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## ROBERT REILLY VARIATION FORMULA TO DETERMINE S4 GRAVITATIONAL FIELD EQUATIONS

The standard Einstein-Hilbert variation involves all possible deformations of a metric on a three-manifold; it was David Hilbert who had derived the Euler-Lagrange equations for the action that is the integral of scalar curvature.  For S4 physics, we want to consider the variation of the integral of the scalar curvature only by deformations that are still hypersurfaces.  In general this is a fairly messy computation (see for example Yano: ).  However, it happens that the the scalar curvature for a hypersurface of a four-sphere can be written as

$S = \bar{S} + cRic(N,N) + c(9H^2 - ||A||^2)$

where $\bar{S}$ is the constant scalar curvature of the ambient four-sphere, $H$ is the mean curvature and $\| A \|^2$ is the length of the second fundamental form.  For this form, we have a functional that can be expressed in terms of the elementary symmetric polynomials of the principal curvatures, i.e. the eigenvalues of the second fundamental form.  Write $S_0 = 1, S_1=H, S_2 = k_1 k_2 + k_2 k_3 + k_1 k_3, S_3 = k_1 k_2 k_3$.  For functions of these elementary symmetric functions, the Euler-Lagrange equation for the action $f(S_1,S_2,S_3)$ have been determined by Robert Reilly (see Reilly)

We have $f(S_1,S_2) = aS_1^2 + bS_2$ and the Euler-Lagrange equation will be

$(2aS_1)_{,ij}T^{ij}_0-S_1 f + (S_1^2 - 2S_2)2 a S_1 + (S_2 S_1 - 3 S_3) b + 2 a S_1 3c + 2bc S_1 + 6 ac S_1 + 6 c a S_1 + 2bc S_1 = 0$

Simplifying, let $x=S_1$ and $y=S_2$ then

$(2ax)_{,ij} T^{ij}_0 = (4a+b)y - c(4b+6a) - ax^2$