Using global positioning systems to test extensions of General Relativity
ABSTRACT We consider the feasibility of using the Galileo Navigation Satellite System
to constrain possible extensions or modifications to General Relativity, by
assessing the impact of the related additions to the Newtonian potential and
comparing with the available observables: the relative frequency shift and the
time delay of light propagation. We address the impact of deviations from
General Relativity based on the parameterized Post-Newtonian parameters due to
the presence of a Cosmological Constant, of a constant acceleration like the
putative Pioneer anomaly, a Yukawa potential term due to massive scalar fields
and a power-law potential term, which can arise from Ungravity or f(R)
theories.
- [Show abstract] [Hide abstract]
ABSTRACT: We consider unparticle inspired corrections of the type ${(\frac{R_{G}}{r})}^\beta$ to the Newtonian potential in the context of the gravitational quantum well. The new energy spectrum is computed and bounds on the parameters of these corrections are obtained from the knowledge of the energy eigenvalues of the gravitational quantum well as measured by the GRANIT experiment. Comment: Revtex4 file, 4 pages, 2 figures and 1 table. Version to match the one published at Physical Review DPhysical review D: Particles and fields 07/2010; - SourceAvailable from: Orfeu Bertolami[Show abstract] [Hide abstract]
ABSTRACT: In the last few years, the so-called flyby anomaly has been widely discussed, but remains still an illusive topic. This is due to the harsh conditions experienced during an Earth flyby as well as due to the limited data available. In this work, we assess the possibility of confirming and characterizing this anomaly by resorting to the scientific capabilities of the future Galileo constellation.09/2011; - SourceAvailable from: Orfeu Bertolami[Show abstract] [Hide abstract]
ABSTRACT: We propose the concept of a space mission to probe the so called flyby anomaly, an unexpected velocity change experienced by some deep-space probes using earth gravity assists. The key feature of this proposal is the use of GNSS systems to obtain an increased accuracy in the tracking of the approaching spacecraft, mainly near the perigee. Two low-cost options are also discussed to further test this anomaly: an add-on to an existing spacecraft and a dedicated mission.International Journal of Modern Physics D 12/2011; 21(4). · 1.03 Impact Factor
Page 1
arXiv:1006.3094v1 [gr-qc] 15 Jun 2010
Using the Galileo satellite constellation to test
Gravitation
O. Bertolami1, J. P´ aramos
Instituto de Plasmas e Fus˜ ao Nuclear, Instituto Superior T´ ecnico
Abstract
We consider the feasibility of using the Galileo Navigation Satellite System to
constrain possible extensions or modifications to General Relativity, by assessing
the impact of the related additions to the Newtonian potential and comparing with
the available observables: the relative frequency shift and the delay time of light
propagation. We address the impact of deviations from General Relativity based
on the parameterized Post-Newtonian parameters, the presence of a Cosmological
Constant, of a constant acceleration like the putative Pioneer anomaly, a Yukawa
potential term due to massive scalar fields and a power-law potential term, which
can arise from ungravity or f(R) theories.
Key words:
PACS: 04.80.Cc, 04.25.Nx, 04.50.+h, 07.87.+v
Galileo Navigation Satellite System, General Relativity, constant acceleration,
Parameterized Post-Newtonian formalism, Yukawa potential, power-law additions,
Ungravity, f(R) theories
Email addresses: orfeu@cosmos.ist.utl.pt (O. Bertolami), paramos@ist.edu
(J. P´ aramos).
URL: http://web.ist.utl.pt/orfeu.bertolami/homeorfeu.html (O.
Bertolami).
1Also at Departamento de F´ ısica, Instituto Superior T´ ecnico
2This work was developed in the context of the first and second conferences Scien-
tific and Fundamental Aspects of the Galileo Programme (respectively in Toulouse,
2007 [1] and Padova, 2009 [2]). The authors thank the organization for the hospi-
tality displayed and Dr. Cl´ ovis de Matos for fruitful discussions.
Preprint submitted to Classical and Quantum Gravity17 June 2010
Page 2
1 Introduction
The Galileo positioning system is an important step towards the improve-
ment and development of new applications in navigation monitoring and re-
lated topics. Its operational use of precision clocks in orbit and comparison
with those on the ground stations enables one to view it broadly as a tim-
ing experiment in outer space. Hence, Galileo offers a great opportunity for
fundamental research in physics: together with the deployed Global Position-
ing System (GPS) and GLONASS systems, satellite navigation are indeed the
first technological application where relativistic effects are taken into account
as a regular engineering constraint on the overall design (see Refs. [3–6] and
references therein).
As such, there are several effects arising from special and General Relativ-
ity (GR) that must be taken into account, i.e. time dilation, gravitational
blueshift and the Sagnac effect. These may yield a clock deviation of as much
as ∼ 40 µs/day, many orders of magnitude above the accuracy of the onboard
clocks considered of the mentioned navigations systems.
Furthermore, the gravitational frequency shift is of the order of VN/c2≃ 10−10
(where VN= GME/REis the Newtonian potential, G is Newton’s constant,
ME≃ 6.0 × 1024kg is the Earth’s mass, RE≃ 6.4 × 106m is its radius and
c is the speed of light), thus falling within the 10−12frequency accuracy of
the current space-certified clocks. In the Galileo Navigation Satellite System,
this correction is accounted by the receiver, while the GPS system accounts
for this mismatch through an offset in the onboard clock frequency.
The Galileo navigation system offers a positioning improvement of one order
of magnitude (from an everyday error margin of ∼ 10 m with the GPS system
to ∼ 1 m); a spatial accuracy of the order of 1 mm is possible using carrier
phase measurements. Given this, a legitimate question arises: what are the
possible implications for fundamental physics that one may extract from this
increased precision?
In this study, we aim to establish bounds on the detectability of extensions
and modifications to General Relativity, by assessing the impact of the related
additions to the Newtonian potential on the observables made available by the
Galileo system, namely the relative frequency shift ǫf≡ f/f0−1 and the light
propagation time delay ∆t. This may be done for a wide variety of already
available phenomenological models (see Refs. [7,8] for updated surveys). We
assume as typical values ǫf= 10−12and ∆t = 10−12s (corresponding to the
1 mm precision).
This paper is organized as follows: firstly, we assess the main relativistic effects
that are present in the Galileo navigation system. We proceed and consider
2
Page 3
the possibility of measuring several corrections to the law of gravity using
Galileo:
• Deviation from GR based on the parameterized Post-Newtonian parameters
• Presence of a Cosmological Constant
• Constant acceleration like the putative Pioneer anomaly
• Yukawa addition mediated by massive scalar fields
• Power-law addition, which can arise from ungravity or f(R) theories
Finally, conclusions are drawn and an outlook is presented.
2 Main relativistic effects
2.1 Frame of reference
One begins by assuming that all time-dependent effects are of cosmological
origin, and thus evolve over a timescale of order H−1
constant; hence, one may discard these as too small within the timeframe of
interest and assume a static, spherically symmetric scenario. Given this, one
considers the standard Schwarzschild metric — which, in isotropic form, is
given by the line element
0, where H0is Hubble’s
ds2=−
?
?
1 +2V
c2
?
?
(c dt)2+
1
1 +2V
1 −2V
c2
dV2
(1)
∼=−1 +2V
c2
(c dt)2+
?
c2
?
dV2,
where dV = dr2+dΩ2is the volume element, and V is the gravitational poten-
tial. In the unmodified GR scenario, this is simply the Newtonian potential
V = VN = −(1 + Σn
perturbations and density profiles.
i=1Jn)GME/r, where the Jnmultipoles reflect geodesic
One now takes into account the rotation of the Earth with respect to this
fixed-axis reference frame, with angular velocity ω = 7.29 × 10−5rad/s; the
so-called Langevin metric may be obtained by performing a coordinate shift
t′= t, r′= r, θ′= θ and φ′= φ − ωt′, yielding the line element
ds2=−
1 +2V
c2−
?ωrsinθ
c
?2
(c dt)2+(2)
3
Page 4
2ωr2sin2θdφdt +
?
1 −2V
c2
?
dV2,
where, for simplicity, primes were dropped.
Clearly, a non-diagonal element appears, plus an addition to the gravitational
potential interpreted as a centrifugal contribution due to the rotation of the
reference frame; this leads to a definition of an effective potential Φ = 2V −
(ωrsinθ)2. The parameterization of the Earth’s geoid is obtained by taking the
multipole expansion of V up to the desired order, calculating the equipotential
lines Φ = Φ0(the value of Φ at the Equator) and solving for r(θ,φ).
In the above, the coordinate time t is equal to the proper time of an observer
at infinity. Given the issue of ground to orbit clock synchronization, the metric
should be rewritten with a time coordinate coincident with the proper time
of clocks at rest on the Earth’s surface.
Since the already discussed geoid provides one with an equipotential surface
Φ = Φ0, all clocks at rest with respect to it beat at the same rate. Hence,
rescaling the time coordinate as t → (1 + Φ0/c2)t yields
ds2= −
?
1 +2(Φ − Φ0)
c2
?
(c dt)2+ 2ωr2sin2θdφdt +
?
1 −2V
c2
?
dΩ2. (3)
Going back to a non-rotating frame, one finally writes the metric as
ds2= −
?
1 +2(V − Φ0)
c2
?
(c dt)2+
?
1 −2V
c2
?
dΩ2. (4)
2.2Constant and periodic clock deviation
Keeping only terms of order c−2, the proper time increment on the moving
clock is approximately given by
dτ = ds/c =
?
1 +V − Φ0
c2
−v2
2c2
?
dt , (5)
so that, assuming an elliptic orbit with semi-major axis a and, for simplicity,
the Newtonian potential generated by a perfectly spherical body V = VN ≃
GME/r, one obtains [3]
dτ = ds/c =
?
1 +3GME
2ac2
+Φ0
c2−2GME
c2
?1
a−1
r
??
dt .(6)
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Page 5
Hence, the constant correction terms are given by
3GME
2ac2
+Φ0
c2= −4.7454 × 10−10, (7)
for Galileo, and −4.4647 × 10−10, for the GPS.
As the above shows, the orbiting clock beats faster by about 41 µs/day
(Galileo) and 39 µs/day (GPS). Other residual periodical corrections are pro-
portional to (1/r − 1/a) and have an amplitude of the order of 49 ns/day
(Galileo) and 46 ns/day (GPS).
2.3 Shapiro time delay and the Sagnac effect
The Shapiro time delay is a second order relativistic effect affecting the prop-
agation of light [3], given by
∆tdelay=Φ0l
c3+2GME
c3
ln
?
1 +
l
RE
?
≃ 6.67 × 10−11s ,(8)
a result obtained after integrating over a radial path of proper length l.
The Sagnac effect is yet another relativistic contribution, which reflects the
rotation of the Earth and the consequent difference between the gravitational
potential V of a non-rotational frame and the effective potential Φ of a rotating
one. From Eq. (3), one gets the additional time delay
∆tSagnac=ω
c2
?
path
r2dφ =2ω
c2
?
path
dAz ,(9)
where dAzis the ortho-equatorial projection of the area element swept by a
vector projecting from the rotation axis. Evaluating the above, one finds that
this amounts to 153 ns (for Galileo) and 133 ns (for the GPS).
As already shown above, a global positioning system is affected by a frequency
shift of order 10−10and a cumulative propagation time delay of the order
of 0.1 µs. The following sections aim at computing other corrections that
should be taken into consideration when computing the clock synchronization
of ground and onboard clocks, and compare the obtained results with the
frequency accuracy of 10−12and the time accuracy of Galileo, of order 10−12s.
5
Page 6
3 Post-Newtonian effects
Before venturing into more speculative and hypothetical effects, arising from
putative extensions to GR, it is natural to first tackle the possibility of mea-
suring Post-Newtonian effects with the Galileo positioning system; these are
naturally much smaller than the previously considered, of higher order in the
Newtonian potential, VN/c2≈ GME/(REc2) ∼ 10−10(again considering a
purely spherical body).
Post-Newtonian effects are suitably addressed by resorting to the so-called
parameterized Post-Newtonian (PPN) formalism, which allows one to de-
scribe higher-order effects induced by metric extensions and alternatives to
GR, which typify any particular model under scrutiny. For simplicity, one fo-
cuses only on the β and γ PPN parameters, thus writing the related PPN
metric [8–10] as
ds2= −
?
1 +2V
c2+ 2β
?V
c2
?2?
(c dt)2+
?
1 + 2γV
c2
?
dV . (10)
The β parameter measures the amount of non-linearity affecting the super-
position law for gravity, while γ is related to the spatial curvature per unit
mass.
Although not evoked here, the full PPN metric includes a total of ten PPN
parameters, which characterize the underlying fundamental theory and its pos-
sible consequences; these may include a violation of momentum conservation,
existence of a privileged reference frame, amongst others deviations from GR.
The PPN formalism is defined so that General Relativity is parameterized
by β = γ = 1, while all remaining parameters vanish; measurements of the
Nordtvedt effect yield |β−1| ≤ 2−3×10−4, while Cassini radiometry indicates
that γ − 1 = (2.1 ± 2.3) × 10−5[11,12].
Having stated the above, and assuming a non-relativistic motion v ≪ c, one
may compute the acceleration as
? a = Γr
00≈ −1
2grrg′
00≈ −GME
r2
?
1 + 2(γ − β)GME
c2r
?
,(11)
where the prime denotes differentiation with respect to the radial coordinate.
Similar expressions may be derived for the time delay and frequency shift,
showing that the post-Newtonian relative corrections are indeed proportional
to VN/c2= GME/(REc2) ∼ 10−10. Recall that the already considered grav-
itational time delay and frequency shift are of the order ∆t ∼ 10−7s and
6
Page 7
ǫf ∼ 10−10, respectively; furthermore, the experimental bounds discussed
above indicate that the difference (γ − β) should be of order ? 10−4. Hence,
comparison with the available precision of Galileo and GPS systems makes
it clear that Post-Newtonian effects signaling deviations from GR are much
below the observation threshold.
4 Detection of the cosmological constant
According to the latest observations, the Universe is currently ongoing a pe-
riod of accelerated expansion; although several proposals exist to account for
this acceleration, the simplest explanation resorts to a cosmological constant
Λ ∼ 10−35s−2, which acts as a fluid with negative pressure (see Ref. [13] and
references therein). The local effect of this component may be evaluated by
matching the outer Friedmann-Robertson-Walker metric with a static, sym-
metric solution given by Birkhoff’s theorem; this yields the Schwarzschild-de
Sitter metric, with a line element [14,15],
ds2= −
?
1 +2(VN+ VΛ)
c2
?
(c dt)2+
1
1 +2(VN+VΛ)
c2
dr2+ dΩ2. (12)
in anisotropic form; the presence of a cosmological constant produces an ad-
ditional potential term VΛ= −Λr2/6.
For simplicity, and since its effect is rather small, a coordinate change to an
isotropic, co-rotating frame of reference may be forfeited. Likewise, one may
safely disregard the redefinition of the time coordinate so to identify proper
time with measurements of clocks at rest on the surface of the Earth’s geoid;
as a side note, one remarks that the identification of the time coordinate t of
the current form as the proper time of an observer at rest at infinity breaks
down, due to the Schwarzschild “bubble” breaking down at a finite distance,
where it matches the exterior FRW metric [16]).
The frequency shift of a signal emitted at a distance from the origin r = RE+h
(for the Galileo system, h = 23.2 × 106m) and received at a distance r = RE
is given by
?fEarth
fSat
?
=
?g00 Earth
g00 Sat
=
?
?
?
?
1 − 2V (RE)/c2
1 − 2V (RE+ h)/c2→ (13)
ǫf≡
?fEarth
fSat
?
− 1 ≃V (RE+ h) − V (RE)
c2
.
7
Page 8
The additional frequency shift induced by this extra potential contribution is
easily obtained,
|ǫΛ| ≃|VΛ(RE+ h) − VΛ(RE)|
c2
=
Λ
6c2h(2RE+ h) ∼ 10−38, (14)
clearly much below the accuracy ǫf= 10−12of the Galileo constellation.
Similarly, the propagational time delay is given by
∆tdelay=1
c
RE+h
?
RE
V (r) dr ,(15)
so that the presence of a cosmological constant results in a further delay of
∆tΛ=1
c
RE+h
?
RE
Λr2
6c2dr =
Λ
18c3h
?
(3RE(RE+ h) + h2?
∼ 10−40s ,(16)
also many orders of magnitude below the time resolution of 10−12s. Therefore,
one concludes that the cosmological constant is completely undetectable by
the Galileo positioning system (as indicated by an analytical study in Ref.
[17]).
5Detection of anomalous, constant acceleration
Although not usually considered, the presence of an anomalous, constant,
acceleration affecting the free-fall of bodies could model effects arising from
some fundamental theory of gravity, perhaps hinting at the existence of a
fundamental threshold between known dynamics and yet undetected exotic
physics.
One widely discussed example is the so-called Modified Newtonian Dynamics
(MOND) model [18–20], which features a departure from the classical Poisson
equation at a characteristic acceleration scale of about 10−10m/s2, and aims
to explain the puzzle of the galaxy rotation curves without evoking any dark
matter component.
From the experimental viewpoint, a constant acceleration a = (8.74±1.33)×
10−10m/s2is reported to affect the Pioneer 10/11 probes [21–23]; its origin,
8
Page 9
either due to an incomplete engineering analysis (see e.g. Ref. [24–28]) or stem-
ming from yet undiscovered fundamental physics [29–31], has been dubbed the
Pioneer anomaly.
An anomalous, constant acceleration, a, would imply on an addition to the
gravitational potential of the form Va= ar; following the procedure depicted
in the previous section, one calculates the related frequency shift as
ǫa≃Va(RE) − Va(RE+ h)
c2
=ah
c2
, (17)
and, comparing with the frequency accuracy ǫf = 10−12, one finds that only
accelerations a ≥ c2ǫf/h ∼ 10−3m/s2may be detected by Galileo.
The propagational time delay due to this extra potential addition is given by
∆ta=1
c
RE+h
?
RE
ar
c2dr =
a
2c3h(2RE+ h) .(18)
By the same token, comparison with a time accuracy of 10−12s indicates that
only accelerations a ? 0.1 m/s2are measurable using time delay.
Since the lowest value for a constant acceleration that may be inferred using
the Galileo system is several orders above (10−10−10−9) m/s2, one concludes
that this range cannot be probed by the satellite constellation; indeed, in order
to detect a constant acceleration of the order of 10−10m/s2, an improvement
of 7 orders of magnitude in frequency accuracy (to ǫf∼ 10−19) and 9 orders
of magnitude in time resolution (to 10−21s) would be in order.
6 Detection of Yukawa potential
A Yukawa potential is one of the more ubiquitous modifications to the law of
gravity, as it may arise from scalar-tensor field “fifth force” models, where its
characteristic range λ is related to the mass m of the scalar or vector field,
λ ∝ m−1[7].
For the case of exchange of a scalar particle, the full potential is given by
V (r) = −G∞ME
r
?
1 + αe−r/λ?
,(19)
where α is the strength of the perturbation and G∞the gravitational coupling
9
Page 10
Fig. 1. Exclusion plot for a Yukawa-type additional force with strength α and range
λ, and superimposed limits obtained for varying frequency accuracy ǫf: 10−10(grey,
full), 10−12(black dash) and 10−19(black full).
for r → ∞; the latter redefines Newton’s constant G through the relation
G = G∞(1 + α); this full potential may be separated into a Newtonian-like
potential and an extra potential VY = −(αGME/(1 + α)r)e−r/λ.
By conjugating several constraints arising from different setups covering a
wide range of distances (from near-millimeter tests to planetary experiments),
stringent bounds have been obtained for the allowed region of parameter space
α, λ, as may be seen in Fig. 1. Thus, both the sub-millimeter λ < 10−3m [32],
as well as the long-range astronomical regimes, λ > 1015m ≈ 0.1 ly remain
unconstrained.
Following the previous steps, one first computes the additional frequency shift
|ǫY|=|VY(RE) − VY(RE+ h)|
c2
= (20)
GME
c2RE
?
α
1 + α
?
e−RE/λ
????e−h/λ
RE
RE+ h− 1
???? .
10
Page 11
The extra time delay is given by
∆tY =1
c
RE+h
?
RE
GME
c2r
?
α
1 + α
?
e−r/λdr .(21)
The above expressions may be considerably simplified if it is assumed that
this additional force is a long-range one, λ ≫ r, or a short-range interaction,
λ ≪ r.
6.1Short-range “fifth force”
If the additional Yukawa interaction is short-ranged, λ ≪ h, RE, one gets
|ǫY| ≃GME
c2RE
?
α
1 + α
?
e−RE/λ. (22)
If this effect is not detected within the frequency accuracy ǫf, one obtains the
constraint for small α
α ?c2RE
GMEeRE/λǫf≈ 1.4 × 10−3eRE/λ≫ 1 ,(23)
which does not provide any further information concerning the unexplored
sub-millimetric regime.
The additional propagation time delay is given by
∆tY = −GMEα
c3
ln
?
1 +
h
RE
?
, (24)
and comparing with the time accuracy of ∆t = 10−12gives, for α ≪ 1
α ≤
?GME
c3
ln
?
1 +
h
RE
??−1
∆t ≈ 0.05 .(25)
One concludes that the short-range regime of a hypothetical Yukawa interac-
tion yields the mild constrain α < 0.05: since there are no current limits on α
for the submillimeter regime (as can be seen in Fig. 1), this is an interesting
result.
11
Page 12
6.2 Long-range “fifth force”
One now examines the opposite assertion concerning the characteristic length-
scale λ, and instead assumes a long range fifth force, λ ≫ h,RE; the exponen-
tial terms may be expanded to first order in r/λ and the additional propagation
time delay reads
∆tY ≃ −GMEα
c3
h
λ
. (26)
If the effect is undetected at a level ∆t ∼ 10−12s, one obtains
α <c3∆t
GME
λ
h≈ 2.9 × 10−9
?
λ
1 m
?
. (27)
For a lower bound of λ ≈ 108m (only one order of magnitude above RE,h),
the result α ? 0.1 is found, which does not advance the already available
bounds (see Fig. 1).
The additional frequency shift is given by
ǫY ≃GMEαh
2c2λ2
, (28)
and comparison with the frequency accuracy level of ǫf∼ 10−12produces, for
α ≪ 1
α <
?
c2
GME
??2λ2
h
?
ǫf≈ 10−5ǫf
?
λ
1 m
?2
,(29)
or, equivalently, the expression
log α < −5 + log ǫf+ 2 log
?
λ
1 m
?
. (30)
The different constraints obtained by varying the frequency accuracy ǫf are
plotted in Fig. 1; as one can see, no new bounds are obtained at the current
level, but the region below the “trough” at λ ∼ 108m (corresponding to
α < 10−8) could be probed if precision is improved down to ǫf∼ 10−19.
12
Page 13
7Detection of a power-law addition to the Newtonian potential
One finally approaches the possibility of additions to the gravitational poten-
tial of the form
VP=GME
r
?R
r
?n
, (31)
where n is a (possibly non-integer) exponent and R is a characteristic length
scale arising from the underlying physical theory.
Phenomenologically, such a modification of the law of gravity is an interesting
alternative to the more usual Yukawa parameterization, and allows one to
investigate a wider range of extensions and modifications of GR.
Such an addition can also be theoretically motivated: it arises from power-law
induced effects at astrophysical scales due to the so-called Ungravity scenario,
which involves the exchange of spin-2 unparticles of a putative scale invariant
“hidden” sector within the Standard Model [33,34]. Bounds for these Ungrav-
ity corrections can be obtained from stellar stability considerations [35] and
cosmological nucleosynthesis [36]. If a power-law addition is related to Un-
gravity, the exponent n follows from the scaling dimension of the unparticle
operators dU, through n = 2dU−2; the lengthscale R reflects the energy scale
of the unparticle interactions, the mass of exchange particles and the type of
propagator involved.
Other possible power-law additions to the Newtonian potential may arise from
f(R) theories of gravity [37–39], which generalize the Einstein-Hilbert action
by considering a non-trivial scalar curvature term and/or a non-minimal cou-
pling of geometry with matter [40]): these extra contribution is obtained in
an astrophysical context, when addressing the puzzle of the galaxy rotation
curves [41,42].
From Eq. (31), one sees that the Newtonian potential ΦN is recovered by
setting R = 0 (for positive n) or R → ∞ (for negative n). The limit n → 0
is ill-defined, since the additional term VP does not vanish, but is instead
equal to the Newtonian potential, VP = ΦN: hence, one should rewrite the
gravitational constant in terms of an effective coupling, leading to the full
potential
Φ = −GPME
r
?
1 +
?R
r
?n?
, (32)
13
Page 14
with
GP=
G
?
1 +
R
R0
?n , (33)
where R0signals the distance at which the full gravitational potential matches
the Newtonian one, Φ(R0) = ΦN(R0).
This additional length scale R0should be an integration constant, obtained
after solving the full field equations that lie behind the considered power-law
correction. For simplicity, one assumes that (R/R0)n≪ 1, so that this term
may be safely discarded — at the cost of neglecting the regime n → 0. With
this in mind, the following paragraphs thus use GP = G freely; one remarks
that this approach is complementary to that considered in Ref. [35].
7.1 Relative frequency shift
As before, the relative frequency shift of an emitted signal is given by
ǫP=VP(RE) − VP(RE+ h)
c2
≃GME
c2RE
?R
RE
?n?
1 −
?
RE
RE+ h
?n+1?
(34)
=6.96 × 10−10ξn?
1 − 0.22n+1?
,
where the dimensionless ratio ξ ≡ R/REis defined. Thus, one must consider
the two asymptotic regimes n ≫ −1 and n ≪ −1; in the former case, one may
approximate Eq. (34) by
ǫP= 6.96 × 10−10ξn.(35)
Given the accuracy ǫf= 10−12of Galileo, one obtains ξn≤ 1.44 × 10−3.
If the limiting case n ≪ −1 is assumed instead, then one obtains
ǫP= −1.53 × 10−10(0.22ξ)n,(36)
which, comparing with the accuracy ǫf= 10−12, yields (0.22ξ)−n≥ 153. Since
the r.h.s. is larger than unity, the rather strong bound ξ > (0.22)−1≃ 4.64 is
obtained for n → −∞ .
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?100
?500
n
50100
?3
?2
?1
0
1
2
3
log Ξ
Fig. 2. Contour plot for the relative frequency deviation ǫ as a function of ξ = R/RE
and n, with contours for ǫ = 10−12(solid line), 10−24(long dash) and 10−36(short
dash), and allowed region grayed out.
In the vicinity of n = −1, one may expand Eq. (34) as
ǫP= 1.07 × 10−9
?n + 1
ξ
?
,(37)
which yields ξ ≥ 1.07 × 106|n + 1|. The different regimes are depicted in Fig.
2.
7.2 Propagation time delay
As before, one also computes the additional propagational time delay,
15
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- Available from Orfeu Bertolami · May 19, 2014
- Available from ArXiv