The first term vanishes as a consequence of the field equations. In the second term, Lxgab = 2X(a. 6). Thus f (^L^d^f ((T°bXa):b-T°b.,bXa)dv. J & J &

The first contribution may be transformed into an integral over the boundary of 3! which vanishes as X is zero there. Since the second term must therefore be zero for arbitrary X, it follows that Tab. b = 0.

We shall now give as examples Lagrangians for some fields which will be of interest later.

Example 1: A scalar field ijf

This can represent, for example, the 77°-meson. The Lagrangian is where m, h are constants. The Euler-Lagrange equations (3.4) are mz fxabf—ftt^ 0.

The energy-momentum tensor is

Example 2: The electromagnetic field

This is described by a one-form A, called the potential, which is defined up to the addition of a gradient of a scalar function. The Lagrangian is i=_iLFfa,FcdgaCffbd'

where the electromagnetic field tensor F is defined as 2dA, i.e. Fab = 2A[b.a1. Varying Aa, the Euler-Lagrange equations (3.4) are

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