Pioneer Anomaly in Perturbed FRW Metric

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by  Hossein Shojaie

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.
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PACSnumbers: 04.20.Cv,95.55.Pe,98.80.-k
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Keywords: Pioneeranomaly,Newtoniangaugetheory
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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.
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nal to the spacecraft such as gravitational and drag
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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-
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tively. Meanwhile, some others consider the possibil-
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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
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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 be-
tween 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),regard-
ing 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. We
from the gravitational force of the sun; that is, the con- would detect it as an anomalous redshift, interpreted as
stant term is not an anomaly anymore but is rather the an acceleration, if the universe would contract.
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