Eddington's Gravity in Immersed Spacetime

TL;DR

This paper extends Eddington's affine gravity to an immersed eight-dimensional spacetime with hypercomplex structure, deriving new gravitational equations that incorporate matter and modify vacuum energy effects.

Contribution

It introduces a novel formulation of affine gravity in an immersed spacetime, leading to modified gravitational equations and insights into vacuum energy without fine-tuning.

Findings

Derived gravitational field equations including matter from immersed Ricci tensors.

Showed vacuum energy gravitates with a different strength, explaining small observed curvature.

Proposed a simple Lagrangian combining Ricci and new tensor fields for modified gravity.

Abstract

We formulate Eddington's affine gravity in a spacetime which is immersed in a larger eight dimensional space endowed with a hypercomplex structure. The dynamical equation of the first immersed Ricci-type tensor leads to gravitational field equations which include matter. We also study the dynamical effects of the second Ricci-type tensor when added to the Lagrangian density. A simple Lagrangian density constructed from combination of the standard Ricci tensor and a new tensor field that appears due to the immersion, leads to gravitational equations in which the vacuum energy gravitates with a different cosmological strength as in Phys. Rev. D {\bf 90}, 064017 (2014), rather than with Newton's constant. As a result, the tiny observed curvature is reproduced due to large hierarchies rather than fine-tuning.