Causal properties of doubly warped spacetimes

XI Encuen o Andaluz de Geome ía IMUS (Uni e sidad de Se illa), 15 de mayo de 2015, págs. 4751 Causal p ope ies o doubly wa ped space imes Luis Aké • José Luis Flo es • Jóna an He e a Abs ac . In his alk we will desc ibe he cha ac e iza ion o he ch onological ela ion on doubly wa ped p oduc space imes o he o m (a, b)×M1×M2 wi h Lo en zian me ic g=−d 2+α1g1+α2g2 , and we will discuss when hese space imes a e causally con inuos and causally simple. This alk is based on he  s sec ion o [1], a ailable a a Xi :1709.00234. 1. In oduc ion In he classical book on Lo en zian geome y [2] he causali y o wa ped space imes p oduc s o he o m ((a, b)×M, −d 2+αgM) , whe e ((a, b),−d 2) is a Lo en zian mani old, (M, gM) is a Riemannian mani old and α:V→(0,∞) is a smoo h unc ion, is s udied and esul s conce ning o he s ably causal, s ongly causal and global hype bolici y o hese space imes a e ob ained, see [2, Lemma 3.55, P op. 3.62 and Thm. 3.66]. Howe e , he s udy o he causally con inuous and causally simple s ages o he causal hie a chy a e no included. Ou aim in his alk is o gi e a cha ac e iza ion o he ch onological and causal ela ions in doubly wa ped space imes and hen we will gi e condi ions o ob ain he causally con inui y and causal simplici y o hese space imes. The s udy o he cha ac e iza ion o he ch onological ela ion on hese space imes will be use ul in o de o s udy he causal comple ion o hese space imes, see [1]. 2. P elimina ies A doubly wa ped space ime can be w i en as: V:= (a, b)×M1×M2 and g=−d 2+α1g1+α2g2. (1) Luis Aké, [email p o ec ed] Depa amen o de Álgeb a, Geome ía y Topología, Uni e sidad de Málaga José Luis Flo es, o [email p o ec ed] Depa amen o de Álgeb a, Geome ía y Topología, Uni e sidad de Málaga Jóna an He e a, [email p o ec ed] Depa amen o de Ma emá icas, Uni e sidad de Có doba 48 L. Aké, J.L. Flo es, J. He e a Whe e αi: (a, b)→(0,∞) a e smoo h unc ions o all i= 1,2 and (Mi, gi) a e Riemannian mani olds o all i= 1,2 . The ch onological ela ion in hese space imes is cha ac e ized in he ollo- wing p oposi ion (see [1]): P oposi ion 2.1. Le (V, g) be a doubly wa ped space ime as in (1), and ( o, xo),( e, xe)∈V wi h xo6=xe . The ollowing condi ions a e equi alen : (i) ( o, xo)( e, xe) ; (ii) he e exis s ic ly posi i e cons an s µ0 1, µ0 2>0 , wi h µ0 1+µ0 2= 1 , such ha Z e opµ0 i αi(s) 2 X k=1 µ0 k αk(s)!−1/2 ds > di(xo i, xe i) o i= 1,2. (2) In o de o gi e a p ope cha ac e iza ion o he causal ela ion in doubly wa ped space imes we need he ollowing deni ion: Deni ion 2.1. A Riemannian mani old (N, h) is L - con ex i any pai o poin s p, q ∈N wi h dh(p, q)< L can be joined by a minimizing geodesic. Now, we can es ablish he announced cha ac e iza ion abou he causal e- la ion (see [3, Thm. 2(2)]) P oposi ion 2.2. Le (V, g) be a doubly wa ped space ime as in (1) whose be s (Mi, gi) a e Li -con ex o i= 1,2 . Conside wo poin s ( o, xo 1, xo 2),( e, xe 1, xe 2)∈ V , wi h o≤ e , sa is ying d(xo i, xe i)< Li , i= 1,2 . Then, he ollowing condi- ions a e equi alen : (i) he poin s a e causally ela ed, ( o, xo 1, xo 2)≤( e, xe 1, xe 2) ; (ii) he e exis s a causal geodesic joining ( o, xo 1, xo 2) wi h ( e, xe 1, xe 2) ; (iii) he e exis cons an s µ0 1, µ0 2≥0 , µ0 1+µ0 2= 1 , such ha Z e opµ0 i αi(s) 2 X k=1 µ0 k αk(s)!−1/2 ds ≥di(xo i, xe i) o i= 1,2. (3) Mo eo e , i he equali ies hold in (3), hen he e is a ligh like and no imelike geodesic joining he poin s. Wi h he cha ac e iza ion o he ch onological ela ion and causali y ela ion, he las unde L− weakly con ex condi ion, we a e eady o s udy he causally con inui y and causally simplici y o doubly wa ped space imes. Causali y on doubly wa ped space imes 49 Causal hie a chy on doubly wa ped space imes Nex we a e going o p o e ha double wa ped space imes a e causally con inuous, and, causally simple i and only i each (Mi, gi) sa is y a no ion o L− con exi y. Fi s we ecall he ollowing deni ions on he causal hie a chy o space imes, see [5] o a comple e lis : Deni ion 2.2. A space ime (V, g) is Causal i i does no con ain closed causal cu es. Dis inguishing i whene e I+(p) = I+(q) and I−(p) = I−(q) , necessa ily p=q . Causally con inuous i i is dis inguishing and he se alued unc ions I+(·) and I−(·) a e ou e con inuous (say, I+(·) is ou e con inuous a some p∈V i , o any compac subse K⊂I+(p) he e exis s an open neighbo hood U3p such ha K⊂I+(q) o all q∈U ). This is equi alen o being dis inguishing and eec ing , i.e. o any pai o e en s p, q ∈V , I+(q)⊂I+(p) i and only i I−(p)⊂I−(q) . Causally simple i i is causal and J±(p) a e closed se s o any p∈V . P op. 2.1 allow us o p o e ha all doubly wa ped space imes a e causally con inuous: Theo em 2.1. Any doubly wa ped space ime (V, g) as in (1) is causally con i- nuous. Idea o he p oo : Since (V, g) is s ably causal, i is also dis inguishing. So, i suces o show ha (V, g) is eec ing. Le ( o, xo 1, xo 2),( e, xe 1, xe 2)∈V be such ha I+(( e, xe 1, xe 2)) ⊂I+(( o, xo 1, xo 2)) , and le us p o e ha I−(( o, xo 1, xo 2)) ⊂I−(( e, xe 1, xe 2)) ( he con e se is analogous). Conside he se- quence {( e+ 1/n, xe 1, xe 2)}n⊂I+(( e, xe 1, xe 2)) and no e ha , by he hypo hesis, his sequence also belongs o I+(( o, xo 1, xo 2)) . The e o e, om P op. 2.1, he e exis cons an s µn 1, µn 2>0 , wi h µn 1+µn 2= 1 , sa is ying he ollowing inequali- ies: Z e+1/n opµn i αi(s) 2 X k=1 µn k αk(s)!−1/2 ds > di(xo i, xe i) o i= 1,2. (4) Up o a subsequence, we can assume ha {µn i}n con e ges o µi , o all i , wi h 0≤µ1, µ2≤1 and µ1+µ2= 1 . Using a gumen s ela ed o con e gence o 50 L. Aké, J.L. Flo es, J. He e a unc ions we deduce ha : Z e o √µi αi(s) 2 X k=1 µk αk(s)!−1/2 ds ≥di(xo i, xe i), o i= 1,2 . I we conside ( o−1/n, xo 1, xo 2) , and modi y sligh ly (µ1, µ2) , by con inui y we ob ain new coecien s (µ0 1, µ0 2) , wi h µ0 1, µ0 2>0 and µ0 1+µ0 2= 1 and p e- ious inequali ies a e s ic . Again om P op. 2.1, we ha e ( o−1/n, xo 1, xo 2) ( e, xe 1, xe 2) o all n . So, aking in o accoun ha I−(( o, xo 1, xo 2)) = ∪n∈NI−(( o− 1/n, xo 1, xo 2)) , we deduce he inclusion I−(( o, xo 1, xo 2)) ⊂I−(( e, xe 1, xe 2)) , as e- qui ed.  Also, he cha ac e iza ion o he causal ela ion gi en in P op. 2.2 allow us o gi e a cha ac e iza ion o he causally simple doubly wa ped p oduc s: Theo em 2.2. A doubly wa ped space ime (V, g) as in (1) is causally simple i and only i (Mi, gi) is Li -con ex o Li=Rb a 1 √αi(s)ds , o all i= 1,2 . Main idea: Fo he implica ion o he igh . Le xo 1 and xe 1 wi h 0< d1(xo 1, xe 1)< L1=Zb a ds pα1(s), hen, he e exis s some c1 and c2 wi h a<c1< c2< b and d1(xo 1, xe 1)< Rc2 c1 ds √α1(s) ake some poin x2=x0 2=xe 2 in M2 and conside he ollowing poin s (c1, xo 1, x2) and (c2, xe 1, x2) , hen, by P op. 2.1 we ha e ha (c2, xe 1, x2)∈ I+((c1, xo 1, x2)) . Since (c1, xe 1, x2)6∈ I+((c1, xo 1, x2)) , he e exis s a poin ( e, 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, he e exis s a null geodesic be ween hese poin s. The p ojec ion o his geodesic on M1 will be a minimizing geodesic be ween xo 1 and xe 1 , he e o e (M1, g1) is L1 -con ex, he same easoning gi es he esul o (M2, g2) . Fo he implica ion o he le , we ha e o show ha J±(( o, xo 1, xo 2)) = J±(( o, xo 1, xo 2)), o all ( o, xo 1, xo 2)∈V . Reasoning o he u u e case, le ( e, xe 1, xe 2)∈J+(( o, xo 1, xo 2)), Causali y on doubly wa ped space imes 51 and conside a sequence {( e+ 1/n, xe 1, xe 2)}n⊂I+(( o, xo 1, xo 2)) , using P op. 2.1 and con e gence o unc ions we show ha 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 -con ex we ha e ha P op. 2.2 implies ha ( e, xe 1, xe 2)∈ J+(( o, xo 1, xo 2)) . The pas case is analogous.  Acknowledgmen s The au ho s a e pa ially suppo ed by he Spanish G an MTM2016-78807- C2-2-P (MINECO and FEDER unds). L. Aké also acknowledges a g an unded by he Consejo Nacional de Ciencia y Tecnología (CONACyT), México. Re e ences [1] L. Aké, J.L. Flo es and J. He e a, Causali y and c-comple ion o doubly wa ped space imes , a Xi :1709.00234. [2] J.K. Beem, P.E. Eh lich and K.L. Easley, Global Lo en zian Geome y . Monog aphs Tex books Pu e Appl. Ma h. 202 , Dekke Inc., New Yo k (1996). [3] J.L. Flo es and M. 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