Proof of a Null GeometryPenrose Conjecture

L

L

Σ

Hubert L. Bray and Henri P. Roesch

Beginning in 1973, Roger Penrose wondered about therelation between the mass of the universe and the contri-butions of black holes ([1], [2]). All we can see are theirouter boundaries or “horizons,” the size of which shoulddetermine their mass contributions. He conjectured thatthe total mass of a spacetime containing black hole hori-zons with combined total area |Σ| should be at least√|Σ|/16𝜋. On the one hand, this conjecture is importantfor physics and our understanding of black holes. On theother hand, Penrose’s physical arguments lead to a fas-cinating conjecture about the geometry of hypersurfacesin spacetimes. For a spacelike “Riemannian” slice (“theuniverse at a particular time”) with zero curvature (zero

Hubert Bray is professor of mathematics and physics at Duke Uni-versity. His email address is hubert.bray@duke.edu.Henri P. Roesch is NSF postdoctoral fellow of mathematics at theUniversity of California, Irvine. His email address is hroesch@uci.edu.For permission to reprint this article, please contact:reprint-permission@ams.org.DOI: http://dx.doi.org/10.1090/noti1629

second fundamental form) the conjecture is known asthe Riemannian Penrose Inequality and was first provedby Huisken–Ilmanen in 2001 (for one black hole) andthen by the first author shortly thereafter, using twodifferent geometric flow techniques. This article concernsa formulation of the conjecture for certain light-cone-like“null” hypersurfaces in spacetimes called the Null PenroseConjecture (NPC). Over the last ten years, there has been agreat deal of progress on the NPC, culminating in a recentproof of the conjecture in a fair amount of generality forsmooth null cones by the second author. One surprisingfact is that these null hypersurfaces, under physicallyinspired curvature conditions on the spacetime, have“monotonic” properties (increasing as expected), includ-ing cross sectional area, notions of energy, and, as we’llsee with Theorem 1 below, a new notion of mass.

The Null Geometry of LightThe theory of General Relativity emerges from AlbertEinstein’s beautiful idea that matter in a physical systemcurves the intertwining fabric of both space and time. A

156 Notices of the AMS Volume 65, Number 2

spacetime is a four-dimensional manifold with a metricof signature (3,1), meaning that the metric on eachtangent plane is isometric to the Minkowski spacetimeℝ4

1 ∶= (ℝ4,−𝑑𝑡2 + 𝑑𝑥2 + 𝑑𝑦2 + 𝑑𝑧2). The minus sign inthe metric implies the existence of null vectors, vectorswith zero length (such as (1,1,0,0)), even though thevectors themselves are not zero. A null hypersurface is

Null geometryis

counterintuitivein a number of

ways.

a codimension-one subman-ifold of a spacetime whosethree-dimensional tangentplanesarenull inonedimen-sion (and hence positivedefinite in the two otherdimensions). For example,if we let 𝑟 = √𝑥2 +𝑦2 + 𝑧2

in theMinkowski spacetime,any translation of the down-ward coneΛ ∶= {𝑡 = −𝑟} asdepicted in Figure 1, is a null hypersurface.

Null geometry is counterintuitive in a number of ways.Since one dimension has zero length, null hypersurfaceshave zero volume. Furthermore, since the metric is notinvertible, the Riemann curvature tensor of the nullhypersurface is not well defined. Also, the vector which isperpendicular to a null hypersurfacemust also be tangent(hence null). This normal-tangent duality complicates thenotion of what the second fundamental form of a nullhypersurface should be defined to be (the classical toolfor analyzing the “shape” of substructures).

t

x,z

t

y

x,z

α

R41

Figure 1. The downward light cone Λ of Minkowski isan example of a null hypersurface.

Fortunately, for a conical null hypersurface Ω ≅ 𝕊2 ×ℝ called a null cone, where 𝕊2 accounts for the twopositive dimensions,Ω can be studied vicariously throughthe geometry of its spherical cross-sections includingtheir Gauss and Codazzi equations. For our sacrifice inintuition to this normal-tangent duality we do gain someadvantages. A direct consequence of a normal vector 𝐿being also tangent is that all curves along 𝐿 must begeodesic. Thus, a null hypersurface can be thought of

Σ0

Σs

Ω

L L

Figure 2. Foliations of Ω

as a collection of light-rays in the framework of GeneralRelativity. Imagine standing on some 2-sphere Σ0 in aspacetime, for example the surface of the Earth, andcollecting all light rays hitting the surface at a particularpoint in time.The resulting set constructs (or recovers) a null coneΩ reducing the usually complicated system of PDEsassociated to flows in spacetimes to an analysis of ODEs.From standard uniqueness of ODEs, any two normal nullflows of surfaces off of Σ0 must result in two foliations ofthe same null cone. Moreover, from standard existence ofODEs, wherever there’s a smooth spacelike 2-sphere Σ0there will be a null cone off of it, at least in a neighborhoodof the sphere.

The Expanding Null ConeThe thermodynamics of black holes rests upon Hawking’sarea theorem (1973), which states that in spacetimessatisfying the Dominant Energy Condition (or DEC) thearea of a cross section of a black hole event horizon isnondecreasing. Similarly in our context, we will brieflyexplore how the DEC, which is a local curvature constraintmodelling nonnegative energy, ensures that null conesreaching null infinity can only be foliated by spheres ofincreasing area. To do so, we first need a quantity alsoneeded to state our main result, Theorem 1.

Definition 1. For a spacelike 2-sphere Σ, the expansion isgiven by the inner product of the null-flow vector 𝐿 withthe mean curvature vector ��:

𝜎 ∶= ⟨−��, 𝐿⟩.

For a null flow off of Σ along 𝐿, 𝜎 comes from𝑑𝑑𝑠𝑑𝐴 = 𝜎𝑑𝐴 where 𝑑𝐴 is the area form (i.e. an elementof area on Σ). Here the DEC comes into play, by way of theso-called Raychaudhuri equation, ensuring 𝑑𝜎/𝑑𝑠 ≤ 0along all geodesics. Hence, the only way to get standardasymptotics at infinity, which by any reasonable notionshould have expanding 2-spheres (i.e “𝜎(∞) > 0”), neces-sarily restricts our choice of Σ0 to have strictly positiveexpansion. In fact, 𝜎 > 0 on all of Ω enforces that allfoliations must have expanding area.

With the null flow vector 𝐿 having zero length we alsoforfeit our intuitive notion of measuring the speed of a

February 2018 Notices of the AMS 157

flow. However, with an expanding null cone Ω, 𝜎 offers aconvenient replacement.

Null cones that reach null infinity also have funda-mental physical significance beyond the monotonicity ofcross-sectional area. In isolated physical systems, such asa cluster of stars or a black hole, we expect the curvatureinduced by localized matter to settle far away back toflat Minkowski spacetime. In our context, Mars and Soria(2015) introduced the notion of an asymptotically flat nullcone whereby the geometry of Ω approaches that of adownward cone of Minkowski at null infinity in a suitablesense. These asymptotically flat null cones allow us theability to measure the total energy and mass of a systemwhich we’ll need in order to state the NPC.

Total Energy and Our Main ExampleIn 1968, Hawking published a new mechanism aimed atcapturing the amount of energy in a given region usingthe curvature of its boundary Σ.

Definition 2. The Hawking Energy is given by

(1) 𝐸𝐻(Σ) = √|Σ|16𝜋(1− 1

16𝜋 ∫Σ⟨��, ��⟩𝑑𝐴).

For example, for any cross-section of the downwardcone Σ ↪ Λ ∶= {𝑡 = −𝑟} in Minkowski the Gauss equationidentifies a beautifully simple relationship between theintrinsic and extrinsic curvature, 𝒦 = 1

4⟨��, ��⟩, where𝒦 is the Gauss curvature of Σ. From the Gauss–BonnetTheorem we therefore conclude that

0 = ∫𝒦− 14⟨��, ��⟩𝑑𝐴 = 4𝜋𝐸𝐻/√|Σ|/16𝜋.

Thus, all cross-sections—no matter how squiggly—envelop matter content of vanishing energy, as expectedof a flat vacuum.

For another and our main example of a null cone, we goto the one-parameter family of Schwarzschild spacetimescharacterized by the metric

−(1− 2𝑀𝑟 )𝑑𝑡2 + 𝑑𝑟2

1− 2𝑀𝑟

+ 𝑟2(𝑑𝜗2 + (sin𝜗)2𝑑𝜑2).

These spacetimes model an isolated black hole with theparameter 𝑀 representing total mass. Note from the lasttwo terms of the metric that the coordinate 𝑟 has beenchosen so that each sphere of fixed (𝑡, 𝑟) is a round sphereof area 4𝜋𝑟2. Also, when 𝑀 = 0 we recover exactly theMinkowski metric in spherical coordinates. The readermay have noticed the singularities 𝑟 = 0 and 𝑟 = 2𝑀.The singularity at 𝑟 = 0 is a curvature singularity calledthe black hole singularity giving rise to the isolated blackhole. We see this black hole is isolated from the factthat the metric approaches the Minkowski metric forlarge values of 𝑟. On the other hand, the singularity at𝑟 = 2𝑀 is superficial, and can be removed with a changeof coordinates. To show this we introduce the ingoingnull coordinate 𝑣, whereby

𝑑𝑣 = 𝑑𝑡 + 𝑑𝑟1− 2𝑀/𝑟,

giving the metric in ingoing Eddington–Finkelstein coordi-nates

(2) − (1− 2𝑀𝑟 )𝑑𝑣2 + 2𝑑𝑣𝑑𝑟+ 𝑟2(𝑑𝜗2 + (sin𝜗)2𝑑𝜑2).

In the transition from 𝑀 = 0 to 𝑀 > 0 the down-ward cones of Minkowski transition to their sphericallysymmetric counterparts in Schwarzschild, referred toas the standard null cones. In Minkowski, Λ = {𝑣 = 0}for 𝑣 = 𝑡 + 𝑟, from (2) we see the analogous three-dimensional sliceΩ𝑆 ∶= {𝑣 = 𝑣0} (i.e. 𝑑𝑣 = 0) inherits themetric 𝑟2(𝑑𝜗2 + (sin𝜗)2𝑑𝜑2) assigning positive lengthsonly forvectorsalong the twospherical coordinates (𝜗,𝜑),not 𝑟. These coordinates (𝑟,𝜗,𝜑) ∈ ℝ×𝕊2 identify pointson the standard null cone Ω𝑆 of Figure 3.

r = 0

r = 2M

ω = r|Σ

L

L

ΩS

Σ

Figure 3.The Standard Null Cone Ω𝑆 of Schwarzschildis the counterpart of Λ in Minkowski.

The 𝑟-coordinate curves are exactly the geodesicsthat rule Ω𝑆, allowing us to identify any cross-sectionby simply specifying 𝑟 as a function on 𝕊2. Similar toMinkowski, the Gauss equation once again simplifiesto an intriguingly simple expression for a cross-sectionΣ ∶= {𝑟 = 𝜔(𝜗,𝜑)} ↪ Ω𝑆 ([3])

(3) 𝒦− 14⟨��, ��⟩ = 2𝑀

𝜔3 .

Σ also inherits the simple metric 𝛾 = 𝜔2�� fromΩ𝑆, where�� is the standard round metric on a sphere. So fromGauss–Bonnet and (3), we conclude that

𝐸𝐻(Σ) = 𝑀√

∫𝜔2 𝑑𝑆4𝜋 ∫ 1

𝜔𝑑𝑆4𝜋,

where 𝑑𝑆 is the area form on a round sphere. Somefascinating observations follow from Jensen’s inequality.We deduce that 𝐸𝐻 ≥ 𝑀, the special relativistic notionthat the energy of a particle is always bounded below byits mass. Jensen’s inequality also ensures that equality

158 Notices of the AMS Volume 65, Number 2

ty

x,z

t

y

x,z

EH(Σs

�

)

→ E

EH(Σs)

→ M

Figure 4. Asymptotically round foliations of Ω𝑆

is reached only if 𝜔 = 𝑟0, corresponding to the 𝑡-sliceintersections with Ω𝑆 (see Figure 4). Moreover, one canshow that Σ is a round sphere if and only if

𝜔(𝜗,𝜑) = 𝑟0√1− |��|2

1 − �� ⋅ ��(𝜗,𝜑)for some 𝑟0, �� inside the unit ball 𝐵3 ⊂ ℝ3, and��(𝜗,𝜑) the unit position vector, giving the energy𝐸𝐻(𝑟0, ��) = 𝑀/√1− |��|2. This is precisely the observedenergy of a particle of mass 𝑀 traveling at velocity ��relative to its observer (with the speed of light set to𝑐 = 1).

Using Schwarzschild as an example we can alsomotivate the notion of total energy and mass for anasymptotically flat null cone Ω. We start by bringing tothe attention of the reader that the intrinsic geometry ofΩ𝑆 is identical to that of the downward cone in Minkowski.This is evident from (2), since the only component givingrise to the Schwarzschild geometry (beyond Minkowski)is 𝑑𝑣2, which vanishes for the induced metric on Ω𝑆. Soinstead, we may actually account for all round spheresof Ω𝑆 by intersecting the downward cone of Minkowskiby Euclidean hyperplanes (see Figure 1). On the one hand,any family of parallel hyperplanes in Minkowski are givenas fixed time slices inside a reference frame (coordinates( 𝑡, 𝑥, ��, 𝑧)) of an observer traveling at velocity ��. On theother hand, the ambient geometry of Schwarzschild set-tles to that of Minkowski, inheriting such characteristicsasymptotically. Therefore, aided by Figure 4, we imaginethat an asymptotically round foliation of Ω𝑆 is inducedby an asymptotically Euclidean slicing of the spacetime(for which 𝐸𝐻 has a verified correlation to total energy).

We conclude that an observer at infinity approximatesour black hole to a particle (similarly to 𝛼 of Figure1) of total energy 𝑀/√1− |��|2. In the general setting,considering an asymptotically flat null cone Ω, it followsthat 𝐸𝐻 approaches a measure of total energy along anasymptotically round foliation called a Bondi Energy, 𝐸𝐵.

Definition 3. For an asymptotically flat null cone Ω, min-imizing over all Bondi Energies 𝐸𝐵 yields the Bondi Mass𝑚𝐵.

Returning to Schwarzschild, we verify that the standardnull cone has Bondi Mass

𝑚𝐵(Ω𝑆) = inf|��|<1

𝑀√1− |��|2

= 𝑀.

Black Holes and the Null Penrose ConjectureFurther inspection of (2) reveals yet another null conegiven by the slice ℋ ∶= {𝑟 = 2𝑀} (see Figure 3). Withinduced metric 4𝑀2(𝑑𝜗2(sin𝜗)2𝑑𝜑2), positive lengthsare once again only assigned to vectors along the sphericalcoordinates, not 𝑣. In contrast to Ω𝑆, any cross sectionΣ ↪ ℋ has metric 𝛾 = 4𝑀2��. Thus, all cross-sectionsexhibit the exact same area 16𝜋𝑀2. In other words, on anycross-section of ℋ, light rays emitted perpendicularlyoff of the surface remain trapped. Since no materialparticles travel faster than the speed of light,ℋ indicatesthe hypersurface from which there is no return uponentering, or the event horizon. As a result of this trapping,the 2-sphere ℋ ∩ Ω𝑆 is the unique cross-section of Ω𝑆satisfying ⟨��, ��⟩ = 0, namely with null mean curvature.

Definition 4. A marginally outer trapped surface (MOTS)Σ0 in an expanding null cone Ω is a surface satisfying thecondition

⟨��, ��⟩ = 0.With a MOTS Σ0 identifying the presence of a black holehorizon in our context, the Null Penrose Conjecture statesthat

√|Σ0|16𝜋 ≤ 𝑚𝐵.

Previous WorkGiven a spacelike 2-sphere Σwith metric 𝛾, every point onits surface has two positive dimensions in the availablefour of spacetime spent on tangent vectors. As a result,we can combine the remaining negative and positivedimensions to form a normal null basis {𝐿, 𝐿}. For a cross-section of a null coneΩwe choose 𝐿 as the normal-tangentto Ω (for example, see Figure 3). This finally allows us tointroduce some final data for our main Theorem below.

Definition 5. For a smooth spacelike 2-sphere Σ of sec-ond fundamental form II and null basis {𝐿, 𝐿} such that⟨𝐿, 𝐿⟩ = 2 we define

𝜒− ∶= ⟨− II, 𝐿/𝜎⟩, 𝜁(𝑉) ∶= 12⟨𝐷𝑉𝐿, 𝐿⟩ = 𝜏(𝑉) +𝑉 log𝜎

where 𝜁 is the connection 1-form (𝑉 a tangent vector fieldof Σ).

February 2018 Notices of the AMS 159

In his PhD thesis, Johannes Sauter (2008) showed forthe special class of shear free null cones (i.e. satisfying𝜒− = 1

2𝛾) inside vacuum spacetimes, one is able to solvea system of ODEs to yield explicitly the geometry of Ω.This then enables a direct analysis of 𝐸𝐻 at null infinitythat allowed Sauter to prove the NPC in this specialcase. An observation of Christodoulou also shows that 𝐸𝐻is monotonically increasing along foliations in vacuumif either the mass aspect function 𝜇 ∶= 𝒦 − 1

4⟨��, ��⟩ −∇ ⋅ 𝜁 or the expansion 𝜎 remain constant functionson each cross-section. For a black hole horizon Σ0, wesee from (1) that 𝐸𝐻(Σ0) = √|Σ0|/16𝜋, making thisobservation particularly interesting. If one is successfulin interpolating 𝐸𝐻 from the horizon to the Bondi Mass𝑚𝐵 along one of these flows, the NPC would follow forvacuum spacetimes √|Σ0|/16𝜋 = 𝐸𝐻(Σ0) ≤ lim

𝑠→∞𝑚(Σ𝑠) =

𝑚𝐵. Sauter was able to show for small pertubations ofΩ off of the shear free condition, one obtains globalexistence of either of these flows and that 𝐸𝐻 converges.Unfortunately, one is unable to conclude that the foliating2-spheres even become round asymptotically let alone 𝐸𝐻approaching 𝑚𝐵. In fact, Bergqvist (1997) noticed thisexact difficulty had been overlooked in an earlier work ofLudvigsen and Vickers (1983) towards proving the weakNPC, namely √|Σ0|/16𝜋 ≤ 𝐸𝐵.

In 2015, Alexakis was able to prove the NPC for vacuumperturbations of the black hole exterior in Schwarzschildspacetime by successfully using the latter of the twoflows in Sauter’s thesis. Alexakis was once again affordedan explicit analysis of 𝐸𝐻 at null infinity. Work by Marsand Soria [4] followed soon afterwards in identifyingthe asymptotically flat condition on Ω to maintain anexplicit limit of lim

𝑠→∞𝐸𝐻(Σ𝑠) along geodesic foliations. In

2016, those authors constructed a new functional on2-spheres and showed for a special foliation {Σ𝜆} off ofthe horizon Σ0 called geodesic asymptotically Bondi (orGAB) that, √|Σ0|/16𝜋 ≤ lim

𝜆→∞𝐸𝐻(Σ𝜆) < ∞. Thus, for GAB

foliations that approach round spheres, this reproducesthe weak NPC of Bergqvist and of Ludvigsen and Vickers.Unfortunately, as in the aforementioned work of Sauter,Bergqvist, Ludvigsen, and Vickers there is no guaranteeof asymptotic roundness.

Mass Not EnergyThese difficultiesmay verywell be symptomatic of the factthat an energy is particularly susceptible to the plethoraof ways boosts can develop along any given flow.

We expect an infinitesimal null flow of Σ to gain energydue to an influx of matter analogous to the additionof 4-velocities in Figure 5, 𝐸3 = 𝐸1 + 𝐸2. However, withenergy being a frame dependent measurement and noway to discern a reference frame, we are left at themercy of distortions along the flow. Without knowing,our measurements could experience either an artificialincrease in energy 𝑃 → 𝑃′ or decrease 𝑃′ → 𝑃, as depictedin Figure 5. Geometrically, this manifests along the flowin a (local) tilting of Σ (recall Figure 4). From the formulaof 𝐸𝐻(Σ) for Σ ↪ Ω𝑆, Jensen’s inequality indicates the

P2 =

=

=

(E

(E

(E

2, �P2)

)

)P1 1, �P1

PPP3 3, �P3 ′

Figure 5. Along a flow in a particular coordinatesystem, energy is subject to boosts and may increaseor decrease.

existence of many flows with increasing 𝐸𝐻 yet only 𝑡-sliceintersections produce the Bondi Mass. Not only is thisflow highly specialized, it dictates strong restrictions onour initial choice of Σ from which to begin the flow.

This is not a problem, however, if appealing instead tomass rather than energy since boosts leavemass invariant,𝑀2 = 𝐸2−|𝑃|2 = (𝐸′)2−|𝑃′|2 = (𝑀′)2.Moreover, byvirtueof the Lorentzian triangle inequality (provided all vectorsare timelike and either all future or all past-pointing),along any given flow the mass should always increase𝑀3 = |(𝐸1+𝐸2, 𝑃1+𝑃2)| ≥ |(𝐸1, 𝑃1)|+|(𝐸2, 𝑃2)| = 𝑀1+𝑀2.One may hope therefore, by appealing to a notion ofmass instead of energy, a larger class of flows willarise exhibiting monotonicity of mass and physicallymeaningful asymptotics.

Recent Progress from a New Quasi-local MassIn search of a mass we return to our favorite modelspacetime, the Schwarzschild spacetime with null conesΩ𝑆. With the tantalizingly simple expression (3), a naturalfirst guess at extracting the black hole mass 𝑀 is to take

��(Σ) = 12(

14𝜋 ∫(𝒦− 1

4⟨��, ��⟩) 23 𝑑𝐴)

32 .

The reason being, irrespective of the cross-sectionΣ ↪ Ω𝑆,this gives

��(Σ) = 12(

14𝜋 ∫(2𝑀𝜔3 )

23 (𝜔2𝑑𝑆))

32 = 𝑀

as desired. Amazingly, Jensen’s inequality also ensuresthat �� ≤ 𝐸𝐻 whenever the integrand is nonnegative, byGauss–Bonnet. Unfortunately, upon an analysis of thepropagation of this mass on a general null cone, no clearmonotonicity properties arise and we’re left needing tomodify it at the very least. However, on a return to thedrawing board Ω𝑆, one finds that all cross-sections alsosatisfy 𝜁 = 𝑑 log𝜎 ↔ 𝜏 = 0 and a sufficient condition.Inspired by this, the second author [3] put forward themodified geometric flux 𝜌 and mass 𝑚(Σ):

𝜌 ∶= 𝒦− 14⟨��, ��⟩ +∇ ⋅ 𝜁−Δ log𝜎

𝑚(Σ) ∶= 12(

14𝜋 ∫𝜌 2

3 𝑑𝐴)32 .

160 Notices of the AMS Volume 65, Number 2

The first thing we observe is that the previously desiredproperties 𝑚(Σ) = 𝑀 for Σ ↪ Ω𝑆 and 𝑚 ≤ 𝐸𝐻 if 𝜌 ≥ 0are maintained (the second property now following fromboth the Gauss–Bonnet and the Divergence Theorems).Even better, from a nine-page calculation followed bythree different integrations by parts most terms combineto ensure this mass function is nondecreasing in greatgenerality, as summarized by our main theorem.

Theorem 1 ([3]). Let Ω be a null cone foliated by spacelikespheres {Σ𝑠} expanding along a null flow vector 𝐿 = 𝜎𝐿−

such that |𝜌(𝑠)| > 0 for each 𝑠. Then the mass 𝑚(𝑠) ∶=𝑚(Σ𝑠) has rate of change

𝑑𝑚𝑑𝑠 =

(2𝑚) 13

8𝜋 ∫Σ𝑠

𝜎𝜌 1

3((|��−|2 +𝐺(𝐿−, 𝐿−))(14⟨��, ��⟩ − 1

3Δ log |𝜌|)

+ 12|𝜈|

2 +𝐺(𝐿−,𝑁))𝑑𝐴,

where 𝐺 = 𝑅𝑖𝑐 − 12𝑅𝑔 (the Einstein tensor), 𝜈 = 2

3 ��− ⋅𝑑 log |𝜌| − 𝜏, and 𝑁 = 1

9 |𝑑 log |𝜌||2𝐿− + 13∇ log |𝜌| − 1

4𝐿+.

From the DEC it follows that 𝐺(𝐿−, 𝐿−),𝐺(𝐿−,𝑁) ≥ 0,so that the convexity conditions

𝜌 > 0(4)14⟨��, ��⟩ ≥ 1

3Δ log𝜌(5)

along a foliation {Σ𝑠}, called a doubly convex foliation,imply

𝑑𝑚𝑑𝑠 ≥ 0.

So how likely is a foliation to be doubly convex? Well,in the case of a cross-section of Ω𝑆 in Schwarzschild,we know (4) holds trivially from (3) since 𝜏 = 0. It alsofollows that

14⟨��, ��⟩ − 1

3Δ log𝜌 = 1𝜔2 (1 − 2𝑀

𝜔 ).

We conclude that all foliations of Ω𝑆 in the black holeexterior (𝜔 ≥ 2𝑀) satisfy (4) and (5). A natural questionfollows as to whether these conditions are physicallymotivated for more general asymptotically flat null cones.

Having found that this mass functional exhibits some-what genericmonotonicity, ournext concern is asymptoticconvergence and whether we obtain a physically signif-icant quantity. From the fact that 𝑚 ≤ 𝐸𝐻, we see thatany doubly convex foliation {Σ𝑠} approaching a geodesicfoliation of an asymptotically flat null cone Ω yields aconverging mass, since 𝐸𝐻(Σ𝑠) converges. However, thisconvergence is an indirect observation insufficient fora direct analysis of the limit. With a fairly standardstrengthening of the decay conditions on the geometryof Ω, called strong flux decay we’re afforded an explicitlimit for lim

𝑠→∞𝑚(Σ𝑠). Amazingly, this limit is independent

of any choice of asymptotically geodesic foliation (as inSchwarzschild), and we conclude with a proof of a NullPenrose Conjecture.

Theorem 2. ([3]) Let Ω be an asymptotically flat null conewith strong flux decay in a spacetime satisfying the DEC.Given the existence of an asymptotically geodesic doublyconvex foliation {Σ𝑠}

lim𝑠→∞

𝑚(Σ𝑠) ≤ 𝑚𝐵.

Moreover, in the case that ⟨��, ��⟩|Σ0 = 0 (i.e. Σ0 is a hori-zon) we prove the NPC. Furthermore, if we have the caseof equality for the NPC and {Σ𝑠} is a strict doubly convexfoliation, then Ω = Ω𝑆.

The first part of the theorem follows from the fact that𝑚 ≤ 𝐸𝐻 and that the limit lim

𝑠→∞𝑚(Σ𝑠) (being independent

of the flow) must therefore bound all Bondi energies frombelow, hence also 𝑚𝐵. In the second part, if ⟨��, ��⟩ = 0along with (5), then its a consequence of the MaximumPrinciple for elliptic PDE that 𝜌 must be constant on Σ0warranting equality in Jensen’s inequality. As a result,

√|Σ0|16𝜋 = 𝐸𝐻(Σ0) = 𝑚(Σ0) ≤ lim

𝑠→∞𝑚(Σ𝑠) ≤ 𝑚𝐵.

Open ProblemsAn interesting condition resulting in a doubly convexfoliation is that 𝜌(𝑠) be constant on each of the leavesof a foliation (i.e. on each Σ𝑠). As a result, this foliationsatisfies 𝑚(Σ𝑠) = 𝐸𝐻(Σ𝑠), representing a rest-frame flowgiven that energy equals mass. Studying the existence ofthis flow is of great interest.

We also invite the reader to recall the dependence of 𝜁and 𝜎 on the null basis {𝐿, 𝐿}. An analogous constructionof data under a role reversal between 𝐿 and 𝐿would resultin a new flux function 𝜌 on our surface Σ. We say a surfaceΣ is time-flat whenever 𝜌 = 𝜌. Time-flat surfaces withinasymptotically flat null cones are particularly interestingas they serve as pivots where a flow can flip direction or“bounce” as in Figure 6 without causing a discontinuousjump in mass (since 𝑚 = ��).

L

L

Σ

Figure 6. On a flat null cone a flow can flip directionor “bounce.”

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This observation is of particular significance in thesearch for more general foliations that inherit the suc-cessesofTheorems1and2.Anotherobjective is toweakenthe underlying smoothness assumption associated withnull cones in Theorem 2, possibly toward broadening itsvalidity to include focal points or multiple black holehorizons.

These open questions are important not only forunderstanding the physics of black holes and mass ingeneral relativity but also for expanding our knowledgeof null geometry. Since null geometry is not particularlyintuitive, physical motivations like these are very usefulfor providing fascinating conjectures to pursue.

References[1] R. Penrose, Naked singularities, Annals of the New York

Academy of Sciences, 224(1):125–134, 1973.[2] , Some unsolved problems in classical general-

relativity,Annals of Mathematics Studies, (102)631–668, 1982.MR 645761

[3] H. Roesch, Proof of a null penrose conjecture usinga new quasi-local mass, Preprint arXiv:1609.02875, 2016.MR 3664854

[4] M. Mars and A. Soria, The asymptotic behaviour of theHawking energy along null asymptotically flat hypersurfaces,Classical and Quantum Gravity, (33)115019, 2016.

Image CreditsFigures 1–6 produced by the authors.Photo of Hubert L. Bray courtesy of Duke Photography.Photo of Henri P. Roesch by Urban South Photo, courtesy of

Henri P. Roesch.

Hubert L. Bray

ABOUT THE AUTHORS

In 2001, Hubert Bray proved theRiemannian Penrose Conjecturefor any number of black holes bydiscovering a new geometric flowthat preserves nonnegative scalarcurvature.

Henri P. Roesch

Henri P. Roesch was born in Pre-toria, South Africa, and movedto the United States to com-plete his graduate studies at DukeUniversity.

162 Notices of the AMS Volume 65, Number 2