We shall say that a timelike geodesic curve y(i) from g to p is maximal if L(Z1, Zs) is negative semi-definite. In other words, if y(t) is not maximal there is a small variation a which yields a longer curve from p to g. Similarly we shall say that a timelike geodesic curve from Jf to p normal to is maximal if L(Zlf Z2) is negative semi-definite, so if y(i) is not maximal there is a small variation which yields a longer curve from to p.
A timelike geodesic curve y(t) from g to p is maximal if and only if there is no point conjugate to g along y(t) in (q, p).
Suppose there is no conjugate point in (q,p). Then introduce a Fermi-propagated orthonorma! basis along y(t). The Jacobi fields along y(t) which vanish at q will be represented by a matrix Aa/}(t) which will be non-singular in (q,p), but which will be singular at q and possibly at p. Since conjugate points are isolated, d(log det A)/ds will be infinite where Aafi is singular. Thus a C°, piecewise C2 vector field Z eTy can be expressed in [q,p\ as
Z« = AafiWf, where Vff is C°, piecewise C1 on [q,p\. Then,
(We take the limit because the second derivative of W' may not be defined at q.) But
Conversely, suppose there is a point re(q,p) conjugate to q along y(t). Let W be the Jacobi field along y which vanishes at q and r. Let K e Ty be such that
Extend W to p by putting it zero in [r,p]. Let Z be eK + e-1W, where e is some constant. Then
U Z, Z) = e2£(K, K) + 2L(K, W) + 2e-2L(W, W) = e2£( K, K) + 2.
Thus by taking e small enough, L(Z, Z) may be made positive. □
One may obtain similar results for the case of a timelike geodesic curve y(t) orthogonal to Jf, from Jf to p.
A timelike geodesic curve y(t) from ¿F to p is maximal if and only if there is no point in (Jf, q) conjugate to along y. □
We shall also consider variations of a non-spacelike curve y(t) from q to p. We shall be interested in the circumstances under which it is possible to find a variation a of y(t) which makes g(8/8t, 8/dt) negative everywhere, or in other words, yields a timelike curve from q to p. Under a variation a:
In order to obtain a timelike curve from q top, one requires this to be less than or equal to zero everywhere on y(t).
lip and q are joined by a non-spacelike curve y(t) which is not a null geodesic they can also be joined by a timelike curve.
If y(t) is not a null geodesic curve from ptoq, there must be some point at which the tangent vector is discontinuous, or there must be some open interval on which the acceleration vector (D/dt) (8/dt) is non-zero and not parallel to 8/dt. Consider first the case where there are no discontinuities. One has
This shows that (Df8t) (8/8t) is a spacelike vector where it is non-zero and not parallel to 8/81. Let W be a C2 timelike vector field along y(t)
such that g(W, 8/8t) < 0. Then one will obtain a timelike curve from p to q under the variation whose variation vector is with a; = c^e6 f e~6(l - &/a2)dt,
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