RENORMALIZABLE CONFORMALLY INVARIANT MODEL FOR THE GRAVITATIONAL-FIELD

We develop a model for the gravitational field which is renormalizable , conformally invariant and integrable in four dimensions. The first t wo conditions can be easily implemented. However, for the latter condi tion we must take recourse to fourth-rank geometry where the line elem ent is defined by a quartic form, ds(4) = G(mu nu lambda rho) dx(mu) d x(nu) dx(lambda) dx(rho). The simplest Lagrangian which can be constru cted in this case depends quadratically on a Ricci tensor constructed only in terms of a connection; therefore a Palatini-like variational p rinciple is applied. The field equations imply that the fourth-rank me tric decomposes into a product of a second-rank metric with itself, an d in this case the geometry becomes Riemannian. The decomposition of t he fourth-rank metric means our field equations become linear in the R icci tensor and thus they are amenable for comparison with the Einstei n field equations. We show that the Einstein field equations are a par ticular case of our field equations. The field equations are solved in the spherically symmetric case. The solution contains the Schwarzschi ld metric and the Kottler metric, corresponding to a massive point par ticle on a Minkowski and a de Sitter background, respectively.