APPENDIX C
THE TRANSFORMATION BETWEEN FERMI NORMAL AND THZ NORMAL
GEOMETRIES
In Step (iii) of Subsection 4.3.3 the transformation between Fermi normal and
THZ normal coordinates was found to be
T = TFN t KLXFNFNPFN + O(XNR ), (C)
168
1X1 X K 12 t K XL XI
XI XN IKX NPN + -oKLXFNXFNXFN
6 3
1 1
K FNPFNTFN + 7-KLXFNX NXFNTFN
6 3
1 1
S.S LXK XL 2 -2XK xL xM x1
4 KL FN FNPFN 1 -KLM NXFNFNXFNXFN
24 12
2 1 L ,,(c xL 2 + V5 /R4).2)
-63 MKX LFN FN FN + (XN ). (C2)
One can verify that Eqs. (C1) and (C2) properly convert the metric of the Fermi
normal geometry into that of the THZ normal geometry with the help of some
properties of the Riemann tensors for vacuum spacetime.
Following Ref. [24], we examine
A'B' AB O9 A' OXB' (3)
9(THZ) (FN) axA axB
^FN FN