Conformal Invariance of the Newtonian Weyl Tensor

Neil Dewar,James Read

doi:10.1007/s10701-020-00386-w

Neil Dewar, James Read

Open Access

https://doi.org/10.1007/s10701-020-00386-w

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Abstract

It is well-known that the conformal structure of a relativistic spacetime is of profound physical and conceptual interest. In this note, we consider the analogous structure for Newtonian theories. We show that the Newtonian Weyl tensor is an invariant of this structure.

Highlights

We begin by introducing a Leibnizian spacetime, which is a triple M, ta, hab, where (i) M is a differentiable manifold; (ii) ta is a non-vanishing, closed 1-form; and (iii) hab is a positive semidefinite symmetric tensor such that habtb = 0
We will confine our attention to spacetimes which are spatially flat: that is, which are such that the Riemann tensor Rabcd of any compatible connection obeys hrbhschtd Rabcd = 0. (One can show that if this holds of any one compatible connection, it holds of all of them.)
If we replace the spatial metric in a Leibnizian spacetime with a conformal equivalence class of spatial metrics, we obtain spatially conformal Leibnizian spacetime

Summary

Conformal Leibnizian Spacetimes

We will confine our attention to spacetimes which are spatially flat: that is, which are such that the Riemann tensor Rabcd of any compatible connection obeys hrbhschtd Rabcd = 0. First, a conformal transformation of the temporal structure ta → ξ 2ta,. To say that ξ is spatially constant means that habdbξ = 0 This is equivalent to ensuring that the conformally transformed temporal 1-form is still closed and that there exists a global time function (and so a notion of Newtonian absolute time) in the conformallytransformed model.. If we replace the temporal 1-form in a Leibnizian spacetime with a conformal equivalence class thereof, we obtain Machian spacetime. We may consider joint conformal transformations of the spatial and temporal structure: ta → λ2 ta,. A spacetime equipped with a conformal equivalence class of (ta, hab) pairs will be referred to as a conformal Leibnizian spacetime.

Invariance of the Newtonian Weyl Tensor

A Degeometrised Weyl Tensor

Applications