Eddington’s gravity in immersed spacetime

Abstract

We formulate Eddington's affine gravity in a spacetime that 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 a 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 Demir (2014 Phys. Rev. D 90 064017), rather than with Newton's constant. As a result, the tiny observed curvature is reproduced due to large hierarchies rather than fine tuning.

Affine inflation

“…This automatically coincides with the action (5) for ξ = 0. Our aim in this paper is to study the gravitational dynamics and the dynamics of the scalar field which is nonminimally coupled to gravity in affine spacetime through action (17). To that purpose, it is important to shed light again on some points concerning the structure of this action 1.…”

Induced affine inflation

“…where M is an integration constant. Obviously, the affine connection Γ λ µν has now reduced to the Levi-Civita connection g Γ λ µν of the emergent metric tensor g µν [3,[20][21][22]…”

Separate Einstein‐Eddington spaces and the cosmological constant

“…The simplest affine theory of gravity is derived from Eddington lagrangian density which is defined by the square root of the determinant of the Ricci tensor [1,2]. Like Einstein's general theory of relativity, Eddington's gravity can be extended to higher dimensions [11].…”

“…In this section we discuss a way to render the cosmological constant to zero by considering the separate Einstein spaces discussed earlier. We will take the case in which the higher dimensional product space discussed in the previous section is a product of four dimensional spaces, i.e, N = 8 [11,17].…”

“…Recently, Eddington's gravity has been formulated in a four dimensional spacetime that was considered plunged in a larger eight dimensional space, and it was shown that matter terms could appear in the new structure [11].…”