New Developments in Cosmology and Gravitation from Extended Theories of General RelativityView this Special Issue
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.
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 spacetime, with as fibre type. Thus, a natural generalization of Kaluza-Klein theory consists in replacing by a nonabelian gauge group (cf. [1–5]). 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  presented the first generalization in this direction. According to it, the local components of the Lorentz metric in a 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 -dimensional spacetime (the “bulk”) with particles and fields trapped on the brane, while gravity is free to access the bulk (cf. ). The other one is called space-time-matter theory and assumes that matter in the 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 space (cf. ). 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 physics (cf. ) and to obtain a classification of the warped spaces satisfying Einstein equations with cosmological constant (cf. ).
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 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 tensor calculus that we introduce via a natural splitting of the tangent bundle of the ambient space. We obtain, in a covariant form and in their full generality, the equations of motion as part of equations of motion in a 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 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 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 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 equations of motions in (cf. (85a) and (85b)). Also, in a particular case, we show that the extra force is orthogonal to the velocity and therefore does not contradict the 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 spacetime. We close the paper with conclusions.
2. 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 are related by the following general transformations:Then, the transformations of the natural frame and coframe fields on have the forms
Throughout the paper we use the ranges of indices: ,. By we denote a function that is locally defined on . Also, for any vector bundle over , we denote by the -module of smooth sections of , where is the algebra of smooth functions on .
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 conditionswhere 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 and put As the transition matrix from to the natural frame field has the form we infer that the matrix is nonsingular. Hence the vector fields form a local basis in , 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 , 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 the Yang-Mills fields corresponding to gauge potentials . 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).
Remark 1. By a different method, the above Yang-Mills fields have been first introduced by Cho . On the other hand, we should stress that we find it more convenient to use and instead of and .
Next, we express the pseudo-Riemannian metric on with respect to the adapted frame field ; that is, we haveThus 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 , 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 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
3. Adapted Tensor Fields on
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 -linear mapping is called a horizontal tensor field of type . Similarly, an -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 -linear mapping is an adapted tensor field of type on . Locally, is given by the functions 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 functions satisfying with respect to the transformations (1a) and (1b). Also, we note that any -linear mapping defines an adapted tensor field of type . Similarly, any -linear mapping defines an adapted tensor field of type . More about adapted tensor fields can be found in the book of Bejancu and Farran .
Next, we will construct some adapted tensor fields which are deeply involved in our study. First, we denote by and the projection morphisms of on and , respectively. Then, we consider the mapping It is easy to check that is -bilinear mapping. Thus is an adapted tensor field of type . 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 . It is easy to verify that both and are -3-linear mappings and therefore define the adapted tensor fields of types and , 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 .
We close this section with a local presentation of the adapted tensor fields and . First, from (33a) and (33b) we deduce that entries (resp., ) of the inverse of the matrix (resp., ) define a horizontal (resp., vertical) tensor field of type (2, 0). Then, we put and by using (44), (45), (47a), (47b), (48a)–(48d), (26a), (26b), and (17a)–(17c), we obtain
Remark 2. If in particular , then represent the local components of the extrinsic curvature used in brane-world theory (cf. ) and in space-time-matter theory (cf. [12, 19]). For this reason we call given by (44) the extrinsic curvature of the horizontal distribution.
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, 4–7]). From (49a) and (49b) we see that this particular case occurs if and only if the extrinsic curvature of vanishes identically on .
4. A Remarkable Linear Connection on
In a previous paper (cf. ), we constructed the Riemannian horizontal connection on the horizontal distribution of a general Kaluza-Klein theory and obtain both the equations of motion and 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. 61]) for all . 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 . Then there exist two linear connections and on and , respectively, given by Conversely, given two linear connections and on and , respectively, there exists an adapted linear connection on given by Also, it is easy to show that an adapted connection is metric; that is, if and only if both and are metric connections; that is, for all . The torsion tensor field of is given by Now, we can prove the following important result.
Theorem 4. Let be a general gauge Kaluza-Klein space. Then there exists a unique metric adapted linear connection whose torsion tensor field is given by
for all .
Proof. First, define and as follows: for all . Then, it is easy to check that given by (59a)–(59d) is a metric adapted linear connection whose torsion tensor field satisfies (58a)–(58c). Next, suppose that is an another metric adapted linear connection satisfying (58a)–(58c). Then, from (58c) we deduce that which implies both (59b) and (59d) for , via (5). Now, we note that (58a) is equivalent to Then by using (56a) and (61) for and taking into account (51), we obtain which proves (59a) for . In a similar way (59c) is proved for . Thus , and the proof is complete.
As and satisfy (56a) and (56b), we call them the Riemannian horizontal connection and the Riemannian vertical connection, respectively. Also, given by (59a), (59b), (59c), and (59d) is called Riemannian adapted connection on .
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 .
Next, we consider given by (59a)–(59d) and put Then, we take into (51), and using (59a), (63a) (26a), and (17b), we obtain Similarly, we take in (51), and by using (59c), (63c), and (26b), we infer that Also, by direct calculations using (59b), (59d), (63b), (63d), (17a), (48b), and (48d), we deduce that According to the splitting in (5), the Riemannian adapted connection defines two types of covariant derivatives. More precisely, if are the local components of an adapted tensor field of type , then we have In particular, from (56a) and (56b) we deduce that
Throughout the paper we use , and for raising and lowering indices of adapted tensor fields as follows: Now, we state the following.
Theorem 7. The Levi-Civita connection on the general gauge Kaluza-Klein space is expressed as follows:
Proof. According to decomposition (5) we put Then by using (59a) and (63a), we deduce that Next, take , and in (51) and by using (26a), (26c), (17a), (17b), (48a), and (70a), we obtain Thus (71a) is obtained from (72). Similarly, we put Then, take , and in (51) and by using (26a), (26c), (17a), (17b), (49a), and (70c), we infer that Also, take , and in (51) and by using (26b), (26c), (17a), and (50b), we deduce that Thus (71b) is obtained from (75). Now, taking into account that is a torsion-free connection and using (17a), (71b), and (66b) we obtain (71c). Finally, (71d) is deduced in a similar way as (71a).
5. 4D Equations of Motion in
In this section we present the first achievement of the new method which we develop on general Kaluza-Klein theories. We obtain, in a covariant form, the 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 . 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 equations of motion in . We justify this name as follows. Suppose that the following conditions are satisfied:for all and . 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, 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 spacetime . Hence, the projections of geodesics of on coincide with the geodesics of the spacetime . This justifies the name 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 physics. First, we define the 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 is the Riemannian horizontal connection. Now, we put and by using (92), (90), (80), and (91), we deduce that Thus, from (89) we obtain Then, by using (90), (92), and (94) and taking into account that are skew symmetric with respect to Greek indices, we infer that Thus the extra force is perpendicular to the velocity, which is a well-known property of the extra force in classical Kaluza-Klein theory. The above result on the extra force enables us to call (89) the Lorentz force equations induced in the space time . Finally, in this particular case, we see that our equations (89) coincide with (44) obtained by Kerner .
6. Motions in and Induced Motions on the Base Manifold
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 velocity given by (90), there exists an velocity given by The whole study is developed in a coordinate neighbourhood around a point . We say that a curve passing through is horizontal (resp.,vertical) if its velocity (resp., velocity) vanishes on . 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.
Theorem 9. (i) A curve is a horizontal geodesic in if and only if (97b) and the following equations are satisfied:
(ii) A curve is a vertical geodesic in if and only if (98b) and the following equations are satisfied:
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 if it is a horizontal curve satisfying 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 , then is just the extrinsic curvature which has been intensively used in both the brane-world theory (cf. ) and space-time-matter theory (cf. ).
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.
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 on ;(b) is an asymptotic line for .
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 and . By (105a) we see that horizontal geodesics in are in fact some particular geodesics of the Lorentz manifolds .
Now, we say that is an oblique geodesic through a point if both the velocity and velocity are nonzero at . 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 given by (63a), (63b), (63c), and (63d), we obtain