Eddington's theory of gravity and its progeny (1006.1769v2)

Abstract: We resurrect Eddington's proposal for the gravitational action in the presence of a cosmological constant and extend it to include matter fields. We show that the Newton-Poisson equation is modified in the presence of sources and that charged black holes show great similarities with those arising in Born-Infeld electrodynamics coupled to gravity. When we consider homogeneous and isotropic space-times we find that there is a minimum length (and maximum density) at early times, clearly pointing to an alternative theory of the Big Bang. We thus argue that the modern formulation of Eddington's theory, Born-Infeld gravity, presents us with a novel, non-singular description of the Universe.

An Examination of Eddington's Theory of Gravity and Its Extensions

The paper under consideration presents a critical analysis and extension of Eddington's original proposal for gravitational action in spacetimes incorporating a cosmological constant. The authors, Maximo Ba~nados and Pedro G. Ferreira, revisit this alternative to the Einstein-Hilbert action by introducing additional matter fields, thereby expanding the theoretical framework and implications of Eddington's gravitational model.

Revisiting Eddington's Proposal

Eddington's approach substitutes the fundamental field in a free de-Sitter space with a connection-based action, expressed as $S_{Edd} = 2\kappa\int d^4x\sqrt{|R|}$, where $R_{\mu\nu}$ is derived solely from the connection. This contrasts with the Einstein-Hilbert action, $S_{EH}$, which hinges on the spacetime metric $g_{\alpha\beta}$ and its Ricci tensor $R_{\alpha\beta}$. The significant distinction lies in the potential duality between $S_{EH}$, directly proportional to the cosmological constant $\Lambda$, and $S_{Edd}$, inversely proportional to $\Lambda$.

Extending to Matter Fields and Born-Infeld Gravity

Recognizing the incompleteness of Eddington's theory—specifically its exclusion of matter fields—the authors adopt an innovative approach by integrating Born-Infeld (BI) structure into the gravitational action. The resulting formulation, expressed as $S_{BI}[g,\Gamma,\Psi]$, accommodates matter in a conventional manner and introduces features indicative of Born-Infeld gravity.

The authors examine the action's limits, asserting that for small values of the parameter $\kappa R$, the theory yields traditional Einsteinian gravity. Notably, however, deviations emerge at high curvatures where Eddington's action presides. This action retains equivalence to the Einstein-Hilbert form in vacuum conditions, demanding exploration in scenarios with substantial matter density.

Key Insights and Novel Predictions

The paper reveals a critical finding regarding the Newton-Poisson equation, modified under substantial matter presence, implying a departure from classical predictions. This behavior suggests that Born-Infeld gravity becomes non-negligible in high density zones, such as in black hole structures or the universe's early epochs.

Application of the theory to a cosmological context shows that it introduces a minimum length and maximum density threshold at very early times. This effectively implies a non-singular model of cosmology that circumvents the Big Bang singularity, suggesting a finite density universe with a uniqueness resting on its capacity to avoid singularities entirely.

Implications and Future Trajectories

The theoretical exploration of Eddington-inspired Born-Infeld gravity presents significant implications for both theoretical physics and cosmology. It raises potential avenues for a singularity-free understanding of the universe—offering a framework that embodies minimal length and density concepts. Such developments may drive future research efforts toward quantum gravity realms, potentially harmonizing with string theory and loop quantum gravity constructs.

Moreover, refocusing gravitational theories to incorporate non-linear dynamics such as those seen in Born-Infeld structures promises to illuminate high curvature and density physics' behavioral nuances, thus refining our understanding of cosmic epochs characterized by Planck-scale phenomena.

In conclusion, the insights provided by revisiting Eddington's gravity through a Born-Infeld lens serve as a pivotal point for advances in gravitational theory, especially in contexts demanding reconciliation with quantum phenomena. This research marks a step forward in seeking a cohesive and comprehensive gravitational interpretation beyond classical Einsteinian boundaries.