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VII. Fermi-Walker Transport
The construction of Fermi coordinates may be generalized to observer
frames moving along arbitrary worldlines. The characterization of a
set of tetrad fields $\left\{ {{{\text{e}}_a}({\mathbf{x}})}
\right\}$ as an accelerated observer frame is given by its velocity
and acceleration along the path ${\mathbf{x}}(\lambda )$ of
the observer:
\[ {\mathbf{u}}: =
{\operatorname{e} _0} {\quad{\mathbf{\dot u}}}: = {\mathbf{u}} \cdot
D{\mathbf{u}} = \frac{{D{\operatorname{e} _0}}}{{d\lambda
}}{\quad}{\mathbf{u}} \cdot {\mathbf{\dot u}} = 0 \]
The motion of the observer frame, relative to parallel transport, is
described by the spin connection bivector (4.5) and can be written
as a rotation (Lorentz transformation) in spacetime:
\[
\frac{{D{\operatorname{e} _a}}}{{d\lambda }} = {\mathbf{u}}
\cdot D{\operatorname{e} _a} = {\mathbf{\Omega }} \cdot
{\operatorname{e} _a}{\quad {\mathbf{\Omega} }} : = {u^\mu
}{{\bs{\omega }}_\mu } = {{\bs{\omega }}_0 } \]
The rotation bivector ${\mathbf{\Omega }}$ has the Ricci
rotation coefficients (4.20) as components:
\[
\begin{gathered}{\mathbf{\Omega }} = \frac{1}{2}{\omega
_{0ab}} \,{\operatorname{e} ^a} \wedge {\operatorname{e} ^b}
{\quad}{\mathbf{\Omega }} \cdot {\operatorname{e} _b} = {\omega
_0}{^c}{_b}\,{\operatorname{e} _c} \\{{\mathbf{\Omega }}_{ab}} =
{\mathbf{\Omega}} _{[ab]} : = {\operatorname{e} _a} \cdot
{\mathbf{\Omega }} \cdot {\operatorname{e} _b} = {\omega _0}_{ab}
\end{gathered} \]
Being anti-symmetric, the rotation bivector has six independent
components. This number agrees with the number of components
in a Lorentz transformation (three parameters for rotations, plus
three parameters for boosts). The components
${\mathbf{\Omega}}_{0i}$ may be identified with the acceleration,
and the spatial components ${\mathbf{\Omega}}_{ij}$ with the frequency
of rotation of the local spatial frame, as measured with
respect to the local non-spinning observer frame.