Causal properties of doubly warped spacetimes

XI Encuentro Andaluz de Geometría IMUS (Universidad de Sevilla), 15 de mayo de 2015, págs. 4751 Causal properties of doubly warped spacetimes Luis Aké • José Luis Flores • Jónatan Herrera Abstract. 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 rst section of [1], available at arXiv:1709.00234. 1. Introduction In the classical book on Lorentzian geometry [2] the causality of warped spacetimes products of the form ((a, b)×M, −dt2+αgM) , where ((a, b),−dt2) is a 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 causally continuous 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 causal relations in doubly warped spacetimes and then we will give conditions to obtain the causally continuity and causal simplicity of these spacetimes. The study of the characterization of the chronological relation on these spacetimes will be useful in order to study the causal completion of these spacetimes, see [1]. 2. Preliminaries A doubly warped spacetime can be written as: V:= (a, b)×M1×M2 and g=−dt2+α1g1+α2g2. (1) Luis Aké, [email protected] Departamento de Álgebra, Geometría y Topología, Universidad de Málaga José Luis Flores, or[email protected] Departamento de Álgebra, Geometría y Topología, Universidad de Málaga Jónatan Herrera, [email protected] Departamento de Matemáticas, Universidad de Córdoba 48 L. Aké, J.L. Flores, J. Herrera Where αi: (a, b)→(0,∞) are smooth functions for all i= 1,2 and (Mi, gi) are Riemannian manifolds for all i= 1,2 . The chronological relation in these spacetimes is characterized in the following proposition (see [1]): Proposition 2.1. Let (V, g) be a doubly warped spacetime as in (1), and (to, xo),(te, xe)∈V with xo6=xe . The following conditions are equivalent: (i) (to, xo)(te, xe) ; (ii) there exist strictly positive constants µ0 1, µ0 2>0 , with µ0 1+µ0 2= 1 , such that Zte topµ0 i αi(s) 2 X k=1 µ0 k αk(s)!−1/2 ds > di(xo i, xe i) for i= 1,2. (2) In order to give a proper characterization of the causal relation in doubly warped spacetimes we need the following denition: Denition 2.1. A Riemannian manifold (N, h) is L - convex if any pair of points p, q ∈N with dh(p, q)< L can be joined by a minimizing geodesic. Now, we can establish the announced characterization about the causal relation (see [3, Thm. 2(2)]) Proposition 2.2. Let (V, g) be a doubly warped spacetime as in (1) whose bers (Mi, gi) are Li -convex for i= 1,2 . Consider two points (to, xo 1, xo 2),(te, xe 1, xe 2)∈ V , with to≤te , satisfying d(xo i, xe i)< Li , i= 1,2 . Then, the following conditions are equivalent: (i) the points are causally related, (to, xo 1, xo 2)≤(te, xe 1, xe 2) ; (ii) there exists a causal geodesic joining (to, xo 1, xo 2) with (te, xe 1, xe 2) ; (iii) there exist constants µ0 1, µ0 2≥0 , µ0 1+µ0 2= 1 , such that Zte topµ0 i αi(s) 2 X k=1 µ0 k αk(s)!−1/2 ds ≥di(xo i, xe i) for i= 1,2. (3) Moreover, if the equalities hold in (3), then there is a lightlike and no timelike geodesic 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 causally continuity and causally simplicity of doubly warped spacetimes. Causality on doubly warped spacetimes 49 Causal hierarchy on doubly warped spacetimes Next we are going to prove that double warped spacetimes are causally continuous, and, causally simple if and only if each (Mi, gi) satisfy a notion of L− convexity. First we recall the following denitions on the causal hierarchy of spacetimes, see [5] for a complete list: Denition 2.2. A spacetime (V, g) is Causal if it does not contain closed causal curves. Distinguishing if whenever I+(p) = I+(q) and I−(p) = I−(q) , necessarily p=q . Causally continuous if it is distinguishing and the set valued functions I+(·) and I−(·) are outer continuous (say, I+(·) is outer continuous at some p∈V if, for any compact subset K⊂I+(p) there exists an open neighborhood U3p such that K⊂I+(q) for all q∈U ). This is equivalent to being distinguishing and reecting , 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 causally continuous: Theorem 2.1. Any doubly warped spacetime (V, g) as in (1) is causally continuous. Idea of the proof: Since (V, g) is stably causal, it is also distinguishing. So, it suces to show that (V, g) is reecting. Let (to, xo 1, xo 2),(te, xe 1, xe 2)∈V be such that I+((te, xe 1, xe 2)) ⊂I+((to, xo 1, xo 2)) , and let us prove that I−((to, xo 1, xo 2)) ⊂I−((te, xe 1, xe 2)) (the converse is analogous). Consider the sequence {(te+ 1/n, xe 1, xe 2)}n⊂I+((te, xe 1, xe 2)) and note that, by the hypothesis, this sequence also belongs to I+((to, xo 1, xo 2)) . Therefore, from Prop. 2.1, there exist constants µn 1, µn 2>0 , with µn 1+µn 2= 1 , satisfying the following inequalities: Zte+1/n topµn i αi(s) 2 X k=1 µn k αk(s)!−1/2 ds > di(xo i, xe i) for i= 1,2. (4) Up to a subsequence, we can assume that {µn i}n converges to µi , for all i , with 0≤µ1, µ2≤1 and µ1+µ2= 1 . Using arguments related to convergence of 50 L. Aké, J.L. Flores, J. Herrera functions we deduce that: Zte to √µi αi(s) 2 X k=1 µk αk(s)!−1/2 ds ≥di(xo i, xe i), for i= 1,2 . If we consider (to−1/n, xo 1, xo 2) , and modify slightly (µ1, µ2) , by continuity we obtain new coecients (µ0 1, µ0 2) , with µ0 1, µ0 2>0 and µ0 1+µ0 2= 1 and previous inequalities are strict. Again from Prop. 2.1, we have (to−1/n, xo 1, xo 2) (te, xe 1, xe 2) for all n . So, taking into account that I−((to, xo 1, xo 2)) = ∪n∈NI−((to− 1/n, xo 1, xo 2)) , we deduce the inclusion I−((to, xo 1, xo 2)) ⊂I−((te, xe 1, xe 2)) , as required.  Also, the characterization of the causal relation given in Prop. 2.2 allow us to 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 if and only if (Mi, gi) is Li -convex for Li=Rb a 1 √αi(s)ds , for all i= 1,2 . Main idea: For the implication to the right. Let xo 1 and xe 1 with 0< d1(xo 1, xe 1)< L1=Zb a ds pα1(s), then, there exists some c1 and c2 with a<c1< c2< b and d1(xo 1, xe 1)< Rc2 c1 ds √α1(s) take some point x2=x0 2=xe 2 in M2 and consider the following points (c1, xo 1, x2) and (c2, xe 1, x2) , then, by Prop. 2.1 we have that (c2, xe 1, x2)∈ I+((c1, xo 1, x2)) . Since (c1, xe 1, x2)6∈ I+((c1, xo 1, x2)) , there exists a point (te, xe 1, x2)) ∈∂I+((c1, xo 1, x2)) = J+((c1, xo 1, x2)) \I+((c1, xo 1, x2)), (because (V, g) is causally simple), so, there exists a null geodesic between these points. The projection of this geodesic on M1 will be a minimizing geodesic between xo 1 and xe 1 , therefore (M1, g1) is L1 -convex, the same reasoning gives the result for (M2, g2) . For the implication to the left, we have to show that J±((to, xo 1, xo 2)) = J±((to, xo 1, xo 2)), for all (to, xo 1, xo 2)∈V . Reasoning for the future case, let (te, xe 1, xe 2)∈J+((to, xo 1, xo 2)), Causality on doubly warped spacetimes 51 and consider a sequence {(te+ 1/n, xe 1, xe 2)}n⊂I+((to, xo 1, xo 2)) , using Prop. 2.1 and convergence of functions we show that di(xo i, xe i)<Zb a √µi αi(s) 2 X k=1 µk αk(s)!−1/2 ds ≤Zb a 1 pαi(s)ds =Li, and, since (Mi, gi) is Li -convex we have that Prop. 2.2 implies that (te, xe 1, xe 2)∈ J+((to, xo 1, xo 2)) . The past case is analogous.  Acknowledgments The authors are partially supported by the Spanish Grant MTM2016-78807- C2-2-P (MINECO and FEDER funds). L. Aké also acknowledges a grant funded by the Consejo Nacional de Ciencia y Tecnología (CONACyT), México. References [1] L. Aké, J.L. Flores and J. Herrera, Causality and c-completion of doubly warped 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 spacetimes. J. Dierential 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.), Recent developments in Pseudo-Riemannian geometry of ESI Lect. Math. Phys., 299358 (2008). [6] Volker Perlick. Gravitational lensing from a spacetime perspective . Living Reviews in Relativity, 7 (1):9 2004.