Condition (i) expresses the principle that all fields have energy. One might possibly object to the' only if' on the grounds that there might be two non-zero fields, one of whose energy-momentum tensor exactly cancelled that of the other. This possibility is related to that of the existence of negative energy which will be discussed in § 3.3.

If the metric admits a Killing vector field K, equations (3.1) can be integrated to give a conservation law. To see this, define Pa to be the vector whose components are P° = TabKb. Then,

The first term is zero by the conservation equations, and the second vanishes as Tab is symmetric and 2jK"(o. w = Zx<7a& = 0, since K is a Killing vector. Thus if 3) is a compact orientable region with boundary d3$, Gauss' theorem (§2.7) shows f Pbdarb = f Pb.bdv = 0. (3.2)

J as Js

This may be interpreted as saying that the total flux over a closed surface of the K-component of energy-momentum is zero.

When the metric is flat, as it is in the Special Theory of Relativity, one may choose coordinates {¡s°} in which the components of the metric are gab = eaSab (no summation) where Sab is the Kronecker delta and ea is — 1 if a = 4 and is + 1 if a — 1,2,3. Then the following are Killing vectors: L = ^ = ^ 2> 3> 4)

(these generate four translations) and & d

(these generate six 'rotations' in space-time). These isometries form the ten-parameter Lie group of isometries of flat space-time known as the inhomogeneous Lorentz group. One may use them to define ten vectors Pa and Pa which will obey (3.2). We may think of P as repre-

a afi 4

senting the flow of energy and P, P, P as the flow of the three compo-

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