Carrollian manifolds and null infinity: A view from Cartan geometry

Abstract

We review three different (conformally) Carrollian geometries and their relation to null infinity from the unifying perspective of Cartan geometry. Null infinity per say comes with numerous redundancies in its intrinsic geometry and the two other Carrollian geometries can be recovered by making successive choices of gauge. This clarifies the extent to which one can think of null infinity as being a (strongly) Carrollian geometry and we investigate the implications for the corresponding Cartan geometries.The perspective taken, which is that characteristic data for gravity at null infinity are equivalent to a Cartan geometry for the Poincaré group, gives a precise geometrical content to the fundamental fact that "gravitational radiation is the obstruction to having the Poincaré group as asymptotic symmetries".

Non-lorentzian spacetimes

“…Then M = ⊔ p∈N M p is the total space of a smooth principal R + -bundle π : M → N called the bundle of scales of the conformal manifold N. Let h = π * g and let ξ be the fundamental vector field of the free right R + -action on M. Then (M, ξ, h) is a (weak) carrollian geometry. This construction of carrollian structures has played a rôle in some recent work [23,24,8,25].…”

“…In this section I discuss the Cartan geometries modelled on the kinematical Klein geometries discussed in Section 4, but before doing so, I review the basic notions of Cartan geometry relevant to our discussion. A good treatment of Cartan geometry is given in [21] and there is a growing list of explicit applications of Cartan geometry to gravitation [22,23,24,25].…”

Magnetic Carrollian gravity from the Carroll algebra

“…This identification endows the tangent spaces to the manifold with the structure of a flat Carroll spacetime, on which the homogeneous Carroll group acts. This approach to Carrollian geometry is quite general and necessary for understanding the gauging procedure to be presented later; see also the reviews [22,23] and references therein for related discussions. We develop in this section the concepts per se, without any reference to a limiting procedure that would regard the Carrollian structure as a contraction of the corresponding Poincaré one.…”

Relativistic fluids, hydrodynamic frames and their Galilean versus Carrollian avatars

“…4 Carrollian symmetry has triggered interest in several directions. On the mathematical side, new geometric structures were discovered dubbed Carrollian manifolds [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46], following patterns similar to those leading to the Galilean duals i.e. the Newton-Cartan geometries.…”