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KALUZA-KLEIN THEORY WITH THE LANCZOS LAGRANGIAN J. M A D O R E 1 Department of Mathematics and Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, Canada Received 6 May 1985; accepted for publication 26 May 1985
A gravitational lagrangian is proposed which is quadratic in the Riemann tensor and which in dimension greater than four yields second-order field equations. Applied to Kaluza-Klein theory, it contains a term which can be used to cancel the vacuum-fluctuation energy and yield a vanishing cosmological constant. The gravitational coupling constant appears naturally as the curvature of the compactified internal dimensions.
1. Introduction. Recall the Casimir effect due to the quantum fluctuations of a massless scalar field within the cavity between two parallel surfaces. I f M is a surface without boundary then the fluctuations within [0, l] × M give rise to an attractive force between the surfaces [1 ]. ff one imposes Dirichlet boundary conditions this force derives from the potential
rt2h Vol[M] + ~ -h~ f RVrg d2x 1440l 3 M
+ o(l~l/L 2) .
The constant L - 2 is characteristic of the curvature of M. With periodic boundary conditions, in which case the cavity can be thought of as S 1 X M, one finds the same formula with different coefficients. A similar formula can be derived with the surface M replaced by a manifold of any dimension, in particular dimension 4, and S 1 replaced by a sphere S n , with the complication that in general the numerical coefficients diverge and are cut-off dependent:
AS d = Cl K2 Vol[S n] Vol[M] +
c2K VoltS"] f R#id4x + ....
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Here ASd is the one-loop quantum correction to the action in a space of dimension d = 4 + n. It is related to the same correction to the classical (euclidean) action in dimension 4 by ASa = Vol[S n ] AS 4 . The constant K is the curvature of the sphere: K = 1/r 2. The expansion is in powers of I[L2K. The coefficients c I (A2/K) and c2(A2/K), depend on a cut-off A. In this formula for ~Sd, the sphere serves mainly to give a mass to the quantum field, a mass of the order h/r. We assume that the equation which defines the quantum fluctuations has no zero modes on Sn . The Sakharov-Klein idea  of induced gravity is to consider AS4 as the entire gravitational action. The first term is the cosmological constant; the second is the Einstein-Hllbert action. There are three problems with this idea: (i) the constants c 1 ~ A4/K 2 and c 2 ~ A 2[K are strongly cut-off dependent; (ii) the curvature K must be chosen to be the Planck mass with no dynamical reason; (iii) there is an enormous cosmological constant which must be suppressed. If we had started from a classical (euclidean) Einstein~Hilbert action S4 = X O V o l t M ] + ~ 1
fM R~/rg d4x
these problems would more or less remain. The bare coupling constants X0 and G O could be used to ab-
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sorb the divergences in (i) but the renormalized gravitational constant would have to be chosen independent of K and the cosmological constant set equal to zero by fiat. There is yet a fourth problem which we shall not consider here. Gravity is not renormalizable and the next term in the series for AS, quadratic in the Riemann tensor, diverges also. The Kaluza-Klein idea [4,6] is to add extra dimensions to space in the form for example of a sphere Sn so as to be able to unify a family of gauge bosons, and eventually a Brans-Dicke scalar field, with the gravitational metric. As the higher-dimensional space came out of the initial big bang one must suppose that it was energetically more favorable because of the Casimir-like quantum effects, for the extra dimensions to remain compact and small rather than expand as did the three space dimensions we live in. We shall later as an example suppose that the internal space is S4. Then the higher-dimensional space would be the sphere S 7. Locally S7 is a product of the internal S4 and a coordinate patch in the observable 3dimensional space. This is the Hopf fibration connected with quaternions. It is traditional to use the Einstein-Hilbert action in the higher-dimensional space since in dimension 4 it is the only one which yields second-order field equations. In higher dimension there are others. In particular one which is quadratic in the Pdemann tensor and which offers the possibility of a solution to problems (ii) and (iii).
gxu = gxu(xa) , gab = e 2 ° ( X a ) g a b ( x a )
If the scale of Sn, and eventually the entire geometry of a more complicated space, is to depend on quantum fluctuations of fields which are functions on the spacetime manifold M then it must also depend on the point x x in M. We have included this point-wise dependence in the conformal function o. With the ansatz and including quantum corrections, the lagrangian becomes £g = £S n + £M + n(n -- 1)KR - n [ 2 A o + (n + 1)~xctaXo]R + 4nRXUoxu
+ n(n - 1)(n - 2)[2Ao + (n -- 1)a~ o~Xu]K + F(e, axo ) + Cl K2 + c2KR + ....
axu = DxDuo + axoauo , the functional F is given by
F(o, Oxo) = 2n(n - 1) [oxuoxu - (Ao) 2] -- ½n (n -- 1)~xoaXa[4nAo + (n 2 -- n + 2)axooxo ] , and the dots designate higher order quantum corrections. In this equation R is the Ricci scalar ofgxu. The action is given by
Sol = V¢ol[Sn ] f £genax/r~ d 4 x . 2. The Lanczos lagrangian. Consider on M X Sn the lagrangian  £g = --½(RiiktRiikl - 4RiJRii + R 2 ) .
This lagrangian generates the Euler class in dimension 4. Its field equations become then the Bach-Lanczos identities. Consider an ansatz for the metric of the form
_ [,gab 0 ) . gi] - ~ 0 g~u
The coordinates x a, 1 ~
The lagrangian for o is degenerate for n ~< 3, and for n t> 4 it has the wrong sign. For simplicity we shall choose n = 4. To lowest order in o then £o = Fe 4 ° ~ - 2 4 K a xoa Xo.
The field o is to be considered only as a phenomenological manifestation of the change in vacuum-fluctuation energy, not as a fundamental field. The term £M, the original lagrangian (2.1) restricted to the manifold M does not contribute to the equations of motion since as we have mentioned, its field equations are the Bach-Lanczos identities. The term £sn is important however since the manifold Sn is compact. We have
£sn = -~n(n - 1)(n - 2)(n - 3)K 2 .
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Neglecting o then we have
Tab = ~T u Ugab •
£g = [c 1 - ~n(n - 1)(n - 2)(n - 3)]K 2
We can offer no explanation for this ad hoe assumption. Notice that Tq is the classical matter tensor. We subtracted off the dominant quantum terms when we supposed that c 1 was given by eq. (2.7). The conservation equations T / / j = 0 imply that
+ [c 2 +n(n - 1)]KR + ....
It is possible that the curvature of Sn and the cut-off arrange themselves so that the cosmological constant vanishes:
c 1 = ~n(n - 1)(n - 2)(n - 3).
We shall assume that they do so. However we cannot show that this is a stable situation. The curvature of the internal and the external spaces indeed both depend on the effective cosmological constant through the field equations but we cannot argue that the former must arrange itself so that this constant vanishes. Newton's constant is given naturally in terms of the curvature K of the internal space (see also ref. ). With n = 4: 1/16rtG~ = (c 2 + 12)K.
3. The field equations. Define the tensor H 0- by (,5/Sgq)(£gx/~) = Hifv/L'g .
Then, neglecting the quantum corrections, Hi/is given
by  Hi/= --Riklm R] klm _ 2RikflRkl + 2RIkRj k - RRij - ~£g gii"
The field equations are
Hi/= - ~ Tij .
The numerical factor is due to a convention. We choose as ansatz for the matter tensor
where we have set ~ = e 20. With the ansatz (2.2), the approximation (3.5) and the assumption (3.6), the field equations become
A~ = 0 ,
G x u = -8~rG T~u - l~hu ,
+ ~ - I ( D x D u # - A~gxu) •
If one defines 4 = #-1 then one finds that this tensor is given by
TSu = (3/242) ( ~ x 4 a u 4 - ½ao4a~4gxu) - 4 -1 ( D x D u 4 - ½ A4gx#) •
These equations are similar to those of Brans and Dicke with two different values of their parameter co: co = oo in the scalar wave equation and co = 3/2 in the gravitational field equations. These two values are permitted here because of the different form of the conservation equations (3.7). The source terms for the field ¢ are all of order IRxupol/K.
4. Cosmological implications. Consider a spacetime metric of the form ds 2 = dt 2 _ a(t)2dx 2 ,
and an energy-momentum tensor of the form In what follows we shall neglect terms of order LRXpal/K. This means that we suppose that ?,
[Rvpol ~ 1/Gh .
= 60 + p ) u x u . - Pgx
The conservation equations (3.7) become (3.5)
In order that to this approximation the field equations (3.3) be consistent with Einstein's equations, we must suppose that the internal-space component of the energy-momentum tensor is given by
d (~3/2a3p)+p ~ (~b3/2a3) = 0 . "~
With p = 0 we look for a solution of the form
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a=ao(t/to) a, $=(t/to)#.
PHYSICS LETTERS (4.3)
There are two solutions to the wave equation (3.8) given by (i)/3 = 0 and (ii)/3 = 1 - 3ix. The first yields from (3.9) the standard k = 0 cosmology; the second yields a new solution given by a = (3 + 2 x / ~ / 1 5 ~- 0 . 5 3 .
This is at the very limit o f what could be tolerated by experiments. The most unusual property o f the new solution is the fact that it is a vacuum solution. The critical density is zero. The variation o f the gravitational constant provides an effective source just strong enough to close the universe without additional matter. Both the standard solution and the new solution are valid in the present theory provided that the curvature o f space-time is small with respect to the Hanck mass, that is, provided that t ~, t c where RxuvoRX~Va ~ K 2 defines tc: (te/tO)2-# ~ GonH 2 .
5. Conclusions.We have considered a new lagrangian, the Lanczos lagrangian, and examined its consequences when applied to Kaluza-Klein theories. As example we have chosen the sphere S4 as internal space. This is the lowest dimension which is not degenerate. The total dimension must be at least 8 for the theory to work, just as the dimension must be at least 4 when the Einstein-Hilbert lagrangian is used.
With an ad hoc assumption on the internal-space components of the classical matter tensor, the theory is compatible with Einstein's equations provided that the local curvature is small with respect to the Planck mass. The theory predicts corrections when this is no longer so but these have not been examined in detail.
This cosmology has a slower expansion rate than the standard one and a decreasing gravitational constant whose rate of change is determined by the Hubble parameter H. At the present epoch t = t 0, IG0/G01 = I/3/alH 0 ~ n 0 ~ 5 X 10 -11 yr -1 .
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The author benefited greatly from discussions with colleagues at the Universities of Edmonton, Syracuse and Toronto. He would especially like to thank J. Goldberg, W. Israel, R. McLenaghan and R. Vanstone. His thanks go also to T. Bloom and D. Sen of the Dept. o f Mathematics of the University of Toronto and to the Canadian Institute for Theoretical Astrophysics for their hospitality and financial support.
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