The curvature on a black hole boundary

Read the full article for free* in Classical and Quantum Gravity:
On the Bartnik mass of apparent horizons
Christos Mantoulidis and Richard Schoen 2015 Class. Quantum Grav. 32 205002
arXiv:1412.0382
*until 04/11/15

Christos Mantoulidis is a graduate student in Mathematics at Stanford University.

In our latest CQG paper we study the geometry (i.e. curvature) of apparent horizons and its relationship with ADM mass.

We were motivated by the following two foundational results in the theory of black holes in asymptotically flat initial data sets (slices of spacetime) satisfying the dominant energy condition (DEC):

Apparent horizons are topologically equivalent to (one or more) two-dimensional spheres.⁽¹⁾
When the initial data set is additionally time symmetric (totally geodesic in spacetime), the apparent horizon’s total area $A$ is bounded from above by the slice’s ADM mass per $A \leq 16\pi m^2$ . This is called the Penrose inequality.⁽²⁾ Equality is only achieved on Schwarzschild data, whose apparent horizon is a single sphere with constant Gauss curvature.

One then naturally wonders:

What can we say about the metric and the curvature of the topological 2-spheres representing apparent horizons?
If the spheres are far from having constant Gauss curvature, is there necessarily a big discrepancy between their area and the ADM mass, $16\pi m^2 - A$ ?

Partially because of the Penrose inequality, we restrict to the setting of initial data sets that are time symmetric and asymptotically flat (hereafter referred to as “initial data sets”). We answer these questions simultaneously by relating them both to the study of the operator -Δ+Κ on 2-spheres (Δ = Laplace-Beltrami operator, K = Gauss curvature). The key construction is:

When -Δ+Κ is positive definite on a 2-sphere, the sphere can be realized as the apparent horizon of an initial data set satisfying the DEC, and where the Penrose inequality is as close to being an equality as we want.

(Note: equality cannot be achieved unless the 2-sphere has constant Gauss curvature.)

How restrictive is the definiteness condition on -Δ+Κ? It isn’t. Nearly all⁽³⁾ apparent horizons in initial data sets with the DEC have positive definite -Δ+Κ.

This construction already answers our first question: 2-spheres that can be apparent horizons are geometrically distinguished from those that cannot precisely by the sign of the first eigenvalue of -Δ+Κ.

It also answers our second question in a manifestly negative way: having a small gap $16\pi m^2 - A$ in the Penrose inequality does not mean our Gauss curvature is nearly constant—it can oscillate wildly and even be very negative in some regions (there are several such metrics with positive definite -Δ+Κ).

The key observation that makes this all work is the following:

The DEC on a time symmetric initial data set (a Riemannian 3-manifold) is equivalent to the geometric requirement that the 3-manifold have non-negative scalar curvature.

Likewise, the positive semi-definiteness of -Δ+Κ on a 2-sphere is equivalent to the existence of a warped product extension into a cylindrical 3-manifold with non-negative scalar curvature.

When -Δ+Κ is strictly positive definite, we can slowly “bend” the cylinder out and glue a Schwarzschild end to it. The discrepancy $16\pi m^2 - A$ is determined by the bending slope, which we can make as small as we like.

Interesting consequences

Our negative answer to the second question posed in the beginning has interesting consequences.

First, among all initial data sets extending a spherical apparent horizon with non-constant Gauss curvature (and positive definite -Δ+Κ), there is none with least ADM mass. That is so because we can get arbitrarily close to the optimal mass $\sqrt{A/16\pi}$ , but we cannot achieve it because our horizon doesn’t have constant curvature.

Note: this is relevant to Bartnik’s static minimization conjecture, which deals with the existence or lack thereof of mass minimizing extensions of 2-spheres that lie strictly outside of horizons.

Second, it rules out a conjecture due to Gibbons in the spirit of Thorne’s hoop conjecture: if an initial data set with the DEC and ADM mass $m$ encloses an apparent horizon with systole $\beta$ , then $\beta/4\pi \leq m$ . The “systole” is the shortest length required of a rubber band so that it can be slid across the 2-sphere.

To disprove this we find a 2-sphere with positive definite -Δ+Κ and $\beta/4\pi > \sqrt{A/16\pi}$ . This is enough because we know we can construct an initial data set with $m \approx \sqrt{A/16\pi}$ and thus $m < \beta/4\pi$ .

Equivalently, let’s arrange for $\beta^2/A > \pi$ . The constant $\pi$ is not an arbitrary constant: it’s the $\beta^2/A$ of a 2-sphere with constant curvature. So we need to exhibit a 2-sphere whose $\beta^2/A$ exceeds that of the round sphere. Here is one:

It’s a slightly fattened up equilateral triangle, whose systole $\beta$ is achieved on a geodesic (in red) going from a vertex to the opposite midpoint and back around the back face. It’s easy to see that $\beta^2/A \approx 2\sqrt{3} > \pi$ .

Note: a famous open problem in geometry says that these triangles are optimal (as the two faces collapse to each other) for $\beta^2/A$ ratios among all topological 2-spheres.

Further research

There are a number of questions that we need to answer to complete our understanding of apparent horizons in time symmetric initial data sets:

Can the discrepancy $16\pi m^2 - A$ be made small when the apparent horizon has two or more components? Our method only works for connected apparent horizons.
Likewise, can the discrepancy $16\pi m^2 - A$ be made small on vacuum spacetimes? Is the Gibbons conjecture true or false for vacuum spacetimes? Our constructions are non-vacuum.
What further constraints do apparent horizons in vacuum initial data sets satisfy?

Of course, lifting the time symmetry requirement would also be very interesting. Presumably, the corresponding Penrose inequality would have to be proved first though.

(1) This is a variant of Hawking’s theorem on the topology of cross-sections of event horizons.

(2) The Penrose Inequality has only been shown (to date) for time symmetric initial data sets.

(3) Actually, it is possible in theory that some only have a positive semi-definite (not definite) -Δ+Κ. This situation can be perturbed into one where -Δ+Κ is positive definite.

Richard Schoen is a Professor of Mathematics at the University of California, Irvine, and at Stanford University.

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