We now describe briefly about the
concept of parallel transport and the geodesics.
Let us imagine a curve parameterized
by $\lambda$,
Consider the change of a vector $\vec{V}$ along the curve;
we then have the following, 
This is variously known
as intrinsic, absolute or the total
derivative. Note that we have made use of the chain rule in the above
expression.
Then
we can now come to a generalization of the concept of parallel transport. Imagine a vector $\vec{V}$ makes
a differential displacement along the curve. We then have equation$(1)$. Now if
the vector is displaced by an infinitesimal amount we can say that vectors
remain parallel. Or in other words “If a vector $\vec{V}$ is “parallel transported” along a line then,
GEODESIC:
The concept of parallel transport
can be used to extend the idea of “straight” lines to curved spaces.
We say that a curve is a straight
line in a particular coordinate system if a vector parallel transports its own
tangent vector. In other words, straight lines in curved spaces are defined by
setting, $V^{\alpha }=du_{\beta }/{dx}$.
We
then have as follows,
This is then equation of the
geodesics. It is interesting to note here that the equation of the geodesics is
of the second order differential equation, similar to those of that of the
Newton’s force law or the equation of the motion. It won’t be surprising to
find that this is in fact the general equation of motion in curved spacetime.
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