On Higher Dimensional Kaluza-Klein Theories

We present a new method for the study of general higher dimensional Kaluza-Klein theories. Our new approach is based on the Riemannian adapted connection and on a theory of adapted tensor fields in the ambient space. We obtain, in a covariant form, the fully general 4D equations of motion in a (4 + n)D general gauge Kaluza-Klein space. This enables us to classify the geodesics of the (4 + n)D space and to show that the induced motions in the 4D space bring more information than motions from both the 4D general relativity and the 4D Lorentz force equations. Finally, we note that all the previous studies on higher dimensional Kaluza-Klein theories are particular cases of the general case considered in the present paper.


Introduction
As it is well known, by the Kaluza-Klein theory, the unification of Einstein's theory of general relativity with Maxwell's theory of electromagnetism was achieved.In a modern terminology, this theory is developed on a trivial principal bundle over the usual 4D spacetime, with (1) as fibre type.Thus, a natural generalization of Kaluza-Klein theory consists in replacing (1) by a nonabelian gauge group  (cf.[1][2][3][4][5]).There have been also some other generalizations wherein the internal space has been considered a homogeneous space of type / (cf.[6,7]).
Two conditions have been imposed in the classical Kaluza-Klein theory and in most of the above generalizations: the "cylinder condition" and the "compactification condition." The former condition assumes that all the local components of the pseudo-Riemannian metric on the ambient space do not depend on the extra dimensions, while the latter requires that the fibre must be a compact manifold.
In 1938, Einstein and Bergmann [8] presented the first generalization in this direction.According to it, the local components of the 4D Lorentz metric in a 5D space are supposed to be periodic functions of the fifth coordinate.Later on, two other important generalizations have been intensively studied.One is called brane-world theory and assumes that the observable universe is a 4-surface (the "brane") embedded in a (4 + )-dimensional spacetime (the "bulk") with particles and fields trapped on the brane, while gravity is free to access the bulk (cf.[9]).The other one is called space-time-matter theory and assumes that matter in the 4D spacetime is a manifestation of the fifth dimension (cf.[10,11]).
Recently, we presented a new point of view on a general Kaluza-Klein theory in a 5D space (cf.[12]).We removed both the above conditions and gave a new method of study based on the Riemannian horizontal connection.This enabled us to give a new definition of the fifth force in 4D physics (cf.[13]) and to obtain a classification of the warped 5D spaces satisfying Einstein equations with cosmological constant (cf.[14]).
The present paper is the first in a series of papers devoted to the study of general Kaluza-Klein theory with arbitrary gauge group.More precisely, our approach is developed on a principal bundle  over the 4D spacetime , with an dimensional Lie group  as fibre type.Moreover, both the cylinder condition and the compactification condition are removed.In other words, the theory we develop here contains as particular cases all the other generalizations of Kaluza-Klein theory that have been presented above.
The whole study is based on the Riemannian adapted connection that we construct in this paper and on a 4D tensor calculus that we introduce via a natural splitting of the tangent bundle of the ambient space.We obtain, in a covariant form Advances in High Energy Physics and in their full generality, the 4D equations of motion as part of equations of motion in a (4 + )D space.We analyze these equations and deduce that the induced motions on the base manifold bring more information than both the motions from general relativity and the motions from Lorentz force equations.Moreover, these equations show the existence of an extra force, which, in a particular case, is perpendicular to the 4D velocity.The general study of the extra force will be presented in a forthcoming paper.Now, we outline the content of the paper.In Section 2 we present the general gauge Kaluza-Klein space (, , ), where  is the total space of a principal bundle over a 4D space time with a Lie group  as fibre type.The pseudo-Riemannian metric  determines the orthogonal splitting (5) and enables us to construct the adapted frame field {/  , /  } (see (12)).Our study is based on a 4D tensor calculus developed in Section 3. The electromagnetic tensor field  = (   ) given by (41) and the adapted tensor fields  and  given by ( 44) and (45), respectively, play an important role in our approach.In Section 4 we construct the Riemannian adapted connection, that is, a metric connection with respect to which both distributions  and  are parallel, and its torsion is given by (58a), (58b), and (58c).Section 5 is the main section of the paper and presents the 4D equations of motions in (, , ) (cf.(85a) and (85b)).Also, in a particular case, we show that the extra force is orthogonal to the 4D velocity and therefore does not contradict the 4D physics.Finally, in Section 6 we show that the set of geodesics in (, , ) splits into three categories: horizontal, vertical, and oblique geodesics.Both, the horizontal and oblique geodesics induce some new motions on the 4D spacetime.We close the paper with conclusions.

General Gauge Kaluza-Klein Space
Let  be a 4-dimensional manifold and  an -dimensional Lie group.The Kaluza-Klein theory we present in the paper is developed on a principal bundle  with base manifold  and structure group .Any coordinate system (  ) on  will define a coordinate system (  ,   ) on , where (  ) are the fibre coordinates.Two such coordinate systems (  ,   ) and ( x , ỹ ) are related by the following general transformations: Then, the transformations of the natural frame and coframe fields on  have the forms respectively.
Next, from (2b) we see that there exists a vector bundle  over  of rank  which is locally spanned by {/  }.We call  the vertical distribution on .Then, we suppose that there exists on  a pseudo-Riemannian metric  whose restriction to  is a Riemannian metric  ⋆ .Denote by  the complementary orthogonal distribution to  in , and call it the horizontal distribution on .Suppose that  is invariant with respect to the action of  on  on the right; that is, we have where   ⋆ is the differential of the right translation   of .
Thus  defines an Ehresmann connection on  (cf.[15, p. 359]).Also, suppose that the restriction of  to  is a Lorentz metric ; that is,  is nondegenerate of signature (+, +, +, −).Thus  is endowed with a Lorentz distribution (, ) and a Riemannian distribution (,  ⋆ ) and admits the orthogonal direct decomposition As we apply the above objects to physics, we need a coordinate presentation for them.First, we recall (cf.[15, p. 359], [16, p. 64]) that the Ehresmann connection defined by  is completely determined by a 1-form  on  with values in the Lie algebra () of , satisfying the conditions where  ⋆ is the fundamental vector field corresponding to  and  denotes the adjoint representation of  in ().Now, suppose that {  } is a basis of left invariant vector fields in () and put where [   ()] is a nonsingular matrix whose inverse we denote by [   ()].Then we put  =     , and by using (6a), (6b), and (7) we deduce that As it is well known,  is the kernel of the connection form . In order to present two other local characterizations of , we consider a local basis {  } in Γ() and put As the transition matrix from {  , /  } to the natural frame field {/  , /  } has the form we infer that the 4 × 4 matrix [   ] is nonsingular.Hence the vector fields form a local basis in Γ(), too.Moreover, from (9) we obtain Note that /  is just the projection of /  on .Also, we define the local 1-forms and by using (8a) and ( 13), we deduce that Hence,  is locally represented by the kernel of the 1-forms {  }.Now, by using the fundamental vector fields { ⋆  } we put and comparing (12) with (15) we obtain via (7).The frame fields {/  , /  } and {/  ,  ⋆  } are called adapted frame fields with respect to the decomposition (5).The commutation formulas for these vector fields will have a great role in the study.First, by direct calculations using (12) we obtain where we put Next, we show that First, according to a general result stated in page 78 in the book of Kobayashi and Nomizu [16], we deduce that the vector fields in the left hand side of ( 19) must be horizontal.
On the other hand, by using ( 7) and (17a), we obtain That is, these vectors fields are vertical, too.This proves (19) via (5).As () is isomorphic to the Lie algebra of vertical vector fields, we have where     are the structure constants of the Lie group .Then, by using (17a)-(17c), (15), (19), (21), and (7), we deduce that where we put Now, taking into account (17c), from (19), we obtain which together with (23) implies By using ( 24) and (25) we are entitled to call  ⋆ℎ  the Yang-Mills fields corresponding to gauge potentials  ℎ  .Also by (18b) we may call    the electromagnetic tensor field corresponding to the electromagnetic potentials    .It is important to note that these objects come from different physical theories, and they are related by ( 22) and ( 16).

Advances in High Energy Physics
Remark 1.By a different method, the above Yang-Mills fields have been first introduced by Cho [4].On the other hand, we should stress that we find it more convenient to use    and    instead of  ⋆ℎ  and  ℎ  .
Next, we express the pseudo-Riemannian metric  on  with respect to the adapted frame field {/  , /  }; that is, we have Thus the local line element representing  has the form Hence  is locally given by the matrices and with respect to the frame fields {/  , /  } and {/  , /  }, respectively.Formally, (29) is identical to(13.31)from [17], but in the latter the local components are supposed to be functions of (  ) alone.So  given by ( 27) is the most general Kaluza-Klein metric considered in any Kaluza-Klein theory.The principal bundle , together with the metric  and the Ehresmann connection defined by the horizontal distribution , is denoted by (, , ), and it is called a general gauge Kaluza-Klein space.Finally, we consider two coordinate systems (  ,   ) and ( x , ỹ ) and by using ( 12), ( 13), (2a), (2b), (3a), and (3b), we obtain Now, we put and by using ( 16) into (30c) we deduce that The transformations (30c) and (32) have a gauge character.
Apart from them we will meet transformations with tensorial character.Here we observe that by using (26a), (26b), (30a), and (2b) we obtain the first such transformations

Adapted Tensor Fields on (𝑀, 𝑔, 𝐻𝑀)
In the present section we develop a tensor calculus on  that is adapted to the decomposition (5).For example, we construct some adapted tensor fields which have an important role in the general Kaluza-Klein theory which we develop in a series of papers.In particular, we show that the electromagnetic tensor field is indeed an adapted tensor field.First, we consider the dual vector bundles  ⋆ and  ⋆ of  and , respectively.Then, an F() − ( + )-linear mapping is called a horizontal tensor field of type (, ).Similarly, an F() − ( + )-linear mapping is called a vertical tensor field of type (, ).For example,  (resp.,  ⋆ ) is a horizontal (resp., vertical) tensor field of type (0, 2).Also,   (resp.,   ) are horizontal (resp., vertical) covector fields, while /  (resp., /  ) are horizontal (resp., vertical) vector fields, locally defined on .More generally, an F() − ( +  +  + )-linear mapping ) . (37) Then by using (2b), (3a), (30a), and (30b) we deduce that there exists an adapted tensor field of type (, ; , ) on , if and only if, on the domain of each coordinate system, there exist with respect to the transformations (1a) and (1b).Also, we note that any F() − ( +  + )-linear mapping defines an adapted tensor field of type (1, ; , ).Similarly, any F() − ( +  + )-linear mapping defines an adapted tensor field of type (, ; 1, ).More about adapted tensor fields can be found in the book of Bejancu and Farran [18].
Next, we will construct some adapted tensor fields which are deeply involved in our study.First, we denote by ℎ and V the projection morphisms of  on  and , respectively.Then, we consider the mapping It is easy to check that  is F()-bilinear mapping.Thus  is an adapted tensor field of type (0, 2; 1, 0).By using (17b) and (41) we obtain where    are given by (18b).Hence the electromagnetic tensor field is indeed an adapted tensor field.Next, we define the mappings: given by for all , ,  ∈ Γ().It is easy to verify that both  and  are F()-3-linear mappings and therefore define the adapted tensor fields of types (0, 2; 0, 1) and (0, 1; 0, 2), respectively.By using  and  and the metrics on  and , we define two adapted tensor fields denoted by the same symbols and given by for all , ,  ∈ Γ().
Remark 3. In all the papers published so far on Kaluza-Klein theories with nonabelian gauge group, the local components   of the Lorentz metric  are supposed to be independent of   (cf.[2,[4][5][6][7]).From (49a) and (49b) we see that this particular case occurs if and only if the extrinsic curvature of  vanishes identically on .

A Remarkable Linear Connection on (𝑀, 𝑔, 𝐻𝑀)
In a previous paper (cf.[12]), we constructed the Riemannian horizontal connection on the horizontal distribution of a 5D general Kaluza-Klein theory and obtain both the 4D equations of motion and 4D Einstein equations.As in that case the vertical bundle was of rank 1, it was not necessary to consider a linear connection on it.On the contrary, the geometric configuration of (, , ) from the present paper requires such connections on both  and .
The construction of these connections is the purpose of this section.First, we denote by ∇ the Levi-Civita connection on (, , ) given by (cf.[20, p. for all , ,  ∈ Γ().Recall that ∇ is the unique linear connection on  which is metric and torsion free.Next, we say that ∇ is an adapted linear connection on (, , ) if both distributions  and  are parallel with respect to ∇; that is, we have for all ,  ∈ Γ().Then there exist two linear connections ℎ ∇ and V ∇ on  and , respectively, given by ℎ Conversely, given two linear connections ℎ ∇ and V ∇ on  and , respectively, there exists an adapted linear connection ∇ on  given by Also, it is easy to show that an adapted connection ∇ = ( if and only if both ℎ ∇ and V ∇ are metric connections; that is, for all , ,  ∈ Γ().The torsion tensor field of ∇ is given by Now, we can prove the following important result.
Next, suppose that ∇  = ( ℎ ∇  , V ∇  ) is an another metric adapted linear connection satisfying (58a)-(58c).Then, from (58c) we deduce that Advances in High Energy Physics 7 which implies both (59b) and (59d) for ∇  , via (5).Now, we note that (58a) is equivalent to Remark 5.It is important to note that both ∇ ℎ  and ∇ V  are adapted tensor fields, where  is an adapted tensor field and ∇ is given by (59a)-(59d).Remark 6.Throughout the paper, all local components for linear connections and adapted tensor fields are defined with respect to the adapted frame field {/  , /  } and the adapted coframe field {  ,   }.

4D Equations of Motion in (𝑀, 𝑔, 𝐻𝑀)
In this section we present the first achievement of the new method which we develop on general (4 + )D Kaluza-Klein theories.We obtain, in a covariant form, the 4D equations of motion induced by the equations of motion in (, , ).This enables us to study the geodesics of the ambient space according to their positions with respect to horizontal distribution.It is noteworthy that the geodesics which are tangent to  must be autoparallel curve for the Riemannian horizontal connection ℎ ∇.The motions on the base manifold are defined as projections of the motions in (, , ).
Let  be a smooth curve in  given by parametric equations Then, we express the tangent vector field / to  with respect to the natural frame field as follows: Taking into account decomposition (5) and using ( 12) into (79), we obtain where we put Next, by direct calculations using (71a)-(71d) and (80), we deduce that where ∇ is the Levi-Civita connection on (, , ).Then, by using (80), (82a), and (82b) and taking into account that    are skew symmetric with respect to Greek indices, we obtain Now, we recall that  is a geodesic of (, , ) if and only if it is a curve of acceleration zero; that is, we have (cf.[20, p. 67]) Thus, using (84), (83), and decomposition (5), we can state the main result of this section.
Theorem 8.The equations of motion in a general gauge Kaluza-Klein space (, , ) are expressed as follows: We call (85a) the 4D equations of motion in (, , ).
We justify this name as follows.Suppose that the following conditions are satisfied: for all ,  ∈ {0, 1, 2, 3} and ,  ∈ {4, . . ., 3 + }.Note that all these conditions have geometrical (physical) meaning, because they are invariant with respect to the transformations (1a) and (1b).Taking into account (86a), (49a), (49b), and (26a), we deduce that the Lorentz metric  on  can be considered as a Lorentz metric on the base manifold .Thus, in this particular case, Γ    given by (64) are functions of (  ) alone and they are given by Moreover, (85a) becomes That is, we obtain the equations of motion in the 4D spacetime (,  =   (  )).Hence, the projections of geodesics of (, , ) on  coincide with the geodesics of the spacetime (, ).This justifies the name 4D equations of motion for (85a).
Next, we suppose that only (86a) and (86c) are satisfied.Then (85a) becomes In this case, we show that there exists an extra force which does not contradict the 4D physics.First, we define the 4D velocity along a geodesic  as the horizontal vector field () given by Then, define the extra force induced by extra dimensions as the horizontal vector field  given by where / is given by (80) and ℎ ∇ is the Riemannian horizontal connection.Now, we put and by using (92), (90), (80), and (91), we deduce that

Motions in (𝑀, 𝑔, 𝐻𝑀) and Induced Motions on the Base Manifold 𝑀
In this section we show that the set of geodesics in (, , ) splits into three categories and state characterizations of each category.Also, we define and study the induced motions on the base manifold.
The study of geodesics of (, , ) is based on their positions with respect to the distributions  and .First, we see from (80) that, apart from the 4D velocity () given by (90), there exists an D velocity () given by The whole study is developed in a coordinate neighbourhood U around a point  0 ∈ .We say that a curve  passing through  0 is horizontal (resp.,vertical)if its D velocity (resp., 4D velocity) vanishes on U. By (80) and (81) we see that  is a horizontal curve if and only if one of the following conditions is satisfied: or Similarly,  is a vertical curve if and only if we have or Then, by using (85a), (85b), (97b), and (98b) we can state the following.
It is noteworthy that the equations in (99a) and (99b) are related to the geometry of the horizontal distribution.To emphasize this, we give some definitions.First, we say that a curve  in  is an autoparallel curve with respect to the

Riemannian horizontal connection
where / is given by (97a).Then, by direct calculations using (97a) and (63a), we deduce that (101) is equivalent to (99a).Now, according to (71a) we may say that are local components of the second fundamental form of the distribution .Note that    are symmetric with respect to Greek indices if and only if  is an integrable distribution.If this is the case and  = 1, then − 1   is just the extrinsic curvature which has been intensively used in both the braneworld theory (cf.[9]) and space-time-matter theory (cf.[19]).
Coming back to the general case, we say that a curve  in  is an asymptotic line for  if it is a horizontal curve satisfying Then taking into account the skew symmetry of    , we deduce that (103) is equivalent to (99b).Summing up this discussion and using assertion (i) in Theorem 9, we can state the following characterization of horizontal geodesics.Remark 11.A similar characterization can be given for vertical geodesics in (, , ).However, we omit it here because as we will see in the last part of the paper the vertical geodesics do not induce any motion on the base manifold.
Next, we consider the case of the integrable horizontal distribution; that is, (86b) is satisfied.Then, any leaf of  is locally given by the equations and it is denoted by ().In this case, any horizontal geodesic must lie in only one leaf of , and by Theorem 9 it is given by the following system of equations: (, ) for all  ∈ {0, 1, 2, 3} and  ∈ {4, . . ., 3 + }.By (105a) we see that horizontal geodesics in (, , ) are in fact some particular geodesics of the 4D Lorentz manifolds ((),   (, )).Now, we say that  is an oblique geodesic through a point  0 if both the 4D velocity and D velocity are nonzero at  0 .By continuity, we deduce that  is an oblique geodesic if and only if both () and () are nonzero for any  ∈ [, ].It is important to note that both velocities () and () are involved in the equations of motion in (, , ).First, by using (90), (96), and the Riemannian adapted connection ∇ = ( ℎ ∇, V ∇) given by (63a), (63b), (63c), and (63d), we obtain Next, we say that  passing through  0 ∈  is a projectable curve around  0 , if its 4D velocity is nonzero around  0 .Taking into account (90), we deduce that through the projection point  0 of  0 on  is passing a smooth curve  in  given by the equations (see (78a) and (78b)) =   () ,  ∈ [, ] ,  ∈ {0, 1, 2, 3} . (108) In case  is a geodesic in (, , ), we call  the induced motion on  by the motion  in .Taking into account the definitions of the above three categories of geodesics in (, , ) we conclude that horizontal geodesics and oblique geodesics are projectable curves, and therefore they will induce some motions in the base manifold .Hence, the vertical geodesics have no influence on the 4D dynamics in .According to the two particular cases considered at the end of Section 5 (see (88) and (89)) we conclude that, in general, the induced motions on  bring more information than both the motions from general relativity and the solutions of the Lorentz force equations.This is due to the existence of extra dimensions and to the action of the Lie group  on .Something interesting can be observed from the particular case, where  is integrable (see (105a), (105b), and (105c)).Let  1 and  2 be two horizontal geodesics in ( 1 ) and ( 2 ), with initial conditions {(  0 ,   1 ), (  , V  1 )} and {(  0 ,   2 ), (  , V  2 )}, respectively.Then the induced motions  1 and  2 on  have the same initial conditions (  0 ,   ), but they come from different systems of equations, and therefore they do not necessarily coincide.This might be used to detect extra dimensions experimentally.

Conclusions
In the present paper we obtain, for the first time in the literature, the fully general equations of motion in a general gauge Kaluza-Klein space (cf.(85a) and (85b)).We pay attention to the 4D equations of motion, which of course modify the well-known motions in 4D Einstein gravity.Comparing (85a) with usual 4D equations of motion (88), we note two important differences.

Corollary 10 .
A curve  is a horizontal geodesic of (, , ) if and only if the following conditions are satisfied: (a)  is an autoparallel curve with respect to the Riemannian horizontal connection ℎ ∇ on ; (b)  is an asymptotic line for .
First, the local coefficients of the Riemannian horizontal connection (Γ    ,    ) do depend on the extra dimensions.Then, there are some extra terms given by the 4D tensor fields (   ,    ), which, in principle, can be used to test the theory.Such terms in is an adapted tensor field of type (, ; , ) on .Locally,  is given by the functions  1 ⋅⋅⋅   1 ⋅⋅⋅   1 ⋅⋅⋅   1 ⋅⋅⋅ defines two types of covariant derivatives.More precisely, if An oblique geodesic of (, , ) is given by the system of equations ( Then, taking into account (106a) and (106b) in (85a) and (85b) we can state the following.Corollary 12.ℎ ∇ / ())