Internal consistency of Kaluza-Klein theories

IL NUOVO CIMENTO VOL. 101 A, N. 3 Marzo 1989

Internal Consistency of Kaluza-Klein Theories.

A. KHEYFETS(*), L. K. NORRIS(*)and A. QADIR(**)

Department of Physics, University of Texas - Austin, Texas 78712

(ricevuto 1'8 Aprile 1988)

Summary. -- We consider the internal consistency of the five-dimensional Kaluza-Klein theories and the implications for its higher-dimensional extensions. In particular we restrict our analysis to the vacuum field equations and the geodesic equation for free test particles. It is shown that a consistent formulation of the five-dimensional Kaluza-Klein theory is not equivalent to the Einstein-Maxwell theory. The difference between these theories should be experimentally testable without building 1019GeV accelerators.

PACS 11.90 - Other topics in general field and particle theory.

1. - I n t r o d u c t i o n .

The earliest attempt at unifying the tensor gravitational field with the vector electromagnetic field was due to Kaluza('). He postulated a five-dimensional pseudo-Riemannian space, the fifteen metric coefficients of which contained the ten gravitational potential components and the four electromagnetic potential components. All metric coefficients were assumed to be independent of the fifth dimension. Klein if) explained the fact that the fifth dimension is not -seen, by postulating that the universe is closed (and very small) in that dimension. He observed that the assumption of compactness automatically leads to

(*) Permanent address, Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205. (**) Permanent address, Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan. (~) Ttl. KALUZA: Sitzu~ger. Preuss. Akad. Wiss. Phys. Math. Kl, 1, 966 (1921). (~) O. KLEIN: Z. Phys., 37, 895 (1926).

367

368 A. KHEYFETS, L. K. NORRIS and A. QADIR

,,quantization, of the charge. After some popularity (3) the theory ,,died, as it appeared to be only a reformulation of Einstein-Maxwell theory with no new physics forthcoming. The Kaluza-Klein (KK) theory was resurrected with the advent of gauge grand unification theories (') (GUTs) and their application to the very early universe (~). In its new incarnation it emerged as a higher-dimensional theory. After repeated ,,deaths, it has repeatedly revived, first for supergravity (~) and later for superstring theory(7), each time attempting to unify gravity with higher non-Abelian gauge fields. In GRll , Duff(~ ended his talk on Kaluza-Klein theories with the statement ,,Kaluza-Klein is dead.., long live Kaluza-Klein!,

In this paper we discuss the internal consistency of the five-dimensional KK theory, and then go on to consider the implications of our five-dimensional analysis for the higher-dimensional extensions. In order to discuss the internal consistency of KK theories it is necessary to make explicit some of their implicit assumptions:

A1) the standard assumption that the five-dimensional metric does not depend on the fifth coordinate;

A2) the vacuum field equations should follow from an unconstrained variation of the five-dimensional scalar curvature with respect to the five- dimensional metric;

A3) free test particles follow five-dimensional timelike geodesics;

A4) the proper time of particles is given by the five-dimensional arc length;

A5) the charge is equal to the fifth component of the covariant momentum.

Assumption A2) needs some clarification. The form stated above is what one would consider in a five-dimensional metric unified theory in the spirit of Einstein's vacuum general relativity. Some authors would regard A2) as a generalization and what we call KK theory as "generalized' KK theory. On the other hand, others consider the constrained form of A2) to yield a 'restricted' theory. This point will be clarified below.

(3) A. PAIS: Subtle is the Lord: The Science and Life of Albert Einstein (Oxford University Press, London, 1982). (4) j . C. PATI and A. SALAM: Phys. Rev. D, 10, 275 (1974); H. GEORGI and S. L. GLASHOW: Phys. Rev. Lett., 32,438 (1974); H. GEORGI: Particles and Fields, edited by C. E. CARLSON (A.I.P., New York, N.Y., 1975); H. FRITZSCH and P. MINKOWSKI: Ann. Phys., 93, 193 (1975). ('~) J. F. LUCIANI: Nucl. Phys. B, 135, 111 (1978); Q. SHAFI and C. WETTERICH: Phys. Lett. B, 29, 387 (1983). (6) E. WIT'rEN: Nucl. Phys. B, 186, 412 (1981). (7) M. J. DUFF: Superstrings and Supergravity, edited by A. T. DAVmS and D. G. SUTHERLAND (SUSSP Publications, 1986). (8) M. J. DUFF: GRll, Stockholm, 1986.

INTERNAL CONSISTENCY OF KALUZA-KLEIN THEORIES 369

It is found that a consistent formulation of KK theory differs from Einstein- Maxwell theory. One difference is that the KK force law has a term quadratic in charge. Another difference is that the measure of time depends on the electromagnetic energy density.

The structure of the paper is as follows. In the next section we briefly review the key features of five-dimensional KK theory needed for our purposes. In sect. 3 we discuss the implications of the five-dimensional geodesic equations. It is necessary to use the super-Hamiltonian formalism (9) to follow throughout all the implications. Symplectic Hamiltonian dynamics is, therefore, briefly reviewed in sect. 4 and applied to 5-dimensional KK theory in sect. 5. In sect. 6 the (4 + 1)- dimensional interpretation of KK theory is analysed and in sect. 7 an alternative 5-dimensional interpretation is discussed. The implications of our analysis for higher-dimensional KK-theories are considered in sect. 8. In sect. 9 we conclude with a discussion of our results and a possible experiment to test KK theory.

2. - R e v i e w o f K a l u z a - K l e i n theory .

Kaluza's innovation was to replace the four-dimensional space-time by a five- dimensional space-time; he chose the five-dimensional metric tensor

(1)

Here a, b = 0 , . . . , 4; ~, v = 0 , . . . , 3; ~,~ is the usual space-time metric tensor in four dimensions; A~ is the electromagnetic potential four-vector, and 2• 2 is the coupling constant which appears in the four-dimensional Einstein field equations. This metric tensor does not depend on the fifth coordinate, x 4. It was hoped that the five-dimensional vacuum Einstein equations would reduce to the four-dimensional Einstein-Maxwell equations and that the five-dimensional free geodesics would give the general relativistic extension of the Lorentz force law. To achieve this goal 5 4 was identified with the charge to mass ratio. By taking the universe to be compact in x 4, Klein quantized this component of the velocity.

An obvious problem is that in five dimensions there are fifteen Einstein field e q u a t i o n s , Rab = 0, for only fourteen fields (ten ~ and four A~). These equations must yield the fourteen Einstein-Maxwell field equations. The fifteenth equation, R44 = 0, is found (,0) to give the algebraic constraint

(2) F ~ F ,~ = 0,

(9) A. KHEYFETS and L. K. NORRIS: preprint from the University of Texas at Austin, April, 1987. (lo) M. J. DUFF, B. E. W. NILSSON and C. N. POPE: Phys. Rep., 130, 1 (1986).


which is not generally satisfied (F~ is the usual Maxwell field tensor). To deal with this problem a fifteenth field ~ is introduced so that the metric tensor takes the form

- ~ ;~A~ "

Another problem, arising from the fifth geodesic equation ("), is not resolved by introducing the additional field ~.

For convenience let us absorb the factor ~-t~ into the definition of the four- dimensional metric tensor. We can then write the KK metric as the sum of the Einstein (gravitational) metric and another term involving the electromagnetic potential four-vector and the extra field as

(4) ds~K = ds~ + ds~.

There are three alternatives: a) the Einstein and KK metrics are identical so that while ds~ contains extra fields, they conspire to make ds~ = 0; b) the physically measured quantity is dS~K ; c) dropping A4), the physically measured quantity is ds~. The problems noted earlier(H) were based on a). The 'conspiracy' places constraints that are too severe. In this paper we reconsider KK theory from the broader point of view just explained.

3. - The geodesic e q u a t i o n .

The Einstein metric gives the proper time as measured in a gravitational field. From eq. (4) it is clear that the KK metric, if it is to be a physically observed quantity, must represent the proper time as measured in gravitational and electromagnetic fields. Also, the quantity to be identified as the gravitational (Einsteinian) part of the KK metric is that which remains in the four-dimensional sector in the absence of electromagnetic fields, namely

(5) g,~+_ ~ - 1 / 3 ~ ,

The reason that the metric is used in the form given by eq. (3) is that it yields the KK Lagrangian in a more convenient form. Though it makes no substantive difference to our results, it is more convenient for our purposes to use g~., and a new field

(6) ~ = ~ .

(~) A. PERVEZ and A. QADIR: Applied Math. Mech. (China) to appear; A. PERVEZ and A . QADIR: preprint from Quaid-i-Azam University, Islamabad, Pakistan, 1987.


Equation (4) can then be writ ten explicitly as

(7) ds~K = g,,, dx ' dx ~ + ~ ( d x 4 - - • A~ dx*) "~ ,

with the five-dimensional KK metric and its inverse

- x ~ A ~ ' • ~-~+x2A~A; '

where four-dimensional indices are raised and lowered by g~.,. The fifteenth field equation gives (,0)

1 2 ,, ~u (9) [-](ln ~) = ~ x ~ - , ~ r ~.

We can now extremize the action using the arc-length Lagrangian. The Lag~ange equation for the fifth dimension yields the fifth geodesic equation

(10) ~(~4 _ xA.~ 2~) = C,

where C is a constant of integration. Using this equation the four-dimensional Lagrange equations yield the geodesic equation

(11) 2 " + l , : x x = x C F t , 2 " + - ~ t ~ g ~un?) , , ,

where /'~ -- 1/2g ."~ (g~,,~ + g~,,, - g~,A. For eq. (11) to yield the usual Lorentz force law we require that ~ be a

constant and that

q (12) C -

x ~ '

where q/m is the electromagnetic charge4o-mass ratio. I t is clear that ~ cannot be constant, as that would lead to eq. (2). As such,

according to eq. (11) there is an additional term in the KK extension of the Lorentz force law that is quadratic in the charge. I f we had some way of assessing what ;~ should be we could make a prediction of how the KK effect should be measured. We shall re turn to this point in the concluding section.

I t is clear that alternative a) leads to an inconsistency as C = 0 when the second te rm on the right side of eq. (11) is zero. From eq. (12) the test particle must be uncharged for C to be zero. In that case eq. (11) reduces to the usual geodesic equation in general relativity. Identifying 5 4 with q/m introduces additional problems noted earlier(l').


Now consider alternative b). Inserting eqs. (10) and (12) in eq. (7), rearranging and taking square roots, we obtain

(13) dSKK-- dSE

•/1 ~ q2

•

This is a variable time scaling due to a nonzero velocity in the fifth dimension. It is not present in Einstein-Maxwell theory and is, therefore, another prediction of KK theory taken physically. If q is measured in electrostatic units x 2 is simply the Newtonian gravitational coupling G. Thus if we knew ~ we should be able to test this prediction of KK theory. As with the previous prediction we shall return to this in the concluding section.

Alternative c) is unattractive as it reduces KK theory to a formalism with no physical significance. Also, it is not clear how to interpret eq. (11) as the generalization of the Lorentz force law requires the use of the physical proper time. This alternative is better discussed with the super-Hamiltonian formalism, which we shall now review briefly.

4. - H a m i l t o n i a n d y n a m i c s : the free part ic le .

Hamiltonian dynamics of a system with a finite number of degrees of freedom, n, is defined by

i) a 2n-dimensional phase space with coordinates (x a, Pb), called dynamical variables;

ii) a Hamiltonian (J((x a, Pb) (a real-valued function on phase space);

iii) a symplectic form S (a closed 2-form of maximal rank on phase space).

If the symplectic form S can be written as

(14) S = dpa n dx a ,

then (x ~, Pb) are called canonical variables. A vector field X on the phase space is called a Hamiltonian vector field

(relative to the symplectic form S and Hamiltonian ,gt') if

(15) S i X = - d •t:.

If (x a, Pb) are canonical variables, then

(16) S I X = d p ~ ( X ) d x ~ - d x ~ ( X ) d p ~ .

I N T E R N A L C O N S I S T E N C Y OF K A L U Z A - K L E I N T H E O R I E S 373

An integral curve of the Hamiltonian vector field can be given by functions x a (),), Pb (~) of a parameter ~. The tangent vector to an integral curve at a point can be expressed in the coordinate basis as

(17) X = 2 ~ a +iSb a 8x ~ 3pb"

Rewritten in components, using eq. (17), eq. (15) gives the 2n Hamiltonian equations

(18) 2~ _ 8 ,.E,~" Spa '

(19) iSb = 3x b ,

in canonical variables. Now consider the phase space to be the cotangent bundle T*M of an n-

dimensional manifold M, with metric ,~ = ,~bdx ~ dx b in local coordinates x ~. A free particle of 'mass' M can be defined by the canonical symplectic form (13) and the super-Hamiltonian

1 2 1 M.~ (20) ,~(=-~-~[M + G~b(xC)papb] = ~ [ +P~Pa].

The constant M is introduced in eq. (20) in super-Hamiltonian formalism in such a way that later the integral of motion gabpapb is identified with - M 2 making ,9"t" a 'zero' integral of motion ( , ~ ' = 0). In the dynamics of a charged particle in general relativity M coincides with the rest mass of the particle. As will be seen, this is not the case in KK theory. One may call the constant M 'Hamiltonian mass'.

The importance of free particles is that test particles in all metric unified theories are assumed to be free.

5. - S u p e r - H a m i l t o n i a n d y n a m i c s o f a test part ic le in K a l u z a - K l e i n theory .

The five-dimensional KK theory is a unified metric theory. The phase space of a test particle in this theory is the cotangent bundle of a five-dimensional space endowed with the metric given in eq. (8).

We will denote canonical variables in KK theory by (xa, Pb). It will be necessary to distinguish between the four-space covariant components of the momentum five-vector and the covariant components of the momentum four- vector. Thus we write

(21) p~ = g.~ p v,

25 - II N u o v o Cimento A.

374

where

(22)

Using eq. (8) for

A. KHEYFETS, L. K. NORRIS and A. QADIR

pa - - _ p , = $ ~ P b = (P~, p4).

~6 ~b we can write the super-Hamiltonian in the form

1 2 (23) , f f s +g~'P~P,, + 2xA~P~P4 + ~ 1 ( I +~•

The Hamiltonian equations are

(24) ~ =- -~[g 'P~ + •

u 2 u (25) 54 M1 [• P~ + ~-1 (1 + ~x A. A~)P4],

(26) p~ = 1 2M [g"~P"P' + 2g'~'~ • + 2g'~xA..,~P.P4 +

+ g~,~• 2A,~A~p ~ + 2g~., • 2A,~A.,,:p ~ _ ~-2 ~,:p~],

(27) P4 = 0.

It follows from eq. (27) that P4 is a constant of motion. We can greatly simplify eqs. (24)-(26) by using eqs. (21) and (22) written in the

f o r m

(28a) P" = g~P~ + • ,

(28b) p4 = x A ~ p , + ~-~ (1 + ~• 2A'~A.~)P4.

The inverted form is

(29a) P4 = - ~xA~P '~ + ~p4 ,

(29b) P , = ( g , + ~• A~A~) P~ - ~• 4 = g~vP~ - •

Equations (24)-(26) then become

(30)

(31)

(32)

~ P ~

M '

. 4 _ P 4

2

+F~.,P. x - -P4• g" ?,~.


If we interpret eq. (32) as an equation on a 4-dimensional Riemannian manifold with metric g,~ and define

(33) q = - P4,

we can rewrite eq. (32) in the form

(34) ~P"= Q x F ~ Q2 ~ + ~_~-2g.~,~ ~.~,

where ~/~ is the covariant derivative with respect to the connection determined by the metric g~. This is the super-Hamiltonian form of eq. (11).

6. - The (4 + 1) in terpre ta t ion o f KK theory .

One way to interpret the equations of motion of the test particle is to assume that all observations are to be interpreted only on the four-dimensional base manifold M4 of the five-dimensional KK space('2), the lat ter considered as a U(1)-bundle over M4. The base manifold M4 is assumed to be endowed with the metric g~.j. This metric, together with a connection on the bundle and a scalar field ~ acting as a field of deformations along the directions of the fibers, defines the bundle metric used above. The field equations then can be considered as five- dimensional Einstein equations without sources.

We have to keep in mind that the parameter ), in eq. (45) is not an affine parameter on the base manifold M 4. Bearing in mind that we are discussing alternative c), we have to replace it with the proper time parameter ~, defined on the base manifold M 4 by the condition

dx ' dxv - 1. (35) g'~'~u"uv = g ~ dr dr

The Hamilton equations readily yield the integral of motion

(36) papa---- Q2 .~ ,~ Q2

+ g~.,P P = __ + M2g,~ U ,~ U y = _ M 2"

Hence

g,~ U '~ U ~ g~.~ (dx#d~)(dx"/d~) (37) - - - 1.

1 + Q2/ M2~ 1 +Q2/M2~

(~) D. BLEECKER: Gauge Theory and Variational Principles (Addison-Wesley, Reading, 1981).


Comparing eqs. (35) and (37) we see that

(1t8) dr = d), 1 + M2----~,

and consequently

(39) i Q2

U ~ = u s 1 -~ . M2~

Likewise, if we introduce the 4-momentum of the particle on the base manifold M 2 in the standard way

(40) p.~ = m u ~ ,

then the comparison of

(41) p~p.~ = -- m 2 ,

with the integral of motion given by eq. (36) provides the relation.

(42) p~ ~ M ~ / Q2

= p . - ~ l+ - - .M2 ~

It is natural (cf. eqs. (28a) and (28b)) to define a transformation for Q

(43) Q = q mM--- 1 + M2p

Alternatively eq. (43) may be considered as a definition of q. Substituting eqs. (38), (39), (42) and (43) into eq. (34) then gives

(44) QZ -4 Q2 d p ~ M +__~_~ =

M2 p dr -m 1

= q m l + M 2 ~ ] Ydr ~ m - - / 1+ ~-2g,~,~.

After simplifications we arrive at the KK force law

q2 (45) ~P~r - qxF% u" + ~ (g.~: + u ~ u v) ~-2 ~,.~.

Equation (45) looks very much like the ordinary Lorentz force law, apart from the additional term on the right side (a force usually taken to be unobserved) and the fact that q is not an integral of motion (cf. eq. (43)). These effects are due to


the fact tha t ~ can vary and disappear when ? is constant. I f ~ is constant the s tatus of eqs. (42) and (43) is reduced to rescaling the momentum and the charge of the particle in al ternat ive c). Ra ther than considering q or M as variable, one may al ternat ively regard the gravitational coupling constant as variable (13).

7 . - A n a l t e r n a t i v e f i v e - d i m e n s i o n a l i n t e r p r e t a t i o n .

Instead of following a (4 + 1) approach we are going to t ry to take the idea of the 5-dimensional space more seriously. We under take an a t t empt to construct a geometr ic description of an observer in 5-dimensional space in such a way that the observer is 4-dimensional and his 4-dimensional space at each point of his t ra jec tory has the metric g , .

To start , we notice tha t the fifth covariant component of the 5-momentum of a neutral particle is zero, i.e. neutral particles move, at each instant, orthogonally to the fibers of M ~. The 1-form e 4 projecting the 5-momentum vectors on the fiber clearly has the following expression in the coordinate basis:

(46) e 4 = ~(dx 4 - xA, dxy).

We can form a basis of the 5-dimensional forms by adding four more 1-forms to the form e 4

(47) e ,~ = dx ' .

The basis of the tangent space of M 2 dual to the basis of 1-forms given in eqs. (46) and (47) is formed by the vectors

a (48) e. = 3x ' + • 3 4,

1 8 ( 4 9 ) e4 = - - - Dx 4 ,

as may be easily verified. The frame, defined by eqs. (48) and (49), has the proper t ies

1 (50) e4" e4 = - ,

(51) e4" e~ = 0,

(52) e.~. e., = g,~.

(~) P. JORDAN: Ann. Phys., 1, 219 (1947).


The metric tensor in this frame has the matrix form

(53) ('d~b) = 1/~ "

The field of such frames over the 5-dimensional space is not holonomic:

(54) [e, e . , ] = x ( A ~ , u - A ~ ) ~ • , ., ~x--~ =- 4 ,

(55) [e, e4]= ~-e , - ~ , , O x a"

Notice, also, that for any function f ( x ~) of space-time coordinates

(56) e. [f] = f ~ ,

(57) et [f] = 0,

i.e. all the partial derivatives ~13x ,~ in the text above can be considered as the action of vectors e~ in 5-space.

The 5-momenta of all the neutral particles at a point belong to the 4-plane spanned by the vectors (e~). The metric of the 5-space induces the metric g~y on such a 4-plane. Consequently, such a 4-plane can be considered as an infinitesimal model of space-time.

At each point of the trajectory of a particle (charged or neutral) we can construct now the 5-frame (e~, e4) and regard the 4-plane spanned by (e~) (the plane of all possible neutral particles) as the infinitesimal model of space-time, and the projection of the 5-momentum of the particle, appropriately rescaled (cf. eq. (40)), as the 4-momentum of the particle. The last operation can be also expressed by the statement that, for the description of the particle motion, we are using an instantaneous comoving neutral observer (particularly, proper time is measured by this neutral comoving observer).

In the case when there is no electromagnetic field, i.e. F~.~ = 0, and hence the field ~ is constant, the distribution of the 4-planes described above is integrable. Thus the 5-dimensional KK space can be fibrated by 4-manifolds each of which is isometrically diffeomorphic to the base manifold. However, this will not be the case if we include nonzero electromagnetic fields. Of course, from a 5- dimensional point of view the described 4-plane distribution can be considered as a model of space-time by virtue of eqs. (52) and (54). Such a description is equivalent to the standard one as long as M 5 is a U(1)-bundle over M 4.

An alternative interpretation can be given if one views the 4-plane distribution as providing infinitesimal models of momentum-energy spaces taken


as four-dimensional affine spaces. The geometric relationship between charged test particles and instantaneously comoving neutral observers discussed above can be used to set up a unified theory of gravitation and electromagnetism based on the Poincar~ group P(4)=O(1,3)| a using a generalized affine connection (9,~).

8. - Impl i f i eat ions for h igher -d imens iona l KK theories .

Any problems or predictions of the five-dimensional KK theory need not persist in higher dimensions. Although there is much literature on KK theories (9.~,~6), most of it is devoted to the quantum aspects of the theory. We are concerned with extending our classical analysis to higher dimensions.

In (4 + d)-dimensions the metric tensor can be written as

(58)

with the inverse

( G.b) = [g~" + • ~ A ~ A ~ - •162 I

\ -•162 ~ ]

(59) ( ~ b ) = ~4~ ~ - - ~ x 2 - ~ ' ~ z y ~, ,~v ,

where a, fl = 4 , . . . , (d + 3) and ~ is the inverse of ~ . In our notation the metric coefficients are independent of all higher-dimensional coordinates. The explicit form of the metric is now

(60) dS~K = g.~v dx '~ dx ~ + ~ (dx ~ - xA; dx')(dx ~ - xA~ dx'~).

The higher-dimensional geodesic equations yield the following generalization of eqs. (10) and (12):

(61) p~ (&~ - ~4~ &~) - q~ • '

where q~ is the gauge 'charge'. The space-time geodesic equations can be identified with the curved space-time generalization of the Yang-Mills force law

(14) L. K. NORRIS: Phys. Rev. D, 31, 3090 (1985). (15) T. APPLEQUIST, A. CHODOS and P. G. O. FREUND: Modern Kaluza-Klein Theories (Addison-Wesley, Menlo Park, 1987). (16) V. M. EMEL'YANOV, YU. P. NIKITIN, I. L. ROZENTAL and A. V. BERKOV: Phys. Rep., 143, 1 (1986).


by requiring that four differential equations hold, relating ~ , A~ and the Lie algebra structure constants. The explicit relationship is not relevant for our purposes. The essential point is that unlike the situation with eq. (11) there is no clear difference of the KK-theory from the postulated force law. It is to be noted that, corresponding to eq. (10), we already have d(d + 1)/2 nonlinear differential equations for the d(d + 1)/2 fields, ~ , in terms of the Yang-Mills field tensor. There is no guarantee that the entire set of equations will be consistent. However, it must be remembered that the Yang-Mills force law has not been tested as well as the Lorentz force law. Thus, even if there are differences between the KK force law and the Yang-Mills-Einstein force law, it would not be easy to observe them experimentally. Of more interest is the other prediction of KK theory that, corresponding to eq. (13), there is a variable time scaling

(62) dSKK ---- dSE

~/1 + T:r q, qz/•

We shall discuss this prediction in more detail in the next section.

9. - D i s c u s s i o n o f r e s u l t s .

We have seen that the five-dimensional KK theory must either make new predictions or there will be problems of internal consistency. In earlier discussions(''.'2,'7) the problems associated with alternative b) and the requirement that ~ = 1 had been considered. One way out in of the problem (=.,8) is to constrain the variation of Gas and ignore the requirement R44 = 0. As pointed out in sect. 7 this reduces the KK force law to the Lorentz force law and rescales the time parameter relevant for charged particles relative to uncharged particles. Since the rescaling is by a constant amount it is not clear how this prediction could make a physical difference and thus be amenable to testing. Apart from this difference we will simply recover Einstein-Maxwell theory. Our objection to this idea is that it violates the general spirit of KK theory as embodied in A2). The resultant theory becomes purely formal. It is worth stressing here that the four-dimensional base manifold cannot generally be identified isometrically with any four-dimensional submanifold of the five- dimensional KK space (provided the four-submanifolds are endowed with the metric induced by the metric of the five-dimensional KK space).

It is interesting to compare a point charge in the KK and Einstein-Maxwell theories. The time rescaling in both theories is identical, but the spatial part of the four-metrics are quite different. Notice, also, that the time rescaling in eq.

(L7) A. EINSTEIN and P. BERGMANN: Ann. Phys., 39, 683 (1938). (18) R. KERNER: Ann. Inst. Henri Poincar~ IX, 143 (1968).

I N T E R N A L CONSISTENCY OF K A L U Z A - K L E I N T H E O R I E S 381

(13) is again of the same type as in the Reissner-Nordstrom metric, except that here it is due to the charge of the test particle.

If KK theory is to be taken seriously as a genuine five-dimensional theory, then not only is there a deviation from the Lorentz force law (due to the quadratic charge term in eq. (11)), but there is also a predicted time contraction given by eq. (13). Disregarding, for the moment, the fact that the predicted deviation from the Lorentz force law has not been seen, let us concentrate on the time rescaling.

For the Einstein equations in the presence of an electromagnetic field to have on the right side the usual Maxwell stress-energy tensor, it is necessary that ~ be unity. Thus it would be necessary that ~ remain approximately 1. First taking ~= 1 we note that for a charge-to-mass ratio of the electron q2/(• 104~ Thus there is a time rescaling difference of 102o between KK theory and Einstein- Maxwell theory. If cosmological time is given any significance for fundamental interactions as, for example, in GUT theory (~) with standard cosmology (~9) or in Dirac's large number hypothesis (.,o), this rescaling would be a very attractive feature of the theory. (Notice that the ratio entering into the above discussion is Dirac's large number.)

Now ~ is given by the nonlinear differential equation (9). Bearing in mind that must be near unity we can write the linearized form of eq. (9)

(63) ( Q _ ~ ) ~ = ~ 2 ,

where ~ = ~ - 1 and

1 2 Ll Lluv (64) 22 = ~ ~ .

Equation (63) is an inhomogeneous Klein-Gordon equation with the electromagnetic field energy density, 22 , playing the role of both the source and mass of the field ~. In particular we could arrange a situation where F, vF ~'~ took given constant values. In this way ~ could be controlled.

A test of KK theory would be possible by comparing dSKK with ~ = 0 (e.g. by taking F~ = 0) and some other controlled value of ~. Since dsF. would remain unaltered we would have

dSKK (~ = 0) ~ ~ . (65) dsKK (~ = ~0)

(,9) G. W. GIBBONS, S. W. HAWKING and S. T. C. SIKLOS: The Very Early Universe (Cambridge University Press, Cambridge, 1983). (~) P. A. M. DIRAC: Nature, 139, 323 (1937); 254, 273 (1975); Proc. R. Soc. London, Ser. A, 165, 199 (1938); 338, 439 (1974).


We could, for example, pass muons through strong, constant, electromagnetic fields and observe the change in their decay rate.

It could be argued that no such experiment is necessary as the violation of the Lorentz force law, predicted by KK theory, already rules it out. This is not necessarily so clear as the extra term may be swamped by radiation reaction and charge renormalization effects. Another point of view might be that the relevant prediction is anyhow of the higher-dimensional KK theory. In that theory the time contraction is given by eq. (62). Presumably, we should require, here, that

(66) ~ ~ ~ .

The time rescaling here is again - 102~ Since it is not so easy to manipulate these gauge fields as it is to handle the Maxwell field, it is not clear how this prediction can be tested. Is it possible that this rescaling of time is responsible for the confinement and asymptotic freedom of the QCD gauge fields via a mechanism like the one suggested in ref. (2~)? Again, does the modified force law correspond better to observation that the usual Yang-Mills force law?

AK is grateful to Prof. J. A. Wheeler for support via h~SF Grants PHY 8205717 and PHY 8503890. LKN wishes to express his gratitude to the Center for Relativity, the University of Texas at Austin, for hospitality. AQ is indebted to the Council for the International Exchange of Scholars for financial support and to Prof. J. A. Wheeler for hospitality at the Physics Department, University of Texas at Austin.

(21) S. M. MAHAJAN, A. QADIR and P. M. VALANJU: Nuovo Cimento, 71, 265 (1982).

�9 RIASSUNTO (*)

Si considera la coerenza interna delle teorie di Kaluza-Klein pentadimensional e le implicazioni per le loro estensioni a pifi dimensioni. In particolare si limita l'analisi alle equazioni del campo nel vuoto e all'equazione geodesica per le particelle test libere. Si mostra che una formulazione coerente della teoria di Kaluza-Klein pentadimensionale non

equivalente alla teoria di Einstein-Maxwell. La differenza tra queste teorie dovrebbe essere saggiabile senza la costruzione di acceleratori a 1019GeV.

(*) Traduzione a cura della Redazione.

INTERNAL CONSISTENCY OF KALUZA-KLEIN THEORIES

BnyTpenn~i cor~acoBaanocTb Teopxfi Ka~IyRa-K~efma.

383

Pe3mMe (*). - - MbI paccMaTpnBaeM BHyTpeHHIOIO corJIaCOBaHHOCTI, n~iTHMepltbIX Teopnfi KaJ~y~a-Ksiefina n npnMeHeHi4e OT~IX TeopIafi B cJIyqae 6 o J ~ m e r o nHcaa ]43MepeHI4fi. B qaCTHOCTII, MhI orpaHnqnsaeMcn aHaJIId30M aaKyyMabIx noJieabix ypaBHeanfi I~ FeoJIe3HqeCKOFO ypaBnemm ~Ia~ CBO6OJIrlbIX rlpO6HblX qaCTttlI. I-IoKa3blBaeTcg, qTO corJmcoBaHHaU qbopMyJmposKa nUTnMepHofi Teopnn Kaayaa -Kae f I aa He aaJIaeTcu OKBHBaJIeHTHOI4 TeopIaH 3fiHtUTefiHa-MaKcaenaa. Pa3Jii4qtle Me)K~y 3THMH Teopn~lMI4 MO)KeT 6~ITJ, aKcnepHMenTa~buO r lpoaepeno 6e3 CTpOnTe~bCTBa ycKopnTenefi Ha a n e p r m o 1019 FoB.

(*) Flepeoec)eno pec)axt~ue~.

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Internal consistency of Kaluza-Klein theories

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