Existence of Black Holes Due to Concentration of Angular Momentum
aa r X i v : . [ g r- q c ] J u l EXISTENCE OF BLACK HOLES DUE TO CONCENTRATION OF ANGULARMOMENTUM
MARCUS A. KHURI
Abstract.
We present a general sufficient condition for the formation of black holes due to concen-tration of angular momentum. This is expressed in the form of a universal inequality, relating thesize and angular momentum of bodies, and is proven in the context of axisymmetric initial data setsfor the Einstein equations which satisfy an appropriate energy condition. A brief comparison is alsomade with more traditional black hole existence criteria based on concentration of mass. Introduction
The Trapped Surface Conjecture [20] and Hoop Conjecture [22] are concerned with the folklorebelief that if enough matter and/or gravitational energy are present in a sufficiently small region,then the system must collapse to a black hole. In more concrete terms, this belief is often realized byestablishing a statement of the following form. Let Ω be a compact spacelike hypersurface satisfyingan appropriate energy condition in a spacetime M . There exists a universal constant C > > C ·
Size(Ω) , then Ω is either enclosed by, or has a nontrivial intersection with, a closed trapped surface. Thepoint is that the presence of a closed trapped surface implies that the spacetime M contains asingularity (or more precisely is null geodesically incomplete) by the Hawking-Penrose SingularityTheorems [11], and assuming Cosmic Censorship [16] must therefore contain a black hole. Althoughthis problem is well-studied, the general case is still open. In particular, previous results [1, 2, 3, 9,13, 14, 15, 24] require special auxiliary conditions, for instance assuming that the spacelike slice isspherically symmetric or maximal, whereas others [8, 19, 25] are not meaningful for slices with smallextrinsic curvature.While it is natural, based on intuition, to suggest that high concentrations of matter and/orgravitational energy should lead to black hole formation, here we will propose a less intuitive criterionfor gravitational collapse. Namely, we will show that high concentrations of angular momentum aloneare enough to induce collapse. In analogy with (1.1), this will be realized by proving the existenceof a universal constant C > |J (Ω) | > C · R (Ω) , then Ω is either enclosed by, or has a nontrivial intersection with, a closed trapped surface. Here J (Ω) represents total angular momentum, and R (Ω) is a certain radius which measures the size ofΩ. Thus, if a rotating body possesses enough angular momentum and is sufficiently small, then thesystem must collapse to form a black hole. To the best of the author’s knowledge, this is a new andpreviously uninvestigated process by which black holes may form. In what follows, we will give aheuristic justification of (1.2) as well as a rigorous proof for axially symmetric rotating systems. The author acknowledges the support of NSF Grant DMS-1308753. Heuristic Evidence and Precise Formulation
In [7], Dain introduced an inequality relating the size and angular momentum for General Rela-tivistic bodies of the form(2.1) R (Ω) & Gc |J (Ω) | , where G is the gravitational constant, c is the speed of light, and & represents an order of magnitude;the precise constant depends on the choice of radius R . He presented heuristic arguments which imply(2.1), and which are based on the following four assumptions and principles:(i) the body Ω is not contained in a black hole,(ii) the speed of light c is the maximum speed,(iii) the reverse inequality of (1.1) holds for bodies Ω which are not contained in a black hole,(iv) inequality (2.1) holds for black holes.Now suppose that (1.2) holds. Then (2.1) is violated, and hence one of the above four statementscannot be valid. Since (ii) is firmly established, (iii) is closely tied to the Trapped Surface/HoopConjecture and is expected to hold, and (iv) has been proven [6], it follows that assumption (i)should be false. In general terms, we conclude that if a body satisfies (1.2), this should indicate thepresence of a black hole.A precise version of this conclusion will now be described, and then proven below. Consider aninitial data set ( M, g, k ) for the Einstein equations. This consists of a 3-manifold M , (complete)Riemannian metric g , and symmetric 2-tensor k representing the extrinsic curvature (second funda-mental form) of the embedding into spacetime, which satisfy the constraint equations16 πGc µ = R + ( T r g k ) − | k | , πGc J i = ∇ j ( k ij − ( T r g k ) g ij ) . (2.2)Here µ and J are the energy and momentum densities of the matter fields, respectively, and R is thescalar curvature of g . In terms of the 4-dimensional stress-energy tensor T ab we have µ = T ab n a n b and J i = T ia n a , where n a denotes the timelike unit normal to the slice M . A body Ω is a connectedopen subset of M with compact closure and smooth boundary ∂ Ω.We say that the initial data are axially symmetric if the group of isometries of the Riemannianmanifold (
M, g ) has a subgroup isomorphic to U (1), and that the remaining quantities defining theinitial data are invariant under the U (1) action. In particular, if η i denotes the Killing field associatedwith this symmetry, then(2.3) L η g = L η k = L η µ = L η J = 0 , where L η denotes Lie differentiation. Furthermore, the total angular momentum of the body Ω isdefined by(2.4) J (Ω) = 1 c Z Ω J i η i dω g . Axisymmetry is imposed primarily to obtain a suitable and well-defined notion of angular momentumfor bodies. Note that in this setting gravitational waves have no angular momentum, so all theangular momentum is contained in the matter sources. Without this assumption quasi-local angular
XISTENCE OF BLACK HOLES DUE TO CONCENTRATION OF ANGULAR MOMENTUM 3 momentum is difficult to define [21]. It will also be assumed that the following version of the dominantenergy condition holds on Ω, namely(2.5) µ ≥ | ~J | + | J ( e ) | where ( e , e , e = | η | − η ) is an orthonormal frame field on M and ~J = J ( e ) e + J ( e ) e . Note thatthis is a stronger version of the classical dominant energy condition which states(2.6) µ ≥ | J | = q | ~J | + J ( e ) . Let us now consider how to measure the size of the body Ω. A particularly pertinent measure inthe current setting, is a homotopy radius defined by Schoen and Yau in [19], which played a crucialrole in their criterion for the existence of black holes due to concentration of matter. The Schoen/Yauradius, R SY (Ω), may be described as the radius of the largest torus that can be embedded in Ω.More specifically, let Γ be a simple closed curve which bounds a disk in Ω, and let r denote thelargest distance from Γ such that the set of all points within this distance forms a torus embeddedin Ω. Then R SY (Ω) is defined to be the largest distance r among all curves Γ. For example, if B ρ isa ball of radius ρ in flat space, then R SY ( B ρ ) = ρ/
2. In analogy with [7], we define the radius thatappears in (1.2) by(2.7) R (Ω) = (cid:0)R Ω | η | dω g (cid:1) / R SY (Ω) . With these definitions of angular momentum J and radius R , we obtain a precise formulationof inequality (1.2), save for the universal constant C to be described below. It will be shown thatthis inequality implies the existence of a closed trapped surface, or more accurately an apparenthorizon. Recall that the strength of the gravitational field in the vicinity of a 2-surface S ⊂ M maybe measured by the null expansions(2.8) θ ± := H S ± T r S k, where H S is the mean curvature with respect to the unit outward normal. The null expansionsmeasure the rate of change of area for a shell of light emitted by the surface in the outward futuredirection ( θ + ), and outward past direction ( θ − ). Thus the gravitational field is interpreted as beingstrong near S if θ + < θ − <
0, in which case S is referred to as a future (past) trapped surface.Future (past) apparent horizons arise as boundaries of future (past) trapped regions and satisfy theequation θ + = 0 ( θ − = 0). Apparent horizons may be thought of as quasi-local notions of eventhorizons, and in fact, assuming Cosmic Censorship, they must generically be contained inside blackholes [23]. 3. Inequality Between Size and Angular Momentum for Bodies
In order to establish the existence of apparent horizons when inequality (1.2) is satisfied, we willutilize a device employed by Schoen and Yau [19]. Namely, they showed that if a certain differentialequation does not possess a regular solution, then an apparent horizon must be present in the initialdata. This so called Jang equation is given in local coordinates by(3.1) (cid:18) g ij − f i f j |∇ f | (cid:19) ∇ ij f p |∇ f | − k ij ! = 0 , where f i = g ij ∇ j f . Geometrically, this expression is equivalent to the apparent horizon equation,but in the 4-dimensional product manifold R × M . When regular solutions do not exist, the graph KHURI t = f ( x ) blows-up and approximates a cylinder over an apparent horizon in the base manifold M (see [10], [18]). Whether or not the solution blows-up, is related to concentration of scalar curvatureor rather matter density for the induced metric, g ij = g ij + ∇ i f ∇ j f , on the graph. In this regard,it is important to have an explicit formula [4, 5, 18] for the scalar curvature of this metric, namely(3.2) R = 16 πGc ( µ − J ( v )) + | h − k | g + 2 | q | g − div g ( q ) , where h is the second fundamental form of the graph, div g is the divergence operator with respectto g , and q and v are 1-forms given by(3.3) v i = f i p |∇ f | , q i = f j p |∇ f | ( h ij − k ij ) . Suppose now that the Jang equation has a regular solution over Ω. One way to measure theconcentration of scalar curvature on this region is to estimate the first Dirichlet eigenvalue, λ , of theoperator ∆ g − R ; here ∆ g = g ij ∇ ij is the Laplace-Beltrami operator. Let φ be the correspondingfirst eigenfunction, then(3.4) λ = R Ω (cid:0) |∇ φ | + Rφ (cid:1) dω g R Ω φ dω g . Notice that in the expression for the scalar curvature (3.2), only the divergence term yields a po-tentially negative contribution to the quotient. However, after applying the divergence theorem, wefind that this term is dominated by the two nonnegative terms |∇ φ | and | q | g φ . It follows that(3.5) λ ≥ πGc R Ω ( µ − J ( v )) φ dω g R Ω φ dω g . In light of the axial symmetry (2.3), if the Jang solution f possesses axially symmetric boundaryconditions, then L η f = 0 on Ω. In particular, the 1-form v has no component in the η direction,which implies that µ − J ( v ) ≥ µ − | ~J | . It follows that(3.6) λ ≥ πGc R Ω ( µ − | ~J | ) φ dω g R Ω φ dω g =: Λ . With a lower bound for the first eigenvalue in hand, we may apply Proposition 1 from [19] toconclude(3.7) R SY (Ω) ≤ r π √ Λ , where R SY denotes the Schoen/Yau radius with respect to the metric g . Observe that since g ≥ g , wehave R SY ≥ R SY . Moreover, multiplying and dividing Λ − by the quantity R Ω | η | dω g (cid:16)R Ω ( µ − | ~J | ) | η | dω g (cid:17) − yields(3.8) Λ − ≤ c C πG R Ω | η | dω g R Ω ( µ − | ~J | ) | η | dω g , where(3.9) C = max Ω (cid:16) µ − | ~J | (cid:17) min Ω (cid:16) µ − | ~J | (cid:17) XISTENCE OF BLACK HOLES DUE TO CONCENTRATION OF ANGULAR MOMENTUM 5 if µ − | ~J | > C = ∞ if µ − | ~J | vanishes at some point of Ω. Hence(3.10) Z Ω ( µ − | ~J | ) | η | dω g ≤ πc C G R Ω | η | dω g R SY (Ω) . All together these arguments produce a general relation between the size and angular momentum ofbodies.
Theorem 3.1.
Let ( M, g, k ) be an axially symmetric initial data set which contains no compactapparent horizons. Assume that either M is asymptotically flat, or has a strongly untrapped boundary,that is H ∂M > | T r ∂M k | . Then for any body Ω ⊂ M satisfying the energy condition (2.5) , the followinginequality holds (3.11) |J (Ω) | ≤ πc C G R (Ω) . Proof.
The conditions on the boundary of M or its asymptotics guarantee the existence of a stronglyuntrapped 2-surface. For instance, if M is asymptotically flat, then a large coordinate sphere in theasymptotic end will be strongly untrapped. This property allows one to solve the Dirichlet boundaryvalue problem [19] for the Jang equation (3.1), with f = 0 on ∂M or on an appropriate coordinatesphere in the asymptotic end. The solution f will then be axisymmetric. Moreover, the absence ofapparent horizons ensures that f is a regular solution. We may then apply the arguments precedingthis theorem to obtain (3.10).Now observe that with the help of the energy condition (2.5), |J (Ω) | ≤ c Z Ω | J ( η ) | dω g = 1 c Z Ω | J ( e ) || η | dω g = 1 c Z Ω h | J ( e ) | + | ~J | − µ + ( µ − | ~J | ) i | η | dω g ≤ c Z Ω ( µ − | ~J | ) | η | dω g . (3.12)Combining (3.10) and (3.12) produces the desired result. (cid:3) This theorem is of independent interest, and generalizes the main result of [7] in two ways. That is,Dain’s inequality between the size and angular momentum of bodies required two strong hypotheses,namely that the initial data are maximal
T r g k = 0 and that the matter density µ is constant. Herewe have removed both of these hypotheses at the expense of a slightly weaker inequality. Moreprecisely, when µ − | ~J | is constant, the two inequalities may be compared directly. The universalconstant obtained by Dain, πc G , is less than the universal constant of (3.11), πc G . Recently, otherrelated inequalities have been derived by Reiris [17], which also require the maximal hypothesis.4. Criterion for the Existence of Black Holes
The proof of Theorem 3.1 naturally leads to a black hole existence result, due to its reliance onsolutions of the Jang equation. As noted, this technique for producing black holes was originallyexploited by Schoen and Yau [19]. Namely, if the reverse inequality of (3.11) holds, then we mustconclude that the Jang equation does not admit a regular solution. This implies that the Jangsolution blows-up and ensures the presence of an apparent horizon. We now state the main result.
KHURI
Theorem 4.1.
Let ( M, g, k ) be an axially symmetric initial data set, such that either M is asymp-totically flat, or has a strongly untrapped boundary, that is H ∂M > | T r ∂M k | . If Ω ⊂ M is a boundedregion satisfying the energy condition (2.5) , with (4.1) |J (Ω) | > πc C G R (Ω) , then M contains an apparent horizon of spherical topology and in particular contains a closed trappedsurface. It should be observed that Theorems 3.1 and 4.1 are independent of the particular matter model,and only require an energy condition which prevents the matter from traveling faster than the speedof light.Whether or not such a process, by which high concentrations of angular momentum leads togravitational collapse, can occur in nature, appears to be an interesting open problem. On thetheoretical side, it is important to understand the types of geometries which admit an inequality ofthe form (4.1). In this regard we note that the inequality cannot hold in the maximal case. This isdue to the fact, explained at the end of the previous section, that the constant in Dain’s inequality[7] is smaller than the constant in (4.1). This is analogous to the situation with Schoen and Yau’scriterion for black hole formation, in which a stronger inequality holds for bodies in the maximalcase, thus preventing such initial data from satisfying their hypotheses for the existence of trappedsurfaces. Thus, the geometries which satisfy the Schoen/Yau criterion require large amounts ofextrinsic curvature; see [15] for a discussion concerning this topic. We expect that the same holdstrue for Theorem 4.1.Let us now compare the above result with more traditional black hole existence criteria based onconcentration of mass. In general terms, we have shown that a body undergoes gravitational collapseif its total angular momentum and radius satisfy an inequality of the form(4.2) Gc |J | & R , whereas a version of the Hoop Conjecture asserts that(4.3) 2 Gmc > R is sufficient for collapse, where m denotes rest mass and the left-hand side is the Schwarzschildradius R s . Consider the fastest spinning pulsar [12] known to date, PSR J1748-2446ad. Its angularvelocity is ω ≈ . × rad s − , it has a radius of R ≈ km , and consists of two solar masses m ≈ M ⊙ = 4 × kg . It follows that(4.4) r Gc |J | = r Gc mω R ≈ . km and 2 Gmc ≈ . km. Upon comparing these values with the radius, we find that although it is close, neither (4.2) nor(4.3) is satisfied, as expected. Moreover, the similarity of the values in (4.4) seems to suggest thatthe criteria (4.2) and (4.3) may apply in similar regimes (at least for astronomical objects), howeverthis is dependent on the magnitude of the optimal constant in (4.2) which is not addressed in thispaper. It is thus a question worthy of further investigation to determine the optimal constant.We may also compare the criteria (4.2) and (4.3) in the realm of elementary particles. It turnsout that here, (4.2) offers additional insight due to the quantization of angular momentum. Moreprecisely, for a particle of spin s its angular momentum is given by(4.5) |J | = p s ( s + 1) ~ , XISTENCE OF BLACK HOLES DUE TO CONCENTRATION OF ANGULAR MOMENTUM 7 where ~ = 1 . × − cm s − kg is Planck’s constant. According to (4.2), in order for such a particleto remain stable gravitationally, its radius should be bounded below by a multiple of [ s ( s + 1)] / l p where l p = q G ~ c is the Planck length. A similar, although somewhat less convincing argumentis known, from which one may derive the same conclusion based on criteria (4.3). Namely, thegravitational stability of a particle implies that its radius should not be less than the Schwarzschildradius, R ≥ R s . Moreover, the length scale at and below which quantum effects become veryimportant is given by the Compton wavelength λ = ~ mc , and hence we should have R s ≥ λ . This,however, implies that R s ≥ √ l p , which again yields a lower bound for the radius of the particle interms of the Planck length. References [1] R. Beig, and N. ´O Murchadha,
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