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HAND 26 ZEITSCHRIFT FÜR NATURFORSCHUNG HE KT 4 Feldtheoretische Konstruktion der Jordan—Brans—Dicke-Theorie H. v o n G r ü n b e r g Institut für Theoretische Physik der Universität zu Köln (Z. Naturforsch. 26 a, 599—622 [1971] ; received 15 December 1970) Fieldtheoretic Construction of the Jordan-Brans-Dicke-Theory In the framework of Lorentz invariant theories of gravitation the fieldtheoretic approach of the generally covariant Jordan-Brans-Dicke-theory is investigated. It is shown that a slight restriction of the gauge group of Einstein’s linear tensor

Abstract

When the Brans-Dicke theory is formulated in terms of the Jordan scalar field φ, the amount of dark energy is related to the mass of this field. We investigate a solution which is relevant to the late universe. We show that if φ is taken to be a complex scalar field, then an exact solution to the vacuum equations requires that the Friedmann equation possesses both a constant term and one which is proportional to the inverse sixth power of the scale factor. Possible interpretations and phenomenological implications of this result are discussed.

gravitational con­ stant is the Jordan-Brans-Dicke theory (JBD), which is also based on quasi-Machian arguments. Recent analyses of lunar laser ranging data 19' 20 give upper limits on a possible elongation of the moon's orbit towards the sun due to a nongeodesical motion of planetary bodies in JBD, the Nordtvedt effect21, implying co> 29 [The parameter co is a measure of the deviation of JBD from general relativity (GR). In the limit io =c, JBD boils down to GR]. Since in JBD - G/G= [2/(3 + 4)] f0_1 (22) the limit co > 29 means — G/G < 10~12 a~4. (23) Note that

.10, and 3.7 2.10 441 SYMBOLS AND NOTATIONS SYMBOL NAME SECTION T trace of extrinsic curvature tensor 4.3, and 5.2 hT* tensor h in the 2.10 transverse-traceless gauge A+, Ax amplitudes of components of 2.10 polarized gravitational waves 4-tj reduced quadrupole moment 2.10 G Newtonian gravitational constant 2.3, 3.2.1, and 3.2.3 U classical gravitational potential 2.3, 3.2.2, and 3.7 a coupling constant of hypothetical 3.2.1 fifth force GPS Global Positioning System 3.2.2 DSN Deep Space Network 3.2.2 0 scalar field of 3.2.3 Jordan-Brans-Dicke theory co

-dependence of Newton’s gravitational constant. My proposal: One should call this theoretical package “Jordan-Brans-Dicke theory”. The historical development of the field theoretical research after the General Relativity Theory (1915) is extremely complicated. I tried to give a short hope- fully understandable review on the most important es- sential ideas and intellectual constructions. I had to be very sparingly in offering quotations. With respect to elder historical facts the reader may look in my text- book [7] and for later material in my monograph on 5-dimensional field

rivals the Jordan- Brans-Dicke theory. As far as we are aware of neutron star models have not been explored in other rival theories but we predict that especially for the so called linearized theories large differences are to be expected from the results presented below. In Einsteins theory the equation for the hydro­ static equilibrium is given (in Schwarzschild coordi­ nates) by dm/dr = 4 ttr2 g , - d P / d r - c , d i ) r~ (l — G m/r c~) and its Newtonian limit by dm/dr = 4 ti r2 g , — dP/dr = G g m/r2 . (12 ) In Eq. (11 ) m ( r ) is the gravitating mass

-61 Boundary, quantities to be fixed at the boundary in variational principle for field equation, 24, 279-282 463 SUBJECT INDEX Boyer-Lindquist coordinates, 41-42 Braginsky and Panov, Moscow, experiment of weak equivalence principle, 93, 95 Braginsky, Polnarev, and Thorne, 347 Brans-Dicke parameter ω and Mercury perihelion advance, 146 limits on, 111, 146, 219 Brans-Dicke theory, 109-111, 113, 146, 219, 275. See also Jordan-Brans-Dicke theory limits on the ω parameter, 111, 146, 219 Bubbles, cosmological, production in old inflationary models, 219 Bubbliness

medium strong form of the equivalence principle, also called the Einstein equivalence principle, in the following way: for every pointlike event of spacetime, there exists a sufficiently small neighbor­ hood such that in every local, freely falling frame in that neighborhood, all the nongravitational laws of physics obey the laws of special relativity. As already remarked, the medium strong form of the equivalence principle is satisfied by Einstein geometrodynamics and by the so-called metric theories of gravity, for example, Jordan-Brans-Dicke theory, etc. (see

, mn . . . must be time independent. In particular, i f we postulate the validity of the very strong equivalence principle, this must be true for all the physical laws, and all the physical constants, including gravity and the gravitational "constant" G . However, one may consider a violation of the very strong equivalence principle, as in the Jordan-Brans-Dicke theory 8 7" 9 3 where only the medium form is valid. In this case, G may change in time and/or space. On the contrary, according to the very strong formulation of the equivalence principle, satisfied

Jordan-Brans-Dicke theory,50 and any of those deviants from standard general relativity which are so numerous (§ 3.7).5 1 Furthermore, for the reasons explained above we demand that the spacetime manifold shall be spatially closed, that is spatially compact and spatially with­ out boundary.52 To formulate space compactness properly requires a further statement about space topology. Space topologies that are compact and natural to consider fall into two classes according to whether they do or do not lead to a model universe of limited duration; for this