Nonstandard Action of Diffeomorphisms and Gravity’s Anti-Newtonian Limit

Max Niedermaier

doi:10.3390/sym12050752

Abstract

A tensor calculus adapted to the Anti-Newtonian limit of Einstein gravity is developed. The limit is defined in terms of a global conformal rescaling of the spatial metric. This enhances spacelike distances compared to timelike ones and in the limit effectively squeezes the lightcones to lines. Conventional tensors admit an analogous Anti-Newtonian limit, which however transforms according to a non-standard realization of the spacetime Diffeomorphism group. In addition to the type of the tensor the transformation law depends on, a set of integer-valued weights is needed to ensure the existence of a nontrivial limit. Examples are limiting counterparts of the metric, Einstein, and Riemann tensors. An adapted purely temporal notion of parallel transport is presented. By introducing a generalized Ehresmann connection and an associated orthonormal frame compatible with an invertible Carroll metric, the weight-dependent transformation laws can be mapped into a universal one that can be read off from the index structure. Utilizing this ‘decoupling map’ and a realization of the generalized Ehresmann connection in terms of scalar field, the limiting gravity theory can be endowed with an intrinsic Levi–Civita type notion of spatio-temporal parallel transport.

Highlights

In a 1+d formulation where the geometry of a foliated manifold M is specified by a lapse-like field ν, a shift-like field νa, and a spatial metric qab with inverse qab, consider the following action S0[q, φ] = 1 2 t ti f dt√ Σ dx q e0(qad qbc −qab qcd )e0 (1) + 1 ν e0( φ)2 2ν[U (
We aim at developing a tensor calculus which bears to Sgravity and its diffeomorphism invariance an analogous relation as Einstein gravity in 1 + d form has to its non-manifest diffeomorphism invariance
The basics of a tensor calculus adapted to the Anti-Newtonian limit of Einstein gravity—Sgravity for short—have been developed

Summary

Introduction

In a 1+d formulation where the geometry of a foliated manifold M is specified by a lapse-like field ν, a shift-like field νa, and a spatial metric qab with inverse qab, consider the following action. For the 1+d block components (G, Ga, Gab) of the Einstein tensor proper, Gμν = Rμν − (1/2)gμν(R − 2Λ), the corresponding gauge transformations differ from (6) This marks on the linearized level the difference between the universal tensorial realization πT and the weight-dependent scaling realization πS. On a foliated pseudo-Riemannian manifold one will naturally require the limits to be compatible with moving indices with the spacetime metric gμν, gμν This turns out to restrict the allowed weights and allows one to define and to compute the transformation law autonomously, using the πS realization only. Appendix A revisits the 1+d tensor calculus of (pseudo-)Riemannian geometry with a focus on (occasionally new) results needed in the bulk of the paper

Sgravity and Its Diffeomorphism Invariance

Kinematical versus Dynamical Gravitational Gradients

A Fully Covariant Carroll Structure for Sgravity

Structure Functions of the SDiff Frame

Stensor Calculus

Limit Construction of Stensors

Examples of Stensors and Their Gauge Variations

Decoupling Maps

Sconnections

Connections in Orthonormal Frames

Conclusions