Non-relativistic limit of the Mielke-Baekler gravity theory

Abstract: In this paper, we present the most general non-relativistic Chern-Simons gravity model in three spacetime dimensions. We first study the non-relativistic limit of the Mielke-Baekler gravity through a contraction process. The resulting non-relativistic theory contains a source for the spatial component of the torsion and the curvature measured in terms of two parameters, denoted by $p$ and $q$. We then extend our results by defining a Newtonian version of the Mielke-Baekler gravity theory, based on a Newtonian like algebra which is obtained from the non-relativistic limit of an enhanced and enlarged relativistic algebra. Remarkably, in both cases, different known non-relativistic and Newtonian gravity theories can be derived by fixing the $\left(p,q\right)$ parameters. In particular, torsionless models are recovered for $q=0$.

• The paper presents the contraction of the relativistic Mielke-Baekler model into a non-relativistic regime, revealing key roles for torsion and curvature parameterized by constants p and q.
• It employs an Inönü-Wigner contraction with added U(1) gauge fields to derive a finite Chern-Simons Lagrangian and obtain algebras like extended Bargmann and torsional Newton-Hooke.
• The study further extends the model to a Newtonian framework, opening avenues for exploring non-relativistic holography and quantum gravity applications.

Non-Relativistic Limit of the Mielke-Baekler Gravity Theory

This paper presents a comprehensive examination of the non-relativistic limit of the Mielke-Baekler (MB) gravity theory, which is one of the most general non-relativistic Chern-Simons (CS) gravity models formulated in three spacetime dimensions. The authors initiate their analysis by applying a contraction process to transition from the relativistic MB gravity framework to its non-relativistic analog. The resulting theory effectively incorporates sources for both the spatial component of the torsion and the curvature, with these properties being parameterized by constants $p$ and $q$.

Overview of Mielke-Baekler Gravity

The MB gravity model, initially laid down by Mielke and Baekler, aligns within the sphere of three-dimensional gravity theories known for their capability to serve as excellent platforms for probing various facets of gravity and quantum gravity. This formulation encapsulates the traditional Einstein-Hilbert action with additional terms representing cosmological constants, translational elements, and rotational aspects, each regulated by distinct coupling constants. Here, curvature and torsion components become explicit through parameters $p$ and $q$. Adjusting these parameters facilitates the retrieval of several notable models like the Einstein-Hilbert, Teleparallel, and Witten's exotic gravity theories.

Transition to Non-Relativistic Limits

Through a meticulous construction process, the authors disassemble the relativistic MB gravity theory, aiming to redefine it in a non-relativistic regime. This involves an Inönü-Wigner contraction where the algebraic structure of the MB model is expanded by the inclusion of additional $\mathfrak{u}(1)$ gauge fields, enabling the resolution of potential singularities and ensuring the preservation of a non-degenerate invariant tensor during contraction. The contraction yields a non-relativistic MB (NRMB) algebra populated by central charges that permit a finite, non-degenerate CS Lagrangian.

Various classifications emerge from differing values of $p$ and $q$, leading to the derivation of several non-relativistic algebras like the extended Bargmann, torsional Newton-Hooke, and more. Each incarnation maintains its characteristics within the new NRMB framework, aligned with unique symmetry properties.

Newtonian Extension

Taking this perspective further, the authors extend the MB framework to embody a Newtonian version, coined the Newtonian MB (NMB) gravity theory. This development is grounded in the enhanced MB algebra, which encompasses additional generators $S_A$ and $L_A$ beyond the conventional MB setup. This extension is pertinent in generating an action principle that aligns with known Newtonian physics and expands the algebraic horizons opened by the relativistic model.

The elaborate algebraic extensions described here communicate not only a broadened view of NR dynamics but also hypothesize the inclusion of non-zero torsional properties into the Newtonian paradigm. This engagement dovetails with significant conceptual shifts, implicating potential applications in fields such as condensed matter systems and non-relativistic holography.

Implications and Future Directions

This study not only widens the scope of three-dimensional CS gravity theories by adopting non-relativistic frameworks but also sets a stage for enriched exploration into torsion and its roles across various gravitational theories. The mathematical formulations lying at the heart of these models promise potential applications extending to holography, higher-spin extensions, and supergravity iterations. Future research should consider these paths, potentially illuminating the complex interplay between non-relativistic physics and more conventional gravitational frameworks. Additionally, examining connections with models supporting non-zero time-like torsion, such as the torsional Newton-Cartan framework, may further enhance our understanding within quantum gravity contexts.