Conformal Fourth-Rank Gravity

Abstract

We consider the consequences of describing the metric properties of space- time through a quartic line element $ds^4=G_{\mu\nu\lambda\rho}dx^\mu dx^\nu dx^\lambda dx^\rho$. The associated "metric" is a fourth-rank tensor $G_{\mu\nu\lambda\rho}$. 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 $\Omega\approx 4y$, where $y={p\over\rho}$; for $\Omega_{small}=0.01$ we obtain $y_{pred}=2.5\times 10^{-3}$. $y$ can be estimated from the mean random velocity of typical galaxies to be $y_{random}=1\times10^{-5}$. For the early universe there is no violation of causality for $t>t_{class}\approx10^{19}t_{Planck}\approx 10^{-24}s$.