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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.