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1 oi ( CERN LIBRHRIES» GENEVF1 A gg (c Q» HE? {E3 ho [INI][ [ [ [ I ] I ] l [ II] tl + C M Peemeosso IASSNS-HEP-93 /86 Violation of Equivalence Principle in Brans-Dicke Theory Y. M. Cho' School of Natural Sciences, Institute for Advanced Study, Princeton, NJ Abstract We argue that the quantum Huctuation should violate the weak equivalence principle in the Brans-Dicke theory, which means that the Jordan frame can no longer be interpreted as physical. The violation of the equivalence principle could be attributed to the dilatonic fifth force which we estimate to be at least 10`3 times weaker than the gravitational attraction. This conclusion applies to all the generalized Brans-Dicke theories. Ever since Brans and Dicke have made the fundamental attempt to generalize the Ein stein s theory of gravitation with the elementary scalar field [1,2], the important question has been what frame should one treat as physical in the theory [3,4]. From the conception the authors have proposed to resolve this issue with the equivalence principle. Indeed by insisting that the ordinary matter should not violate the weak equivalence principle, they have effectively selected the Jordan frame as physical. This is because by construction only the Jordan metric couples minimally to the ordinary matter, and thus guarantees the equiv alence principle for the ordinary matter. With this minimal coupling the Jordan frame has always been treated as physical in the theory. On the other hand it has become clear that, even in the Jordan frame, the theory should violate the strong equivalence principle. This is because the modification of the Newton s potential by the Brans-Dicke scalar field alters the gravitational energy (and thus OCR Output

2 the gravitational mass) of a self gravitating body. This has been noticed from the beginning by Dicke himself [5], and has generally been referred as the Nordtvedt effect But notice that this violation of the equivalence principle is a purely classical phenomenon due to the finite size of the self gravitating test body, which is absent in the ideal point-like test particles [7]. The purpose of this letter is to argue that, even in the Jordan frame, the quantum effect should force the point-like test particles to violate the weak equivalence principle. This brings back the old que tions and raises new ones. What is the physical frame, and how can one identify that? What is the mechanism for the violation of the equivalence principle, and how can one verify this? We will address on these questions in the followings, and argue that the violation of the equivalence principle is due to the existence of the dilatonic fifth force which could have had a deep impact in cosmology. Let us start from the Brans-Dicke theory [1] where 7,,,, is the Jordan metric, 45 is the elementary scalar field, and w is the Brans-Dicke coupling constant. Notice that here we have chosen a scalar field I' and the electromagnetic field F,,,, as the matter fields for simplicity. But the important point here is that the Jordan metric couples minimally to the ordinary matter. In other words the Brans-Dicke scalar field does not couple directly to the ordinary matter. Notice that this minimal coupling holds true only in the Jordan frame because, if we choose another frame by making a conformal transformation of the Jordan metric, the new metric no longer couples minimally to the ordinary matter. So the theory guarantees the equivalence principle for the ordinary matter only in the Jordan frame. L = Lg + L1, in = \/ Y[ R + 1' '(@» )(@» ) (1) B1 =»/ Y I v""(6. I )(@» I ) + V( I ) + r' "r F» F I» Unfortunately the theory can not really guarantee the weak equivalence principle, even if we treat the Jordan metric as physical. To see this notice first that the Jordan metric does OCR Output

3 not couple minimally to the Brans Dicke scalar field. This implies that the elementary scalar particle will not follow the geodesic determined by the Jordan metric, and thus violate the equivalence principle But the real problem is that the weak equivalence principle should be violated not only by the Brans-Dicke scalar particle but also by the ordinary matter, because the minimal coupling of the Jordan metric to the ordinary matter can not be maintained when the quantum effect is taken into account. When the quantum correction takes place the ordinary matter must couple to the Bran -Dicke scalar field through the Jordan metric, as shown in Fig.1. This means that the minimal coupling is not stable under the quantum fluctuation. So already at this first order onéfban not avoid a direct coupling between the ordinary matter and the Brans Dicke scalar field, which will obviously spoil the minimal coupling of the Jordan metric to the ordinary matter. Notice that this argument does not depend on the renormalizability of the theory, because the direct coupling arises from the first order quantum correction which is certainly finite. Another way to see the existence of the direct coupling is to consider the effective action of the theory. The equations of motion of the Lagrangian (1) tells that the Brans-Dicke scalar field should couple to the trace of the energy momentum tensor of the ordinary matter. This implies that the Lagrangian has an effective interaction of the following type, sf = q T, (2) where T is the trace of the energy momentum tensor of the ordinary matter. This should repel any doubt on the existence of the direct coupling of the ordinary matter to the Brans Dicke scalar field in the Jordan frame. With the direct coupling, of course, the ordinary matter will no longer respect the weak equivalence principle. If the minimal coupling of the Jordan metric to the ordinary matter can not be enforced, there is no reason why one should treat the Jordan frame as physical If so, what frame should one choose as physical? Unfortunately it appears that a priori there is no fundamental principle which can guide us to choose the correct conformal frame. But the simplicity of the logic favors the Pauli frame [3,4]. To see this notice that the Jordan metric is not a OCR Output

4 mass eigenstate, but is in fact a strange superposition of the massless spin-two graviton and the Brans-Dicke scalar field. Indeed there is only one conformal frame, the Pauli frame, in which the metric describes the massless spin two graviton. To show this we introduce the Brans-Dicke dilaton field 0 and the Pauli metric gw, by [4] so = =" ms, (3) where Now in terms of the Pauli metric the Lagrangian (1) has the following form, B = -R 1%% + y""(@» )(6 ) [% t/5 % g "<6» ><6» v> + * *v< v> + ;g *g F,...F. (4) < > Obviously this confirms that the Pauli metric describes the massless graviton. But there are two more things to be noticed here. First, in this Pauli frame it is the dilaton which couples minimally to the Einstein s gravitation. Here the ordinary matter, with the exception of the gauge field, couples non-minimally to the gravitation. The reason that the gauge field keeps the minimal coupling is, of course, that the gravitational coupling of the gauge field is conformally invariant in the 4-dimensional space-time. This means that the dilaton and the photon would still respect the weak equivalence principle, but the ordinary matter must violate it, in the Pauli frame. Secondly, the Bra.ns Dicke coupling constant w is replaced by a new coupling constant a. What is nice about this is that a unnaturally large coupling constant w (w > 600) is replaced by a perfectly reasonable coupling constant a (cz < 0.01), which specifies the interaction of the dilaton with the ordinary matter. So the physics becomes more transparent in the Pauli frame. The Einstein s gravitation is described by the Pauli metric (also called the Einstein metric), which still respect the weak equivalence principle. But now the dilaton creates a universal fifth force [11,12] which OCR Output

5 will couple to all the ordinary matter except the gauge Held, and thus will inevitably mix with the gravitation. This mixing, of course, spoils the weak equivalence principle. Notice that in the old Brans-Dicke theory the gravitation is supposed to be described by the dilaton as well as the by the Jordan metric. Nevertheless only the Jordan metric is responsible for the gravitational interaction of the ordinary matter. In retrospect this looks very strange, because the Lagrangian Lg in (1) which describes the Brans-Dicke gravitation is conformally invariant when the conformal transformation is accompanied by a reparametrization of w This means that, in the absence of the matter Held, there is no way of telling how much of gravitation comes from the scalar and how much from the metric. In the new interpretation there is no such ambiguity, because here the massless graviton is solely responsible for the gravitation. At this point it is perhaps important for us to discuss the similarity between the Brans Dicke theory and the Kaluza-Klein theory, and to clarify some of the misunderstandings on the Kaluza-Klein theory. It is well known that the Kaluza-Klein theory reduces to a generalized Brans-Dicke theory after the dimensional reduction in the zero modes approxi mation. It must be emphasized, however, that this becomes true only after one chooses the proper frame [4,8]. This is because in the Jordan frame (which one arrives at after the naive dimensional reduction) the (4 + n)-dimensional Kaluza-Klein theory has a negative w given n 1 w = _ (6) So the Kaluza Klein dilaton becomes a ghost in the Jordan frame, which is hardly acceptable. To cure this defect one must first choose the Pauli frame before discussing the physics. Unfortunately many of the literatures on the Kaluza-Klein theory fail to do so. This failure is evident in the popular Kaluza Klein cosmology, where the cosmological potential has no true minimum Only in the Pauli frame one obtains a well-behaved cosmological potential [8,10]. Now we discuss the dilatonic Hfth force. The important issues here are how strong is the OCR Output

6 fifth force, and how far does it act. The strength is determined by the size of the coupling constant cx, but the range is determined by the mass p of the dilaton. Let F5 be the fifth force between the two mass points ml and mq separated by r. From the dimensional argument one may express the total force in the Newtonian approximation as F = F, + F5 z _` (1 + pe"'), (7) where F, is the gravitational force, oz, is the gravitational fine structure constant, and B 2 :12. Now, with (4) we can easily estimate Experimentally [7] we have w > 600, so that B 2 a2 < 10'. So we expect the fifth force to be at least 10`3 times weaker than the gravitation. Unfortunately there is no simple way to estimate pz. Of course the dilaton is massless in the tree approximation, so that it could acquire a small mass only through the quantum correction. But it is extremely difficult to estimate this quantum correction, espesially when the renormalizability of the theory is not guaranteed. Under this circumstance we have the following possibilities : a) y. = 0 (long range). Experimentally we have B < 10', from which one might conclude that there is no such fifth force. At this point, however, it is good to remember that the ratio between the gravitational and the electromagnetic couplings is extremely small, oz,/cx, 2 10', at the atomic scale. b) p. 2 2 x 10`1 ev (1km range). Experimentally we have Be"" l0". This is perfectly consistent with our estimate of B. Of course B could still be much smaller than 10 * here, in which case a best way to measure B is the laboratory (small size) experiments. c) y 2 1eV (2 x 10' cm range). In this case there is no experimental constraint yet, and the possibility B 2 10'3 can not be ruled out. In fact at this short distance B could be considerably larger than 10', because our estimate of,8 is based on the long range experiments. d) p GeV (2 X 10" cm range). In this case there is practically no way to detect the fifth force in the present universe, even though it may very well exist at this short distance. In the early universe, however, this fifth force could have played an important role to create OCR Output

7 In this Letter we have shown that the Brans-Dicke theory must violate the weak equiva lence principle, even in the Jordan frame. The violation is due to the presence of the dilatonic fifth force. Clearly our conclusions should also apply to the generalized Brans-Dicke theories [14] with only minor modifications, which again has to be re-interpreted. We conclude with the following remarks: 1) It must be emphasized that, although the Pauli frame is the most natural candidate for the preferred physical frame, one can not rule out the other possibilities purely from the logical point of view. It is entirely possible that the nature could have chosen another frame for some unknown reason. However, this does not mean that one can avoid the violation of the weak equivalence principle, which must be present in any frame. Only after the future precision experiments tell us exactly how this violation occurs, one could decide what is the correct physical frame. This is why we need more precision experiments. 2) The importance of the dilaton in cosmology can not be over-emphasized. The dilaton is responsible for the generalized indation [10], or the extended indation [13]. Indeed with the dilatonic fifth force, one could finally have a. dynamical explanation of the iniation. Furthermore the existence of the dilaton should have a crucial impact on the dark matter problem, because the dilatonic matter could account for the dark matter of the universe which one need for the inflationary cosmology [4,10]. 3) Finally what makes our discussions so important is that the Brans-Dicke theory (or a generalization of it) appears in all the higher-dimensional unified field theories, including the supergravity and superstring. This tells that the violation of the equivalence principle and the existence of the dilatonic fifth force is a unique characteristics of the unified field theories which could actually be tested by the experiment. Indeed the confirmation of the fifth force by experiments could be interpreted as a confirmation of the higher dimensional unification itself. From the above discussions it must be clear that one must look at the equivalence prin ciple, the fifth force, the dark matter, and the iniation from a single unified point of view. OCR Output

8 In particular one should look for the fifth force not only in the present universe, but also within the context of the cosmology. The details of the above discussions will be published elsewhere. ACKNOWLEDGMENTS Many discussions with C. Brans, M. Bucher, F. Dyson, W. T. Ni, K. Nordtvedt, F. Wilczek, and Y. S. Wu are gratefully acknowledged. The work is supported in part by the Ministry of Education, the Korea Science and Engineering Foundation, and the SeoAm Foundation. OCR Output

9 REFERENCES Permanent address: Center for Theoretical Physics, Seoul National University, Seoul , Korea. [1] C. Brans and R. Dicke, Phys. Rev. 124, 921 (1961); R. Dicke, Phys. Rev. 125, 2163 (1962). [2] P. Jordan, Ann. Phys. (Leipzig) 1, 218 (1947); Schwerkraft und Weltall Vieweg und Sohn, Braunschweig, 1955); Y. Thirry, C. R. Acad. Sci. (Paris) 22, 216 (1948). [3] W. Pauli, in Schwerkraft und Weltall (see Ref. [2]); M. Fierz, Helv. Phys. Acta. 29, 128 (1956). [4] Y. M. Cho, Phys. Rev. Lett. 68, 3133 (1992). See also Y. M. Cho, in Evolution of Uni verse and Its Observational Quest, Proceedings of the XXXVII-th Yamada Conference, Tokyo, 1993, edited by K. Sato (Universal Publishing, Tokyo, 1993). [5] R. Dicke, in Gravitation and Relativity, edited by H. Y. Chiu and W. F. Hoffman, (Benjamin, New York, 1964). [6] K. Nordtvedt, Phys. Rev. 169, 1014 (1968); C. M. Will, J. Astrophys. 163, 611 (1971); W. T. Ni, J. Astrophys. 176, 769 (1972). [7] C. M. Will, Theory and Experiment in Gravitational Physics, (Cambridge University Press, 1993). [8] Y. M. Cho, J. Math. Phys. 16, 2029 (1975); Y. M. Cho and P. G. O. Freund, Phys. Rev. D12, 1711 (1975); Y. M. Cho, Phys. Lett. B199, 358 (1987); Y. M. Cho, Phys. Rev. D35, 2628 (1987); Y. M. Cho and D. S. Kimm, J. Math. Phys. 30, 1570 (1989). [9] E. Kolb and M. Turner, The Early Universe, (Addison Wes1ey, New York, 1990). [10] Y. M. Cho, Phys. Rev. D.{1, 2462 (1990). See also Y. M. Cho, in Proceedings of the Fifth Marcel Grossmann Meeting, edited by D. Blair and M. Buckingham (World Scientific, OCR Output

10 Singapore, 1989). [11] E. Fishbach, D. Sudarsky, A. Szafer, C. Talmadge, and S. Arouson, Phys. Rev. Lett. 56, 3 (1986); Y. M. Cho, Phys. Rev. D35, 2628 (1987); Y. Fujii, Int. J. Mod. Phys. A6, 3505 (1991). [12] Y. M. Cho and D. H. Park, Gen. Rel. Grav. 23, 7441 (1991). [13] D. La and P. Steinhardt, Phys. Rev. Lett. 62, 376 (1989); P. Steinhardt and F. Accetta., Phys. Rev. Lett. 64, 2740 (1990). [14] T. Damour, G. W. Gibbons, and C. Gundlach, Phys. Rev. Lett. 64, 123 (1990); T. Damour and K. Nordtvedt, Phys. Rev. Lett. 70, 2217 (1993). 10 OCR Output

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