- Email: [email protected]
The index form of a warped product
Nonlinear Analysis 63 (2005) e435 – e445 www.elsevier.com/locate/na
The index form of a warped product Paul E. Ehrlicha , Seon-Bu Kimb,∗,1 a Department of Mathematics, The University of Florida, Gainesville, FL 32611, USA b Department of Mathematics, Chonnam National University, Kwangju 500-757, Republic of Korea
Abstract We construct the index form along time-like geodesics on a Lorentzian warped product manifold and apply this index form to generalized Robertson–Walker (GRW) space–times. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Index form of a timelike geodesic; Warped product; Generalized Robertson–Walker space–time
1. Preliminaries Since many space–times in relativity may be represented by Lorentzian warped product manifolds ([1–7,9]), a study of the time-like index form for this class of metrics is relevant. Because Hess(f 2 )(v, w)=2f Hess(f )(v, w)+2∇f, v∇f, w and calculations shows that the index form contains Hessian terms, we employ the formalism g¯ = gB ⊕ f g F , f : B → (0, +∞) for the warped product as originally treated in [3,1], rather than the currently more common formalism g¯ = gB ⊕ f 2 gF which has been more popular since the publication of O’Neill’s monograph [8]. Thus let (B, gB ) be a Lorentzian manifold (or interval ((a, b), −dt 2 )) and let (F, gF ) be a Riemannian manifold. For M = B × F, let : M → B and : M → F denote the projection maps. Then given any curve : I → M, we may set B = ◦ , F = ◦ and so obtain (t) = (B (t), F (t)). ∗ Corresponding author.
E-mail addresses: [email protected]fl.edu (P.E. Ehrlich), [email protected] (S.-B. Kim). 1 The second author was supported by Chonnam National University in Korea, 2004.
0362-546X/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.09.022
(1)
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Correspondingly, since T(b,f ) M = Tb B ⊕ Tf F , we may decompose V ∈ () as V = (VB , VF ).
(2)
If : I ×(−, ) → M is a variation of (t), then setting B =◦, F =◦ : I ×(−, ) → M, we obtain (t, s) = (B (t, s), F (t, s))
(3)
so that a variation of gives rise to a variation B of B and F of F . Conversely, given such variations of B and F , Eq. (3) defines a corresponding variation of . In the language of O’Neill [8], but with the convention on the warping function as in [1], for f : B → (0, +∞) smooth, (B, gB ) Lorentzian (or ((a, b), −dt 2 )) and (F, gF ) ¯ is the product manifold M = B × F Riemannian, the warped product (M, g) ¯ = (B×f F, g) furnished with the Lorentz metric g¯ = ∗ (gB ) + (f ◦ )∗ (gF )
(4)
g(x, ¯ x) = gB (∗ x, ∗ x) + f (p)gF (∗ x, ∗ x)
(5)
or
for x ∈ T(p,q) M. As is common notation nowadays, we identify X ∈ (B) with (X, 0) = X¯ in TM and V ∈ (F ) with (0, V ) = V¯ in TM in the sequel. Also, we may write (4) more succinctly as in [1] as g¯ = gB ⊗ f g F .
(6)
With convention (6) employed, the basic Proposition 35 in [8, p. 206] translates as Proposition 1.1. For X, Y ∈ L(B), V , W ∈ L(F ) (1) (2) (3) (4)
DX Y ∈ L(B) is the lift of DX Y on B, DX V = DV X = (X(f )/2f )V , nor(DV W ) = − 21 gF (V , W )∇f , tan(DV W ) ∈ L(F ) is the lift of ∇V W on F.
Applying the techniques of Proposition 38 of [8, p. 208] and Proposition 1.1, we obtain the following formula for the acceleration vector of a smooth curve (t) in (M, g): ¯ (t) = ∇*/*t (f ◦ B ) (t) F (t) . = ∇*/*t B − 21 gF ( F (t), F (t))∇f, ∇*/*t F + (f ◦ B )(t)
(7)
Or if one prefers the sum notation, (t) = ∇*/*t B − 21 gF ( F (t), F (t))∇f + ∇*/*t F +
(f ◦ B ) (t) (t). (f ◦ B )(t) F
(8)
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Hence, the geodesic equations corresponding to Proposition 38 in [8, p. 208] ensures (t) is a smooth geodesic in (M, g) ¯ iff ⎧ 1 ⎨ ∇*/*t B = 2 gF (F (t), F (t))∇f, (9) (f ◦ B ) (t) ⎩ ∇*/*t F = − F (t) (f ◦ B )(t) and so the well-known result follows that F (t) is always a pregeodesic in (F, gF ) if (t) is a geodesic in (M, g). ¯ Employing again the proof idea of Proposition 38 of [8] and Proposition 1.1 above, the formula for the covariant derivative of a smooth vector field V = (VB , VF ) along the smooth curve (t) may be obtained V (t) = ∇*/*t V
= (VB (t) − 21 gF ( F , VF )∇f ) (t) gB (∇f, VB ) + VF (t) + B VF (t) + F (t) . 2f 2f
(10)
This may be recast in the direct sum language as V (t) = (VB (t), VF (t)) B (t) VB (f ) 1 + 2 −gF (F , VF )∇f, VF (t) + F (t) f f
(11)
which manifests how far the warped product covariant derivative formula is from the simple product case f = 1 for which V (t) = (VB (t), VF (t)).
2. Constructions of the index form With some effort, assuming that (t) is a time-like geodesic, the curvature formula g(R(V ¯ , ) , V ) of the index form may be calculated. However, given the rather complicated formula for this term as well as the somewhat complicated formula (11) for the covariant differentiation, the direct approach of calculating these terms directly in the general index form formula b I (X, Y ) = − [X , Y − R(X, ) , Y ] dt t=a
seems less helpful than deriving formula from a variational approach. Thus we shall not present the curvature formula for the warped product here. Now let : I → (M, g) ¯ be a unit time-like curve. Further, let : I × (−, ) → (M, g) ¯ be a variation of (t). Then from general results, for a compact interval I =[a, b], the curves (0, s) will be time like if is taken to be sufficiently small. As above, decompose s (t) := (t, s) = (B (t, s), F (t, s))
(12)
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and define corresponding variation vector fields *B * * * = , = , WB = B∗ *s *s *s *s *F * WF = F∗ = , *s *s
W = ∗
(13)
and V (t) = W (t, 0),
VB (t) = WB (t, 0),
VF (t) = WF (t, 0).
(14)
Also, *B *F * * s = B∗ , F∗ = , . *t *t *t *t Since the curves s (t) are time like, if *F *F *B *B h(t, s) = − gB , − (f ◦ B )(t, s)gF , *t *t *t *t * * = − g¯ , *t *t
(15)
(16)
and *F *F 1 * *B *B F (t, s) = gB ∇*/*s , , + (f ◦ B )(t, s)gF 2 *s *t *t *t *t *F *F + (f ◦ B )(t, s)gF ∇*/*s , , *t *t
(17)
then h(t, 0) = 1,
(18)
*h = −2F (t, s) *s
(19)
and L(s ) =
b t=a b
=
t=a
−g¯
* * , dt *t *t
(h(t, s))1/2 dt.
(20)
Thus
L (s ) = −
b t=a
(h(t, s))1/2 F (t, s) dt
(21)
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and one obtains the first variation formula (for smooth, unit time like) 1 L (0) = − g(V gB ¯ , ) + − gF (F , F )∇f, VB 2 t=a (f ◦ B ) (t) (t), VF dt + (f ◦ B )(t)gF ∇*/*t F + (f ◦ B )(t) F b = − g(V ¯ , ) + g( ¯ , V ) dt
b
∇*/*t B
t=a
(22)
consistent with Eqs. (8) and (9) above. Still assuming only that (t) is a unit time-like curve,
b
* [(h(t, s))−1/2 F (t, s)] dt t=a *s b −3/2 2 −1/2 *F = − dt. (h(t, s)) (F (t, s)) + (h(t, s)) *s t=a
L (s ) = −
Thus recalling (18),
L (0) = −
b t=a
*F (t, 0) dt. (F (t, 0)) + *s
2
(23)
Lemma 2.1. Assuming that (t) is a unit time-like geodesic in (M, g), ¯ then F (t, 0) =
d ¯ , ). [g(V ¯ , )] = g(V dt
Proof. Since ∗ [ **s , **t ] = 0, etc., one has
*B *B
*F *F
1 * , (f ◦ B )(t, 0)gF , F (t, 0) = gB ∇*/*s s=0 + 2 *s *t *t
*t *t s=0
*F *F
, + (f ◦ B )(t, 0)gF ∇*/*s *t *t s=0 = gB (VB , B ) + 21 gB (∇f, VB )gF ( F , F ) + (f ◦ B )(t)gF (VF , F ) d = [gB (VB , B )] − gB (VB , B ) + 21 gB (∇f, VB )gF ( F , F ) dt + (f ◦ B )(t)gF (VF , F ).
Substituting for B from formula (9), the second and third terms cancel so that d [gB (VB , B )] + (f ◦ B )(t)gF (VF , F ) dt d d = [gB (VB , B )] + (f ◦ B )(t) (gF (VF , F )) − gF (VF , F ) . dt dt
F (t, 0) =
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Now applying formula (9) for F we obtain F (t, 0) =
d d [gB (VB , B )] + (f ◦ B )(t) [gF (VF , F )] dt dt + (f ◦ B ) (t)gF (VF , F )
=
d [gB (VB , B ) + (f ◦ B )(t)gF (VF , F )] dt
=
d ¯ , ) + g(V ¯ , ) [g(V ¯ , )] = g(V dt
= g(V ¯ , ) since = 0 is assumed.
b Thus L (0) = − t=a [(g(V ¯ , ))2 + **Fs (t, 0)] dt. It remains to calculate (*F /*s)(t, 0). Since *F *F 1 * *B *B + , F (t, s) = gB ∇*/*s , (f ◦ B )(t, s)gF 2 *s *t *t *t *t *F *F + (f ◦ B )(t, s)gF ∇*/*s , , *t *t
commuting the differentiation, we have *F *B *B *B *B = gB ∇*/*s ∇*/*t , + gB ∇*/*t , ∇*/*s *s *s *t *s *t 2 *F *F 1 * , (f ◦ )(t, s)g + B F 2 *s 2 *t *t * *F *F + (f ◦ B )(t, s)gF ∇*/*s , *s *t *t * *F *F + (f ◦ B )(t, s)gF ∇*/*t , *s *s *t *F *F + (f ◦ B )(t, s)gF ∇*/*s ∇*/*t , *s *t *F *F + (f ◦ B )(t, s)gF ∇*/*s , ∇*/*s . *t *t
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Hence, commuting more derivatives,
*F
*B *B
, ∇ = g ∇ + gB (VB , VB ) B */*s */*t *t s=0 *s *s s=0 2
1 * (f ◦ B )(t, 0)gF ( F (t), F (t)) 2 *s 2 * +2 (f ◦ B )(t, 0)gF (VF , F ) *s
*F *F
, +(f ◦ B )(t)gF ∇*/*s ∇*/*t *s *t s=0 + (f ◦ B )(t)gF (VF , VF ) = gB (VB , VB ) − gB (R(VB , B ) B , VB )
*B *B
+ gB ∇*/*t ∇*/*s , + (f ◦ B )(t)gF (VF , VF ) *s *t
+
s=0
− (f ◦ B )(t)gF (R(VF , F ) F , VF )
*F *F
, + (f ◦ B )(t)gF ∇*/*t ∇*/*s *s *t s=0 2
+
1 * * (f ◦ B )(t, 0)gF (VF , F ). (f ◦ B )(t, 0)gF ( F , F ) + 2 2 *s 2 *s
Now again we bring the geodesic Eq. (9) to bear
*B *B
gB ∇*/*t ∇*/*s , *s *t s=0 d [gB (∇*/*s WB , B )] − gB (∇*/*s WB , ∇*/*t B ) = dt d = [gB (∇*/*s WB , B )] − gB (∇*/*s WB , 21 gF ( F , F )∇f ) dt d = [gB (∇*/*s WB , B )] − 21 gB (∇*/*s WB , ∇f )gF ( F , F ). dt Thus
2 *B *B
1 * gB ∇*/*t ∇*/*s , + (f ◦ B )(t, 0)gF ( F (t), F (t)) *s *t s=0 2 *s 2 d = WB , B )] [gB (∇ dt */*s
2 1 * + (f ◦ B )(t, 0) − (∇*/*s WB )(f ) gF ( F (t), F (t)) 2 *s 2 d = [gB (∇*/*s WB , B )] + 21 Hess(f )(VB , VB )gF ( F (t), F (t)), dt recalling Hess(f )(X, Y )=∇X ∇f, Y =X(∇f, Y )−∇f, ∇X Y =X(Y (f ))−(∇X Y )(f ).
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Applying the geodesic Eq. (9) also to the term (f ◦B )(t)gF (∇*/*t ∇*/*s *F /*s, *F /*t), we obtain
*F *F
, (f ◦ B )(t)gF ∇*/*t ∇*/*s *s *t s=0 d = (f ◦ B )(t) [gF (∇*/*s WF , F )] − (f ◦ B )(t)gF (∇*/*s WF , ∇*/*t F ) dt d = (f ◦ B )(t) [gF (∇*/*s WF , F )] dt −(f ◦ B ) (t) F − (f ◦ B )(t)gF ∇*/*s WF , (f ◦ B )(t) d = (f ◦ B )(t) [gF (∇*/*s WF , F )] + (f ◦ B ) (t)gF (∇*/*s WF , F ) dt d = [(f ◦ B )(t)gF (∇*/*s WF , F )]. dt Thus *F (t, 0) = gB (VB , VB ) − gB (R(VB , B ) B , VB ) *s + (f ◦ B )(t)[gF (VF , VF ) − gF (R(VF , F ) F , VF )] d + [gB (∇*/*s WB , B ) + (f ◦ B )(t)gF (∇*/*s WF , F )] dt + 21 Hess(f )(VB (t), VB (t))gF ( F (t), F (t)) * + 2 (f ◦ B )|s=0 gF (VF (t), F (t)). *s If desired, the last term may be rewritten as 2gB (∇f, VB )gF (VF (t), VF (t)) since (*B /*s) (t, 0) = VB (t) so that (*/*s)(f ◦ B )|s=0 = VB (t)(f ) = gB (∇f, VB (t)). The second variation formula may now be stated. Proposition 2.2. Let : [a, b] → (M, g)=(B× ¯ f F, gB ⊕f g F ) be a unit time-like geodesic and let : [a, b] × (−, ) → (M, g) ¯ be a smooth variation of (t) with variation vector field V = (VB , VF ) and W (t, s) = ∗ **s |(t,s) . Then
L (0) = −
b t=a
[(g(V ¯ , ))2 + (gB (VB , VB ) − gB (R(VB , B ) B , VB ))
+ (f ◦ B )(t)(gF (VF , VF ) − gF (R(VF , F ) F , VF ))
+ 21 Hess(f )(VB (t), VB (t))gF ( F (t), F (t)) + 2gB (∇f, VB (t))gF (VF (t), F (t))] dt
− [gB (∇*/*s WB , B (t)) + f ◦ B (t)gF (∇*/*s WF , F (t))]|bt=a . Remark 2.3. The second and fourth terms could be combined as g((V ¯ B , VF ), )
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with similar expressions for the two curvature terms and the last evaluation term. But g((V ¯ B , VF ), ) and the curvature term g((R(V ¯ B , B )B , R(VF , F )F ), (VB , VF ))
are so far removed from g(V ¯ , ) (cf. Eq. (11)) and the warped product curvature, that such a reworking of the formula in Proposition 2.2 does not seem warranted. Note that this result bears some resemblance (with = −1, c = 1 and the opposite sign convention on the curvature tensor) to what O’Neill [8, p. 266] terms “Synge’s formula for the second variation”. As is well known, in studying the second variation and index form, it suffices to consider vector fields perpendicular to the given geodesic (t). But since (t) is a geodesic, differentiating g(V ¯ , ) = 0 implies g(V ¯ , ) = 0 as well, and hence in this setup, the first term in the formula for L (0) in Proposition 2.2 vanishes. As in [2], let V ⊥ () denote the vector space of piecewise smooth vector fields V along with g(V ¯ , ) = 0 and let V0⊥ () = {V ∈ V ⊥ () | V (a) = V (b) = 0}. Then guided by the result of Proposition 2.2, the index form I : V0⊥ () × V0⊥ () → R should be given by I (V , V ) = − − −
b
t=a b t=a b t=a
[gB (VB , VB ) − gB (R(VB , B ) B , VB )] dt f ◦ B (t)[gF (VF , VF ) − gF (R(VF , F ) F , VF )] dt [ 21 Hess(f )(VB (t), VB (t))gF ( F (t), F (t))
+ 2gB (∇f, VB (t))gF (VF (t), F (t))] dt
(24)
and I (V , W ) could be obtained from (24) by polarization. All the terms in this formula seem satisfactory except for the term containing gF (VF (t), F (t)), where it would be desirable to get rid of the derivative of VF (t).
3. Comments on the GRW case Let us specialize the case that (M, g) ¯ = ((a, b)×f F, g¯ = −dt 2 ⊕ f g F ) and (t) = (I (t), F (t)) is a unit time-like geodesic for I = (a, b) and consider variation vector fields V with g(V ¯ , ) = 0. First for a “stationary” maximal time-like geodesic with I (t) = t − t0 and F (t) = q for a fixed q in F, it is easy to check that VI (t) = 0.
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Consider now a “non-stationary” unit time-like geodesic = (I , F ) : (, ) → M and to avoid ambiguity, write I (t) = (t) so we have the decompositions ⎧ * ⎪ ⎨ I = (t) ◦ I |t , *t (25) ⎪ ⎩ V = (t) * ◦ | , I I t *t where we let (t) = I (t) : (, ) → R. Regarding (t) as being prescribed, seek to find VF along F so that with VI as in (25), we have g(V ¯ , ) = 0. Since F (t) is a pregeodesic in (F, gF ) and −1=g( ¯ , )=−( (t))2 +f ◦I (t)gF ( F , F ), we may suppose that (t) > 1 and gF (F , F ) > 0. Let us try to find VF of the form VF (t) = (t) F (t),
(26)
where : (, ) → R is an unknown function. Writing out 0 = g(V ¯ , ) with (26), we obtain (t) (t) = (t)(f ◦ I )(t)gF ( F , F ) and we may solve for (t) as
(t) :=
(t) (t) (f ◦ I )(t)gF ( F (t), F (t))
since the expression in the denominator is positive. We conclude with several preliminary comments on the time-like index form (24) in the generalized Robertson–Walker case. Given the space limitations of these conference proceedings, a more detailed analysis will appear elsewhere. Recall first that for general space–times, since B is rarely a pregeodesic in B, that the interpretation of the first index style term in (24) is problematic. But in the generalized Robertson–Walker case, gI (R(VI , I ) I , VI ) = 0 so that the first integral term in (24) is semi-definite. By contrast, since F is always a pregeodesic in F, the second integral term in (24) may always be viewed as the index form of a pregeodesic.
References [1] J.K. Beem, P.E. Ehrlich, Global Lorentzian Geometry, Pure and Applied Mathematics, vol. 67, Marcel Dekker, New York, 1981. [2] J. K. Beem, P.E. Ehrlich, P.E. Easley, Global Lorentzian Geometry, second ed., Pure and Applied Mathematics, vol. 202, Marcel Dekker, New York, 1996. [3] J.K. Beem, P.E. Ehrlich, Th.G. Powell, in: Warped Product Manifolds in Relativity, North-Holland,Amsterdam, 1982, pp. 41–56. [4] J.K. Beem, Th.G. Powell, Geodesic completeness and maximality in Lorentzian warped products, Tensor N.S., 1982. [5] K.L. Easley, Local existence of warped product metrics, Doctoral Thesis, University of Missouri, Columbia, 1991.
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[6] P.E. Ehrlich, Yoon-Tae Jung, Seon-Bu Kim, Constant scalar curvatures on warped product manifolds, Tsukuba J. Math. 20 (1996) 239–256. [7] H.-J. Kim, Index form of the pregeodesics, Thesis for M.S., Chonnam National University, 2002. [8] B. O’Neill, in: Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, London, 1983, pp. 204–211. [9] T.G. Powell, Lorentzian manifolds with non-smooth metrics and warped products, Ph.D. Thesis, University of Missouri, Columbia, 1982.
The index form of a warped product Paul E. Ehrlicha , Seon-Bu Kimb,∗,1 a Department of Mathematics, The University of Florida, Gainesville, FL 32611, USA b Department of Mathematics, Chonnam National University, Kwangju 500-757, Republic of Korea
Abstract We construct the index form along time-like geodesics on a Lorentzian warped product manifold and apply this index form to generalized Robertson–Walker (GRW) space–times. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Index form of a timelike geodesic; Warped product; Generalized Robertson–Walker space–time
1. Preliminaries Since many space–times in relativity may be represented by Lorentzian warped product manifolds ([1–7,9]), a study of the time-like index form for this class of metrics is relevant. Because Hess(f 2 )(v, w)=2f Hess(f )(v, w)+2∇f, v∇f, w and calculations shows that the index form contains Hessian terms, we employ the formalism g¯ = gB ⊕ f g F , f : B → (0, +∞) for the warped product as originally treated in [3,1], rather than the currently more common formalism g¯ = gB ⊕ f 2 gF which has been more popular since the publication of O’Neill’s monograph [8]. Thus let (B, gB ) be a Lorentzian manifold (or interval ((a, b), −dt 2 )) and let (F, gF ) be a Riemannian manifold. For M = B × F, let : M → B and : M → F denote the projection maps. Then given any curve : I → M, we may set B = ◦ , F = ◦ and so obtain (t) = (B (t), F (t)). ∗ Corresponding author.
E-mail addresses: [email protected]fl.edu (P.E. Ehrlich), [email protected] (S.-B. Kim). 1 The second author was supported by Chonnam National University in Korea, 2004.
0362-546X/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.09.022
(1)
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Correspondingly, since T(b,f ) M = Tb B ⊕ Tf F , we may decompose V ∈ () as V = (VB , VF ).
(2)
If : I ×(−, ) → M is a variation of (t), then setting B =◦, F =◦ : I ×(−, ) → M, we obtain (t, s) = (B (t, s), F (t, s))
(3)
so that a variation of gives rise to a variation B of B and F of F . Conversely, given such variations of B and F , Eq. (3) defines a corresponding variation of . In the language of O’Neill [8], but with the convention on the warping function as in [1], for f : B → (0, +∞) smooth, (B, gB ) Lorentzian (or ((a, b), −dt 2 )) and (F, gF ) ¯ is the product manifold M = B × F Riemannian, the warped product (M, g) ¯ = (B×f F, g) furnished with the Lorentz metric g¯ = ∗ (gB ) + (f ◦ )∗ (gF )
(4)
g(x, ¯ x) = gB (∗ x, ∗ x) + f (p)gF (∗ x, ∗ x)
(5)
or
for x ∈ T(p,q) M. As is common notation nowadays, we identify X ∈ (B) with (X, 0) = X¯ in TM and V ∈ (F ) with (0, V ) = V¯ in TM in the sequel. Also, we may write (4) more succinctly as in [1] as g¯ = gB ⊗ f g F .
(6)
With convention (6) employed, the basic Proposition 35 in [8, p. 206] translates as Proposition 1.1. For X, Y ∈ L(B), V , W ∈ L(F ) (1) (2) (3) (4)
DX Y ∈ L(B) is the lift of DX Y on B, DX V = DV X = (X(f )/2f )V , nor(DV W ) = − 21 gF (V , W )∇f , tan(DV W ) ∈ L(F ) is the lift of ∇V W on F.
Applying the techniques of Proposition 38 of [8, p. 208] and Proposition 1.1, we obtain the following formula for the acceleration vector of a smooth curve (t) in (M, g): ¯ (t) = ∇*/*t (f ◦ B ) (t) F (t) . = ∇*/*t B − 21 gF ( F (t), F (t))∇f, ∇*/*t F + (f ◦ B )(t)
(7)
Or if one prefers the sum notation, (t) = ∇*/*t B − 21 gF ( F (t), F (t))∇f + ∇*/*t F +
(f ◦ B ) (t) (t). (f ◦ B )(t) F
(8)
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Hence, the geodesic equations corresponding to Proposition 38 in [8, p. 208] ensures (t) is a smooth geodesic in (M, g) ¯ iff ⎧ 1 ⎨ ∇*/*t B = 2 gF (F (t), F (t))∇f, (9) (f ◦ B ) (t) ⎩ ∇*/*t F = − F (t) (f ◦ B )(t) and so the well-known result follows that F (t) is always a pregeodesic in (F, gF ) if (t) is a geodesic in (M, g). ¯ Employing again the proof idea of Proposition 38 of [8] and Proposition 1.1 above, the formula for the covariant derivative of a smooth vector field V = (VB , VF ) along the smooth curve (t) may be obtained V (t) = ∇*/*t V
= (VB (t) − 21 gF ( F , VF )∇f ) (t) gB (∇f, VB ) + VF (t) + B VF (t) + F (t) . 2f 2f
(10)
This may be recast in the direct sum language as V (t) = (VB (t), VF (t)) B (t) VB (f ) 1 + 2 −gF (F , VF )∇f, VF (t) + F (t) f f
(11)
which manifests how far the warped product covariant derivative formula is from the simple product case f = 1 for which V (t) = (VB (t), VF (t)).
2. Constructions of the index form With some effort, assuming that (t) is a time-like geodesic, the curvature formula g(R(V ¯ , ) , V ) of the index form may be calculated. However, given the rather complicated formula for this term as well as the somewhat complicated formula (11) for the covariant differentiation, the direct approach of calculating these terms directly in the general index form formula b I (X, Y ) = − [X , Y − R(X, ) , Y ] dt t=a
seems less helpful than deriving formula from a variational approach. Thus we shall not present the curvature formula for the warped product here. Now let : I → (M, g) ¯ be a unit time-like curve. Further, let : I × (−, ) → (M, g) ¯ be a variation of (t). Then from general results, for a compact interval I =[a, b], the curves (0, s) will be time like if is taken to be sufficiently small. As above, decompose s (t) := (t, s) = (B (t, s), F (t, s))
(12)
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and define corresponding variation vector fields *B * * * = , = , WB = B∗ *s *s *s *s *F * WF = F∗ = , *s *s
W = ∗
(13)
and V (t) = W (t, 0),
VB (t) = WB (t, 0),
VF (t) = WF (t, 0).
(14)
Also, *B *F * * s = B∗ , F∗ = , . *t *t *t *t Since the curves s (t) are time like, if *F *F *B *B h(t, s) = − gB , − (f ◦ B )(t, s)gF , *t *t *t *t * * = − g¯ , *t *t
(15)
(16)
and *F *F 1 * *B *B F (t, s) = gB ∇*/*s , , + (f ◦ B )(t, s)gF 2 *s *t *t *t *t *F *F + (f ◦ B )(t, s)gF ∇*/*s , , *t *t
(17)
then h(t, 0) = 1,
(18)
*h = −2F (t, s) *s
(19)
and L(s ) =
b t=a b
=
t=a
−g¯
* * , dt *t *t
(h(t, s))1/2 dt.
(20)
Thus
L (s ) = −
b t=a
(h(t, s))1/2 F (t, s) dt
(21)
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and one obtains the first variation formula (for smooth, unit time like) 1 L (0) = − g(V gB ¯ , ) + − gF (F , F )∇f, VB 2 t=a (f ◦ B ) (t) (t), VF dt + (f ◦ B )(t)gF ∇*/*t F + (f ◦ B )(t) F b = − g(V ¯ , ) + g( ¯ , V ) dt
b
∇*/*t B
t=a
(22)
consistent with Eqs. (8) and (9) above. Still assuming only that (t) is a unit time-like curve,
b
* [(h(t, s))−1/2 F (t, s)] dt t=a *s b −3/2 2 −1/2 *F = − dt. (h(t, s)) (F (t, s)) + (h(t, s)) *s t=a
L (s ) = −
Thus recalling (18),
L (0) = −
b t=a
*F (t, 0) dt. (F (t, 0)) + *s
2
(23)
Lemma 2.1. Assuming that (t) is a unit time-like geodesic in (M, g), ¯ then F (t, 0) =
d ¯ , ). [g(V ¯ , )] = g(V dt
Proof. Since ∗ [ **s , **t ] = 0, etc., one has
*B *B
*F *F
1 * , (f ◦ B )(t, 0)gF , F (t, 0) = gB ∇*/*s s=0 + 2 *s *t *t
*t *t s=0
*F *F
, + (f ◦ B )(t, 0)gF ∇*/*s *t *t s=0 = gB (VB , B ) + 21 gB (∇f, VB )gF ( F , F ) + (f ◦ B )(t)gF (VF , F ) d = [gB (VB , B )] − gB (VB , B ) + 21 gB (∇f, VB )gF ( F , F ) dt + (f ◦ B )(t)gF (VF , F ).
Substituting for B from formula (9), the second and third terms cancel so that d [gB (VB , B )] + (f ◦ B )(t)gF (VF , F ) dt d d = [gB (VB , B )] + (f ◦ B )(t) (gF (VF , F )) − gF (VF , F ) . dt dt
F (t, 0) =
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Now applying formula (9) for F we obtain F (t, 0) =
d d [gB (VB , B )] + (f ◦ B )(t) [gF (VF , F )] dt dt + (f ◦ B ) (t)gF (VF , F )
=
d [gB (VB , B ) + (f ◦ B )(t)gF (VF , F )] dt
=
d ¯ , ) + g(V ¯ , ) [g(V ¯ , )] = g(V dt
= g(V ¯ , ) since = 0 is assumed.
b Thus L (0) = − t=a [(g(V ¯ , ))2 + **Fs (t, 0)] dt. It remains to calculate (*F /*s)(t, 0). Since *F *F 1 * *B *B + , F (t, s) = gB ∇*/*s , (f ◦ B )(t, s)gF 2 *s *t *t *t *t *F *F + (f ◦ B )(t, s)gF ∇*/*s , , *t *t
commuting the differentiation, we have *F *B *B *B *B = gB ∇*/*s ∇*/*t , + gB ∇*/*t , ∇*/*s *s *s *t *s *t 2 *F *F 1 * , (f ◦ )(t, s)g + B F 2 *s 2 *t *t * *F *F + (f ◦ B )(t, s)gF ∇*/*s , *s *t *t * *F *F + (f ◦ B )(t, s)gF ∇*/*t , *s *s *t *F *F + (f ◦ B )(t, s)gF ∇*/*s ∇*/*t , *s *t *F *F + (f ◦ B )(t, s)gF ∇*/*s , ∇*/*s . *t *t
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Hence, commuting more derivatives,
*F
*B *B
, ∇ = g ∇ + gB (VB , VB ) B */*s */*t *t s=0 *s *s s=0 2
1 * (f ◦ B )(t, 0)gF ( F (t), F (t)) 2 *s 2 * +2 (f ◦ B )(t, 0)gF (VF , F ) *s
*F *F
, +(f ◦ B )(t)gF ∇*/*s ∇*/*t *s *t s=0 + (f ◦ B )(t)gF (VF , VF ) = gB (VB , VB ) − gB (R(VB , B ) B , VB )
*B *B
+ gB ∇*/*t ∇*/*s , + (f ◦ B )(t)gF (VF , VF ) *s *t
+
s=0
− (f ◦ B )(t)gF (R(VF , F ) F , VF )
*F *F
, + (f ◦ B )(t)gF ∇*/*t ∇*/*s *s *t s=0 2
+
1 * * (f ◦ B )(t, 0)gF (VF , F ). (f ◦ B )(t, 0)gF ( F , F ) + 2 2 *s 2 *s
Now again we bring the geodesic Eq. (9) to bear
*B *B
gB ∇*/*t ∇*/*s , *s *t s=0 d [gB (∇*/*s WB , B )] − gB (∇*/*s WB , ∇*/*t B ) = dt d = [gB (∇*/*s WB , B )] − gB (∇*/*s WB , 21 gF ( F , F )∇f ) dt d = [gB (∇*/*s WB , B )] − 21 gB (∇*/*s WB , ∇f )gF ( F , F ). dt Thus
2 *B *B
1 * gB ∇*/*t ∇*/*s , + (f ◦ B )(t, 0)gF ( F (t), F (t)) *s *t s=0 2 *s 2 d = WB , B )] [gB (∇ dt */*s
2 1 * + (f ◦ B )(t, 0) − (∇*/*s WB )(f ) gF ( F (t), F (t)) 2 *s 2 d = [gB (∇*/*s WB , B )] + 21 Hess(f )(VB , VB )gF ( F (t), F (t)), dt recalling Hess(f )(X, Y )=∇X ∇f, Y =X(∇f, Y )−∇f, ∇X Y =X(Y (f ))−(∇X Y )(f ).
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Applying the geodesic Eq. (9) also to the term (f ◦B )(t)gF (∇*/*t ∇*/*s *F /*s, *F /*t), we obtain
*F *F
, (f ◦ B )(t)gF ∇*/*t ∇*/*s *s *t s=0 d = (f ◦ B )(t) [gF (∇*/*s WF , F )] − (f ◦ B )(t)gF (∇*/*s WF , ∇*/*t F ) dt d = (f ◦ B )(t) [gF (∇*/*s WF , F )] dt −(f ◦ B ) (t) F − (f ◦ B )(t)gF ∇*/*s WF , (f ◦ B )(t) d = (f ◦ B )(t) [gF (∇*/*s WF , F )] + (f ◦ B ) (t)gF (∇*/*s WF , F ) dt d = [(f ◦ B )(t)gF (∇*/*s WF , F )]. dt Thus *F (t, 0) = gB (VB , VB ) − gB (R(VB , B ) B , VB ) *s + (f ◦ B )(t)[gF (VF , VF ) − gF (R(VF , F ) F , VF )] d + [gB (∇*/*s WB , B ) + (f ◦ B )(t)gF (∇*/*s WF , F )] dt + 21 Hess(f )(VB (t), VB (t))gF ( F (t), F (t)) * + 2 (f ◦ B )|s=0 gF (VF (t), F (t)). *s If desired, the last term may be rewritten as 2gB (∇f, VB )gF (VF (t), VF (t)) since (*B /*s) (t, 0) = VB (t) so that (*/*s)(f ◦ B )|s=0 = VB (t)(f ) = gB (∇f, VB (t)). The second variation formula may now be stated. Proposition 2.2. Let : [a, b] → (M, g)=(B× ¯ f F, gB ⊕f g F ) be a unit time-like geodesic and let : [a, b] × (−, ) → (M, g) ¯ be a smooth variation of (t) with variation vector field V = (VB , VF ) and W (t, s) = ∗ **s |(t,s) . Then
L (0) = −
b t=a
[(g(V ¯ , ))2 + (gB (VB , VB ) − gB (R(VB , B ) B , VB ))
+ (f ◦ B )(t)(gF (VF , VF ) − gF (R(VF , F ) F , VF ))
+ 21 Hess(f )(VB (t), VB (t))gF ( F (t), F (t)) + 2gB (∇f, VB (t))gF (VF (t), F (t))] dt
− [gB (∇*/*s WB , B (t)) + f ◦ B (t)gF (∇*/*s WF , F (t))]|bt=a . Remark 2.3. The second and fourth terms could be combined as g((V ¯ B , VF ), )
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with similar expressions for the two curvature terms and the last evaluation term. But g((V ¯ B , VF ), ) and the curvature term g((R(V ¯ B , B )B , R(VF , F )F ), (VB , VF ))
are so far removed from g(V ¯ , ) (cf. Eq. (11)) and the warped product curvature, that such a reworking of the formula in Proposition 2.2 does not seem warranted. Note that this result bears some resemblance (with = −1, c = 1 and the opposite sign convention on the curvature tensor) to what O’Neill [8, p. 266] terms “Synge’s formula for the second variation”. As is well known, in studying the second variation and index form, it suffices to consider vector fields perpendicular to the given geodesic (t). But since (t) is a geodesic, differentiating g(V ¯ , ) = 0 implies g(V ¯ , ) = 0 as well, and hence in this setup, the first term in the formula for L (0) in Proposition 2.2 vanishes. As in [2], let V ⊥ () denote the vector space of piecewise smooth vector fields V along with g(V ¯ , ) = 0 and let V0⊥ () = {V ∈ V ⊥ () | V (a) = V (b) = 0}. Then guided by the result of Proposition 2.2, the index form I : V0⊥ () × V0⊥ () → R should be given by I (V , V ) = − − −
b
t=a b t=a b t=a
[gB (VB , VB ) − gB (R(VB , B ) B , VB )] dt f ◦ B (t)[gF (VF , VF ) − gF (R(VF , F ) F , VF )] dt [ 21 Hess(f )(VB (t), VB (t))gF ( F (t), F (t))
+ 2gB (∇f, VB (t))gF (VF (t), F (t))] dt
(24)
and I (V , W ) could be obtained from (24) by polarization. All the terms in this formula seem satisfactory except for the term containing gF (VF (t), F (t)), where it would be desirable to get rid of the derivative of VF (t).
3. Comments on the GRW case Let us specialize the case that (M, g) ¯ = ((a, b)×f F, g¯ = −dt 2 ⊕ f g F ) and (t) = (I (t), F (t)) is a unit time-like geodesic for I = (a, b) and consider variation vector fields V with g(V ¯ , ) = 0. First for a “stationary” maximal time-like geodesic with I (t) = t − t0 and F (t) = q for a fixed q in F, it is easy to check that VI (t) = 0.
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Consider now a “non-stationary” unit time-like geodesic = (I , F ) : (, ) → M and to avoid ambiguity, write I (t) = (t) so we have the decompositions ⎧ * ⎪ ⎨ I = (t) ◦ I |t , *t (25) ⎪ ⎩ V = (t) * ◦ | , I I t *t where we let (t) = I (t) : (, ) → R. Regarding (t) as being prescribed, seek to find VF along F so that with VI as in (25), we have g(V ¯ , ) = 0. Since F (t) is a pregeodesic in (F, gF ) and −1=g( ¯ , )=−( (t))2 +f ◦I (t)gF ( F , F ), we may suppose that (t) > 1 and gF (F , F ) > 0. Let us try to find VF of the form VF (t) = (t) F (t),
(26)
where : (, ) → R is an unknown function. Writing out 0 = g(V ¯ , ) with (26), we obtain (t) (t) = (t)(f ◦ I )(t)gF ( F , F ) and we may solve for (t) as
(t) :=
(t) (t) (f ◦ I )(t)gF ( F (t), F (t))
since the expression in the denominator is positive. We conclude with several preliminary comments on the time-like index form (24) in the generalized Robertson–Walker case. Given the space limitations of these conference proceedings, a more detailed analysis will appear elsewhere. Recall first that for general space–times, since B is rarely a pregeodesic in B, that the interpretation of the first index style term in (24) is problematic. But in the generalized Robertson–Walker case, gI (R(VI , I ) I , VI ) = 0 so that the first integral term in (24) is semi-definite. By contrast, since F is always a pregeodesic in F, the second integral term in (24) may always be viewed as the index form of a pregeodesic.
References [1] J.K. Beem, P.E. Ehrlich, Global Lorentzian Geometry, Pure and Applied Mathematics, vol. 67, Marcel Dekker, New York, 1981. [2] J. K. Beem, P.E. Ehrlich, P.E. Easley, Global Lorentzian Geometry, second ed., Pure and Applied Mathematics, vol. 202, Marcel Dekker, New York, 1996. [3] J.K. Beem, P.E. Ehrlich, Th.G. Powell, in: Warped Product Manifolds in Relativity, North-Holland,Amsterdam, 1982, pp. 41–56. [4] J.K. Beem, Th.G. Powell, Geodesic completeness and maximality in Lorentzian warped products, Tensor N.S., 1982. [5] K.L. Easley, Local existence of warped product metrics, Doctoral Thesis, University of Missouri, Columbia, 1991.
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[6] P.E. Ehrlich, Yoon-Tae Jung, Seon-Bu Kim, Constant scalar curvatures on warped product manifolds, Tsukuba J. Math. 20 (1996) 239–256. [7] H.-J. Kim, Index form of the pregeodesics, Thesis for M.S., Chonnam National University, 2002. [8] B. O’Neill, in: Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, London, 1983, pp. 204–211. [9] T.G. Powell, Lorentzian manifolds with non-smooth metrics and warped products, Ph.D. Thesis, University of Missouri, Columbia, 1982.