Connections and dynamical trajectories in generalised Newton-Cartan gravity. II. An ambient perspective Available to Purchase

PLATO “The Republic” Book VII.

By homogeneous Bargmann group, we mean the homogeneous Galilei group in a d + 2 dimensional representation inherited from that of the Bargmann group, cf. Ref. 22.

The term (Galilean, Carrollian, etc.) “manifold” will stand for a smooth manifold endowed with a metric structure and a compatible connection.

We denote $igld+2=gld+2∈Rd+2$ the affine Lie algebra in d + 2 dimensions.

Let $g$ be a Lie algebra with $h⊆g$ being a Lie subalgebra (i.e., $h,h⊆h$⁠) and $i⊆g$ an ideal (i.e., $i,g⊆i$⁠). The algebra $g$ is the semidirect sum of $i$ with $h$ if it admits the decomposition $g=h⊕i$ as $h$-modules. The subalgebra $h$ will be called the homogeneous part of $g$⁠.

The leaf t = 0 is special in that only the homogeneous Carroll algebra acts faithfully.

Actually, when dealing with non-Riemannian geometries, the definition of the vector space $W(M)$ will sometimes necessitate a piece of the metric structure (e.g., $W(M,ψ)$ for Galilean connections and $W(M,ξ)$ for Carrollian and ambient Galilean connections).

The isomorphism is not necessarily canonical but may depend on the choice of an element of an affine space (cf., e.g., Lemma 4.2 of Ref. 5 or Subsection 4 of the  Appendix).

Indeed, an important discrepancy between Riemannian and non-Riemannian geometries lies in the fact that, in the latter case, only a subset of “metric” structures admits torsionfree compatible connections (e.g., Leibnizian structures with closed absolute clock in the Galilean case or invariant Carroll structures in the Carrollian case).

We pass on an important subtlety that arises in nonrelativistic cases (in contradistinction with the relativistic case), namely, that there may not be a canonical choice of a natural origin (and thus of affine map Θ) but rather a class of such origins (dubbed special connections in our terminology). The class of special connections forms an affine space which can be “resolved,” i.e., which is isomorphic to a “simpler” affine space [e.g., the affine space of fields of observers to define special Galilean connections, cf. Proposition (3.9) in Ref. 5, or the one of principal connections to define special Carrollian connections, cf. the  Appendix].

In the relativistic case, the presence of a canonical origin (the Levi-Civita connection) ensures that the classification is a vector space. However, in the nonrelativistic cases, the lack of a canonical origin (cf. footnote 57) prevents the classification to possess a natural structure of vector space. In the latter case, a classification is thus an affine space, isomorphic to $DM,m$⁠, obtained by taking the product of $W(M)$ by the affine space used in the definition of Θ and then quotienting by the model vector space of the latter (cf. Proposition B.4 in Ref. 5 for precise statements).

Note that the second relation of Proposition 3.10 is invariant under the transformation V ↦V + fξ, for all $f∈C∞M$ (and similarly for W) so that for a given field of observers N, the bilinear form defined by relation 2 really acts on $ΓKerψ/Span ξ$ and then constrains $dd+12$ components.

We pursue with the notation convention used in Ref. 5 and make use of the same symbols for the various spaces and maps encountered in the torsionfree and torsional cases in order to emphasise the similitude in the logic of the arguments.

Note that the tensor $Uλ|μν$ in (3.25) is independent of the choice of field of observers N, hence the absence of superscript on $U$⁠.

Recall that in the Lorentzian case, the arbitrariness is encoded in the unconstrained torsion tensor, i.e., in a section of the vector bundle $∧2T*M⊗TM$ whose fibers have dimension $d+1d+222$⁠.

Note that relations 1–2 of (^*) together with $UN,Aλ|μν=0$ ensure that $FNμν∈Γ∧2Annξ$⁠. Similarly, the fact that $ΣA[μν]=0$ guarantees that $ΣA(μν)∈Γ∨2Annξ$⁠.

Note that the invariance of γ and N ensure $LξγN=0$ so that the second component of $ΘN,A$ does not depend on A and will then be denoted as $UN$⁠.