### The “Hoop Conjecture” of Kip Thorne and Spherically Symmetric Space-times

Abstract. (This being a rather long post, I feel the need to write one.) In the post I first gather some miscellaneous thoughts on what the hoop conjecture is and why it is difficult to prove in general. After this motivation, I show also how the statement becomes much easier to state and prove in spherical symmetry: the entire argument collapses to an exercise in ordinary differential equations. In particular, I demonstrate a theorem that is analogous, yet slightly different, from a recent result of Markus Khuri, using much simpler machinery.

The Hoop conjecture is a proposed criterion for when a black-hole will form under gravitational collapse. Kip Thorne, in 1972 [see Thorne, Nonspherical Gravitational Collapse: a Short Review in Magic without Magic] made the conjecture that (I paraphrase here)

Horizons form when and only when a mass $M$ gets compacted into a region whose circumference $C$ in EVERY direction is bounded by $C \lesssim M$.

This conjecture, now widely under the name of “Hoop conjecture”, is deliberately vague. (This seemed to have been the trend in physics, especially in general relativity. Conjectures are often stated in such a way that half the effort spent in proving said conjectures are used to find the correct formulation of the statement itself.) Especially I wish to point out that

1. The notion of horizons is not defined. Classically a black hole is defined to be the interior of the event horizon. But as is well known, the event horizon is a non-local object: we need a global causal picture to determine where the event horizon lies, so it seems rather unlikely that the formation of the event horizon can depend on purely local information. A perhaps more natural formulation is in terms of apparent horizons or trapped surfaces.
2. The notion of a mass is not clear. If we take the very restrictive model where we consider some sort of dust/fluid model for the matter, and take the mass to be the rest mass of the matter field, we immediately run into problems: if we allow other forms of interaction, such as electromagnetism, which can provide repulsive forces (say assume the dust particles are all statically charged positively), then it is easy to imagine a situation where a sufficiently large mass is restricted to a compact spatial region, but the electric forces are strong enough to prevent the collapse into a black hole. Such an example was provided by Ponce de Leon. So one may consider a better notion of mass to be the the “energy” term in the stress-energy-momentum tensor. This, however, has also its problems
1. It is well known that black holes exist in pure vacuum, without any matter present. This is represented by the Schwarzschild solution, which is later generalized to the Kerr-Newman family of solutions. So a measure of mass purely by stress-energy-momentum tensor cannot satisfy the “only when” condition in the hoop conjecture. Furthermore, such solutions cannot be ruled out even if we postulate that the Hoop conjecture is “generic” (in a similar sense to the cosmic censorship conjectures), in view of the recent work of Demetrios Christodoulou and the improvements by Sergiu Klainerman and Igor Rodnianski, which showed that black holes can form “dynamically” from very generic data via the focusing of gravitational waves without the presence of matter.
2. Even ignoring the problem with the vacuum solutions, the notion of mass-energy will depend on the choice of a preferred time-like vector field. This is already evident in special relativity, where the mass-energy of a particle is dependent on the reference frame in which we make the measurement, and that only the energy-momentum 4-vector is a Lorentz invariant. The problem sort of gets worse in general relativity.

One proposed way out of this conundrum, is via the study of quasi-local masses. The pioneering work of Bondi, Sachs, and many others in the 60s through 80s lead to the understanding that gravitational waves also carry energy. But it is also well known in general relativity that because of general covariance, there cannot be a purely local formulation of “gravitational potential energy”. A quasi-local mass, then, is an attempt to define a total energy confined in a region, including both matter contributions and an approximation of the gravitational potential energy. To date many quasi-local masses have been proposed, and the applicability of some of them in terms of the Hoop conjecture has recently been examined. And now we run into a chicken-and-egg problem: a quasi-local mass is, by its own very nature, not a natural construct. So we can define it as an object that satisfies certain properties (sort of a definition in the Grothendieck sense). The question then is: should we add, as an axiom of definition, that a tenable quasi-local mass is one that satisfies the Hoop conjecture (and then, how can we show that such a quasi-local mass exists?); or should we re-formulate the Hoop conjecture as dependent on the quasi-local mass chosen (and then, which quasi-local mass will work? and will some work better than others?)

3. The notion of circumference is not clear, for several reasons.
1. If we were to speak of a region in Euclidean 3-space, then the “hoop” picture can be used: the circumference of a region is the circumference of the smallest hoop which can be passed about the region in any direction. More precisely: let $\omega$ be any arbitrary unit vector in $\mathbb{R}^3$. Given a region $U\subset \mathbb{R}^3$, we can take its projection in the direction $\omega$:

$\displaystyle \mathbb{R}^2 \cong \omega^\perp \supset P_\omega U := \{ \eta \in \omega^\perp | \exists c\in \mathbb{R}, c\omega + \eta \in U\}$

In words, the projection $P_\omega U$ is the shadow cast by the region $U$ when a light is shone from the direction of $\omega$. The circumference $C(\omega,U)$ of $P_\omega U$ can be determined as the circumference of the smallest circle that contains all of $P_\omega U$. We can then take the circumference $C(U)$, which fits in any direction, as the largest of the $C(\omega,U)$ over all possible directions $\omega$. This definition, however, makes heavy use of the vector space structure of Euclidean space (an affine structure that is not available on the general manifolds considered in relativity) as well as the fact that circles are well-defined in the Euclidean plane. (Implicitly and morally, Bernstein’s Theorem, which says that the only minimal surfaces [and hence the only totally geodesic surfaces] in Euclidean 3-space are the Euclidean planes, is also used in the definition.)

2. One may propose that instead of using a perfectly round circle (an object which is hard to come by in general geometries), we slide a measuring tape around the shape we are interested in. After all, in four-dimensional general relativity, the topology theorem of Stephen Hawking [also see Greg Galloway and Rick Schoen’s generalization] tells us that the apparent horizon (should we choose to use it as the horizon in the conjecture) must be a topological sphere. The immediate problem with this is that there is no preferred way to wrap a measuring tape around the horizon. One can imagine a situation in which the tape is wrapped in a complicated shape in a very inefficient manner. One proposed method by Gary Gibbons is to study the Birkhoff invariant of a surface (see the cited paper for a definition of what the invariant is). The problem with the Birkhoff invariant is that it rids us of the notion of every direction. Take a long skinny rod for example. An intuitive understanding of a hoop fitting around the rod in every direction requires a circle that has diameter as long as the length of the rod. The Birkhoff invariant, however, will give a much smaller number that corresponds to the cross-sectional diameter of the rod: so a hoop with a circumference given by the Birkhoff invariant can only pass through the hoop lengthwise, not sideways.
3. Another possible problem is the question of how to measure the circumference of a hoop. If we treat the statement of the conjecture as a gedanken experiment, we can actually try to physically pass a hoop around an object. But are we to measure the circumference of the hoop in the rest frame of the object? or in the frame of the hoop? Just by special relativistic arguments the two maybe very different.

In relation to this difficulty, there is an interesting recent proposal by Senovilla which seeks to re-formulate the Hoop conjecture in terms of trapped hoops rather than trapped surfaces.

4. The last bit of uncertainty relates to the symbol $\lesssim$ in the statement of the conjecture. The symbol suggests an uncertainty about the constant of proportionality in the inequality. Does the constant depend on the model of physical matter? Is the constant the same for the “when” and the “only when” part of the statement?

In spite of these apparent difficulties, certain progress has been made. Thorne made his original conjecture after some preliminary computations shows that making a long-skinny rod skinnier does not cause a black hole, whereas making it shorter does. But the approximate computations used is no proof in itself. The most success toward the conjecture had been found in the spherically symmetric case, in particular the series of works of Bizon, Malec, and Murchadha that culminated in their 1990 report.

There are also many works by mathematicians — especially those studying Riemannian geometry — on this subject. In the general, non-spherically symmetric case, among the most well-known is the result of Rick Schoen and S-T Yau which bounds the square of the “smallest radius” of a region from below by the inverse of the minimum of the energy density in the region. (Notice that if we take a ball, the smallest radius is the actual radius. So the energy contained in the ball is at least as much as the cube of the radius $R$ times the minimum energy density $\mu$: $M \sim \mu R^3$. Noting that $C \sim R$, the inequality $C \lesssim M$ is morally the same as $R \lesssim \mu R^3$ or $1 / \mu \lesssim R^2$, which is what Schoen and Yau proved.)

Recently there is an effort to approach the version of the problem in Riemannian geometry using techniques from the study of the Bartnik extension problem, the (modified) Jang equation, and the Riemannian Penrose inequality. The goal is to improve the result of Schoen and Yau. A model version of such a proof has recently been published by Markus Khuri, which unfortunately can only deal with the spherically symmetric case at the moment. (I’m not sure how easy it is to extend the approach to the non-symmetric case; in the paper Khuri asserts that the only real missing ingredient is a suitable existence theorem for a modified Jang equation in general symmetry, so I cannot decide whether this approach is actually viable or whether the complications are just swept under the rug of the “suitable existence theorem”.) (The one caveat that I can think of, however, is the problem of focusing gravitational waves. The measure of energy content in Khuri’s result [Theorem 1 of the linked paper] is via the matter energy-momentum density. In Christodoulou’s dynamical black hole formation situation, the matter energy-momentum density is identically 0. So certain additional contributions must factor into a suitable inequality in the non-spherical case. It remains to be seen what such terms should be.)

To finish this post, I will sketch how using a space-time approach (rather than a Riemannian geometry approach), a Schoen-Yau style radius–energy-density version of the hoop conjecture can be proven in spherical symmetry. The conclusion reached will be similar to the main result of Khuri’s, but sufficiently different that they are not directly comparable.

A few remarks before I state the theorem and its proof: the hoop conjecture is a lot more approachable in spherical symmetry due to various simplifications resulting from the symmetry reductions. We will take the approach that the horizon to consider is the apparent horizon. First, the definition of mass is no-longer indistinct: a natural mass to take in spherical symmetry is the Hawking mass (a.k.a. the Misner-Sharp mass). It approaches the ADM mass at spatial infinity, and it approaches the Bondi mass at future null infinity. It is “positive” in the sense that it cannot become negative from evolution of data with positive mass. And it has very nice monotonicity properties provided that the region of space-time under consideration does not contain a white-hole. (The monotonicity properties are even nicer if you also assume the region does not contain a black-hole… but that is what we are trying to address, so we’d be begging the question if we assume it.) Secondly, the notion of a circumference is also natural: we can just take $2\pi$ times the area-radius of the orbits of spherical symmetry. The region we wish to consider is naturally the “interior” to some symmetry orbit. Thirdly, by Birkhoff’s theorem, the only vacuum, spherically symmetric solutions in general relativity are Minkowski space and the Schwarzschild solution. So in a sense we can safely rule out the pure gravitational-waves problem. With these preferred objects, the statement of the conjecture becomes more tractable.

Well then: here is the theorem. (The result of this theorem, or at least the computations leading up to it, is certainly well-known to at least Christodoulou and Mihalis Dafermos. Knowledge of it likely dates to even earlier. I will not make an effort to establish historical primacy.)

Theorem A
Let $(M,g, \Phi)$ be a spherically-symmetric space-time with matter fields $\Phi$. Assume Einstein’s equation $R_{ab} - \frac12 R g_{ab} = 8\pi T_{ab}$ is satisfied. Let $\Sigma$ be a space-like hypersurface in $M$ diffeomorphic to $S^3$ with two disjoint balls removed. Assume $\Sigma$ is invariant under the spherical symmetry, and that its boundary is the disjoint union of two spherical orbits $S_i, S_f$, with $r(S_i) < r(S_f)$ where $r$ is the area radius function. And lastly assume that $\Sigma$ contains no trapped surfaces. Then the following inequality must be satisfied:
$\displaystyle \frac{ m(S_f) - m(S_i) }{ \frac{4\pi}{3} (r(S_f)^3 - r(S_i)^3)} > \inf_{s\in\Sigma}\inf_{v\in T_s\Sigma, g(v,v) = 1} T(\tau,\tau + v)$
where $m$ is the Hawking mass function mapping two surfaces in $M$ to the real numbers, and $\tau$ is a unit-normal vector field to $\Sigma$.

Remark B
A quick remark on the content of the theorem. The left hand side is purely geometric: it can be seen as an average Hawking mass density (the denominator is roughly the volume contained in the three-dimensional surface $\Sigma$). The right hand side is physical. The first infimum, taken over all directions on $\Sigma$, can be seen as establishing at a point the smallest mass density as seen by all possible observers [basically the one who is co-moving with the matter]; this sort of “normalizes” the mass to be independent of local Lorentz transformations.

Before I present the proof, it will be necessary to first define the terms already used in the theorem. Below I will give a quick account of some facts about general spherically-symmetric space-times. The computations were carried out by Christodoulou in section 3 of his paper on the “two-phase model”; a summary of the results can also be found in Dafermo’s paper on a general extension principle in spherical symmetry. (See Dafermo’s paper for some nice pictures of what’s going on here; for more basic geometric things, see O’Neill’s Semi-Riemannian Geometry.)

If $(M,g)$ is a spherically symmetric Lorentzian manifold of (1+3)-dimensions, we can quotient by the action of $SO(3)$ and obtain a (1+1)-dimensional Lorentzian manifold $(Q,h)$, where $Q$ may have a boundary corresponding to the symmetry axis of $M$ under rotations. $(M,g)$ can be obtained from the quotient manifold (modulo a little bit of delicacy at the symmetry axis) as the warped product of $(Q,h)$ with the standard two-sphere. The square root of the warp function is the area-radius of the spherical orbits (because in the induced metric on the orbits, the area of the orbit is given by $4\pi$ times the warp function). We write the area-radius function as $r$.

Now, each symmetry orbit (away from the axis) is a topological 2-sphere embedding in $M$. So we can compute its mean curvature vector $H$. Now, for an arbitrary 2-surface $\Sigma$ embedded in a space-time $M$, the mean curvature vector gives us some information about the local area evolution under a causal flow of $\Sigma$. More precisely, if $H$ is future causal at some point $s\in\Sigma$, we say that surface $\Sigma$ is locally contracting at $s$. If $H$ is past causal, we say the surface is locally expanding. (The terminology is based on the following intuition: the inner product $g(n,H)$ between some element $n$ of the normal bundle of $\Sigma$ and the mean curvature vector can be interpreted as the rate of change of the logarithm of the area element of $\Sigma$ under a flow by the vector $n$. If $H$ is future causal, then for any future causal normal vector $n$, the flow will result in a local decrease of area, since $g(n,H) \leq 0$.) The 2-surface $\Sigma$ is said to be future-trapped if it is everywhere contracting, and past-trapped if it is everywhere expanding. Furthermore, the trapping is said to be strict if $H$ is always time-like, and marginal if $H$ is always light-like. Morally speaking, a strictly future-trapped 2-surface is, for our intents and purposes, “inside” a black-hole; whereas the “boundary” of a black hole is a marginal future-trapped surface.

Now, given an arbitrary closed space-like 2-surface $\Sigma$, we can define its Hawking mass by the formula

Equation 1: General definition for Hawking mass
$\displaystyle m(\Sigma) := \sqrt{\frac{\mathop{Area}(\Sigma)}{16\pi}}\left( \frac{\chi(\Sigma)}{2} - \frac{1}{16\pi}\int_\Sigma g(H,H) dvol_\Sigma\right)$

where $dvol_\Sigma$ is the induced area form on $\Sigma$ by the metric $g$, and $\chi(\Sigma)$ the Euler characteristic for $\Sigma$. In general this quantity need not be positive. But we can now impose spherical symmetry and compute for $\Sigma$ being the symmetry orbits. As $\Sigma$ are spheres, the Euler index is 2. A computation (perhaps using the interpretation that the mean curvature is the rate of change of logarithm of area) shows that $H = \frac{2}{r}\nabla r$, and so the integrand $g(H,H) = \frac{4}{r^2} g(\nabla r,\nabla r)$. By spherical symmetry, the integrand is constant along the orbits, and so the integral evaluates to, using the fact that the area of the orbits is $4\pi r^2$, just $g(\nabla r,\nabla r)$. Lastly using the warped product structure, each orbit $\Sigma$ corresponds to an interior point of $q\in Q$, and $g(\nabla r, \nabla r) = h(\nabla r, \nabla r)$. So the mass $m$ descends to a function on $(Q,h)$:

Equation 2: Hawking mass in spherical symmetry
$\displaystyle m = \frac{r}{2}(1 - h(\nabla r,\nabla r))$

Rearranging the terms, and notice that $H$ is causal if and only if $h(\nabla r,\nabla r) \leq 0$, we have that a point $q\in Q$ represents a trapped surface in $M$ if and only if $1 - \frac{2m}{r} \leq 0$ at $q$.

In view of spherical symmetry and warped product structure, the Ricci tensor of $(M,g)$ must split into parts purely tangential to the symmetry orbits, and parts purely orthogonal to it. By Einstein’s equation, the stress energy tensor $T$ must also split in the same way. We will use $\tilde{T}$ to denote the purely orthogonal part, which naturally restricts to a tensor on $(Q,h)$. The main computation I want to claim is the following (see equation (3.18) on page 362 in Christodoulou’s derivation) expression which takes place on the quotient manifold:

Equation 3
$X(m) = 4\pi r^2 ( \tilde{T}(X,\nabla r) - X(r) \mathop{tr}_h \tilde{T})$

where $X$ is an arbitrary vector field on $Q$. A bit of linear algebra shows that the terms inside the parentheses can be written as $\tilde{T}(*X,-*\nabla r)$ where $*$ is the Hodge star operator associated to the metric $h$.

Proof of theorem. Let $\gamma(s): s\in [a,b]$ be a smooth, non-self-intersecting curve in $Q$, so that it lifts to a hypersurface in $M$. Let $\dot{\gamma}$ be its tangent vector field as parametrized by $s$. The fundamental theorem of calculus, along with Equation 3, gives

Equation 4
$\displaystyle m\circ\gamma(b) - m\circ\gamma(a) = 4\pi \int_a^b r^2 \left( \tilde{T}(\dot\gamma,\nabla r ) - \dot\gamma(r)\mathop{tr}\tilde{T}\right) \circ\gamma ds$

Now impose the assumption that $\gamma$ is space-like, and the parametrisation is by arc-length. Then along $\gamma$ the tangent space is spanned by the orthonormal frame ${e_0, e_1}$ where $e_1 = \dot\gamma$ is space-like, and $e_0$ is its future-directed time-like normal. Now $\nabla r = - e_0 \nabla_0r + e_1\nabla_1r$. In this frame, the integrand in Equation 4 becomes $r^2( \tilde{T}_{00}\nabla_1r - \tilde{T}_{01}\nabla_0r)$.

If we assume $\nabla r$ is space-like (which is the case where no trapped surfaces form along $\gamma$), then $|\nabla_1 r| > |\nabla_0 r|$. Furthermore, we’ll take $\gamma(b)$ to correspond to $S_f$ and $\gamma(a)$ to $S_i$, so $r\circ\gamma(b) > r\circ\gamma(a)$. Then $\nabla_1 r > 0$. So we can estimate the integrand in Equation 4 from below by $\frac13 (\tilde{T}_{00} - |\tilde{T}_{01}|) \nabla_1r^3$. From which we get

Equation 5
$m\circ\gamma(b) - m\circ\gamma(a) > \frac{4\pi}{3}[r^3\circ\gamma(b) - r^3\circ\gamma(a)] \inf_{x\in\gamma} (\tilde{T}_{00} - |\tilde{T}_{01}|)$

By a simple algebraic manipulation, and observing that $\tilde{T}_{00} = T(\tau,\tau)$ and $T(\tau,v) = 0$ if $v$ is tangent to the symmetry orbits, Q.E.D.

Now, we also see that the proof still carries through if $M$ has a symmetry axis, and we take $S_i$ to approach that axis. Regularity of space-time will require that the Hawking mass and area radius vanish there. Now, we can re-express the Hawking mass in terms of the mean curvature of $S_f$, so the left hand side of the inequality in Theorem A becomes

$\frac12 \frac{r(S_f)}{\frac43\pi r(S_f)^3} - \frac{3}{32\pi}g(H(S_f),H(S_f))$

So as a corollary of Theorem A, we establish an inequality reminiscent of that of Theorem 1 in Khuri’s paper, which is phrased in a form similar to the statement of Schoen and Yau.

Corollary C
Let $(M,g, \Phi)$ be a spherically-symmetric space-time with matter fields $\Phi$. Assume Einstein’s equation $R_{ab} - \frac12 R g_{ab} = 8\pi T_{ab}$ is satisfied. Let $\Sigma$ be a space-like hypersurface in $M$ diffeomorphic to a 3-dimensional ball. Assume $\Sigma$ is invariant under the spherical symmetry, and that its boundary is a spherical orbits $S$, then if
$\displaystyle \inf_{s\in\Sigma}~\inf_{v\in T_s\Sigma, g(v,v) = 1} T(\tau,\tau + v) + \frac{3}{32\pi}g(H(S),H(S)) > \frac{3}{8\pi r(S)^2}$
where $H$ is the mean curvature vector, $\tau$ the unit normal field to $\Sigma$, and $r$ the area radius, we conclude that $\Sigma$ must contain a trapped sphere.