P.d.e.'s which imply the Penrose conjecture
aa r X i v : . [ m a t h . DG ] M a y P.D.E.’s which Imply the Penrose Conjecture
Hubert L. Bray ∗ & Marcus A. Khuri † November 5, 2018
Abstract
In this paper, we show how to reduce the Penrose conjecture to theknown Riemannian Penrose inequality case whenever certain geometri-cally motivated systems of equations can be solved. Whether or not thesespecial systems of equations have general existence theories is thereforean important open problem. The key tool in our method is the deriva-tion of a new identity which we call the generalized Schoen-Yau identity,which is of independent interest. Using a generalized Jang equation, wepropose canonical embeddings of Cauchy data into corresponding staticspacetimes. In addition, our techniques suggest a more general Penroseconjecture and generalized notions of apparent horizons and trapped sur-faces, which are also of independent interest.
In addition to their intrinsic geometric appeal, the Penrose conjecture [25] andthe positive mass theorem [28] are fundamental tests of general relativity as aphysical theory. In physical terms, the positive mass theorem states that thetotal mass of a spacetime with nonnegative energy density is also nonnegative.The Penrose conjecture, on the other hand, conjectures that the total mass ofa spacetime with nonnegative energy density is at least the mass contributedby the black holes in the spacetime. In this section, we will explain how thesesimple physical motivations translate into beautiful geometric statements.After special relativity, Einstein sought to explain gravity as a consequence ofthe curvature of spacetime caused by matter. In contrast to Newtonian physics,gravity is not a force but instead is simply an effect of this curvature. As ananalogy, consider a heavy bowling ball placed on a bed which causes a significantdimple in the bed. Now roll a small golf ball off to one side of the bowling ball.Note that the path of the golf ball curves around the bowling ball because of thecurvature of the surface of the bed. In this analogy, the bowling ball represents ∗ Mathematics Department, Duke University, Box 90320, Durham, NC 27708. Supportedin part by NSF grant † Mathematics Department, Stony Brook University, Stony Brook, NY 11794. Supportedin part by NSF grant N , g N ) is a Lorentzian manifold, meaning that themetric g N has signature ( − + ++) at each point. Note that at each point,time-like vectors (vectors v with g N ( v, v ) <
0) are split into two connectedcomponents, one of which we will call future directed time-like vectors, and theother of which we will call past directed time-like vectors.Next, define T ( v, w ) to be the energy density going in the direction of v as measured by an observer going in the direction of w , where v, w are future-directed unit time-like vectors at some point p ∈ N . In addition, suppose that T is linear in both slots so that T is a tensor. Then the physical statement thatall observed energy densities are nonnegative translates into T ( v, w ) ≥ v and w at allpoints p ∈ N , known as the dominant energy condition.The goal, then, is to set T , which is called the stress-energy tensor, equal tosome curvature tensor. A natural first idea is to consider the Ricci curvaturetensor since it is also a covariant 2-tensor. In fact, this was Einstein’s first idea.However, the second Bianchi identity on a manifold N with metric tensor g N implies that div( G ) = 0 , where G = Ric N − R N g N , Ric N is the Ricci curvature tensor, and R N is thescalar curvature. This geometric identity led Einstein to propose G = 8 πT, (1)known as the Einstein equation, since as an added bonus we automatically geta conservation-type property for T , namely div( T ) = 0. Naturally this is a verynice feature of the theory since energy and momentum (the spatial componentsof the energy vector) are conserved in every day experience.2he next step in pursuing this line of thought is to try to find examplesof spacetimes which satisfy the dominant energy condition, the simplest caseof which would be spacetimes with G = 0 which are naturally called vacuumspacetimes. Taking the trace implies that such spacetimes (in 2+1 dimensionsand higher) have zero scalar curvature and therefore zero Ricci curvature aswell. The first example (in 3+1 dimensions) is clearly Minkowski space (cid:0) R , − dt + dx + dy + dz (cid:1) which has zero Riemann curvature tensor. The second simplest example of aspacetime with G = 0, R × ( R \ B m/ (0)) , − (cid:18) − m r m r (cid:19) dt + (cid:16) m r (cid:17) ( dx + dy + dz ) ! , (2)where r = p x + y + z , is a one parameter family of spacetimes called theSchwarzschild spacetimes. When m >
0, these spacetimes represent static blackholes in a vacuum spacetime.While the Schwarzschild spacetime can be covered by a single coordinatechart (see Kruskal coordinates described in section 2), the coordinate chartabove only covers the exterior region of the black hole and has a coordinate sin-gularity (not an actual metric singularity) on the coordinate cylinder r = m/ (cid:0) R × M, − φ ( x ) dt + g (cid:1) , where t ∈ R , x ∈ M , and g is a Riemannian (positive definite) metric on M ,are called static spacetimes. This name is appropriate since we see that thecomponents of the spacetime metric in this coordinate chart do not depend on t but instead are entirely functions of x . Note also that static metrics are definednot to have any time/spatial cross terms. (Spacetimes which allow time/spatialcross terms but where the metric components still only depend on x are calledstationary spacetimes.)An important result, first proved by Bunting and Masood-ul-Alam [6] us-ing a very clever argument involving the positive mass theorem, is that theonly complete, asymptotically flat static vacuum spacetimes with black holeboundaries (or no boundary) are the two spacetimes that we have listed so far,Minkowski and Schwarzschild. This fact suggests that a thorough understand-ing of these two spacetimes, including what makes them special as comparedto generic spacetimes, may be important for understanding some of the mostfundamental properties of general relativity.In fact, the Minkowski and Schwarzschild spacetimes are the extremal space-times for the positive mass theorem and the Penrose conjecture, respectively.3hat is, the case of equality of the positive mass theorem states that any space-like hypersurface of a spacetime satisfying the hypotheses of the positive masstheorem which has m = 0 can be isometrically embedded into the Minkowskispacetime. Similarly, the case of equality of the Penrose conjecture (which, whilestill a conjecture, has no known counter-examples in spite of much examination)states that any space-like hypersurface of a spacetime satisfying the hypothesesof the Penrose conjecture which has m = p A/ π (or to be more precise, theregion outside of the outermost minimal area enclosure of the apparent horizons)can be isometrically embedded into the Schwarzschild spacetime.Before we can state these theorems, though, we need to define a few terms.The basic object of interest in this paper is a space-like hypersurface M of aspacetime N , along with the induced metric g on M and its second funda-mental form k in the spacetime.From this point on we will assume that M has a global future directedunit normal vector n future in the spacetime. This standard assumption is notnecessary for stating the dominant energy condition or, as we will see in section3, for defining generalized apparent horizons or stating the generalized Penroseconjecture, but it is necessary for the traditional definition for apparent horizonsof black holes, as we will see. So for convenience, we will abuse terminologyslightly and also call k ( V, W ) = −h∇ V W, n future i (3)the second fundamental form of M , where V, W are any vector fields tangentto M and ∇ is the Levi-Civita connection on the spacetime N . In this mannerwe are defining k to be a real-valued symmetric 2-tensor, where the true secondfundamental form, which takes values in the normal bundle to M , is k · n future . Definition 1
The triple ( M , g, k ) is called the Cauchy data of M for anypositive definite metric g and any symmetric 2-tensor k . This name is appropriate because this is the data required to pose initialvalue problems for p.d.e.’s such as the vacuum Einstein equation G = 0, or theEinstein equation coupled with equations which describe how the matter evolvesin the spacetime. Note when M is flowed at unit speed orthogonally into thefuture that ddt g ij = 2 k ij , so that k is in fact the first derivative of g in the time direction (up to a factor).Curiously, as we will see, the positive mass theorem and the Penrose con-jecture reduce to and are fundamentally statements about the Cauchy data ofspace-like hypersurfaces of spacetimes, not the spacetimes themselves. Definition 2
At each point on M , define µ = T ( n future , n future ) to be theenergy density and the covector J on M to be the momentum density, where J ( v ) = T ( n future , v ) , where v is any vector tangent to M .
4y the Einstein equation (equation 1) and the Gauss-Codazzi identities [23],it follows that µ and J can be computed entirely in terms of the Cauchy data( M , g, k ). In fact,(8 π ) µ = G ( n future , n future ) = ( R + tr( k ) − k k k ) / π ) J = G ( n future , · ) = div ( k − tr( k ) g ) , (5)where R is the scalar curvature of ( M , g ) at each point, and the above traces,norms, and divergences are naturally taken with respect to g and the Levi-Civitaconnection of g . Then the dominant energy condition on T implies that we musthave µ ≥ | J | , (6)which we will call the nonnegative energy density condition on ( M , g, k ), whereagain the norm is taken with respect to the metric g on M .Equations 4 and 5 are called the constraint equations because they imposeconstraints on the Cauchy data ( M , g, k ) for each initial value problem. Forexample, we clearly need to impose µ = 0 and J = 0 on any Cauchy data whichis meant to serve as initial conditions for solving the vacuum Einstein equation G = 0. However, for our purposes throughout the rest of this paper, we will beinterested in Cauchy data ( M , g, k ) which only needs to satisfy the nonnegativeenergy density condition in inequality 6. Since the assumption of nonnegativeenergy density everywhere is a very common assumption, the theorems we provewill apply in a very broad set of circumstances.Next we turn our attention to the definition of the total mass of a spacetime.Looking back at the Schwarzschild spacetime, time-like geodesics (which rep-resent test particles) curve in the coordinate chart as if they were acceleratingtowards the center of the spacetime at a rate asymptotic to m/r in the limitas r goes to infinity. Hence, to be compatible with Newtonian physics (with theuniversal gravitational constant set to 1) in the low field limit, we must define m to be the total mass of the Schwarzschild spacetime.More generally, consider any spacetime which is isometric to the Schwarzschildspacetime with total mass m for r > r and which is any smooth Lorentzianmetric satisfying the dominant energy condition on the interior region. Of coursethe Schwarzschild spacetime satisfies the dominant energy condition since it has G = 0. Then the same argument as in the previous paragraph applies to thisspacetime, so its total mass must be m as well. This last example inspires the fol-lowing definition, which comes from considering the t = 0 slice of Schwarzschildspacetimes. Definition 3
The Cauchy data ( M , g, k ) will be said to be Schwarzschild atinfinity if M can be written as the disjoint union of a compact set K and afinite number of regions E i (called ends), where k = 0 on each end and each ( E i , g ) is isometric to (cid:16) R \ ¯ B R i (0) , (cid:0) m i r (cid:1) ( dx + dy + dz ) (cid:17) for some m i and some R i > max(0 , − m i / . In addition, the mass of the end E i will bedefined to be m i .
5e refer the reader to [27] and [32] for more general definitions of asymp-totically flat
Cauchy data, but for this paper the special case of being preciselySchwarzschild at infinity is sufficiently interesting.Typically we will be interested in Cauchy data with only one end. However,sometimes it is convenient to allow for the possibility of multiple ends. Eachend represents what we would normally think of as a spatial slice of a universe,and the positive mass theorem and the Penrose conjecture may be applied toeach end independently. In fact, since ends can be compactified by adding apoint at infinity and then using a very large spherical metric on the end withoutviolating the nonnegative energy density assumption, without loss of generalitywe may assume that any given Cauchy data has only one end for the problemswe will be considering.
Theorem 1 (The Positive Mass Theorem, Schoen-Yau, 1981 [27]; Witten,1981 [33])Suppose that the Cauchy data ( M , g, k ) is complete, satisfies the nonnegativeenergy density condition µ ≥ | J | , and is Schwarzschild at infinity with totalmass m . Then m ≥ , and m = 0 if and only if ( M , g, k ) is the pullback of the Cauchy data inducedon the image of a space-like embedding of M into the Minkowski spacetime. The above theorem has an important special case when k = 0 which is al-ready extremely interesting. Note that the nonnegative energy condition reducesto simply requiring ( M , g ) to have nonnegative scalar curvature. Theorem 2 (The Riemannian Positive Mass Theorem, Schoen-Yau, 1979 [26];Witten, 1981 [33])Suppose that the Riemannian manifold ( M , g ) is complete, has nonnegativescalar curvature, and is Schwarzschild at infinity with total mass m . Then m ≥ , and m = 0 if and only if ( M , g ) is isometric to the flat metric on R . The adjective Riemannian was introduced by Huisken-Ilmanen in [14] sincethe theorem is a statement about Riemannian manifolds as opposed to Cauchydata in the more general case. We remind the reader that Cauchy data ( M , g, k )is still required to have a positive definite metric g .Notice that the Riemannian positive mass theorem is a beautiful geometricstatement about manifolds with nonnegative scalar curvature. In fact, Schoen-Yau were studying such manifolds [29] for purely geometric reasons when theyfirst realized that they could use minimal surface techniques to prove the Rie-mannian positive mass theorem. They then observed [27] that theorem 1 (whichis quite mysterious from a geometric point of view without the physical moti-vation) reduced to theorem 2 after solving a certain elliptic p.d.e. on ( M , g, h )6alled the Jang equation, named after the theoretical physicist who first intro-duced the equation in [15].Witten’s proof of the positive mass theorem uses spinors and proves bothof the above statements by applying the Lichnerowicz-Weitzenbock formula toa spinor which solves the Dirac equation, and then integrating by parts. Thisproof has a strong appeal because it computes the total mass as an integral of anonnegative integrand. However, so far it has not been clear how to generalizethis approach to achieve the Penrose conjecture, although very interesting worksin this direction include [11] and [19].Before we can state the Penrose conjecture, we need several more definitions.For convenience, we modify the topology of M by compactifying all of the endsof M except for one chosen end by adding the points {∞ k } . (However, themetric will still not be defined on these new points.) Definition 4
Define S to be the collection of surfaces which are smooth compactboundaries of open sets U in M , where U contains the points {∞ k } and isbounded in the chosen end. All of the surfaces that we will be dealing with in this paper will be in S . Also, we see that all of the surfaces in S divide M into two regions, aninside (the open set) and an outside (the complement of the open set). Thus,the notion of one surface in S enclosing another surface in S is well defined asmeaning that the one open set contains the other. Definition 5
Given any Σ ∈ S , define ˜Σ ∈ S to be the outermost minimal areaenclosure of Σ . That is, in the case that there is more than one minimal area enclosure of thesurface Σ, choose the outermost one which encloses all of the others. The factthat an outermost minimal area enclosure exists and is unique roughly followsfrom the following: if ∂A and ∂B are both minimal area enclosures of somesurface, then so are ∂ ( A ∪ B ) and ∂ ( A ∩ B ) since | ∂ ( A ∪ B ) | + | ∂ ( A ∩ B ) | = | ∂A | + | ∂B | = 2 A min and both have area at least A min . A rigorous proof thatthe outermost minimal area enclosure of a surface in an asymptotically flatmanifold exists and is unique is given in [14]. Definition 6
Define Σ ∈ S in ( M , g, k ) to be an apparent horizon if it is oneof the following three types of horizons,a future apparent horizon if H Σ + tr Σ ( k ) = 0 on Σ , (7) a past apparent horizon if H Σ − tr Σ ( k ) = 0 on Σ , (8) and a future and past apparent horizon if H Σ = 0 and tr Σ ( k ) = 0 on Σ , (9)7 here H Σ is the mean curvature of the surface Σ in ( M , g ) (with the signchosen to be positive for a round sphere in flat R ) and tr Σ ( k ) is the trace of krestricted to the surface Σ . Note that equation 9 follows from assuming both equations 7 and 8 everywhereon Σ. Also note that Σ is not required to be connected, although from a physicalpoint of view each component of Σ is usually thought of as the apparent horizonof a separate black hole. Finally, observe that all three types of horizons aresimply minimal surfaces (surfaces with zero mean curvature) in the importantspecial case when k = 0.Physically, the only relevant apparent horizons for a spacecraft flying aroundin a spacetime are future apparent horizons, because spacecraft are only con-cerned about being trapped inside black holes in the future. Mathematically,however, merely changing the choice of global normal vector n future to M in N to − n future changes the sign on k which causes past apparent horizons tobecome future apparent horizons, and vice versa.Equations 7, 8, 9 are actually all conditions on the mean curvature vectorof Σ in the spacetime. Note that at each point of Σ , the normal bundle, ofwhich the mean curvature vector is a section, is a 2-dimensional vector spacewith signature ( − +). Naturally, a basis for this vector space is any outwardfuture null vector along with any outward past null vector. Since Σ boundsa region in M , outward is well-defined, and since there exists a global normalvector n future to M , the future direction is well-defined.Geometrically, if one flows a submanifold in the normal directions ~η , thenthe rate of change of the area form of the submanifold is given by ddt dA = −h ~η, ~H i dA where ~H is the mean curvature vector. It turns out that the mean curvature ofa surface Σ contained in a slice with Cauchy data ( M , g, k ) has coordinates(tr Σ ( k ) , − H ), where the first coordinate is in the unit future normal directionto the slice and the second component is in the unit direction outward perpen-dicular to the surface and tangent to the slice. With this convention, then thevector with components (1,1) is an outward future null vector, and the vectorwith components (-1,1) is an outward past null vector.Hence, equation 7 is equivalent to requiring that, at each point of Σ, thedot product of the mean curvature vector with any outward future null vectoris zero (which implies that the mean curvature vector is a real multiple of theoutward future null direction). Similarly, equation 8 is equivalent to saying thatthe dot product of the mean curvature vector with any outward past null vectoris zero. Thus, future apparent horizons have the property that their areas do notchange to first order when flowed in outward future null directions. The sameis true for past apparent horizons when flowed in outward past null directions.We are now able to state the Penrose conjecture. An excellent survey of thisconjecture is found in [20]. 8 onjecture 1 (The Penrose Conjecture, 1973 [25] - Standard Version)Suppose that the Cauchy data ( M , g, k ) is complete, satisfies the nonnegativeenergy density condition µ ≥ | J | , and is Schwarzschild at infinity with totalmass m in a chosen end. If Σ ∈ S is a future apparent horizon, then m ≥ r A π , (10) where A is the area of the outermost minimal area enclosure ˜Σ = ∂U of Σ .Furthermore, equality occurs if and only if ( M \ U , g, k ) is the pullback of theCauchy data induced on the image of a space-like embedding of M \ U intothe exterior region of a Schwarzschild spacetime (which maps ˜Σ to a futureapparent horizon). Penrose’s heuristic argument for a future apparent horizon in this conjectureis described in more detail in [4] and [20] but roughly goes as follows: If, as isgenerally thought, asymptotically flat spacetimes eventually settle down to aKerr spacetime [12], then in the distant future inequality 10 will be satisfiedsince explicit calculation verifies this fact for Kerr spacetimes, where A is thearea of the event horizon. Given that some energy may radiate out to infinity,the total mass of these slices of Kerr may be less than the original total mass.Also, by the Hawking area theorem [10] (made more rigorous in [7]), and thusby the cosmic censor conjecture [24] as well, the area of the event horizon isnondecreasing in the spacetime evolution. Hence, this leads us to conjectureinequality 10 in the initial Cauchy data slice, but where A is the total areaof the event horizons of all of the black holes. The problem, though, is that,unlike apparent horizons, event horizons are not determined by local geometrybut instead are defined in terms of which points in spacetime can eventuallyescape out to infinity along future directed time-like curves. Thus, in principle,there is no way to know which points this includes without looking at the entireevolution of the spacetime into the future. However, in [25] Penrose arguedusing the cosmic censor conjecture that future apparent horizons, which aredefined in terms of local geometry, must be enclosed by event horizons. Thus,the area of ˜Σ serves as a lower bound for the total area of the event horizons [16],[13], and the Penrose conjecture follows. This same argument, but run in theopposite time direction, yields the same conjecture for past apparent horizonsas well. Thus, in the conjecture one could replace “future apparent horizon”with simply “apparent horizon.”It is also important to note, as Penrose did originally, that a counterexampleto the Penrose conjecture would be a very serious issue for general relativity sinceit would imply that some part of the above reasoning is false. The consensusamong many is that the cosmic censor conjecture is the weakest link in the aboveargument. If the cosmic censor conjecture turns out to be false, and nakedsingularities (singularities not enclosed by the event horizons of black holes)do develop in generic spacetimes, then this would present a very interestingchallenge to general relativity as a physical theory.9owever, like the positive mass theorem, setting k = 0 yields another beau-tiful geometric statement about manifolds with nonnegative scalar curvature,which is known to be true. Theorem 3 (The Riemannian Penrose Inequality, Bray, 2001 [2])Suppose that the Riemannian manifold ( M , g ) is complete, has nonnegativescalar curvature, and is Schwarzschild at infinity with total mass m in a chosenend. If Σ ∈ S is a zero mean curvature surface, then m ≥ r A π , (11) where A is the area of the outermost minimal area enclosure ˜Σ = ∂U of Σ . Furthermore, equality occurs if and only if ( M \ U , g ) is isometric to theSchwarzschild metric (cid:16) R \ B m/ (0) , (cid:0) m r (cid:1) ( dx + dy + dz ) (cid:17) . In 1997, Huisken-Ilmanen proved a slightly weaker version of the above resultwith the modification that A is the area of the largest connected component ofthe outermost minimal area enclosure of Σ and with the additional assumptionthat H ( M ) = 0. (This last topological condition can be replaced by assumingthat Σ is already a connected component of the outermost minimal area surfaceof ( M , g ) by Meeks-Simon-Yau [22].) Their method of proof, first proposedby the theoretical physicists Geroch [9] and Jang-Wald [16], uses a parabolictechnique called inverse mean curvature flow. Starting with a connected zeromean curvature surface, Huisken-Ilmanen found a weak definition of inversemean curvature flow, where the surface is flowed out at each point in ( M , g )with speed equal to the reciprocal of the mean curvature of the surface at thatpoint, for almost every surface in the flow. Then they showed that the Hawkingmass of the surface is nondecreasing under this flow, equals the right hand sideof the Riemannian Penrose inequality initially, and limits to the left hand sideof the Riemannian Penrose inequality as the surface flows out to large roundspheres going to infinity. Both the physicists’ insight into proposing this ideaand the mathematicians’ cleverness at generalizing the argument to somethingwhich could be made rigorous are remarkably beautiful.Bray’s proof also involves a flow, but of the Riemannian manifold ( M , g ).The flow of metrics stays inside the conformal class of the original metric andeventually flows to a Schwarzschild metric (shown as the case of equality metric).The conformal flow of metrics is chosen so as to keep the area of the outermostminimal area enclosure of Σ constant. Also, the total mass of the Rieman-nian manifold is nonincreasing by a clever argument (first used by Bunting andMasood-ul-Alam in [6]) using the positive mass theorem after a reflection ofthe manifold along a zero mean curvature surface and a conformal compacti-fication of one of the resulting two ends. Then since the Schwarzschild metricgives equality in inequality 11, the inequality follows for the original Riemannianmanifold ( M , g ). 10ll three systems of equations discussed in this paper which imply the Pen-rose conjecture are based on a new geometric identity which we call the gen-eralized Schoen-Yau identity. The identity is proved in section 5, but withthe lengthy computations relegated to the appendices for readability. This newidenity is a generalization of equation 2.25 in Schoen-Yau’s paper [27]. The orig-inal Schoen-Yau identity was used to reduce the positive mass theorem to theRiemannian positive mass theorem by solving a p.d.e. called the Jang equation.For all three systems, our technique will involve a generalization of the Jangequation to solving a system of two equations, the first of which is a generalizedJang equation in all three cases. Rather than spending time explaining the Jangequation, we will go straight to our proof since the Jang equation appears as aspecial case of our method (which for future reference is the case φ = 1).As a final comment, the Penrose conjecture can be generalized to a statementabout Cauchy data on n -manifolds motivated by considering ( n +1)-dimensionalspacetimes, where n ≥
3. In fact, the positive mass theorem was proved bySchoen-Yau in dimensions n ≤ M n is spin. The Riemannian Penroseinequality was proved by Bray [2] in dimension 3 using a proof which that authorand Dan Lee [5] have generalized to manifolds in dimensions n ≤
7, and in aslightly weaker form by Huisken-Ilmanen [14] in dimension 3. Since we will bereducing the general case of the Penrose conjecture to the Riemannian Penroseinequality, the techniques presented here have, at a minimum, the potential toaddress the Penrose conjecture for manifolds with dimensions n ≤
7. However,we will focus on n = 3 for simplicity. In this section we carefully study the case of equality of the Penrose conjecturefor the obvious reason that all of our estimates used to prove the conjecture mustgive equality in these cases. Also, we want to make sure that our techniquesapply to all of the case of equality examples as a check that we are not makingunjustified assumptions.We refer the reader to [23] for a discussion of the Schwarzschild spacetime inKruskal coordinates and follow those conventions (except for the names of thetwo functions α and β defined in a moment). Understanding the Schwarzschildspacetime in Kruskal coordinates is essential since this is the simplest global co-ordinate chart for the spacetime. In Kruskal coordinates, the entire Schwarzschildspacetime is expressed as the subset uv > − m/e of R × S with coordinates( u, v, σ ∈ S ) and line element2 β ( r ) dudv + r dσ , (12)where dσ is the standard round unit sphere metric on S , r > u, v determined by uv = α ( r ) = ( r − m ) e ( r/ m ) − , β ( r ) = (8 m /r ) e − ( r/ m ) . The first quadrant region described by u, v > R × ( R \ B m (0)) , − (cid:18) − mr (cid:19) dt + (cid:18) − mr (cid:19) − dr + r dσ ! (13)under the isometry u = p α ( r ) e − t/ m , v = p α ( r ) e t/ m , which we leave as an exercise for the interested reader to check. Note thatwe have now defined three different coordinate chart representations for theSchwarzschild spacetime, the two above in equations 12 and 13, and our originalone in equation 2.A key point is that two of these coordinate chart representations of theexterior region of the Schwarzschild spacetime are written in the form of astatic spacetime. For example, using the coordinates in equation 13, the exteriorregion can be expressed as (cid:0) R × M , − φ dt + g (cid:1) , (14)where φ = 1 − mr , which of course gives us r = 2 m − φ and r − m = 2 mφ − φ . Hence, if we think of a slice of the static spacetime expressed as the graph of t = f ( x ) in the static spacetime, x ∈ M , we see that f = 2 m log( v/u ) , (15)2 mφ − φ exp (cid:18) φ − φ (cid:19) = α ( r ) = uv. (16)The reason that these last two equations are important is that it allowsus to understand the behavior of f and φ as they approach the boundary ofthe exterior region { x | u > , v > } of the Schwarzschild spacetime. Ourslice (intersected with the exterior region of the Schwarzschild spacetime) has afuture apparent horizon boundary if u = 0 everywhere on the boundary, a pastapparent horizon boundary if v = 0 everywhere on the boundary, and a futureand past apparent horizon boundary if u = v = 0 everywhere on the boundary.The mixed case where u = 0 on part of the boundary and v = 0 on the restof the boundary does not represent a traditional apparent horizon boundary.12owever, we note that whenever this boundary is area-outerminimizing, we arein fact in a case of equality of the Penrose conjecture. This observation helpsmotivate the definition of a generalized apparent horizon in the next section.Also, while the u = 0 level set and the v = 0 level set on M are both smooth(since the gradients of u and v on M are never zero since M is space-like),the boundary of { x ∈ M | u ( x ) > , v ( x ) > } in M need not be smooth sincethe zero level sets of u and v do not need to intersect smoothly. However, whenthe boundary has corners it is never area outerminimizing and thus not a caseof equality of the Penrose conjecture.We also note that apparent horizons outside of the exterior region, saywith u = 0 but with v < u and v are smooth onany slice, even up to the apparent horizon boundary, f and φ are not necessarily.In fact, we see that f goes to ±∞ logarithmically at the apparent horizonboundary typically (when u or v goes to zero and the other stays positive). Also, φ vanishes on the apparent horizon boundary only linearly if either u or v isstrictly positive, which means that the derivative of φ is going to ∞ . However,in the future and past apparent horizon boundary case where u, v both go tozero, then φ vanishes quadratically and φ is smooth up to the boundary. Itis also true that f is smooth up to the boundary in this case by L’Hopital’srule since u and v , which equal zero on the future and past apparent horizon,have nonzero derivatives there (since the hypersurface is space-like). Theseobservations are helpful since we will be dealing with slices of the exterior regionof the Schwarzschild spacetime viewed in static coordinates for the rest of thispaper. In this section we describe the most general version of the Penrose conjecturethat we believe to be true. Naturally this is an important question to considersince proofs of a conjecture may be more easily found when the most naturalversion of the conjecture is understood.
Definition 7
Define the smooth surface Σ ∈ S in ( M , g, k ) to be a generalizedapparent horizon if H Σ = | tr Σ ( k ) | (17)13 nd a generalized trapped surface if H Σ ≤ | tr Σ ( k ) | . (18)In terms of the mean curvature vector ~H of Σ in the spacetime, a generalizedtrapped surface is one where ~H is not strictly inward space-like anywhere onΣ. Also note that this definition of a generalized apparent horizon does notneed a globally defined future directed unit normal to M since the definitionis unaffected by a change of sign of the second fundamental form k . A relatedclass of surfaces, referred to as “ ∗ -surfaces”, appears in a different context in[30].Referring to the previous section, note that any smooth slice M of theSchwarzschild spacetime which smoothly intersects (which is often not the case)with the boundary of the first quadrant { u ≥ , v ≥ } of Kruskal coordinatesintersects in a generalized apparent horizon. These generalized apparent hori-zons also give equality in the Penrose conjecture, so it is natural to include themin the statement of a generalized Penrose conjecture. Also note that traditionalapparent horizons, if they are not already generalized apparent horizons, are atleast always generalized trapped surfaces.Another consideration which leads to this definition of generalized apparenthorizons is the case when a surface with multiple connected components is afuture apparent horizon on some connected components and a past apparenthorizon on the others. While Penrose’s original heuristic argument does notapply to this surface, the techniques that we develop in this paper seem toapply perfectly well. Thus, we would like a generalized Penrose conjecturewhich includes this case as well.After a talk on generalized apparent horizons by the first author at theNiels Bohr International Academy’s program “Mathematical Aspects of GeneralRelativity” in April 2008, Robert Wald posed the following insightful question:Is it possible for generalized trapped surfaces to exist as boundaries of space-likeslices of Minkowski space? (A similar query was posed by Mars and Senovillain [21].) This question raises the issue of whether or not generalized trappedsurfaces always yield a positive contribution to the ADM mass, which of courseis a prerequisite for a generalized version of the Penrose conjecture. If one couldfind a generalized trapped surface which was the boundary of a space-like sliceof Minkowski space, then the total mass of the slice would be zero, making aPenrose-type inequality for the surface impossible.In response to this question, the second author of this paper showed thatno such generalized trapped surface in Minkowski space exists [18]. Further-more, he showed that Witten’s proof of the positive mass theorem also worksfor asymptotically flat manifolds with generalized trapped surface boundary andgives a positive lower bound on the total mass. This result suggests that gen-eralized trapped surfaces and generalized apparent horizons have some physicalsignificance in that such surfaces, along with nonnegative energy density µ ≥ | J | everywhere in the spacetime, always imply that the total mass is positive. Find-ing the best possible lower bound on the total mass motivates conjecturing ageneralized Penrose inequality. 14n addition, a discussion between the first author and Tom Ilmanen led to twomore conjectures about generalized apparent horizons and generalized trappedsurfaces, which are known to be true in the special case k = 0. We are pleasedthat Michael Eichmair [8] has announced proofs of these two conjectures (exceptfor the topological part of conjecture 3) using elliptic techniques (whereas Ilma-nen’s original ideas used parabolic techniques). We omit the n = 2 case in thesenext two conjectures because they are less relevant for our present purposes,but we understand that Eichmair’s results apply there as well. Conjecture 2 (Tom Ilmanen, 2006)Given complete, asymptotically flat Cauchy data ( M n , g, k ) , ≤ n ≤ , with ageneralized trapped surface Σ n − , then there exists a unique outermost general-ized trapped surface ¯Σ which is a generalized apparent horizon. Conjecture 3 (Tom Ilmanen, 2006)Furthermore, ¯Σ is strictly area outerminimizing (every other surface which en-closes it has larger area), and for n = 3 , the region exterior to ¯Σ is diffeomorphicto R minus a finite number of disjoint closed balls. The above conjecture is a generalization of Meeks-Simon-Yau [22], which isthe case when k = 0. The topological conclusions of this last conjecture, like theoriginal Meeks-Simon-Yau result, make this conjecture particularly interestingfor its own sake as well as important for the Jang-IMCF system of equationswe will describe later in the paper. We also encourage the reader to study therelated theorems of Andersson and Metzger [1] on future and past apparenthorizons, which are relevant for this discussion.All together, these considerations lead us to make the following generalizedPenrose conjecture. Since traditional apparent horizons are always generalizedtrapped surfaces, this conjecture implies the original Penrose conjecture. Conjecture 4 (The Generalized Penrose Conjecture)Suppose that the Cauchy data ( M , g, k ) is complete, satisfies the nonnegativeenergy density condition µ ≥ | J | , and is Schwarzschild at infinity with totalmass m in a chosen end. If Σ ∈ S is a generalized trapped surface, then m ≥ r A π , (19) where A is the area of the outermost minimal area enclosure ˜Σ = ∂U of Σ .Furthermore, equality occurs if and only if ( M \ U , g, k ) is the pullback of theCauchy data induced on the image of a space-like embedding of M \ U into theexterior region of a Schwarzschild spacetime (which maps ˜Σ to a generalizedapparent horizon). We note that this conjecture is true when k = 0 by [2]. In this case, Σ hasnonpositive mean curvature and acts as a barrier to imply the existence of anoutermost minimal area enclosure of Σ which is minimal.15t is important to note that conjectures 2 and 3 imply that the generalizedPenrose conjecture (and hence the original Penrose conjecture) follows from thefollowing important case of the generalized Penrose conjecture. Conjecture 5 (The Generalized Penrose Conjecture - Outermost Case)Suppose that the Cauchy data ( M , g, k ) is complete, satisfies the nonnegativeenergy density condition µ ≥ | J | , and is Schwarzschild at infinity with totalmass m in a chosen end. Suppose also that Σ = ∂U ∈ S is a strictly areaouterminimizing generalized apparent horizon, that no other generalized trappedsurfaces enclose it, and that M \ U is diffeomorphic to R minus a finitenumber of disjoint closed balls. Then m ≥ r A π , (20) where A is the area of Σ . Furthermore, equality occurs if and only if ( M \ U , g, k ) is the pullback of the Cauchy data induced on the image of a space-like embedding of M \ U into the exterior region of a Schwarzschild spacetime(which maps Σ to a generalized apparent horizon). Most of the remainder of this paper will focus on proving the above con-jecture. Naturally, when attempting a difficult conjecture, it makes sense toconsider the simplest case which still captures the essential subtleties of the prob-lem, and the above conjecture is arguably that case. We will focus on the threedimensional case in this paper, but the above conjecture is the same in higherdimensions up to and including seven, but without any hypothesis on the topol-ogy of M n \ U n , where the conjectured inequality becomes m ≥ c n A ( n − / ( n − - see [5].Furthermore, the condition of not having any generalized trapped surfacesoutside of Σ may turn out to be very important. There is reason to believethat the generalized Jang equation, which is a p.d.e. we will define later in thepaper which is central to all of our approaches, may blow up on surfaces with | H Σ | = | tr Σ ( k ) | (which are clearly generalized trapped surfaces). Since we areleaving the existence theory of our proposed p.d.e.’s which imply the Penroseconjecture open, those who are considering studying these existence theorieswill have to understand this possible behavior carefully. In this section we will prove the Penrose conjecture, conjecture 1, with twoextra assumptions, and show how the conjecture follows from the RiemannianPenrose inequality, theorem 3. This special case, where a correct approach isquite clear, will help us motivate the general case which is not so obvious.A major hint in the statement of the Penrose conjecture is the case of equal-ity. Since the Penrose conjecture is an equality for any slice (space-like hyper-surface) of the exterior region of the Schwarzschild spacetime with an apparent16orizon boundary, we know that all of our techniques must preserve this equalityin every estimate we derive.On the other hand, if we are given some Cauchy data ( M , g, k ) which comesfrom a slice of the Schwarzschild spacetime, it may be difficult to recognize itas such. However, our techniques must absolutely be able to recognize theseCauchy data as the instances where we get equality in all of our inequalities.More generally, suppose ( M , g, k ) comes from a slice of the static spacetime (cid:0) R × M , − φ dt + ¯ g (cid:1) , (21)where φ is a real-valued function on M and ¯ g is some other Riemannian metricon M . Notice that the Schwarzschild spacetime can be expressed in the form ofequation 21. However, while the Schwarzschild spacetime is vacuum (meaningit has zero Einstein curvature and consequently zero Ricci curvature), we aremaking no such requirement on (cid:0) R × M , − φ dt + ¯ g (cid:1) .Given a real-valued function f on M , define the graph map F : M R × M (22)where F ( x ) = ( f ( x ) , x ). Then a short calculation reveals that the pullback ofthe induced metric on the image of F in a coordinate chart is ¯ g ij − φ f i f j , sosetting ¯ g ij = g ij + φ f i f j , guarantees that the pullback of the induced metric on the image of the graphmap F is precisely g . A similar type of calculation (but which is much longerand so is carried out in the appendices) yields that the pullback of the secondfundamental form of the image of the graph map F in the static spacetime to( M , g ) is h ij = φHess ij f + φ i f j + f i φ j (cid:0) φ | df | g (cid:1) / , (23)where subscripts on f and φ represent coordinate chart partial derivatives andthe Hessian of f is taken with respect to the metric g (or the Levi-Civita con-nection of g if one prefers). These considerations lead us to the following specialcase of the Penrose conjecture which has an elegant and relatively short proofusing the Gauss-Codazzi identities and the Riemannian Penrose inequality. Theorem 4
The Penrose conjecture as stated in conjecture 1 follows for ( M , g, k ) if there exist two smooth functions f and φ on M such that k ij = h ij = φHess ij f + φ i f j + f i φ j (cid:0) φ | df | g (cid:1) / outside of Σ (24) and φ = 0 on Σ , (25) where φ > outside of Σ and f has compact support. roof: We will reduce the Penrose conjecture on ( M , g, k ) to the Rieman-nian Penrose inequality on ( M , ¯ g ). To do this we need to show that • the scalar curvature ¯ R of ¯ g is nonnegative and that ◦ Σ has zero mean curvature ¯ H in ( M , ¯ g ).Then the fact that ¯ g measures areas to be at least as large as g does impliesthat the area of any surface in ( M , ¯ g ) is at least as large as the area of thatsame surface in ( M , g ). Thus,¯ A := | ˜Σ ¯ g | ¯ g ≥ | ˜Σ ¯ g | g ≥ | ˜Σ g | g =: A, where ˜Σ ¯ g and ˜Σ g are the outermost minimal area enclosures of Σ in ( M , ¯ g )and ( M , g ), respectively. Since f has compact support, the masses of the twomanifolds are the same. Then by the Riemannian Penrose inequality m = ¯ m ≥ r ¯ A π ≥ r A π , which proves that Penrose conjecture on ( M , g, k ). Thus, all that is left toprove are the two bullet points ( • ) and ( ◦ ). Proof of ( ◦ ): Since ¯ g ij = g ij + φ f i f j and φ = 0 on Σ and φ and f are smooth,the two metrics are the same up to first order on Σ. But the mean curvatureof a surface, which is the main term in the first variation of area formula, onlydepends on the metric and the first derivatives of the metric. Hence, ¯ H = H .Since Σ is an apparent horizon, H = ± tr Σ ( k ). But since φ = 0 on Σ,derivatives along Σ of φ are zero as well, so our assumption on the special formof k in equation 24 implies that tr Σ ( k ) = 0. Hence,¯ H = H = tr Σ ( k ) = 0 . Proof of ( • ): Working inside of the static spacetime in equation 21, let n bethe future pointing normal vector to the image of M under the graph map F from equation 22 and let ¯ n be the future pointing normal vector to M viewedas the t = 0 slice of the static spacetime. Then these two vector fields onhypersurfaces can be extended to the entire spacetime by requiring that theseextended vector fields are invariant under translation in the time coordinate(which is an isometry of the spacetime).The trick is to compute G ( n, ¯ n ) using the Gauss-Codazzi identities, but intwo different ways. We are given the nonnegative energy density condition on( M , g, k ) that µ ≥ | J | . Since we are in the very special case that k actuallyequals the second fundamental form h of the graph, ( M , g, h ) has µ ≥ | J | too.This is equivalent to saying that G ( n, w ) ≥ w in the spacetime. Letting w = ¯ n thus implies that G ( n, ¯ n ) ≥ . (26)18n the other hand, applying the Gauss-Codazzi identities to the t = 0 slice ofthe static spacetime gives us(8 π ) ¯ µ = G (¯ n, ¯ n ) = ( ¯ R + tr(¯ p ) − k ¯ p k ) / π ) ¯ J = G (¯ n, · ) = div (¯ p − tr(¯ p )¯ g )where ¯ p is the second fundamental form of the t = 0 slice, which of course iszero by the time symmetry of the spacetime. Hence, (8 π )¯ µ = ¯ R/ J = 0.Thus, if we let n = α ¯ n + (vector tangent to t = 0 slice) , where α is a positive function on M , we have that G (¯ n, n ) = αG (¯ n, ¯ n ) = α ¯ R/ . (27)But G is symmetric, so by inequality 26, ¯ R ≥
0, which completes the proof of( • ) and the proof of the Penrose inequality in this special case.The case of equality of the above theorem would follow from conjecture 7in section 7. We refer the reader to that section for discussion on the case ofequality since the main purpose of this section was to motivate the identitiescomputed in the next section. The proof of the Penrose conjecture in the special case presented above suggestshow the Gauss-Codazzi identities can be used to compute a formula for the scalarcurvature ¯ R of ¯ g = g + φ df in terms of the scalar curvature R of g , the graphfunction f , and the warping factor φ . In this section we will derive this formulaand then show how this formula leads to an identity central to our approach tothe Penrose conjecture.From this point on we will abuse terminology slightly and always refer tothe image of the graph map F ( M ) simply as M and the t = 0 slice of theconstructed spacetime as ¯ M . This notation is convenient since then ( M, g ) and( ¯
M , ¯ g ) are space-like hypersurfaces of the spacetime (cid:0) R × M , − φ dt + ¯ g (cid:1) . Let π : M ¯ M be the projection map π ( f ( x ) , x ) = (0 , x ) to the t = 0 slice of thespacetime.Establishing some notation, let ¯ ∂ = ∂ t and { ¯ ∂ i } be coordinate vectorstangent to ¯ M . Define ∂ i = ¯ ∂ i + f i ¯ ∂ (28)to be the corresponding coordinate vectors tangent to M so that π ∗ ( ∂ i ) = ¯ ∂ i .Then in this coordinate chart, we have that g ij = ¯ g ij − φ f i f j . (29)It is convenient to write g ij = ¯ g ij + v i v j , (30)19 ¯ ∂ ( R × M , − φ ( x ) dt + ¯ g ( x )) t ∈ R x ∈ M R height = f M ( M , g, h )( ¯ M , ¯ g, n ∂ ∂ ∂ ¯ n = φ ∂ t ∂ ¯ ∂ ( , x )( f ( x ) , x )( , x ) F ( x ) = ( f ( x ) , x ) π Figure 1: Schematic diagram of the constructed static spacetimewhere v i = φf ¯ i (1 − φ | df | g ) / = φf i (1 + φ | df | g ) / . (31)We also define ¯ v = v i ¯ ∂ i and v = v i ∂ i (32)so that π ∗ ( v ) = ¯ v , and observe the useful identity(1 − φ | df | g ) · (1 + φ | df | g ) = 1 , (33)which is evident by looking at the ratios of the volume forms. See appendix Cfor more discussion on these calculations.In this paper we use the convention that a barred index (as in f ¯ i above)denotes an index raised (or lowered) by ¯ g as opposed to g . That is, f ¯ i =¯ g ij f j , where as usual f j = ∂f /∂x j in the coordinate chart. In general, barredquantities will be associated with the t = 0 slice ( ¯ M , ¯ g ) and unbarred quantitieswill be associated with the graph slice ( M, g ).In appendix D we compute that the second fundamental form of the graphslice (
M, g ) in our constructed static spacetime is h ij = φ Hess ij f + ( f i φ j + φ i f j ) − φ h df, dφ i ¯ g f i f j (1 − φ | df | g ) / (34)= φ Hess ij f + ( f i φ j + φ i f j )(1 + φ | df | g ) / , (35)20hich we list now for future reference.Finally, we extend h and k trivially in our constructed static spacetimeso that h ( ∂ t , · ) = 0 = k ( ∂ t , · ) and such that these extended 2-tensors equalthe original 2-tensors when restricted to M . Note that this gives h ( ∂ i , ∂ j ) = h ( ¯ ∂ i , ¯ ∂ j ), so we can call this term h ij without ambiguity. The same is true for k ij and components of 1-forms like f i and φ i . However, we remind the reader thatthe Hessian of a function, which is the covariant derivative of the differential ofa function, depends on the connection and hence the metric since we will alwaysbe using the respective Levi-Civita connections on ( M, g ) and ( ¯
M , ¯ g ).Now we are ready to proceed to compute a formula for ¯ R . It is a shortcalculation to verify that, in the constructed spacetime, h n, ¯ n i = − (1 − φ | df | g ) − / = − (1 + φ | df | g ) / (36)Thus, ¯ n = (1 + φ | df | g ) / n + tangraph(¯ n ) (37)where another short calculation reveals thattangraph(¯ n ) = − φf j ∂ j = − φ ∇ f. (38)As in the previous section, the trick is to compute G ( n, ¯ n ) two different waysusing the Gauss-Codazzi identities. As before, applying these identities to the t = 0 slice ( ¯ M , ¯ g ) of the constructed spacetime gives us G ( n, ¯ n ) = (1 + φ | df | g ) / G (¯ n, ¯ n )= (1 + φ | df | g ) / · ¯ R/ t = 0 slice has zero second fundamental form. On the other hand,applying the Gauss-Codazzi identities to the graph slice ( M, g ) yields G ( n, ¯ n ) = (1 + φ | df | g ) / G ( n, n ) + G ( n, tangraph(¯ n ))= (1 + φ | df | g ) / [ R + (tr g h ) − k h k g ] / h − (tr g h ) g )( − φ ∇ f ) . Combining the two previous equations, we get our first desired result¯ R = R + (tr g h ) − k h k g + 2( d (tr g h ) − div( h ))( v ) . (39)Of course, what we are given in the hypotheses of the Penrose conjecture isthat µ ≥ | J | g , where(8 π ) µ = G ( n, n ) = ( R + tr g ( k ) − k k k ) / π ) J = G ( n, · ) = div( k ) − d (tr g ( k )) , for some symmetric 2-tensor k . Hence,¯ R = 16 π ( µ − J ( v )) + (tr g h ) − (tr g k ) − k h k g + k k k g +2 v (tr g h ) − v (tr g k ) − h )( v ) + 2 div( k )( v ) . (40)21ote that µ − J ( v ) ≥ | v | g ≤
1. Hence, as we saw in the previoussection, if we can choose a φ and an f so that h = k , then we immediatelyget that ¯ R ≥
0. However, we are interested in investigating if a more generalrelationship between h and k can give a similar result.Our procedure is to convert our formula for ¯ R to an expression in terms of the¯ g metric. Arguably ¯ g is more natural than g since it is the metric induced on the t = 0 slice of the static spacetime. To perform the conversion, we need severalidentities for arbitrary symmetric 2-tensors k which are proven in appendix Dand which we list here. Identity 1 ( tr g ( k )) − k k k g = ( tr ¯ g k ) − k k k g + 2 k (¯ v, ¯ v ) tr ¯ g k − | k (¯ v, · ) | g Identity 2 v ( tr g k ) = ¯ v ( tr ¯ g k + k (¯ v, ¯ v )) Identity 3 Γ kij − Γ kij = h ij v k − φf i f j φ ¯ k Identity 4 div ( k )( v ) = div ( k )(¯ v ) + ( ∇ ¯ v k )(¯ v, ¯ v ) − | ¯ v | g k (cid:18) ¯ v, ∇ φφ (cid:19) + h h (¯ v, · ) , k (¯ v, · ) i ¯ g + 2 h (¯ v, ¯ v ) k (¯ v, ¯ v ) + ( tr ¯ g h ) k (¯ v, ¯ v ) Identity 5 v ¯ ı ¯; j = h ij + v ¯ ı h (¯ v, · ) j − φ i v ¯ φ Identity 6 div ( k )(¯ v ) = div ( k (¯ v, · )) − h h, k i ¯ g − h h (¯ v, · ) , k (¯ v, · ) i ¯ g + k (cid:18) ¯ v, ∇ φφ (cid:19) Identity 7 ( ∇ ¯ v k )(¯ v, ¯ v ) = ¯ v ( k (¯ v, ¯ v )) − h h (¯ v, · ) , k (¯ v, · ) i ¯ g − h (¯ v, ¯ v ) k (¯ v, ¯ v ) + 2 | ¯ v | g k (cid:18) ¯ v, ∇ φφ (cid:19) Identity 8 div ( k )( v ) = div ( k (¯ v, · )) + ¯ v ( k (¯ v, ¯ v )) + k (cid:18) ¯ v, ∇ φφ (cid:19) −h h, k i ¯ g − h h (¯ v, · ) , k (¯ v, · ) i ¯ g + ( tr ¯ g h ) k (¯ v, ¯ v )Identities 1 and 2 are short calculations. Identity 3 is used in the proof ofidentity 4. Plugging identities 6 and 7 (which are proved using identity 5) intoidentity 4 results in identity 8. Finally, plugging identities 1, 2, and 8 into ourformula for ¯ R results in the main identity of this paper.22 dentity 9 ( The Generalized Schoen-Yau Identity ) ¯ R = 16 π ( µ − J ( v )) + k h − k k g + 2 | q | g − φ div ( φq )+( tr ¯ g h ) − ( tr ¯ g k ) + 2¯ v ( tr ¯ g h − tr ¯ g k ) + 2 k (¯ v, ¯ v )( tr ¯ g h − tr ¯ g k ) where q = h (¯ v, · ) − k (¯ v, · ) = h ( v, · ) − k ( v, · ) . Note that the two definitions of q exist on the entire constructed static spacetimeand are equal since both h and k are extended trivially in the constructed staticspacetime. We also observe that1 φ div( φq ) = div ST ( q ) , where div ST is the divergence operator in the constructed static spacetime.In the special case that φ = 1, the above identity was derived by a differentmethod by Schoen-Yau as equation 2.25 of [27] (in fact the procedure in [27]may also be used to obtain identity 9 and will be presented in a future paper).In that paper, they used the Jang equation,0 = tr ¯ g ( h − k )to reduce the positive mass theorem to the Riemannian positive mass theorem.While imposing the Jang equation in the special case that φ = 1 does not implythat ¯ R ≥ R ≥ | q | g − q ) implies that thereexists a conformal factor on ¯ g such that the conformal metric has nonnegativescalar curvature and total mass less than or equal to that of ¯ g and g . Then theRiemannian positive mass theorem applied to the metric conformal to ¯ g impliesthe positive mass theorem on ( M, g ). This approach does not quite work for thePenrose conjecture because the conformal factor needed to achieve nonnegativescalar curvature changes the area of the horizon in a way which is difficult tocontrol.
All of the approaches to the Penrose conjecture that we consider in this paperuse the generalized Schoen-Yau identity. This identity plays a central role in theremainder of our discussions because it directly relates the nonnegative energycondition on ( M , g, k ) (which implies that µ ≥ J ( v ) since | v | g <
1) to thescalar curvature of ( M , ¯ g ).Furthermore, this generalized Schoen-Yau identity strongly motivates thegeneralized Jang equation, 0 = tr ¯ g ( h − k ) , (41)23hich on the original manifold ( M , g ) with Cauchy data ( M , g, k ) is the equa-tion 0 = (cid:18) g ij − φ f i f j φ | df | g (cid:19) φHess ij f + φ i f j + f i φ j (cid:0) φ | df | g (cid:1) / − k ij ! (42)when one substitutes the formulas for h and ¯ g ij in a coordinate chart. (In thispaper we adopt Einstein’s convention that whenever there are both raised andlowered indices, summation is implied, so the above formula is a summationover i, j both ranging from 1 to 3.)Of course the original Jang equation, which again is the special case φ ( x ) = 1,only had one free function, f , whereas the generalized Jang equation has twofree functions, f and φ . Hence, to get a determined system of equations, weneed to specify one more equation. Later in the paper we will propose variouschoices for this second equation, but our choice for the first equation will alwaysbe the generalized Jang equation above.Once the generalized Jang equation is specified, the generalized Schoen-Yauidentity simplifies greatly to¯ R = 16 π ( µ − J ( v )) + k h − k k g + 2 | q | g − φ div( φq ) . (43)It is important to note that the first three terms of the right hand side of theabove equation are all nonnegative since µ ≥ | J | g and | v | g < Examining the case of equality slices of the Schwarzschild spacetime describedin section 2 leads us to propose the following boundary conditions on generalizedapparent horizons. At a minimum, these boundary conditions are satisfied al-most everywhere for slices of the exterior region of the Schwarzschild spacetimewith generalized apparent horizon boundaries.
Boundary Conditions on Generalized Apparent Horizons
Given a generalized apparent horizon Σ with mean curvature H Σ = | tr g Σ ( k ) | andoutward unit normal ν in ( M , g ) , we require that φ = 0 and h ν, v i g = sign (tr g Σ ( k )) (44) on Σ , where as usual v = φ ∇ f (1 + φ | df | g ) / and v is extended to the boundary Σ by continuity. Note that these boundary conditions are consistent with f blowing up to + ∞ where tr g Σ ( k ) <
0, blowing down to −∞ where tr g Σ ( k ) >
0, and f stayingbounded where tr g Σ ( k ) = 0 on Σ. 24he hope is that these or similar boundary conditions imply that Σ, whichwas a generalized apparent horizon in ( M , g ), becomes a minimal surface with¯ H Σ = 0in ( M , ¯ g ). We discuss the general case of this question in appendix E. Fornow, we observe two important special cases.The first important special case is when Σ is a traditional apparent horizon,either future or past, and H Σ >
0. If we also assume that f goes to ±∞ oneach connected component of Σ in a reasonable fashion, then the level sets of f converge to Σ. The formula for the mean curvature of the level sets of f in thenew metric ¯ g is ¯ H = (1 + φ | df | ) − / H = (1 − | v | g ) / H since ¯ g = g + φ df does not change the metric on the level sets of f , stretcheslengths perpendicular to the level sets of f by a factor of (1 + φ | df | ) / , andby the first variation formula for area. Then if we assume that f and φ behavesimilarly to the case of equality slices of Schwarzschild, we get the followinglemma. Lemma 1
Suppose that ( M , g, k ) has a smooth interior boundary Σ whichis a future [past] apparent horizon with H Σ > . Then if f blows up [blowsdown] logarithmically, | df | blows up asymptotic to /s , and φ goes to zeroasymptotic to s (where s is the distance to Σ in ( M , g ) ), then the limit of themean curvatures ¯ H of the level sets of f in ( M , ¯ g ) is zero. The second important special case is the case where the boundary is a futureand past apparent horizon. In this case, based on the case of equality slices ofSchwarzschild, we expect f to stay bounded and smooth and φ to stay smoothas well. Lemma 2
Suppose that ( M , g, k ) has a smooth interior boundary Σ whichis a future and past apparent horizon (which by definition has H Σ = 0 and tr g Σ ( k ) = 0 ). Then if f is bounded and smooth and φ is smooth and equals zeroon Σ , then ¯ H Σ = H Σ = 0 . The proof of this lemma appeared in this paper already in section 4. The pointis that since ¯ g = g + φ df , both metrics ¯ g and g are the same up to first orderon Σ since f and φ are smooth and φ = 0 on Σ. Then since the mean curvatureis only a function of the metric and first derivatives of the metric, the two meancurvatures are equal, and since H Σ = 0, both are zero. One phenomenon of the original Jang equation ( φ = 1) is that f can blowup to ∞ on future apparent horizons or blowdown to −∞ on past apparent horizons,25nd this feature is still present in the generalized Jang equation (given plausibleassumptions about the behavior of f and φ ). More importantly, according to[27], blowups and blowdowns of f with the original Jang equation can only occuron apparent horizons.An important question, then, is to understand when blowups can occurwith the generalized Jang equation. Certainly blowups and blowdowns can stilloccur on traditional apparent horizons. A reasonable conjecture is that theblowup properties of the generalized Jang equation are the same as the originalJang equation as long as φ is smooth and strictly positive. However, if φ isallowed to go to zero, then we have already seen that f can have a mixtureof blowup, blowdown, and bounded behavior on generalized apparent horizonsin case of equality slices of the Schwarzschild spacetime. Those who study theexistence theories of the systems of equations proposed in this paper will needto understand these issues.A reasonable hope, however, is that as long as our boundary Σ is already anoutermost generalized apparent horizon, so that no other generalized trappedsurfaces enclose Σ, then f stays bounded away from Σ as long as φ stays strictlypositive. A relevant calculation which is useful for studying the question of when f can blowup or blowdown is the following.Recall the standard identity∆ f = Hess f ( ν, ν ) + H Σ · ν ( f ) + ∆ Σ f for the Laplacian of a function in terms of the Laplacian of that function re-stricted to a hypersurface with mean curvature H and outward unit normal ν .If we let Σ be any level set of f , then we get thattr g Σ (Hess f ) = ∓| df | g H Σ for blowup and blowdown respectively. Thus, the generalized Jang equationimplies that 0 = tr ¯ g ( h − k )= ¯ g ij ( h ij − k ij )= (cid:2) ( g ij − ν i ν j ) + ( ν i ν j − v i v j ) (cid:3) ( h ij − k ij )= tr g Σ ( h − k ) + ( h − k )( ν, ν )1 + φ | df | g = φ tr g Σ (Hess f )(1 + φ | df | g ) / − tr g Σ ( k ) + ( h − k )( ν, ν )1 + φ | df | g = ∓ φ | df | g (1 + φ | df | g ) / H Σ − tr g Σ ( k ) + ( h − k )( ν, ν )1 + φ | df | g = h ν, v i g H Σ − tr g Σ ( k ) + ( h − k )( ν, ν )1 + φ | df | g
26n level sets of f , where we have used the facts that v and ν are collinear, | v | g = 1 −
11 + φ | df | g , and the formulas for ¯ g ij and h ij from section 5. Lemma 3
When f is blowing up or blowing down, the term h ( ν, ν ) is boundedif φ df is assumed to be smooth and nonzero in the limit up to the boundary and φ = 0 on the boundary, which is true in case of equality slices of Schwarzschild.Proof: Referring back to section 2, in a smooth slice of Schwarzschild φ df can be expressed in terms of the smooth Kruskal coordinate variables u, v . Sinceby equation 15 df = 2 m (cid:18) dvv − duu (cid:19) and by equation 16 φ = uvγ ( uv )for some smooth function γ = 0 for φ <
1, we have that φ df = 2 mγ ( uv ) ( udv − vdu ) . The fact that our case of equality slices of Schwarzschild are spacelike impliesthat | du | g = 0 = | dv | g , and the fact that we are assuming blowup or blowdownimplies that exactly one of u, v is going to zero on the boundary. Hence, notonly is φ df smooth up to the boundary, it is also nonzero in the limit up tothe boundary.Then the fact that h ( ν, ν ) is bounded up to the boundary assuming thissmoothness follows from the short calculation that h ( ν, ν ) = ∇ ν ( φ df )( ν )( φ + | φ df | g ) / , (45)which completes the proof of the lemma.Thus, referring back to our calculation before the lemma, given blowup orblowdown with behavior on f and φ as seen in the case of equality slices ofSchwarzschild in section 2, φ goes to zero linearly, | df | g goes to infinity like 1 /s so that φ | df | g goes to infinity like 1 /s , and h ( ν, ν ) stays bounded. Then since k is given to be smooth and therefore bounded, we conclude that the generalizedJang equation implies that 0 = ∓ H Σ − tr g Σ ( k )on surfaces with this type of blowup or blowdown of f , which of course are theequations for future and past apparent horizons, respectively.27 The Jang - Zero Divergence Equations
Looking at equation 43, the most direct way to get ¯ R ≥ φq ) = 0.We will call the resulting system of equations, equations 46 and 47, the Jang- zero divergence equations. The following existence conjecture for these equa-tions implies the outermost case of the generalized Penrose conjecture, conjec-ture 5, using the Riemannian Penrose inequality, and is therefore an importantopen problem. Conjecture 6
Given asymptotically flat Cauchy data ( M, g, k ) with an outer-most generalized apparent horizon boundary Σ , there exists a solution ( f, φ ) tothe system of equations tr ¯ g ( h − k ) (46)0 = div ( φq ) , (47) with lim x →∞ f ( x ) = 0 , φ |∇ f | = o ( r − ) , ∇ ( φ |∇ f | ) = o ( r − ) ,and lim x →∞ φ ( x ) = 1 , where ¯ g = g + φ df , h = φ Hess f + ( df ⊗ dφ + dφ ⊗ df )(1 + φ | df | g ) / , (48) q = h ( v, · ) − k ( v, · ) , and v = φ ∇ f / (1 + φ | df | g ) / , such that Σ has zero meancurvature in the ¯ g metric. The boundary conditions on f which lead to Σ having zero mean curvature in( M , ¯ g ) are discussed in the previous section and in appendix E.Also, we comment that while equation 47 is third order in f , subtractingderivatives of equation 46 can remove the third order terms of f in favor of Riccicurvature terms. While the resulting system has quadratic second order termsin f in the second equation, the system is degenerate elliptic.The above system may also be reduced to a system of 1st order equationsby introducing new variables. If we let α = df , then the above system hasa solution whenever the first order system with variables φ (a 0-form), α (a1-form), and β (a 2-form) 0 = dα (49)0 = dβ (50)0 = tr ¯ g ( h − k ) (51) φq = d ¯ ∗ β (52)has a solution, where d ¯ ∗ is the d star operator with respect to the ¯ g metricwhich sends 2-forms to 1-forms. In these variables, ¯ g = g + φ α , h = φ ∇ α + ( α ⊗ dφ + dφ ⊗ α )(1 + φ | α | g ) / , (53) q = h ( v, · ) − k ( v, · ), and v = φ~α/ (1 + φ | α | g ) / , where ~α is the dual vector to α with respect to g . 28 heorem 5 Conjectures 6 and 7 (defined below to handle the case of equality)imply conjecture 5, the outermost case of the generalized Penrose conjecture.Proof:
As was discussed in the previous section, the point of requiring Σ to beoutermost in the above conjecture is so that f does not blowup or blowdown onthe interior of M . Then the method of proof assuming conjecture 6 is basicallythe same as the proof of the Penrose conjecture in the special case in section4. The total mass of ( M , g ) is the same as the total mass of ( M , ¯ g ) since thetotal mass is defined in terms of the 1 /r rate of decay of the metrics which areequal since || ¯ g − g || g = φ | df | g . Also, since ¯ g measures lengths, areas, etc. tobe greater than or equal to that measured by g ,¯ A = | ˜Σ ¯ g | ¯ g ≥ | ˜Σ ¯ g | g ≥ | ˜Σ g | g = A. (54)Hence, the Penrose conjecture on ( M , g, k ) follows from the Riemannian Pen-rose inequality on ( M , ¯ g ).Thus, all that remains is to show that the Riemannian Penrose inequality canbe applied to ( M , g ). The existence theorem already gives us that ¯ H = 0, so thelast thing to check is that ¯ R ≥
0, which follows directly from the generalizedSchoen-Yau identity and equation 43. This proves the inequality part of thePenrose conjecture on ( M , g, k ).In the case of equality of the Penrose conjecture, clearly we must have equal-ity in all of our inequalities. Since the case of equality of the Riemannian Pen-rose inequality is solely when ¯ g is the Schwarzschild metric which has zero scalarcurvature ¯ R , equation 43 gives us0 = 16 π ( µ − J ( v )) + k h − k k g + 2 | q | g . (55)Since each of these three terms is nonnegative, each must be zero. Hence, k = h . If we could argue that φ = φ , where φ is the warping factor fromthe Schwarzschild spacetime, then we would have that k = h is the secondfundamental form and g = ¯ g − φ df is the induced metric of a slice of aSchwarzschild spacetime, as desired.However, there is a delicate point here. In fact, φ does not have to equal φ for ( M , g, k ) to be the Cauchy data from a slice of a Schwarzschild space-time. If φ = cφ for some constant c >
0, then defining df = c df (which canbe integrated to recover f ) implies that ( f , φ ) and ( f, φ ) produce the samemetrics and second fundamental forms. This may seem like a minor point atfirst, but in fact this statement is still true if (and only if which we leave as anexercise) dc = 0 on the open region D where df = 0. Thus, c may be differentconstants on each connected component of D . Again, df , which is still closed,may be integrated to recover f since the t = 0 slice of Schwarzschild is simplyconnected. Thus, we have the following lemma. Lemma 4 If φ = cφ , where dc = 0 on { x | df = 0 } , then ( M , g, k ) comesfrom a slice of the Schwarzschild spacetime.
29o prove the case of equality of the Penrose conjecture then, we need toprove the hypotheses of the above lemma. Looking back at equation 55, we seethat we must also have0 = µ − J ( v ) = ( µ − | J | ) + | J | (1 − | v | ) + ( | J || v | − J ( v )) , where all norms are with respect to g . Again, since each of the three groupedterms is nonnegative, all must be zero. Since | v | <
1, the second term equalingzero implies that | J | = 0 so that the first term equalling zero implies that µ = 0.In the appendices we compute that in the static spacetime ¯ g − φ dt , n = (1 − φ | df | g ) / (¯ n + φ ∇ f )and that if ¯ R = 0, J = G ( n, · ) = (1 − φ | df | g ) / (cid:20) Ric − Hess φφ + ¯∆ φφ ¯ g (cid:21) ( φ ¯ ∇ f, · )where · is a tangent vector to the graph slice ( M , g ) in the first instance and itscomponent tangent to the t = 0 slice in the second. Then since the Schwarzschildspacetime has G = 0, ¯ R = 0, ¯∆ φ = 0, andRic = Hess φ φ ,J = 0 in the case of equality implies the overdetermined equation (when df = 0)for φ that 0 = (cid:20) Hess φ φ − Hess φφ + (cid:18) ¯∆ φφ − ¯∆ φ φ (cid:19) ¯ g (cid:21) ( ¯ ∇ f, · ) . (56) Conjecture 7
Equation 56 implies the hypotheses of lemma 4.
Clearly the hypotheses of lemma 4 imply equation 56, but we need theconverse to be true as well. Assuming conjecture 6 is true, a proof of conjecture7 would finish the case of equality part of the outermost case of the generalizedPenrose conjecture.
Equation 43 is a remarkable equation which deserves very careful consideration.Since we need a lower bound on the scalar curvature ¯ R of ( M , ¯ g ), the onlytroublesome term in that equation is the last one, the divergence term. In theprevious section, we dealt with this last term by setting it equal to zero. Inthis section, we make the natural observation that divergence terms can also bedealt with by integrating them. 30 heorem 6 If ¯ g = g + φ df on M with boundary Σ and Cauchy data ( M , g, k ) satisfying the nonnegative energy condition µ ≥ | J | , the generalizedJang equation ¯ g ( h − k ) is satisfied, and f and φ behave at infinity and on the boundary Σ such thatequations 60 and 61 are satisfied as expected, then Z M ¯ Rφ dV ≥ , (57) where ¯ R is the scalar curvature of ¯ g and dV is the volume form of ¯ g . In other words, no matter what φ ( x ) is (as long as certain boundary condi-tions are satisfied), the generalized Jang equation by itself already gives a lowerbound on the integral of the scalar curvature of ¯ g , weighted by φ . Of course thechoice of φ affects f since φ appears in the generalized Jang equation.In the next couple of sections we discuss two different inequalities of theform ¯ m − r ¯ A π ≥ Z M Q ( x ) ¯ R ( x ) dV , (58)for some Q ( x ) ≥
0, where each inequality is based on one of the two proofs ofthe Riemannian Penrose inequality. The expression for Q ( x ) differs in the twocases and will be described later. However, if we then choose φ ( x ) = Q ( x ) tobe our second equation to be coupled with the generalized Jang equation, thenexistence of such a system implies m − r A π ≥ ¯ m − r ¯ A π ≥ Z M Q ( x ) ¯ R ( x ) dV = Z M ¯ Rφ dV ≥ , (59)proving the corresponding form of the Penrose conjecture for the original Cauchydata ( M , g, k ). We will call any method of proof as above an Einstein-Hilbertaction method. So far we know of only two methods of this form, the Jang-IMCF equations presented in the next section, and the Jang-CFM equationsdiscussed in the section after that.Note that any inequality of the form of inequality 58 proves the RiemannianPenrose inequality for ( M , ¯ g ) as a special case since then ¯ R ≥ Proof of theorem 6:
Applying the divergence theorem to equation 43 givesus that Z M ¯ Rφ dV ≥ Z Σ − S ∞ φq (¯ ν ) dA = 2 Z Σ φ ( h − k )(¯ v, ¯ ν ) dA, k is assumed to converge to zero at infinity (or have compact support),and df is assumed to decay at least as fast as 1 /r at infinity (with reasonablebounds on Hess f as well) so that0 = lim r →∞ Z S r φ ( h − k )(¯ v, ¯ ν ) dA, (60)where ¯ ν is the unit outward normal vector to Σ and the sphere at infinity in( M , ¯ g ).In appendix E, we observe that¯ ν = φ | df | g φ | ( df | Σ ) | g ! / ( ν − h ν, v i v )and dA = (1 + φ | ( df | Σ ) | g ) / dA. Then since 1 + φ | df | g = 1 / (1 − | v | g ), we get that Z M ¯ Rφ dV ≥ Z Σ φ ( h − k )(¯ v, ¯ ν ) dA = 2 Z Σ φ ( h − k )( v, ν − h ν, v i v )(1 − | v | g ) / dA = 0if we assume that 0 = lim Σ ǫ → Σ Z Σ ǫ φ ( h − k )( v, ν − h ν, v i v )(1 − | v | g ) / dA (61)for some smooth family of surfaces Σ ǫ converging to Σ, proving the theorem.In the case that f is blowing up (or down) everywhere on Σ, then choosingΣ ǫ to be the level sets of f simplifies things even more since then v = ±| v | ν .Then the integrand becomes φ (1 − | v | g ) / ( h − k )( v, ν ) and equals zero with theusual boundary behavior since φ = 0 on the boundary, | v | g is going to one onthe boundary, and both h ( ν, ν ) and k ( ν, ν ) are bounded by lemma 3. Equation61 is also clearly satisfied in the case of a future and past apparent horizonwhere we assume that f and φ stay smooth and bounded, since all of the termsin the integrand will be bounded, and φ = 0 on the boundary Σ.As a final comment on theorem 6 before moving on, we fully admit thata better understanding of the boundary behavior of f and φ is needed. Thisbetter understanding should be able to be achieved when the existence theoriesfor the equations we are proposing are discovered.Before we get into applications of this theorem, it is worth noting that E (¯ g, φ ) = Z M ¯ Rφ dV (cid:0) S × M, − φ dt + ¯ g (cid:1) , where we have turned the usual R time coordinate into an S of length one to geta finite integral. The Einstein-Hilbert action is defined to be the total integralof the scalar curvature R ST of the spacetime. In the appendix we observe that R ST = ¯ R − φφ , and since dV ST = φ dV , the Einstein Hilbert action of the quotiented spacetimeis Z S × M R ST dV ST = Z M ( ¯ Rφ − φ ) dV = Z M ¯ Rφ dV − Z ∂M h∇ φ, ¯ ν i ¯ g dA. We further observe that the boundary term vanishes when φ is harmonic on( M, ¯ g ), as is the case in the Schwarzschild spacetime.Finally, the vacuum Einstein equation G = 0 is the Euler-Lagrange equationwhich results from requiring a spacetime to be a critical point of the Einstein-Hilbert action. Since the Minkowski and Schwarzschild spacetimes are the onlyvacuum static spacetimes (with no boundary or black hole boundary) [6], itfollows that they are the only two static spacetimes which are critical points ofthe Einstein-Hilbert action, or equivalently E ( g, φ ), since boundary terms areirrelevant for variations away from the boundary. In this section we show how inverse mean curvature flow in ( M , ¯ g ) can be usedto determine a warping factor φ for the generalized Jang equation to get a systemof equations which, when there are solutions, implies the Penrose conjecture fora single black hole when H ( M ) = 0. Alternatively, the method presentedhere has the potential to address the outermost case of the Penrose conjectureas stated in conjecture 5, with the additional assumption that the outermostgeneralized apparent horizon is connected. We will call the system of equationswe are proposing in this section the Jang-IMCF equations. An important openproblem is to find an existence theory for these equations.Before we state the Jang-IMCF equations, we need to review inverse meancurvature flow. As introduced by Geroch [9] and Jang-Wald [16], a smoothfamily of surfaces Σ( t ) in ( M , ¯ g ) is said to satisfy inverse mean curvature flow ifthe speed in the outward normal direction of the family of surfaces as t increasesat each point is equal to 1 / ¯ H , where ¯ H > t ) is nondecreasing when ( M , ¯ g ) has nonnegative scalarcurvature ¯ R .To be more precise, define the Hawking mass of a surface Σ in ( M , ¯ g ) to be m H (Σ) = r | Σ | ¯ g π (cid:18) − π Z Σ ¯ H dA (cid:19) , where all quantities are computed in ( M , ¯ g ). Then we can compute the rateof change of the Hawking mass of a surface when flowed out orthogonally withspeed η = 1 / ¯ H in ( M , ¯ g ) by using the first variation formula ddt ( dA ) = ( η ¯ H ) dA = dA, the second variation formula ddt ¯ H = − ∆ η − k II k g η − Ric(¯ ν, ¯ ν ) η, and the Gauss equationRic(¯ ν, ¯ ν ) = 12 ¯ R − ¯ K + 12 ¯ H − k II k g , where II is the second fundamental form of Σ in ( M , ¯ g ), Ric is the Ricci cur-vature of ( M , ¯ g ), and ¯ K is the Gauss curvature of Σ, to get ddt ( m H (Σ( t ))) = r | Σ( t ) | ¯ g π "
12 + 116 π Z Σ( t ) |∇ ¯ H | g ¯ H + ¯ R − K + k II k g −
12 ¯ H . The essential assumption that Σ( t ) is connected is used to conclude that Z Σ( t ) ¯ KdA = 2 πχ (Σ( t )) ≤ π by the Gauss-Bonnet formula, which, along with k II k g ≥
12 tr(II) = 12 ¯ H allows us to conclude that ddt ( m H (Σ( t ))) ≥ r | Σ( t ) | ¯ g π Z Σ( t ) ¯ R π dA. If ( M , ¯ g ) has nonnegative scalar curvature, then the above equation impliesthat the Hawking mass of the smooth family of surfaces determined by inversemean curvature flow is nondecreasing. However, we get a more general result ifwe integrate the above equation in t and use the co-area formula dAdt = 1 η dV = ¯ HdV
34o conclude that for a smooth family of surfaces satisfying inverse mean curva-ture flow which foliates M , that m H (Σ( ∞ )) − m H (Σ(0)) = Z M ¯ H r | Σ( t ) | ¯ g π (cid:18) ¯ R π (cid:19) dV where at each point x ∈ M , ¯ H is the mean curvature of the surface Σ( t ) throughthe point x . Then assuming that Σ(0) has ¯ H = 0 and is area outerminimizingand that ( M , ¯ g ) is sufficiently asymptotically flat, we conclude our main resultthat ¯ m − r ¯ A π ≥ Z M Q ( x ) (cid:18) ¯ R ( x )16 π (cid:19) dV ( x ) (62)where ¯ A is the area of Σ = Σ(0), the area outerminimizing minimal boundaryof M , and Q ( x ) = ¯ H r | Σ( t ) | ¯ g π . More generally, Huisken-Ilmanen [14] observed that there exists a weak no-tion of inverse mean curvature flow in which the surfaces Σ( t ) jump outwardto their outermost minimal area enclosures whenever they are not already thatsurface. A key step in their approach is to represent the family of surfacesΣ( t ) as the levels sets of a real-valued function u ( x ) on M called the level setfunction. Then if Σ( t ) = ∂ { x | u ( x ) ≤ t } , it follows that η = 1 / |∇ u | ¯ g and¯ H = div (cid:18) ∇ u |∇ u | ¯ g (cid:19) , so that inverse mean curvature flow on the level sets of u ( x ) is equivalent todiv (cid:18) ∇ u |∇ u | ¯ g (cid:19) = |∇ u | ¯ g . (63)Huisken-Ilmanen then proceed to define a notion of weak solutions to the abovelevel set equation using an energy minimization technique. These solutions have“jump regions” where ∇ u = 0 corresponding to where the family of surfaces Σ( t )is not continuously varying but instead “jumps” over these regions. Further-more, Huisken-Ilmanen, using elliptic regularization, proved that weak solutionsof their inverse mean curvature flow always exist. We refer the reader to theirbeautiful work [14]. However, using their generalized inverse mean curvatureflow, we achieve the following theorem. Theorem 7
Given an asymptotically flat ( M , ¯ g ) with H ( M ) = 0 and a min-imal connected boundary Σ which bounds an interior region, then ¯ m − r ¯ A π ≥ Z M Q ( x ) (cid:18) ¯ R ( x )16 π (cid:19) dV ( x ) (64)35 here ¯ A is the area of the outermost minimal area enclosure ˜Σ = ∂U of Σ , ¯ R is the scalar curvature and dV is the volume form of ¯ g , and Q = |∇ u | ¯ g r ¯ Ae u π , (65) where u ( x ) is a weak solution to Huisken-Ilmanen inverse mean curvature flowequalling zero on Σ .Proof: As described in Huisken-Ilmanen’s paper, if Σ is not already its ownoutermost minimal area enclosure, it immediately jumps to it. During this initialjump, but only on this first jump, the area of the surface may decrease. Hence,¯ A must be defined to be the area of the outermost minimal area enclosure ˜Σ ofΣ, which also has zero mean curvature by the maximum principle using Σ asa barrier. Also, since H ( M ) = 0, it follows that ˜Σ is connected. Since eachcomponent of ˜Σ bounds a region, it follows that if ˜Σ did have more than oneconnected component, all of the components except for one could be removed(by either filling in holes or removing disconnected regions), thereby decreasingthe area. Then starting the flow at ˜Σ, our previous calculations generalize.The condition that H ( M ) = 0 is also used to guarantee that each Σ( t ) isconnected after each jump and therefore has Euler characteristic ≤ t ).By the first variation formula mentioned earlier in this section, inverse meancurvature flow grows the area form exponentially. Hence, | Σ( t ) | ¯ g = ¯ Ae t . Thus, we have that Q ( x ) = div (cid:18) ∇ u |∇ u | ¯ g (cid:19) r ¯ Ae u π , which equals the desired result by equation 63.Theorem 7 deserves careful consideration. In the case that ¯ R ≥
0, we re-cover a Riemannian Penrose inequality for a single black hole. More generally,however, since ¯ R/ π is energy density, we see that we have a kind of integralof energy density on the right hand side of equation 64, modified by the factor Q ( x ). On the flat metric on R and starting inverse mean curvature flow at apoint, Q ( x ) = 1, and on the Schwarzschild metric, Q ( x ) is the harmonic functiongoing to one at infinity and equally zero on the minimal neck. This last fact,which can be verified by direct calculation, will turn out to be important sincethis harmonic function also equals the warping factor φ ( x ) in the Schwarzschildmetric. 36heorem 7 and theorem 6 together motivate the system of equations,0 = tr ¯ g ( h − k ) (66) |∇ u | ¯ g = div (cid:18) ∇ u |∇ u | ¯ g (cid:19) (67) Q = |∇ u | ¯ g r ¯ Ae u π (68) c φ = Q, (69)where for our later convenience we choose c = q ¯ A π . The first equation is thegeneralized Jang equation again. The second equation is the level set formu-lation of inverse mean curvature flow on ( M , ¯ g ). The third equation is thedefinition of Q ( x ) in terms of the inverse mean curvature flow level set function u ( x ). The new equation, then, is the fourth equation, which sets φ ( x ) equal to Q ( x ), up to a constant. Then by theorems 7 and 6, we conclude that¯ m − r ¯ A π ≥ Z M Q ( x ) (cid:18) ¯ R ( x )16 π (cid:19) dV ( x )= c π Z M ¯ RφdV ≥ . Then recalling that ¯ g measures areas at least as large as g does, we havethat ¯ A = | ˜Σ ¯ g | ¯ g ≥ | ˜Σ ¯ g | g ≥ | ˜Σ g | g = A, where again ˜Σ ¯ g is the outermost minimal area enclosure of Σ in ( M , ¯ g ) and˜Σ g is the outermost minimal area enclosure of Σ in ( M , g ). Recall also thatsince H ( M ) = 0, Σ connected (and bounding a region) implies that both ˜Σ ¯ g and ˜Σ g are also connected (and bound a region). Hence, if we can solve theabove system with boundary conditions so that Σ has zero mean curvature in( M , ¯ g ) and so that the total masses of ( M , g ) and ( M , ¯ g ) are the same, thenwe would be able to conclude that m = ¯ m ≥ r ¯ A π ≥ r A π , which would prove the Penrose conjecture for a single black hole in the casethat H ( M ) = 0.In the case of equality in the above inequalities, ( M , ¯ g ) has to be a timesymmetric slice of the Schwarzschild spacetime by the original Huisken-Ilmanenresult. Thus, inverse mean curvature flow yields precisely the spherically sym-metric spheres of Schwarzschild, so u is easy to compute. Direct computationthen reveals that Q ( x ) is the harmonic function in ( M , ¯ g ) which equals zeroon Σ(0) and goes to one at infinity. Since φ equals Q (up to a multiplica-tive constant, which is irrelevant after a constant rescaling of the time coor-dinate in what follows), we get that (cid:0) R × M , − φ dt + ¯ g (cid:1) is isometric to a37chwarzschild spacetime. Hence, g = ¯ g − φ df is the induced metric on aslice of Schwarzschild with graph function f ( x ). Finally, examining the case ofequality of theorem 6 (and that theorem’s use of the generalized Schoen-Yauidentity) forces k h − k k g = 0, which of course implies that k ij = h ij . Hence,the original Cauchy data ( M , g, k ) is the induced Cauchy data on a slice of aSchwarzschild spacetime with graph function f ( x ).Thus, understanding this system of equations, and whatever existence theorymight be associated with it, is a very interesting and important open problem.A first step is to observe that Q does not need to be defined in the system.Hence, our system is equivalent to The Jang - Inverse Mean Curvature Flow Equations ¯ g ( h − k ) (70) |∇ u | ¯ g = div (cid:18) ∇ u |∇ u | ¯ g (cid:19) (71) φ = |∇ u | ¯ g e u/ , (72)where we recall that ¯ g = g + φ df and h = φ Hess f + ( df ⊗ dφ + dφ ⊗ df )(1 + φ | df | g ) / , which can be thought of as three equations and three free functions f , u , and φ on the original Cauchy data ( M , g, k ).In fact, the third equation, equation 72, can be used to solve for φ in terms of u , du , and df . The purpose of this is to recognize that the Jang-IMCF equationsmay also be thought of as two equations and two free functions f and u once wesubstitute for φ . Since only first derivatives of f and u appear in the expressionfor φ below, the resulting equivalent system is two second order equations in f and φ .Unfortunately, the expression for φ in terms of f and u on ( M , g ) is a bitmessy, but at least it is explicit. From equation 72, we get φ = | du | ¯ g e u/ (73)which is simple enough except that φ also appears in the expression for ¯ g . Nextwe note that | du | g = ¯ g ij u i u j = (cid:18) g ij − φ f i f j φ | df | g (cid:19) u i u j = | du | g − φ h df, du i g φ | df | g , φ = e u | du | g − φ h df, du i g φ | df | g ! . It follows that φ solves the quadratic equation, | df | g · φ + B · φ − e u | du | g = 0 , where B = 1 + e u (cid:0) h df, du i g − | df | g | du | g (cid:1) . Thus, φ = − B + q B + 4 e u | df | g | du | g | df | g , (74)which is clearly always nonnegative (and where we disregard the negative squareroot in the quadratic formula since that solution is nonpositive). Thus, anequivalent formulation of the Jang-IMCF equations is0 = tr ¯ g ( h − k ) (75) |∇ u | ¯ g = div (cid:18) ∇ u |∇ u | ¯ g (cid:19) (76)where ¯ g = g + φ df , (77) h = φ Hess f + ( df ⊗ dφ + dφ ⊗ df )(1 + φ | df | g ) / , (78) φ = r − B + q B + e u | df | g | du | g | df | g , (79)and B = 1 + e u (cid:0) h df, du i g − | df | g | du | g (cid:1) , (80)which can be thought of as two equations and two free functions f and u on theoriginal Cauchy data ( M , g, k ). (In equation 79, φ = | du | g e u/ when | df | g = 0by equation 73).We end this section with a general discussion of the some of the challengesinvolved in finding an existence theory for the Jang-IMCF equations. First,note that these equations reduce to the Huisken-Ilmanen IMCF equation on( M , g ) when tr g ( k ) = 0 since then we can choose f = 0 (which implies ¯ g = g )to satisfy the generalized Jang equation (equation 75). Thus, clearly a notion ofa weak solution to this system of equations is required. Furthermore, the notionof “jumps” must still be involved when there are regions in which du = 0. Notethat when du = 0, then φ = 0. Thus, if f stays smooth and bounded, it wouldfollow that h = 0, which means that the generalized Jang equation cannot be39olved unless tr g ( k ) = 0 in this region as well. If tr g ( k ) = 0 in this region, thenthis would suggest that f needs to be unbounded or undefined in this region.Clearly this is an important issue to understand.Given these and other considerations, one might be tempted to be pessimisticabout finding a general existence theory for the Jang-IMCF equations. In fact,it was once the case that most were pessimistic about the original inverse meancurvature flow proposed by Geroch [9], right up until Huisken-Ilmanen [14] foundan amazingly beautiful and natural existence theory for a generalized version ofinverse mean curvature flow. Thus, there is also precedent for optimism.
10 The Jang-CFM Equations
In this section we comment that there is at least one other Einstein-Hilbertaction method in addition to the Jang-IMCF equations. So far we have seenhow the Penrose conjecture would follow from a general existence theory forthe Jang-Zero Divergence equations presented in section 7 or, for a single blackhole in dimension three, from a general existence theory for the Jang-IMCFequations presented in section 9. In this section, we briefly discuss a third systemof equations whose existence theory would also imply the Penrose conjecture.The precise statement of this third system is a bit laborious and so we do notstate it here, but only describe it and the additional considerations it involves.In section 8, we explained how any inequality of the form¯ m − r ¯ A π ≥ Z M Q ( x ) ¯ R ( x ) dV , (81)for some Q ( x ) ≥
0, leads to a system of equations which implies the Penroseconjecture. The first equation in the system is the generalized Jang equationand the second equation in the system is simply φ ( x ) = Q ( x ) (times a constantif one likes). In section 9 we pursued this approach in detail for the Huisken-Ilmanen inverse mean curvature flow.Bray’s proof [2] of the Riemannian Penrose inequality, when revisited, alsoyields an inequality of the form of equation 81. This proof of the Rieman-nian Penrose inequality involves a conformal flow of metrics (CFM) which flowsan initial asymptotically flat metric with nonnegative scalar curvature to aSchwarzschild metric in the limit as the flow parameter goes to infinity. Further-more, the area of the horizon stays constant, and (by the Riemannian positivemass theorem it turns out that) the total mass is nonincreasing during the flow.To generalize the conformal flow of metrics (CFM) proof to get an inequalityas in equation 81, we first need to generalize the positive mass theorem to getan inequality of the form ˜ m ≥ Z M Q ( x ) ˜ R ( x ) ˜ dV , (82)for some Q ( x ) ≥ M, ˜ g ). Witten’s spinor proof [33] of the Riemannianpositive mass theorem provides such a result, for example, at least whenever a40pinor solution to the Dirac equation exists (since we are not assuming ˜ R ≥ g by a conformal factor to achieve zero scalar curvatureglobally (when such a factor exists), and then measuring how much the masschanges. This last idea is made precise by Jauregui in [17]. Finally, one can alsouse inverse mean curvature flow starting from any point to prove an inequality ofthe above form in equation 82. This third approach currently has the advantageover the first two in that it is known to work in all cases in dimension three bythe previous section and the work of Huisken and Ilmanen [14] and Streets [31].In the conformal flow of metrics ( M, g t ) with total masses m ( t ), m ′ ( t ) = −
12 ˜ m ( t ) . (83)Hence, m − r A π ≥ Z ∞
12 ˜ m ( t ) dt (84)since the areas A ( t ) of the horizons of ( M, g t ) stay constant and the flow of met-rics converges to Schwarzschild where m − q A π = 0. Then plugging equation82 into the above equation and accounting how the scalar curvature transformsconformally gives a result of the desired form in equation 81. Hence, modulothe existence questions needed to get an equality of the form of equation 82, weget a generalization of the Riemannian Penrose inequality.One difference between the Jang-CFM equations and the Jang-IMCF equa-tions, however, is that the Jang-CFM equations are not local. That is, Q ( x ) inthis case does not satisfy a local p.d.e. at each point and instead has a morecomplicated expression. Hence, for the Jang-CFM equations to have an exis-tence theory, the theory would have to work for a wide range of possible Q .On the other hand, the Q ( x ) from the Jang-CFM equations has the potentialto have better regularity than the Q ( x ) from the Jang-IMCF equations. Forthe Jang-IMCF equations, cφ = Q is not necessarily continuous or even posi-tive, and in fact equals zero in jump regions of the inverse mean curvature flowon ( M , ¯ g ), which as discussed at the end of the previous section, introducesadditional analytical challenges.
11 Open Problems
The two most interesting and important open problems discussed in this paperare finding an existence theory for the Jang-Zero Divergence Equations (whichwould prove the Penrose conjecture) and finding an existence theory for theJang-IMCF equations (which would prove the Penrose conjecture for a singleblack hole when H ( M ) = 0). Another interesting problem is to find a generalexistence theory for any Einstein-Hilbert action method as long as the associated Q ( x ) has certain properties. If the Q ( x ) from the Jang-CFM equations qualifiedfor such a theory, this would also prove the Penrose conjecture.41nother interesting problem is to find additional Einstein-Hilbert actionmethods by finding new inequalities of the form of equation 81. Since thespecial case of ¯ R = 0 implies the Riemannian Penrose inequality, one wouldeither have to find a new proof of the Riemannian Penrose inequality or use theRiemannian Penrose inequality itself to find such a generalization. There maybe reasonable ideas to try in this latter approach.There is also the question of the physical interpretation of inequalities of theform of equation 81. Rewriting the inequality gives us¯ m ≥ r ¯ A π + Z M Q ( x ) ¯ R ( x ) dV , (85)which could be interpreted as saying that the total mass of a time-symmetricslice of a spacetime (not necessarily with nonnegative energy density) is at leastequal to the mass contributed by the black holes (the first term) plus a weightedintegral of the energy density (the second term), since energy density at eachpoint can be interpreted as ¯ µ = ¯ R/ π . The purpose of Q can be interpreted asthe need to account for potential energy. Also, Q should go to zero (and doesin the IMCF and CFM cases) at and inside the horizons of the black holes sincematter inside black holes should not affect the total mass.We also believe that the generalized Schoen-Yau identity and the general-ized Jang equation have much potential for many possible applications in thestudy of general relativity. One point of view, for example, is that the Jang-Zero Divergence equations give a canonical way of embedding Cauchy data( M , g, k ) into a static spacetime. If one is interested in understanding how theinitial Cauchy data evolves under the vacuum Einstein equations, or some otherequation coupled with the Einstein equation, then one could compute how thecanonical static metrics associated with the evolving Cauchy data slices evolve.One nice property of this approach is that if the initial Cauchy data is a sliceof the Schwarzschild spacetime, and we are solving the vacuum Einstein equa-tions for example, then while the Cauchy data is evolving in what might appearto be complicated ways, the associated canonical static spacetime remains theSchwarzschild spacetime. Also, since the generalized Jang equation blows upon horizons, this method could only be used to study the exterior region ofspacetimes outside the apparent horizons of black holes. There may be someadvantages to this restriction if this becomes a natural way to avoid spacetimesingularities. A Introduction to the Appendices
The target audience of these appendices are graduate students and other re-searchers who are interested in entering geometric relativity as a field to study.As such, we have included more detail in these calculations than is typical. Wejustify this choice in part with the fact that there are so many computations,many people would have a hard time duplicating all of these computations in areasonable amount of time, even with well chosen hints. We also hope that these42ppendices will be useful to students and researchers who are interested in prac-ticing their computational skills. We recommend the book “Semi-RiemannianGeometry with Applications to Relativity” by Barrett O’Neill as an excellentintroduction to the differential geometry of general relativity, and we mostlyfollow that book’s notation here. Readers should go through the calculations inthese appendices in order since notational conventions which are established inone appendix apply to the appendices which follow as well.The authors would like to thank Alan Parry for helping with the TeXingand Jeff Jauregui for helpful comments improving the readability of these ap-pendices.
B Curvature of Static Spacetimes
In this section we compute the Einstein curvature, Ricci curvature, and scalarcurvature of the general static spacetime metric˜ g = − φ ( x ) dt + ¯ g on R × M , where t ∈ R , x ∈ M and ¯ g is a positive definite metric on M .First, choose a coordinate chart on ( M , ¯ g ) with coordinates ( x , x , x ) andlet x = t . Then { ¯ ∂ α = ∂∂x α } α =0 is a basis of the tangent plane at each pointof the spacetime. Let { ¯ ω α } α =0 be the corresponding dual basis of one forms ateach point of the spacetime so that ¯ ω α ( ¯ ∂ β ) = δ αβ . (We use bars over these basesinstead of tildes to be consistent with section 5 and subsequent appendices).Finally, we define the components of ˜ g (which are the same as the componentsof ¯ g for tangent vectors to M ) to be˜ g αβ = h ¯ ∂ α , ¯ ∂ β i ˜ g (= ˜ g ( ¯ ∂ α , ¯ ∂ β ) by convention)so that ˜ g = − φ ( x ) , ˜ g i = ˜ g i = 0, and ˜ g ij = h ¯ ∂ i , ¯ ∂ j i ¯ g for 1 ≤ i, j ≤ (Notation: We adopt the convention that Greek indices always rangefrom to and Latin indices always range from to . Also, we adoptEinstein’s convention that any time an index is both an upper andlower index in an expression, summation over that index is implied.) (Recall also that ˜ g αβ are the components of the inverse matrix of ˜ g expressedas a matrix at each point of the coordinate chart, and that indices of a tensormay be raised or lowered by contracting with ˜ g αβ or ˜ g αβ , respectively [23]).By the Koszul formula [23], the Levi-Civita connection ˜ ∇ of ˜ g can be ex-pressed in terms of its components as˜ ∇ ¯ ∂ α ¯ ∂ β = ˜Γ γαβ ¯ ∂ γ , where ˜Γ γαβ = 12 ˜ g γθ (˜ g αθ,β + ˜ g βθ,α − ˜ g αβ,θ ) (86)are called the Christoffel symbols of ˜ ∇ .43 Notation: Commas denote differentiation with respect to the coor-dinate chart so that ˜ g αβ,θ = ∂ ˜ g αβ ∂x θ ). Plugging in our expressions for ˜ g αβ , short calculations reveal that0 = ˜Γ = ˜Γ j i = ˜Γ ji = ˜Γ ij , ˜Γ i = ˜Γ i = φ i φ , and that ˜Γ i = φ · φ ¯ ı , where φ i = φ ,i = ∂φ ( x ) ∂x i and, as previously stated, φ ¯ ı = ¯ g ij φ j = ˜ g ij φ j = ˜ g iα φ α = φ ˜ ı since φ is only a function of x and does not depend on t . Note that weare using the convention that a raised index with a bar or tilde over it denotesraising the index with ¯ g or ˜ g , respectively.Since the Lie bracket of coordinate vector fields is zero, it follows from thedefinition of the Riemann curvature tensor that R lijk = ¯ ω l ( ˜ ∇ ¯ ∂ i ˜ ∇ ¯ ∂ j ¯ ∂ k − ˜ ∇ ¯ ∂ j ˜ ∇ ¯ ∂ i ¯ ∂ k ) . Hence, R lijk = ¯ ω l (cid:16) ˜ ∇ ¯ ∂ i (cid:16) ˜Γ αjk ¯ ∂ α (cid:17) − ˜ ∇ ¯ ∂ j (cid:16) ˜Γ αik ¯ ∂ α (cid:17)(cid:17) = ¯ ω l (cid:18)(cid:16) ˜Γ αjk (cid:17) ,i ¯ ∂ α + ˜Γ αjk ˜Γ βiα ¯ ∂ β − (cid:16) ˜Γ αik (cid:17) ,j ¯ ∂ α − ˜Γ αik ˜Γ βjα ¯ ∂ β (cid:19) = (cid:16) ˜Γ ljk (cid:17) ,i − (cid:16) ˜Γ lik (cid:17) ,j + X α (cid:16) ˜Γ αjk ˜Γ liα − ˜Γ αik ˜Γ ljα (cid:17) For the beginner, we note that about half of the text books define the Riemanncurvature tensor to be the negative of what we used above. However, all textseventually end up with the same definition of the Ricci curvature (defined in amoment) which is agreed to be a positive multiple of the metric on the standardsphere.Plugging in our formulas for the Christoffel symbols, we thus compute that˜ R jk = (cid:16) ˜Γ jk (cid:17) , − (cid:16) ˜Γ k (cid:17) ,j + X α (cid:16) ˜Γ αjk ˜Γ α − ˜Γ α k ˜Γ jα (cid:17) = (cid:18) − φ k φ (cid:19) ,j + X m ˜Γ mjk · φ m φ − φ k φ · φ j φ = − Hess jk φφ from which it follow that˜ R jj = φ ∆ φ M , ¯ g ). The second com-putation follows from the first by first lowering the raised 0 index (introducinga factor of − φ ), using the antisymmetry of the Riemann curvature tensor toswitch indices, and then taking the trace of the Hessian. We remind the readerthat the Latin letters j , k , m range from 1 to 3. The beginning student shouldreview the definition of the Hessian, the Laplacian, and the use of normal co-ordinates. In this case, we note that we may choose normal coordinates on M such that ˜Γ mjk = 0 at a single point. Similarly, it is straightforward to verifythat ˜ R j = 0 = ˜ R kkj . It turns out that the above components of the Riemann curvature tensor areall that we need to compute the Ricci curvature of the spacetime. For example,˜Ric jk = ˜ R ααjk = ˜ R aajk + ˜ R jk = ¯ R aajk + ˜ R jk = Ric jk + ˜ R jk . The first equality is the definition of the Ricci curvature as the trace of the Rie-mann curvature tensor. For the second equality recall our summation conventionfor Latin and Greek indices stated above. The third equality is a consequenceof the Gauss equation for submanifolds since the t = 0 slice of our spacetimehas zero second fundamental form by symmetry. The fourth equality is simplythe definition of the Ricci curvature Ric of ( M , ¯ g ). Also,˜Ric = ˜ R αα = ˜ R + R jj = ˜ R jj by antisymmetry of the Riemann curvature tensor. Finally,˜Ric j = ˜ R ααj = ˜ R j + ˜ R kkj = 0 . Thus, putting it all together, we have formulas for the components of theRicci curvature of the static spacetime metric ˜ g = − φ ( x ) dt + ¯ g on R × M ,namely ˜Ric = φ ∆ φ ˜Ric jk = Ric jk − Hess jk φφ ˜Ric j = ˜Ric j = 0in terms of the Ricci curvature Ric of ( M , ¯ g ) and the Hessian and Laplacian of φ on ( M , ¯ g ).Next we can compute the scalar curvature of the spacetime by taking thetrace of the Ricci curvature,˜ R = ˜ g jk ˜Ric jk = ¯ R − φφ . G = ˜Ric − ˜ R ˜ g , the components of the Einstein curvature tensorof the static spacetime metric are˜ G = 12 ¯ Rφ ˜ G jk = Ric jk − Hess jk φφ + (cid:18) ∆ φφ − ¯ R (cid:19) ¯ g ˜ G j = ˜Ric j = 0as desired. C The Second Fundamental Form of the Graph
In this section we will compute the second fundamental form of a space-like sliceof the static spacetime (cid:0) R × M , ˜ g (cid:1) , where˜ g = − φ dt + ¯ g, (87) φ is a real-valued function on M , and ¯ g is a Riemannian metric on M . Given areal-valued function f on M , define the graph map F : M R × M (88)where F ( x ) = ( f ( x ) , x ).As we established in section 5, we will abuse terminology slightly and alwaysrefer to the image of the graph map F ( M ) simply as M and the t = 0 slice ofthe constructed spacetime as ¯ M . This notation is convenient since then ( M, g )and ( ¯
M , ¯ g ) are space-like hypersurfaces of the spacetime (cid:0) R × M , − φ dt + ¯ g (cid:1) (given appropriate bounds on the gradient of f ). Let π : M ¯ M be theprojection map π ( f ( x ) , x ) = (0 , x ) to the t = 0 slice of the spacetime.We repeat some notation and definitions from section 5 for clarity. Let¯ ∂ = ∂ t and { ¯ ∂ i } be coordinate vectors tangent to ¯ M . Define ∂ i = ¯ ∂ i + f i ¯ ∂ (89)to be the corresponding coordinate vectors tangent to M so that π ∗ ( ∂ i ) = ¯ ∂ i .Then in this coordinate chart, the components of the metrics g and ¯ g inducedfrom the spacetime are g ij = h ∂ i , ∂ j i and ¯ g ij = h ¯ ∂ i , ¯ ∂ j i where the angle brackets refer to the spacetime metric. Then it follows imme-diately that g ij = ¯ g ij − φ f i f j . (90)We comment here that the reader should think of g , φ , and f as the variablesthat we get to choose which determine ¯ g . The metric g comes from the initial46 ¯ ∂ ( R × M , − φ ( x ) dt + ¯ g ( x )) t ∈ R x ∈ M R height = f M ( M , g, h )( ¯ M , ¯ g, n ∂ ∂ ∂ ¯ n = φ ∂ t ∂ ¯ ∂ ( , x )( f ( x ) , x )( , x ) F ( x ) = ( f ( x ) , x ) π Figure 2: Schematic diagram of the constructed static spacetimeCauchy data (
M, g, k ) and φ and f are functions which will satisfy a system ofequations of our choosing.The inverse of { g ij } turns out to be g ij = ¯ g ij + v i v j , (91)where v i = φf ¯ i (1 − φ | df | g ) / = φf i (1 + φ | df | g ) / . (92)The above computation is most easily verified at each point in normal coordi-nates of ¯ g at that point, where the gradient of f is assumed to lie in the firstcoordinate direction. The second part of equation 92 can be computed in thesame manner, but where we consider that ¯ g ij = g ij + φ f i f j and then use normalcoordinates as before, but this time for the metric g . (For the beginner, the useof normal coordinates is exemplified in more detail in a moment.)We also define ¯ v = v i ¯ ∂ i and v = v i ∂ i (93)so that π ∗ ( v ) = ¯ v , and observe the useful identity(1 − φ | df | g ) · (1 + φ | df | g ) = 1 , (94)which follows directly from computing the ratio of the volume forms of g and ¯ g two different ways, namely with respect to g and then ¯ g .47s established in section 5, we use the convention that a barred index (asin f ¯ i above) denotes an index raised (or lowered) by ¯ g as opposed to g . Thatis, f ¯ i = ¯ g ij f j , where as usual f j = ∂f /∂x j in the coordinate chart. In general,barred quantities will be associated with the t = 0 slice ( ¯ M , ¯ g ) and unbarredquantities will be associated with the graph slice ( M, g ).Our next step is to compute the unit normal vector n to the graph slice( M, g ) defined by f . It is straightforward to verify that n = ¯ ∂ + φ f ¯ k ¯ ∂ k φ (cid:16) − φ | df | g (cid:17) / (95)has unit length in the spacetime metric, is perpendicular to the tangent vectors ∂ i = ¯ ∂ i + f i ¯ ∂ to the graph slice, and hence must be the correct expression.Following our convention for the definition of the second fundamental formdefined in equation 3, we thus have that the components of the second funda-mental form h are h ij = h ( ∂ i , ∂ j )= −h ˜ ∇ ∂ i ∂ j , n i = h ˜ ∇ ∂ i n, ∂ j i = * ˜ ∇ (¯ ∂ i + f i ¯ ∂ ) ¯ ∂ + φ f ¯ k ¯ ∂ k φ (cid:16) − φ | df | g (cid:17) / , ¯ ∂ j + f j ¯ ∂ + = D ˜ ∇ (¯ ∂ i + f i ¯ ∂ ) h ¯ ∂ + φ f ¯ k ¯ ∂ k i , ¯ ∂ j + f j ¯ ∂ E φ (cid:16) − φ | df | g (cid:17) / where ˜ ∇ is the Levi-Civita connection on our spacetime (cid:0) R × M , ˜ g (cid:1) . The thirdand fifth equalities follow from the fact that h n, ∂ j i = 0 on M .From the form of the above expression, we see that the Christoffel symbolsof the spacetime, defined and computed in appendix B, are going to come intoplay. From those computations, it follows that˜ ∇ ¯ ∂ ¯ ∂ = ˜Γ ¯ ∂ + ˜Γ k ¯ ∂ k = φφ ¯ k ¯ ∂ k ˜ ∇ ¯ ∂ ¯ ∂ i = ˜Γ i ¯ ∂ + ˜Γ k i ¯ ∂ k = φ i φ ¯ ∂ ˜ ∇ ¯ ∂ i ¯ ∂ = ˜Γ i ¯ ∂ + ˜Γ ki ¯ ∂ k = φ i φ ¯ ∂ where we remind the reader that Latin indices, when summation is implied byone raised and one lowered, only sum from 1 to 3, and a bar over a raised indexindicates that the index was raised with ¯ g as opposed to g .It now becomes convenient to use normal coordinates on ( ¯ M , ¯ g ). Note thatthese are not normal coordinates on the whole spacetime, just on the t = 048lice of the spacetime ( ¯ M , ¯ g ). Since by symmetry this slice has zero secondfundamental form, ˜ ∇ ¯ ∂ i ¯ ∂ k = ∇ ¯ ∂ i ¯ ∂ k = 0 at a single point of our choosing. Innormal coordinates, derivatives of the metric components ¯ g ij and ¯ g ij are zeroat the chosen point, thereby making the Christoffel symbols zero at that pointas well. In addition, ( f ¯ k ) i = ( f m ¯ g mk ) i = f im ¯ g mk = f ¯ ki at this single arbitrarypoint. Note that f im = ∂ f∂x i ∂x m is simply a coordinate chart second derivativein our notation.Hence,˜ ∇ (¯ ∂ i + f i ¯ ∂ ) h ¯ ∂ + φ f ¯ k ¯ ∂ k i = φ i φ ¯ ∂ + (2 φφ i f ¯ k + φ f ¯ ki ) ¯ ∂ k + f i ( φφ ¯ k ¯ ∂ k + φ f ¯ k · φ k φ ¯ ∂ )= (cid:18) φ i φ + φf i f ¯ k φ k (cid:19) ¯ ∂ + (cid:16) φφ i f ¯ k + φ f ¯ ki + φf i φ ¯ k (cid:17) ¯ ∂ k . Since h ¯ ∂ , ¯ ∂ i = − φ and h ¯ ∂ k , ¯ ∂ j i = ¯ g kj (which then lowers indices on otherterms), we have computed that h ij = − φ f j (cid:16) φ i φ + φf i f ¯ k φ k (cid:17) + (cid:16) φφ i f ¯ k + φ f ¯ ki + φf i φ ¯ k (cid:17) ¯ g kj φ (cid:16) − φ | df | g (cid:17) / = − φ i f j − φ f i f j f ¯ k φ k + 2 φ i f j + φf ij + f i φ j (cid:16) − φ | df | g (cid:17) / = φf ij + f i φ j + φ i f j − φ h df, dφ i ¯ g f i f j (cid:16) − φ | df | g (cid:17) / (96)at the chosen point in normal coordinates for ( ¯ M , ¯ g ). Since Hess ij f = f ij aswell at the chosen point in these normal coordinates, we have that h ij = φ Hess ij f + f i φ j + φ i f j − φ h df, dφ i ¯ g f i f j (cid:16) − φ | df | g (cid:17) / (97)at the chosen point. However, the above equation represents the components ofthe tensorial equation h = φ Hess f + df ⊗ dφ + dφ ⊗ df − φ h df, dφ i ¯ g df ⊗ df (cid:16) − φ | df | g (cid:17) / (98)by which we mean both the left and right sides of the equation are tensors. Sincetensorial equations may be verified in any coordinate chart, we conclude thatequation 98 is true at our chosen point. Since our chosen point was arbitrary,49t follows that equation 98 is true at every point. Thus, equation 97 is true atevery point as well, in any coordinate chart.The beginner differential geometer should take note on how using normalcoordinates simplified these computations substantially. However, the compu-tation can also be done straightforwardly without using normal coordinates,just not as elegantly.As we mentioned originally in section 5, we extend h trivially in our con-structed static spacetime so that h ( ∂ t , · ) = 0. Note that this gives h ( ∂ i , ∂ j ) = h ( ¯ ∂ i , ¯ ∂ j ), so we can call this term h ij without ambiguity. Our next goal is toconvert the above formula for h ij expressed with respect to ¯ g to one expressedwith respect to g .To convert the tensor Hess f , we must recall that it is defined to be thecovariant derivative ∇ of the 1-tensor df . Note that df does not involve anymetric, since df ( W ) = W ( f ) by definition. However, ∇ does involve the metric¯ g when applied to tensors. For example, in coordinates,Hess ij f = ∇ ( df )( ¯ ∂ i , ¯ ∂ j )= ¯ ∂ i ( df ( ¯ ∂ j )) − df ( ∇ ¯ ∂ i ¯ ∂ j )= ¯ ∂ i ( ¯ ∂ j ( f )) − ( ∇ ¯ ∂ i ¯ ∂ j )( f )= f ij − ¯Γ kij f k which involves the metric ¯ g and its first derivatives by equation 86. Hence,Hess ij f − Hess ij f = − (¯Γ kij − Γ kij ) f k . (99)Thus, we need to compute the difference of the Christoffel symbols of ¯ g and g .By equations 90 and 91,2Γ kij = g km ( g im,j + g jm,i − g ij,m )= ¯ g km + φ f ¯ k f ¯ m − φ | df | g ! · (cid:0) ¯ g im,j + ¯ g jm,i − ¯ g ij,m − ( φ f i f m ) ,j − ( φ f j f m ) ,i + ( φ f i f j ) ,m (cid:1) = (cid:0) ¯ g km + v k v m (cid:1) (cid:8) ¯ g im,j + ¯ g jm,i − ¯ g ij,m − φφ j f i f m − φ f m f ij − φ f i f mj − φφ i f j f m − φ f m f ji − φ f j f mi + 2 φφ m f i f j + φ f j f im + φ f i f jm (cid:9) = (cid:0) ¯ g km + v k v m (cid:1) (cid:8) ¯ g im,j + ¯ g jm,i − ¯ g ij,m − φ f m f ij − φφ j f i f m − φφ i f j f m + 2 φφ m f i f j } (100)50o that in normal coordinates on ( ¯ M , ¯ g ),Γ kij − Γ kij = φ f ij f ¯ k + φf i φ j f ¯ k + φφ i f j f ¯ k − φf i f j φ ¯ k + v k (cid:2) φ v ( f ) f ij + φv ( f ) φ j f i + φv ( f ) φ i f j − φv ( φ ) f i f j (cid:3) = φf ¯ k ( φf ij + f i φ j + φ i f j ) − φf i f j φ ¯ k + v k φv ( f ) ( φf ij + f i φ j + φ i f j ) − φv ( φ ) f i f j v k = φf ¯ k + φ | df | g f ¯ k − φ | df | g ! ( φf ij + f i φ j + φ i f j ) − φ h df, dφ i ¯ g f i f j f ¯ k − φ | df | g − φf i f j φ ¯ k = φf ¯ k − φ | df | g (cid:2) φf ij + f i φ j + φ i f j − φ h df, dφ i f i f j (cid:3) − φf i f j φ ¯ k = h ij v k − φf i f j φ ¯ k . (101)(Note that the above equation is a proof of identity 3 from section 5.) Pluggingthis equation into equation 99 gives usHess ij f − φf i f j h dφ, df i ¯ g = Hess ij f − h ij v ( f ) . (102)Hence, using the above formula and equation 94, we can transform equation97 to h ij = φ Hess ij f + f i φ j + φ i f j − φ h df, dφ i ¯ g f i f j (cid:16) − φ | df | g (cid:17) / = (cid:16) φ | df | g (cid:17) / ( φ Hess ij f − φh ij v ( f ) + f i φ j + φ i f j )so that by the definition of v in equation 92 (cid:16) φ | df | g (cid:17) h ij = (cid:16) φ | df | g (cid:17) / ( φ Hess ij f + f i φ j + φ i f j ) . Thus, we see that the second fundamental form of the graph slice expressedwith respect to the metric g is h ij = φ Hess ij f + ( f i φ j + φ i f j )(1 + φ | df | g ) / , (103)as claimed in section 5. Technically, we’ve shown that the above equation istrue at an arbitrary point in normal coordinates. Again, the above equationrepresents the components of the tensorial equation h = φ Hess f + ( df ⊗ dφ + dφ ⊗ df )(1 + φ | df | g ) / , (104)which is therefore true at the arbitrary point, and thus is true everywhere.Hence, equation 103 is true at every point as well, in any coordinate chart.51 Derivation of Identities
In this appendix we finish the proof of the generalized Schoen-Yau identitysketched out in section 5. We begin with equation 40 derived in that section,which we repeat for clarity:¯ R = 16 π ( µ − J ( v )) + (tr g h ) − (tr g k ) − k h k g + k k k g +2 v (tr g h ) − v (tr g k ) − h )( v ) + 2 div( k )( v ) . We will convert this formula for ¯ R to an expression in terms of the ¯ g metric.To perform the conversion, we need several identities (originally listed in section5) for arbitrary symmetric 2-tensors k which we now prove. We continue withthe notation established in the previous appendix, which might be thought ofas an introduction to this appendix. Identity 1 (tr g ( k )) − k k k g = (tr ¯ g k ) − k k k g + 2 k (¯ v, ¯ v )tr ¯ g k − | k (¯ v, · ) | g Proof: (tr g k ) − k k k g = ( g ij k ij ) − g ik g jl k ij k kl = (cid:2)(cid:0) ¯ g ij + v i v j (cid:1) k ij (cid:3) − (cid:0) ¯ g ik + v i v k (cid:1) (cid:0) ¯ g jl + v j v l (cid:1) k ij k kl = [tr ¯ g k + k (¯ v, ¯ v )] − k k k g − k (¯ v, ¯ v ) − | k (¯ v, · ) | g = (tr ¯ g k ) − k k k g + 2 k (¯ v, ¯ v ) tr ¯ g k − | k (¯ v, · ) | g . The first equality is true by definition. The second equality follows from equation91. For the third equality, remember that k is defined to have zero time-timecomponents and time-spatial components. Hence, k ( v, w ) = k (¯ v, w ) for all w since v projects to ¯ v (and consequently v and ¯ v are equal except for their timecomponents). Identity 2 v (tr g k ) = ¯ v (tr ¯ g k + k (¯ v, ¯ v )) Proof: v (tr g k ) = v (tr ¯ g k + k (¯ v, ¯ v ))= ¯ v (tr ¯ g k + k (¯ v, ¯ v )) . The first equality was shown in the proof of identity 1. The second equalityfollows since v and ¯ v only differ by their time components, and the functionbeing differentiated does not depend on the time coordinate, by definition. Identity 3 Γ kij − Γ kij = h ij v k − φf i f j φ ¯ k roof: See equation 101.
Identity 4 div( k )( v ) = div( k )(¯ v ) + ( ∇ ¯ v k )(¯ v, ¯ v ) − | ¯ v | g k (cid:18) ¯ v, ∇ φφ (cid:19) + h h (¯ v, · ) , k (¯ v, · ) i ¯ g + 2 h (¯ v, ¯ v ) k (¯ v, ¯ v ) + (tr ¯ g h ) k (¯ v, ¯ v ) Proof: div( k )( v ) = g ij ( ∇ ∂ i k )( ∂ j , v )= g ij [ ∂ i ( k ( ∂ j , v )) − k ( ∇ ∂ i ∂ j , v ) − k ( ∂ j , ∇ ∂ i v )]= g ij (cid:2) ( k jα v α ) ,i − Γ mij k mα v α − k ( ∂ j , ( v α ) ,i ∂ α + v α ∇ ∂ i ∂ α ) (cid:3) = g ij (cid:2) ( k jα v α ) ,i − Γ mij k mα v α − ( v α ) ,i k jα − v α Γ miα k jm (cid:3) = g ij (cid:0) k jα,i − Γ mij k mα − Γ miα k jm (cid:1) v α = (cid:0) ¯ g ij + v i v j (cid:1) h k jα,i + k mα (cid:16) − Γ mij + h ij v m − φf i f j φ ¯ m (cid:17) + k jm (cid:16) − Γ miα + h iα v m − φf i f α φ ¯ m (cid:17)i v α = div( k )(¯ v ) + (tr ¯ g h ) k (¯ v, ¯ v ) − φ | df | g k ( ∇ φ, ¯ v ) + h h (¯ v, · ) , k (¯ v, · ) i ¯ g − φ ¯ v ( f ) k ( ∇ f, ∇ φ ) + ( ∇ ¯ v k )(¯ v, ¯ v ) + h (¯ v, ¯ v ) k (¯ v, ¯ v ) − φ ¯ v ( f ) k ( ∇ φ, ¯ v ) + h (¯ v, ¯ v ) k (¯ v, ¯ v ) − φ ¯ v ( f ) k (¯ v, ∇ φ )= div( k )(¯ v ) + (tr ¯ g h ) k (¯ v, ¯ v ) + h h (¯ v, · ) , k (¯ v, · ) i ¯ g + ( ∇ ¯ v k )(¯ v, ¯ v )+ 2 h (¯ v, ¯ v ) k (¯ v, ¯ v ) − k (cid:18) ¯ v, ∇ φφ (cid:19) n φ | df | g | ¯ v | g + 2 φ | df | g o = ( ∇ · k )(¯ v ) + (tr ¯ g h ) k (¯ v, ¯ v ) + h h (¯ v, · ) , k (¯ v, · ) i ¯ g + ( ∇ ¯ v k )(¯ v, ¯ v )+ 2 h (¯ v, ¯ v ) k (¯ v, ¯ v ) − | ¯ v | g k (cid:18) ¯ v, ∇ φφ (cid:19) . The first equality is the definition of divergence. The second equality is thedefinition of the covariant derivative of a tensor. The third and fourth equalitiesuse Christoffel symbols as defined in the previous appendix. The sixth equalityuses equation 91 and then identity 3. The seventh equality is most easily seenby using normal coordinates with respect to ¯ g . The eighth equality combinesterms using the fact that ¯ v is parallel to ∇ f in ( M , ¯ g ) by the definition of ¯ v inequations 92 and 93. The ninth equality is the simplification2 φ | df | g | ¯ v | g + 2 φ | df | g = 2 φ | df | g (cid:16) | ¯ v | g + 1 (cid:17) = 2 φ | df | g − φ | df | g = 2 | ¯ v | . Identity 5 v ¯ ı ¯; j = h ij + v ¯ ı h (¯ v, · ) j − φ i v ¯ φ roof: First, let us clarify our notation. Recall that bars refer to the metric ¯ g .Hence, for example, v ¯ ı = ¯ g ik v k = h v, ¯ ∂ i i ¯ g , where v k is defined in equation 92. Asis standard, semicolons refer to covariant differentiation (whereas commas referto coordinate chart derivatives). Of course in our case, we need to specify withrespect to which metric are we performing covariant differentiation. Hence, weplace a bar over the semicolon to denote covariant differentiation with respectto ¯ g . Hence, v ¯ ı ¯; j = h∇ ¯ ∂ j v, ¯ ∂ i i ¯ g .All of our computations in the proof of this identity and the two that followonly involve the metric ¯ g , so it is notationally convenient (though not reallynecessary) to use normal coordinates with respect to this metric. Then at thatpoint, v ¯ ı ¯; j = v ¯ ı,j = φf i (cid:16) − φ | df | g (cid:17) / ,j = φf ij + f i φ j (1 − φ | df | g ) / + (cid:16) φφ j | df | g + φ f αj f ¯ α (cid:17) ( φf i )(1 − φ | df | g ) / = h ij + φ h df, dφ i ¯ g f i f j − φ i f j (1 − φ | df | g ) / + φ j φ | ¯ v | g + φf αj v α (cid:16) − φ | df | g (cid:17) / · v ¯ ı = h ij + | ¯ v | g · v ¯ ı φ j φ + φ h df, dφ i ¯ g f i f j − φ i f j + φv ¯ ı f αj v α (1 − φ | df | g ) / = h ij + | ¯ v | g · v ¯ ı φ j φ + v ¯ ı h (¯ v, · ) j + (cid:16) − φ | df | g (cid:17) − / · (cid:8) v ¯ ı v α (cid:0) φ h df, dφ i ¯ g f α f j − f α φ j − φ α f j (cid:1) + φ h df, dφ i ¯ g f i f j − φ i f j (cid:9) = h ij + v ¯ ı h (¯ v, · ) j − φ i v ¯ φ + | ¯ v | g v ¯ ı φ j φ − v ¯ ı v α (cid:18) v ¯ α φ j φ + φ α v ¯ φ (cid:19) + φ ¯ v ( φ ) (cid:16) f i f j + | ¯ v | g f i f j (cid:17) = h ij + v ¯ ı h (¯ v, · ) j − φ i v ¯ φ + ¯ v ( φ ) φ (cid:16) (1 + | ¯ v | g ) φ f i f j − v ¯ ı v ¯ (cid:17) = h ij + v ¯ ı h (¯ v, · ) j − φ i v ¯ φ . The above calculations follow from our formula for h in equation 96, our defini-tion of ¯ v in equations 92 and 93, and the substitution v ¯ ı = φf i (cid:16) − φ | df | g (cid:17) / which we use a number of times. 54 dentity 6 div( k )(¯ v ) = div( k (¯ v, · )) − h h, k i ¯ g − h h (¯ v, · ) , k (¯ v, · ) i ¯ g + k (cid:18) ¯ v, ∇ φφ (cid:19) Proof:
By identity 5,div ( k (¯ v, · )) = div( k )(¯ v ) + h k ij , v ¯ ı ¯; j i ¯ g = div( k )(¯ v ) + h k, h i ¯ g + h k (¯ v, · ) , h (¯ v, · ) i ¯ g − k (cid:18) ∇ φφ , ¯ v (cid:19) . Identity 7 ( ∇ ¯ v k )(¯ v, ¯ v ) = ¯ v ( k (¯ v, ¯ v )) − h h (¯ v, · ) , k (¯ v, · ) i ¯ g − h (¯ v, ¯ v ) k (¯ v, ¯ v ) + 2 | ¯ v | g k (cid:18) ¯ v, ∇ φφ (cid:19) Proof:
By identity 5, (cid:0) ∇ ¯ v ¯ v (cid:1) ¯ ı = v j v ¯ ı ¯; j = h (¯ v, · ) i + h (¯ v, ¯ v ) v ¯ ı − | ¯ v | g φ i φ so that by the definition of covariant differentiation of a symmetric 2-tensor,( ∇ ¯ v k )(¯ v, ¯ v ) = ¯ v ( k (¯ v, ¯ v )) − k (¯ v, ∇ ¯ v ¯ v )= ¯ v ( k (¯ v, ¯ v )) − h k (¯ v, · ) , h (¯ v, · ) i ¯ g − k (¯ v, ¯ v ) h (¯ v, ¯ v ) + 2 | ¯ v | g k (cid:18) ¯ v, ∇ φφ (cid:19) proving the identity. Identity 8 div( k )( v ) = div( k (¯ v, · )) + ¯ v ( k (¯ v, ¯ v )) + k (cid:18) ¯ v, ∇ φφ (cid:19) −h h, k i ¯ g − h h (¯ v, · ) , k (¯ v, · ) i ¯ g + (tr ¯ g h ) k (¯ v, ¯ v ) Proof:
Plugging identities 6 and 7 into identity 4 and simplifying proves theidentity.
Identity 9 ( The Generalized Schoen-Yau Identity )¯ R = 16 π ( µ − J ( v )) + k h − k k g + 2 | q | g − φ div( φq )+(tr ¯ g h ) − (tr ¯ g k ) + 2¯ v (tr ¯ g h − tr ¯ g k ) + 2 k (¯ v, ¯ v )(tr ¯ g h − tr ¯ g k )where q = h (¯ v, · ) − k (¯ v, · ) = h ( v, · ) − k ( v, · ) . roof: First we recall equation 40 derived in section 5,¯ R = 16 π ( µ − J ( v )) + (tr g h ) − (tr g k ) − k h k g + k k k g +2 v (tr g h ) − v (tr g k ) − h )( v ) + 2 div( k )( v ) . Next, we plug in identities 1, 2, and 8. Note that these identities are true forarbitrary symmetric 2-tensors k and hence are true for h as well. Thus,¯ R = 16 π ( µ − J ( v ))+(tr ¯ g h ) − k h k g + 2 h (¯ v, ¯ v )tr ¯ g h − | h (¯ v, · ) | g − (tr ¯ g k ) + k k k g − k (¯ v, ¯ v )tr ¯ g k + 2 | k (¯ v, · ) | g +2¯ v (tr ¯ g h ) + 2¯ v ( h (¯ v, ¯ v )) − v (tr ¯ g k ) + 2¯ v ( k (¯ v, ¯ v )) − h (¯ v, · )) − v ( h (¯ v, ¯ v )) − h (cid:18) ¯ v, ∇ φφ (cid:19) +2 k h k g + 4 | h (¯ v, · ) | g − ¯ g h ) h (¯ v, ¯ v )+2div( k (¯ v, · )) + 2¯ v ( k (¯ v, ¯ v )) + 2 k (cid:18) ¯ v, ∇ φφ (cid:19) − h h, k i ¯ g − h h (¯ v, · ) , k (¯ v, · ) i ¯ g + 2(tr ¯ g h ) k (¯ v, ¯ v ) . Simplifying and combining terms then gives us that¯ R = 16 π ( µ − J ( v ))+ k h k g − h h, k i ¯ g + k k k g +2 | h (¯ v, · ) | g − h h (¯ v, · ) , k (¯ v, · ) i ¯ g + 2 | k (¯ v, · ) | g − h (¯ v, · )) + 2div( k (¯ v, · )) − h (cid:18) ¯ v, ∇ φφ (cid:19) + 2 k (cid:18) ¯ v, ∇ φφ (cid:19) +(tr ¯ g h ) − (tr ¯ g k ) + 2¯ v (tr ¯ g h − tr ¯ g k ) + 2 k (¯ v, ¯ v )(tr ¯ g h − tr ¯ g k )so that¯ R = 16 π ( µ − J ( v )) + k h − k k g + 2 | q | g − q ) − q (cid:18) ∇ φφ (cid:19) +(tr ¯ g h ) − (tr ¯ g k ) + 2¯ v (tr ¯ g h − tr ¯ g k ) + 2 k (¯ v, ¯ v )(tr ¯ g h − tr ¯ g k )where q = h (¯ v, · ) − k (¯ v, · ) = h ( v, · ) − k ( v, · ) . Note that these two definitions of q exist on the entire constructed static space-time and are equal since both h and k are extended trivially in the time directionof the constructed static spacetime and v and ¯ v differ only in their time com-ponents. By the product rule the above equation proves the identity.56 The Transformation of the Mean Curvatureof the Boundary
In this appendix we derive the transformation formula for the mean curvature ofthe apparent horizon boundary Σ of (
M, g ) as approximated by the level sets of φ . Hence, to be useful, we need to assume that φ = 0 on Σ, is strictly positiveelsewhere, and has level sets converging smoothly to Σ. The discussion heresupplements the discussion in section 6.1.More precisely, given ¯ g = g + φ df , we compute the mean curvature ¯ H of the level sets of φ in ( M, ¯ g ) in terms ofthe mean curvature H of those level sets in ( M, g ) and f and φ . Identity 10
Transformation of Mean Curvature Identity
The mean curvature with respect to ¯ g = g + φ df of a level set Σ of φ isgiven by ¯ H = φ |∇ f | g φ |∇ Σ f | g ! / [( H − II ( T, T )) − (tr g Σ h − h ( T, T )) h ν, v i g ]+ φ |∇ Σ f | g φ |∇ f | g ! / ν ( φ ) |∇ Σ f | g (1 − | T | g ) where ∇ is the gradient with respect to g , ∇ Σ is the gradient with respect to g restricted to Σ , II is the second fundamental form of Σ in ( M, g ) (so that H = tr Σ ( II ) ), ν is the outward unit normal vector to Σ in ( M, g ) , and v = φ ∇ f (1 + φ |∇ f | g ) / ,T = φ ∇ Σ f (1 + φ |∇ f | g ) / = tan Σ ( v ) ,h = φ Hess f + ( df ⊗ dφ + dφ ⊗ df )(1 + φ | df | g ) / . Proof:
To derive this identity, it is convenient to let γ = φdf so that ¯ g = g + γ ⊗ γ .We also define γ tan = tan g Σ γ . Then it is a short exercise to verify that¯ ν = | γ | g | γ tan | g ! / ν − γ ( ν ) γ ∗ g | γ | g ! (105)is the unit normal to Σ in ( M, ¯ g ), where γ ∗ g is defined to be the vector dual tothe covector γ with respect to g (which of course is φ ∇ f ). The above formulafor ¯ ν has the property that | ¯ ν | ¯ g = 1 and h ¯ ν, T i ¯ g = 0 for all vectors T tangentto Σ. 57o compute the mean curvature of Σ with respect to ( M n , ¯ g ) at a pointp, choose a coordinate chart in a neighborhood of p so that the first n − R n with the standard flat metric, we choose the normal vector to be the onepointing outwards and the mean curvature to be positive. Hence, by identity 3,¯ H = − n − X i,j =1 ¯ g ij (cid:10) ¯ ν, ∇ ¯ ∂ i ¯ ∂ j (cid:11) ¯ g = − n − X i,j =1 n X θ =1 ¯ g ij D ¯ ν, Γ θij ∂ θ E ¯ g = − X ¯ g ij D ¯ ν, (cid:16) Γ θij + h ij v θ − φf i f j φ ¯ θ (cid:17) ∂ θ E ¯ g = − X ¯ g ij (cid:16) Γ θij + h ij v θ − φf i f j φ ¯ θ (cid:17) [ h ¯ ν, ∂ θ i g + γ (¯ ν ) γ ( ∂ θ )]= − X ¯ g ij (cid:16) Γ θij + h ij v θ − φf i f j φ ¯ θ (cid:17) | γ | g | γ tan | g ! / · * ν − γ ( ν ) γ ∗ g | γ | g , ∂ θ + g + γ ν − γ ( ν ) γ ∗ g | γ | g ! γ ( ∂ θ ) = − X ¯ g ij (cid:16) Γ θij + h ij v θ − φf i f j φ ¯ θ (cid:17) | γ | g | γ tan | g ! / · " h ν, ∂ θ i g − γ ( ν ) γ ( ∂ θ )1 + | γ | g + γ ( ν ) γ ( ∂ θ ) − | γ | g | γ | g ! so that ¯ H = − X ¯ g ij (cid:16) Γ θij + h ij v θ − φf i f j φ ¯ θ (cid:17) h ν, ∂ θ i g | γ | g | γ tan | g ! / . Substituting φ ¯ θ = ¯ g θk φ k = g θk − γ θ γ k | γ | g ! φ k = φ θ − h γ, dφ i g | γ | g γ θ
58e get that¯ H = − | γ | g | γ tan | g ! / X g ij − γ i γ j | γ | g ! · Γ θij + h ij v θ − φf i f j φ θ − h γ, dφ i g | γ | g γ θ !! h ν, ∂ θ i g = − | γ | g | γ tan | g ! / X g ij − γ i γ j | γ | g ! · * ν, ∇ ∂ i ∂ j + h ij v − φf i f j ∇ φ − h γ, dφ i g | γ | g γ ∗ g !+ g = | γ | g | γ tan | g ! / " H − II( γ tan , γ tan )1 + | γ | g ! − (tr g Σ h ) − h ( γ tan , γ tan )1 + | γ | g ! h ν, v i g + φ (cid:12)(cid:12) ∇ Σ g f (cid:12)(cid:12) g − h df, γ tan i g | γ | g ! ν ( φ ) − γ ( ν ) h γ, dφ i g | γ | g ! . Substituting γ = φ df , we get¯ H = φ |∇ f | g φ |∇ Σ f | g ! / ( H − φ II( ∇ Σ f, ∇ Σ f )1 + φ |∇ f | g ! − tr g Σ h − φ h ( ∇ Σ f, ∇ Σ f )1 + φ |∇ f | g ! h ν, v i g + φ |∇ Σ f | g − φ |∇ Σ f | g φ |∇ f | g ! ν ( φ ) − φ ν ( f ) h df, dφ i g φ |∇ f | g !) (106)So far we have not used the assumption that Σ is a level set of φ , so theabove formula would be of interest if one wanted to analyze the mean curvaturesof a family of surfaces converging to the boundary other than the level sets of φ .Since in our case Σ is a level set of φ , it follows that ∇ φ || ν and ∇ φ = ν ( φ ) ν .Hence, ν ( φ ) − φ ν ( f ) h df, dφ i g φ |∇ f | g = ν ( φ ) − φ ν ( f ) φ |∇ f | g ! = ν ( φ ) φ |∇ Σ f | g φ |∇ f | g ! . Thus, if we recall that v = φ ∇ f (cid:16) φ |∇ f | g (cid:17) / T = φ ∇ Σ f (cid:16) φ |∇ f | g (cid:17) / (= tan Σ ( v ))we get that¯ H = φ |∇ f | g φ |∇ Σ f | g ! / [( H − II(
T, T )) − (tr g Σ h − h ( T, T )) h ν, v i g ]+ φ |∇ Σ f | g φ |∇ f | g ! / · ν ( φ ) · |∇ Σ f | g · (cid:16) − | T | g (cid:17) (107)as claimed in the identity.The purpose of including this identity in this paper is to help those who wantto study the existence theories of one of the systems of equations described inthis paper, such as the Jang - zero divergence equations. The above identitymay be useful for understanding boundary behavior.In particular, to reduce the Penrose conjecture to the Riemannian Penrosecase, it is necessary for ¯ H ≤ H = 0. The cases of blowup, blowdown, or bounded behavioreverywhere are discussed in section 6.1 from a different point of view. In thecases of blowup or blowdown the level sets of f were used instead of those of φ .However, if we are going to allow mixed blowup and blowdown behavior ongeneralized apparent horizons, then we can no longer use the level sets of f everywhere. However, since φ is always assumed to go to zero on the boundary,it is natural to study the mean curvatures of the level sets of φ . Since anexistence theory is necessary before we can make very many conclusions aboutboundary behavior, we restrict this final discussion to a few observations.At points on the boundary in the interior of a blowup region, a blowdownregion, or a bounded behavior region, it is plausible that v = − ν , v = ν , or v = 0, respectively, in the limit as the level sets of φ approach the boundary. Inthose three cases, it follows that T = 0. If we further assume that the secondterm in the formula for ¯ H can be shown to be zero, we then get¯ H = φ |∇ f | g φ |∇ Σ f | g ! / [ H − (tr g Σ h ) h ν, v i g ] . The term in brackets is then [ H + (tr g Σ h )], [ H − (tr g Σ h )], and [ H ] in thosethree respective cases. Modulo possible issues with taking limits, one could thenuse the generalized Jang equation to conclude that0 = tr ¯ g ( h − k ) = tr g ( h − k ) − ( h − k )( v, v ) = tr g Σ ( h − k ) (108)in the case of either blowup or blowdown, since ¯ g ij = g ij − v i v j . Hence, the termin brackets equals zero in the three respective cases of a local future apparenthorizon, a local past apparent horizon, or a local future and past apparent60orizon, as desired. One would then need to show that the term in front of thebrackets still allows one to conclude ¯ H = 0 in the limit, even when it divergesas the level sets of φ approach the boundary.Of course the really tricky part is understanding points on the boundarywhere every open set around the point contains two or more of blowup, blow-down, and bounded behavior. We offer the above formula for the mean curvature¯ H of the level sets of φ in case it is helpful to others who approach this problem.61 eferences [1] L. Andersson, and J. Metzger, The area of horizons and the trapped region ,preprint, arXiv:0708.4252, 2007.[2] H.L. Bray,
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