Spacetime Penrose inequality

An initial data set $(M,g,k)$ consists of a smooth $3$-manifold $M$ with compact boundary $\partial M$, a complete Riemannian metric $g$ and a symmetric $2$-tensor $k$. Its energy and momentum densities are

$$\mu=\tfrac12\bigl(R_g+(\operatorname{tr}_gk)^2-|k|_g^2\bigr),\qquad J=\operatorname{div}_g\bigl(k-(\operatorname{tr}_gk)\,g\bigr),$$

and it satisfies the dominant energy condition if $\mu\ge|J|_g$ on $M$.

The data are asymptotically flat if, outside a compact set, $M$ is diffeomorphic to $\mathbb R^3$ minus a ball and, for some $q>\tfrac12$,

$$g_{ij}-\delta_{ij}=O_2(|x|^{-q}),\qquad k_{ij}=O_1(|x|^{-q-1}),\qquad \mu,\ J\in L^1(M).$$

The ADM energy and momentum are

$$E=\lim_{r\to\infty}\frac1{16\pi}\int_{|x|=r}\bigl(\partial_jg_{ij}-\partial_ig_{jj}\bigr)\nu^i\,dA,\qquad P_i=\lim_{r\to\infty}\frac1{8\pi}\int_{|x|=r}\bigl(k_{ij}-(\operatorname{tr}_gk)\,g_{ij}\bigr)\nu^j\,dA ,$$

and the ADM mass is $m=\sqrt{E^2-|P|^2}$, which is well defined because $E\ge|P|$ under the dominant energy condition.

For a closed surface $\Sigma\subset M$ with unit normal $\nu$ pointing toward the asymptotic end and mean curvature $H$, the null expansions are $\theta_\pm=H\pm\operatorname{tr}_\Sigma k$. A component of $\Sigma$ is a marginally outer trapped surface if $\theta_+=0$ and a marginally inner trapped surface if $\theta_-=0$. Assume that every component of $\partial M$ is of one of these two kinds, and that $\partial M$ is outermost: no closed surface in $M\setminus\partial M$ that encloses $\partial M$ has all of its components satisfying $\theta_+\le0$ or $\theta_-\le0$. Let

$$A_{\min}=\inf\bigl\{|\Sigma|_g:\ \Sigma\subset M\text{ a closed surface enclosing }\partial M\bigr\}.$$

Conjecture (spacetime Penrose inequality). Every asymptotically flat initial data set as above satisfies

$$m\ \ge\ \sqrt{\frac{A_{\min}}{16\pi}} ,$$

and equality holds only if $(M,g,k)$ arises as a spacelike hypersurface, with induced metric $g$ and second fundamental form $k$, in a Schwarzschild spacetime of mass $m$.

The problem is to prove this inequality with its rigidity statement, or to construct data satisfying all the hypotheses that violate it. The minimal enclosing area cannot be replaced by the area $|\partial M|$ of the horizon itself, because that version has counterexamples [6]. [1, 7, 15]

Background

Penrose derived the inequality in 1973 from a heuristic argument [1]. A spacetime that collapses to a black hole should settle down to a Kerr spacetime. The horizon area cannot decrease by Hawking's area theorem, and the Bondi mass cannot increase. Comparing the initial data with the final Kerr state yields $m\ge\sqrt{A/16\pi}$. A counterexample would therefore cast doubt on weak cosmic censorship, which the heuristic assumes. A proof would sharpen the positive mass theorem [5], which is the case $A_{\min}=0$.

In the time-symmetric case $k=0$, the dominant energy condition becomes $R_g\ge0$. Apparent horizons are then minimal surfaces, and an outermost one minimizes area among enclosing surfaces, so $A_{\min}=|\partial M|$. This is the Riemannian Penrose inequality. For general $k$ the Jang equation and generalized Jang equations relate the spacetime problem to the Riemannian one, but the reduction is not known to preserve the sharp constant [7]. Bray and Khuri proposed a variant using generalized apparent horizons, and Carrasco and Mars found a counterexample to it [8].