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V. Parallel Transport
If a geodesic curve is timelike, it is a possible world line
for a particle. Its uniformly ticking parameter $\lambda = a\tau +
b$ is a multiple of the particle's proper time defined by
normalizing the tangent vector ${\mathbf{v}}: = d{\mathbf{x}}/d\tau
$, ${\mathbf{v}} \cdot {\mathbf{v}} = 1$. In a spacetime with
Lorentzian metric the timelike/null/spacelike character of the
geodesic (relative to a metric-compatible connection) never changes
and timelike geodesics are maxima of the proper time.
In the absence of gravity, the equation of motion of a free
particle is given by:
\[
\frac{{d{\mathbf{v}}}}{{d\tau }} = \frac{{\partial {\mathbf{v}}}}
{{\partial {x^\mu }}}\frac{{d{x^\mu }}}{{d\tau }} = {v^\mu
}{\partial _\mu }{\mathbf{v}} = 0 \]
In the presence of gravity this becomes an equation that
nullifies the proper acceleration
\[ {\mathbf{a}}: =
\frac{{D{\mathbf{v}}}}{{d\tau }}: = {v^\mu }{D_\mu }{\mathbf{v}} =
{\mathbf{v}} \cdot D{\mathbf{v}} = 0 \]
That is, in GR a ‘freely falling’ particle follows a geodesic which
is a curve that parallel-transports its tangent vector
${\mathbf{v}} = d{\mathbf{x}}/d\tau $ along itself, preserving the
norm ${\mathbf{v}} \cdot {\mathbf{v}} = 1$.
To calculate the geodesic, one may use the component formalism, in
particular equation (5.6). Equation (5.12) then gives the geodesic
equation describing a (point) particle in free fall
\[ \frac{{d{v^\mu
}}}{{d\tau }} + \Gamma _{\nu \kappa }^\mu {v^\nu }{v^\kappa } =
0 \]
It is a second-order differential system in the unknowns ${x^\mu
}(\tau )$ involving coordinate rates of a change of the metric
tensor describing gravity. The equation can be solved (in principle)
when initial data have been specified.