DEAN ALLISON

G E O D E S I C C O M P L E T E N E S S IN S T A T I C

S P A C E - T I M E S

ABSTRACT. Let (H,h) be a Riemannian manifold and assume f : H - ~ (0, oo) is a smooth function. The Lorentzian warped product (a, b) : x H, - oo ~< a < b ~< oo, with metric ds 2 = (__f 2 dt 2) • h is called a standard static space-time. A study is made of geodesic completeness in standard static space-times. Sufficient conditions on the warping function f: H -~ (0, oo) are obtained for (a,b): x H to be timelike and null geodesically complete. In the timelike ease, the sufficient condition is independent of the completeness of the Riemannian manifold (H, h).

1. I N T R O D U C T I O N

Geodesic completeness of space-times has played a crucial role in singu- larity theory within general relativity. Even though it is known that large classes of physically realistic space-times are nonspacelike incomplete [7], many questions remain as to the nature of the incompleteness in an arbitrary space-time. For example, the relationship between timelike, null, and spacelike completeness is not well understood. In this paper, geodesic completeness is studied in a large class of space-times called warped products.

A standard static space-time (a,b): x H is a Lorentzian warped product manifold (a,b) x H, - ~ ~< a < b ~< ~ , with metric ~ = ( - f 2 d t 2 ) O h , where (H,h) is a Riemannian manifold and f: H ~ (0, oo) is a smooth function. Kobayashi and Obata [9] posed the following problem: Given timelike geodesic completeness of a standard static space-time of the form •: x H, find conditions o n f such that (H, h) is necessarily complete. We say that a function f : I t -~ (0, ~ ) satisfies the K-growth condition if for all :~ > 0 there is a compact set K in H such that f(x)>1 • for all x~HNK. In Theorem 3.7 we show that a sufficient condition for timelike geodesic completeness of R: × H is that f satisfies the K-growth condition, inde- pendent of the completeness of the Riemannian space (H, h). Thus, if f satisfies the K-growth condition, then the problem posed by Kobayashi and Obata will fail to have a solution.

Bishop and O'Neill [5] proved that a Riemannian warped product M: x H is complete if and only if both (M,g) and (H,h) are complete. A Lorentzian warped product is obtained from the product M x H of a Lorentzian manifold (M,g) and a Riemannian manifold (H,h), using a warped metric 9. Several authors have studied nonspacelike geodesic com- pleteness in Lorentzian warped products of the type (M x ell, g O e 2 h ) ,

wheree: M ~ (0, oo) is a smooth function on the Lorentzian manifold (M, g) ([2], [31, [4], [-10], [ l l ]) . In this paper we give some new results on

Geometriae Dedicata 26 (1988)~ 85-97. © 1988 by Kluwer Academic Publishers.

86 DEAN A L L I S O N

geodesic completeness in Lorentzian warped products My × H which have the warping function f: H ~ (0, oo) defined on the Riemannian factor. The metric is given by ~ = f 2 g @ h . We are primarily interested in the case (a,b): x H, - ~ ~< a < b < ~ , of standard static space-times (Section 3), although geodesic completeness of M: x H, dim M >/2, is discussed in Section 4. Results on the causal structure of Lorentzian warped products, which are needed in Sections 3 and 4, are given in Section 2, though stated in the more general setting of Lorentzian doubly warped products. Kemp [8] has previously considered geodesic completeness in Lorentzian warped products of the type (a, b): x H.

Examples of standard static space-times include Minkowski space-time, Schwarzschild space-time, universal anti-de Sitter space-time, and the Einstein static universe. The exterior Schwarzschild space-time is given by a warped product of ( ~ , - dt 2) with the Riemannian manifold (2m, oo) x S 2

using the metric

( ds 2= 1 - dr2 + r2(dO2 + sin2Od~o2), r > 2m,

with f : (2re, o e ) x S 2 ~(0, oo) given by f(r, 0, q))= (1-(2m/r)) 1/2. The universal covering space of anti-de Sitter space-time is the Lorentzian analogue of the Riemannian hyperbolic space of constant negative sectional curvature. It may be viewed as a warped product of the form (N: x H, - f 2 d t 2 Oh), where (It, h) is hyperbolic 3-space with constant negative sectional curvature and f: H ~ ( 0 , oo) is defined by f(r,O,~o)= coshr. Universal anti-de Sitter space-time is an example of a space-time used in general relativity which may be written as a warped product of the type M: x t f but not as a warped product of the type M x ell.

After previous studies on Riemannian warped products and Lorentzian warped products of the type M x ell, one expects that geodesic complete- ness will depend in general on the warping function f and completeness of the factor manifolds. In Section 3 it is shown that standard static space- times of the form (a, b): x H, with (a, b) an interval and a or b finite, are always timelike, null, and spacelike geodesically incomplete, independent of the warping function f: H ~ (0, ~) (Propositions 3.4 and 3.6). However, nonspacelike geodesic completeness in a standard static space-time R: x H does depend on the warping function. As noted earlier, a sufficient con- dition for timelike geodesic completeness of R: x H is that f satisfy the K- growth condition (Theorem 3.7). However, the K-growth condition is not sufficient for null geodesic completeness (Example 3.8). In Theorems 3.10 and 3.12 we find that a sufficient condition for both timelike and null

G E O D E S I C C O M P L E T E N E S S IN STATIC S P A C E - T I M E 87

geodesic completeness of IRf × H is (1) completeness of (H,h) and (2) that f : H -~ [m, oo) be bounded from below by a constant m > 0.

2. LORENTZIAN DOUBLY WARPED PRODUCTS

In this section we shall discuss some properties of Lorentzian doubly

warped products.

D E F I N I T I O N 2.1. Let (M,9) and (H,h) be pseudo-Riemannian manifolds and let e: M - ~ (0, oo) and f : H - ~ ( 0 , oo) be smooth functions. Let

n: M x H -~ M and r/: M x H ~ H be the projections. The doubly warped product M: x eH is the product manifold M x H furnished with the metric

tensor ~ defined by

~(v, w) = ( f o r/)E(p)g(n, v, n , w) + (e o n)2 (p)h(~l, v, q, w)

for v, we Tp(M x H).

If (M,9) is an n-dimensional manifold (n >~ 1) with signature ( - , + , ' " , + ) and (H,h) is a Riemannian manifold, the resulting warped product M: × eH will be a Lorentzian manifold.

The causal structure and completeness properties for the Lorentzian case

with f - I are known. The case with e - 1 and f arbitrary has also been considered ([8], [9]). We shall primarily be interested in Lorentzian singly

warped products of the form My × H, and in particular, in the following

special case.

D E F I N I T I O N 2.2. Let (H,h) be a Riemannian manifold, (a,b) an open

interval, - oe ~< a < b ~< oo, and f : H ~ (0, oo) a smooth function. Let t and

be the projections of (a,b) x H onto (a,b) and H. The standard static space-time (a,b) I × H is the manifold (a,b) x H with the metric

= - - ( f o r / ) 2 d t 2 O h .

O'Neill [10] proves that any static space-time is locally isometric to a

standard static space-time. For each (p ,q)eM x H, we shall call q - l (q) = M × {q} afiber, and call

~-a(p) = {p} × H a leaf. Vectors tangent to leaves will be called horizontal. Vectors tangent to fibres will be called vertical.

Given vector fields X 1 and Y1 on M and vector fields X 2 and Y2 o n H ,

we may lift them to vector fields X = (X1,0) + (0 ,X2)= (X1,X2) and Y = (YI,0) + (O, Y2) = (Y1,Y2). Now we will proceed to determine the Levi- Civita connection V for a doubly warped product (M: x ell,(7) for X and Y

as above. Let V ~ denote the Levi-Civita connection for (M,9) and V 2 denote the

88 DEAN ALLISON

Levi-Civita connection for (H,h). We denote the lifts of f and e to M x H

b y f = f o r/and ~ = eo n. To simplify expressions, we shall sometimes u se /~

for M: × ell. The connection V for (Mf x e H, f 2 g ® e 2 h ) is related to the metric ~ = f 2 g 0 e2h by the Koszul formula

2~(VxY, Z) = X~(Y, Z) + Y~(X, Z) - Z~(X, Y) + ~([X, Y], Z)

- ~ ( [X,Y] , Z) - ~([Y, Z] , X)

[6]. A calculation yields the following formula for the Levi-Civita con-

nection V for M: x eH.

X z ( f ) y Y2(f) x (1) V x Y = V~,Y, + V2xY2 + - - f ~ - , + ~ f - - 1

+ X, (e )Y2 + Y,(e) x : _ ~(Xl,rl) g r a d e r

e e f

- - g ( X 2 , Y2) g r a d ~ e

Here we are identifying the vector V~xlYI~TpM with the vector (V~IY 1 ]p,0q)e T(p,q)(M X H), etc.

From formula (1) we can immediately deduce the following:

P R O P O S I T I O N 2.3. In a doubly warped product My x e H each leaf zt-l(p) = p x H and fiber rl- l(q)= M x q is totally umbilic. Furthermore, for a pseudo-Riemannian warped product of the form M f x H, each leaf rt-l(p) is totally 9eodesic.

In order to discuss causality, we first need to discuss a time orientation

for My x e H. The proof is similar to Lemma 2, pp. 39-40 of [8].

L E M M A 2.4. The Lorentzian doubly warped product My x eH is time- orientable if and only if(M,9) is time-orientable (if dim M >/2) or (M,9) is a one-dimensional manifold with a negative definite metric.

We begin our discussion of causality by considering the case when M is a one-dimensional manifold with a negative definite metric - d t 2. M is

diffeomorphic to S 1 or ~. In case M ~ S 1 the integral curves of the time- orienting vector field are closed timelike curves. Hence S} x ¢H is never

chronological. In the case M ~ R one can show that n: (a,b) x H ~ (a,b) has a timelike

gradient and is thus a time function. This yields the following result.

P R O P O S I T I O N 2.5. Let (H,h) be an arbitrary Riemannian manifold and let M = (a,b), - oo -K< a < b -K< oo, be 9iven the negative definite metric - d t 2.

GEODESIC COMPLETENESS IN STATIC SPACE-TIME 89

Then the Lorentzian doubly warped product (a,b) f x ~H is stably causal, and consequently, strongly causal, distinguishing, causal, and chronological.

COROLLARY 2.6. Every standard static space-time is stably causal.

Further results on causality in Lorentzian doubly warped products M I × e H with d imM >/2 may be found in [1].

3. GEODESIC COMPLETENESS

In this section we discuss geodesic completeness in standard static space- times (a,b)yx H, - ~ ~<a<b~<oo .

Let 7z: (a,b) × H ~(a,b) and 0: (a,b) × H ~ H denote the projection maps. The Levi-Civita connections on ((a,b)i x H, ( - f 2 d t 2 ) G h ) and (H,h) will be denoted by V and V 2, respectively. The gradient of a function ~0 on (H,h) will be denoted by grad q~.

Let y be a geodesic in ( a , b ) z x H and let ~ = ~ o ? ' , f l= r /oy . If (dct/ds)(s) # 0 for an affine parameter s, the geodesic differential equations become

(2) ~ ( s ) = c f - 2(fl(s)), c = const # 0

(3) 2 V¢'tsl fl'(s) = 1c2 grad f - 2 (fl(s)).

Also, we note the following result by Kobayashi and Obata [9].

PROPOSITION 3.1 . Let 1~1 = (a,b)f × H with static metric ~ = ( _ f 2 dt 2) • h. Let y(s) be a timelike geodesic in M with fl = ;7 o y. Then

(i) f = const on M implies fl is a geodesic in H. (ii) I f f :~ const on f/l then fl(s)e {x ~ H: f (x ) ~< c} for some constant c.

Furthermore, fl is a pregeodesic in H if and only if it is an orthogonal trajectory of f .

One approach to studying nonspacelike geodesic completeness in (a, b) z × H is suggested by the treatment of Beem et al. [3] in their study of Lorentzian warped products of the form M x ell.

THEOREM 3.2. Let M = ( a , b ) y × H, - o o ~<a<b~<oo , be a standard static space-time. I f Yo is a future-directed inextendible nonspacelike geodesic in (]if'I, {I) we will denote by y the reparametrization of Yo to an inextendible pregeodesic of the form y(t)= (t, p(t)) defined on the interval (Wo, wl) with

90 D E A N A L L I S O N

a ~< w o < w 1 ~< b. Then 7o is past [resp. future] incomplete if and only if

t

lim i ( f o p)2 t ~Wo+ (u) d u > - c~

i /

t o

t ?.. ,,m q ,or . , . . ' t ~ w 1 -

t o

Proof. If 7 and Yo are as in the s tatement of the theorem, then letting

~ = rtO?o , fl = qo?o, we have de/ds > 0 by (2). So we have ? = 7o ° ~ - 1 and p = fl o ~- 1. N o w 7o is future incomplete if and only if lim t ~wl- ~ - 1 (t) < oo. Since (d/dt)c~-1(0 = [f(p(t))2]/c for some cons tant c, by (2), it follows that

7o is future incomplete if and only if lim,~wl_ j lof(p(t)) 2 du < oo for a fixed

t o G (Wo, wl). Past incompleteness follows similarly. []

R E M A R K 3.3. In the null case, Theorem 3.2 can be reformulated in terms

of the energy of the projection p = / ? ° 7 of the null geodesic into the Riemannian manifold (H, h). Here the energy functional Ep(t) of the curve

pjtto is defined by Ea(t)= ~'toh(p'(u), p'(u)) du. If 7 ( t ) = (t,p(t)) is a null

pregeodesic, then

0 = 0(7'(t), 7'(0) = - ( f ° P)2(t) + h(p'(t),p'(t)).

Thus, a future-directed inextendible (to the right) null pregeodesic 7: [to, wl) --* ~ of the form 7(0 = (t, p(t)) is future incomplete if and only if

the energy Eo(w :) of p lto ~ is finite. In the case of a or b finite, we can obtain the following results for

(a ,b)s x H.

P R O P O S I T I O N 3.4. Let Y4 = (a, b)i x H be a standard static space-time. I f b < oo (resp. a > -oo) , then all future-directed timelike 9eodesics are

future (resp. past) incomplete. Proof. Suppose b < ~ and let ?o: [0, d)--* M be an arbi t rary future-

directed inextendible timelike geodesic. Reparametr ize ?o to an (inexten-

dible) timelike pregeodesic of the form 7(0 = (t, p(t)) where 7: [to, w:) --* M, a ~ < t o < W l - K < b < ~ , and p = r / ° y is the projection of 7 into H. By

Proposi t ion 3.1,f(p(t) ~< c for some constant c and all t. The incompleteness of 7o follows immediate ly by an applicat ion of Theorem 3.2. The case

a > - oo is similar. [ ]

In c o n t r a s t , / ~ = (a,b) I x H with a or b finite can contain null inextendible

complete geodesics, as is shown by the following example.

G E O D E S I C C O M P L E T E N E S S IN STATIC S P A C E - T I M E 91

E X A M P L E 3.5. Let H = (0, oo) and (a, b) = (0, or). Let f : H -o (0, oo) be f (x) = 1/x. So ds 2 = - ( 1 / x e ) d t 2 + dx 2 on the space ,M = (0, oo)z x (0, oo).

Then the curve 7: R-o h7"/ defined by y ( s )= (½e 2', e *) is a comple te null

geodesic.

However , if a or b is finite, M = (a, b ) / x H will always contain at least

one incomplete null geodesic and at least one incomplete spacelike geodesic.

P R O P O S I T I O N 3.6. Let h4 = (a, b)i x H be a standard static space-time. I f b < oo (resp. a > - oo), then M is future (resp. past) null and spacelike

geodesically incomplete. Proof. We shall p rove the case b < oo. The case a > - oo is similar. Let 7

be a null future-directed inextendible geodesic in ~/. If y is incomplete we

are done so assume y is defined on the interval [0, oo). Let a = rt o y and

fl = r/o y be the project ions of y into (a, b) and H, resp. Define a new curve

in .M by ~(s) = (a(s) + b - a(1), fl(s)). Clearly ~ is also a null geodesic. But l im,~ 1 _ ~(s) does not exist so ~ is future incomplete. This a rgument applies

as well to any spacelike geodesic not of the form y(t) = (t o, fl(t)) for a fixed

to ~ (a, b). [ ]

To study the nonspacel ike geodesic completeness of a spacet ime •y × H,

Theorem 3.2 is difficult to apply since the integrals depend on the para-

metr izat ion of the geodesic in question. In fact, this paramet r iza t ion is not

even the affine parametr izat ion. To avoid this difficulty, we will discuss

other approaches to obtain sufficient condit ions for nonspacel ike geodesic completeness of ~y x H.

If for all ~ > 0 there is a compac t set K c H such that f ( x ) >1 ~ for all x E H N K , then f : H--* (0, oo) is said to satisfy the K-growth condition.

N o w we can state the following sufficient condi t ion for M = Ey x H to be

t imelike geodesically complete.

T H E O R E M 3.7. Let M = [~f × H be a standard static space-time with metric ~ = ( - f 2 dt2) 0 h. I f f : H --* (0, or) satisfies the K-growth condition, then M is timelike geodesically complete.

Proof Let y be a t imelike future-directed inextendible geodesic in 2['/and assume without loss of generali ty that ~(~,],') = - 1. If we let ~ = n o ~ and

fl = r/oy, then (2) and Proposi t ion 3.1 imply

d ~ 0 < fl~ --< dss for some constant fl~.

On the other hand the K-growth condit ion implies that (d~/ds)(s)=

92 DEAN ALLISON

c/[f(f l(s)) 2] is bounded above. Hence,

(4) 0 < 131 -<< ~ ( s ) ~</3 2 <

for all s for constants fll and/3 2. We need to show y(s) is defined for arbitrari ly large positive and negative

values of s. Suppose by way of contradic t ion that 7(s) is only defined for

- oo ~< a < s --< b < o0. Fix s o ~ (a, b). By an integrat ion of (4), we can find

constants B 1 and B 2 such that - oo < B 1 -<< ~(s) -<< B 2 < oo for s o -<< s < b.

Fur thermore , we can find a compac t K o c H such that fl(s)~ K o for all s. T o choose K o, recall that f(fl(s)) -,< c for some constant c and use the K-

growth condi t ion to choose K o such that f (x ) >/c + 1 for all x e H N K o. So

we have shown ;~(s) = (~(s), fl(s)) ~ [B1, B2] x K o for all s, and hence, that y is an inextendible t imelike geodesic curve which is future imprisoned in a

compac t set. This is impossible in a strongly causal space-t ime [7]. A similar contradic t ion is obtained if y(s) cannot be defined for arbitrari ly

large negative values. Since 7 was arbitrary, M must be t imelike geodesically complete. []

The K-growth condi t ion is not sufficient for null geodesic completeness, as

the following example indicates.

E X A M P L E 3.8. Let 37/= •I x ( - 1 , 1 ) with metric ds 2 -- ( - f ( x ) 2 dt 2) +

dx 2. Let f : ( - 1 , 1 ) ~ (0, oo) be defined by f ( x ) = 1 / x / 1 - x 2. Then the curve 7 given by y ( s ) = (½s + ¼ sin2s, sins) is an incomplete null geodesic.

On the other hand, incompleteness of (H, h) does not imply null geodesic

incompleteness of ~ I × H.

E X A M P L E 3.9. Let 1~ = IRy x ( - 1 , 1 ) with metric ds 2 = ( - f ( x ) Z d t 2 ) +

dx 2. Let f : ( - 1 , 1 ) ~ (0, oo) be defined by f ( x ) = 1 / ( 1 - x 2 ) . Then

((--1, 1),dx 2) is an incomplete Riemannian manifold and (/~,ds 2) is null geodesically complete.

If we require that (H, h) be complete as a Riemannian manifold, then we

can state sufficient condi t ions on the warping function f to ensure both timelike and null geodesic completeness of ~j. x H, as we discuss next.

T H E O R E M 3.10. Let ]fl = N I x H be a standard static spacetime with (H, h) a complete Riemannian manifold. I f 0 < m ~ f ( x ) for some constant m and all x ~ H, then (MI, ~) is timelike oeodesically complete.

Proof. Let 7 be an arbi t rary t imelike future-directed inextendible (to the right) geodesic. Let c~ = n o 7, fl = 17 o y. We suppose by way of contradic t ion that 7(s) is only defined for - co ~< a < s < b < oo. Fix So ~ (a, b). As in the

G E O D E S I C C O M P L E T E N E S S I N S T A T I C S P A C E - T I M E 93

proof of Theorem 3.7, now using Proposi t ion 3.1 and the hypothesis that

f ( x ) is bounded from below, we obtain

- oo < B1 -<< ~(s) --< B 2 < oO for s o < s < b

for constants B 1 and B2. To obtain the desired contradiction, we need to

show that fl]~0 is also trapped in a compact set.

Consider the timelike character of y.

, ,

0 >/g(v (s), 7 (s)) = - f ( f l ( s ) )2Ld s J + h(fl'(s), if(s))

C 2

f(fl(s))2 + h(ff(s), if(s)),

for a constant c > 0, us ing (2). Hence, h(fl'(s), if(s)) --< c2 /m 2 for all s. Thus, the length of fll~o is given by

s 0

£

ds "-< ~ (b - So) < oo. m

The H o p f - R i n o w theorem 1-2] implies that there exists a Po E H such that

fl(s)--,po as s ~ b - . Thus, fl(s) is contained in a compact set D for

So ~< s < b, and 7(s) is future imprisoned in the compact set 1-B 1, B2] × D. The case a > - o o is similar.

For null geodesic completeness, we obtain a similar result but via a different proof.

L E M M A 3.11. Let ffl = R s × H be a standard static space-time with (H, h) a complete Riemannian manifold. I f y: [0, d)--* M1 is a smooth inextendible (to the right) null geodesic with x o 7 = ct, r/0 7 = fl, then the Riemannian length of fl[~ is infinite.

Proof. Suppose V is future-directed. By (2), ~: [0, d) ~ ~ is an increasing

function and consequently, either lims_ ~_ ~(s) exists or lims_~ ~(s) -- + ~ . We consider the two cases separately.

CASE 1. l i m ~ _ ~(s) = t o < ~ . Suppose L(fl[~) < ~ . Then there exists a

Po ~ H such that f l ( s )~ Po as s ~ d - by the H o p f - R i n o w theorem I-2]. Thus, y ( s )~ (to, po)~ E x H as s ~ d - . But this contradicts the inextendi- bility of 7. Hence,

L(fl[~) = oo.

CASE 2. lims~a ~(S) = o0. Again, we suppose L(f l l~)< oe so there

94 D E A N A L L I S O N

exists p o e H such that f l ( s ) ~ p o as s - - , d - . Since f is smooth,

f(fl(s)) ~ f ( P o ) > 0 as s ~ d - . N o w we shall show that d = oe.

By (2),

; jc lim dc~ = lim . . . . dt.

~_~- ~-~- f ( f l ( t )) 2 0 0

Hence,

s

oo = s-a-lim o~(s)- o~(0)= s~-lim f f(fl~t))~ dt.

0

Since f ( f l ( s ) ) ~ f ( P o ) as s - d - , we can find an So such that

c ~< 2c ~< d. ? ( p o f for So < s

Therefore,

s s o d

oo = lira dt ~< dt + dt. ~_~_ f ( - p ~ 0 0 s o

But this inequality is possible only if d = ~ . N o w we can derive a

contradiction. No te that

0 = O(~(s), ~(s)) = - f ( f l ( s ) ) 2 f(fl~-s))2 + h(I~(s),/~(s)).

Also, we can find an N such that f ( f l (s)) ~< 2f(po) for s >i N. Thus,

;c ic L(fl]~) = f ( - ~ ds >/ 2f (~0) ds = ~ . o N

So, in either case, we must have L(fll~) --- ~ . If 7 is past-directed, a similar

argument applies. [ ]

It is clear that a similar lemma can be proven in case a null geodesic

7: (d, 0-] ~ M is inextendible to the left, in which case q °7 = fll ° would have

infinite length. N o w we can state sufficient condit ions for E I × H to be null geodesically

complete.

G E O D E S I C C O M P L E T E N E S S I N S T A T I C S P A C E - T I M E 95

T H E O R E M 3.12. Let f/l = ~ y x H be a standard static space-time with (H, h) a complete Riemannian manifold. I f 0 < m ~< f (x) for some constant m and all x ~ H, then (R/I, ~) is null geodesically complete.

Proof. To show (M, ~) must be future null geodesically complete, we let

Y: [0, A) --, fi7/be an arbitrary null future-directed inextendible (to the right) geodesic. Let ct = n o 7, fl = t/o 7. We must show ~, is defined for arbitrarily large values of s, so suppose not.

Using (2) we obtain

C 2

h(fl(s), fi(s)) - f (fl(s))2 for all s.

Now compute the length of fl[oA:

L(fllc~) =

A A

f x/h(t~(s), fi(s)) ds = f ~ s ~ d s

0 0

A fcd, cA = - - < 0 0

m

o

since m > 0 , c > 0 , and we are assuming A < o v . But by Lemma 3.7,

L(fll6 ~) = or. So ~ must be future complete. A similar argument with an arbitrary null geodesic 7: ( - A , 0]--+ M shows (]~, ~) is past null geodesi- cally complete. []

C O R O L L A R Y 3.13. I f ~ f × H is a standard static space-time with (H, h) a compact Riemannian manifold, then R: x H is nonspacelike geodesically complete.

4. GEODESIC COMPLETENESS IN M : × H (dim M >/2)

In this section we shall discuss some results on geodesic completeness in a Lorentzian warped product of the form 1~/= M s × H where (M, g) is a

space-time (dim M >~ 2) and (H, h) is a Riemannian manifold, and we use the Lorentzian metric .~ = f2g ® h on hT/.

We begin with the observation due to Ehrlich [11] that if 7 is a geodesic of M = M: × H, then n ° 7 is a pregeodesic of M.

LEMMA 4.1. Let ~ be a geodesic in M with ~(0)=p . For any [o=(p,q)~M × H and for ~=(7'(O),v)~Tv(M x H) there exists a pre- geodesic ~ such that n o y = ~ and 7'(0) = f~.

96 D E A N A L L I S O N

N o w we can state sufficient condit ions for timelike geodesic completeness

of M I x H to imply timelike geodesic completeness of M.

T H E O R E M 4.2. Let (M,g) be a space-time and (H,h) be a Riemannian manijbld. Let f : H ~ (0, N] be a bounded smooth function. I f ~I = M s × H is timelike oeodesically complete, then (M, g) is also timelike 9eodesically complete.

Proof. Suppose (M, 9) is t imelike geodesically incomplete. Let ~t: [0, b) --, M be a unit speed inextendible t imelike geodesic with b < oo, ct(0) = p c M. Fix q e H and let ~ = (ct'(0), 0q)e Ttp, ql(M x H). Using L e m m a

4.1 we can find a pregeodesic 7 in M with y ' ( 0 ) = ~ and rco7 = ~. Let

fl = r/o7. Since ~(y'(0), 7 ' (0) )= f(fl(0))20(~'(0), ~' (0)) < 0, 7 is timelike.

N o w 7 can be maximal ly extended to some interval [0, b') with b' ~< b. T o

show M is timelike geodesically incomplete, it suffices to show tha t the length of the pregeodesic 71 b' is finite.

b' / I

L(y I~') = | x / f ( f l ( S ) ) 2 - - h(fl'(s), if(s)) ds I L l

0

b t

~< f f(fl(s)) ds 0

G O O .

Interestingly, the analogous theorem for null geodesic completeness does not require any condi t ions on the warping function.

T H E O R E M 4.3. Let (M,g) be a space-time and (H,h) a Riemannian ma- nifold. I f the Lorentzian warped product ~1 = My × H is null geodesically complete, then (M, g) is also null geodesically complete.

Proof. Suppose (M, g) is null geodesically incomplete, so there exists an

inextendible null geodesic ~: [0, b) ~ M with b < oo. Define a curve Y in by 7(s) = (ct(s), q) for a fixed q e H. Since

g (7 ' (S ) , 7 ' ( s ) ) = f (q )2g(~ ' (S ) , O~'(s)) + h(Oq, Oq) = O,

7 is a null curve. The connect ion formula (1) yields

, = Vc,,(s ) 7 - - - V~,(s)7 (S) 1 ='(s) - ½O(ct'(s), ~t'(s)) grad f 0

since a is a null geodesic. Hence, 7 is a null geodesic in M. But, 7: [0, b) --, 5q t

is not extendible to b. If it were, then we would be able to extend 7 = n o 7 to

GEODESIC COMPLETENESS" IN STATIC SPACE-TIME 97

b, contradicting the definition of b. Hence, (~r, ~) is null geodesically incomplete. []

A CKNOWLEDGEMENTS

These results were part of the author's doctoral disseration written under the direction of Professor John K. Beem at the University of Missouri- Columbia. The author wishes to thank Professor Beem for his assistance and encouragement, as well as Professor Paul Ehrlich for his valuable comments. In addition, the author wishes to thank the referee for helpful suggestions, including a shorter proof of Theorem 3.2.

R E F E R E N C E S

1. Allison, D. E., 'Lorentzian Warped Products and Static Space-Times', Doctoral Thesis, University of Missouri-Columbia, 1985.

2. Beem, J. K. and Ehrlich, P. E., Global Lorentzian Geometry, Marcel Dekker, New York, 1981, pp. 55-79.

3. Beem, J. K., Ehrlich, P. E., and Powell, T. G., 'Warped Product Manifolds in Relativity', in Selected Studies: Physics-Astrophysics, Mathematics, History of Science (eds. Th. M Rassias and G. M. Rassias), North-Holland, Amsterdam, 1982, pp. 41-56.

4. Beem, J. K. and Powell, T. G., 'Geodesic Completeness and Maximality in Lorentzian Warped Products', Tensor (N. S.) 39 (1982), 31-36.

5. Bishop, R. L. and O'Neill, B., 'Manifolds of Negative Curvature', Trans. Amer. Math. Soc. 145 (1969), 1-49.

6. Cheeger, J. and Ebin, D., Comparison Theorems in Riemannian Geometry, North-Holland, Amsterdam, 1975, p. 2.

7. Hawking, S. W. and Ellis, G. F. R., The Large Scale Structure of Space-time, Cambridge University Press, Cambridge, 1973.

8. Kemp, M. H., 'Lorentzian Warped Products of a Second Type', Doctoral Thesis, University of Missouri-Kansas CitY, 1981.

9. Kobayashi, O. and M. Obata, 'Certain Mathematical Problems on Static Models in General Relativity', Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 3, (eds. S. S. Chern and W. Wen-tsiin), 1980, pp. 1333- 1344.

10. O'Neill, B., Semi-Riemannian Geometry, Academic Press, New York, 1983. 11. Powell, T. G., 'Lorentzian Manifolds with Non-Smooth Metrics and Warped Products',

Doctoral Thesis, University of Missouri-Columbia, 1982.

Author's address: Dean Allison Department of Mathematics, Southern Illinois University at Carbondale, Carbondale, IL62901, U.S.A.

(Received, November 18, 1985; revised version, January 24, 1987).