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 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
).
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
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 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
As a special case of Equations (119) and (120
) we find that
Inversion of Equations (119) and (120
) gives
with
This is the metric nearNotice 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.
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 coordinatesThe 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
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 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 follows that , which is just the same statement as in
Equation (131
).
Alternative expressions for the components of the Thorne-Hartle metric are