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Pioneer Anomaly in Perturbed FRW Metric ∗ Hossein Shojaie Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 1983963113, Iran In this manuscript, it is shown that the Pioneer anomaly is the local evidence for an expanding universe. In other words, its value is a direct measure of the Hubble constant while its sign shows the expanding behavior of the dynamics of the universe. This analysis is obtained by studying the radial geodesic deviation of the light rays in theperturbed Friedmann-Robertson-Walkermetric in theNewtonian gauge. 2 PACSnumbers: 04.20.Cv,95.55.Pe,98.80.-k 1 Keywords: Pioneeranomaly,Newtoniangaugetheory 0 2 n I. INTRODUCTION It is worth noting that there is an approximate coin- a cidence between the value of this anomaly and the cross J Radio-metricDopplerdatafromthePioneer10and11 product of the Hubble constant and the speed of light, 3 spacecraft, when they were at distances 20AU to 70 AU that is 1 fromthesun,showsasmallanomalousblueshiftwhichis m ] consistent with a sunwardconstantdeceleration in these H0c0 ≈6.99×10−10s2, (2) O spacecraft C m where the Hubble constant is assumed to be H0 = a =(8.74±1.33)×10−10 , (1) 100hKm/s/Mpc with h ≈ 0.72. It should also be men- h. P s2 tioned that no such anomaly has been detected in the p as reported first by Anderson et al. [2] and confirmed orbits of planets yet [3]. - o further by more accurate data analysis [3, 40, 41]. This Inthismanuscript,thePioneeranomalyisinvestigated r apparentdeviationfromthe Newtoniangravitationalin- in the Perturbed Friedmann-Robertson-Walker Metric t s versesquarelaw is knownasthe Pioneeranomaly. Since (FRW) in the Newtonian Gauge. A motivation for this a then,manyproposalshavebeenconsideredtoexplainthe is provided in Section 2. In Section 3, it is shown that [ effect. According to Turyshev & Toth [42], these expla- the Pioneer anomaly, at a distance far enough from the 1 nations,coveringfieldsfromconventionaltonewphysics, sunandthe planets,is the firstcorrectionto the conven- v can be broadly classified into many categories. tionalNewtoniangravitationalforce. Section4,contains 9 For instance, some can be classified as forces exter- summary and remarks. 8 nal to the spacecraft such as gravitational and drag 8 3 forces. These can be due to, for example, the solar II. SCHWARZSCHILD METRIC VS. FRW . wind [2], unknown mass distributions and the Kuiper 1 METRIC belt [7, 14, 16, 29], and interplanetary dust [28], respec- 0 tively. Meanwhile, some others consider the possibil- 2 1 ity for new physics, such as dark matter [31], modified The universe is expanding and there is no restriction : Newtoniandynamics (MOND) [23], modified Newtonian abouttheleastscaleinwhichthisexpansionproceeds[1]. v gravity [5, 6, 11, 17, 18, 25, 33, 36, 39], scalar-tensor However,thegravityoflocallynon-uniformlydistributed i X theories of gravitation [12, 43], scalar-tensor-vector the- massive systems is dominant in local scales and this r ories of gravitation [26], and f(R) theories of gravita- causes structures within these scales not to feel the ex- a tion [13]. Another class contains cosmologically origi- pansion but, in fact, to be contracting under their inter- nated effects such as the cosmologicalconstant [22], and nalgravity. Inotherwords,differentscaleshavedifferent theprobableeffectoftheexpansionoftheuniverseonlo- geometriesand consequentlydifferent metrics. Two best cal systems [20]. Yet, the other explanations are related candidatesforthispurposearetheSchwarzschildandthe to on-board systematic effects [8], such as thermal recoil FRW metrics. force[9,10,19,27,37,38],andgasleakage. Andfinally,a The Schwarzschildmetric is a vacuum solutionof Ein- class regards probable effects on radio signals [4, 21, 35]. stein’s equation. It is applicable to the outer part of any But up to now, there is no universally accepted descrip- sphericallysymmetric mass distribution, andit can even tion for this phenomena. A detailed review of this phe- be an approximation for non-globally distributed celes- nomenon can be found in Turyshev & Toth [42] and ref- tial structures like galaxies and clusters of galaxies, at erences therein. least at distances far enough from them. On the other hand, the FRW metric is the result of applying the cosmological principle. It is applicable to the early universe as well as the largest scales in late ∗ [email protected] time. Nonetheless, the whole universe, is bulky at small 2 scales,andhomogeneousatlargeones. Thismeansthat, III. PIONEER ANOMALY IN THE a suitable metric, if any, would rather be a combination PERTURBED FRW METRIC of these two metrics, that is, it should asymptotically tend to the FRW in large scales, while having the form The lack of an exact metric in this case, can lead one of the Schwarzschild metric in local scales. Also, the to study the effect of the universal expansion on local conjunction conditions between two regions of validity systems in local inertial frames, whereas, the effect of should be carefully specified. A need for such a metric the background metric is shown in deviation from the is in, for instance, studying the evolution of black holes geodetic motion, that is and collapsing stars in an expanding universe. To be more specific, three fundamental differences between these two metrics which make the construction of d2xi dxj dxk dx0 2 a hybrid metric difficult, can be summarized as follows: dt2 +Γijk dt dt =−Ri0j0xj(cid:18) dt (cid:19) . (6) • The Schwarzschild metric is static while the FRW Now, regarding the perturbed FRW metric in the New- is a dynamical one. tonian gauge, namely (5), and assuming k = 0, one has for a radialmotion(or rather,the radialcomponent of a • The Schwarzschild metric is necessarily a vacuum motion) solution but the FRW can concern a non-zero energy-momentum tensor. d2r dr 2 dr dt2 =−Γ100−Γ111(cid:18)dt(cid:19) −2Γ101dt −g11R0101r , (7) • The Schwarzschild metric describes the spacetime outside a spherically symmetric object and there- which after substitution, it leads to fore it is inhomogeneous. Homogeneity, on the other hand, is a key assumption in deriving the d2r 1 mc2 1 mr˙2 = − (8) FRW metric. dt2 2a3r2(1−m/ar) 2ar2(1−m/ar) a˙ (2−m/ar) 1 mc2(2−m2/a2r2) McVittie was the first who introduced a metric which − r˙− a(1−m/ar) 2a3r2(1−m/ar)(1−m2/a2r2) had different behaviors in local and global scales, each compatiblewithoneoftheabovemetrics[24]. Hisworks 1 a˙ 2 m(1−2m/ar) + was followed by the others [15, 32, 34]. The McVittie 2(cid:18)a(cid:19) a(1−m/ar)(1−m2/a2r2) metric is the best prototype for a hybrid metric, em- a¨ m 1 m(1+m2/a2r2) bedding a spherically symmetric object in a dynamical + + . a(cid:20)(1−m/ar) 2a(1−m/ar)(1−m2/a2r2)(cid:21) universe. Generally, it can be written as ds2 = 1−µ(t,r) 2dt2−(1+µ(t,r))4a2(t)dr2+r2dΩ2 , derItiviesdwfoorrthnnoont-innugllthgaetodaeltshicosu,ghoneequcaatnionsp(e6c)ifhyasabneuelnl (cid:18)1+µ(t,r)(cid:19) (1+ 41kr2)2 geodesic with the limit of the direction of a time- or (3) space-likegeodesictendingarbitrarilyclosetowardanull where direction. Hence, this equation is also applicable for null geodesics. µ(t,r)= m 1+ 1kr2 . (4) For a signal from the spacecraft to the earth (or the 2a(t)rr 4 sun), one has r˙ =c. This reduces the last equation to with m ≡ GM/c2 and k being the (geometrized) mass d2r =−1 mc2(2−m2/r2) (9) of the central object and the curvature of the space, re- dt2 2r2(1−m/r)(1−m2/r2) spectively. The parameter a(t) is the asymptotic cosmo- (2−m/r) 1 m(1−2m/r) logical scale factor. The metric is exactly the FRW for −(1−m/r)H0c+ 2H02(1−m/r)(1−m2/r2) m → 0 and tends to the Schwarzschild metric when the m 1 m(1+m2/r2) curvatureofthespaceandthetimeevolutionofthescale +a¨ + . factor are assumed to be negligible. Between these two (cid:20)(1−m/r) 2(1−m/r)(1−m2/r2)(cid:21) extremes, where m≪r, the metric is approximated as wheretheconventiona0 =1hasbeenused. Byapplying ds2 =(1+2µ)dt2−(1−2µ)a2(t) dr2+r2dΩ2 , (5) the Taylor expansion, the above equation becomes (cid:0) (cid:1) d2r mc2 m m which is the perturbed FRW metric in the Newtonian dt2 =− r2 1+ r +··· −H0c 2+ r +··· (10) (cid:16) (cid:17) (cid:16) (cid:17) gauge. In this sense, µ can be considered as the New- 1 m 3 m tonian gravitational potential. For an ordinary star, the + mH02 1− +··· + ma¨ 1+ +··· . 2 r 2 r metric (5) is applicable from just outside the star up to (cid:16) (cid:17) (cid:16) (cid:17) 2 distanceswhere either the sphericalsymmetry stillholds Here, m = GM/c , with M being the mass of the sun, or m→0. is of order of 103 in MKS units. After rearranging the 3 terms, one has dominant term of the spacecraft’s deceleration. d2r GM 1 3 dt2 =(cid:18)− r2 −2H0c+ 2mH02+ 2ma¨(cid:19) (11) IV. SUMMARY AND REMARKS GM 1 3 GM +(cid:18)− r2 −H0c− 2mH02+ 2ma¨(cid:19) rc2 In this manuscript, the geodesic deviation has been derived in the perturbed FRW metric in the Newtonian 2 2 +O (Gm/rc ) . gauge. The derived relation when applied to a light ray, (cid:0) (cid:1) has a constant term, 2H0c, as its first correction to the In the first line, the first term, which is simply due to conventionalNewtoniangravitationaldeceleration,inter- the NewtonianGravitationalforce,is oforderof 10−5 at preted as an additional radial deceleration. r ∼ 20AU, and the second term, which is a constant, is Three important hints seem necessary to be empha- oforderof10−9,allinMKSunits. Thisdecelerationmay sized: be responsible for the Pioneer anomaly. The third and forth terms have the same order as the cross product of 1. This constant term can best be measured far the critical density of the universe and m, that is 10−23. enough away from the central object, where the In the second line, after multiplying by m/r, the first value of the other terms which are depend on 1/r term is of order of 10−15 at at r ∼ 20AU, which is six and its powers have been sufficiently decreased. orders of magnitude less than the constant anomaly in the first line, far beyond the precession of the observed 2. The coefficientofH0c,ifthere onlyexistsacentral Pioneer anomaly; it is ever decreasing as the spacecraft celestialobjectwithmassM inanotherwiseempty goes further. dynamicuniverse,isexactly“two”. Thiscoefficient will surelybe affectedby the existenceofother ob- As aresult,the firsttwoleadingtermsin(11),regarding the order of the terms, are GM/r2 and 2H0c, both jects which evidently change the metric. Besides, it shouldn’t be forgotten that the metric (5) is an with negative sign corresponding to a deceleration. Re- approximation, too; that is, it is not valid in the spectively,thesetwotermsareresponsiblefortheconven- whole interval r=0 to infinity. tionalblueshiftfortheNewtoniangravitationalpotential and the extra blueshift dubbed as the Pioneer anomaly. 3. Even the spacecraft themselves, experience such a Indeed, this anomaly is induced by the expansion of the deceleration, but its value is proportional to H0r˙ universe and can be regarded as a local consequence of as can be seen in (8). Considering the fact that the global expansion of the universe. r˙ ≪c, its value is sufficiently less than that of the To be sure that this anomaly will not be canceled out light ray and can be neglected up to this order. bytherealdecelerationofthespacecraftaccordingtothe metric (5), one should compare it with the deceleration Regardingallthe aboveremarkstogether,the Pioneer induced on the spacecraft. As can be easily checked, the anomaly can be explained in the context of the the per- induced acceleration on the spacecraft is at least four turbed FRW metric in the Newtonian gauge. It should orders of magnitude less than the predicted deceleration be kept in mind that the detected anomalous blueshift (the third term in (8)). in the signals received from the spacecraft is not due to It is worth noting that using a purely Schwarzschild the sun,the spacecraftand/ortheirrelativemotions. Its metric does not leadto such an anomalyin the equation value is completely determined by the current value of of motion. On the other hand, although the FRW met- the Hubble parameter and its sign is an indication of ric has this term, it does not contain the terms arising whether the universe is expanding or contracting. 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