The transformation via Eqs. (4-113) and (4-114) reproduces the desired
geometry of Fermi normal coordinates. We examine this in Appendix B.
Step (iii).
Out of any I .i.. ,l Inertial Cartesian coordinates, we can develop the three
geodesic equations as described by Eqs. (4-97)-(4-99). Thus the coordinates XA in
Step (ii) may be replaced by another locally inertial coordinates XA to make initial
coordinates. Then by combining the integral solutions of the geodesic equations
(4-97)-(4-99) we now construct a new family of geodesics XA(s, A1, A2, A3)
XA(, ', A2, 3) ds ds us + I dA' I1 ds ( 0
I a9s 1-1 M as ) X_
3 1 2 3 A I) \
+ dA dA a (4-119)
I 1 (M) J=1 ()
where s becomes the time coordinate TFN and A' (I = 1,2, 3) become the spatial
coordinates X1N when the equation is inverted.
each vector of the tetrad. These vectors constitute a local orthonormal frame at
each point along the timelike world-line F of a particle, in which the timelike vec-
tor uA, the four-velocity of the particle is tangent to F and gives the direction for
the time-axis, and the spacelike vectors n() (I 1,2, 3) serve as the basis vec-
tors for the space triad. Then it follows that (i) gABV p)V = rpQ and that (ii)
gABV (P) rl Q and thQ)ii
IPQV Ap V)B AB. Splitting the tetrad VA into uA and n), the relation (ii)
can be rewritten as -uA B + I n1AI) (I)B gAB, which proves the identity in
Eq. (4-118).