### 3.2 Fermi normal coordinates

#### 3.2.1 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 if 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 Equation (112) 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 Equation (112).

#### 3.2.2 Tetrad and dual tetrad on

Let be an arbitrary reference point on . At this point we erect an orthonormal tetrad where, contrary 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 is also FW transported on . The tetrad and its dual give rise to the completeness relations

#### 3.2.3 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 Equation (115) 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.

Figure 6: | Fermi normal coordinates of a point relative to a world line . The time coordinate selects a particular point on the word line, and the disk represents the set of spacelike geodesics that intersect orthogonally at . The unit vector selects a particular geodesic among this set, and the spatial distance selects a particular point on this geodesic.

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 Equations (113), yields
where is determined by .

#### 3.2.4 Coordinate displacements near

The relations of Equation (118) can be expressed as expansions in powers of , the spatial distance from to . For this we use the expansions of Equations (88) and (89), 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 Equation (118), 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 Equations (119) and (120) 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 the FNC; orthogonality between and means that . Similarly, , , and are the components of the Riemann tensor (evaluated on ) in the Fermi normal coordinates.

#### 3.2.5 Metric near

Inversion of Equations (119) and (120) 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

with

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 Equations (125, 126, 127) 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.

#### 3.2.6 Thorne-Hartle coordinates

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

In such circumstances, a transformation from the Fermi normal coordinates to the Thorne-Hartle 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 [58] and the coordinate transformation was later produced by Zhang [64].

The key to the simplification comes from Equation (128), which dramatically reduces the number of independent components of the Riemann tensor. In particular, Equation (128) 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 Equation (115) and take frame components of Equation (128). This produces the three independent equations

the last of which states 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 Equation (132) into Equation (127) gives

and we have not yet achieved the simple form of Equation (131). The missing step is the transformation from the FNC to the Thorne-Hartle coordinates . This is given by

It is easy to see that this transformation affects neither 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 Equation (131).

Alternative expressions for the components of the Thorne-Hartle metric are