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
õi
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
õi
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. Sánchez,
Geodesic connec edness o mul iwa ped spa-
ce imes.
J. Die en ial Equa ions,
186
(1):1-30, 2002.
[4] J.L. Flo es and M. Sánchez,
The causal bounda y o wa e- ype space imes
.
J. High. Ene gy Phys., (3):036,
43
, 2008.
[5] E. Minguzzi and M. Sánchez,
The causal hie a chy o space imes
. Zu ich:
Eu . Ma h. Soc. Publ. House, ol. H. Baum, D. Aleksee sky (eds.), Recen
de elopmen s in Pseudo-Riemannian geome y o ESI Lec . Ma h. Phys.,
299358 (2008).
[6] Volke Pe lick.
G a i a ional lensing om a space ime pe spec i e
. Li ing
Re iews in Rela i i y,
7
(1):9 2004.