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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 (4.4) and can be written as a rotation (Lorentz
transformation) in spacetime:
\[ \frac{{D{\operatorname{e}
_a}}}{{d\lambda }} = {\mathbf{u}} \cdot D{\operatorname{e} _a} =
{\mathbf{\Omega }} \cdot {\operatorname{e} _a}{\quad {\mathbf{\Omega} }} : =
{u^\mu }{{\mathbf{\omega }}_\mu } = {{\mathbf{\omega }}_0 } \]
The rotation bivector ${\mathbf{\Omega }}$ has the Ricci rotation
coefficients (4.19) 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.