- SPACE-TIME POINT PICKING WITH RESTRICTIONS 61
be the maximal time interval such that for any t E ( t$, t*],(18.46) dh(t) (w,p) < dh(t*) (y*,p) + ~A^1 /^2 R-^112 (y*, t*),
where either
Case 1. t$ = t* - DR-^1 (y*, t*) or
Case 2. t$ > t*-DR-^1 (y*, t*) and equality in (18.46) holds when t = t$.
(Clearly there is such a t$.)
If Case 1 is true, then since by (18.43), s E [t* - DR-^1 (y*, t*), t* J, we
have
w E Bh(s) (p, dh(t*) (y*,p) + A^1 /^2 R-^1 l^2 (y*, t*))
and we obtain claim (18.45).
If Case 2 is true, then we shall obtain a contradiction. Since at time t*
we have
dh(t*) (w,p) :::; dh(t*) (w, y*) + dh(t*) (y*,p)< d h(t) ( y,p ) + 10 l Al/2 R-1/2( y, t ) '
there exists a smallest time t# E ( t$, t*) such that
(18.47) dh(t) (w,p) > dh(t) (y,p) + tA^1 l^2 R-^1!^2 (y, t) fort E [t$, t#),
(18.48) dh(t#) (w,p) = dh(t) (y,p) + tA1/2 R-1/2(y, t).
For any t E [t$, t#], to estimate the change of the distance dh(t) (w,p)
using the changing distances inequality (Theorem 18.7(2)), we consider the
balls Bh(t) (w,ro) and Bh(t) (p,ro), for some
(18.49) r 0 < - _!_A 10 112 R-^112 (y ' t )
to be chosen later. By (18.47), we have 2ro:::; dh(t) (w,p) and by (18.46) and
(18.49) we have
(18.50)
Bh(t) (w,ro) U Bh(t) (p,ro) c Bh(t) (p,dh(t) (y,p) +A^1 l^2 R-^112 (y, t)).
Now we verify the curvature assumption for z E Bh(t) ( w, ro) UBh(t) (p, ro) in
Theorem 18.7(2). If (z, t) tJ. NB,c, then, using R(y, t) > C + B(t* - to)-^1 ,
we have
R (z, t) :::; C + B (t - to)-^1 :::; C + B (t$ - t 0 )-^1
:::; c + B (t - DR-^1 (y, t*) - tor
1
:::; C + 2B (t - to)-^1 :::; 2R(y, t*).
On the other hand, if (z, t) E NB,c, then by (18.50) and Lemma 18.24 we
get