Let be a timelike curve described by parametric relations in which is proper time. Let be the curve’s normalized tangent vector, and let be its acceleration vector.
A vector field is said to be Fermi–Walker transported on if it is a solution to the differential equation
The operation of Fermi–Walker (FW) transport satisfies two important properties. The first is that is automatically FW transported along ; this follows at once from Eq. (9.1) and the fact that is orthogonal to . The second is that if the vectors and are both FW transported along , then their inner product is constant on : ; this also follows immediately from Eq. (9.1).
Let be an arbitrary reference point on . At this point we erect an orthonormal tetrad where, as a modification to former usage, the frame index runs from 1 to 3. We then propagate each frame vector on by FW transport; this guarantees that the tetrad remains orthonormal everywhere on . At a generic point we have
To construct the Fermi normal coordinates (FNC) of a point in the normal convex neighbourhood of we locate the unique spacelike geodesic that passes through and intersects orthogonally. We denote the intersection point by , with denoting the value of the proper-time parameter at this point. To tensors at we assign indices , , and so on. The FNC of are defined by
Suppose that is moved to . This typically induces a change in the spacelike geodesic , which moves to , and a corresponding change in the intersection point , which moves to , with . The FNC of the new point are then and , with determined by . Expanding these relations to first order in the displacements, and simplifying using Eqs. (9.2), yields
The relations of Eq. (9.7) can be expressed as expansions in powers of , the spatial distance from to . For this we use the expansions of Eqs. (6.7) and (6.8), in which we substitute and , where is a dual tetrad at obtained by parallel transport of on the spacelike geodesic . After some algebra we obtain
where are frame components of the acceleration vector, and are frame components of the Riemann tensor evaluated on . This last result is easily inverted to give
Proceeding similarly for the other relations of Eq. (9.7), we obtain
As a special case of Eqs. (9.8) and (9.9) we find thatcomponents of the acceleration vector in FNC; orthogonality between and means that . Similarly, , , and are the components of the Riemann tensor (evaluated on ) in Fermi normal coordinates.
Inversion of Eqs. (9.8) and (9.9) gives
Notice that on , the metric of Eqs. (9.14) – (9.16) reduces to and . On the other hand, the nonvanishing Christoffel symbols (on ) are ; these are zero (and the FNC enforce local flatness on the entire curve) when is a geodesic.
The form of the metric can be simplified when the Ricci tensor vanishes on the world line:Thorne–Hartle–Zhang (THZ) coordinates brings the metric to the form  and the coordinate transformation was later produced by Zhang .
The key to the simplification comes from Eq. (9.17), which dramatically reduces the number of independent components of the Riemann tensor. In particular, Eq. (9.17) implies that the frame components of the Riemann tensor are completely determined by , which in this special case is a symmetric-tracefree tensor. To prove this we invoke the completeness relations of Eq. (9.4) and take frame components of Eq. (9.17). This produces the three independent equations
the last of which stating that has a vanishing trace. Taking the trace of the first equation gives , and this implies that has five independent components. Since this is also the number of independent components of , we see that the first equation can be inverted – can be expressed in terms of . A complete listing of the relevant relations is , , , , , and . These are summarized by
We may also note that the relation , together with the usual symmetries of the Riemann tensor, imply that too possesses five independent components. These may thus be related to another symmetric-tracefree tensor . We take the independent components to be , , , , and , and it is easy to see that all other components can be expressed in terms of these. For example, , , , and . These relations are summarized by
Substitution of Eq. (9.21) into Eq. (9.16) gives
and we have not yet achieved the simple form of Eq. (9.20). The missing step is the transformation from the FNC to the THZ coordinates . This is given by
It follows that , which is just the same statement as in Eq. (9.20).
Alternative expressions for the components of the THZ metric are
Living Rev. Relativity 14, (2011), 7
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