On Twisted Spacetimes: a new class of Galilean cosmological models

1 Introduction

The geometric formulation of Newton’s theory of gravity began with E. Cartan in the early twentieth century [1], [2]. However, it was not until the second half of the century that the so-called Newton-Cartan theory began to receive significant attention from various authors. In the 1960s, Malament studied the Lorentzian and Galilean groups, proving that Newtonian gravity constitutes a limit of General Relativity and how free particles are represented by geodesic curves of certain connections in both theories [3]. Additionally, Trautman showed the existence of a symmetric connection in both theories [4], arising due to the local nature of physical laws and simultaneously encoding the inertia principle and gravitational forces. Concretely, classical Newtonian gravitation is formulated as a covariant theory, showing that certain results previously considered characteristic or singular of the theory of Relativity are also shared by the (geometric) Newton-Cartan gravitational theory. In fact, Newtonian gravity is explained by the curvature of a connection in the spacetime, although in this case it no longer arises from any semi-Riemannian metric. In the geometric formulation of gravity, the spacetime structure is dynamic in the sense that it participates in the development of physics rather than being a fixed backdrop (see [3] and references therein).

Thereafter, several steps were taken towards the generalization of this theory, such as the introduction of new Newtonian models satisfying the cosmological principle [5]. Thus, obtaining the Galilean analogue of relativistic Robertson-Walker spacetimes. In [6], Newtonian theory achieves the status of a gauge theory and is shown to be a limit of the Einstein-Klein-Gordon theory. This view as a limit of General Relativity continued to develop during these years [7], concluding also that the results obtained from Newton-Cartan cosmology were similar on not too large scales to those derived from Einstein’s theory [8], [9].

In the last few decades, the simplicity of certain Newtonian models in Newton-Cartan theory, in contrast to the computational complexity of General Relativity, has led to new applications of this non-relativistic theory in Cosmology [10], Hydrodynamics [11], quantum collapse [12], AdS/CFT correspondence [13], condensed matter systems [14], quantum Hall effect [15] and other related phenomena. The many applications have motivated a review of the geometric structures associated with generalized Newton-Cartan theory [16] and new extensions such as Newton-Cartan Supergravity [17] and non-relativistic Strings [18], [19]. Furthermore, new models have been obtained, including Galilean Generalized Robertson-Walker spacetimes [20], the Newtonian Gödel spacetime [21], standard stationary Galilean spacetimes [22], as well as the embedding of these geometries in higher dimensional Lorentzian manifolds [23], [24], [25], [26], [27].

In this manuscript (Sect. 3), the cosmological models of Galilean Generalized Robertson-Walker spacetimes (GGRW) [20] are introduced in a wider family of Galilean model spacetimes. Definition 3.1 states

Given a real interval I ⊆ R , a Riemannian manifold (F, h) and a positive smooth function f ∈ C ^∞ (I × F), a Galilean Twisted (GT) spacetime is given by the tuple I × F , Ω = d t , g = f 2 π F * h , ∇ where t : = π _I and ∇ is the only symmetric Galilean connection on I × F such that ∇ ∂ t ∂ t = 0 and Rot∂_t = 0.

According to the cosmological principle, classical Robertson-Walker spacetimes (see, e.g., [28]) model a spatially homogeneous and isotropic universe composed of matter (galaxies) subject only to the action of its own gravity. Thus, the “absolute space at each moment in absolute time” must be locally Euclidean with constant sectional curvature. Furthermore, as large-scale experimental measurements (redshift) indicate, the expansion is assumed to be homogeneous, i.e., the expansion rate is the same at all points at a fixed (absolute) time. However, while the hypothesis of spatial homogeneity and isotropy may be a reasonable first approximation to the large scale structure of the universe, it could not be appropriate when we consider a more accurate scale. This led to the introduction of the family of GGRWs spacetimes, where the absolute space is modeled by an arbitrary Riemannian manifold, which does not necessarily have to be of constant curvature. Finally, the Twisted spacetimes presented here assume a non-homogeneous expansion rate, which means that it depends not only on the moment in time but also on the point in space. In other words, the “Hubble constant” is no longer constant.

Twisted products have been studied in the relativistic setting [29], [30], [31], [32]. They have been analyzed in relation to geometric models of fluids, resulting in a characterization in terms of the stress-energy tensor of an imperfect fluid [33] and obtaining the properties of static perfect fluids in Twisted Lorentzian spacetimes [34]. Similar results could be explored in the non-relativistic setting introduced here.

A Galilean Twisted (GT) spacetime possesses an infinitesimal symmetry given by the existence of a timelike irrotational and (spatially) conformally Leibnizian field of observers. Several geometrical properties, as well as physical interpretations of this family of spacetimes, are given in Section 4 and Section 5. We concretely focus on the completeness of its free falling observers. Our main result (Theorem 4.9) asserts

Let (F, h) be a complete Riemannian manifold and let f ∈ C ∞ ( R × F ) be a positive function. If inf  f > 0 and inf(f _t /f) > −∞ along finite times, then the GT spacetime M = R × F , Ω = d π R , g = f 2 π F * h , ∇ is geodesically complete.

From a physical perspective, this result is relevant since the completeness of the timelike trajectories is crucial in order to have a consistent theory (without observers that appear or disappear suddenly).

Section 6 is devoted to the study of Galilean spacetimes admitting a torqued vector field (see Definition 6.2), which is closely related to the previously introduced family of irrotational conformally Leibnizian vector fields [20]. The main result in this section is Theorem 6.5, which gives the following statement.

Let (M, Ω, g, ∇) be a Galilean spacetime. If it admits a Galilean torqued vector field K ∈ Γ(TM), then for each p ∈ M, there exists an open neighborhood of p, U , and a Galilean diffeomorphism Ψ : N → U , where N is a GT spacetime.

Finally, Section 7 is devoted to face splitting-type problems, i.e., under which geometrical assumptions a Galilean spacetime decomposes globally into a Galilean Twisted spacetime. Theorem 7.5 states

Let (M, Ω, g, ∇) be a simply connected Galilean spacetime. It admits a global decomposition as a GT spacetime if and only if there exists a Galilean torqued vector field K ∈ Γ(TM) such that the associated field of observers, Z  : = K/Ω(K), is a uniform global generator.

Moreover, if the leaves of the foliation induced by Ω are compact, we can ensure again a global splitting result (Theorem 7.6).

2 Preliminaries

Let M be a (n + 1)-dimensional smooth manifold. As usual, it is assumed connected, Hausdorff and paracompact. A Leibnizian spacetime [16] is a triad (M, Ω, g) where (Ω, g) is a Leibnizian structure defined by a global non-vanishing one-form on M, Ω ∈ Λ¹(M), and a positive definite metric g on the vector bundle defined by the kernel of Ω. Concretely, if AnΩ := {v ∈ TM   : Ω(v) = 0} is the n-distribution induced by Ω, then g is a smooth, bilinear, symmetric and positive definite 2-covariant tensor

where Γ(AnΩ) denote the subset of sections V ∈ Γ(TM) such that V _p ∈ AnΩ, ∀p ∈ M.

Given a Leibnizian spacetime (M, Ω, g), the points p ∈ M are called events. If p ∈ M, Ω _p is known as the absolute clock at p and An Ω p , g p is called the absolute space at p (see for example [35]). Adopting relativistic terminology, a tangent vector v ∈ T _p M is said to be spacelike (resp. timelike) if Ω _p (v) = 0 (resp. Ω _p (v) ≠ 0). Moreover, a timelike vector v is called future-pointing (resp. past-pointing) if Ω _p (v) > 0 (resp. Ω _p (v) < 0).

On the other hand, an observer is defined as a smooth curve γ : I ⊆ R → M , being I an interval, such that its velocity is a normalized timelike future-pointing vector (standard timelike unit [35]), i.e., Ω_γ(s)(γ′(s)) = 1, ∀s ∈ I. The parameter s ∈ I is the proper time of the observer γ. A field of observer Z ∈ Γ(TM) is a smooth vector field whose integral curves are observers, equivalently, Ω(Z) = 1.

In this non-relativistic setting, the synchronizability of observers is intrinsic to the Leibnizian structure. If Ω ∧dΩ = 0 (i.e., the smooth distribution AnΩ is integrable), the spacetime is called locally synchronizable. In this case, Frobenius Theorem (see [36]) states that there exists a foliation consisting of spacelike hypersurfaces. Moreover, locally Ω = λdT, for certain smooth functions λ > 0 and T, and the hypersurfaces of the foliation are exactly {T = constant}. Thus, every observer can locally synchronize its proper time with the so-called “compromise time” T. If the absolute clock is closed, dΩ = 0, the Leibnizian spacetime is said to be proper time locally synchronizable. Under this assumption, locally we can express Ω = dT, for certain smooth function T, and the observers can synchronize their proper times up to a constant. In the case Ω = dT, for T ∈ C ^∞ (M), we will suppose every observer is parameterized by the absolute time T. Notice that, if M is simply-connected and Ω is closed, we can ensure the existence of an absolute time function making use of the Poincaré Lemma (see, for instance, [37]).

If the integrability condition Ω ∧dΩ = 0 does not hold, then the causal structure is not well-defined. In fact, if Ω(p) ∧dΩ(p) ≠ 0 at some p ∈ M, Caratheodory’s Theorem ensures that there exists an open neighborhood of p in M where all events are connected by some spacelike curve (see, e.g., [38]). Thus, even locally the notion of absolute space cannot be considered.

Let (M, Ω, g) and ( M ̃ , Ω ̃ , g ̃ ) be two Leibnizian spacetimes. A diffeomorphism λ : M → M ̃ is called Leibnizian if it preserves the absolute clock and space, that is, λ * Ω ̃ = Ω and λ * g ̃ = g .

To give complete physical meaning to a Leibnizian spacetime, the principle of inertia has to be included via an affine connection ∇. It is necessary to consider ∇ compatible with the absolute clock Ω and the absolute space (AnΩ, g), i.e.,

1. ∇Ω = 0, equivalently, X(Ω(Y)) = Ω(∇ _X Y), for all X, Y ∈ Γ(TM),

2. ∇g = 0, that is, Z(g(V, W)) = g(∇ _Z V, W) + g(V, ∇ _Z W), for all Z ∈ Γ(TM) and V, W ∈ Γ(AnΩ).

Note that, thanks to condition (1), the right-hand side of item (2) is well defined. Any such ∇ is called a Galilean connection. Consequently, the Leibnizian spacetime does not determine a canonical Galilean connection. The tuple (M, Ω, g, ∇) is known as a Galilean spacetime. Recall that if its torsion tensor vanishes, a connection is said to be symmetric, i.e.,

From a physical viewpoint, it is often convenient to choose a symmetric connection because it is determined by the associated geodesics [[39], Add. 1, Ch. 6]. Observe that the closeness of the absolute clock and the torsion tensor are related [[16], Lemma 13]

From now on, we will only consider symmetric Galilean connections, then using equation (1), dΩ = 0.

Given two Galilean spacetimes (M, Ω, g, ∇) and ( M ̃ , Ω ̃ , g ̃ , ∇ ̃ ) , a diffeomorphism λ : M → M ̃ is called Galilean if it is Leibnizian and additionally λ * ∇ ̃ = ∇ , i.e., ∇ ̃ d λ ( X ) d λ ( Y ) = ∇ X Y , for all X, Y ∈ Γ(TM).

On the other hand, given a Galilean spacetime (M, Ω, g, ∇) and a field of observers Z ∈ Γ(TM), it is well-known that the Galilean connection ∇ can be characterized using Z. The gravitational field induced by ∇ in Z is the spacelike vector field G = ∇ Z Z and similarly, the vorticity or Coriolis field is the 2-form ω = 1 2 Rot ( Z ) . The space of symmetric Galilean connections is mapped bijectively onto Γ(AnΩ) ×Λ²(AnΩ), giving ∇ ↦ ( G Z , ω Z ) (see [[16], Cor. 28]). Moreover, the converse is obtained through a formula ’à la Koszul’. From the tangent space splitting TM = ⟨Z⟩⊕AnΩ, we can consider

where P ^Z X = X − Ω(X)Z is the natural spacelike projection for Z, then for each V ∈ Γ(AnΩ) [[16], eq. (13)],

3 Galilean Twisted Spacetimes

We introduce a new class of Galilean spacetimes, which extends the family of cosmological models known as Galilean Generalized Robertson-Walker spacetimes (see [5], [20]).

Abusing the notation, we will denote g = f 2 π F * h . Analogously to the Riemannian twisted product, the Riemannian manifold (F, h) is called fiber.

The coordinate vector field ∂_t is a field of observers in M, and the observers in ∂ _t are called commovil observers in analogy with both the Galilean and relativistic Generalized Robertson-Walker spacetimes [20], [40]. Notice that the conditions (7) determine the Galilean connection ∇ (see equation (2)).

4 Geodesic completeness of GT spacetimes

In this section we are going to study the completeness of the geodesics in a GT spacetime. In particular, this will allow us to know if the free falling observers live forever.

Given a geodesic γ : J → M in a GT spacetime (M = I × F, Ω, g, ∇), where J ⊂ R denotes an interval in the real line, we will be able to derive the corresponding equation of its projection π _F (γ) on the fiber F. The following lemma gives us the necessary computations.

The previous formulae (8) allow us to obtain the equations of the projections of geodesics onto the fiber of a GT spacetime.

For the rest of the section we suppose I = R since it is necessary to ensure the completeness of commovil observers.

Let (M, Ω, g, ∇) be a GT spacetime with M = R × F and g = f 2 π F * h . We can give a unique vector field G on T ( R × F ) such that, the timelike geodesic in M are represented by integral curves of G in T ( R × F ) . Indeed, let γ(s) = (s, σ(s)) be a free falling observer, s₀ ∈ J and (U, x¹, …, x ⁿ ) a coordinate system in F, such that σ(s₀) ∈ U. If we identify x(σ(s)) ≡ x(s) = (x¹(s), …, x ⁿ (s)), equation (9) in R × U is given by

where k = 1, …, n and the Γ’s denote the Christoffel symbols of the Riemannian metric h. Making use of the standard procedure, we can introduce auxiliary variables v i = x ̇ i to convert the previous second-order system, in the following equivalent first-order system in twice the number of variables,

for k = 0, …, n.

Treating (x¹, …, x ⁿ , v¹, …, v ⁿ ) as coordinates on T ( R × U ) , we recognize (12) as the equations for the flow of the vector field G ∈ X ( R × U ) given by

To show that G define a global well-defined vector field on T ( R × F ) , it is enough to see that G acts on a smooth function φ by

where γ _v (s) denote a geodesic satisfying γ(0) = p and γ ̇ v ( 0 ) = v , p ∈ R × F and v ∈ T p ( R × F ) . This last statement is clear from the previous coordinates expressions.

Making use of [[28], Lem. 1.56], we can assert that an integral curve ξ of G on an interval [ 0 , b [ ⊂ R , for b < + ∞, can be extended to b (as an integral curve) if and only if there exists a sequence in [0, b[, {s _n }↗ b, such that {ξ(s _n )} converges. Therefore, we obtain the following technical lemma.

The following result gives natural conditions to ensure completeness of the free falling observers in a GT spacetime.

In the case of spacelike geodesics, we obtain an analogous result.

Note that, if the fiber (F, h) is complete, Theorem 4.8 assures the geodesic completeness of any Riemannian manifold (F, g _t ), for g t = f 2 ( t , ⋅ ) π F * h , under a natural assumption on f.

We can ensure the geodesic completeness of a GT spacetime by unifying the results on the completeness of free falling observers and spacelike geodesics.

If F is compact, the hypothesis for f ∈ C ∞ ( R × F ) in Theorem 4.9 are fulfilled trivially.

The following example prove that the hypotheses in Theorem 4.9 are optimal.

5 Physical interpretation of geodesics

Several interesting physical observations can be obtained from the expressions in the previous section.

Now, equation (18) allows us to heuristically deduce the behaviour of the complete timelike geodesics in a GT spacetime in relation to certain properties of the twisting function. Consider two simultaneous events (s₀, p), (s₀, q) ∈ M. Their spacelike distance (in the absolute space) is given by

where C _F ([0, 1], p, q) is the set of admissible curves in F from p to q (see, for instance, [41]). Note that the function dist F s 0 ( ( s 0 , p ) , ( s 0 , q ) ) determine the distance between the commovil observer θ _p (s₀) = (s₀, p) and θ _q (s₀) = (s₀, q) in the absolute space F s 0 . Moreover, if β(r) = (s₀, ξ(r)), r ∈ [0, 1], is a minimizing geodesic in F s 0 from p to q, then we have

where ‖ ⋅‖ is the norm induced by g in F s 0 .

Obviously, if f _t > 0 (resp. f _t < 0) the absolute space is expanding (resp. contracting) over time. If for instance f _t > 0, then the spacelike component of the velocity γ′(s) of a free falling observer (no commovil) γ, relative to the corresponding instant commovil observer, decreases in norm. Taking this into account, if we assume that the twisting function f _t tends to +∞ for t ↗ + ∞, each free falling observer γ evolves asymptotically to commovil observer directions (see Figure 1(a)). If otherwise f _t < 0 and we assume f _t tends to −∞ for t ↗ + ∞ and inf  f > 0, then every free falling observer γ evolves asymptotically to some spacelike direction (see Figure 1(b)).

Figure 1:

Evolution of free falling observers in a Galilean Twisted spacetime: (a) f _t (s₀, p) > 0: Every free falling observer γ evolves asymptotically to a commovil observer. (b) f _t (s₀, p) < 0: Each free falling observer γ evolves to some spacelike direction.

Now, it is natural to ask how free falling observers measure the speed of each other. If two points in the fiber p, q ∈ F are closely enough, free falling observers passing through one of them can measure the velocity of the others. If the leaves of the foliation induced by Ω are affine Euclidean spaces, the classical notion of relative velocity of an observer with respect to another is determined by the existence of a position vector between them. As a generalization, we can give the following.

Recall that in a Riemannian manifold, the existence and uniqueness of geodesics between two given points is locally ensured. Globally, the existence result holds in the complete case. Moreover, the non-existence of conjugate points globally guarantees uniqueness.

Let us deduce some interesting geometric properties of this definition in a GT spacetime. The fixed reference observer will be taken as a commovil observer. Indeed, given a free falling observer γ in a GT spacetime, denote γ ′ ( s 0 ) = θ p ′ ( s 0 ) + v , for v ∈ ( A n d t ) ( s 0 , p ) . The parallel transport P ( s 0 , p ) , ( s 0 , q ) β ( γ ′ ( s 0 ) ) = V ( 1 ) is defined by a parallel vector field V along β with V(0) = γ′(s₀). This can be split into

where a is a smooth function on [0,1] and U is a spacelike vector field along β. The initial value condition V(0) = γ′(s₀) turns out

From the fact that V is parallel and β can be lifted from a curve in F, it is easy to obtain that a′ = 0 and D U d r = − u t β ′ . Therefore, a ≡ 1 and U satisfy the initial value problem

The modulus of the relative spacelike velocity of γ′(s₀) measured by θ _q (s₀), V γ ′ ( s 0 ) , θ q ( s 0 ) = ‖ U ( 1 ) ‖ = g ( U ( 1 ) , U ( 1 ) ) , can be explicitly obtained in this case. It is enough to compute

using β is a geodesic. Obviously, ‖β′‖ is a constant. Taking into account the initial values, we can integrate both two differential equations, obtaining

where v = P ∂ t γ ′ ( 0 ) and the vector w: = β′(0). This expression, takes into account the geometric structure of the slice F s 0 , as well as, the distance between the points (s₀, p) and (s₀, q) (see equations (20) and (21)). In particular, if γ is a commovil observer, the velocity of θ _p (s₀) measured by θ _q (s₀) is

where |⋅| is the absolute value function. We see that it is proportional to the spacelike distance between the points (s₀, p) and (s₀, q).

Finally, if the GT spacetime can be reduced to a GGRW spacetime, f = f(t), then expression (23) turns out ‖ V θ p ( s 0 ) , θ q ( s 0 ) ‖ = | u t ( s 0 ) | dist F s 0 ( s 0 , p ) , ( s 0 , q ) .

6 Galilean torqued vector fields

Along this section, we will see that there exists a natural generalization of GT spacetimes in a rich geometrical structure characterized by the existence of a distinguished vector field on the Galilean spacetime. In fact, the main theorem will state that these Galilean spacetimes are locally GT spacetimes.

As we announced, a generalization of GT spacetimes is defined by means of a vector field satisfying an analogous expression to equation (24).

The torqued function and form are fully determined by the function Ω(K), as we prove in the following lemma.