Embedding Galilean and Carrollian geometries. I. Gravitational waves

Namely, in the sense that trajectories of particles submitted to such forces are geodesics with respect to a (suitable) connection.

Or in the words of the French philosopher J. S. Partre: “Le cinématisme est un dynamisme.”

We restrict here to the subset of geometric kinematical groups i.e. kinematical groups whose corresponding Lie algebra admits a faithful representation on the space of vector fields on a manifold of same dimension as the subspace of transvections, cf. Ref. 47 for details.

The Carroll group was originally introduced – mostly for pedagogical purpose – by Lévy-Leblond in Ref. 144 as a “degenerate cousin of the Poincaré group.” The rationale behind the reference to L. Carroll is justified in Ref. 144 as originating from the lack of causality in a Carrollian universe (or flat Carroll spacetime) as well as for the arbitrariness of time intervals (cf. Chapter VII - A Mad Tea-Party in Ref. 145). Later, F. Dyson¹⁴⁶ further justified the reference to Carroll by appealing to the immobility of (“timelike”) Carrollian observers as reminiscent of the following dialog between Alice and the Red Queen (Chapter II - The Garden of Live Flowers¹⁴⁷): “Well, in our country,” said Alice, still panting a little, “you’d generally get to somewhere else-if you ran very fast for a long time as we’ve been doing.” “A slow sort of country!” said the Queen. “Now, here, you see, it takes all the running you can do, to keep in the same place.”

Note that the recognition of the rôle played by the Bargmann group – the central extension of the Galilei group – in connection with nonrelativistic structures dates back to Ref. 148 where Newtonian connections were geometrically characterized as connections on the affine extension of the frame bundle by the Bargmann group (whose associated curvature only takes values in the homogeneous Galilei algebra). The connection between the Bargmann group and Galilean geometry was further explored in Ref. 149 and recently readdressed in Ref. 98.

Here and throughout the rest of this paper (Part I), the notion of parallelism on Lorentzian manifolds is provided by the Levi–Civita connection for the associated metric.

In the following, we will maintain a terminological distinction between manifolds endowed with metric structures (referred to as structures) and structures supplemented with a compatible connection (hereafter referred to as manifolds). For example, a Lorentzian structure will refer to a pair $(M,g)$ consisting of a manifold $M$ endowed with a Lorentzian metric g while a Lorentzian manifold will denote a triplet $(M,g,∇)$ where the Lorentzian structure is supplemented with a compatible connection ∇. A torsionfree manifold will therefore refer to a manifold for which the connection has vanishing torsion.

Referred to hereafter as the Duval–Künzle condition.

Whose respective isometry group is the (A)dS Carroll kinematical group, cf. Ref. 56.

Note that (A)dS Carroll spacetimes are the only maximally symmetric non-Riemannian manifolds that do not admit an embedding inside a Bargmann–Eisenhart wave. In particular, their nonrelativistic counterparts (Newton-Hooke spacetimes) are known to be embedded inside a Hpp wave,¹⁰⁷ as reviewed below.

Apart from the “horizontal” character of the embedding of Carrollian manifolds – as opposed to the vertical Eisenhart lift –, the terminology “Carroll train” refers to the short-lived comic journal The Train, edited by Edmund Yates, who published in 1856 the first piece of work – the romantic poem Solitude – of the Oxford college mathematics lecturer Charles Lutwidge Dodgson to be signed under his more well-known pen name Lewis Carroll. In other words, The Train welcomed the transition from Dodgson to Carroll, hence our choice to refer to it in the present context.

Galilean structures were referred to as Leibnizian structures in Refs. 60 and 61. In order to avoid confusion with the ambient notion of Leibnizian manifolds,^47,52 we did not retain this terminology in the present work.

Although two timelike observers sharing the same starting and ending points will generically measure different proper times, the time unit allows them to synchronize their proper time with the absolute time, cf. e.g. Ref. 61 for details.

In this case, any timelike observer is automatically synchronized with the absolute time.

This terminology is justified by the following fact: in a spacetime endowed with a non-Frobenius clock, all points in a given neighbourhood are simultaneous to each other, in the sense that any pair of points can be joined by a spacelike curve, cf. e.g. Ref. 61.

Or equivalently [ψ, γ].

An index free “Koszul-like” formula is displayed in Refs. 60 and 61.

The proof follows straightforwardly from the Koszul-like formula for Galilean connections.^60,61

Despite the seemingly non-linear nature of the Duval–Künzle condition.

Note that this constraint is holonomic if ψ is closed, cf. e.g. Ref. 61.

Let us emphasize that the connection is part of the structure and, as such, should be preserved by an isometric transformation. Technically, the affine Killing condition $LX∇=0$ is necessary in order to ensure that the isometry algebra is finite-dimensional, cf. Ref. 79.

Cf. e.g. Table I of Ref. 52 for a summary of this duality at the level of geometric structures.

Recall that sections of Ann ξ are 1-forms $α∈Ω1M$ satisfying α(ξ) = 0.

An index free “Koszul-like” formula for Carrollian connections can be found in Ref. 52.

The affine Killing condition is tensorial and reads more explicitly as $(Lξ∇)XY=0$ for all $X,Y∈ΓTM$ where $(Lξ∇)XY≔ξ,∇XY−∇ξ,XY−∇Xξ,Y$ or in components $Lξ∇μνλ≔ξρ∂ρΓμνλ+Γρνλ∂μξρ+Γμρλ∂νξρ−Γμνρ∂ρξλ+∂μ∂νξλ$⁠. Note that, in the torsionfree case, the latter expression can be recast as $Lξ∇μνλ=∇μ∇νξλ−Rν|μρλξρ$⁠.

Recall that any Killing vector field ξ for a non-degenerate metric g is automatically affine Killing for the associated Levi–Civita connection ∇ i.e. $Lξg=0⇒Lξ∇=0$⁠. However, there is no such implication in the Carrollian case, so that ξ is not necessarily affine Killing for ∇ despite being Killing for γ, i.e. $Lξγ=0̸⇒Lξ∇=0$⁠. This is related to the fact that, in contradistinction with the non-degenerate case, the Carrollian metric structure $(M,ξ,γ)$ does not entirely determine torsionfree compatible connections, cf. Proposition 3.14.

In other words $(M,ξ¯,γ,∇)$ is not a Carrollian manifold.

We restrict ourselves to considerations at the kinematical level i.e. none of the definitions below involve equations of motion.

We pursue with the terminology used in Sec. III which distinguishes between manifolds endowed with metric structures (referred to as structures) and structures supplemented with a compatible connection (referred to as manifolds), cf. Ref. 46.

The square in (4.2) is introduced for later convenience.

Where $d^:Ω•(M^)→Ω•+1(M^)$ denotes the de Rham differential on $M^$⁠.

Note, however, that the variables appearing in Sec. IV B are a priori defined over the whole ambient spacetime $M^$⁠.

Cf. Refs. 48 and 61 for an interpretation of the transformation (4.9) both in a purely nonrelativistic and ambient context.

Recall that a metric is called universal⁹³ if the following condition is satisfied: all conserved symmetric rank-2 tensors constructed from the metric, the Riemann tensor (of the associated Levi–Civita connection) and its covariant derivatives are themselves multiples of the metric. Hence, universal metrics provide vacuum solutions for all gravitational theories whose Lagrangian is a diffeomorphism invariant density constructed from the metric, the Riemann tensor and its covariant derivatives. In other words, universal spacetimes are solutions to Einstein equations while being immune to any higher-order – or “quantum” – corrections.

Recall that a distribution $D⊂TM$ is said to be autoparallel with respect to the connection ∇ if $∇XY∈ΓD$ for all $X,Y∈ΓD$ If ∇ is torsionfree, then the autoparallel distribution $D$ is involutive (i.e. $X,Y∈ΓD$ for all $X,Y∈ΓD$⁠) hence integrable and the leaves of the induced foliation are said to be autoparallel submanifolds.

Recall that the Bargmann algebra is the central extension of the Galilei algebra, cf. Ref. 41.

48 90.

More precisely, a rescaling of $ξ^$ cannot be compensated by a mere shift of η, as was the case in (4.22).

We are grateful to S. Hervik for a useful comment regarding this point.

Note that the invariance of the scaling factor of a Platonic wave (i.e. $Lξ^Ω=0$⁠) ensures that $η∈ΓAnnξ^$⁠.

I.e. it is neither Walker, Platonic nor Bargmann–Eisenhart. Note however that the Anti de Sitter spacetime (but not the de Sitter spacetime) can define a Platonic wave.⁴⁸ Explicitly, using the Poincaré coordinates, the line element $ds^2=1z22dudt+dz2+δabdxa⁡dxb$⁠, with $a,b∈1,…,d−1$⁠, admits the Killing hypersurface-orthogonal vector field $ξ^≔∂u$⁠.

Note that a non-trivial inclusion of Platonic waves within the class of Dodgson waves exists, modulo a rescaling of the vector field $ξ^$⁠, cf. Proposition 4.20.

Or torsionfree Bargmannian manifolds, cf. Remarks 4.3 and 4.11.

Or equivalently as (lightlike) Kaluza–Klein dimensional reduction.

Since any representative $ξ^$ is assumed to be nowhere vanishing, the induced flow action is free. We also assume that the latter is proper so that the quotient manifold theorem applies (cf. e.g. Theorem 21.10 in Ref. 150) and $M$ is hence a manifold.

Explicitly, given a projectable Bargmannian structure $M^,ξ^,ĝ$⁠, the 1-form $ψ^≔ĝ(ξ^)$ dual to $ξ^$ projects as an absolute clock ψ on $M$ whereas the contravariant metric ĝ⁻¹ projects as a contravariant Galilean metric h (or absolute rulers) compatible with ψ on $M$ (equivalently, the Leibnizian metric $γ^$ induced by ĝ on $Kerψ^$ – cf. Remark 4.3 – projects as the covariant Galilean metric γ on $M$ acting on Ker ψ), cf. Ref. 52 for details.

Cf. e.g. footnote 7 of Ref. 61.

Heuristically, the first two properties follow from the torsionfreeness of the ambient Levi–Civita connection together with its compatibility with both the ambient metric and vector field. The Duval–Künzle condition in turn follows from the symmetry of the associated Riemann tensor under exchange of the first and last pair of indices.

We should emphasize that relaxing the closedness condition dψ = 0 does not lead to a projectable connection as the components $Γμuλ$ do not vanish, hence Condition (b) of Proposition 5.9 would not be satisfied in this case.

The condition m = 0 ensures that $M2=−γ^(ẋ,ẋ)⩽0$ with $γ^$ the Leibnizian metric induced by ĝ, cf. Remark 4.3.

Irrespectively of the sign of M² i.e. whether the relativistic trajectory is timelike, spacelike or lightlike.

The fact that the proof requires the ambient vector field to be Killing allows to generalize the Eisenhart Theorem to Platonic waves – cf. Theorem 6.12 – but prevents further extension to larger classes of gravitational waves.

A lightlike hypersurface $i:Σ↪M^$ of a Lorentzian manifold $(M^,ĝ)$ is said to be totally geodesic if, for any geodesic x(τ) of the induced connection ∇, the curve i○x(τ) is a geodesic of the Levi–Civita connection $∇^$ associated with ĝ, cf. e.g. Ref. 121.

More precisely, $x(τ)∈it(Mt)$ for all τ.

Cf. Definition 4.12.

Cf. Definitions 5.9 and 5.11.

We will denote $Kerψ^$ the canonical involutive distribution induced by $(M^,[ξ^],ĝ)$ – cf. Remark 4.3 – and Ker ψ its projection on $M$⁠.

Cf. Ref. 58 for details.

Recall that the Riemannian metric γ is obtained as projection of the Leibnizian metric $γ^≔g|Kerψ^$⁠, cf. Ref. 113.

The notion of invariant lifts (cf. Remark 5.8) is well-defined here since the definition of a Platonic wave involves a distinguished vector field $ξ^$⁠, so that $(M^,ξ^)$ is a (parameterized) ambient structure.

I.e. ψ(X) ≠ 0 or ψ(Y) ≠ 0.

Cf. Ref. 58 for a more detailed geometric proof.

Recall that $dψ=2d⁡lnΛ¯∧ψ$⁠, so that $dψ¯=0$⁠, i.e. $(ψ¯,γ¯)$ is the special Galilean structure associated with (ψ, γ), cf. Remark 3.2.

The condition m = 0 ensures that $M2=−γ^(ẋ,ẋ)⩽0$ with $γ^$ the Leibnizian metric induced by ĝ.

Irrespectively of the sign of M² i.e. whether the relativistic trajectory is timelike, spacelike or lightlike.

Performing a redefinition of τ allows to put Eq. (6.17) in the parameterized form $∇¯wẋẋ=0$⁠.

Note that the induced Newtonian connection $∇¯w$ for timelike trajectories also defines a connection on the absolute spaces. The latter coincides with the Levi–Civita connection $∇γ¯$ associated with the Riemannian metric $γ¯$⁠, cf. Remark 3.5. However, in contrast to the Eisenhart lift, the spacelike trajectories are controlled by a different connection, namely the Levi–Civita connection ∇_γ associated with γ.

Upon setting ω = 1.

Recall that a pseudo-invariant connection satisfies by definition the following relation $Lξ∇μνλ=−ξλΩ−1∇μ∇νΩ$ with Ω a nowhere vanishing invariant function, cf. Remark 3.22.

The invariance of $ΣA$ (i.e. $LξΣA$ = 0) follows from (3.31) and (7.5), cf. Remark (3.22).

That is, if one trusts the modus ponens, cf. Ref. 151.

Note that, even for a generic Dodgson wave, the solution m = 0 for all τ is compatible with Eq. (5.11) and thus defines a subclass of geodesic curves for which the second statement of Theorem 7.6 holds globally.

Whenever $ξ^$ is Killing, the underlying wave is Bargmann–Eisenhart – cf. Proposition 4.20 – so that one recovers the case covered by Theorem 5.29.