Non-relativistic limit of gravity theories in the first order formalism

We consider the non-relativistic limit of gravity in four dimensions in the first order formalism. First, we revisit the case of the Einstein-Hilbert action and formally discuss some geometrical configurations in vacuum and in the presence of matter at leading order. Second, we consider the more general Mardones-Zanelli action and its non-relativistic limit. The field equations and some interesting geometries, in vacuum and in the presence of matter, are formally obtained. Remarkably, in contrast to the Einstein-Hilbert limit, the set of field equations is fully determined because the boost connection appears in the action and field equations. The fate of the cosmological constant in the non-relativistic limit is analyzed. It is found that the cosmological constant must disappear in the non-relativistic Mardones-Zanelli action at leading order. The conditions for Newtonian absolute time be acceptable are also discussed. It turns out that Newtonian absolute time can be safely implemented with reasonable conditions.


Introduction
One of the most important features of a new larger theory is that it must be reduced to the known, well established, smaller theory at some particular limit. General Relativity (GR) is a celebrated example [1][2][3]. In fact, Newtonian gravity can be obtained from GR by a suitable weak field limit. In the context of GR described as a gauge theory for the Poincaré group 1 [4][5][6], the so-called first order formalism, GR can be reduced to Newtonian gravity by a appropriate Inönü-Wigner (IW) contraction [9], see for instance [10] and references therein. On the one hand, one can take the ultra-relativistic limit of the Poincaré group and end up in the so-called Carrol gravity. On the other hand, the non-relativistic limit of the Poincaré group leads to the Galilei group and Galilei gravity. Galilei gravity is then a gauge theory based on the Galilei group and describes a geometrodynamical theory which, under appropriate assumptions [11][12][13][14], is equivalent to Newton theory of gravity. An important condition is known as twistless torsion constraint which is necessary to define the Newtonian absolute time. Nevertheless, Galilei gravity is far more general than Newtonian gravity, carrying, for instance, torsion degrees of freedom [15]. For the earlier works in NC geometrodynamics we refer to the seminal papers of E. Cartan [16,17] and also [1,[18][19][20][21][22]. In contrast to the non-relativistic limit of GR as the gauge theory for the Galilei group, one can start directly from the Galilei algebra and its extensions in order to construct more general Newton-Cartan (NC) gravity theories [23][24][25]. Two renowned examples are the Bargmann and Shrödinger groups. A particularly appealing feature of these extensions [11,13,26] is that it can be used as a base for constructing Hořava-Lifshitz gravity models [27]. Moreover, NC geometries have renowned importance in string and brane scenarios, see for instance [28][29][30][31] and references therein.
The gauge approach to describe GR allows torsion degrees of freedom to enter in the context in a natural way [4][5][6][7][8]. Moreover, the differential form language shows itself to be a useful and natural framework in this scenario because it allows to work in a coordinate independent manner. This approach enables one to write down more general actions, going beyond the Einstein-Hilbert (EH). In fact, in [7] Mardones and Zanelli were able to generalize Lovelock gravity theory [32] in order to account for torsion terms in any spacetime dimension. These theories, known as Lovelock-Cartan (LC) gravities, are described by first order, and polynomially local actions with explicit Lorentz gauge symmetry, called Mardones-Zanelli (MZ) actions 2 . Moreover, in differential form language, MZ actions do not depend explicitly on the metric tensor. The aim of the present work is to develop the non-relativistic limit of such theories and go beyond EH action in the study of non-relativistic gravity systems. In this first endeavor, we stick to four dimensions. In four-dimensional MZ action, besides the EH term, gravity gains topological terms (which do not affect the classical dynamics of gravity), a cosmological constant term and two extra terms associated with torsion. Thus, the present study opens the door in the understanding of the possible effects of the cosmological constant and torsion (or if the lack of them can be accommodated in the non-relativistic MZ theory) in non-relativistic gravitational systems.
In LC gravities, the EH action is a particular case. Hence, before considering the four-dimensional LC theory we review the EH case. Galilei gravity is obtained from the EH action by employing the appropriate IW contraction of the Poincaré group. The field equations are derived and solutions are formally discussed. We analyze torsionless and torsional solutions in vacuum and in the presence of matter in a quite general way. Particularly, we discuss twistless torsion conditions in the presence of other torsion components. Furthermore, an explicit example beyond Newtonian absolute time is worked out. In this example, we find a Weintzenböck-like geometry [33][34][35][36] with conformally flat metrics.
As mentioned before, the main target of the present work is the four-dimensional full MZ action. Its non-relativistic limit is performed and the field equations derived. Obviously, the residual gauge symmetry is again described by the Galilei group. To distinguish from the Galilei gravity obtained from the EH action, we call the non-relativistic theory obtained from the MZ action by Galilei-Cartan (GC) gravity. The first interesting result is that, differently from the EH case, the boost connection is explicitly present at the action and field equations. Thence, the system of field equations are fully determined. Second, it is shown that for the non-relativistic limit of the maximally symmetric vacuum solution of the MZ action be a solution of the GC gravity in vacuum, the cosmological constant must vanish in the non-relativistic limit. This result reduces the final form of the GC action to one which has no cosmological constant. The third result is the establishment of the properties that the GC theory must fulfill in order to accept Newtonian absolute time. Hence, a formal discussion about torsionless and torsional solutions in the presence of matter is provided.
This work is organized as follows: In Section 2 we review the construction of the LC theory of gravity in four dimensions and the maximally symmetric vacuum solution is derived. In Section 3 a discussion about the IW contraction from the Poincaré group towards Galilei group is provided. As an intermediate illustrative step, the four-dimensional non-relativistic limit of the EH action is obtained in Section 4. Also in this section, some novel solutions are discussed. The non-relativistic limit of the four-dimensional LC theory of gravity is considered in Section 5. Formal solutions in vacuum and in the presence of matter distributions are obtained. The fate of the cosmological constant and the acceptance of Newtonian absolute time are also analyzed in this section. Finally, our conclusions are displayed in Section 6.

Lovelock-Cartan gravity
In this section we review the construction of the LC theory of gravity in four dimensions [7,8] which generalizes Lovelock gravity theories [32] within Einstein-Cartan first order formalism by including torsional terms in the gravity action. The scenario is a four-dimensional manifold M with local Minkowski metric, η = diag(−, +, +, +). The construction is fundamentally based on the 10 parameter Poincaré group ISO(1, 3) = SO(1, 3) × R 1,3 , whose algebra is given by The vierbein E A ensures the equivalence principle by defining a local isomorphism between generic spacetime coordinates x µ (Lowercase Greek indices refer to spacetime indices and run through µ, ν, α . . . ∈ {0, 1, 2, 3}) with inertial coordinates x A . The latter can be identified with coordinates in the tangent space T (M ) at the point x ∈ M . The Lorentz connection Y AB carries information about the parallel transport on M because it is directly related to the affine connection on M . Since E and Y are independent fields, the geometric properties of metricity and parallelism are also independent concepts in the first order formalism. The corresponding 2-form field strengths constructed out from E and Y are the curvature and torsion, respectively defined by The gauge (local Lorentz) transformations are defined through where u = exp α ≈ 1 + α is a Lorentz group element and α = α AB Σ AB is assumed to be an infinitesimal algebra-valued parameter 4 .
The inverse of the vierbein E A is assumed to exist in such a way that Moreover, spacetime and locally invariant metrics, g µν and η AB , respectively, obey with g µα g αν = δ ν µ and η AC η CB = δ B A . Moreover, besides δ B A , we have the gauge invariant skew-symmetric object ǫ ABCD , the Levi-Civita symbol at our disposal.
The MZ theorem [7,8] states that the most general four-dimensional action which is gauge invariant, polynomially local, explicitly metric independent, and that depends only on first order derivatives is given by The first term is the usual EH action while the second is the cosmological constant term. Hence, κ is related to the inverse of Newton's constant and Λ is the cosmological constant. Third and fourth terms are essentially the same, up to surface terms. The parameters z 1 and z 2 carry mass squared dimension and when z 2 = −z 1 these terms are reduced to the Nieh-Yan topological term [7,8,[37][38][39]. The fifth and six terms are, respectively, Gauss-Bonnet and Pontryagin topological terms [7,8,40,41]. Being topological, these terms do not contribute to the field equations. Clearly, z 3 and z 4 are dimensionless topological parameters. Finally, S m stands for the matter content of fields and particles which, up to some considerations, we will keep as general as possible in this paper.
The field equations are quite easily obtained by varying the MZ action (2.8) with respect to E A and Y AB , respectively, For further use, we can derive a torsionless vacuum solution of the field equations (2.9), with a 0 ∈ R. It can be easily checked that the substitution of solutions (2.11) in Bianchi identities (2.4) leads to a 0 = −1. Hence, the torsionless vacuum solution (2.10) of the MZ action is a maximally symmetric spacetime For a solution with general a 0 , one should, perhaps, consider non-trivial torsion.
The specific case of GR can be obtained as a particular case of the MZ action (2.8) by setting Λ = z i = 0, namely (2.12) Before we discuss the non-relativistic limit of MZ action in Section 5, we will revisit the non-relativistic limit of the Einstein-Hilbert (2.12) in Section 4. But first, let us take a look at the Inönü-Wigner contraction [9] from the Poincaré group to the Galilei group in the next Section.

Inönü-Wigner contraction of the Poincaré group
The first step towards the non-relativistic limit of actions (2.8) and (2.12) is to split the Poincaré group as s to space translations, and R t to time translations. The corresponding algebra is easily obtained by projecting the algebra (2.1) in spatial and temporal components. We define and zero otherwise. The consequences for the fields are where e a , called here simply by space vierbein, relates coordinates in spacetime M with coordinates in the tangent space of 3-manifolds T 3 (M ). The time-projected vierbein, here called time vierbein q, connects coordinates in spacetime with T (M )/T 3 (M ). In the same spirit, ω ab is called spin connection while θ a is the boost connection. For the field strengths we have and In the above expressions, the covariant derivative D· ≡ d · +[ω, ·] is taken with respect to the SO (3) sector. The field R ab will be called space curvature while S a is the boost curvature. The fields T a , K a , and Q are named, respectively, space torsion, reduced torsion, and time torsion.
The non-relativistic limit of the Poincaré group is obtained following reference [10]. First, the boost and temporal translational generators are redefined by together with the rescalings in such a way that definitions (3.3) remain unchanged. At this point, the limit ζ −→ ∞, which is equivalent to consider 1/c −→ 0, can be performed. The consequence for the Poincaré algebra (3.2), at leading order, is that it is contracted down to the Galilei algebra, namely, and zero otherwise. Thence, the limit implies on an Inönü-Wigner contraction [9] of the form being the Galilean boosts, see for instance [10,25]. From the algebra (3.9), one easily observes that the Galilei group is not a semi-simple Lie group since the annexes B(3) × R 3 s and R t are normal Abelian subgroups of the Galilei group. Hence, there is no invariant Killing form. In fact, the isometries of the Galilei group imply on degenerate metric tensors: Clearly, η s ≡ δ ab is the 3-dimensional Euclidean flat metric while η t is its equivalent at time axis. These metrics are orthogonal to each other. The local invariant metrics (3.10) induces two sets of metrics on M, and Moreover, due to the relations (2.6) and (2.7), the following inverse relations hold Metrics (3.11) and (3.12) relate through (3.14) The fields (3.3) remains unchanged but the fields (3.5) and (3.6) reduce to The infinitesimal gauge transformations (2.5) of the fields are contracted down to 5 Galilei gauge transformations, where α = α ab Σ ab + α a G a . For completeness, we write down the gauge transformations of curvatures and torsions, Up to tautological relations, the hierarchy relations (2.4) lead to the following set of self-consistent equations The consequences of the Inönü-Wigner contraction setup here discussed are the basis to study the non-relativistic limit of gravity theories. We begin with the EH action as a first example in the following section.

Galilei gravity
What we call in this work Galilei gravity can be obtained from the non-relativistic limit of the Einstein-Hilbert action (4) at leading order in the 1/c expansion. This task is achieved by considering the Inönü-Wigner contraction of the Poincaré group to the Galilei group, as described in Sect. 3. See for instance [10]. Thence, employing decompositions (3.3) and (3.4), the Einstein-Hilbert action (2.12) reads Performing the rescalings (3.8) we get 2) The non-relativistic limit is then obtained by taking ζ −→ ∞ together with the coupling rescaling κ −→ ζ −1 κ. The result is the Galilei gravity action [10,25], In this procedure, we assume that the corresponding limit of the matter action S m is consistent.
It is worth mentioning that the pure gravitational sector of action (4.3), S pG = S G − S m , is invariant under local scale invariance [10,25] with φ = φ(x) being a local gauge parameter. In functional form, symmetry (4.4) implies on Identity (4.5) assigns a global charge ±1 for e and q, respectively.
The field equations can be easily computed from the action (4.3) for the fields q, e, ω, and θ, respectively given by The last equation implies on two important features of Galilei gravity: First, it establishes a constraint saying that the non-relativistic limit of the matter content must not couple with the boost connection; And second, that the field θ a remains free, since it does not appear in the field equations (nor in the action (4.3)). Typically, this problem is solved by considering higher order corrections in ζ −1 at the nonrelativistic limit of the Einstein-Hilbert action (4.1). Otherwise, a constraint for θ a must be implemented by hand. The second equation in (4.6) implies on another constraint on the matter content, namely, Moreover, combining the first and second equations in (4.6) we get saying that symmetry (4.4) must be obeyed by the non-relativistic limit of the matter content as well. Constraint (4.7) states that whenever e a appears at the matter action, a q must be there as well, i.e., e a and q always come in pairs qe a . Constraint (4.8) reinforces constraint (4.7) by saying that the matter action must be linear in e a .

Vacuum solutions
The field equations (4.6) in vacuum are of particular interest. The first two equations imply on We focus in this equation to complement the curvature solution (4.9) with possible torsional solutions in vacuum (other than the trivial inconsistent solution q = 0). It is important to keep in mind that boost connection and boost curvatures remain free.

A no-go constrained solution
As a no-go result, let e a and q be related by with n a being an arbitrary algebra-valued 0-form. This tentative ansatz is a solution of equation (4.10), as one can easily check. Nevertheless, such attempt is inconsistent with the degenerate nature of the metric. For instance, this is evident by contracting the ansatz (4.11) with q µ , resulting in q µ q µ = 0, which contradicts relations (3.13). In other words, q and e a are orthogonal by definition while the proposal (4.11) enforces these fields to be parallel.

Weak twistless torsion solution
In tensor notation, equation (4.10) can be manipulated to provide the following geometric relations with 0 standing for the time direction in tangent space. The first equation in (4.12) is the well known twistless torsion condition. It fixes a specific spacetime foliation for which time torsion Q has no projection on the locally inertial leaves. This condition allows the implementation of Newtonian absolute time and spatial causality. Equations (4.9) and (4.12) synthesize the well-known results of Galilei gravity in vacuum [10,11]. We call this solution weak twistless torsion solution since it still gives room for non-vanishing time torsion solutions. Notice that, imposition of K a bc = 0 leads to Q 0a = 0 in tangent projections. Moreover, Q ab = Q 0a = 0 does not imply on Q = 0. The latter constitutes what we call here strong twistless condition, which we discuss in the next section.

Strong twistless torsion solution
As just mentioned, strong twistless torsion solution is given by which is clearly a stronger condition than the first equation in (4.12). The consequence of the strong twistless torsion condition (4.13) is that the time vierbein can be chosen as an exact field, q = dT . Thus, time vierbein is defined by an arbitrary scalar function T (x) which, among an infinity number of possibilities, can be fixed as the time coordinate t. Since q is gauge invariant, this solution is absolute up to time translations t −→ t + constant. Hence, t is identified with the absolute Newtonian time. If no other assumption is made over q, equation (4.10) is satisfied for If S a remains free, equations (4.13) and (4.14) compose a torsional solution case, T a ∝ qθ a . For a torsionless solution, one might choose θ a = 0 −→ S a = 0. Nevertheless, (4.14) is not a requirement. Equation (4.10) together with (4.13) imply on the softer condition for the reduced torsion with θ a and S a still free to be fixed.

A Weitzenböck-like solution
An ansatz for Q can be made, with n being an arbitrary gauge invariant 1-form field. Combining (4.10) with (4.16) provides q(ne a − K a ) = 0, which is satisfied, for instance, by The first Bianchi identy in (3.18) states that qn must be closed, Using now the second Bianchi identity of (3.18), we find the curvature associated to the proposed solution (4.16) and (4.17), which is inconsistent due to the symmetry properties of the indices a and b. Hence, inevitably, space curvature must vanish and n must be closed where a is another constant 1-form field and b a constant scalar field. It can be easily checked that, with the help of equation (4.20), expression (4.21) satisfies equation (4.18). To be even more specific in this example, we consider ADM formalism in the temporal gauge [1]. In the first order formalism this means that E 0 i = 0 and E 0 0 = N , with N being the lapse function. Then, q = E 0 0 dt + E 0 i dx i = N dt, with t being the Newtonian absolute time. Also, choosing 7 n = ndt and a = adt, with n and a being positive constants, we get N = a exp(−nt) + bn . (4.22) Since N = N (t) = 1, the lapse function characterizes a non-trivial time different from the absolute time t. On the other hand, (4.22) also says that q −→ bn as time evolves. Hence, bn is the asymptotic lapse function. We can set then b = n −1 in order to identify the asymptotic time with the Newtonian absolute time where N abs = 1. Hence, N = a exp(−nt) + 1 . Notice that for this specific case, Q = 0 because q is parallel to n, even though q = dt, except asymptotically. Nevertheless, it is possible to write q = dT where T = − a n exp(−nt) + t + c , (4.24) with c a constant to be fixed. In fact, if we interpret T as a non-Newtonian time, we can fix c by demanding T to be positive definite in the domain t ∈ [0, ∞]. Since T ∈ [−a/n + c, ∞), for T ≥ 0 we get that c ≥ a/n. Thence, we can set T = a n [1 − exp(−nt)] + t , (4.25) to have the range T ∈ [0, ∞). Note that T (t) t→∞ ∼ t + a/n. Moreover, N = dT /dt characterizes the time flow rate with respect to the Newtonian time. Thus, non-Newtonian and Newtonian time intervals, respectively ∆T = T 2 − T 1 and ∆t = t 2 − t 1 , are related by This is an explicit example of time dilation in NC gravity. A nice discussion about time dilation in NC gravity can be found in [42].
Finally, still in this example, space torsion reads (see (4.17)) with lowercase Latin indices i, j, . . . , s running through {1, 2, 3}. Thence, while R ab µν = S a µν = Q µν = K a ij = 0, we have K a 0i = 0. The non-triviality of spacetime geometry is encoded in (4.27). We remark that we can easily set the Newtonian absolute time by choosing a = 0. Nevertheless, expression (4.27) remains valid since it does not depend on the 1-form a. To find the components of e a i , we can set ω ab = 0, which solves (4.9)à la Weitzenböck while giving room for torsional degrees of freedom. Thence, equation (4.27) can be directly integrated to provide with c a = c a i dx i being a constant 1-form field. Relations (3.13) allow to construct the full vierbein solution whose non-vanishing components are It can be checked using (3.11) and (3.12) that this solution induces conformally flat metrics of the form and Henceforth, the space metric trivializes while the new temporal metric reads (4.34) One can also trivialize temporal metric while keeping space metric conformally flat by performing an appropriate conformal transformation. To trivialize all metrics simultaneously, Newtonian time must be evoked by setting a = 0.

Solutions in the presence of matter
Going back to equations (4.6) in the presence of matter, we rewrite them as with τ and τ a being 3-forms associated with the energy-momentum tensor of the matter content while σ a and σ ab are 3-forms describing the spin-density of the matter content. In terms of τ and τ a , constraints (4.7) and (4.8) are rewritten as These equations suggest that τ and τ a can be chosen as with m a being a 2-form describing the matter content. The factor 4 is just a convenient normalization factor. We notice that m a does not depend on q nor e a (See the argument below (4.8)). Moreover, there is no need in defining m, the m a time counterpart, since the second of (4.37) (which is just (4.8)) defines a constraint between τ and τ a . The matter content is then written as 8 Thus, Imposing strong twistless torsion condition (4.13), applying D on the second of (4.35), and using Bianchi identities (3.18), one also finds Dτ a = 0 .
Clearly, reduced torsionless geometry requires that the matter content does not depend on the spin connection or, at least, that K a has a part independent of q. For fully torsionless geometry, one also has to set θ a = 0, see (3.5). The first two equations in expressions (4.42) are satisfied for We can fix θ a or S a at will as long as (4.47) is respected. For instance, the trivial solution θ a = 0 ⇒ S a = 0 is allowed. Non-trivially, we may set, for example, where a ∈ R. Thus, from (4.45), we get the constraint The fieldθ a is introduced because θ a and e a transform differently under Galilei gauge transformations. Thus,θ a must obey the following properties, Thence, up to an arbitrary dimensionless non-vanishing real constant a, θ a is totally fixed. Obviously, a = 0 sets θ a = 0 and S a = 0.
The main conclusion of this section is that Galilei gravity accepts consistent solutions in the presence of matter within Newtonian absolute time. Moreover, we found spaces for strong twistless torsion condition while having non-trivial curvatures and reduced torsion.

Galilei-Cartan gravity
We now consider the non-relativistic limit of the Mardones-Zanelli action (2. From the rescaling of the fields (3.8), we get The corresponding non-relativistic limit is obtained by taking ζ −→ ∞ together with the coupling rescalings 9 Thus, the MZ action (5.2) reduces to its non-relativistic limit at leading order, It is a straightforward exercise to check that this action is invariant under the Galilei gauge transformations (3.16) and (3.17). For now, we will refer to this theory as Galilei-Cartan gravity. The first term in the action (5.4) is the same that appears in the non-relativistic limit of the Einstein-Hilbert action (4.3). The second term is a non-relativistic cosmological relic. The terms accompanied by z 1 are new and will affect the dynamics. Terms with factors z 2 and z 4 are of topological nature (Abelian and non-Abelian Pontryagin actions, respectively) and do not contribute to the field equations. Moreover, it is assumed that the non-relativistic limit of S m is consistent. One can readily note that transformations (4.4) do not constitute a symmetry of action (5.4).
The field equations generated by the action (5.4) are 10 2κǫ abc R ab e c + Λe a e b e c − z 1 e a S a = −τ , We may have lost the scale symmetry (4.4), but now we have an extra equation (the last of equations (5.5)) associated with θ a . Moreover, the boost connection appears explicitly in the field equations. This is a welcome feature of the non-relativistic limit of the MZ action: The system of equations is now fully determined already at leading order. Combining the first and second equations in (5.5) we get 4κΛǫ abc qe a e b e c + 2z 1 R ab e a e b = qτ − e a τ a . (5.6) In the same spirit, we can combine the third and fourth equations in (5.5) to get In the next section we show that, for consistency, the cosmological constant term in (5.4) must disappear already at leading order in the non-relativistic limit.

Vacuum solution and the fate of the cosmological constant
We consider first the field equations (5.5) in vacuum, 2κǫ abc R ab e c + Λe a e b e c − z 1 e a S a = 0 , Qe a − qK a = 0 . with b and c being dimensionless parameters and a a mass squared parameter. Note that, due to the presence of the cosmological constant, vanishing curvatures do not satisfy the first two field equations (5.8). Substitution of (5.9) in the third Bianchi identity in (3.18) implies on the relation where the fourth equation in (5.8) was employed. On the other hand, curvatures (5.9) relate to each other through Applying the covariant derivative in relation (5.11) one finds Now, for the theory to be physically consistent, a torsionless vacuum solution should be acceptable, This solution satisfies the third and fourth field equations in (5.5). For this solution be consistent with the relation (5.10), one can quickly verify that we need to set However, (5.13) together with (5.14) do not satisfy (5.12). In fact, (5.12) will only be satisfied if the cosmological constant is set to zero. This is important if one wishes torsionless vacuum solutions together with non-trivial curvatures which might be associated with the mass parameters of the theory. The present analysis suggests that is safer to consider that the cosmological constant scale as (5.15) in the non-relativistic limit of the MZ action. The parameter ξ is some dimensionless positive real number to be determined (see below). In this case, the cosmological constant term vanishes in the non-relativistic limit. It is worth mentioning that there may exist solutions different of (5.9) which are consistent with vanishing torsion. Nevertheless, as we will show in the sequence, the vanishing of the cosmological constant is in fact needed if one requires that the non-relativistic limit of the maximally symmetric solution (2.11) is also a solution of the non-relativistic MZ field equations.
The action (5.4) with vanishing cosmological constant simplifies to From now on, we will refer to the theory described by the action (5.16) and field equations (5.17) as Galilei-Cartan gravity and we consider it for the rest of the paper. Torsionless configurations (5.13) and the curvatures We still need to fix ξ and a. This task can be achieved by looking at the non-relativistic limit of the maximally symmetric relativistic solution (2.11). Its decomposed version reads To obtain the non-relativistic limit of these relations we first perform the rescalings (3.7), (3.8), and (5.15) in (5.19), If we demand that, at the limit ζ −→ ∞, the solution ( In summary, for the non-relativistic limit of the maximally symmetric solution (2.11) be a solution of the non-relativistic limit of the MZ action, the cosmological constant must scale according to Λ −→ ζ −2 Λ in the non-relativistic limit process; Thence, the non-relativistic limit of the MZ action is actually independent of Λ; Nevertheless, Λ could still be probed in non-relativistic limits of gravitational systems through the solution (5.18); We remark that the configurations (5.18) are not solutions of the field equations (5.5) in vacuum. See also [43,44] for discussions about the non-relativistic limit of de Sitter and anti-de Sitter spacetimes.

Newtonian absolute time
An important feature that a Newton-Cartan gravity theory must have is to accept twistless torsion condition. This means that Newtonian absolute time can be safely employed. To show that this property is well accepted by the full set of field equations (5.17), let us impose Q = 0 to them. This imposition has no direct effect on the first two equations of the field equations (5.17). On the fourth equation, however, implies on For this equation to be true, K a must have a part which is independent of q. If not, then S m must not depend on the boost connection. Applying Q = 0 and expression (5.22) on the third equation of the field equations (5.17), we get The first two equations of equations (5.17) together with equations (5.22) and (5.23) are the field equations for strong twistless torsion condition. In other words, the set of equations to be solved if one wishes to work within Newtonian absolute time.

Solutions in the presence of matter
To find solutions of equations (5.17) in the presence of matter we first constrain S m to be of the form (4.39). Notice that, differently from the Galilean case, m a can depend on e, ω, and θ, but not on q. This form of S m is a bit restrictive, but sufficient for our initial purposes in probing the existence of consistent solutions in the presence of matter, at least at formal level. The field equations (5.17) reduce to 11 2κǫ abc R ab e c − z 1 e a S a = 4e a m a , We found earlier that for a strong twistless torsion solution be acceptable either S m must not depend on the boost connection or K a must have a piece which is independent of q. The first of (5.28) suggests that this independent part could be X a = 4 z 1 e c δm c δθ a . (5.29) Hence, the solution for K a can be obtained by combining the first of (5.27) with (5.29), which suggests again that the only way to accommodate Newtonian absolute time in the presence of matter for a matter content in the form (4.39) is indeed to impose that m c does not depend on the boost connection. It is also possible to show that expression (5.31) is satisfied for δm a /δθ a = 0.

Conclusions
In this work we have studied the non-relativistic limit of gravity theories in the first order formalism. Particularly, the Einstein-Hilbert and Mardones-Zanelli actions were considered [7,8]. The differential form language was extensively employed. This formalism allowed us to formally obtain some interesting new results. We remark that our main strategy to solve the field equations was to look for solutions for the curvatures and torsions 2-forms. This strategy is very useful because the field equations are algebraic with respect to curvatures and torsions.
In the case of the EH action, the non-relativistic limit is widely known, see for instance [10,15,25]. Nevertheless, we were able to obtain some novel results which are listed below: • Starting from the field equations (4.6) in vacuum, we have firstly obtained a solution constraining the space and time vierbeins, see (4.11). Even though this proposal solves the field equations (4.10), it is inconsistent with the degenerate Galilean metric nature; • Next, in the strong twistless torsion context, characterized by equation (4.13), non-trivial space curvature R and reduced torsion K were found, see equations (4.9) and (4.15). In agreement with the known literature [15]. Of course, in all cases boost curvature S a is free, since the boost connection θ does not appear in the action; • We have considered the possibility of non-trivial torsional solutions in vacuum by defining a 1form field n and the solutions (4.16) and (4.17). By setting R = S = 0, the field equations are satisfied and a kind of Newtonian Weitzenböck spacetime is obtained. In this context, by setting n to be constant, an explicit solution was found for the time vierbein q as displayed in expression (4.21). A non-Newtonian time was then derived as a function of the Newtonian absolute time, see (4.24). Asymptotically, the non-Newtonian and Newtonian time coincide. Nevertheless, the obtained solution still describes a strong twistless torsion situation since the time vierbein is parallel to n. In this example, by setting the spin connection to vanish, we were able to find a complete solution in terms of the explicit expressions of the vierbeins, see (4.29). Interestingly, the Although many physical insights were obtained from our formal analysis, it remains to apply the results to actual physical systems. These tests could provide some interesting bounds for some parameters of the MZ action, specifically z 1 . Moreover, new effects could emerge from these generalized gravity theory which could be tested in the non-relativistic context. Another point to be explored is the fact that MZ theorem holds for any spacetime dimension. So, generalization of the present results in other dimensions could be developed in the future. Furthermore, one could consider the usual extensions of the Galilei group, such as Bargmann and Schrödinger groups, in order to construct more general theories [25]. As mentioned in the Introduction, these theories could be relevant in Hořava-Lifshits [13] and string [30,31] contexts.