Gauging the Carroll Algebra and Ultra-Relativistic Gravity (1505.05011v1)

Abstract: It is well known that the geometrical framework of Riemannian geometry that underlies general relativity and its torsionful extension to Riemann-Cartan geometry can be obtained from a procedure known as gauging the Poincare algebra. Recently it has been shown that gauging the centrally extended Galilei algebra, known as the Bargmann algebra, leads to a geometrical framework that when made dynamical gives rise to Horava-Lifshitz gravity. Here we consider the case where we contract the Poincare algebra by sending the speed of light to zero leading to the Carroll algebra. We show how this algebra can be gauged and we construct the most general affine connection leading to the geometry of so-called Carrollian space-times. Carrollian space-times appear for example as the geometry on null hypersurfaces in a Lorentzian space-time of one dimension higher. We also construct theories of ultra-relativistic (Carrollian) gravity in 2+1 dimensions with dynamical exponent z<1 including cases that have anisotropic Weyl invariance for z=0.

Overview of "Gauging the Carroll Algebra and Ultra-Relativistic Gravity"

The paper "Gauging the Carroll Algebra and Ultra-Relativistic Gravity" explores the intricate geometrical and algebraic frameworks underlying ultra-relativistic limits, specifically focusing on the Carroll algebra. This paper presents an exploration of the Carrollian space-times and their implications for gravity theories, especially in the context of holography.

Carroll Algebra Gauging

The research investigates the consequence of gauging the Carroll algebra, an ultra-relativistic limit derived from the Poincaré algebra by contracting the speed of light to zero. This leads to a geometry characterized by Carrollian space-times, where the light cone collapses to a line, significantly altering its structure. The paper elaborates on the methodology for constructing the affine connection compatible with Carrollian metrics, which are shown to be realized as the geometry on null hypersurfaces embedded in Lorentzian space-times. This is contrasted with Newton–Cartan geometries, highlighting the distinct algebraic properties and transformations involved.

Strong Numerical Results and Novel Claims

Significantly, the paper outlines frameworks to develop theories of ultra-relativistic Carrollian gravity in 2+1 dimensions. It formulates these theories using an effective action approach with dynamical exponent $z < 1$, extending the notion of Weyl invariance for cases with $z = 0$. These formulations introduce a comprehensive architectural paradigm for constructing actions that respect the ultra-relativistic Carrollian transformations.

Implications and Future Prospects

The practical and theoretical implications of this paper are profound, particularly for the field of flat space holography. The Carrollian structures posited offer potential avenues for understanding holography beyond the AdS/CFT frameworks, suggesting configurations where boundary theories exhibit non-Riemannian geometries. Considering the embedding capabilities on null hypersurfaces, there exists a parallel narrative for exploring dual geometries within the realms of quantum field theories coupled to these backgrounds.

Furthermore, the work fosters prospective inquiries into the coupling of warped conformal field theories (WCFTs) to curved backgrounds with Carrollian symmetries, broadening the scope of holography in ultra-relativistic scenarios. It also sets the stage for scrutinizing the phase space, degrees of freedom, and perturbative properties of ultrasymmetric gravity theories, exemplified by gapless and scale-invariant models.

In conclusion, the paper opens an investigative frontier for gauging the Carroll algebra to formulate and understand gravity theories under ultra-relativistic conditions, advocating rich interactions between algebraic methods, holographic principles, and innovative gravitational models. Such researches are promising pathways towards decoding gravitational dynamics in non-standard relativistic limits.