Newtonian Gravity and the Bargmann Algebra (1011.1145v3)

Abstract: We show how the Newton-Cartan formulation of Newtonian gravity can be obtained from gauging the Bargmann algebra, i.e., the centrally extended Galilean algebra. In this gauging procedure several curvature constraints are imposed. These convert the spatial (time) translational symmetries of the algebra into spatial (time) general coordinate transformations, and make the spin connection gauge fields dependent. In addition we require two independent Vielbein postulates for the temporal and spatial directions. In the final step we impose an additional curvature constraint to establish the connection with (on-shell) Newton-Cartan theory. We discuss a few extensions of our work that are relevant in the context of the AdS-CFT correspondence.

• The paper derives Newton-Cartan gravity by gauging the Bargmann algebra, establishing a formal link between algebraic methods and classical Newtonian gravity.
• The paper employs a strategy that assigns gauge fields to algebra generators and imposes curvature constraints to recover the Poisson and geodesic equations.
• The paper underscores the potential of algebraic approaches to advance non-relativistic gravitational theories, paving the way for future explorations in supersymmetry and holography.

A Study on Newtonian Gravity through the Bargmann Algebra

The paper "Newtonian Gravity and the Bargmann Algebra" by Roel Andringa, Eric Bergshoeff, Sudhakar Panda, and Mees de Roo addresses the Newton-Cartan formulation of Newtonian gravity through the framework of the Bargmann algebra. The authors aim to show how Newtonian gravity, traditionally expressed through the Newton-Cartan theory, can be derived from the gauging of the Bargmann algebra, a centrally extended form of the Galilean algebra.

Key Insights

The paper demonstrates a formal method of deriving the Newton-Cartan theory, which is a non-relativistic counterpart of general relativity. Building upon the well-established method of deriving Einstein's theory of relativity through the gauging of the Poincaré algebra, the paper investigates the possibility of achieving similar results with the Bargmann algebra. The main objective is to formalize the connection between non-relativistic gravity and Newton-Cartan theory using algebraic structures.

Methodology and Findings

1. Introduction to the Bargmann Algebra: The Bargmann algebra serves as a crucial element of this paper. It is described as the non-relativistic version of the Poincaré algebra, obtained by defining a contraction process and augmenting with a central generator, crucial for including massive representations and ensuring Galileo invariance.
2. Gauging the Algebra: The process involves associating gauge fields with each algebra generator and imposing specific curvature constraints to convert spatial and temporal translational symmetries into general coordinate transformations. This allows the authors to derive a system where gauge fields related to spin connections become dependent rather than independent fields.
3. Connection with Newton-Cartan Gravity: The authors make a clear comparison between the derived system and Newton-Cartan gravity, showing that the derived algebraic system aligns closely with Newton-Cartan theory. They highlight similarities in the derivation of both the Poisson and geodesic equations in the context of a massive particle, indicating a successful translation from the gauge-theoretic framework to a more classical gravitational framework.
4. Curvature and Metric Compatibility: By imposing constraints on curvature tensors and metrics, the authors ensure that the derived equations remain consistent with physical expectations from Newtonian gravity. Specifically, they resolve for the dependent spin connection fields and perform a detailed analysis of the curvature tensors, ensuring they align with the familiar geometrical picture of Newton-Cartan theory.

Theoretical and Practical Implications

The implications of this research are twofold. Theoretically, it solidifies our understanding of how non-relativistic gravity can be geometrically formulated through algebraic means, emphasizing the robustness and flexibility of using algebraic structures to describe gravitational theories. Practically, it opens pathways for further explorations into non-relativistic gravity theories, potentially leading to new insights in fields such as the AdS-CFT correspondence, particularly in scenarios where non-relativistic limits or descriptions are beneficial.

Directions for Future Work

Future research could explore extensions into supersymmetric versions of the Bargmann algebra and their implications for Newtonian versions of Poincaré supergravity. Additionally, the methods in this paper could be applied to other algebras relevant in the AdS-CFT correspondence, like the Galilean Conformal, Schrödinger, and Lifshitz algebras. Such explorations could extend the understanding of non-relativistic versions of conformal gravity and potentially develop a non-relativistic conformal tensor calculus.

In conclusion, the paper presents a detailed and formal approach to deriving Newton-Cartan theory from the Bargmann algebra, providing valuable insights into the algebraic roots of non-relativistic gravity. The work serves as a foundational step towards further theoretical developments in non-relativistic gravitational theories and their potential applications in broader fields such as quantum gravity and holography.