A simple proof of the recent generalisations of Hawking s black hole topology theorem

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1 A simple proof of the recent generalisations of Hawking s black hole topology theorem arxiv: v3 [gr-qc] 25 Jun 2010 István Rácz RMKI, H-1121 Budapest, Konkoly Thege Miklós út Hungary June 29, 2010 Abstract A key result in four dimensional black hole physics, since the early 1970s, is Hawking s topology theorem asserting that the cross-sections of an apparent horizon, separating the black hole region from the rest of the spacetime, are topologically two-spheres. Later, during the 1990s, by applying a variant of Hawking s argument, Gibbons and Woolgar could also show the existence of a genus dependent lower bound for the entropy of topological black holes with negative cosmological constant. Recently Hawking s black hole topology theorem, along with the results of Gibbons and Woolgar, has been generalised to the case of black holes in higher dimensions. Our aim here is to give a simple self-contained proof of these generalisations which also makes their range of applicability transparent. iracz@sunserv.kfki.hu, This research was supported in part by OTKA grant K

2 1 Introduction The notion of a trapped surface was introduced by Penrose [11]. In 4-dimensional spacetime the spacelike boundary,, of a 3-dimensional spatial region is called a future trapped surface if gravity is so strong there that even the future and outwards directed normal null rays starting at are dragged back so much that their expansion is non-positive everywhere on. Careful analysis justified that trapped surfaces necessarily occur whenever sufficient amount of energy is concentrated in a small spacetime region [12]. Intuitively a black hole region is considered to be a part of a spacetime from which nothing can escape. Therefore a black hole region is supposed to be a future set comprised by events that individually belong to some future trapped surface. The boundary of such a black hole region, referred to usually as the apparent horizon, H, is then supposed to be comprised by marginal future trapped surfaces. As one of the most important recent results in black hole physics the existence of an apparent horizon was proved in [1, 2]. More specifically, it was shown there that given a strictly stable marginally outer trapped surface (MOT) 0 Σ 0 in a spacetime with reference foliation {Σ t }, then, there exists an open tube, H = t t, foliated by marginally outer trapped surfaces, with t Σ t, through 0. Let us merely mention, without getting into details here, that the applied strict stability assumption is to exclude the appearance of future trapped surfaces in the complementer of a black hole region. Hawking s black hole topology theorem [9] is proven by demonstrating that whenever the dominant energy condition (DEC) holds a MOT can be deformed, along the family of null geodesics transverse to the apparent horizon, yielding thereby on contrary to the fact that is a MOT a future trapped surface in the complementer of the black hole region, unless the Euler characteristic χ of is positive. Whenever is a codimension two surface in a 4-dimensional spacetime the Euler characteristic and the genus, g, of can be given, in virtue of the Gauss-Bonnet theorem, via the integral of the scalar curvature R q of the metric q ab induced on as 2πχ = 4π(1 g ) = 1 R q ǫ q. (1.1) 2 The main difficulty in generalising Hawking s argument to the higher dimensional case originates from the fact that whenever is of dimension s = n 2 3 in an n-dimensional spacetime, the integral of the scalar curvature R (s) by itself is not informative, as opposed to the case of n = 4, therefore the notion of Euler characteristic has to be replaced by the Yamabe invariant. The latter is known to be a fundamental topological invariant and it is defined as follows. Denote by [q] the conformal class of Riemannian metrics on determined by q ab. It was conjectured by Yamabe, and later proved by Trudinger, Aubin, and choen that to every conformal class on any 2

3 smooth compact manifold there exists a metric q ab of constant scalar curvature so that ( ) 2 s 2 R q = Y(,[q]) ǫ q, (1.2) where the Yamabe constant Y(,[q]), associated with the conformal class [q], is defined as [ 4 s 1 s 2 (Da u)(d a u)+r q u 2] ǫ q Y(,[q]) = inf Rˆqǫˆq ) s 2 ˆq [q] ( ǫˆq s = inf u C (),u>0 ( ) u 2s s 2 s s 2 ǫq. (1.3) In the later case, the metric ˆq [q] can be given as ˆq ab = u 4 s 2 qab, and, moreover, D a and R q denote the covariant derivative operator and the scalar curvature associated with the metric q ab on. The Yamabe invariant is defined then as Y() = supy(,[q]). (1.4) [q] ome of the recent generalisation of Hawking s [9] black hole topology theorem, and also that of Gibbons [7] and Woolgar s [15] results, proved by Galloway, choen, O Murchadha and Cai, that are covered by Refs.[3, 4, 6] may be formulated then as. Theorem 1.1 Let (M,g ab ) be a spacetime of dimension n 4 satisfying the Einstein equations R ab 1 2 g abr+λg ab = 8πT ab, (1.5) with cosmological constant Λ, and with matter subject to DEC. uppose, furthermore, that is a strictly stable MOT in a regular spacelike hypersurface Σ. (1) If Λ 0 then is of positive Yamabe type, i.e., Y() > 0. (2) If Y() < 0 and Λ < 0 then for the area A() = ǫ q of the inequality ( Y() A() 2 Λ ) s 2 (1.6) holds. The significance of these results get to be transparent if one recalls that in the first case, i.e., when Y() > 0, cannot carry a metric of non-positive sectional curvature which immediately restricts the topological properties of [8]. Whereas, in the second case the a lower bound on the entropy of a black hole, that is considered to be proportional to the area A(), is provided by (1.6). 3

4 Before proceeding we would like to stress on an important conceptual point. Most of the quoted investigations of black holes, see Refs.[1, 2, 3, 4, 5, 6], starts by assuming the existence of a reference foliation{σ t } of the spacetime by (partial) Cauchy surfaces Σ t. In this respect it is worth recalling that by a non-optimal choice of {Σ t } one might completely miss a black hole region as it follows from the results of [14] where it was demonstrated that even in the extended chwarzschild spacetime one may find a sequence of Cauchy surfaces which get arbitrarily close to the singularity such that neither of these Cauchy surfaces contains a future trapped surface. Hence, one of the motivations for the present work besides providing a reduction of the complexity of the proof of Theorem 1.1, and also a simultaneous widening of its range of applicability was to carry out a discussion without making use of any reference foliation. 2 Preliminaries As it will be seen below the simplicity of the presented argument allows the investigation of black holes essentially in arbitrary metric theory of gravity. Thereby, we do not restrict our considerations to either of the specific theories. Accordingly, a spacetime is assumed to be represented by a pair (M,g ab ), where M is an n- dimensional (n 4), smooth, paracompact, connected, orientable manifold endowed with a smooth Lorentzian metric g ab of signature (,+,...,+). It is assumed that (M,g ab ) is time orientable and that a time orientation has been chosen. The only restriction, concerning the geometry of the allowed spacetimes, is the following generalised version of DEC. A spacetime (M,g ab ) is said to satisfy the generalised dominant energy condition if there exists some smooth real function f on M such that for all future directed timelike vector t a the combination [G a bt b +f t a ] is a future directed timelike or null vector, where G ab denotes the Einstein tensor R ab 1 2 g abr. It is straightforward to see that in Einstein s theory of gravity, where g ab is subject to (1.5), the generalised dominant energy condition, with the choice f = Λ, is equivalent to requiring the energy-momentum tensor, T ab, to satisfy DEC [13]. To restrict our considerations to black hole spacetimes, we shall also assume the existence of future trapped surfaces in (M,g ab ) which are defined as follows. Let us consider a smooth orientable (n 2)-dimensional compact manifold with no boundary in M. Let l a and k a be smooth future directed null vector fields on scaled such that k a l a = 1, and that are also normal to, i.e., g ab l a X b = 0 and g ab k a X b = 0 for any vector field X a tangent to. Consider then the null hypersurfaces generated by geodesics starting on with tangent l a and k a. These null hypersurfaces are smooth in a neighbourhood of, and, by making use of the associated synchronised affine parametrisations of their null generators, the 4

5 vector fields l a and k a can be, respectively, extended to them. The level surfaces of the corresponding synchronised affine parametrisations do also provide foliations of these null hypersurfaces by (n 2)-dimensional compact manifolds homologous to. Denote by ǫ q the volume element associated with the metric, q ab, induced on these (n 2)-dimensional surfaces. Then the null expansions θ (l) and θ (k) with respect to l a and k a are defined by l ǫ q = θ (l) ǫ q, k ǫ q = θ (k) ǫ q, (2.1) where l and k denotes the Lie derivative with respect to the null vector fields l a and k a. The (n 2)-dimensional surface is called future trapped surface if both of the null expansions θ (l) andθ (k) are non-positive on. In the limiting case, i.e., whenever either of these null expansions (say θ (l) ) vanishes on identically, is called future marginally trapped surface. In case of a generic (n 2)-dimensional surface the quasi-local concept of outwards and inwards directions is undetermined. Nevertheless, these concepts get to be welldefined for non-minimal marginally trapped surfaces. It will be said that l a and k a point outwards and inwards, respectively, provided that θ (l) = 0 and θ (k) 0, and that θ (k) is not identically zero. (If bothθ (l) andθ (k) vanish identically is a minimal surface and the notions outwards and inwards become to be degenerate.) To see that this, quasi-local, concept of outer direction is not counter intuitive, consider the null hypersurface N generated by null geodesics starting at the points of with tangent n a = k a (see Fig.1). ince N is smooth in a neighbourhood O N k a n a l a H Figure 1: The black hole, represented by the shaded region, is bounded by horizon H that is foliated by MOT homologous to. of it can be smoothly foliated by (n 2)-dimensional surfaces, u, defined as the u = const cross-sections of N, where u is the affine parameter along the generators of N such that n a = ( / u) a and u = 0 on. Then, it seems to be natural to consider k a as inward pointing if the area A( u ) = u ǫ q of the cross-sections u 5

6 is non-decreasing in the direction of n a which, in the case under consideration, follows from θ (k) 0 as da( u ) u=0 = k ǫ q = θ (k) ǫ q 0. (2.2) du Accordingly, is called future marginally outer trapped surface (MOT) if θ (l) = 0 and θ (k) 0 on. In deriving our results we shall also apply a stability assumption. Before formulating it let us recall first that the above imposed conditions do not uniquely determine the pair of null vector fields l a and k a on. In fact, together with l a and k a any pair of null vector fields l a and k a that is yielded by the boost transformation k a k a = e v k a, l a l a = e v l a, (2.3) where v : R is an arbitrary smooth function on, will be suitable. It is well-known, however, that the signs of θ (l) and θ (k) are invariant under such positive rescaling of l a and k a. uppose then that is a future MOT with respect to a null normal l a. Then, will be called strictly stably outermost if there exists a rescaling of the type (2.3) so that k θ (l ) 0 for the yielded vector fields l a and k a, and also k θ (l ) < 0 somewhere on. Obviously, this definition is independent of the use of any sort of reference foliation. Moreover, as it will be indicated below, it can be seen to be equivalent to the corresponding criteria applied in [1, 2]. 3 The proof of the main result The main argument of the present paper can then be given in the following simple geometric setup. We have already defined n a = k a to be a smooth past directed null vector field on that is also normal to. imilarly, the smooth null hypersurface N, spanned by the (n 2)-parameter congruence of null geodesics starting at with tangent n a, and the affine parameter u along the geodesics, synchronised such that u = 0 on, have already been introduced. Denote by n a the tangent field ( / u) a on the null hypersurface N, that is foliated by the smooth u = const cross-sections, u. Then, there exists a uniquely defined future directed null vector field l a on N such that g ab n a l b = 1, and that l a is orthogonal to each u. Denote by r the affine parameter of the null geodesics determined by l a which are synchronised such that r = 0 on N. ince l a is, by construction, smooth on N the null geodesics starting with tangent l a on N do not meet within a sufficiently small open elementary spacetime neighbourhood O of. Extend, then, the function u from N onto O by keeping its value 6

7 to be constant along the geodesics with tangent l a. Then the vector fields n a and l a, defined so far only on N, do also extend onto O such that the relations n a = ( / u) a and l a = ( / r) a hold there which do also imply that n a and l a commute on O. Note that O is smoothly foliated by the 2-parameter family of (n 2)-dimensional u = const, r = const level surfaces u,r. The spacetime metric in O can then always be given (see, e.g., [10] for a justification) as g ab = 2 ( (a r rα (a u rβ (a ) b) u+γ ab (3.1) where α, β a and γ ab are smooth fields on O such that β a and γ ab are orthogonal to n a and l a. Recall also that γ ab and the positive definite metric q ab on the (n 2)- dimensional surfaces u,r are related as q ab = r 2 β c β c l a l b 2rβ (a l b) +γ ab. (3.2) This latter relation implies thatq ab = γ ab onn, represented by ther = 0 hypersurface in O, i.e., γ ab is the metric on the cross-sections u of N. ince the vector fields n a and l a are null and normal to the cross-sections u the expansions of the associated null congruences at u can be given as θ (n) u = 1 2 qef ( n q ef ) = 1 2 γef ( n γ ef ) and θ (l) u = 1 2 qef ( l q ef ) = 1 2 γef ( l γ ef ), (3.3) where n and l denote the Lie derivative with respect to the vector fields n a and l a, respectively, and (here and elsewhere) all the indices are raised and lowered with the spacetime metric g ab. The second equalities in (2.1) follow from the fact that β a and γ ab are orthogonal to n a and l a and also that n a and l a commute in O. By making use of (3.1), and the definition of the Einstein tensor, it is straightforward to see that G ab n a l b = R ab n a l b 1 2 R [ ef 2n (e l f) +γ ef] = 1 2 γef R ef (3.4) holds on N. Then, in virtue of equation (82) of [10], and by the coincidence of q ab and γ ab, and also of the projectors q a b, γ a b and p a b on N (for the definition of p a b see (76) of [10]) we also have that on N [ G ab n a l b = 1 2 γ ab ( l n γ ab ) αγ ab ( l γ ab )+R q +D a β a 1 2 βa β a +γ ab γ cd ( l γ ac )( n γ bd ) 1 ] 2 γab ( l γ ab )γ cd ( n γ cd ), (3.5) where D a and R q denote the covariant derivative operator and the scalar curvature associated with the metric q ab = γ ab on the (n 2)-dimensional surfaces u on N. 7

8 By making use of the fact that the vector fields n a and l a commute in O and that they are orthogonal to γ ab a direct calculation justifies then the relation γ ab ( l n γ ab ) = l ( γ ab n γ ab ) γ ab γ cd ( l γ ac )( n γ bd ). (3.6) ince n a and l a commute we also have that l θ (n) = n θ (l) (3.7) in O, where the expansion θ (n) is defined with respect to the volume element ǫ q associated with the metric q ab on the surfaces u,r as in (2.1). imilarly, it can be verified that k θ (l) = n θ (l) (3.8) onn, i.e., whenever r = 0, where k a denotes the unique future directed null extension [ k a = n a + (rα+ 12 ] r2 β e β e )l a +rβ a (3.9) of k a = n a on onto O which is normal to the surfaces u,r and is scaled such that k a l a = 1 in O. Then, by making use of the vanishing of θ (l) on, the above relations in particular, equations (2.1), (3.5), (3.6) and (3.7) imply that n θ (l) = G ab n a l b + 1 [R q +D a β a 12 ] 2 βa β a. (3.10) ince n a and l a are both future directed null vector fields on N, and also the generalised dominant energy condition holds, i.e., there exists a real function f on M such that the vector field [G a bl b +f l a ] is future directed and causal, the inequality G ab n a l b + f 0 holds on N. Finally, since was assumed to be a strictly stable MOT, in virtue of (3.8), the null normals n a = k a and l a may be assumed, without loss of generality, to be such that n θ (l) 0, and also that n θ (l) > 0 somewhere on. To see that the stability condition applied here is equivalent to the one used in [1, 2] note that β a = q e an b e l b and it transforms under the rescaling (2.3) of the vector fields k a = n a and l a on as β a β a = β a +D a v. By making use of the notation ψ = e 2v and s a = 1 2 β a, it can be verified then that ψn θ (l ) = D a D a ψ +2s a D a ψ + ψ 2 [ Rq +2G ab n a l b +2D a s a 2s a s a ] (3.11) holds, which is exactly the expression δ q θ given in Lemma3.1 of [2] whenever is a MOT and the variation vector field q a is chosen to be ψn a. This justifies then 8

9 that the strict stability conditions applied here and in [1, 2] (see, e.g., Definition 5.1 and the discussion at the end of ection 5 of [2] for more details) are equivalent. In returning to the main stream of our argument note that whenever is a strictly stable MOT and the generalised DEC holds then, in virtue of (3.10), R q +D a β a 1 2 βa β a 2f, (3.12) so that the inequality is strict somewhere on. ince q ab is positive definite we also have that for any smooth function u on u 2 D a β a = D a (u 2 β a ) 2u(D a u)β a D a (u 2 β a )+2(D a u)(d a u)+ 1 2 u2 β a β a. (3.13) Thus, multiplying (3.12) by u 2, where u > 0 is arbitrary, we get, in virtue of (3.13), that 2(D a u)(d a u)+r q u 2 +D a (u 2 β a ) 2fu 2, (3.14) so that the inequality is strict somewhere on. To get the analogue of the first part of Theorem1.1 assume now that f is such that f 0 throughout. Then, by taking into account the inequality 4 s 1 > 2, s 2 which holds for any value of s 3, we get from (3.14) that [ 4 s 1 s 2 (Da u)(d a u)+r q u 2] ǫ q ( ) u 2s s 2 > 0, (3.15) s s 2 ǫq for any smooth u > 0, i.e., Y(,[q]) > 0, which implies that is of positive Yamabe type. imilarly, whenever the minimal value fmin of f on is negative, on one hand, 2 fu 2 ǫ q 2 fmin u 2 ǫ q (3.16) while, on the other hand, by applying the Hölder inequality ( φ 1 φ 2 ǫ q )1 ( )1 φ 1 a a ǫ q φ 2 b b 1 ǫ q, a + 1 b = 1 (3.17) to the functions φ 1 = u 2 and φ 2 = 1 with a = s, b = s, we get that s 2 2 ( u 2 ǫ q ) s 2 u 2s s s 2 ǫq [A()] 2 s. (3.18) 9

10 The combination of (3.16) and (3.18), along with (3.14), justifies then that [ 4 s 1 s 2 (Da u)(d a u)+r q u 2] ǫ q ( ) u 2s s 2 2 fmin [A()] 2 s. (3.19) s s 2 ǫq Assuming finally that Y() < 0 we get that, for any conformal class [q], the Yamabe constant Y(,[q]) Y() < 0. This, along with (3.19), implies then Y() Y(,[q]) = Y(,[q]) 2 f min [A()] 2 s, (3.20) which leads to the variant of the inequality (1.6) yielded by the replacement of Λ by f min. 4 Final remarks What has been proven in the previous section can be summarised as. Theorem 4.1 Let (M,g ab ) be a spacetime of dimension n 4 in a metric theory of gravity. Assume that the generalised dominant energy condition, with smooth real function f : M R, holds and that is a strictly stable MOT in (M,g ab ). (1) If f 0 on then is of positive Yamabe type, i.e., Y() > 0. (2) If Y() < 0 and fmin < 0, where f min denotes the minimal value of f on, then ( ) s Y() 2 A(). (4.1) 2 fmin We would like to mention that the argument of the previous section does also provide an immediate reduction of the complexity of the original proof of Hawking and that of Gibbons and Woolgar. To see this recall that, in virtue of (1.1), (1.3) and (1.4), Y() = 4πχ, whenever is of dimension s = 2, and also that f = Λ, if attention is restricted to Einstein s theory with matter satisfying the dominant energy condition. Thereby, as an immediate consequence of Theorem4.1, we have that whenever Λ 0 on, while χ > 0, (4.2) A() 4π(1 g ) Λ (4.3) 10

11 whenever both χ and Λ are negative. Clearly the above justification of Theorem4.1 is free of the use of any particular reference foliation of the spacetime. Note also that in the topological characterisation of an (n 2)-dimensional strictly stable MOT only the quasi-local properties of the real function f : M R are important. In particular, as the conditions of Theorem4.1 do merely refer to the behaviour of f on it need not to be bounded or to have a characteristic sign throughout M. imilarly, it would suffice to require the generalised dominant energy condition to be satisfied only on. Finally, we would also like to emphasise that Theorem 4.1 provides a considerable widening of the range of applicability of the generalisation of Hawking s black hole topology theorem, and also that of the results of Gibbons and Woolgar. As its conditions indicate, Theorem 4.1 applies to any metric theory of gravity and the only restriction concerning the spacetime metric is manifested by the generalised dominant energy condition and by the assumption requiring the existence of a strictly stable MOT. Accordingly, Theorem4.1 may immediately be applied in string theory or in various other higher dimensional generalisations of general relativity. 11

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