Carroll versus Galilei gravity

Abstract

We consider two distinct limits of General Relativity that in contrast to the standard non-relativistic limit can be taken at the level of the Einstein-Hilbert action instead of the equations of motion. One is a non-relativistic limit and leads to a so-called Galilei gravity theory, the other is an ultra-relativistic limit yielding a so-called Carroll gravity theory. We present both gravity theories in a first-order formalism and show that in both cases the equations of motion (i) lead to constraints on the geometry and (ii) are not sufficient to solve for all of the components of the connection fields in terms of the other fields. Using a second-order formalism we show that these independent components serve as Lagrange multipliers for the geometric constraints we found earlier. We point out a few noteworthy differences between Carroll and Galilei gravity and give some examples of matter couplings.

Dynamics of Carroll strings

“…In contrast, in the non-relativistic case the string and particle limit lead to different non-relativistic models, one being the vibrating string (with the string NR limit) [1,3,4] and the other a non-vibrating string with a fixed lenght (particle limit) [38,39]. If we consider the coupling of the Carroll extended objects to Carroll gauge fields as it is done for the case the particle in [27], the dynamics becomes non-trivial [40] because the interaction with the Carroll background fields opens the light cone. Extended objects with a Carroll supersymmetry that generalize the Carroll superparticle [28] can also be studied.…”

Canonical analysis of non-relativistic particle and superparticle

“…We see that the Hamiltonian constraint (20) coincides with the constraint (7). Using (20) we find the canonical form of the Newton-Cartan particle action…”

“…There is also an interesting connection between Hořava-Lifshitz gravity and NewtonCartan gravity [4,5]. Newton-Cartan gravity is covariant and geometric reformulation of Newton gravity that is now very intensively studied, see for example [6][7][8][9][10][11][12][13]. 1 Concept of non-relativistic physics also emerged in string theory when strings and branes were analyzed at special backgrounds.…”

Remark about non-relativistic p-brane

“…There is also an interesting connection between Hořava-Lifshitz gravity and Newton-Cartan gravity [9,10]. In fact, Newton-Cartan gravity and its relation to different limits was also studied recently in a series of papers [11][12][13][14][15][16][17].…”