Suppose now there is some subdivision tQ < <x < <2 < ••• < such that the tangent vector 8/dt is continuous on each segment [ij, i1+1]. If a segment <i+1] is not a null geodesic curve, it can be varied to give a timelike curve between its endpoints. Thus one has only to show that one can obtain a timelike curve from a non-spacelike curve y(t) made up of null geodesic segments whose tangent vectors are not parallel at points of discontinuity y(<i). The parameter t can be taken to be an affine parameter on each segment [<f, <1+1]. The discontinuity [d/dt]|t. will be a spacelike vector, as it is the difference between two non-parallel null vectors in the same half of the null cone. Thus one can find a C2 vector field W along [if_x, t1+1] such that g(W, 8/8t) < 0 on ['<-i> and g(W, 8/8t) > 0 on <i+1]. Then a timelike curve between •y{if_i) and y(ti+1) will be obtained from the variation with variation vector field Z =xW, where x = c-1{i1+1 — tt) (i — tw) for <i_1 < t < tt, and a; = c"1^-^) (i1+1 -1) for t{ < t < <1+1, wherec = -g(W,8/8t). □ Thus if y{t) is not a geodesic curve, it can be varied to give a timelike curve. If it is a geodesic curve, the parameter t may be taken to be an affine parameter. One then sees that a necessary, but not sufficient, condition for a variation to yield a timelike curve is that the variation vector 8/8u should be orthogonal to the tangent vector 8/8t everywhere on y(t), since otherwise {8/8t) g(8l8u, 8j8t) would be positive somewhere on y(t). For such a variation the first derivative {8/8u)g(dl8t, 8j8t) will be zero and so one will have to examine the second derivative.

We shall therefore consider a two-parameter variation a of a null geodesic y(t) from q to p. The variation a will be defined as before except that, for the reason given above, we shall restrict ourselves to variations whose variation vectors

^¡htzl and are orthogonal to the tangent vector 8/8t on y(t).

It is not convenient to study the behaviour of L under such a variation since (-g(dldt,dldt))l is not differentiable when g(d/a<,a/di) = 0. Instead we shall consider the variation in:

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