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 that
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:
In such circumstances, a transformation from the Fermi normal coordinates 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
http://www.livingreviews.org/lrr-2011-7 |
Living Rev. Relativity 14, (2011), 7
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