9 - Curved Spacetime

Summary

So far we have concentrated on a region of spacetime where gravitational effects may be neglected. Such a region could be the interior of a spaceship hurtling toward the earth over a period of a few seconds, or some vast region of interstellar space. The basic idea is that gravitational tidal effects may be made arbitrarily small by restricting attention to a sufficiently small region of spacetime. This idea is known as the principle of equivalence.

We shall now impose no such restriction on the size of our region and consider the geometry of spacetime as a whole, taking into account gravitational tidal effects. This means that we no longer have a physically defined affine structure applicable to the whole of spacetime, and hence no notion of parallel spacetime displacements. We do, however, retain the notion of spacetime points, world lines, null rays, and null cones. Using these physical notions we shall in the next few chapters consider the physical geometry of spacetime in the presence of gravity.

Spacetime as a Manifold

At its most basic level, spacetime is no more than a set, M, whose points represent the spacetime positions of physical events. A real-valued function f on M assigns a number f (p) to each point p of M. A curve on M may be represented by a one-to-one mapping c : I → M, where I is either an interval (open curve) or a circle (closed curve), which gives a point, c(t)∈M for each t∈I. Given a function f and a curve c, we have a function fc : I → ℝ, given by f_c(t) = f (c(t)).