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Solar System Tests of the Equivalence Principle and Constraints on Higher-Dimensional Gravity

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Solar System Tests of the Equivalence Principle and Constraints on Higher-Dimensional Gravity Solar system tests of the equivalence principle and constraints on higher-dimensional gravity J. M. Overduin Department of Physics, University of Waterloo, ON, Canada N2L 3G1 and Gravity Probe B, Hansen Experimental Physics Laboratory, Stanford University, California (July 21, 2000) the theory, ξ and αk refer to possible preferred-location In most studies of equivalence principle violation by solar and preferred-frame effects, and the ζk allow for possible system bodies, it is assumed that the ratio of gravitational to violations of momentum conservation (see [1] for discus- inertial mass for a given body deviates from unity by a param- sion). One has η = 0 in standard general relativity, where eter ∆ which is proportional to its gravitational self-energy. γ = β = 1. In 4D scalar-tensor theories, by contrast, Here we inquire what experimental constraints can be set on γ =(1+ω)/(2 + ω)andβ =1+ω /[2(2 + ω)(3 + 2ω)2], ∆ for various solar system objects when this assumption is re- 0 laxed. Extending an analysis originally due to Nordtvedt, we where ω = ω(φ) is the generalized Brans-Dicke parame- obtain upper limits on linearly independent combinations of ter and ω0 = dω/dφ. ∆ for two or more bodies from Kepler’s third law, the position The relative gravitational self-energy U can be cal- of Lagrange libration points, and the phenomenon of orbital culated for most objects in the solar system, subject polarization. Combining our results, we extract numerical to uncertainties in their mass density profiles. Thus, 5 upper bounds on ∆ for the Sun, Moon, Earth and Jupiter, for example, US 10− for the Sun, while Jupiter 8 ∼− using observational data on their orbits as well as those of the has UJ 10− [3]. More precise estimates are avail- ∼− 10 Trojan asteroids. These are applied as a test case to the the- able for the Earth and Moon, UE = 4.64 10− and 11 − × ory of higher-dimensional (Kaluza-Klein) gravity. The results UM = 1.9 10− respectively [10]. With U known, con- are three to six orders of magnitude stronger than previous straints− on× EP violation (in theories of this kind) take the constraints on the theory, confirming earlier suggestions that form of upper limits on the PPN parameter η. The lat- extra dimensions play a negligible role in solar system dynam- est such bounds (from lunar laser ranging, or LLR) are 3 ics and reinforcing the value of equivalence principle tests as of order η 10− [9–12]. a probe of nonstandard gravitational theories. There| exist,|≤ however, theories of gravity in which EP violations are not necessarily related to gravitational self- 04.80.-y,04.50+h,96.35.Fs energy. In such theories, analysis of solar system data can lead only to upper limits on ∆ for the individual bodies involved. These limits will in general differ from object I. INTRODUCTION to object. They will, however, typically be orders of mag- nitude stronger than the above-mentioned experimental bound on η (since they are not diluted by the small fac- We investigate the consequences of possible violations tor U). In what follows, we obtain numerical constraints of the equivalence principle (EP) for solar system bodies on ∆ for the Sun, Moon, Earth and Jupiter, and ap- whose ratio of gravitational mass m to inertial mass m g i ply these as an example to higher-dimensional (Kaluza- is given by Klein) gravity, in which EP violations are related to the curvature of the extra part of the spacetime manifold. mg/mi =1+∆. (1.1) The study of this problem has a long history stretching back to Newton [1,2]. In modern times it has been most II. EQUIVALENCE PRINCIPLE VIOLATIONS IN closely associated with Nordtvedt [3–9], who investigated KALUZA-KLEIN GRAVITY the possibility that ∆ is proportional to the body’s rela- tive gravitational self-energy U Theories of gravity in more than four dimensions are older than general relativity [13], and most often associ- ∆=ηU , (1.2) ated with the names of Kaluza [14] and Klein [15] (for G d3xd3x U = ρ(x) ρ(x ) 0 / ρ(x) d3x, recent reviews, see [16,17]). The extra dimensions have 2 0 −c Z x x0 Z traditionally been assumed to be compact, in order to | − | explain their nonappearance in low-energy physics. The where η is a universal constant made up of parametrized past few years, however, have witnessed an explosion post-Newtonian (PPN) parameters, η 4β γ 3 of new interest in non-compactified theories of higher- 10ξ/3 α +2α /3 2ζ /3 ζ /3 [4].≡ In the− standard− − − 1 2 − 1 − 2 dimensional gravity [18–20]. In such theories the dimen- PPN gauge, γ is related to the amount of space curva- sionality of spacetime may in principle manifest itself at ture per unit mass, β characterizes the nonlinearity of 1 experimentally accessible energies. We examine here the bounds on the metric parameter b, which characterizes theory of Kaluza-Klein gravity [16], and focus on the pro- the departure from flatness of the fifth dimension in the totypical five-dimensional (5D) case, although the exten- vicinity of the central mass. sion to higher dimensions is straightforward in principle. In 5D Kaluza-Klein gravity, objects such as stars and planets are modelled by a static, spherically-symmetric III. THE MODIFIED THIRD LAW OF KEPLER analog of the 4D Schwarzschild solution known as the soliton metric. This may be written (following [21], but Let us now proceed to see what experimental con- switching to nonisotropic form, and defining a 1/α, straints can be placed on the parameter ∆ for solar sys- ≡ b β/α and M 2m) tem bodies. Given any inverse-square law central force ∗ ≡ ≡ 2 2 a 2 a b 2 1 a b (F = κ/r ), it may be shown that the two-body prob- dS = A dt A− − dr A − − − − − × lem admits as solutions elliptical orbits satisfying r2 dθ2 +sin2 θdφ2 Abdy2 , (2.1) − τ =2πa3=2 µ/κ , (3.1) where y is the fifth coordinate, A(r) 1 2M /r, M ≡ − ∗ ∗ p is a parameter related to the mass of the object at the where τ is the orbital period, a the semi-major axis, and center of the geometry, and the constants a, b satisfy a µ mi; mi; /(mi; + mi; ) the reduced mass of the sys- consistency relation a2 + ab + b2 =1,or ≡ 1 2 1 2 tem [27]. (Here mi;k is the inertial mass of the kth body.) a = b/2 (1 3b2/4)1=2 , (2.2) We are concerned in this paper with the case in which − ± − κ = Gmg;1mg;2, where the gravitational masses mg;k are which follows from Einstein’s field equations. Equa- related to the inertial ones mi;k by equation (1.1). Re- tion (2.1) reduces to the 4D Schwarzschild solution on arranging equation (3.1), one then obtains the following hypersurfaces y = const as b 0anda +1. modified form of Kepler’s third law Solitons differ in interesting→ ways from→ 4D black holes, 2 3 as discussed by many authors [16,17,21–23]. Of most rel- G(m1 + m2 + m2∆1 + m1∆2)=ω a , (3.2) evance for our purposes is the fact that a soliton’s gravita- where ω 2π/τ is the mean orbital angular frequency, tional mass (as identified from the asymptotic behaviour ≡ and mk refers (here and elsewhere in the remainder of of g00)isgivenbymg = aM , while its inertial mass (as obtained using the Landau-Lifshitz∗ energy-momentum this paper) to the gravitational mass of the kth body. Equation (3.2) was first obtained by Nordtvedt using a pseudotensor) turns out to be mi =(a + b/2)M .Us- ing equations (1.1) and (2.2), it follows that ∗ somewhat different argument [3]. If all bodies have the same value of ∆, we will not see 2 1=2 ∆= b/2(1 3b /4)− . (2.3) violations of the EP. The common part of ∆1 and ∆2 can ∓ − be absorbed into a rescaled gravitational constant. One It is important to recognize that b is not a universal con- way to see this explicitly is to rewrite equation (3.2) in stant like η, but depends in principle on local physics [23] the form and may vary from soliton to soliton. (Birkhoff’s theorem 2 3 in its usual form does not hold in higher-dimensional gen- G(1 + ∆1)(m1 + m2)+Gm1(∆2 ∆1)=ω a . (3.3) eral relativity [24].) To constrain the theory, one there- − fore hopes to apply as many tests as possible to a given There are two modifications to Kepler’s third law here: astrophysical system. a rescaling of the value of G in the first term, and a The classical tests of general relativity have been completely new second term, which depends only on the worked out for the soliton metric. Data from long- difference ∆ ∆ . This latter term is a clear manifes- 2 − 1 baseline radio interferometry, ranging to the Mars Viking tation of EP violation in the system. lander, and the perihelion precession of Mercury imply Nevertheless, we can put experimental constraints on that b 0.07 in the solar system [25,26]. In what fol- the values of ∆1 and ∆2, even in cases where they are | |≤ lows, therefore, we keep only terms of second order or equal. To see this, divide equation (3.3) through by Gm1 lower in b. We also drop the negative roots in equa- and rearrange to obtain tion (2.2) [ie, the positive roots in equation (2.3)], in 2 3 order to eliminate the possibility of negative gravita- ω a m2 1+ =(∆2 ∆1) tional and/or inertial mass, and to ensure that the 4D Gm1 − m1 − Schwarzschild solution is recovered (for y =const)as m2 b 0 [16].
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