n-dimensional Weyl conformal tensor in a 4-index gravitational field theory

TL;DR

This paper introduces a novel four-index gravitational field equation incorporating the n-dimensional Weyl conformal tensor, revealing new degrees of freedom and naturally deriving the cosmological constant as an integration constant.

Contribution

It proposes a new four-index gravitational field equation involving the Weyl tensor, providing insights into additional constraints and the natural emergence of the cosmological constant.

Findings

Weyl tensor components relate to tidal gravitational fields

Cosmological constant appears as an integration constant

New constraints on the metric from the four-index theory

Abstract

The Weyl conformal tensor is the traceless component of the Riemann tensor and therefore, as is known, the information it contains does not appear explicitly in Einstein's equation. Following a rigorous mathematical treatment based on the variational principle, we will suggest that there exists a four-index gravitational field equation linearly containing the Weyl tensor closely related to a tidal gravitational field tensor whose components will be calculated. The new degrees of freedom, introduced via the n-dimensional Weyl tensor, will therefore clearly appear as additional constraints on the metric and we will demonstrate, among other things, that the cosmological constant appears as a natural solution of the four-index theory in the form of an integration constant which therefore does not need to be introduced ad hoc into a Lagrangian.