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.
Highlights
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
It would be interesting to extend this work in the following directions. It has been known for a long time that the asymptotic symmetry algebra of asymptotically flat space-times is given by the Bondi-Metzner-Sachs (BMS) algebra [50,51,52]
In 3 bulk dimensions it has been shown that the BMS algebra is isomorphic to the 2-dimensional Galilean conformal algebra [55, 56]
Summary
The Carroll algebra is obtained as a contraction of the Poincare algebra by sending the speed of light to zero [11, 12]. For an earlier discussion of gauging the Carroll algebra see [38]. We would like to think of ξμ as the generator of diffeomorphisms and Σ as the internal (tangent) space transformations. To this end we introduce a new local transformation denoted by δthat is defined as δAμ = δAμ − ξν Fμν = LξAμ + ∂μΣ + [Aμ, Σ] ,. The tangent space has a Carrollian light cone structure by which we mean that the light cones have collapsed to a line. This can be seen from the fact that there are no boost transformations acting on the spacelike vielbeins eaμ.
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