## 9 Fermi normal coordinates

### 9.1 Fermi–Walker transport

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

Notice that this reduces to parallel transport when and is a geodesic.

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

From the tetrad on we define a dual tetrad by the relations
this also is FW transported on . The tetrad and its dual give rise to the completeness relations

### 9.3 Fermi normal coordinates

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

the last statement determines from the requirement that , the vector tangent to at , be orthogonal to , the vector tangent to . From the definition of the FNC and the completeness relations of Eq. (9.4) it follows that
so that is the spatial distance between and along the geodesic . This statement gives an immediate meaning to , the spatial Fermi normal coordinates, and the time coordinate is simply proper time at the intersection point . The situation is illustrated in Figure 6.

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

where is determined by .

### 9.4 Coordinate displacements near

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

and
where and are additional frame components of the Riemann tensor evaluated on . (Note that frame indices are raised with .)

As a special case of Eqs. (9.8) and (9.9) we find that

because in the limit , the dual tetrad at coincides with the dual tetrad at . It follows that on , the transformation matrix between the original coordinates and the FNC is formed by the Fermi–Walker transported tetrad:
This implies that the frame components of the acceleration vector, , are also the components 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.

### 9.5 Metric near

Inversion of Eqs. (9.8) and (9.9) gives

and
These relations, when specialized to the FNC, give the components of the dual tetrad at . They can also be used to compute the metric at , after invoking the completeness relations . This gives
This is the metric near in the Fermi normal coordinates. Recall that are the components of the acceleration vector of – the timelike curve described by – while , , and are the components of the Riemann tensor evaluated on .

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.

### 9.6 Thorne–Hartle–Zhang coordinates

The form of the metric can be simplified when the Ricci tensor vanishes on the world line:

In such circumstances, a transformation from the Fermi normal coordinates to the Thorne–Hartle–Zhang (THZ) coordinates brings the metric to the form
We see that the transformation leaves and unchanged, but that it diagonalizes . This metric was first displayed in Ref. [174] and the coordinate transformation was later produced by Zhang [187].

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

and satisfies .

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

where is the three-dimensional permutation symbol. The inverse relation is .

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 is easy to see that this transformation does not affect nor at orders and . The remaining components of the metric, however, transform according to , where

It follows that , which is just the same statement as in Eq. (9.20).

Alternative expressions for the components of the THZ metric are