180
GAUGE FIELDS AND STRINGS
limit it is a local, invariant expression depending on the metric tensor
If the metric were Euclidean then
(L " Leo), =
- V \V ,eo, + V,co, - h^.Vco,)
(ftab ~ à ab)
and
(9.127)
dV _
{2ny
2nt
(9.128)
where the coefficient 2 comes from the number of the components of a> .
As i -► 0 the external gravitational field does not have time to influence
(9.128). As we see from this formula the characteristic intervals for the
motion of a “particle” described by the wave equation (9.127) are
A(^ ~ \/p y/t which is just a diffusion law. The curvature R(^) will
have a considerable effect only if :
R(0(A0" - tR(è) - 1
(9.129)
This consideration makes plausible the statement that for i -► 0 the
expansion parameter in
is t • R. We expect that:
^ {1 +
+ 0(ñ}
IO = ^ {1 + a^tRiO + 0(t^)}
(9.130)
Later we shall derive these formulas explicitly and compute
2 - Now,
substituting (9.130) into (9.124) we get
iVo(L)-iVo(LO = c.z
where c = 2(a^ — ü2 ) is still to be determined and
(9.131)
(9.132)
is the Euler character of our maniford. Here, in order to find c, it is
sufficient to consider a sphere as an example in which Nq{L) and
Nq{L'^) can be explicitly found. Using formulas of Riemannian geometry one has for g^b =
r/fc =
- d.cpòj
(9.133)