B2 Equations for VA

In what follows, I first consider equations relating to the region VA and deal with Vv afterwards (see B11). We can express the transformation law of the Einstein (and Ricci) tensor as (P&R 6.8.24)

i>ABA-B-- Oaba-b-=HVa(a'Vb')bH-1 = - H-1V a(a-Vb-)bH

together with (P&R 6.8.25)

This last equation has considerable pure-mathematical interest, being an instance of what is referred to as the Calabi equation.iB3] But it also has physical interest, being the equation for a conformally invariant self-coupled scalar field w which, with R=4A, we can write as

Every solution of this 'w-equation', as I shall henceforth refer to it, provides us with a new metric w2gab whose scalar curvature has the constant value 4A. The conformal invariance of the w-equation is expressed in the fact that if we choose a new conformal factor ñ and transform from Qab to a new conformally related metric gab gab — Qab = íf2bab then the conformal scaling for the ra-field ra = if-1ra gives us (as has been remarked upon earlier, in A8; see P&R 6.8.32)

(□ + f)ra = ñ-3(d + f)ra, from which the required conformal invariance of the non-linear ra-equation immediately follows. (Note that when ñ=ñ and ra = ñ we simply revert to Einstein's gab metric, with ra = 1, and the equation becomes the identity §A = §A.)

We have seen in A8 that the energy tensor for such a physically regarded ra-field, when the ra3 term is absent, would be (P&R 6.8.36)

where C is some constant. Moreover, we find that the ra3 term in the ra-equation does not disturb the conservation equation VaTab[ra] = 0, so we adopt this expression for the energy tensor for the ra-field also, and for consistency with what follows, I shall choose

Comparing this with (P&R 6.8.24, B2) above, we find

Tab[ñ] = ñ2 ^ABAB = ñ2TTab from Einstein's equation

&ABA-B- = 4nGTTab, which holds for the gab metric. For a trace-free energy tensor, we find that the scaling 'Tab = ñ-2Tab (A8, P&R 5.9.2) preserves the conservation equation, so we are led to the somewhat remarkable re-formulation of Einstein's theory, as referred to the gab metric, for massless sources Tab.

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