Separate Einstein‐Eddington spaces and the cosmological constant

Abstract

Based on Eddington affine variational principle on a locally product manifold, we derive the separate Einstein space described by its Ricci tensor. The derived field equations split into two field equations of motion that describe two maximally symmetric spaces with two cosmological constants. We argue that the invariance of the bi-field equations under projections on the separate spaces, may render one of the cosmological constants to zero. We also formulate the model in the presence of a scalar field. The resulted separate Einstein-Eddington spaces maybe considered as two states that describe the universe before and after inflation. A possibly interesting affine action for a general perfect fluid is also proposed. It turns out that the condition which leads to zero cosmological constant in the vacuum case, eliminates here the effects of the gravitational mass density of the perfect fluid, and the dynamic of the universe in its final state is governed by only the inertial mass density of the fluid

Affine inflation

“…This theory was shown to be equivalent to general relativity for V (φ) = m 2 φ 2 /2, where the metric tensor arises as the momentum canonically conjugate to the connection [14]. This proof can be straightforwardly extended to a general potential V (φ) [15,18].…”

“…In other words, the metrical description of the GR might have arisen dynamically as the universe evolves. To this end, the affine gravity (AG) [13][14][15][16][17][18][19], based solely on connection with no notion of metric, stands out a viable framework to study. The AG framework necessitates scalar fields to have nonvanishing potentials, and thus, studying inflation in the AG is important by itself.…”

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]…”

Are there really conformal frames? Uniqueness of affine inflation

“…Other affine approach to gravity has been proposed later as a different formulation of general relativity where the metric tensor appears as a momentum canonical conjugate to the affine connection, and the derived field equations are equivalent to those of GR with scalar and possibly gauge fields [49,63]. In the recent few years, attempts have been made to consider general and different approaches to pure affine gravity, in vacuum and in the presence of matter and even in higher dimensions [64][65][66][67][68][69][70].…”