On Born-Infeld Gravity in Weitzenbock spacetime (0812.1981v1)

Abstract: Using the Teleparallel Equivalent of General Relativity formulated in Weitzenb\"{o}ck spacetime, we thoroughly explore a kind of Born-Infeld regular gravity leading to second order field equations for the vielbein components. We explicitly solve the equations of motion for two examples: the extended BTZ black hole, which results to exist even if the cosmological constant is positive, and a cosmological model with matter, where the scale factor results to be well behaved, giving so a singularity-free solution.

• The paper introduces a Born-Infeld deformation of TEGR to smooth GR singularities using a non-Riemannian Weitzenböck framework with zero curvature and non-zero torsion.
• The paper demonstrates that extended BTZ black holes, typically bound to a negative cosmological constant in GR, can exist with positive values in this modified theory.
• The paper shows that the BI teleparallel framework yields regular cosmological models by capping high-energy divergences, providing new insights for early universe behavior.

On Born-Infeld Gravity in Weitzenböck Spacetime

The paper "On Born-Infeld Gravity in Weitzenböck spacetime" by Rafael Ferraro and Franco Fiorini investigates a variant of gravity theory that introduces modifications to address certain limitations inherent in General Relativity (GR). The focal point of this research is a teleparallel theory of gravity derived from the Teleparallel Equivalent of General Relativity (TEGR) augmented by a Born-Infeld (BI) type action.

Core Concepts

The paper leverages the Weitzenböck spacetime formulation, which, unlike the conventional GR dependent on the Levi-Civita connection, employs the Weitzenböck connection characterized by zero Riemann curvature but non-zero torsion. This approach underscores the difference between the two theories, with TEGR being formulated in terms of the vielbein components, allowing the field equations to remain second-order even under deformation.

The authors propose to smooth singularities in the framework of GR by adapting a Born-Infeld inspired deformation to TEGR. This deformation is designed to yield second-order field equations by considering a Lagrangian that deforms the conventional action while remaining sensitive to high-energy corrections, which are characterized using a scale parameter, $\lambda$, that controls deviations from classical solutions.

Numerical Results and Theoretical Implications

The paper presents two main exemplar studies utilizing this modified teleparallelism framework:

1. Extended BTZ Black Hole: The spinning Ba~{n}ados-Teitelboim-Zanelli (BTZ) black hole, typically confined to a negative cosmological constant in GR, is analyzed to show how BI teleparallelism allows for existence with positive cosmological constant. The deformation modifies the nature of the horizons and ergospheres, categorized under three types based on the value of $\epsilon = 4\Lambda/\lambda$. The deformation does not substantially alter core qualitative structures but allows compatibility with different cosmological constant signs, an impossibility in standard GR.
2. Regular Cosmological Models: The potential of BI-teleparallelism to offer non-singular cosmological solutions is explored. In contrast to standard GR where solutions may diverge towards singularities, here the cosmological scale factor is shown to be well-behaved under high-energy conditions, avoiding singularities in solutions such as the Friedmann-Robertson-Walker (FRW) universe. This suggests potential applications for early universe modeling, where standard GR predictions face challenges.

The paper demonstrates that the Born-Infeld framework yields regular solutions by bounding the dynamics of relevant parameters, such as the Hubble parameter, ensuring these do not diverge even at high energies typical of the early universe.

Future Developments

The research opens potential avenues for further refinement and exploration of gravity theories incorporating Born-Infeld-type corrections to other dynamics frameworks, including broader classes of non-linear field theories. Given the indication that alternative approaches can yield better-behaved solutions without altering the fundamental geometric structure excessively, the investigation of other invariant-based deformations could lead to a comprehensive suite of models capable of bridging quantum and classical gravity.

In conclusion, the paper provides insightful perspectives into how deformation theories can impact our understanding of gravitational phenomena, especially under conditions where classical theories like GR face intrinsic limitations. This framework could inform future studies, particularly in cosmology and black hole physics, in addressing and potentially circumventing classical singularities.