A Schrödinger approach to Newton-Cartan and Hořava-Lifshitz gravities

Abstract

Abstract:We define a 'non-relativistic conformal method', based on a Schrödinger algebra with critical exponent z = 2, as the non-relativistic version of the relativistic conformal method. An important ingredient of this method is the occurrence of a complex compensating scalar field that transforms under both scale and central charge transformations. We apply this non-relativistic method to derive the curved space Newton-Cartan gravity equations of motion with twistless torsion. Moreover, we reproduce z = 2 Hořava-Lifshitz gravity by classifying all possible Schrödinger invariant scalar field theories of a complex scalar up to second order in time derivatives.

Dynamics of Carroll strings

“…Recently these theories have been constructed from the gauging of the Bargmann algebra [9], from Lihshitz holography [10] and from the use non-relativistic conformal methods [11].…”

Three-dimensional extended Lifshitz, Schrödinger and Newton-Hooke supergravity

“…Therefore, it would be very useful to study supermultiplet representations of the extended Schrödinger superalgebra. In particular, based on the bosonic construction [18], it is natural to expect that a multiplet with a complex scalar field Ψ as the lowest element can be used to construct a super-Schrödinger invariant gravity model. Taking multiple number of such multiplets and gauge fixing the Schrödinger supergravity action would give rise to the supergravity coupling of such a multiplet to off-shell supergravity, which is an important step towards non-relativistic localization.…”

“…In the relativistic context, this is most straightforwardly achieved by applying superconformal tensor calculus [14][15][16][17], where one first constructs superconformal models then gauge fix the redundant conformal symmetries to obtain a super-Poincaré invariant theory. In the case of non-relativistic gravity, a methodology for a conformal tensor calculus was established for the bosonic and supersymmetric conformal extension of the Galilei algebra, known as the Schrödinger algebra, in [18] and [19], respectively.…”

Covariant Poisson’s equation in torsional Newton-Cartan gravity

“…The gauging procedure is a powerful tool specially for constructing models of gravity which they do not seem to occupy any limiting corner of the general relativity. 1 Newton-Cartan gravity has extensions to include twistless and arbitrary torsion. The presence of torsion in Newton-Cartan geometry refers to the case where a specific non-relativistic diffeomorphism is allowed along the the time coordinate implying that the time is not absolute any more.…”

“…The analogue of the conformal algebra in the Galilean setting is the Schrödinger algebra with Lifschitz scaling z = 2 [59][60][61][62]. In [1] a similar approach denoted as the 'non-relativistic conformal method' has been introduced for constructing local Galilean invariants in the Newton-Cartan geometry with twistless torsion by exploring and classifying classical Schrödinger field theories in flat background. This classification was carried out off-shell by obtaining the Hořava-Lifshitz gravity Lagrangian at z = 2 and on-shell by obtaining the Newton-Cartan gravity equations of motion with twistless torsion.…”