perturbations of g(FN)AB derived from these equations. From Ref. [24] we have
A'B' ]AB8X^FN B'FN
9(FN)-N IN (B7)
OXA OXB
Combining this with Eqs. (B1)-(B4) and (4-90) one finds
00 AB OTFN aTFN
g(FN) O9 A xB
-1 [-HABCD + 2 (ABCD + ACD B CD )] ioAoBCD
SAB D + 2 (AB CDE + C DE + CDB E + C'DE
XUoAoB X X' + O(X4/R4), (B8)
or AB aTFN a9XN
9(FN) XA xB
[ABCD 2 (ABCD + ACB + ADB)] (A ) XCXD
+ ACDE- +2 (CDE + ( C DE+ CD BE + B CDE )
xni( O)t C DXE" + o(X4/-4), (B9)
IJ N AB+ X FN /FX N
9(FN) O kA 0F
I : + ABCD + 2 ACD ACBD CD+ I) A Ct D
+ [tAB CDE + 2 ((AB CDE + AACB DE + ( CD BE + A CDEB B
Xh h(J) _CDE + 0(14/R4). (10)
To view the new geometry properly, one should express its metric perturbations in
terms of the coordinates of the new geometry itself. Thus, we take the inverse of
XA via Eqs. (Bl) and (B2) and specify it to the first order
A ~TFN (I)X N O(XN FN2), (B11)
which is essentially the same expression as Eq. (4-104) considering that the
quadratic-order terms are absent from the expansions as a result of the integration