(^292) 7. THE REDUCED DISTANCE
above approach of renormalizing the Riemannian length functional, is as
follows. We define the space-time graph
ry: [Ti,T2] -+N X [Ti,T2]
of the path / by ;y (T) ~ (r (T), T), so that ~2 (T) = ( ~; (T), 1). Note
that the parameter T, of which / is a function, also serves as time; so it is
natural to consider its graph. Define the space-time metric h ~ h + RdT^2.
In general, this metric is indefinite since R may be negative somewhere. We
easily compute
1
T2 Id- 12
c (r) = Tl VT d; ( T) h dT.
Using(}= 2VT, we may rewrite the £-length as
where (Ji~ 2y'Ti, i = 1, 2. That is, C (r) is the energy of the space-time path
;y with respect to the space-time metric h and the new time parameter(}.
If a: [T1, T2] --+ N and {3: b, T3] --+ N are paths with a (T2) = {3 (T2),
then we define the concatenated path a'---' {3: [T1, T3] --+ N by
We have the following additivity property.LEMMA 7.7 (Additivity of the £-length).(7.20) C (a'---' {3) =·C (a)+ C ({3).
However, the £-length of a path I is not invariant under reparametriza-tions of f. We leave it to the reader to make the easy verification of this
fact.
The following bound on C is elementary.
LEMMA 7.8 (Lower bound for the £-length).(7.21) C (r) ~ ~ (T~/^2 - Tf 12 ) inf R.
3 Nx[Ti,T2]1
T2
This follows directly from C ( /) ~ VT Rinf ( T) dT, where Rinf ( T) ~
Tl
infNx{ 7 } R. The Riemannian counterpart of estimate (7.21) is the obvious
fact that the length of a path is nonnegative.