Generalized Galilean Algebras and Newtonian Gravity

The non-relativistic versions of the generalized Poincar\'{e} algebras and generalized $AdS$-Lorentz algebras are obtained. This non-relativistic algebras are called, generalized Galilean algebras type I and type II and denoted by $\mathcal{G}\mathfrak{B}_{n}$ and $\mathcal{G}\mathfrak{L}_{_{n}}$ respectively. Using a generalized In\"{o}n\"{u}--Wigner contraction procedure we find that the generalized Galilean algebras type I can be obtained from the generalized Galilean algebras type II. The $S$-expansion procedure allows us to find the $\mathcal{G}\mathfrak{B}_{_{5}}$ algebra from the Newton--Hooke algebra with central extension. The procedure developed in Ref. \cite{newton} allow us to show that the non-relativistic limit of the five dimensional Einstein--Chern--Simons gravity is given by a modified version of the Poisson equation. The modification could be compatible with the effects of Dark Matter, which leads us to think that Dark Matter can be interpreted as a non-relativistic limit of Dark Energy.


I. INTRODUCTION
In Refs. [2,3] was shown that the S-expansion procedure allows to construct Chern-Simons gravities in odd dimensions invariant under an algebra referred as B m algebra and Born-Infeld gravities in even dimensions [4][5][6][7] invariant under a certain subalgebra of the B m algebra, leading to general relativity in a certain limit. The B m algebras, which could be also called 'generalized Poincaré algebras', was constructed from AdS-algebra and a particular semigroup denoted by S α=0 , which is endowed with the multiplication rule λ α λ β = λ α+β if α + β ≤ N + 1; λ α λ β = λ N +1 if α + β > N + 1.
In Ref. [8] was shown that the so-called AdS-Lorentz algebra so (D − 1, 1)⊕ so (D − 1, 2) algebra [9][10][11] in D dimensions can be obtained from AdS-algebra so (D − 1, 2) by means of the S-expansion procedure with a semigroup which is kown as S (2) M . This AdS-Lorentz algebra is related to the so called Maxwell algebra [12,13] via a contraction process [14].
Recently was shown in Ref. [15] that the resonant S-expansion of the AdS Lie algebra leads to a generalization of the AdS-Lorentz algebra when it is used S as semigroup, which is endowed with the multiplication rule λ α λ β = λ α+β if α + β ≤ N; λ α λ β = λ α+β−2[(N +1)/2] if α + β > N. These algebras are called generalized AdS-Lorentz algebras. In this same Ref. [15] was found that a generalized Inönü-Wigner contraction of the generalized AdS-Lorentz algebras provides the so called generalized Poincaré algebras, B m .
On the other hand, in Ref. [1] was shown how the Newton-Cartan formulation of Newtonian gravity can be obtained from gauging the Bargmann algebra, i.e. the centrally extended Galilean algebra. This paper is organized as follows: In Section II it is shown that, using an analogous procedure to that used in Ref. [16] it is possible to obtain the non-relativistic versions of the generalized Poincaré algebras and generalized AdS-Lorentz algebras. The nonrelativistic algebras will be called, generalized Galilean algebras type I and type II and denoted by GB n and GL n respectively. In Section III it is shown that the generalized Galilean algebras type I can be obtained by a generalized Inönü-Wigner contraction of generalized Galilean algebras type II. In this section it is also shown that the procedure of S-expansion allows us to find the GB 5 algebra from the Newton Hooke algebra with central extension. In Section IV it is show that the non-relativistic limit of Einstein-Chern-Simons gravity is given by a modified version of the Poisson equation. In Section V it is found that, using an analogous procedure to that used in Ref. [1], it is possible to find a generalization of the Newtonian gravity. Finally our conclusions are presented in Section VI.
The use of the procedure developed in Ref. [16], allow us to show that it is possible to obtain the non-relativistic versions of the generalized Poincaré algebras and of the generalized AdS-Lorentz algebras. The nonrelativistic algebras will be called, generalized Galilean type I and type II algebras and denoted by GB n and GL n respectively. We consider the particular cases n = 4, 5.
Consider now the non relativistic versions of the Maxwell and B 5 algebras. Separating the spatial temporal components in the generators {P a , J ab , Z a , Z ab }, performing the rescaling N and then taking the limit c, R → ∞, we find that: (i) the generators of the non-relativistic version of the Maxwell algebra, which we will denote by GB 4 , satisfy the following commutation relations and (ii) the generators of the non-relativistic version of the B 5 algebra [2], [15] which we will denote by GB 5 , satisfy the commutation relations where ν = c/R is a finite constant with c the speed of light and R the universe radius.
Following the same procedure used previously, it is possible to find non-relativistic versions of the generalized AdS-Lorentz algebras, which will be called generalized Galilean type II algebras and denoted and GL n . It is direct to show that using a Inönü-Wigner contraction procedure we can obtain the GB n from GL n .

III. GB n ALGEBRAS
A. GB 4 algebra from the GL 4 algebra From Ref. [15] we know that the generalized Poincaré algebras can be obtained from the generalized AdS-Lorentz algebras by means of a generalized Inönü-Wigner contraction.
This property of these relativistic algebras is inherited by their corresponding nonrelativistic algebras. We consider, as an example, the contraction of the GL 4 algebra. In fact, performing the following rescaling the GL 4 algebra provides in the limit λ −→ 0 the GB 4 algebra (1). Similarly we can get the GB 5 algebra from the GL 5 algebra.

B. GB 5 algebra from the Newton Hooke algebra
In Refs. [2,3] was shown that the S-expansion procedure allows to obtain the 'generalized Poincaré algebras' B n from AdS-algebra. In this subsection it is shown that the procedure of S-expansion allows us to find the GB 5 algebra from the Newton Hooke algebra with central extension. A representation of AdS algebra is given by the matrices where the matrices γ µ satisfy the Clifford algebra γ µ γ ν +γ ν γ µ = 2η µν , with µ, ν : 0, 1, ···, 5 and it leads to the commutation relations of the Newton Hooke algebra with central extension Ref. [17].
From Ref. [18] we find that the non-vanishing components of the invariant tensor for so(4, 2) are given by Following an analogous procedure to that used in Ref. [16], we find that the only nonzero components of the invariant tensor for the 5-dimensional Newton Hooke algebra with central extension.
Following the definitions of Ref. [19] (see also [20]) let us consider the S-expansion of Newton Hooke algebra with central extension using as semigroup S After extracting a resonant and reduced subalgebra, one finds the GB 5 algebra, given by (2). The invariant tensors for GB 5 can be obtained from Newton Hooke algebra with central extension. Using V II.2 from Ref. [19] we find where the constants α 1 and α 3 are dimensionless and the factors l,v are introduced to display the dimension of · · · , where l and v are parameters of dimension length and velocity respectively.

IV. NON-RELATIVISTIC LIMIT OF EINSTEIN-CHERN-SIMONS GRAVITY
The five dimensional Chern-Simons lagrangian for the B 5 algebra is given by [2] L (5) where α 1 , α 3 are parameters of the theory, l is a coupling constant, R ab = dω ab + ω a c ω cb corresponds to the curvature 2-form in the first-order formalism. In Ref. [21] was considered that in the presence of matter the lagrangian is given by ChS is the five-dimensional Chern-Simons lagrangian given by (4), L M = L M (e a , h a , ω ab ) is the matter lagrangian and κ is a coupling constant related to the effective Newton's constant.
The Lagrangian (4) shows that standard, five-dimensional General Relativity emerges as the l → 0 limit of a Chern-Simons theory for the generalized Poincaré algebra B 5 . Here l is a length scale, a coupling constant that characterizes different regimes within the theory.
The variation of the lagrangian (4) w.r.t. the dynamical fields vielbein e a , spin connection ω ab , h a and k ab , leads to the following field equations [22] where we have considered that the torsion vanishes T a = 0 ( δL M δω ab = 0) and k ab = 0, while the field h a is associated, in the context of Einstein-Chern-Simons cosmology, with the dark energy, as shown in Refs. [21,22].
In the case where the equations (5-7) satisfy the cosmological principle and the ordinary matter is negligible compared to the dark energy, we find that the equations (5 -7) take the The field equations (8-12) were completely resolved for the age of Dark Energy in Ref. [22], where was find that the field h a has a similar behavior to that of a cosmological constant.
In fact, in Section 3 of Ref. [22] has been found solutions that describes accelerated expansion for the three possible cosmological models of the universe. Introducing (6) in (5) where δL M δh a , and * is the Hodge star operator.
In the limit of weak gravitational field one assumes that the world metric tensor g µν is not very much different from the Minkowski metric η µν = diag (−1, 1, ..., 1). In fact, it can be then written in the form g µν = η µν + h µν , where h µν represents the small corrections to the flat space-time metric η µν due to the presence of a weak gravitational field. In this approximation |h µν | << 1, so that terms of order higher than the first in h µν can be neglected in the field equations. So that, Introducing an orthonormal basis and using the first and second structural equations From (13) we can see In the limit of weak gravitational field one assumes that the leading term in the energymomentum tensors are Υ 00 = ρ and Υ (h) 00 = ρ (h) so that On the another hand the motion of a particle described by the geodesic equation where x µ = {x 0 , x i } = (t, x i ). In the nonrelativistic limit eq. (16) becomes and in the limit of weak gravitational field we can put g µν = η µν + h µν , with |h µν | << 1, and we can neglect terms of order h 2 and higher. From the definition of the Christoffel symbols we have Γ i 00 = − 1 2 δ i j ∂ j h 00 , where we have assumed that the field is static, i.e., ∂ 0 g µν = 0. So that the geodetic equation is given by which coincides with Newton equation of motion d 2 x i dt 2 = −∂ i φ provided that h 00 = −2φ, therefore Γ i 00 = δ ij ∂ j φ. This means that the only non-zero component of the Riemann tensor corresponding to connection Γ i 00 is given by R i 0j0 = δ ik ∂ k ∂ j φ, so from (15) we can conclude that the nonrelativistic limit of the five dimensional Einstein-Chern-Simons gravity is a modified version of the Poisson equation given by where k 1 = β 1 8α 3 , k 2 = β 2 24α 3 and α = 3α 1 α 3 [21]. If in (19) we choose α = 0 or k 2 = 0 we obtain the Poisson equation in five dimensions provided that k 1 = 8πG.

V. NEWTON-CHERN-SIMONS GRAVITY
In Ref. [1] was shown how the Newton-Cartan formulation of Newtonian gravity can be obtained from gauging the Bargmann algebra. In Refs. [2] was shown that the gauging of B 5 lead to a five-dimensional Chern-Simons gravity which empties into general relativity in a certain limit. On the other hand, we have seen that the non-relativistic version of the B 5 algebra is given by the GB 5 algebra and that the procedure of S-expansion allows us to find the GB 5 algebra from the Newton Hooke algebra with central extension. In this Section we show that, using an analogous procedure to that used in Ref. [1], it is possible to find a generalization of the Newtonian gravity.
A. Gauging the GB 5 algebra We start with a one-form gauge connection A valued in the GB 5 algebra is given by where l and v are parameters of dimension length and velocity respectively. The corresponding two-form curvature is given by where with T i = de i + ω ij e j , R ij = dω ij + ω i k ω kj . Since the gauge connection A transforms as we find, using the GB 5 algebra, that the variations of the gauge fields are given by where the derivative D is covariant with respect to the J-transformations.
Following Ref. [1] we impose now several curvature constraints. These constraints convert the P and H transformations into general coordinate transformations in space and time.
We write the parameter of the general coordinate transformations ξ λ as Here we have used the inverse spatial vielbein e λ i and the inverse temporal vielbein τ λ defined by [1] e i µ e µ j = δ i j , τ µ τ µ = 1, τ µ e i µ = 0 From (23) we can see that only the gauge fields e i µ , τ µ , m µ , h i µ , h 0 µ and n µ transform under the P and H transformations. These are the fields which should remain independents, while the remaining fields will be dependent upon the aforementioned fields. This can be achieved with the following constraints An analogous procedure to that used in Ref. [1] allows us to obtain the k µ ij and k µ i fields. In fact, using the constraints (26) we find

B. Newton-Chern-Simons Lagrangian
A Chern-Simons lagrangian form L ChS (A, 0) ≡ Q 2n+1 (A, 0) is a diferential form defined for a connection, whose exterior derivative yields a Chern class. Although the Chern classes are gauge invariant, the Chern-Simons forms are not; under gauge transformations they change by a closed form. A transgression form Q 2n+1 (A 1 , A 2 ) on the other hand, is an invariant differential form whose exterior derivative is the difference of two Chern classes. It generalizes the Chern-Simons form with the additional advantage that it is gauge invariant.
To obtain the lagrangian for 5-dimensional Chern-Simons gravity we use subspaces separation method introduced in Ref. [23] and write L ChS in terms of a transgression form, a Chern-Simons form and a total exact form where, where now A t = tA 2 = tω.
So that if we don't consider boundary terms the Chern-Simons lagrangian is given by:

VI. COMMENTS
In the present work we have shown that: (i) it is possible to obtain the non-relativistic versions of both generalized Poincaré algebras and generalized AdS-Lorentz algebras. These non-relativistic algebras are called generalized Galilean type I and type II algebras and denoted by GB n and GL n respectively. (ii) The procedure of S-expansion allows us to find the GB 5 algebra from the Newton-Hooke algebra with central extension. (iii) Using an analogous procedure to that used in Ref. [1], it is possible to find the non-relativistic limit of the five dimensional Einstein-Chern-Simons gravity which lead us to a modified version of the Poisson equation.
It is interesting to note that the B 5 algebra is a generalization of the Poincaré algebra which includes the extra generators Z ab and Z a . This algebra leads to a Chern-Simons lagrangian which coincides with the Einstein-Hilbert lagrangian in a certain limit, even if the new gauge field vanishes and therefore leads to newtonian gravity in the non relativistic limit.
The generators Z i0 , Z ij , are the space-time components of the Z ab = (Z i0 , Z ij ) relativistic generators, whose gauge field k ab = (k i , k ij ) we fix to k ab = 0 in the field equations.
On the other hand the gauge field h a = (h 0 , h i ) associated to the generators Z a generates modifications in the Einstein equations which can be interpretated, in the cosmological contex, as an effect due to the dark energy [21,22]. This modifications leads, in the non- where φ is the gravitational potencial (meaning that for a test particle a = − ∇φ), and ρ denotes the matter mass density. The corresponding equation for φ is given by where µ( √ y) = df (y)/dy, which can be written as Comparing this last equation with equation (43), we can see that in some particular cases the MOND approach to gravity could coincide with the modified Poisson equation (