via Eqs. (B23)-(B25) together with (C1) and (C2). Then, the metric for the THZ

coordinates takes the form

 o00 o o K L 1 o vK vL vM
 g(THZ) -+ R(FN)0 KL oXFNXFN + 3 (FN)0K L,M oFNFN FN

 +O(X4N/74), (C4)
 2 1
 g(THZ) 3 R(FN)0K L XFNAFN R(FN)0K L, Mo FN FNF
 ,3 *11
 + KFNPFN 2KLXKFNX FN N + O(X N/F4), (C5)
 iJ 61 J 1 L 1 K L
 g(THZ) + 3 R(FN) K L o KNAX N + R(FN) K L,M oKFN FN FN
 1lJ 2 2( KvX J) 2 Si vK vL
 SPFN + 0IKXFNFN -CL FN FN
 3 3 3
 1,I U K 2 5 ,(1 IKIILIXvJ)
 SK NPFN + 1 S KLXEFN NXFN

 4 (1 19J)PXK 2 8 (I K|P| lIKIyILIyJ)
 PKB FNPFN PK LX FNXFNFN

 +O(XN /R4). (C6)

 It is clear from Eqs. (C1) and (C2) that the inverse transformation is identity

at the linear order,

 XAN XA + O(X3/2). (C7)

Then, our transformation factors, which are equivalent to the set of local tetrad

vectors in the THZ coordinates, are simply


 o(B) (C8)

The Riemann tensors and their derivatives evaluated at the location of the particle

follow covariant transformation,

 R(FN)PQRS o R(THZ)ABCD oA C D (R)S(C9)

 R(FN) PQRS,U o R(THZ)ABCD,E o ) oQ) (R)V OS) OU), (C10)