Modified newtonian dynamics and non-relativistic ChSAS gravity

In the context of the non-relativistic theories, a generalization of the Chern--Weil-theorem allows us to show that extended Chern--Simons actions for gravity in d=4 invariant under some specific non-relativistic groups lead to modified Poisson equations. In some particular cases, these modified equations have the form of the so-called MOND approach to gravity. The modifications could be understood as due to the effects of dark matter. This result could leads us to think that dark matter can be interpreted as a non-relativistic limit of dark energy.


I. INTRODUCTION
In Ref. [1] it was shown that: (i) it is possible to obtain non-relativistic versions of both generalized Poincaré algebras B n [2][3][4] and generalized AdS-Lorentz algebras [3,[5][6][7]. (ii) Using an analogous procedure to that used in Ref. [8], it is possible to find the non-relativistic limit of the five-dimensional Einstein-Chern-Simons gravity which leads to a modified version of the Poisson equation.
On the other hand, in the context of the so-called extended gauge theory, Antoniadis, Konitopoulos and Savvidy introduced background-free gauge invariants including gauge potentials described by higher degree differential forms [9][10][11][12]. This construction has allowed to study in Refs. [13][14][15] a generalization of the Chern-Weil theorem, which made possible to construct generalized 2n and (2n + 2)-dimensional trangression forms, to reproduce the (2n + 2)-dimensional Chern-Simons forms obtained in Refs. [11,12] and to show that the 2n-dimensional Chamseddine's topological gravity [16][17][18] corresponds to a Chern-Simons-Antoniadis-Savvidy (ChSAS) form. These mathematical results were then used to study the construction of an off-shell invariant ChSAS action for gravity in d = 4 which is gauge quasi-invariant under the generalized gauge transformations for the Maxwell algebra. In Ref. [19] it was shown that the extended invariants found by Antoniadis, Konitopoulos and Savvidy can also be obtained by gauging free differential algebras.
It is the purpose of this paper to consider the non-relativistic versions of the generalized Poincaré algebra B 4 (Maxwell algebra [20,21]) denoted by GB 4 and the AdS-Lorentz algebra AdSL 4 denoted by GL 4 to find the non-relativistic limit of the four dimensional ChSAS action for gravity. This paper is organized as follows. In Section 2 we present a short review: (i) on the non-relativistic versions of the Maxwell algebra B 4 and the AdS-Lorentz, known as galilean algebra type I GB 4 and galilean algebra type II GL 4 respectively; (ii) on ChSAS gravity. In Sections 3 and 4, using an analogous procedure to that used in Ref. [8], generalizations of the Newtonian gravity are found by gauging the GB 4 and GL 4 algebras. In section 5 we study the possible relations between the Newtonian gravities found in the previous sections and the so called Modified Newtonian Dynamics (MOND) models. Finally our conclusions are presented in Section 6.

II. NON-RELATIVISTIC GB 4 AND GL 4 ALGEBRAS AND CHSAS GRAVITY
In [1] were found the non-relativistic versions of the generalized Poincaré algebras B n [2,4] and the generalized AdS-Lorentz algebras AdSL n [3,[5][6][7]22] using a generalized Inönü-Wigner contraction. These non-relativistic algebras were called generalized Galilean algebras type I and type II and denoted by GB n and GL n respectively.
A. Galilean algebra type I GB 4 In Ref.
[1] was found in that, separating the spatial and temporal components in the and taking the limits c, R → ∞ (with c the speed of light and R is the cosmic radius), the generators of the GB 4 algebra satisfy the following non-vanishing commutation relations where i, j, k, l = 1, 2, 3. This algebra correspond to a S-expansion of the Newton-Hooke algebra with central extension [23][24][25], whose representation can be obtained from the SO (3,2) algebra using the gamma matrices where γ µ satisfied the Clifford algebra γ µ γ ν + γ ν γ µ = 2η µν with µ, ν = 0, 1, . . . , 4 and and Γ * = 2M with Γ * = γ 0 γ 1 γ 2 γ 3 γ 4 gives us the commutation relations of Newton-Hooke algebra with central extension and allows to know the invariant tensors of generalized Galilean algebras by means of the S-expansion procedure [23,24].
B. Galilean algebra type II GL 4 The GL 4 algebra corresponds to the nonrelativistic limit of the AdSL 4 ≡ so(D − 1, 1) ⊕ so(D − 1, 2) algebra [1]. This algebra was introduced in Refs. [5][6][7], reobtained from the Maxwell algebra in Ref. [31] using a method known as deformation of Lie algebras and later from de AdS algebra in Ref. [3] using the so called S-expansion procedure.
In Ref.
[1] it was found that separating the spatial and temporal components in the generators {P a , J ab , Z ab } of AdSL 4 algebra, performing the rescaling and taking the limit c, R → ∞, the generators of the GL 4 algebra satisfy the following non-vanishing commutation relations This algebra can be also written as the direct sum

Newton-Hooke with central extension and E(3) is the Euclidean algebra in three dimensions.
In fact, carrying out the base changẽ in the GL 4 algebra (3), we find that the only non vanishing commutator are: (a) the Newton-Hooke algebra with central extension (c) J ij conmutators are given by From where we can see that the Newton Hooke algebra with central extension is subalgebra of GL 4 which correspond to the non-relativistic limit of AdS algebra. It is interesting to note that in (5) the generatorZ ij corresponds to a rotation in so (3) and On the other hand, in (6) J ij corresponds to a rotation in so (3),Z i is a boost, M corresponds to the center of the algebra and P i , H are space and time translation operators respectively.

C. Pontryagin-Chern and ChSAS forms
The idea of extending the Yang-Mills fields to higher rank tensor gauge fields was used in Refs. [10][11][12] to construct gauge invariant and metric independent forms in higher dimensions. These forms are analogous to the Pontryagin-Chern forms in Yang-Mills gauge theory and are given by, where F 3 , F 4 , F 6 , F 8 are the field-strength tensors for the gauge fields A 2 , A 3 , A 5 , and A 7 respectively and where C (2n+2) ChSAS are the corresponding Chern-Simons-Antoniads Savvidy forms which are given by In Ref. [19] were shown that the ChSAS invariants found in Refs. [10][11][12] can be constructed from a algebraic structure known as free differential algebra (FDA).
These mathematical results were used to construct a four-dimensional action for gravity in whose Lagrangian L ChSAS = F, A 2 ≡ F, B is a ChSAS form.

III. NEWTONIAN CHSAS GRAVITY FOR THE GB 4 ALGEBRA
In Ref. [8] was shown how the Newton-Cartan formulation of Newtonian gravity can be obtained from gauging the Bargmann algebra. In Refs. [13] it was shown that the gauging of B 4 allows us to construct a four-dimensional ChSAS gravity which leads general relativity in a certain limit. On the other hand, we have seen that the non-relativistic version of the B 4 algebra is given by the GB 4 algebra. In this Section we show that, using an analogous procedure to that used in Ref. [8], it is possible to find a generalization of the Newtonian gravity.
A. Gauging the GB 4 algebra The one-form gauge connection A valued in the GB 4 algebra is given by where ν, l are parameters of dimensions velocity and length respectively. The corresponding where T i = De i and R ij = dω ij + ω ik ω j k . Here D is the covariant derivative with respect to the SO(3) transformations. For the two-form gauge potential B we can write whose associated 3-form curvature is given by where H 0 = dB 0 , These equations are analogous to Eq. (2.13) of Ref. [26] and Eq. (III.6.47) of Ref. [27] and therefore they are not a FDA. However, when the condition The problem now is to express the form B in terms of the one-forms τ, e i , m, ω ij , ω i , k ij , k i of the non-relativistic Maxwell algebra.
To express the 2-forms as the wedge product of the 1-forms, we follow a procedure developed in Refs. [26,28]. We impose the ansatz where a i , . . . , a 6 , b 1 , b 2 , c 1 , c 2 , d 1 , . . . , d 9 , f 1 , . . . , f 6 , g 1 , . . . , g 9 and h 1 , . . . , h 6 are arbitrary constants. Introducing (15) in the corresponding FDA for the fields B i , B 0 , B (m) , B ij , G i , β ij , β i , we find relations between these constants. These relations lead to the following form to (15) There are 14 arbitrary constants in the FDA expansion in terms of 1-forms; the fields given by Eqs. (16) represent the most general solution that can be built with the fields τ, e i , m, ω ij , ω i , k ij , k i . Any choice of the constants represent a solution to the FDA.

B. Non-relativistic ChSAS Lagrangian
Using the theorem VII.2 of Ref. [23] it is possible to show that the invariant tensors for GB 4 are given by being α 0 and α 2 arbitrary constants. The ChSAS Lagrangian is given then by the following Chern-Simons form whose explicit form is Introducing the FDA expansion given by Eqs. (16) in (19), we find that when v, l → ∞ we find that the non-relativistic ChSAS Lagrangian for the GB 4 algebra takes the form In presence of matter, the complete Lagrangian of the theory is given by The variation of L leads to the following equations of motion we find that the field equation (27) can be written as The contraction of this equation with g σδ leads to Taking the components 00 of (30) and using (31) we find Following the procedure of Ref. [8], we find R 00 = ∇ 2 φ, g 00 = τ 0 τ 0 = 1, T 00 = ρ.

IV. MOND THEORY CONNECTION
The modified form of Poisson equation (34) suggests a possible connection with the socalled MOND approach to gravity interactions. In fact, the first complete MOND theory was constructed by Milgrom and Bekenstein in Ref. [30]. It involves a modification of the Poisson equation and can be derived from the following Lagrangian where ϕ is the gravitational potential (meaning that for a test particle a = − ∇ϕ), ρ denotes the matter mass density, and F(x 2 ) is an arbitrary function. The variation of L Mond with respect to ϕ leads to the following field equations . A little bit of algebra allows us to see that this equation takes the form Comparing this last equation with (34) we can see that in some particular cases the MOND approach to gravity could coincide with such modified Poisson equation. In fact, if we consider the case where the k αβ β field does not depend on time and its non zero components are given by k i0 where D α is the covariant derivative. Following Ref. [8] with Γ i 00 = δ ij ∂ j φ(x) we can see that Introducing (37,38) in (34) we find Comparing (39)  Let us now consider the one and two forms gauge fields A, B valued in the GL 4 algebra Following the same procedure used in the previous section, it is found that the corresponding non-relativistic ChSAS Lagrangian leads to the following generalized Poisson equation Comparing (42) with (36) we can see that in some particular cases, the MOND approach to gravity could coincide with such modified Poisson equation (42). If we consider again the case where the k αβ β field does not depend on time and its non-zero components are given by we find k αγ α k β γ β − k αγ β k β γ α = 0, k 0γ 0 k β γ β − k 0γ β k β γ 0 = 0, k αγ 0 k 0 γ α − k αγ α k 0 γ 0 = 0. So that, Eq. (42) takes the form where α = − 1 α 0 (a 4 − a 2 + a 3 ) , β = − α 2 α 0 .

VI. COMMENTS
In the present work we have studied the non-relativistic versions of the generalized Poincaré algebra B 4 denoted by GB 4 and the AdS-Lorentz algebra AdSL 4 denoted by GL 4 to find the non-relativistic limit of the four dimensional ChSAS action for gravity.
We have shown that the gauging of non-relativistic algebras GB 4 and GL 4 permits to construct generalizations of the Newtonian gravity which leads to modified versions of the Poisson equation. In some particular cases, it is possible to find relations between the generalized Newtonian gravities and the so called MOND model. These modifications to the Poisson equation, could be compatible with dark matter and would allow us to conjecture that dark matter could be interpreted as the non-relativistic limit of dark energy.