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VII. Fermi-Walker Transport
Given a
geodesic curve and a non-spinning tetrad frame defined by (7.1), any vector
${\mathbf{Y}}$ can be expanded as ${\mathbf{Y}}(\lambda ) = {Y^a}(\lambda
){\operatorname{e} _a}(\lambda )$ in terms of the basis vectors of that frame.
Such local coordinates adapted to a geodesic are called Fermi normal
coordinates. These coordinates are the natural coordinate system of an
observer in general relativistic spacetimes.
The
F-derivative of any vector function ${\mathbf{Y}}(\lambda )$ along a
curve ${\mathbf{x}}(\lambda )$ is defined as the vector function given by the
rate of change relative to the geodesic frame:
\[
\frac{{{d_{\text{F}}}{\mathbf{Y}}}}{{d\lambda }}: = \frac{{d{Y^a}}}{{d\lambda
}}{\operatorname{e} _a} \]
In other words, the F-derivative of ${\mathbf{Y}}(\lambda )$ is evaluated
with respect to the family of Lorentz frames, whereby the basis vectors
$\left\{ {{\operatorname{e} _a}({\mathbf{x}})} \right\}$, which are parallel
transported according to (7.1), are supposed to be constant, as if
they represented an inertial Lorentz frame.
The
F-derivative (7.2) is related to the coderivative (5.2) by the formula
\[
\frac{{{d_{\text{F}}}{\mathbf{Y}}}}{{d\lambda }} =
\frac{{D{\mathbf{Y}}}}{{d\lambda }} - {Y^a}\frac{{D{\operatorname{e}
_a}}}{{d\lambda }} = \frac{{D{\mathbf{Y}}}}{{d\lambda }} \]
In other words, using these coordinates, all connection coefficients $\Gamma
$ vanish exactly on the geodesic curve ${\mathbf{x}}(\lambda )$. If
the F-derivative of some vector field vanishes, that is, its components
${Y^a}$ are constant, this vector it said to experience Fermi transport
or to be Fermi-transported along the world line of the observer.
Fermi-transport is a special instance of Fermi-Walker transport defined next.