Characterization of the null energy condition via displacement convexity of entropy

We characterize the null energy condition for an $(n+1)$-dimensional Lorentzian manifold in terms of convexity of the relative $(n-1)$-Renyi entropy along displacement interpolations on null hypersurfaces. More generally, we also consider Lorentzian manifolds with a smooth weight function and introduce the Bakry-Emery $N$-null energy condition that we characterize in terms of null displacement convexity of the relative $N$-Renyi entropy. As application we then revisit Hawking's area monotonicity theorem for a black hole horizon and the Penrose singularity theorem from the viewpoint of this characterization and in the context of weighted Lorentzian manifolds.


Introduction
Many classical theorems in general relativity rest on local geometric constraints for the underlying Lorentzian space-time.These local constraints often have the form of lower bounds for curvature quanitities like the Ricci tensor and via the Einstein equation they are interpreted as energy conditions.One such condition is the null energy condition that requires nonnegativity of the Ricci tensor in direction of null vectors.The null energy condition plays a crucial role in the Penrose Singularity Theorem about incompleteness of null geodesics [Pen65] which forshadowed the existence of black holes and in Hawking's Area Monotonicity Theorem [Haw72] which asserts that the area of cross-sections of a black hole horizon is non-decreasing towards the future provided the horizon is future null complete.
In this article we present a characterization of the null energy condition for a time-oriented Lorentzian manifold in terms of entropy convexity along the future-directed geodesic null flow on null hypersurfaces.
A null hypersurface H in a Lorentzian manifold (M n+1 , g) is a submanifold of dimension n such that the metric g restricted to H degenerates.The entropy is the relative (n − 1)-Renyi entropy where µ is a probability measure on M , vol H is the degenerated volume of the Lorentzian metric g on H and ρ is the density of µ in the Lebesgue decomposition w.r.t.vol H .A null flow on H consists of null geodesics that foliate the hypersurface H.We say that a probability measure π on M × M is a null coupling if there exists a null hypersurface H ⊂ M such that π is concentrated on R H = (x, y) ∈ H 2 : ∃ flow curve γ x,y s.t.γ x,y (s) = x, γ x,y (t) = y & x ≤ y .
Two probability measures µ 0 , µ 1 ∈ P(H, vol H ) are called acausal if they are supported on acausal submanifolds S 0 and S 1 .Here P(H, vol H ) is the set of vol H -absolutely continuous probability measures concentrated on H.We say that two acausal probability measures µ 0 and µ 1 are null connected if there exists a null coupling π such that the marginal measures of π are µ 0 and µ 1 .For x, y ∈ R H let t ∈ [0, 1] → γx,y (t) be the affine reparametrization of γ x,y .If µ 0 and µ 1 are null connected, we define µ t = (e t ) # π for t ∈ [0, 1] where e t : (x, y) → γx,y (t) is the evaluation map.We will call (µ t ) t∈[0,1] null displacement interpolation.
1.1.Statement of main result.Our main result is the following theorem.
Theorem 1.1.Let (M n+1 , g) be a Lorentzian manifold.The null energy condition holds if and only if for all µ 0 , µ 1 ∈ P(M, vol H ) that are acausal and null connected, one has We also introduce a Bakry-Emery N -null energy condition for weighted Lorentzian manifolds.We say (M, g, e −V ) with V ∈ C ∞ (M ) satisfies the Bakry-Emery N -null condition for N ≥ n − 1 if for any null vector v ∈ T M and any N ′ > N .We obtain the following.
Theorem 1.2.Let (M n+1 , g, e −V ) be a weighted Lorentzian manifold for V ∈ C ∞ (M ).The Bakry-Emery N -null energy condition holds if and only if for all µ 0 , µ 1 ∈ P(M, m H ) that are acausal and null connected, one has where m H = e −V vol H .
Theorem 1.1 and Theorem 1.2 are in the spirit of similar results for the strong energy condition by R. McCann [McC20] and for the Einstein equation by A. Mondino and S. Suhr [MS22b] for which these authors proved charactizations via entropy convexity on the space of probability measures equipped with a Wasserstein distance derived from the time separation function.Such results are motivated by the desire to formulate local energy conditions in a low regularity framework where the classical notion of Ricci curvature breaks down.In particular, this was the starting point for the development of a synthetic definition of the strong energy condition via entropy convexity [CM20] in the framework of Lorentzian length spaces [KS18] (see also [Bra22,MS22a] for further developments).The characterization of lower Ricci curvature bounds via entropy convexity on the Wasserstein space of probability measures was first established for Riemannian manifolds [CEMS01,vRS05,OV00].Optimal transport of probability measures that are supported on submanifolds inside of a Riemannian manifold in relation to curvature bounds has been studied in [KM18].
We use the characterization of Theorem 1.2 to define a synthetic N -null energy condition for a time-oriented Lorentzian manifold (M, g) equipped with a continuous weight of the form e −V (Definition 3.11).As an application of our main results we revisit Hawking's Area Monotonicity Theorem and the Penrose Singularity Theorem in the setting of such weighted, time-oriented Lorentzian manifolds that satisfy the synthetic N -null energy condition.First we prove the following version of the monotonicity theorem.
Theorem 1.3 (Hawking Monotonicity).Let (M n+1 , g, e −V ) be a weighted space-time that satisfies the synthetic N -null energy condition for N ≥ n − 1 and let H ⊂ M be a null hypersurface.Assume H is future null complete.Let Σ 0 , Σ 1 ⊂ M two acausal spacelike hypersurfaces.Define Σ i ∩ H = S i , i = 0, 1, and assume S 0 ⊂ J − (S 1 ).Then We also introduce the notion of synthetically future converging for a codimension 2 submanifold S in a weighted Lorentzian manifold (M, g, e −V ) (Definition 4.8) and we prove the following version of the Penrose Singularity Theorem.
Theorem 1.4.For V ∈ C 0 (M ) let (M, g, e −V ) be a weighted, time-oriented Lorentzian manifold that is globally hyperbolic.Assume there exists a noncompact Cauchy hypersurface Σ ⊂ M , M contains a synthetically future converging, compact, oriented codimension two submanifold S and the synthetic N -null energy condition for N ≥ n − 1 holds.Then M is future null incomplete.
The rest of this article is structured as follows.
In Section 2 we will briefly recall important notions about smooth, timeorientied Lorentzian manifolds and null hypersurfaces.We will define the Ricci and the Bakry-Emery Ricci tensor and the corresponding null energy conditions.We also introduce the degenerated volume and the corresponding relative Renyi entropy.
In Section 3 we define the notion of null coupling, null connectedness and null displacement interpolation.We prove that null displacement interpolations are induced by maps.Then, we prove that the null energy condition is necessary and sufficient for null displacement convexity for the Renyi entropy along null hyupersurfaces.
In Section 4 we present consequences that we can derive directly from the null displacement convexity.This includes the Hawking area monotonicity and a corollary that leads to the Penrose singularity theorem.
Acknowledgments.The main part of this work was done when the author stayed at the Fields Institute in Toronto as Longterm Visitor during the Thematic Program on Nonsmooth Riemannian and Lorentzian Geometry.CK wants to thank the Fields Institute for providing an excellent research environment.In particular many thanks go to Robert McCann, Clemens Saemann and Mathias Braun for stimulating discussions about topics in general relativity.This work was completed during a research visit of the author at UNAM Oaxaca, Mexico.

Preliminaries
2.1.Lorentzian manifolds.A space-time is a smooth, connected, timeoriented Lorentzian manifold (M, g) with dim M = n + 1.The signature of M is (−, +, . . ., +).We also write g = •, • .A vector v ∈ T x M is timelike, spacelike, causal or null if v, v is negative, positive, non-positive or 0, respectively.A C 1 curve γ : I → M is then timelike, spacelike, causal or null if γ ′ (t) is so for every t ∈ I.For x, y ∈ M we write x ≤ y if there exists a future directed causal curve from x to y and x ≪ y if there exists a future directed timelike curve from x to y.For A ⊂ M let J + (A) := {y ∈ M : ∃x ∈ A s.t.x ≤ y} be the causal future of A and similarly, let J − (A) := {y ∈ M : ∃x ∈ A s.t.y ≤ x} be the causal past.A is called achronal (acausal) if no timelike (causal) curve intersects A twice.An achronal hypersurface Σ is a Cauchy hypersurface if every inextendible timelike curve intersects Σ exactly once.A space-time (M, g) which possesses a Cauchy hypersurface, is called globally hyperbolic.
Definition 2.1.The Ricci tensor of (M, g) is (0, 2)-tensor defined as where v, w ∈ T x M for x ∈ M and e 0 , e 1 , . . ., e n an orthonormal basis of T x M w.r.t.g x .If | v, v | = 1, one can pick an orthonormal basis e 0 , . . ., e n such that v = e j for some j ∈ {0, . . ., n} and hence Given N ∈ [N + 1, ∞) and K ∈ R the weighted space-time (M, g, e −V ) satisfies the timelike Bakry-Emery N -Ricci curvature condition if for any timelike vector v ∈ T M and ∀N ′ > N. In particular, if N = n + 1, it follows that V is constant along any timelike curve.A characterization of the timelike Bakry-Emery N -Ricci curvature condition in term of entropy displacement convexity was given in [McC20,MS22b].
We say (M, g, e −V ) satisfies the Bakry-Emery N -null condition if for any null vector v ∈ T M and any N ′ > N .
If N = n−1, it follows that V is constant along any lightlike curve.Hence ∇V, v = ∇V 2 (v, v) = 0 for every null vector v and the condition reduces to the null energy condition ric(v, v) ≥ 0 for any null vector v ∈ T M.

Null hypersurfaces.
A null hypersurface H in M is an embedded C 2 submanifold of codimension one such that the pull-back metric of g on H is degenerated.There exists a non-vanishing C 1 future-directed, null vector field K ∈ Γ(T H) such that K ⊥ x = T x H for all x ∈ H.The flow curves of K in H are null geodesics and one says that H is null geodesically generated (ruled).A flow curve admits an affine reparametrization.More precisely, let γ x : (α, β) → H be the flow curve of K with γ x (0) = x.There exists a reparametrization ϕ : (a, b) → (α, β) of γ x such that γ x •ϕ(t) = exp x (tK(x)).The vectorfield K is unique up to a positive factor.For further details about the geometry on null hypersurface we refer to [CDGH01, Appendix A].
Example 2.3.Let S be a codimension 2 smooth hypersurface that is spacelike.Assume also that S is an oriented manifold.At any point x ∈ S we can pick two future-directed orthogonal null vectors L x and L x .Assume L x projects to the exterior of S. We will call L x an outer null normal to S at x and L x an inner null normal to S at x.We choose the map x ∈ S → L x to be smooth.For every x ∈ S there exists a null geodesic γ x (t) = exp x (tL x ) with γ x (0) and γ ′ x (0) = L x and we consider the set C = x∈S Imγ x formed by these geodesics.C is also called a null geodesic congruence.By replacing L with L we also define C. A neighborhood H of S in C is a smooth submanifold of codimension two and the vector field L extends to a smooth vector field Then H is a null hypersurface.Similarly there exists a corresponding null hypersurface H for C.
We write P(H, m H ) for the set of m H -absolutely continuous probability measures.

Relative entropy.
The relative N -Renyi entropy w.r.t.m H of a probability µ ∈ P(M ) is defined as where ρd m H +µ s is the Lebesgue decomposition of µ.Basic properties of 3. Proof of the main results 3.1.Null couplings.
Definition 3.1.Let (M, g) be a space-time and H ⊂ M a null hypersurface with tangent vector field K that generates a geodesic null flow.We define a transport relation R H on H by
Let (µ t ) t∈[0,1] be a null displacement interpolation.For µ t -almost every x ∈ H there exists a unique γt x ∈ G(H) such that γt x (t) = x.Then we also have the µ t -almost everywhere defined map Λ t : x → γt x .One can check that (Λ t ) # µ t = Π.
We call a probability measures µ acausal if there is a space-like acausal C 2 submanifold Σ such that µ is concentrated in Σ.
We call two acausal probability measures µ 0 and µ 1 null connected if there exists a coupling π between µ 0 and µ 1 that is null.In particular, it holds supp µ i ⊂ H ∩ Σ i , i = 0, 1, for a null hypersurface H and two spacelike, acausal submanifolds Σ 0 and Σ 1 .Lemma 3.2.A null coupling π between acausal probability measures µ 0 and µ 1 as above is induced by a C 1 map T : U → H where U ⊂ S 0 open such that supp µ 0 ⊂ U .More precisely, there exists r ∈ C 1 (U ) such that T (x) = exp x (r(x)K(x)) and (id supp(µ 0 ) , T ) # µ 0 .We call T transport map.
Proof. 1.Since Σ 1 is acausal, for every x ∈ Σ 0 there exists a most one y ∈ Σ 1 such that (x, y) ∈ R H .Moreover, by existence of a null coupling π for µ 0 -almost every x there exists at least one such y ∈ Σ 1 .Hence y is the unique intersection of γ x with Σ 1 .So we set µ 0 -almost everywhere x → y =: T (x), and for π-almost every (x, y) it holds T (x) = y.2. Let Φ : S 0 ×(0, ∞) → M be the map given by Φ(x, t) = exp x (tK(x)).For every (x, t) ∈ S 0 × (0, ∞) with Φ(x, t) ∈ H there is exactly one inextendible flow curve γ y of K in H passing through Φ(x, t).Hence Φ(x, s) = γ y • ϕ x (s) for a regular reparametrization ϕ x : (0, ω x ) → R. In particular Φ(x, s) ∈ H for s ∈ (0, ω x ).Hence, by Lemma 4.15 in [CDGH01] there are no focal points along t ∈ (0, ω x ) → Φ(x, t).Consequently DΦ (x,t) : Since T (x) is already the unique intersection point of the flow curve γ x with S 1 , we have T (x) = exp(g(x)K(x)) for all x ∈ supp µ 0 ∩ O. Hence, the map T is the restriction of the C 1 map T : Example 3.3.Consider a space-time (M, g) and let Σ 0 , Σ 1 ⊂ M be spacelike, acausal submanifolds and let S i = H ∩ Σ i such that S 0 ⊂ J − (S 1 ).Since Σ 1 is acausal, for every x ∈ S 0 there exists at most one y ∈ S 1 such that J − (y) ∋ x.Then, exactly as in the proof of the previous lemma we can construct a C 1 map T : S 0 → S 1 of the form T (x) = exp x ( K(x)).In particular, if vol H (S 0 ) < ∞ (for the definition of vol H see Section 2.3.1 below), we can define µ 0 = 1 vol H (S 0 ) vol H | S 0 and µ 1 = (T ) # µ 0 .Then µ 0 and µ 1 are null connected.
Remark 3.4.The proposition implies that exp x (t K(x)) = γx,T (x) (t) =: γx (t) where K(x) = r(x)K(x) and γx,T (x) = Υ(x, T (x)).We set T t : U → H via T t (x) := exp x (t K(x)) = γx (t).In particular, we have for the induced null displacement interpolation that . Since, by Lemma 4.15 in [CDGH01], there are no focal points along γx (t) unless there is an endpoint of γ x in H, DT t,x v is not lightlike for any v ∈ T x S 0 .(Since H 2 is C 2 submanifold and K a nonvanishing C 1 vectorfield on H, there are no endpoints in H.) Hence T t is a C 1 diffeomorphism with T t (S 0 ) = S t spacelike and acausal.In particular g| St is non-degenerated.
Remark 3.5.This setup is also meaningful in a more general context.A general null hypersurface H is a topological codimension 1 submanifold that is the union of null geodesics.The concept of transport relation and null coupling are defined analogously.A measure µ is acausal if it is supported on a n − 1-rectifiable subset that is acausal.The definition of r and the map T is the same and by [CDGH01] T and r are Lipschitz continuous.

The null energy condition implies null displacement convexity.
Let µ 0 , µ 1 ∈ P(M, m H ) be acausal and null connected and let (µ t ) t∈[0,1] be the induced null displacement interpolation.Each µ t is supported on the set T t (U ) ⊂ S t where S t is the image of S 0 under T t : U → H. Lemma 3.6.Let V ∈ C 0 (M ) and m H = e −V vol H . Then µ t ∈ P(M, m H ) ∀t ∈ (0, 1).We denote the densities of µ t w.r.t.m H with ρ t .
Proof.Let N ⊂ S t with m St (N ) = 0. T t is a C 1 diffeomorphism and hence T −1 t is a C 1 .It follows (T t ) −1 (N ) has 0 measure w.r.t.vol H and therefore w.r.t.m H . Let Π be the dynamical null coupling and π = (e 0 , e t ) # Π.Then Hence µ t (N ) = 0 and µ t is m H absolutely continuous.
Proof.Let W ⊂ S 0 be an measurable and arbitrary.S 0 and S t are equipped with the restricted metric g.Then where we used the area formula for the map T t : S 0 → S t .Since W was arbitrary, the claim follows.
(1) If the null energy condition holds and µ 0 , µ 1 ∈ P(M, vol H ), then it (2) If the weighted Lorentzian manifold (M, g, e −V ) satisfies the Bakry-Emery N -null energy condition for N ≥ n−1 and µ 0 , µ Proof.Let K(x) = r(x)K(x), x ∈ U ⊂ S 0 , the vector field given by Lemma 3.2 and let T t (x) = exp x (t K(x)) = γx (t).The family of maps from T x S 0 to T γx(t) S t satisfies the Jacobi equation H and check that it satifies the Riccatti equation . This equation is also known as the Optical Equation [CDGH01].
Let E 1 (t), . . ., E n−1 (t) ∈ T γx(t) S t be the solutions of the linear equation for an orthonormal Basis e 1 , . . ., e n−1 of T x S 0 .The map P t : T γx(t) H → T γx(t) S t is the linear projection.
Claim.E 1 (t), . . ., E n−1 (t) is an orthonormal basis of T γx(t) S t .Proof of the claim.We observe Claim.The derivative This proves the claim.
Claim.tr(U x ) ′ = tr(U ′ x ).Proof of the claim.We compute Hence taking the trace of the optical equation we obtain x ) ≥ 0 and the last inequality follows from the null energy condition.
Proof of Claim.Note that U x (t) = A ′ x (t)A −1 x (t) and A x (t 0 + s)z = J(s) satisfies the Jacobi equation with J(0) = v ∈ T γx S t 0 where vA −1 x (t 0 ) = z and J ′ (0) = w ∈ T γx(t0) H.By standard Riemannian calculus we can write where α : (−ǫ, ǫ) → H is C 2 and satisfies α(0) = γx (t 0 ) =: y, α ′ (0) = v and K is a C 1 vectorfield on H in a neighborhood of y such that K(0) = γ′ x (t 0 ) and Since K is also a normal vectorfield, this is the null Weingarten map of H in y = γx (t 0 ).Hence U x (t 0 ) is self-adjoint.
By an application of the Cauchy-Schwarz inequality it follows tr(U 2 x ) ≥ Hence tr(U x ) ′ + 1 n−1 tr(U x ) 2 ≤ 0. We set det(A x ) = y x .It follows that (log y x ) ′ = tr(A ′ x A −1 x ) = tr(U x ) and therefore Equation (3) also follows from the Raychaudhuri equation.However for this paper we gave a derivation of (3) that is closer to ideas in optimal transport.We have µ t = (T t ) # µ 0 with T t .Then, together with Lemma 3.7, This finishes the proof of (1).
Similarly, we can consider z x (t) = e −V •γx(t) y x (t).It follows that where we used 2 and the Bakry-Emery N -null energy condition.
Then we also have x ≤ 0. Again using Lemma 3.7, it follows This finishes the proof of (2).
(1) Let (M, g) be a (n + 1)-dimensional, time-oriented Lorentzian manifold.Assume for every null hypersurface H and for every µ 0 , µ 1 ∈ P(M, vol H ) that are null connected via a null coupling π, it holds where µ t is the induced null displacement interpolation.Then (M, g) satisfies the null enery condition.
(2) Let (M, g, e −V ) be a weighted, (n + 1)-dimensional, time-oriented Lorentzian manifold with V ∈ C ∞ (M ) and assume for every null hypersurface H and for every µ 0 , µ 1 ∈ P(M, m H ) that are null connected via a null coupling π, it holds Then (M, g, e −V ) satisfies the Bakry-Emery N -null energy condition.
Proof.We argue by contradiction in both cases.
For (2) assume ∃ N ′ > N and a null vector v ∈ T p M with In the following by a slight abuse of notation we write N ′ = N .1. Consider the exponential map exp p : U ⊂ T p M → M in p where U is an open subset of T p M such that exp p is a diffeomorphismus.Let I : R n+1 1 → (T p M, g p ) an isometrie between the Minkowski space R n+1 1 and the tangent space at p. Hence φ := exp p •I : We choose an orthonormal basis e 0 , e 1 , . . ., e n in R n+1 1 with e 0 timelike and e 1 , . . ., e n spacelike such that e 0 + e 1 = σv for some σ > 0. The vectors e 1 , . . ., e n span a spacelike linear subspace Λ ′ = e 1 , e 2 , . . ., e n in R n+1 In Λ ′ ≃ R n we choose a codimension 1 submanifold S such that 0 ∈ S and such that the second fundamental form in 0 satisfies Π p = λ •, • R n−1 for λ ∈ R that we choose later.
Let N 0 : S → T Σ be the smooth normal unit vectorfield of S as submanifold in Σ, and let N 1 : Σ → T M be the future pointing normal unit vector field on Σ in M such that we have Dφ 0 (e 1 ) = N 0 (p) and Dφ 0 (e 0 ) = N 1 (0).Hence σv = N 0 (p) + N 1 (p).We define N : S → T M as N 0 + N 1 | S .It follows that N is a null vector field along S and since the covariant derivative of N 1 at p in direction of vectors in T p Σ vanishes.Here we use the properties of normal coordinates at the base point.Hence, we also have S is the second fundamental form of S in the ambient space Σ.We can extend N : S → T S to a smooth null vector field on Σ that we denote as N .We define the map Φ : Σ×(−ǫ, ǫ) → M via Φ(x, t) = exp x (t N (x)) and the restriction Φ| S×(−ǫ,ǫ) = Ψ.We can compute the differential of Ψ in (p, 0) and see it is injective.Hence, we can choose a neighborhood O ′ ∩ Σ of p in Σ and δ such that Ψ| O×(−δ,δ) is a diffeomorphism to its image where O = S ∩ O ′ .By the definition of the map Ψ the image Ψ(O × (−δ, δ)) =: H is a null hypersurface in M that contains O.
Because N (q) is a null vector in T H we also have that N (q) ∈ T q H ⊥ for q ∈ S. Now, we extend N to a smooth, non-vanishing null vector field N on a neighborhood of p ∈ H in H. Therefore the second fundamental form of H in p can be computed as Remark 3.10.Let us make a few comments at this point.If K is a nonvanishing, smooth null vector field along H, the null Weingarten map is defined as ∇ X K mod K = b(X) where X is the equivalence class of spacelike vector X modulo K.If K = f K is another such null vector field for a smooth function f , then ∇v K = f ∇ v K mod K. Hence the Weingarten map b is a tensor field and depends at a given point p ∈ H only on the value of K at p.The null second fundamental form of H is then defined as H (X, Y ) for X, Y spacelike.This is consistent with the above.
Repeating the calculation as in Theorem 3.8 we obtain 2. First we prove (1).For this we choose λ = 0. Recall γ ′ p (0) = v.By continuity of ric and (x, t) → γ ′ x (t) we can choose r and η such that ric . By continuity of (x, t) → U x (t) we can choose η and r small enough to obtain Then we compute Hence, setting y x (t) = det A x (t) we get and therefore (y x ) Here we used that strict positivity of the second derivative is sufficient for strict convexity.
We can proceed as in the end of the proof of Theorem 3.8 and obtain This contradicts null displacement convexity.

Finally we treat (2). For this we choose
. Provided η > 0 and r > 0 are small enough, by continuity of ric V and (x, t) → γ x (t) we have and by continuity of (x, t) → A x (t) and (x, t) , N } and since (x, t) → U x (t) and (x, t) → ∇V • γ x (t), γ ′ x (t) are continuous, we can choose η and r even smaller such that . With this we compute first, using (1) and (3), and obtain (⋆) Exactly as before it follows that is a contradiction.
Definition 3.11.A weighted space-time (M n+1 , g, e −V ) with V ∈ C 0 (M ) satisfies the synthetic N -null energy condition for some N ≥ n − 1 if for every null hypersurface H and for every µ 0 , µ 1 ∈ P(M, m H ) that are null connected via a null coupling π, it holds where µ t is the corresponding null displacement interpolation. 3.4.Localisation.
Proposition 3.12.Consider a weighted, time-oriented Lorentzian manifold (M n+1 , g, e −V ).Then the synthetic N -null energy condition for N ≥ n − 1 holds if and only if for every µ 0 , µ 1 ∈ P(M, m H ) that are acausal and null connected via a dynamical null coupling Π supported on a null hypersurface H one has for every t ∈ [0, 1] that where ρ t is the density of the m H -absolutely continuous part of the measures µ t of the corresponding null displacement interpolation.
Proof.By Hölder's inequality it is possible to replace N in (4) with N ′ > N .Integrating (4) w.r.t. the dynamical null coupling Π that comes from π yields Let us now assume the synthetic N -null energy condition.Let (µ t ) t∈[0,1] be a null displacement interpolation between two acausal measures µ 0 , µ 1 ∈ P(H, m H ) supported on S 0 and S 1 .µ t is induced by a C 1 diffeomorphism T t : S 0 → H and T t (S 0 ) = S 1 is a spacelike C 1 submanifold.We fix τ ∈ [0, 1] and x ∈ S τ , and we define where B δ (x) is the ball of radius δ > 0 in S t .Assume Π(Γ) > 0 and set Π ′ = 1 Π(Γ) Π| Γ .In particular µ τ (B δ (x)) = Π(Γ).(et ) # Π ′ = µ ′ t is still a null displacement interpolation that is induced by the same family of maps T t .Moroever, for B ⊂ S t where the first inequality follows from Jensen's inequality.Now let x be a density point of the measure µ t w.r.t.m H and also a density point of Here we use and by Jensen inequality also Now, pick t 0 ≫ 1 and let Tτ (x) = exp x (τ t 0 K(x)) and define ν τ := ( Tτ ) For t 0 ≫ 1 large enough we have a contradiction.4.2.Penrose singularity theorem.Definition 4.4.Let (M n+1 , g, e −V ) be a weighted space-time and let H ⊂ M be a null hypersurface.Let Σ ⊂ M be a acausal spacelike hypersurface and we set Σ∩H = S.For every p ∈ S there exists a geodesic ball B η (p) ⊂ S w.r.t.g| S and a null vectorfield K : S → T H such that x ∈ S → T t (x) = exp x ( K(x)) is a C 1 diffeomorphism for all t ∈ (0, t p ) and for some t p > 0.
We say S is synthetically future converging in H if for every p ∈ S there exists η > 0, T t as above and ǫ(p) ∈ (0, 1) s.
Differentiating this formula at t = 0 yields for alle x ∈ B δ (p) where e 1 , . . ., e n−1 is an orthonormal bases in T x S. Hence For every future-directed null vector w ∈ T p H there exists λ ∈ R such that λ K(p) = w.It follows that H V,w (p) > 0.
If we assume (5), then for every p ∈ S we find η > 0 and ǫ(p) > 0, such that H K(x) (x) + ∇V (x), K(x) ≥ ǫ(p) for x ∈ B δ (p) and for all δ ∈ (0, η) where K is the vector field that we find according to Definition 4.4.With the previous computations we see that S is future convergin in H.
Remark 4.7.The definition of future converging for a codimension two submanifold S in a Lorentzian manifold (M, g) is that H, v > 0 for every future directed null vector v normal to S [O'N83, Chapter 14].For a weighted space-time we make the following definition.Definition 4.8.Let (M, g, e −V vol g ) be a weighted Lorentzian manifold with V ∈ C ∞ (M ).We call a codimension two submanifold S ⊂ M Bakry-Emery future converging (or Bakry-Emery trapped) if H, v + ∇V, v > 0 for every future-directed normal null vector at S.
If V ∈ C 0 (M ), consider C, C, H, H, L and L as in Example 2.3.We call the codimension two submanifold S ⊂ M synthetically future converging (or synthetically trapped) if S is future converging in H and future converging in H. Corollary 4.9.Consider a weighted space-time (M, g, e −V ) that satisfies the synthetic N -null energy condition and let H be a null hypersurface.Let Σ ⊂ M be an acausal spacelike hypersurfaces and define Σ ∩ H = S. Assume S is future converging in H. Then H is not future null complete.Proof.For p ∈ S we choose η > 0 and K as in Definition 4.4.We argue by contradiction.Assume H is future complete.It follows that T t (x) = exp x (t K(x)), x ∈ B η (p), is a C 1 diffeomorphism for all t ∈ (0, ∞).We define µ 0 = 1 m H m H | B δ (p) and µ t = (T t ) # µ 0 .Since H is future complete, it follows, similarly as in the Hawking Monotonicity Theorem, that for all N ′ > N .Then it follows with the synthetic null energy condition 0 > −(N ′ − 1) m H (B δ (p)) This is a contradiction.
Remark 4.10.We can refine the analysis of the previous proof as follows.
Let p ∈ S and η > 0 and K as above such that T t (x) = exp x (t K(x)), x ∈ B η (p), is a C 1 diffeomorphism for all t ∈ (0, t p ) and for some t p > 0. We define µ 0 = 1 m H m H | B δ (p) for δ ∈ (0, η) and µ t = (T t ) # µ 0 .Recall that area formula implies Since t → (ρ t • T t (p)) − 1 N ′ is concave, we have t p < 1 ǫ(x) by Riccatti comparison.Now, we will prove a Penrose Singularity theorem for the synthetic N -null energy condition.
Theorem 4.11.Let (M, g, e −V ) be a weighted space-time that is globally hyperbolic and V ∈ C 0 (M ).Assume there exists a non-compact Cauchy hypersurface Σ ⊂ M , M contains a synthetically future converging, compact, oriented codimension two submanifold S and the synthetic N -null energy condition for N ≥ n − 1 holds.Then M is future null incomplete.
Proof.Let C, C, H, H, L and L as in Example 2.3.We define the map T t (x) = exp x (tL(x)) and the map T t (x) = exp x (tL(x)) on S. Assume now that (M, g) is future null complete.Then it follows that for a x ∈ S exp x (tL(x)) = γ x (t) and exp x (tL(x)) = γx (t) are defined on [0, ∞).
Let p ∈ S and define µ 0 = 1 m H (B δ (p)) m H | B δ (p) Let t p > 0 such that T t is a diffeomorphism on B δ (p) for all t ∈ (0, t p ) and µ t := (T t ) # µ 0 .
Let ρ t be the density of µ t w.r.t.m H .By the previous proof we have that t p ∈ (0, 1 ǫ ).Hence, there is a focal point τ before 1 ǫ .For t > 1 ǫ ≥ τ there exists a time-like geodesic from S to γ x (t) and therefore γ x (t) / ∈ ∂J + (S).Similarly we argue for γx (t).
It follows that ∂J + (S) is a closed and bounded, hence compact, subset of C ∪ C.
From here we can follow the proof of the classical Penrose Singularity Theorem.Let us outline the argument.
In view of the time-orientability of (M, g), there is a global timelike vector field T whose integral curves are timelike, foliate M and intersect the Cauchy hypersurface Σ exactly once.Since J + (S) is a future set, the topological boundary ∂J + (S) is an achronal (n − 1)-dimensional closed Lipschitz submanifold without boundary.Hence, every integral curve of T intersects ∂J + (S) at most once.The projection P of ∂J + (S) to the Cauchy Hypersurface Σ along the flow lines of T is continuous and bijective.Since ∂J + (S) is compact, P is a homeomorphism.This is contradiction, since Σ was assumed to be non-compact.
t. Let (M n+1 , g, e −V ) be a weighted space-time with V ∈ C ∞ (M ) and let H ⊂ M be a null hypersurface.Let Σ ⊂ M be an acausal and spacelike hypersurface.Σ ∩ H = S is synthetically future converging in H if and only if H V,w (p) := H w (p) + ∇V (p), w > 0, p ∈ S and for every future-directed null vector w ∈ T p H where H w (p) = H(p), w and H is the mean curvature vector of S. We call H V,w the weighted mean curvature in direction of w.