Consider the action of the particle along the orbit in a flat space parametrized with D rectilinear, Cartesian coordinates:
It may be transformed to curvilinear coordinates , via some functions
is the induced metric for the curvilinear coordinates. Repeated indices are understood to be summed over, as usual. For Cartesian coordinates, upper and lower indices i are the same. The indices of the curvilinear coordinates, on the other hand, are lowered by contraction with the metric or raised with the inverse metric .
The length of the orbit in the flat space is given by
Both the action (3) and the length (5) are invariant under arbitrary reparametrizations of space .
Einstein's equivalence principle amounts to the postulate that the transformed action (3) describes directly the motion of the particle in the presence of a gravitational field caused by other masses. The forces caused by the field are all a result of the geometric properties of the metric tensor.
The equations of motion are obtained by extremizing the action in Eq. (3) with the result
is the Riemann connection or Christoffel symbol of the first kind. Defining also the Christoffel symbol of the second kind
we can write
The solutions of these equations are the classical orbits. They coincide with the extrema of the length l of a curve in (5). Thus, in a curved space, classical orbits are the shortest curves, called geodesics. The reason for the name shortest lines is that they minimize the invariant length (5) of all lines connecting two given points and .
The same equations can also be obtained directly by transforming the equation of motion from
to curvilinear coordinates , which gives
At this place it is useful to employ the so-called basis triads
and the reciprocal basis triads
which satisfy the orthogonality and completeness relations
The induced metric can then be written as
Using the basis triads, Eq. (11) can be rewritten as
The subscript separated by a comma denotes the partial derivative , i.e., . The quantity in front of is called the affine connection:
Due to (14), it can also be written as
Thus we arrive at the transformed flat-space equation of motion
The solutions of this equation are called the straightest lines or autoparallels.
If the coordinate transformation is smooth and single-valued, it is integrable. This property is expressed by Schwarz's integrability condition, according to which derivatives in front of such a function commute:
Then the triads satisfy the identity
implying that the connection is symmetric in the lower indices. In this case it coincides with the Riemann connection, the Christoffel symbol . This follows immediately after inserting into (7) and working out all derivatives using (23). Thus, for a space with curvilinear coordinates which can be reached by an integrable coordinate transformation from a flat space, the autoparallels coincide with the geodesics.