nents of linear momentum. The P can be interpreted as the flow of aft angular momentum.

If the metric is not flat there will not, in general, be any Killing vectors and so the above integral conservation laws will not hold. However, in a suitable neighbourhood of a point q one may introduce normal coordinates {¡c0}. Then at q the components gab of the metric are eaSab (no summation), and the components T0^ of the connection are zero. One may take a neighbourhood 3t of q in which the gab and P*^ differ from their values at q by an arbitrarily small amount; then the L(a. b) and M(a. b) will not exactly vanish in but will in this neigh-

a ' aft bourhood differ from zero by an arbitrarily small amount. Thus f Pbdab and f P»d<rb

J 33 a J will still be zero in the first approximation; that is to say, one still has approximate conservation of energy, momentum and angular momentum in a small region of space-time. Using this it can be shown that a small isolated body moves approximately on a timelike geodesic curve independent of its internal constitution provided that the energy density of matter in it is non-negative (for an account of the motion of a small body in relativity, see Dixon (1970)). This may be thought of as Galileo's principle that all bodies fall equally fast. In Newtonian terms one would say that the inertial mass (the m in F = ma) and the passive gravitational mass (the mass acted on by a gravitational field) are equal for all bodies. This has been verified to a high order of accuracy in experiments by Eotvos and by Dicke (1964).

Postulate (a) enables one to measure the metric up to a conformal factor at each point. Using postulate (6) one may relate these factors at different points, for the conservation equations Tab. b = 0 would not in general hold for a connection derived from a metric g = ijsg. One way of doing this would be to observe the paths of small 'test' particles and so to determine the timelike geodesic curves. Then if y(t) is such a curve with tangent vector K = {8/dt)y, one has from (2.29)

Was this article helpful?

0 0

Post a comment