However, due to Eq. (C8) we have
R(FN)ABCD = R(THZ)ABCD cI (Cil)
R(FN)ABCD,E R(THZ)ABCD, EL (C12)
The THZ coordinates describe the external multiple moments of a vacuum
solution of the Einstein equations as we can see from the metric in Eqs. (4-49)-
(4-51). Then, using these equations, we can also express the Riemann tensors
and their derivatives in the THZ coordinates in terms of these external multiple
moments. At the location of the particle they turn out to be
R(THZ)OIOJ0 = Si, (C13)
R(THZ)OIOJ, K SIJK + CJKPI + CIKP (C4)
R(THZ) IJKO o = JPBPK (C15)
4 cp (I3KL +- JIKSJL), (6
R(THZ) IJKO, L o P KL + KIL KL(C )
R(THZ) IJKL o = IK8JL + 6jLSIK 6IL8JK 6JK8IL, (C17)
R(THZ)IJKL, M J6IK rSJLM + 7 (cLMPB J + J.MIP L)
+~JL rSIKM + 3 (KMP I + CIMPI3 K)
-6IL SJKM + 3 (KIMPS J + .JMPIP K
-6J SLM + 3 QLMP + IMP L) (C18)
where I, J, K, M, P 1, 2, 3. These are all the consequences of the solution of
the vacuum Einstein equations with de Donder gauge conditions. The general
algorithm for finding the solution is discussed in the Appendix of Ref. ([20]) and
Ref. ([21]).