Conformal Invariance of the Newtonian Weyl Tensor
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Conformal Invariance of the Newtonian Weyl Tensor Neil Dewar1 · James Read2 Received: 30 March 2020 / Accepted: 21 September 2020 © The Author(s) 2020
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. Keywords Weyl tensor · Newton–Cartan theory · Conformal invariance · Conformal Newtonian spacetimes
1 Conformal Leibnizian Spacetimes We begin by introducing a Leibnizian spacetime, which is a triple M, ta , h ab , where (i) M is a differentiable manifold; (ii) ta is a non-vanishing, closed 1-form; and (iii) h ab is a positive semidefinite symmetric tensor such that h ab tb = 0. A connection ∇ on M is said to be compatible with this spacetime if and only if ∇a tb = 0,
(1a)
∇a h bc = 0.
(1b)
We will confine our attention to spacetimes which are spatially flat: that is, which are such that the Riemann tensor R a bcd of any compatible connection obeys h r b h sc h td R a bcd = 0. (One can show that if this holds of any one compatible connection, it holds of all of them.) Because of the separation of the spatial and temporal metrical structure, we have scope to vary conformally the spatial and temporal structure independently of one
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James Read [email protected] Neil Dewar [email protected]
1
Munich Centre for Mathematical Philosophy, Ludwigstraße 31, 80539 München, Germany
2
Faculty of Philosophy, University of Oxford, Oxford OX2 6GG, UK
123
Foundations of Physics
another (although as we shall see, there are reasons to couple the two kinds of conformal transformation). Consider, first, a conformal transformation of the temporal structure ta → ξ 2 ta ,
(2)
where ξ is a nowhere-vanishing and spatially constant scalar field. To say that ξ is spatially constant means that h ab db ξ = 0. This is equivalent to ensuring that the conformally transformed temporal 1-form is still closed and thus that there exists a global time function (and so a notion of Newtonian absolute time) in the conformallytransformed model.1 If we replace the temporal 1-form in a Leibnizian spacetime with a conformal equivalence class thereof, we obtain Machian spacetime. Second, consider a conformal transformation of the spatial structure, h ab → λ2 h ab ,
(3)
where λ is, again, a nowhere-vanishing and spatially constant scalar field. This time, we require that λ be spatially constant in order to preserve spatial flatness of the spacelike hypersurfaces. If we replace the spatial metric in a Leibnizian spacetime with a conformal equivalence class of spatial metrics, then we obtain spatially conformal Leibnizian spacetime. Finally, we may consider joint conformal transformations of the spatial and temporal structure: 1 ta , λ2 → λ2 h ab .
ta → h ab
(4a) (4b)
where λ is a nowhere-vanishing and spatially constant scalar field. As we will show in the next section, it is conformal tra
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