aa r X i v : . [ m a t h . DG ] F e b Answer to a question asked by Gregory Galloway
Olaf M¨uller ∗ February 15, 2021
Abstract
Considering the problem of the existence of CMC Cauchy surfaces, Galloway [2] asked whetherfor any globally hyperbolic, spatially compact and future timelike geodesically complete mani-fold (
M, g ) satisfying the strong energy condition (SEC), the future causal boundary of (
M, g )consists of one point. We show that the answer to this question is ”no” in any dimension greateror equal to 3, and that there are plenty of counterexamples, more precisely: On a manifold ofdimension ≥ every globally hyperbolic spatially compact conformal class contains future com-plete metrics satisfying the strong energy condition. In the spatially noncompact case, the sameis true in the future of any Cauchy surface. On p. 675 of [2], Galloway writes (where ”SEC” stands for ”strong energy condition”):”
Conjecture 2.
Let ( M, g ) be a spacetime with compact Cauchy surfaces. If ( M, g ) is futuretimelike geodesically complete and satisfies the SEC then ( M, g ) contains a CMC surface. A question related to this conjecture is whether the assumptions imply that the future causalboundary consists of one point.”The author learned about the relevance of this question when Galloway presented it at the GeLoCorconference in February 2021, pointing out that the conjecture already figures in an earlier articleof his with Eric Ling [3], and that, if the dimension of M is 2, then the answer to the question is”yes”, as in this case the SEC is equivalent to assuming nonpositive timelike sectional curvature,and then the statement is implied by [4], Prop. 5.11.This short note is intended to show that the combination of the two conditions in the question donot restrict the conformal structure of the spacetime at all in dimension ≥
3, answering Galloway’squestion in the negative: We just consider a semi-Riemannian product ( R × N, − dt + g N ) whosefuture causal boundary is homeomorphic to N and equip it with an appropriate conformal factor,which does ot change the future causal boundary. The result is some sort of combination of twoearlier results of the author ([5], Th. 16, and [6], Th.8), ensuring the existence of one conformalfactor satisfying both requirements at once, whose proof however requires a different method. ∗ Institut f¨ur Mathematik, Humboldt-Universit¨at zu Berlin, Unter den Linden 6, D-10099 Berlin,
Email:[email protected] heorem 1 Let ( M, g ) be a globally hyperbolic Lorentzian manifold of dimension ≥ . Then thefollowing holds:1. For every spacelike Cauchy surface S of ( M, g ) , there is u ∈ C ∞ ( M ) such that ( M, e u · g ) is causally future geodesically complete and b.a.-complete and satisfies the strong energycondition on J + ( S ) .2. If ( M, g ) is spatially compact, then there is u ∈ C ∞ ( M ) such that ( M, e u · g ) is causallyfuture geodesically complete and b.a.-complete and satisfies the strong energy condition. Proof.
We consider first the case that (
M, g ) is spatially compact. Let t be a smooth Cauchytemporal function on M . We will choose u := a ◦ t for some φ : R → R . By its gradient flow, t inducesan isometric diffeomorphism F : ( M, g ) → ( R × N, − k · dt + pr ∗ ( g N ( t )) where pr : R × N → N isthe projection to the second component, k ∈ C ∞ ( R × N, (0; ∞ )) and t → g N ( t ) is a smooth curve ofRiemannian metrics on N . Using the freedom to choose conformal factors, we can assume w.l.o.g.that k is identically 1. For any conformal multiple U g of g , the very same isometry F induced bythe function t is an isometry between U g and − U dt + U g t . The geodesic equation in ( M, U g ) forthe t coordinate along a geodesic c (where q ′ := ddt q for a differentiable function q on R × N ) readsafter calculation of the Christoffel symbol: d tds = − n − X m =1 U ,m U dx m ds dtds − U ′ U ( dtds ) − n − X m,n =1
12 ( g ′ mn + U ′ U g mn ) dx m ds dx n ds , (1)where s is the affine parameter of c . We want to show that for the appropriate growth of U , Eq. 1implies convexity of t ◦ c . With the above ansatz U = A ◦ t the first term vanishes. If we considera b.a. curve we get a bounded real function D as an additional additive term on the right handside. For the geodesic case just replace D by 0 from now on. Because of compactness of the t -levelsets, there is an f ( t ) such that g ′ + f ( t ) · g is positive definite. Now, if β ( t ) := d (ln( A ( t ))) dt = a ′ ( t ) ≥ f ( t ) , (2)then both remaining terms on the RHS of Eq. 1 are nonnegative, thus d tds ≤ D , but as dtds >
0, bythe mean value theorem we get s ≥ ( dtds ) − | s =0 · t (geodesics) or s ≥ D ( dtds ) − | s =0 · t (b.a. curves),so in either case, for bounded s , t is bounded as well, thus ( I + ( S ) , U · g ) is future geodesicallycomplete and b.a.-complete.On the other hand, for the energy condition, we consider the well-known equation f ric = ric − ( n − Ddu − du ⊗ sym du ) + (∆ u − ( n − g (grad g u, grad g u )) · g (3)2or the Ricci curvature f ric of the metric ˜ g := e u g (see for example Eq. 1.159 d) from [1]). In ourcase we recall that u = (ln U ) / U = A ◦ t for some A : R → (0; ∞ ). Let τ M : T M → M be thetangent bundle projection, let H b be the compact set of causal vectors v in D b := ( τ M ) − ( t − ( b ))such that dt ( v ) = 1, then every causal vector in D b is a multiple of a vector in H b . Focussing onthe quadratic terms du ⊗ sym du and g (grad g u, grad g u )) · g we see that the latter is nonnegative on H b × H b whereas the former is positive on H b × H b if A ′ ( x ) > ∀ x ∈ R , and due to compactnessof H b it is even bounded below by a positive constant. This implies that there is C ( b ) ∈ (0; ∞ )such that the timelike convergence condition is satisfied on D b as soon as for β := a ′ > C ( b ) · β ′ ( b ) < β ( b ) . As C : R → R is locally bounded above, we can choose C ∈ C ∞ ( R ) with C > C . If β satisfies C · β ′ ( b ) < e β ( b ) (4)this implies the TCC condition on t − ( b ). Even more, the function Φ : R R → C ( R ) defined byΦ( β ) := ( b inf { ric e u g ( v, v ) | v ∈ H gb } )where a ( s ) := R s β ( σ ) dσ and u := a ◦ t , is, due to Eq. 4, a flatzoomer in the terminologyintroduced [8]. So we can use Th. 4.1 of [8] (with M = R , ε i = 0 for all i ∈ N and w = f ) toget the existence of a β ∈ C ∞ ( R ) pointwise greater than f and satisfying Eq. 4. This finishes theproof of the spatially compact case.For the general case, we realize that the only thing we need to transfer the proof above to the newsituation is properness of t | J + ( S ) , i.e., compactness of t − (( −∞ ; D )) ∩ J + ( S ) for every D ∈ R . Andindeed, we can construct such a ’future-proper’ Cauchy temporal function in the general case: Lemma 1
Let ( M, g ) be globally hyperbolic and let S ⊂ ( M, g ) be a spacelike hypersurface. Thenthere is a smooth steep Cauchy temporal function t with t − (( −∞ ; D )) ∩ J + ( S ) compact ∀ D ∈ R . Proof of the lemma.
Let t be a smooth steep Cauchy temporal function with S = t − ( { } ),whose existence is ensured by [7]. We denote for U ⊂ M by D ± ( U ) the future resp. past domain ofdependence of U , D + ( U ) := { y ∈ M | Any C − inextendible past causal curve intersects U nontrivially } .For n ∈ N , let U n ⊂ S be compact with U n ⊂ int( U n +1 ) for all n ∈ N , and S ∞ n =0 U n = S . For every n ∈ N , we define K n := D + ( U n ) ∩ t − (( −∞ ; n ]). As the U n are part of a Cauchy surface, we have D + ( U n ) := { x ∈ M | J − ( x ) ⊂ U n } . The K n are compact, K n ⊂ I − ( K n +1 ) = int( K n +1 , J + ( S )) forall n ∈ N by continuity of J − , and S n ∈ N K n = J + ( S ) by Cauchyness of t and S . We want toobtain a Cauchy temporal function t on M with t | J + ( S ) \ K n ≥ n. (5)3e have ∂K n ∩ I + ( S ) = ( ∂ + K n ) ∩ I + ( S ) ⊂ int K n +1 , for every p ∈ ∂ + K n there are p −− ≪ p − ≪ p with p −− ∈ D − ( U n +1 ), and we cover the compact sets ∂K n ∩ J + ( S ) with finitely many of thesets A i,n := I + ( p i,n − ). Then let ψ ∈ C ∞ ( R ) with ψ ( x ) = 0 ∀ x ≤ ψ ′ ( x ) > ∀ x > ψ ′ ( x ) ≥ ∀ x ≥
1. Let t i,n be a smooth steep Cauchy temporal function on B i,n := I + ( p i,n −− ) with t i,n ( p i,n − ) = 1. Then we define τ i,n := ψ ◦ t i,n on B i,n and τ i,n | M \ B ni = 0.Then t n := P a i,n · τ i,n ∈ C ∞ ( M, R ) is, for a i,n >
0, a smooth steep temporal function on an opensubset containing J + ( S ) \ K n , supp( t n +1 ) ∈ J + ( U n +1 ) by the condition p i,n −− ∈ D − ( U n +1 ) aboveand, for a i,n sufficiently large, we have t n | J + ( s ) \ K n ≥ n .Finally, t := t + P ∞ i =1 t i is well-defined and smooth on M by standard arguments (for any fixedpoint q there are only finitely many nonzero contributions from the sum, due to compactness of J − ( q ) ∩ S and the condition p i −− ∈ M \ J − ( K n )), it is a Cauchy temporal function being the sum of aCauchy temporal function and a temporal function (recall that the latter is defined by monotonicityalong future causal curves and the former additionally by surjectivity along C -inextendable futurecausal curves), and it satisfies Eq. 5, proving the statement. (cid:4) (Lemma)Applying the lemma (even without the steepness part) concludes the proof of the theorem. (cid:4) For the semi-Riemannian product above, all t -level sets in the conformally rescaled metric are stillCMC, providing no counterexample to the conjecture, but only to the question in [2]. Acknowledgements:
The author would like to thank Gregory Galloway and Eric Ling for helpfulcomments on a first version of this note.
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