10 - Curvature and Gravity

Summary

We come now to the physical interpretation of the curvature tensor R_abcd. As we have seen, R_abcd arises from the spacetime metric, which in turn arises from the properties of inertial world lines and null rays. If there are no gravitational tidal effects, then spacetime is flat and hence R_abcd = 0. Conversely, if R_abcd = 0, then, apart from the possibility of topologies other than ℝ⁴, spacetime is flat and there are no gravitational tidal effects. A nonvanishing curvature tensor is thus an indication and a measure of gravity. In this chapter we shall show that a gravitational field is completely described by R_abcd. But first we need the concept of a geodesic.

Geodesics

The world line of an inertial particle has the remarkable property of being uniquely determined by its initial four-velocity vector. If we take proper time to be the parameter along the world line, then its tangent vector at any point will be a four-velocity vector. Similarly, a null ray is uniquely determined by specifying a null vector at some point, O say, and the set of all null rays through O forms a null cone N(O). We recall that it was precisely these properties of inertial world lines and null rays that allowed us to define a unique spacetime metric. Furthermore, the metric led to a unique connection and, finally, to the curvature tensor.