S 9ZSds SgZ[V

where V = DV/Ss is the acceleration. From this one sees again that a necessary condition for y(t) to be the longest curve from q to p is that it should be an unbroken geodesic curve as otherwise one could choose a variation which would yield a longer curve.

One may also consider a timelike curve y(t) from a spacelike three-surface to a point p. A variation a of this curve is defined as before except that condition (3) is replaced by:

(3) a(u, 0) lies on Jf, a(u, tp) = p. Thus at Jf the variation vector Z = 8/du lies in Jif.

Lemma 4.5.5 8L 8u

=V r^rftf,Z)da+Zg(Z,\V])+g(Z,V)|,.0. 0 < = ljti ¿-2

Prom this one sees that a necessary condition for y(i) to be the longest curve from JF to p is that it is an unbroken geodesic curve orthogonal to ¿if.

We have seen that, under a variation a, the first derivative of the length of a timelike geodesic curve is zero. To proceed further we shall calculate the second derivative. We define a two-parameter variation a of a geodesic curve y(i) from q to p as a C1 map:

such that

(2) there is a subdivision 0 = ^ < t2 < ... < tn = tp of [0,tp] such that a is C3 on each

(4) for all constant ult u2, a(ult w2, t) is a timelike curve. We define

as the two variation vectors. Conversely given two continuous, piece-wise C2 vector fields Zj and Z2 along y(i) one may define a variation for which they will be the variation vectors, by:

Lemma 4.5.6

Under the two-parameter variation of the geodesic curve y(i), the second derivative of the length will be:


£-0 = Z1 j^'g (zx, (g (Z2+g(V, Z2) V) - R(V, Z2) vj) ds + sV(zi,g(Z2+g(V,Z2)V)]).

8u1 Therefore 8*L

du2 du1

The first and third terms vanish as y(t) is an unbroken geodesic curve. In the second term one can write:

8ut 8t 8t ~ \8t' 8uJ 8t + 8t8u28t = -r(1 AU + - —

\8t'duj8t 8t* 8u2


In the fourth term:

Then taking t to be the arc-length s, one obtains the required result. □

Although it is not immediately obvious from the appearance of the expression, one knows from its definition that it is symmetric in the two variation vector fields Zx and Z2. One sees that it only depends on the projections of Zy and Z2 into the space orthogonal to V. Thus we can confine our attention to variations a whose variation vectors are orthogonal to V. We shall define Ty to be the (infinite-dimensional) vector space consisting of all continuous, piecewise C2 vector fields along y(t) orthogonal to V and vanishing at q and p. Then 82L/8u28u1

will be a symmetric map of Ty x Ty to R1. One may think of it as a symmetric tensor on Ty and write it as:

8u2 8u1

One may also calculate the second derivative of the length from Jif to p of a geodesic curve y(i) normal to 34?. One proceeds as before except that one endpoint of y(t) is allowed to vary over Jf instead of being fixed.

Lemma 4.5.7

The second derivative of the length of y(i) from JF to p is: 82L


where Zx and Zg have been taken orthogonal to V and x(Zlt Z2) is the second fundamental tensor of .

The first two terms are as for lemma 4.5.6. The extra terms are:

The second term vanishes as is orthogonal to 8/8t. If one takes < to be the arc-length s, then 8j8t will be equal to the unit normal N at 3/?. Since the endpoint of y(i) is restricted to varying over 8\8ux will always be orthogonal to N. Thus

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