The physical significance of curvature

In this chapter we consider the effect of space-time curvature on families of timelike and null curves. These could represent flow lines of fluids or the histories of photons. In §4.1 and §4.2 we derive the formulae for the rate of change of vorticity, shear and expansion of such families of curves; the equation for the rate of change of expansion (Raychaudhuri's equation) plays a central role in the proofs of the singularity theorems of chapter 8. In § 4.3 we discuss the general inequalities on the energy-momentum tensor which imply that the gravitational effect of matter is always to tend to cause convergence of timelike and of null curves. A consequence of these energy conditions is, as is seen in §4.4, that conjugate or focal points will occur in families of non-rotating timelike or null geodesies in general space-times. In §4.5 it is shown that the existence of conjugate points implies the existence of variations of curves between two points which take a null geodesic into a timelike curve, or a timelike geodesic into a longer timelike curve.

4.1 Timelike curves

In chapter 3 we saw that if the metric was static there was a relation between the magnitude of the timelike Killing vector and the Newtonian potential. One was able to tell whether a body was in a gravitational field by whether, if released from rest, it would accelerate with respect to the static frame defined by the Killing vector. However, in general, space-time will not have any Killing vectors. Thus one will not have any special frame against which to measure acceleration; the best one can do is to take two bodies close together and measure their' relative acceleration. This will enable one to measure the gradient of the gravitational field. If one thinks of the metric as being analogous to the Newtonian potential, the gradient of the Newtonian field would correspond to the second derivatives of the metric. These are described by the Riemann tensor. Thus one would expect that the relative

acceleration of two neighbouring bodies would be related to some components of the Riemann tensor.

In order to investigate this relation more precisely we shall examine the behaviour of a congruence of timelike curves with timelike unit tangent vector V V) = — 1). These curves could represent the histories of small test particles, in which case they would be geodesies, or they might represent the flow lines of a fluid. If this were a perfect where = Va. b Vb is the acceleration of the flow lines and hab = VaVb is the tensor which projects a vector XeTq into its component in the subspace Hq of Tq orthogonal to V. One may also think of hab as the metric in Hq (cf. §2.7).

Suppose X(t) is a curve with tangent vector Z = (d/St)^ Then one may construct a family X(t, s) of curves by moving each point of the curve A(£) a distance s along the integral curves of V. If one now defines Z as (djdt)AftfS) it follows from the definition of the Lie derivative (see § 2.4) that LfZ = 0 or in other words that

One may interpret Z as representing the separation of points equal distances from some arbitrary initial points along two neighbouring curves. If one adds a multiple of V to Z then this vector will represent the separation of points on the same two curves but at different distances along the curves. It is really only the separation of neighbouring curves that one is interested in, not the separation of particular points on these curves. One is thus concerned only with Z modulo a component parallel to V, i.e. only with the projection of Z at each point q into the space Qq consisting of equivalence classes of vectors which differ only by addition of a multiple of V. This space can be represented as the subspace Hq of Tq consisting of vectors orthogonal to V. The projection of Z into Hq will be denoted by Zfl = hab Zb. In the case of a fluid one can regard JL as the distance between two neighbouring particles of the fluid as measured in their rest frame.

From (4.2) it follows that fluid, then by (3.10) ^ +p) fa =

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