Conformal Fourth-Rank Gravity

We consider the consequences of describing the metric properties of space- time through a quartic line element $ds^4=G_{\mu\nu\lambda\rho}dx^\mu dx^\nu dx^\lambda dx^\rho$\mathrm{.}Theassociated\mathrm{"}metric\mathrm{"}isafourth-ranktensor$G_{\mu\nu\lambda\rho}$. We construct a theory for the gravitational field based on the fourth-rank metric $G_{\mu \nu \lambda \rho }$ which is conformally invariant in four dimensions. In the absence of matter the fourth-rank metric becomes of the form $G_{\mu \nu \lambda \rho }=g_{(\mu \nu }{g}_{\lambda \rho )}$ 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 $k_{obs}=0$ the field equations predict $\mathrm{\Omega }\approx 4y$, where $y=\frac{p}{\rho }$; for ${\mathrm{\Omega }}_{small}=0.01$ we obtain $y_{pred}=2.5\times {10}^{-3}$. $y$ can be estimated from the mean random velocity of typical galaxies to be $y_{random}=1\times {10}^{-5}$. For the early universe there is no violation of causality for $t>t_{class}\approx {10}^{19}{t}_{Planck}\approx {10}^{-24}s$.