and these quantities are essential for transforming the initial coordinates XA into
Fermi normal coordinates and then finally into THZ coordinates through Steps (ii)
and (iii).
Step (ii).
To construct Fermi normal coordinates out of the initial coordinates XA,
first we evolve the particle's motion along F from the initial point XA = 0,
which corresponds to x' in the Schwarzschild coordinates. Since F is a geodesic
of the particle's motion, its tangent vector iA, the four-velocity of the particle is
transported parallel to itself along F
BVB A = 0. (4-97)
We call F the timelike geodesic, and the time-axis of an observer's frame that is co-
moving with the particle is tangent to this geodesic. While an observer is traveling
with the particle along F, his space triad remains orthogonal to F-parallel
transport preserves orthogonality to F [25], i.e.
UBVBB ) = 0, (4-98)
where iA (I = 1, 2, 3) are basis vectors for the space triad, spanning the hyper-
surface orthogonal to F. Along each direction of ,the physical measurement
made by the observer should not be affected by where it is made, thus each of
n A should be transported parallel to itself. At the same time, each of A) should
alv--,- remain orthogonal to the others Aj) (J $ I). Then, altogether
(J) VB() 0, (4-99)
which gives three spacelike geodesics, F(I) (I = 1,2, 3).
The set of vectors { A, if), fA(), nf (} above form an orthonormal basis for
the co-moving observer's frame. Now having this basis we may construct a family