In this talk we will describe the characterization of the chronological relation on doubly warped product spacetimes of the form (a, b) × M1 × M2 with Lorentzian metric g = −dt2 + α1g1 + α2g2, and we will discuss when these spacetimes are causally continuos and causally simple. This talk is based on the first section of L. Aké, J.L. Flores and J. Herrera, Causality and c-completion of doubly warped spacetimes, available at arXiv:1709.00234.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalConsejo Nacional de Ciencia y Tecnología (México

Aké Hau, Luis

Carriazo Rubio, Alfonso (Coordinador)

Flores Dorado, José Luis

Fuente Benito, Daniel de la (Coordinador)

Herrera Fernández, Jónathan

Márquez García, María del Carmen (Coordinador)

Romero Sarabia, Alfonso (Coordinador)

idUS. Depósito de Investigación Universidad de Sevilla

XI Encuentro Andaluz de GeometríaIMUS (Universidad de Sevilla), 15 de mayo de 2015, págs. 4751Causal properties of doubly warped spacetimesLuis Aké • José Luis Flores • Jónatan HerreraAbstract. In this talk we will describe the characterization of the chronologicalrelation on doubly warped product spacetimes of the form (a, b) ×M1 ×M2with Lorentzian metric g = −dt2 +α1g1 +α2g2, and we will discuss when thesespacetimes are causally continuos and causally simple. This talk is based on theﬁrst section of [1], available at arXiv:1709.00234.1. IntroductionIn the classical book on Lorentzian geometry [2] the causality of warpedspacetimes products of the form ((a, b)×M,−dt2+αgM ), where ((a, b),−dt2) isa Lorentzian manifold, (M, gM ) is a Riemannian manifold and α : V → (0,∞)is a smooth function, is studied and results concerning to the stably causal,strongly causal and global hyperbolicity of these spacetimes are obtained, see[2, Lemma 3.55, Prop. 3.62 and Thm. 3.66]. However, the study of the causallycontinuous and causally simple stages of the causal hierarchy are not included.Our aim in this talk is to give a characterization of the chronological and causalrelations in doubly warped spacetimes and then we will give conditions to obtainthe causally continuity and causal simplicity of these spacetimes. The study ofthe characterization of the chronological relation on these spacetimes will beuseful in order to study the causal completion of these spacetimes, see [1].2. PreliminariesA doubly warped spacetime can be written as:V := (a, b)×M1 ×M2 and g = −dt2 + α1g1 + α2g2. (1)Luis Aké, luisake@uma.esDepartamento de Álgebra, Geometría y Topología, Universidad de MálagaJosé Luis Flores, ﬂoresj@uma.esDepartamento de Álgebra, Geometría y Topología, Universidad de MálagaJónatan Herrera, jonatanhf@uma.esDepartamento de Matemáticas, Universidad de Córdoba48 L. Aké, J.L. Flores, J. HerreraWhere αi : (a, b)→ (0,∞) are smooth functions for all i = 1, 2 and (Mi, gi) areRiemannian manifolds for all i = 1, 2.The chronological relation in these spacetimes is characterized in the follo-wing proposition (see [1]):Proposition 2.1. Let (V, g) be a doubly warped spacetime as in (1), and(to, xo), (te, xe) ∈ V with xo 6= xe. The following conditions are equivalent:(i) (to, xo) (te, xe);(ii) there exist strictly positive constants µ′1, µ′2 > 0, with µ′1 + µ′2 = 1,such that∫ teto√µ′iαi(s)(2∑k=1µ′kαk(s))−1/2ds > di(xoi , xei ) for i = 1, 2. (2)In order to give a proper characterization of the causal relation in doublywarped spacetimes we need the following deﬁnition:Deﬁnition 2.1. A Riemannian manifold (N,h) is L-convex if any pair of pointsp, q ∈ N with dh(p, q) < L can be joined by a minimizing geodesic.Now, we can establish the announced characterization about the causal re-lation (see [3, Thm. 2(2)])Proposition 2.2. Let (V, g) be a doubly warped spacetime as in (1) whose ﬁbers(Mi, gi) are Li-convex for i = 1, 2. Consider two points (to, xo1, xo2), (te, xe1, xe2) ∈V , with to ≤ te, satisfying d(xoi , xei ) < Li, i = 1, 2. Then, the following condi-tions are equivalent:(i) the points are causally related, (to, xo1, xo2) ≤ (te, xe1, xe2);(ii) there exists a causal geodesic joining (to, xo1, xo2) with (te, xe1, xe2);(iii) there exist constants µ′1, µ′2 ≥ 0, µ′1 + µ′2 = 1,such that∫ teto√µ′iαi(s)(2∑k=1µ′kαk(s))−1/2ds ≥ di(xoi , xei ) for i = 1, 2. (3)Moreover, if the equalities hold in (3), then there is a lightlike and no timelikegeodesic joining the points.With the characterization of the chronological relation and causality relation,the last under L−weakly convex condition, we are ready to study the causallycontinuity and causally simplicity of doubly warped spacetimes.Causality on doubly warped spacetimes 49Causal hierarchy on doubly warped spacetimesNext we are going to prove that double warped spacetimes are causallycontinuous, and, causally simple if and only if each (Mi, gi) satisfy a notion ofL−convexity. First we recall the following deﬁnitions on the causal hierarchy ofspacetimes, see [5] for a complete list:Deﬁnition 2.2. A spacetime (V, g) isCausal if it does not contain closed causal curves.Distinguishing if whenever I+(p) = I+(q) and I−(p) = I−(q), necessarilyp = q.Causally continuous if it is distinguishing and the set valued functionsI+(·) and I−(·) are outer continuous (say, I+(·) is outer continuous atsome p ∈ V if, for any compact subset K ⊂ I+(p) there exists an openneighborhood U 3 p such that K ⊂ I+(q) for all q ∈ U). This is equivalentto being distinguishing and reﬂecting, i.e. for any pair of events p, q ∈ V ,I+(q) ⊂ I+(p) if and only if I−(p) ⊂ I−(q).Causally simple if it is causal and J±(p) are closed sets for any p ∈ V .Prop. 2.1 allow us to prove that all doubly warped spacetimes are causallycontinuous:Theorem 2.1. Any doubly warped spacetime (V, g) as in (1) is causally conti-nuous.Idea of the proof: Since (V, g) is stably causal, it is also distinguishing. So, itsuﬃces to show that (V, g) is reﬂecting. Let (to, xo1, xo2), (te, xe1, xe2) ∈ V be suchthat I+((te, xe1, xe2)) ⊂ I+((to, xo1, xo2)), and let us prove thatI−((to, xo1, xo2)) ⊂ I−((te, xe1, xe2)) (the converse is analogous). Consider the se-quence {(te + 1/n, xe1, xe2)}n ⊂ I+((te, xe1, xe2)) and note that, by the hypothesis,this sequence also belongs to I+((to, xo1, xo2)). Therefore, from Prop. 2.1, thereexist constants µn1 , µn2 > 0, with µn1 + µn2 = 1, satisfying the following inequali-ties:∫ te+1/nto√µniαi(s)(2∑k=1µnkαk(s))−1/2ds > di(xoi , xei ) for i = 1, 2. (4)Up to a subsequence, we can assume that {µni }n converges to µi, for all i, with0 ≤ µ1, µ2 ≤ 1 and µ1 + µ2 = 1. Using arguments related to convergence of50 L. Aké, J.L. Flores, J. Herrerafunctions we deduce that:∫ teto√µiαi(s)(2∑k=1µkαk(s))−1/2ds ≥ di(xoi , xei ), for i = 1, 2.If we consider (to − 1/n, xo1, xo2), and modify slightly (µ1, µ2), by continuity weobtain new coeﬃcients (µ′1, µ′2), with µ′1, µ′2 > 0 and µ′1 + µ′2 = 1 and pre-vious inequalities are strict. Again from Prop. 2.1, we have (to − 1/n, xo1, xo2)(te, xe1, xe2) for all n. So, taking into account that I−((to, xo1, xo2)) = ∪n∈NI−((to−1/n, xo1, xo2)), we deduce the inclusion I−((to, xo1, xo2)) ⊂ I−((te, xe1, xe2)), as re-quired.Also, the characterization of the causal relation given in Prop. 2.2 allow usto give a characterization of the causally simple doubly warped products:Theorem 2.2. A doubly warped spacetime (V, g) as in (1) is causally simple ifand only if (Mi, gi) is Li-convex for Li =∫ ba1√αi(s)ds, for all i = 1, 2.Main idea: For the implication to the right. Let xo1 and xe1 with0 < d1(xo1, xe1) < L1 =∫ bads√α1(s),then, there exists some c1 and c2 with a < c1 < c2 < b and d1(xo1, xe1) <∫ c2c1ds√α1(s)take some point x2 = x02 = xe2 in M2 and consider the followingpoints (c1, xo1, x2) and (c2, xe1, x2), then, by Prop. 2.1 we have that (c2, xe1, x2) ∈I+((c1, xo1, x2)). Since (c1, xe1, x2) 6∈ I+((c1, xo1, x2)), there exists a point(te, xe1, x2)) ∈ ∂I+((c1, xo1, x2)) = J+((c1, xo1, x2)) \ I+((c1, xo1, x2)),(because (V, g) is causally simple), so, there exists a null geodesic between thesepoints. The projection of this geodesic on M1 will be a minimizing geodesicbetween xo1 and xe1, therefore (M1, g1) is L1-convex, the same reasoning givesthe result for (M2, g2).For the implication to the left, we have to show thatJ±((to, xo1, xo2)) = J±((to, xo1, xo2)),for all (to, xo1, xo2) ∈ V . Reasoning for the future case, let(te, xe1, xe2) ∈ J+((to, xo1, xo2)),Causality on doubly warped spacetimes 51and consider a sequence {(te + 1/n, xe1, xe2)}n ⊂ I+((to, xo1, xo2)), using Prop. 2.1and convergence of functions we show thatdi(xoi , xei ) <∫ ba√µiαi(s)(2∑k=1µkαk(s))−1/2ds ≤∫ ba1√αi(s)ds = Li,and, since (Mi, gi) is Li-convex we have that Prop. 2.2 implies that (te, xe1, xe2) ∈J+((to, xo1, xo2)). The past case is analogous.AcknowledgmentsThe authors are partially supported by the Spanish Grant MTM2016-78807-C2-2-P (MINECO and FEDER funds). L. Aké also acknowledges a grant fundedby the Consejo Nacional de Ciencia y Tecnología (CONACyT), México.References[1] L. Aké, J.L. Flores and J. Herrera, Causality and c-completion of doublywarped spacetimes, arXiv:1709.00234.[2] J.K. Beem, P.E. Ehrlich and K.L. Easley, Global Lorentzian Geometry.Monographs Textbooks Pure Appl. Math. 202, Dekker Inc., New York(1996).[3] J.L. Flores and M. Sánchez, Geodesic connectedness of multiwarped spa-cetimes. J. Diﬀerential Equations, 186(1):1-30, 2002.[4] J.L. Flores and M. Sánchez, The causal boundary of wave-type spacetimes.J. High. Energy Phys., (3):036, 43, 2008.[5] E. Minguzzi and M. Sánchez, The causal hierarchy of spacetimes. Zurich:Eur. Math. Soc. Publ. House, vol. H. Baum, D. Alekseevsky (eds.), Recentdevelopments in Pseudo-Riemannian geometry of ESI Lect. Math. Phys.,299358 (2008).[6] Volker Perlick. Gravitational lensing from a spacetime perspective. LivingReviews in Relativity, 7(1):9 2004.

Causal properties of doubly warped spacetimes

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Causal properties of doubly warped spacetimes

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idUS. Depósito de Investigación Universidad de Sevilla