110 Theory of Relativity- The tensor Rij[g] — {l/2)R[g)gij, which is the left-hand side of (2.4.22),
is called Einstein's tensor.
Even though equations (2.4.22) are complicated, let us assume that
we were able to solve them and found the metric tensor g. In classical
Newtonian mechanics, the ultimate goal is to compute the trajectory of a
moving object under the action of forces. The equation of the trajectory is
provided by the Second Law of Newton. In general relativity, the Second
Law of Newton is replaced by a postulate that a point mass in a curved
space moves along a geodesic. A geodesic in a curved space is what a
straight line is in a flat space. Once you think about it, a straight line in
a fiat space has two special properties: (a) it defines the path of shortest
distance between two points; (b) it is straight in the sense that the tangent
vectors at each point are parallel, all being parallel to the direction vector
of the line. Accordingly, each of these properties could be used to define
a geodesic in a curved space, and it turns out that the second property
results in a more convenient definition. By this definition, it is shown in
differential geometry that x% = xl{s), s > 0, i = 1,...,4, is the vector
representation of a geodesic if and only if
where the r*-fc are from (2.4.27) on page 107. Below, we outline the proof
that a curve defined by (2.4.30) is a path of shortest distance between two
points.
Therefore, a trajectory of a moving object in a gravitational field in
general relativity is computed as follows:
(a) Find the metric tensor g by solving the field equations (2.4.22) for the
specified stress-energy tensor Ty;
(b) Compute the functions Tljk according to (2.4.27);
(c) Solve the system of equations (2.4.30).This summarizes the general relativity mechanics.
Let us now look more closely at equations (2.4.30). Without any gravi-
tation, the space is flat so that each fly is independent of x, which makes
each Tljk — 0. Then the geodesic equations (2.4.30) become d^2 xl(s)/ds^2 = 0
so that xl (s) — xl(0) + vxs, which is a straight line, corresponding to the
motion with constant velocity.