Conformal Fourth-rank Gravity

We consider the consequences of describing the metric properties of space-time through a quartic line element ds = Gμνλρdx dxdxdx. The associated ”metric” is a fourth-rank tensor Gμνλρ. We construct a theory for the gravitational field based on the fourth-rank metric Gμνλρ which is conformally invariant in four dimensions. In the absence of matter the fourth-rank metric becomes of the form Gμνλρ = g(μνgλρ) therefore we recover a Riemannian behaviour of the geometry; furthermore, the theory coincides with General Relativity. In the presence of matter we can keep Riemannianicity, but now gravitation couples in a different way to matter as compared to General Relativity. We develop a simple cosmological model based on a FRW metric with matter described by a perfect fluid. Our field equations predict that the entropy is an increasing function of time. For kobs = 0 the field equations predict Ω ≈ 4y, where y = p ρ ; for Ωsmall = 0.01 we obtain ypred = 2.5 × 10. y can be estimated from the mean random velocity of typical galaxies to be yrandom = 1× 10. For the early universe there is no violation of causality for t > tclass ≈ 10tPlanck ≈ 10s. Short title: Conformal fourth-rank gravity. Classification Number: 0450 Unified field theories and other theories of gravitation. ”The next case in simplicity includes those manifoldnesses in which the line-element may be expressed as the fourth-root of a quartic differential expression.” B. Riemann, 1854