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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.