. It can be viewed as a kind of blow up of the geometry of a maximally spinning Kerr black hole near its horizon, and is therefore sometimes called a “near horizon geometry”. This near horizon geometry is a rather interesting, and highly symmetric, object in its own right. It has an isometry group of $\mathbb{R} \times SL(2,\mathbb{R})$, which contains the symmetry group of 2-dimensional AdS-space. (This relationship with AdS-space is also the geometrical basis for attempts to apply the AdS/CFT correspondence to the understanding of the quantum properties of black hole horizons.)

We have recently been able to establish a higher dimensional version of the A-J inequality, which based on an earlier classification of certain higher dimensional near horizon geometries. The theorem is proved for, possibily time-dependent, spacetimes that are “axi-symmetric” in a generalized sense, the assumption being that the isometry group is, or contains, $U(1)^{n-3}$, where $n$ is the dimension of spacetime. The inequality corresponding to Dain’s inequality in $n \ge 4$ is

\begin{equation}
|B | \ge 8\pi \sqrt{|J_+ J_-|}
\end{equation}

where $J_{\pm}$ are distinguished components associated with the angular momentum invariant arising from the symmetry group $U(1)^{n-3}$. In $n=4$ dimensions, we have $J_\pm = J$, so our inequality reduces to that of Dain.The details how $J_\pm$ are defined depend on the topology of $B$. In fact, it turns out in our case $B \cong S^2 \times T^{n-4}, S^3 \times T^{n-5},$ or $\cong L(p,q) \times T^{n-5}$.

Some references are: