L

hold for all i. These are the equations of the fields.

We obtain the energy-momentum tensor from the Lagrangian by considering the change in the action induced by a change in the metric.

81 8u

Suppose a variation gab(u, r) leaves the fields vPV"'6c...d unchanged but alters the components gab of the metric. Then

The last term arises because the volume measure dv depends on the metric, and so will vary when the metric is varied. To evaluate this term, recall that dv is in fact the four-form (4!)—1 tj whose components are Vabcd = (-#*! WW. where g = det (&,„). Therefore

cffab

The first term in (3.5) arises because A(vF(i)°—6c d;,.) will not necessarily be zero even though Ay¥l{)a—bc d is, since the variation in the metric will induce a variation in the components r0^ of the connection. As the difference between two connections transforms like a tensor, AT"^ may be regarded as the components of a tensor. They are related to the variation in the components of the metric by

(The easiest way to derive this formula is to note that since it is a tensor relation, it must be valid in any coordinate system. In particular, one could choose normal coordinates about a point p. For these coordinates the components r0^ and the coordinate derivatives of the components gab vanish at p. The formula given can then be verified to hold aty.) Using this relation, Ax¥({,a—bc^d.e may be expressed in terms of (^ffbc)-,d and the usual integration by parts employed to give an integrand involving Agab only. Thus we may write 81 ¡8u as where Tab are the components of a symmetric tensor which is taken to be the energy-momentum tensor of the fields. (See Rosenfeld (1940)

for the relation between this tensor and the so-called canonical energy-momentum tensor.)

This energy-momentum tensor satisfies the conservation equations as a consequence of the field equations obeyed by the x¥(aa "bc...d- For suppose one has a diffeomorphism <j>: which is the identity everywhere except in the interior of Then, by the invariance of integrals under a differential map,

If the diffeomorphism \$ is generated by a vector field X (non-zero only in the interior of 3) it follows that f,J><*J>- o. iJsLx(Ln) = 2 \9 (^„A.-.JJ