Experimental tests of extra dimensions

Volume 186, number 1

PHYSICS LETTERS B

26 February 1987

EXPERIMENTAL TESTS OF EXTRA DIMENSIONS ~ J.A. CASAS l, C.P. MARTIN and A.H. VOZMEDIANO 2 Departamento de Fisica Te6rica, C-XI, Universidad Aut6noma de Madrid, Cantoblanco, 28049 Madrid, Spain Received 24 July 1986

Spherically symmetric solutions of general relativity in higher dimensions are tested. We find that the existence of extra dimensions is compatible with the classical tests of general relativity, which place almost no restrictions on the parameters involved. The models studied fail to explain the geological data concerning the apparent variability of the gravitational constant with the distance.

Kaluza-Klein theories (KK) [ 1,2] ~, whose recent popularity has been reinforced by superstrings [ 3 ], are now a part of our common understanding of physics. Today the generic name of Kaluza-Klein stands for a wide variety of different approaches all having in common the existence of a D > 4-dimensional spacetime. Two main lines of KK can be traced, the old "pure" KK where extra dimensions serve to generate symmetries of the four-dimensional world, and the recent versions where extra dimensions are needed to give a sense to some theories as in supergravity [4 ] and superstrings [ 3 ]. The later approach could well be called "anti K K " since compactification of the extra dimensions not only should not give rise to new symmetries but it should break some of the preexistent ones. Despite their popularity, KK theories have remained so far untested for well known reasons. The natural scale of the theory lies at the Planck mass where quantum gravity must be invoked. Physics below this level is quite accurately described by general relativity (GR) and the standard model so the KK phenomenology is almost restricted to reproduce the known results of the standard model in the

* Work supported by the CAICYT, Spain. J Present address: Physics Department, Oxford University, 1 Keble Road, Oxford OX 1 3NP, UK. 2 Present address: Jadwin Hall, Princeton University, Princeton, NJ 08540, USA. ~t For a review see ref. [2].

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case of superstrings, and to be compatible with GR in the old approach. The "physical reality" of the extra dimensions has been lately considered by Chodos and Detweiler [ 5 ] who opened the way to higherdimensional cosmologies [6] and gravity [ 7-9]. String inspired cosmologies are now being developed [ 10] while the problem of getting gravity - other than Minkowski space - in four dimensions out of the string models is much more involved. All these works provide indirect tests of extra dimensions. In this paper we are retaking the old approach in the spirit of refs. [ 5,7] and work with some simple high-dimensional models that can be directly confronted with the experiments. First we provide a compatibility test for the existence of extra dimensions by applying the standard tests of GR to fourdimensional metrics obtained by dimensional reduction of higher-dimensional models. In view of the perfect agreement of GR with the data [ 11 ], it should be expected that the values of the KK parameters involved would come very close to their GR limiting values. We rather find that the effective four-dimensional models are compatible with the tests with the parameters essentially unconstrained. Next we study the possibility that this models could explain the apparent variability of the gravitational constant G with the radial distance suggested by the geological data of ref. [ 12 ]. Although this data has been used by Fischbach et al. [ 13 ] as a support for the necessity of the fifth force, they can be described by an appropriate modification of the Newton law. KK gives the 29

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correction in the models studied but it is seen that the value prescribed for the K K parameter lies outside the range allowed by the previous result. Some comments are added at the end. Let us first note that most of the G R tests examine the weak gravitational field exterior to a spherically symmetric mass distribution with no internal charges. The generalization of this situation including extra dimensions has been treated in the literature and there exists a general solution in five dimensions [ 7 ] and particular solutions with any number of extra dimensions [8,9]. The most general static metric which is spherically Symmetric in the usual threespace in the absence of internal charges and with n extra dimensions is g = - A 2 ( r ) dt 2 +B2(r) 6ij dx i dx j + C 2 ( r ) c~,~a dx '~ dx p.

(1)

A solution to the Einstein equations with any number n of extra dimensions is [ 10]

1 - 2 r / m ,k

A(O=- ~

B(r)= C(r)=-

,

m 2 11+2r/ml ~k-~+l 4r 2 11_2r/ml~k_~_l, (2)

where the parameters ¢ and k are related by the constraint e2[k 2 - k + ½(1 + l/n)] = 1.

(3)

From this solution an effective four-dimensional metric is obtained as described in ref. [ 8 ], to be used as an input in the classical tests of GR:

g~ = d i a g [ - A 2,

BE(~ij].

(4)

Notice that such an effective metric will only be a solution to the Einstein equations in four dimensions with empty spacetime, when C(r) is strictly a constant [8]; in this case, the metric (2) is the Schwarzchild metric. This limiting case occurs when the parameter p, defined as

p=¢(k-l/2),

(5)

takes the value 1. Otherwise a scalar field is coupled 30

to gravity in four dimensions. The scalar charge, given by [9]

s= [(n+ 2)/n] 1/2 mE~2,

(6)

measures the influence of the extra dimensions and is the only free parameter in the theory for this particular case ,2. ~.mong the classical tests of G R [ 11 ] we have chosen the radar echo delay to demonstrate the result as it is the most accurately measured one. The standard computation [ 14 ] gives after some algebra: A t - 2 ( p 2 -- 1 M E - \pS 2r ° \ × [ c o s - l ( r o / r . ) + cos-l(ro/ro)] ) ,

(7)

where At measures the difference between the G R result (p = 1 ) and that of KK. The various quantities involved are the sun's radius r o, the earth and satellite distance to the sun, re and ro, and the physical mass parameter of the theory identified as [8] M = mp made equal to the sun's mass. The experimental data given by the Viking [ 11 ] allows us to extract a lower bound on p, IPl>~10 -2.

1 +2r/m ~n ~ ,

elf gv~=C(r)g~,

26 February 1987

(8)

The bound is not improved by any other classical test. Noticing that the absolute value o f p is restricted by (3) to be less than one, we see that only the extreme values o f p close to zero, i.e., scalar charge going to infinite, are suppressed; otherwise p can take values quite far away from the G R limit p = 1. Let us mention that (8) excludes the existence of massless black holes suggested in ref. [ 8 ], putting a limit on the ratio m / M involved. We find again the phenomenological impossibility of having a nonzero charge (scalar charge in this case) sitting in a massless particle. We now analyze the effective four-dimensional gravitational constant coming from KK. As is known, the variability of the fundamental coupling constants is one of the most genuine predictions of K K which has severely constrained the cosmological models. This variability is apparent in the case of the gauge coupling constants which are given in terms of the mean radius of the extra dimensions. In the case :2 For a discussion on this parameter see ref. [7 ].

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o f the gravitational constant the variability is not so obvious because the Weyl factor in front o f the higherdimensional metric is usually chosen so that the fourdimensional G is really a constant (for a discussion o f this point see ref. [ 15 ]). The effective G can then be extracted from the effective four-dimensional g0o of eq. (4) by performing a post-newtonian approximation. We will proceed as follows. The authors ofref. [ 9 ] show a discrepancy between the value o f G measured in the laboratory G l a b = [ 6 . 6 7 2 6 ( 5 ) ] × 1 0 -11 m 3 kg -1 s -2,

(9)

and the G obtained by geophysical measures. What they really measure is the gradient o f the acceleration towards the earth, g, with the depth. The relationship o f g with the metric coefficient goo is

g= ½dgoo/dr.

(10)

The gradient o f g is proportional to G

Gocdg/dr.

26 February 1987

on a single parameter and approaching G R smoothly will hardly fit these data. On the other hand, the analysis done does in no way exhaust the K K possibilities. Perhaps a different type o f solution involving more free parameters - for instance nonstatic solutions - would present a more suitable behavior. Some work along this lines is in progress. To conclude we have s e e n how, contrary to the usual expectations and also to what happens in theories departing smoothly from GR, the free parameter o f K K remains basically unconstrained by the classical G R tests. In other words, the existence o f extra dimensions does not seem to be noticeable in the solar system. Although the models tested may not be very realistic, they are good approximations to the geometry surrounding a spherical mass distribution. The possibility o f getting curved four-dimensional spacetime in the low-energy limit of superstring theories with geometries appropriated to describe the physics o f black holes is currently under study.

(11 )

The value G obtained through this measurements is Gs~o = (6.720 + 0.002 + 0.024) × I 0 - l l m 3 kg -1 s -2.

(12)

As we have seen, the effective four-dimensional metric discussed previously approaches the Schwarzchild metric when the radius o f the extra dimensions C(r) goes to a constant, i.e., p goes to 1. Then G is strictly a constant. When the radius C(r) really depends on r, the four-dimensional goo defined in (4) gives through (10) and (1 1 ) an effective G(r):

(

5 p2 _ 1 M ~ Go b R ~ ,]'

G(Re)ocGo 1-F 2 p2

(13)

where Go is the newtonian gravitational constant as calculated in G R with the known Schwarzchild metric, M s is the earth mass and Ra~ is the earth mean radius. We see from (13) that the K K correction to Go has the wrong sign if we want to fit the value o f Gg~o given in (12). Even if the sign in (13) would have been correct, the value of p needed to fit the data lies outside the range o f values allowed by (8). A few comments are in order. First we see that the discrepancy among the experimental values o f G quoted is huge if it is to be evaluated in units of M/r for a body o f the solar system. Any theory depending

One o f us (A.H.V.) is thankful to Professor A. Davidson for calling her attention on the solutions discussed and to J.M. Labastida for carefully reading the manuscript and making useful comments.

References [ 1] T. Kaluza, Sitz. Preuss. Akad. Wiss. Berlin, Math. Phys. K 1 (1921) 966; O. Klein, Z. Phys. 37 (1926) 895. [2] E. Witten, Nucl. Phys. B 186 (1981) 412, and references therein. [3] See e.g.J.H. Schwarz, Superstrings, Vols. 1,2 (World Scientific, Singapore, 1985). [4] M.J. Duff, B.E. Nilsson and C.N. Pope, Phys. Rep. 130 (1986) 1. [5] A. Chodos and S. Detweiler, Phys. Rev. D 21 (1980) 2167. [6] See e.g.P.G.O. Freund, Nucl. Phys. B 209 (1982) 146; S. Randjbar-Daemi, A. Salam and J. Strathdee, Phys. Lett. B 135 (1984) 388; E. Alvarez and M.B. Gavela, Phys. Rev. Len. 51 (1983) 931; D. Sahdev, Phys. Lett. B 137 (1984) 155; A. Davidson, J. Sonnenschein and A.H. Vozmediano, Phys. Rev. D 32 (1985) 1330; Phys. Lett. B 168 (1986) 183. [7] A. Chodos and S. Detweiler, Gen. Rel. Grav. 14 (1982) 879. [8] A. Davidson and D.A. Owen, Phys. Lett. B 155 (1985) 247. [9] T. Dereli, Phys. Lett. B 161 (1985) 307. [ 10] K. Huang and S. Weinberg, Phys. Rev. Lett. 25 (1970) 895; N. Matsuo, Osaka University, preprint OU-HET-96 (1986); H. Nishimura and M. Tabuse, Kobe University preprint KOBE-86-10 (1986). 31

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[ 11 ] C.M. Will, General Relativity, eds. S.W. Hawking and W. Israel (Cambridge U.P., Cambridge, UK, 1979). [ 12] F.D. Stacey and G.J. Tuck, Nature 292 (1981) 230; S.C. Holding and G.J. Tuck, Nature 307 (1984) 714; S.C. Holding, F.D. Stacey and G.J. Tuck, Phys. Rev. D 33 (1986) 3487.

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[ 13] E. Fischbach et al., Phys. Rev. Lett. 56 (1986) 3, 1427(E), 2424. [ 14 ] S. Weinberg, Gravitation and cosmology (Wiley, New York, 1972). [ 15] T. Appelquist and A. Chodos, Phys. Rev. D 28 (1983) 772.