Equations of Motion with Respect to the $(1+1+3)$ Threading of a $5D$ Universe

We continue our research work started in"Kinematic Quantities and Raychaudhuri Equations in a $5D$ Universe"(Eur. Phys. J. C, 2015), and obtain in a covariant form, the equations of motion with respect to the $(1+1+3)$ threading of a $5D$ universe $(\bar{M}, \bar{g})$. The natural splitting of the tangent bundle of $\bar{M}$ leads us to the study of three categories of geodesics: spatial geodesics, temporal geodesics and vertical geodesics. As an application of the general theory, we introduce and study what we call the $5D$ Robertson-Walker universe.


Introduction
This paper is a continuation of our previous paper [1] on kinematic quantities and Raychaudhuri equations in a 5D universe. According to the new approach presented in [1], the 5D universeM = M × K is studied by means of the submersion ofM on the 4D spacetime M . Note that in all the other theories of a 5D universe the study was performed via an immersion of M inM (cf. [2][3][4][5]).
The kinematic quantities together with the spatial tensor fields and the Riemannian spatial connection enable us to obtain, in a covariant form, the equations of motion in (M ,ḡ). By using the natural splitting of the tangent bundle ofM we introduce into the study three categories of geodesics: spatial geodesics, temporal geodesics and vertical geodesics. We apply the general theory to what we call the 5D Robertson-Walker universe, which can be thought as a disjoint union of 4D Robertson-Walker spacetimes. In this case, the above three categories of geodesics are completely determined. Now, we outline the content of the paper. In Section 2 we recall from [1] the kinematic quantities with respect to the (1 + 1 + 3) threading of the 5D universe (M ,ḡ), and the Riemannian spatial connection ∇ on the spatial distribution SM . The complete characterization of the Levi-Civita connection on (M ,ḡ) (cf. (2.18)) enables us to write down in Section 3, for the first time in literature, the splitting of the equations of motions into three groups (cf.(3.6)). As an example, we present the 5D Robertson-Walker universe (see (3.7)) together with its equations of motion (cf. (3.14)). In Section 4 we introduce spatial, temporal and vertical geodesics and state their characterizations via the geometric objects defined on (M ,ḡ) (cf. Theorems 4.1 and 4.5). In case TM ⊕ VM is a Killing vector bundle, we show that spatial geodesics coincide with autoparallel curves of ∇ (cf. Theorem 4.3). Finally, we describe explicitly the above three categories of geodesics in a 5D Robertson-Walker universe (cf. Theorem 4.4 and Corollary 4.2). The conclusions on the research developed in the paper are presented in Section 5.
2 Kinematic quantities and the Riemannian spatial connection in a 5D universe In this section we describe the geometric configuration of a 5D universe that has been presented in [1]. LetM = M × K be a product bundle over M , where M and K are manifolds of dimensions four and one, respectively. The existence of two vector fields η and U onM and M respectively, induces a coordinate system (x a ) onM such that η = ∂ ∂x 4 and U = ∂ ∂x 0 . The coordinate transformations onM are given by Throughout the paper we use the ranges of indices: a, b, c, ... ∈ {0, 1, 2, 3, 4}, i, j, k, ... ∈ {0, 1, 2, 3}, α, β, γ, ... ∈ {1, 2, 3}. Also, for any vector bundle E overM denote by Γ(E) the F(M )-module of smooth sections of E, where F(M ) is the algebra of smooth functions onM .
Next, suppose thatM is endowed with a Lorentz metricḡ such that Denote by VM the line bundle overM spanned by η, and by HM its complementary orthogonal vector bundle in TM . Then, suppose that the lift of ∂/∂x 0 toM is timelike with respect toḡ, and denote by δ/δx 0 its projection on H(M ), that is, we have It is proved that exists a globally defined vector field ξ onM which is locally given by δ/δx 0 , and we haveḡ Thus the tangent bundle ofM admits the orthogonal decomposition where TM is the line bundle spanned by ξ, and SM is the complementary orthogonal distribution to TM in HM . We call TM , SM and VM the temporal distribution, the spatial distribution and the vertical distribution, respectively. According to (2.5) there exists an adapted frame field {δ/δx 0 , δ/δx α , ∂/∂x 4 } onM , where we put (2.6) Its dual frame field {δx 0 , dx α , δx 4 }, where we put is called an adapted coframe field onM . The pair (M ,ḡ) with the geometric configuration presented above is called a 5D universe, and it is the main object of study in the present paper. Now, denote by h the Riemannian metric induced byḡ on SM , and put Then, the line element with respect to the adapted coframe field is given by The 4D vorticity ω αβ and 5D vorticity η αβ in the 5D universe are given by (2.10) Also, the 4D expansion tensor field Θ αβ and the 5D expansion tensor field K αβ are given by The 4D expansion function Θ and the 5D expansion function K are the traces of the spatial tensor fields from (2.11), expressed as follows (2.12) Remark 1.1 It is worth mentioning that ω αβ , η αβ , Θ αβ and K αβ define spatial tensor fields of type (0, 2) on the 5D universe (M ,ḡ). According to the study presented in [1], this means that with respect to the transformations (2.1) they change like tensor fields of type (0,2) on a 3-dimensional manifold. ✷ An important geometric object introduced in [1] is the Riemannian spatial connection onM , which is a linear connection ∇ on the spatial distribution SM , given by where∇ is the Levi-Civita connection onM and S is the projection morphism of TM on SM with respect to (2.5). Locally, ∇ is given by where we put (2.15) Next, we express the Lie brackets of vector fields from the adapted frame field, as follows: where we put Note that a α , b α , c α and d α define spatial tensor fields of type (0, 1) onM . Finally, the Levi-Civita connection∇ on (M ,ḡ), is expressed as follows: where we put 3 Equations of motion in a 5D universe In this section we write down, in a covariant form, the equations of motion in the 5D universe (M ,ḡ). It is first time in literature when these equations are expressed by three groups of equations (cf. (3.6)), and in terms of kinematic quantities and of the local coefficients of the Riemannian spatial connection.
As an example of such 5D universe we present the 5D Robertson-Walker universe (cf.(3.7)), and state its equations of motion (cf.(3.14)). LetC be a smooth curve inM given by the equations Then by direct calculations using (2.3) and (2.6), we deduce that the tangent vector field d dt toC is expressed with respect to the adapted frame field where we put Next, after some long calculations by using (3.2) and (2.18) we obtain Now, by using (3.2), (3.4), (2.15b) and (2.15c), and taking into account that ω αβ and η αβ are skew-symmetric spatial tensor fields onM , we deduce that Finally, sinceC is a geodesic of (M ,ḡ), if and only if, the left hand side in (3.5) vanishes identically onM , we can state the following theorem.
Theorem 3.1 Let (M ,ḡ) be a 5D universe with kinematic quantities {ω αβ , η αβ , Θ αβ , K αβ } and with the Riemannian spatial connection ∇ given by (2.14) and (2.15). Then the equations of motion in (M ,ḡ) are expressed as follows: It is the first time in literature when the equations of motion in a 5D universe are expressed in terms of kinematic quantities and of some spatial tensor fields. The first type of equations of motion was presented in formula (5.28) of [2], wherein the natural frame field {∂/∂x a }, a ∈ {0, 1, 2, 3, 4}, has been used. In this way, no differences were noticed between temporal variable x 0 , the spatial variable (x α ) and the vertical variable x 4 . Also, in [6] the author stated another form of equations of motion, wherein the temporal distribution was not taken into consideration. The main difference between (5.6) of [6] and (3.6), is that the latter can relate physics and geometry with observations, via the kinematic quantities. Next, we construct an example of 5D universe and write down its equations of motion. Suppose that the line element of the Lorentz metricḡ has the particular form where f is a positive smooth function on an open region R of R 2 1 , and g αβ define a positive definite symmetric spatial tensor field g onM . Taking into account thstḡ given by (3.7) satisfies ḡ( ∂ ∂x 0 ) = 0, and using (2.3) and (2.6), we obtain (3.8) By using (3.8a) we see that the distributions SM , TM ⊕SM and SM ⊕VM are integrable, and as a consequence of (2.16) we deduce that In order to obtain the other kinematic quantities, we note that Then, by using (3.10) into (2.11) and (2.12), we infer that and respectively. Also, note that the local coefficients Γ γ α β of the Riemannian spatial connection ∇ given by (2.15a) become the usual Christoffel symbols with respect to the Riemannian metric g = (g αβ ). Finally, by using (3.7) -(3.13) into (3.6), we obtain the following equations of motion in a 5D universe (M ,ḡ) whose Lorentz metric is given by (3.7): (3.14) As leaves of TM ⊕ SM are locally given by x 4 =const., from (3.7) we see that the metric induced on them is of Robertson-Walker metric type (cf. [7], p.343), provided the leaves of SM are 3-dimensional manifolds of the same constant curvature. This happens in the case we takeM = I × S × K, where I is an open interval in R and S is a 3-dimensional Riemannian manifold of constant curvature k = 1, 0 or −1. Thus, we may think such a 5D universe as a disjoint union of Robertson-Walker spacetimes. For this reason we call the 5D universe (M ,ḡ) whose metric is given by (3.7), a 5D Robertson-Walker universe, with the warping function f.

Special geodesics in a 5D-universe
This section is devoted to the study of some particular classes of geodesics in a 5D universe (M ,ḡ). The existence of these geodesics is due to the splitting (2.5) of TM , which has been considered first in [1]. LetC be a curve inM given by (3.1). Then, we say thatC is a spatial curve, if it is tangent to the spatial distribution at any of its points. Thus, by (3.2) and (3.3), we deduce thatC is a spatial curve, if and only if, we have or equivalently If moreover, a spatial curveC is a geodesic of (M ,ḡ), we say that it is a spatial geodesic. Taking into account of (4.2) into (3.6), we state the following theorem.
Theorem 4.1 A spatial curveC is a spatial geodesic, if and only if, the following equations are satisfied: Next, we say thatC is an autoparallel curve inM with respect to the Riemannian spatial connection ∇, if it is a spatial curve satisfying Then, by using (4.1) and (2.14a) into (4.4) we obtain the following.
Theorem 4.2 A spatial curveC is an autoparallel curve with respect to ∇, if and only if, the equations (4.3a) are satisfied.
Thus, the relationship between spatial geodesics and autoparallel curves with respect to ∇, can be stated in the next corollary.
Corollary 4.1 A spatial geodesic of (M ,ḡ) must be an autoparallel curve with respect to ∇. Conversely, an autoparallel curve with respect to ∇ is a spatial geodesic, if and only if, (4.3b) and (4.3c) are satisfied.
Next, we define the Lie derivative of the Riemannian metric h on SM with respect to a vector field Z ∈ Γ(TM ⊕ VM ) as follows:  Theorem 4.3 Let (M ,ḡ) be a 5D universe such that TM ⊕ VM is a Killing vector bundle. Then a spatial curveC inM is a spatial geodesic, if and only if, it is an autoparallel with respect to the Riemannian spatial connection ∇.
Theorem 4.4 Let (M ,ḡ) be a 5D Robertson-Walker universe whose metric is given by (3.7). Then a curveC inM is a spatial geodesic, if and only if, the following conditions are satisfied: (i) The parametric equations ofC have the form where c and k are constants, and x γ = x γ (t), γ ∈ {1, 2, 3}, define a geodesic of a leaf of SM with respect to the Riemannian metric g = (g αβ ).
(ii) The warping function f admits (c, k) as critical point, that is, The above theorem says that spatial geodesics in (M ,ḡ) exist, if and only if, the warping function has at least one critical point (c, k). In that case, if S is the leaf of SM given by equations x 0 = c, x 4 = k, then the lifts of geodesics of (S, g) are spatial geodesics of (M ,ḡ). Finally, we say that a geodesicC of (M ,ḡ) is a temporal geodesic (resp. vertical geodesic) if it is tangent to TM (resp. VM ) at any of its points. Then, by using (3.2), (3.3) and (3.6) we state the following theorem.
where k α are constants.
(ii) A curveC is a vertical geodesic in (M ,ḡ), if and only if, we have: where λ i are constants.
Corollary 4.2 Let (M ,ḡ) be a 5D Robertson-Walker universe. Then we have the following assertions: (i) The temporal geodesics of (M ,ḡ) exists, and they are portions of lines given by x u = k u , u ∈ {1, 2, 3, 4}, where k u are constants.

Conclusions
The present paper has its roots in [1] and [8], wherein we developed new approaches on the (1 + 1 + 3) threading of a 5D universe and on the (1 + 3) threading of a spacetime, respectively. The main geometric objects used in the paper are: the adapted frame and coframe fields, the kinematic tensor fields, and the Riemannian spatial connection. By using these geometric objects, we state in a 5D covariant form, the equations of motion in (M ,ḡ). The splitting of such equations in three groups (see (3.6)) enables us to consider the spatial, temporal and vertical geodesics. We note the interrelations between spatial geodesics and autoparallel curves with respect to the Riemannian spatial connection (cf. Corollary 4.1). In particular, if TM ⊕ VM is a Killing vector bundle, we show that spatial geodesics coincide with autoparallel curves of ∇ (cf. Theorem 4.3). This shows that ∇ has an important role in the study of geometry and physics of a 5D universe.
As a new example of 5D universe in the sense considered in [1], we present what we call the 5D Robertson-Walker universe, whose metric is given by (3.7). We show that such a universe can be thought as a disjoint union of 4D Robertson-Walker spacetimes. The equations of motion have the simple form (cf. (3.14)), wherein the first two groups remind us of the equations of motion in a 4D Robertson-Walker spacetime (cf. [7], p.353). Also, we show that the projections of spatial geodesics of (M ,ḡ) on the leaves of SM are just geodesics of the leaves with the Riemannian metric g (cf. Theorem 4.4).
Finally, we note that throughout the paper, the spatial tensor fields enable us to apply the principle of covariance, which is one of the most powerful ideas in modern physics, This will be seen more evidently in a forthcoming paper on the splitting of Einstein equations in a 5D universe.