4. ASYMPTOTICS OF THE HEAT KERNEL FOR A STATIC METRIC 253
since J'VrJ^2 = 1. On the other hand, the space derivative of (23.115) holds
in the sense that fro(r (x)^4 ) = o(r (x)^3 )' so that
since oxP or = xP r • Hence the Laplacian of the distance squared function has
the expansion
LEMMA 23.31.
(23.118)
.6.x (r^2 ) (x) = 2n - ~Rpq (y) xPxq - ~'VrRpq (y) xPxqxr + 0 (r (x)^4 ).
We compute the expansion for (.6.x</>o) (x,y) = .6.x (a-^112 ) (x,y) for x
near y. For any function f we have
.6.f =^1 >::i o • ( ydetgke l..^1 ~ of)
v det 9k£ uxi uxJ- -^1 a ( iJ .. of)
- a oxi ag oxj '
We now compute the RHS of (23.119). Taking the derivative of the
formula for a^112 which is analogous to (23.116), we have
(23.120)
(^0) ( al/2) 1 1 ( )
oxj = -6Rjq (y) xq - 24 ('VjRpq + 2\7pRjq) (y) xPxq + O r (x)^3.