Kinematic Quantities and Raychaudhuri Equations in a $5D$ Universe

Based on some ideas emerged from the classical Kaluza-Klein theory, we present a $5D$ universe as a product bundle over the $4D$ spacetime. This enables us to introduce and study two categories of kinematic quantities (expansions, shear, vorticity) in a $5D$ universe. One category is related to the fourth dimension (time), and the other one comes from the assumption of the existence of the fifth dimension. The Raychaudhuri type equations that we obtain in the paper, lead us to results on the evolution of both the $4D$ expansion and $5D$ expansion in a $5D$ universe.


Introduction
As it well known, the (1+3) threading of a 4D spacetime was developed in order to relate physics and geometry to the observations. This theory is based on the existence of a congruence of timelike curves and it was successfully applied to: the relativistic cosmology [1], the study of gravitoelectromagnetism [2], the splitting of Einstein equations [3], and to some other physical theories.
In the present paper, we extend the above theory to a (1+1+3) threading of a 5D universe. As far as we know, there are two important fivedimensional gravity theories: the brane-world gravity [4], and the spacetime-matter theory [5]. In the theory we develop in the present paper, the 4D spacetime M is the base manifold of a submersion on the 5D universē M , while in the above mentioned theories, M is an embedded submanifold ofM . The new geometric configuration ofM enables us to consider two orthogonal line bundles: the temporal distribution TM which defines a congruence of timelike curves, and the vertical distribution VM which is tangent to a congruence of spacelike curves. The spatial distribution SM is the complementary orthogonal distribution to TM ⊕ VM in the tangent bundle ofM . The kinematic quantities (expansion, shear, vorticity) inM , are introduced as spatial tensor fields onM , which roughly speaking, behave like tensor fields on a three-dimensional manifold. In order to define a covariant derivative of such tensor fields with respect to any vector field on M , we introduce the Riemannian spatial connection, which is a metric linear connection on SM . Finally, we obtain three Raychaudhuri type equations which enable us to study the evolution of the expansion in the 5D universē M . Now, we outline the content of the paper. In Section 2 we present the geometric structure of the 5D universe (M ,ḡ) (see (2.12)), and construct the adapted frame and coframe fields {δ/δx 0 , δ/δx α , ∂/∂x 4 } and {δx 0 , dx α , δx 4 }, respectively. Then, in Section 3 we consider the 4D velocity ξ = δ/δx 0 and the 5D velocity η = ∂/∂x 4 , and obtain the expression of the line element ofḡ with respect to the adapted coframe field (cf. (3.4)). The 4D and 5D kinematic quantities: expansion, shear and vorticity tensor fields are defined in Section 4. They are geometric objects onM , whose local components with respect to adapted frame and coframe fields behave as tensor fields on a three-dimensional manifold. The Riemannian spatial connection, which we introduce in Section 5, has an important role into the study. Here, we show that the Levi-Civita connection of (M ,ḡ) is completely determined by the kinematic quantities, the Riemannian spatial connection, and the spatial tensor fields we introduced in Section 4 (cf. (5.12)). In Section 6 we show that each spatial tensor field defines a tensor field of the same type on M , and evaluate the covariant derivatives of both velocities (cf. (6.9) and (6.10)). In Section 7 we derive the Raychaudhuri equations with respect to the expansions Θ and K of the 5D universe (cf. (7.14), (7.16), (7.22)). In spite of the full generality approached in the paper, we can see that due to the spatial tensor fields and to the Riemannian spatial connection, the calculations can be easily handled. By using Raychaudhuri equations we state two results on the evolution of Θ and K in (M ,ḡ) (cf. Theorems 7.1, 7.2). Conclusions on the theory we develop in the paper are presented in Section 8.

Adapted Frames and Coframes in a 5D Universe
Let M and K be manifolds of dimension four and one respectively, and M = M ×K be the product bundle over M with fibre K. Then, a coordinate system (x i ) on M determines a coordinate system (x a ) = (x i , x 4 ) onM , where x 4 is the fibre coordinate. A general Kaluza-Klein theory onM is developed with respect to the gauge group U (1), and the coordonate transformations onM are given by As a consequence of (2.1) we deduce that the natural frame fields ∂/∂x a and ∂/∂ x a are related by Thus, the coordinate transformations onM are given by (2.5) and (2.1b).
where {∂/∂x i } is the lift of the natural frame filed on M toM . Next, suppose that the lift of ∂/∂x 0 toM is timelike with respect toḡ and denote by δ/δx 0 its projection on HM with respect to (2.4). Thus there exists locally onM a unique function A 0 , such that (2.7) By direct calculations, using (2.7), (2.6b) and (2.2b), we deduce that with respect to the coordinate transformations onM . From (2.8a) we conclude that there exists a globally defined horizontal vector field ξ onM , which is locally given by δ/δx 0 . Moreover, ξ is a timelike vector field onM .
To show this, we consider the line element ofḡ given by Then, taking into account that ξ is orthogonal to η, and using (2.7), (2.9b) and (2.3), we obtain By similar calculations we infer that Asḡ 00 < 0, there exists a globally defined non-zero function Φ onM such that ḡ( where A α and B α are locally defined functions onM . By direct calculations, using (2.6a), (2.8a), (2.2b) and (2.7) into (2.13), we deduce that (2.14) By the above geometric construction, we introduce into the study the orthogonal frame field {δ/δx 0 , δ/δx α , ∂/∂x 4 }, which we call adapted frame field onM . 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 described in this section is called a 5D universe, and is going to be the main object studied in the present paper. It is important to note that the 5D universe that we introduce in this paper is different from the ones considered in the well known theories: brane-world theory [4] and space-time-matter theory [5]. This is because in the present theory the 4D spacetime is the base manifold of a submersion defined on the 5D universe, while in the above theories the 4D spacetime is considered embedded in the 5D universe.

3
4D and 5D Velocities in a 5D Universe Let (M ,ḡ) be a 5D universe with the line element given by (2.9). From the previous section we conclude thatM admits a double threading by two orthogonal congruences of curves. These congruences are defined by the timelike vector field ξ and by the spacelike vector field η, which we call the 4D velocity and 5D velocity, respectively. The name 4D velocity for ξ is justified by the fact that its integral curves are tangent on their entire length to the horizontal distribution, whose fibres are four-dimensional. On the contrary, the integral curves of η are orthogonal to HM , and therefore they are intimately related to the fifth dimension. Moreover, by using (2.7), we obtain Thus, the distribution TM ⊕VM is an integrable distribution, and therefore, the 5D universe admits also a foliation by surfaces, whose transversal bundle ( [6], p.7) is the spatial distribution SM . Next, denote by h the Riemannian metric induced byḡ on SM , and put Then, by using (2.9b), (2.13), (3.2), (2.11) and (2.3), we deduce that Due to (2.3), (2.11) and (3.3), the line element with respect to the adapted coframe field has the simple form Now, in order to find the covariant local components of the above velocities we consider the 1-forms ξ ⋆ and η ⋆ given by Then we put and by using (3.6), (3.5), (2.7), (2.11) , (2.13) and (2.3), we obtain By using (3.8a) and (3.7b) into (2.7) and (2.13), we infer that Similarly, (2.15) becomes In what follows, we also call ξ ⋆ = (ξ a ) and η ⋆ = (η a ) the 4D velocity and 5D velocity in (M ,ḡ), respectively.

Spatial Tensor Fields and Kinematic Quantities in a 5D Universe
In this section we introduce spatial tensor fields in (M ,ḡ) as geometric objects whose local components behave as the ones of tensor fields on a threedimensional manifold. In particular, we define the expansion, shear and vorticity tensor fields as spatial tensor fields. First, by using (2.14a) into (3.2), we deduce that the local components of the Riemannian metric h on SM satisfy with respect to the coordinate transformations onM . Also, the entries of the inverse of the 3 × 3 matrix [h αβ ] satisfy Thus, h αβ and h αβ are locally functions on the five-dimensional manifold M , but they are transformed as the local components of some tensor fields of type (0, 2) and (2, 0) on a three-dimensional manifold. This leads us to an important category of geometric objects onM . Namely, we say that the functions T with respect to the coordinate transformations onM . By using (4.1) and (4.2), we see that h αβ and h αβ define spatial tensor fields of types (0,2) and (2,0), respectively. Next, by direct calculations using (2.7) and (2.13), we obtain (4.5) By using (2.2b), (2.8a) and (2.14a) into (4.4), it is easy to check that a α , b α , c α , d α define spatial tensor fields of type (0, 1), while ω αβ and η αβ define skew-symmetric spatial tensor fields of type (0, 2). Moreover, from (4.4c) we deduce that the spatial distribution is integrable, if and only if, Thus, extending the terminology from (1+3) threading of a 4D spacetime ( [1], p.81), we call ω αβ and η αβ the 4D vorticity tensor field and the 5D vorticity tensor field, respectively. The prefix 4D is placed in front of vorticity to emphasize that this object comes from the structure of the 4D spacetime M . On the contrary, the 5D vorticity is determined by the existence of the fifth dimension. Throughout the paper, we use this rule for some other geometric objects. Now, we define the spatial tensor fields of type (0,1): and by using (3.7b) and (3.8a) into (4.5e) and (4.5f), we deduce that Next, we denote by L the Lie derivative onM and define the functions Then, by using (3.2), (4.4a) and (4.4b) into (4.9), we obtain Moreover, applying δ/δx 0 and ∂/∂x 4 to (4.1), and taking into account (2.5a), (2.8a) and (2.2b), we deduce that Θ αβ and K αβ define symmetric spatial tensor fields of type (0,2). We call Θ αβ and K αβ the 4D expansion tensor field and 5D expansion tensor field, respectively. Taking the traces of these tensor fields, we obtain the 4D expansion function Θ and the 5D expansion function K, given by Finally, we define the trace-free symmetric spatial tensor fields Then, inspired by the terminology from the kinematic theory in a 4D spacetime, we call σ αβ and H αβ the 4D shear tensor field and 5D shear tensor field, respectively. As a conclusion of this section, we may say that {ω αβ , Θ αβ , Θ, σ αβ } and {η αβ , K αβ , K, H αβ } are the 4D kinematic quantities and the 5D kinematic quantities in the 5D universe (M ,ḡ), with respect to the congruence of curves defined by ξ and η, respectively.

The Riemannian Spatial Connection in a 5D Universe
LetM ,ḡ) be a 5D universe, and SM ⋆ be the dual bundle of the spatial distribution SM . Suppose that is a (p + q) − F(M )-multilinear mapping, and locally define the functions T γ 1 ···γp Then, it is easy to check that T γ 1 ···γp α 1 ···αq satisfy (4.3), and therefore they define a spatial tensor field of type (p, q). Conversely, suppose T The purpose of this section is to define covariant derivatives of the spatial tensor fields given either as in (5.1), or by their local components from (5.2). First, we consider the Levi-Civita connection∇ on (M ,ḡ) given by (cf. [7], p.61) for all X, Y, Z ∈ Γ(TM ). Then define the operator where s is the projection morphism of TM on SM with respect to (2.12).
It is easy to check that ∇ is a metric linear connection on SM , that is we have where h is the Riemannian metric on SM . We call ∇ the Riemannian spatial connection in the 5D universe (M ,ḡ) .
Taking into account that all the kinematic quantities have been defined by their local components, we need to characterize ∇ by its local coefficients with respect to an adapted frame field. First, we put Then, take X = δ/δx β , Y = δ/δx α and Z = δ/δx µ in (5.3), and by using (5.4), (5.6a), (3.2) and (4.4c), we obtain Note that formally, Γ γ α β look like the Christoffel symbols for a Levi-Civita connection on a three-dimensional manifold, but two main differences should be pointed out: (i) In general, h αβ are functions of all five variables (x a ), (ii) The usual partial derivatives are replaced here by the operators defined by (2.13).

(5.10)
As ∇ is a metric connection on SM , we have: Finally, by using (5.3), the spatial tensor fields introduced in the previous section, and the local coefficients of the Riemannian spatial connection, we express the Levi-Civita connection of the 5D universe (M ,ḡ), as follows: where we put (5.13)

Covariant Derivatives of 4D and 5D Velocities
In this section we show that the covariant derivatives of both the 4D velocity ξ ⋆ and 5D velocity η ⋆ are completely determined by the kinematic quantities and the spatial tensor fields we introduced in Sections 4 and 5. Also, we compare the results with what is known in the (1+3) threading of the 4D spacetime. First, by using (3.5) and taking into account that∇ is a metric connection, we obtain Then, consider the 4D acceleration . ξ a and the 5D acceleration . η a , given by Next, by using (3.9) we express the natural frame field {∂/∂x a } in terms of the adapted frame field, as follows: Then, by using (6.2), (6.1), (6.3), (5.12h) and (5.12i), we infer that Now, by using (6.1), (6.3), (5.12) and (5.9), we obtain Finally, taking into account (6.3)-(6.6), we deduce that (6.8) Next, we note that each spatial tensor field defines a tensor field of the same type onM . Here, we give some examples: Taking into account this transformation process of spatial tensor fields into tensor fields, we express (6.7) and (6.8) as follows: (6.10) Now, we consider some particular cases. First, suppose that both ξ and η are unit vector fields, that is Φ 2 = Ψ 2 = 1. Then, (6.9) and (6.10) becomē Also, from (5.12h), (5.12i) and (6.4), we deduce that condition (ii) is equivalent to . η a = 0. (6.14) Then, by using (4.6), (6.13) and (6.14) into(6.9) and (6.10), we obtain It is an interesting (and difficult as well) question, to find solutions for Einstein equations in a 5D universe satisfying the conditions (i) and (ii).

Raychaudhuri Equations in a 5D Universe
As it is well known, the evolution of the expansion in a 4D spacetime is governed by Raychaudhuri equation, which also plays an important role in the proof of Penrose-Hawking singularity theorems. So, it is a need for a study of the evolutions of both the 4D and 5D expansions Θ and K, given by (4.11). Such a study leads us to some equations of Raychaudhuri type, expressing the derivatives of Θ and K with respect to both variables x 0 (time) and x 4 (fifth dimension).
Finally, we shall state some Raychaudhuri equations which involve Θ |4 = ∂Θ ∂x 4 , and K |0 = δK δx 0 . First, by using (7.1), (5.12), (4.4a) and (2.3), we infer that (7.17) Contracting (7.17) by h αβ , and using (7.4c), (4.11), (5.9a) and (5.11b), we obtain On the other hand, we calculateR from (7.4c) by using (7.1), (5.12), (4.4b) and (2.11), and deduce that Then, by using (7.19) into (7.4c), and taking into account (4.11), (5.11b) and (5.9b), we infer that (7.20) By elementary calculations using (4.7), we obtain Finally, by using (7.21a) into (7.20), and (7.21b) into (7.18), we deduce that We call (7.22) the mixed Raychaudhuri equation in the 5D universe (M ,ḡ). The classical Raychaudhuri equation in a 4D spacetime has ben generalized in several directions. We recall here two of such generalizations. First, in [8] there were stated Raychaudhuri equations for single and two non-normalized vector fields in a 4D spacetime. Our Raychaudhuri equations are also with respect to two non-normalized vector fields, but in a 5D universe. Then, in [9] there have been obtained generalizations of Raychaudhuri equation for the evolution of deformations of a relativistic membrane of arbitrary dimension in an arbitrary background spacetime. The approach we developed in this paper is totally different from the one presented in [9].
Note that the above Raychaudhuri equations have been obtained in the most general 5D universe. In particular, suppose that the lift of ∂/∂x 0 from the 4D spacetime to the 5D universe is orthogonal to ∂/∂x 4 . Then, by (2.9b) and (2.10) we deduce that A 0 = 0, which implies δ/δx 0 = ∂/∂x 0 , (7.23) via (2.7). Now, each point P ofM can be considered as intersection point of an integral curve of ∂/∂x 0 on which we choose x 0 as parameter, with an integral curve of ∂/∂x 4 parametrized by x 4 . Since in this case a 0 = 0 (see (3.1b)), from (7.22) and (7.23) we deduce that at any point P ∈M . Thus, the rate of change of the 4D expansion Θ in the direction of the fifth dimension, is equal to the rate of change of the 5D expansion K in the time direction. Finally, we prove the following important results for the kinematic theory of a 5D universe.
The Theorems 7.1 and 7.2 might be useful in an attempt to prove singularity theorems for a 5D universe. Also, the three types of Raychaudhuri equations (7.14), (7.16) and (7.22) might have an important role in a study of the evolution of a concrete 5D universe.

Conclusions
In the present paper, for the first time in literature, we develop a kinematic theory in a 5D universe. The main tools in our approach are the spatial tensor fields and the Riemannian connection defined on the spatial distribution. It is worth mentioning that all the kinematic quantities (acceleration, expansion, shear and vorticity) are defined as spatial tensor fields, and therefore they should be considered as 3D geometric objects in a 5D universe. Now, we stress on the novelty brought by our paper into the study of a 5D universe. First, we mention the (1+1+3) splitting determined by the two vector fields ξ and η, which has an important role in relating geometry and physics to the observations. In this way, the 5D universe is filled up by nets determined by integral curves of both ξ and η. So, apart from the 4D kinematic quantities related to ξ, should be taken into consideration those determined by the assumption on the existence of the fifth dimension. Also, it is important to mention the three types of Raychaudhuri equations, which we obtained in the 5D universe (M ,ḡ). They describe the evolution of both the 4D expansion and 5D expansion along the two congruences determined by ξ and η. The Theorems 7.1 and 7.2 prove the existence of the singularities in both expansions Θ and K. Actually, these theorems state that caustics will develop in both congruences if convergence occurs anywhere. Taking into account the methods used to prove the Penrose-Hawking singularity theorems in a 4D spacetime, we think that such theorems might play an important role in a proof of the existence of singularities in a 5D universe.
Finally, we should mention that our study is mainly developed on the mathematical part of the kinematic theory in a 5D universe. This must be followed by detailed studies which might bring new insights on the 4D physics in the presence of the fifth dimension.