Curvatures and Non-metricities in the Non-Relativistic Limit of Bosonic Supergravity

TL;DR

This paper develops a covariant, metric-like formulation of the non-relativistic limit of bosonic supergravity, incorporating non-metricities and enabling tensor decomposition in a geometrically consistent way.

Contribution

It introduces a purely geometrical, covariant approach to non-relativistic supergravity with non-metricities, linking it to string Newton--Cartan geometry and facilitating tensor analysis.

Findings

Constructed a covariant decomposition of the Riemann tensor, Ricci tensor, and scalar curvature.

Established an equivalence with the vielbein approach of string Newton--Cartan geometry.

Proposed applications include rewriting supergravity Lagrangians and deriving corrections.

Abstract

We construct a metric-like formulation of the non-relativistic (NR) limit of bosonic supergravity at the Lagrangian level. This formulation is particularly useful for decomposing relativistic tensors, such as powers of the Riemann tensor, in a manifest covariant form with respect to infinitesimal diffeomorphisms. The construction is purely geometrical and is based on a torsionless connection, mimicking the construction of the relativistic theory. The formulation contains non-vanishing non-metricities, which are associated with the gravitational fields of the theory ($\tau_{\mu\nu}$, $h_{\mu\nu}$, $\tau^{\mu\nu}$, $h^{\mu\nu}$). The non-metricities are fixed by requiring compatibility with the relativistic metric, before taking the NR expansion. We provide a fully covariant decomposition of the relativistic Riemann tensor, Ricci tensor, and scalar curvature. Our results establish an equivalence between the vielbein approach of string Newton--Cartan geometry at the level of the Lagrangian and the proposed construction. We also discuss potential applications, including a pure metric rewriting of the two-derivative finite bosonic supergravity Lagrangian under the NR limit, a powerful simplification in deriving NR bosonic $\alpha'$-corrections and extensions to more general $f(R,Q)$ Newton--Cartan geometries.