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Mass, Inertia, and Gravitation 521

In the solar system, tests of gravity are usually performed by comparing

observations with the predictions obtained from a family of parametrized post-

Newtonian (PPN) metrics which extend GR metric by introducing additional

parameters [25, 103]. In the approximation of a stationary point-like gravitational

source, ignoring effects due to size and rotation, PPN metrics may be written in

terms of a single potential  (which takes in isotropic coordinates the same form

as Newton potential ˚

in Schwarzschild coordinates (58)) and of two constant

parameters ˇ and  [104]







 g

C g

fdr

C r

.d

C sin

d'

/g;

D 1 C 2 C 2ˇ

C;D

;

D1 C 2 C: (62)

Eddington parameters  and ˇ; respectively, describe linear effects on light de-

ﬂection and nonlinear effects on perihelia of planets. They are deﬁned so that GR

corresponds to the particular values ˇ D  D 1.

It is easily seen that PPN metrics (62) correspond to particular cases of the gen-

eral metric extensions of GR that have been previously introduced

ı˚

D .ˇ  1/

C O.

/; ı˚

D.  1/ C O.

ıE

O.

/; ıE



2.  1/ C O.



ŒPPN; (63)

PPN metrics span a two-dimensional subspace, labeled by .ˇ; /, of the neighbor-

hood of GR corresponding to solutions of generalized gravitation equations (55),

which are labeled by two functions ı˚

and ı˚

. According to Eqs. 63,this

subspace corresponds to metrics with Einstein curvatures which vanish at large dis-

tances of the gravitational source. Alternatively,the metric extensions of GR (58, 61)

may be seen as generalizations of PPN metrics, where the two Eddington parameters

ˇ and  are replaced by two arbitrary functions ı˚

and ı˚

of the radial distance

to the gravitational source.

Assuming that the metric associated with the gravitational ﬁeld of the Sun takes

the form of a PPN metric, predictions can be made for motions in this gravitational

ﬁeld and compared with observations [105]. Observations in the solar system pro-

vide constraints on the form of the single potential  and on the values of Eddington

parameters ˇ and . Modiﬁcations of Newton potential  are usually parametrized

in terms of an additional Yukawa potential depending on two parameters, a range 

andanamplitude˛ measured with respect to 

ı.r/ D ˛e





.r/; (64)

Corrections behaving as (64) have been looked for at various values of  rang-

ing from the millimeter scale [1] to the size of planetary orbits [20]. For ranges of

522 M.-T. Jaekel and S. Reynaud

the order of the Earth–Moon [106] to Sun–Mars distances [42, 66, 85]testshave

been performed by following the motions of planets and artiﬁcial probes. Although

agreeing with GR, all these results still show [20,86] that windows remain open for

violations of the Newton force law at short ranges , below the millimeter, as well

as long ones, of the order of or larger than the size of the solar system. Similarly,

experiments performed up to now have conﬁrmed that, assuming that ˇ and  take

constant values, the latter should be close to 1. From Doppler ranging on Viking

probes in the vicinity of Mars [42] to deﬂection measurements using VLBI astrom-

etry [91] or radar ranging on the Cassini probe [9], the allowed values for   1

have reduced with time. Precessions of planet perihelion [94] and polarization by

the Sun of the Moon orbit around the Earth [78] constrain linear superpositions of

ˇ and . As a result, in order to remain compatible with gravity tests, PPN metrics

must satisfy rather stringent constraints

j  1j10

5

; jˇ  1j10

4

: (65)

Obviously, gravity tests performed in the solar system provide evidence for a

metric theory lying very close to GR. In case of a PPN metric (62), the latter should

correspond to values of ˇ and  close to 1. These constraints however result from an

analysis performed with the assumption of constant Eddington parameters, which

only covers a small subspace of potential deviations from GR (63). Dependences

of ˇ and  on the length scale are not excluded and deviations at very short or

very large scale are loosely constrained. Gravity tests still leave room for alternative

metric theories, such as metric extensions of GR (61), provided they satisfy these

criteria. They should correspond to small anomalous potentials ı˚

and ı˚

that to remain close to GR and thus compatible with present gravity tests.

As remarked in the introduction, deviations from GR may have already been

observed in the very domain where classical tests have been performed, namely

within the solar system. The Pioneer 10/11 probes, which were launched in the

1970s and tracked during their travel to the outer part of the solar system indeed

showed anomalous behaviors. The trajectories of the probes where determined from

the radio frequency signals sent to them, transponded on board and received by

stations on Earth. The Doppler shifts affecting the signals received on Earth, when

compared with the emitted ones, delivered the velocity and hence, by integration,

the distance separating the probe from the Earth. In fact, Doppler data provide the

trajectory once a modelization of all gravitation effects on light propagation and on

geodesic trajectories has been made. It appeared that the modelization based on GR

led to anomalies, taking the form of Doppler residuals, that is, differences between

observed and modelized velocities v

obs

 v

model

, which could not be reduced [5].

The latter furthermore took the form of a linear dependence in time t, that is, of

a roughly constant acceleration a

, over distances ranging from 20 to 70 A.U.,

directed toward the Sun

obs

 v

model

'a

.t  t

/; a

' 0:8 nm s

2

: (66)

Mass, Inertia, and Gravitation 523

No satisfactory explanation in terms of systematic effects taking their origin on the

probe itself or in its spatial environment has been found up to now [76]. Recently,

a larger set of data, covering parts of the Pioneer 10/11 missions which had not

been analyzed, has been recovered and put under scrutiny [82]. Several teams have

engaged in an independent reanalysis of the Pioneer data [71, 73, 79]. Proposals

have also been made for missions dedicated to a study of deep space gravity [24].

At present, the possibility that the Pioneer anomaly points at the necessity to change

the theoretical framework cannot be ignored. In that case, the existence of an ex-

tended framework having the ability to account for the Pioneer anomaly while

remaining consistent with all gravity tests constitutes a crucial element.

PPN metrics which are compatible with classical gravity tests practically reduce

to GR metric (see Eq. 65). Furthermore, the radial dependence of their curvatures

(63) does not allow PPN metrics to reproduce the properties of the Pioneer anomaly

(66). On another hand, metric extensions of GR (58, 61) provide a phenomeno-

logical framework which enlarges the PPN neighborhood of GR, with corrections

brought to curvatures which may remain signiﬁcant at large distances from the grav-

itational source.

When performed within the framework of a general metric extension of GR,

an explicit computation of the Doppler signal, in the physical conﬁguration corre-

sponding to the Earth and a remote probe, exhibits a discrepancy with the similar

signal computed using GR [60, 62]. Computation is more easily achieved by using

the time delay function, that is, a two-point function which describes the elapsed

time between emission and reception of a lightlike signal propagating between two

spatial positions. The time delay function is easily determined from the metric com-

ponents by using a reference frame where the extended metric takes the form of a

stationary isotropic metric [63]. The Doppler signal corresponding to a tracking of

the probes by stations on Earth is then obtained as the time derivative of the time

delay function evaluated on the trajectories of the Earth and probes. The time deriva-

tive of the Doppler signal itself may be written under the form of a time-dependent

acceleration, as the observed Pioneer anomaly (66)[5]. For an extended metric (61)

which is close to GR metric, dependences on metric components may be treated

perturbatively and only the ﬁrst order in metric perturbation may be kept. Metric

perturbations may be seen to modify the expression for the Doppler signal in two

ways, through a perturbation of the time delay two-point function itself, and through

a perturbation of the trajectories on which this function is evaluated. A difference

then emerges between the acceleration obtained for an extended metric and that

entailed by GR metric, which corresponds to the discrepancy which would appear

when comparing an observed signal with the similar signal predicted using GR. This

difference can be considered as representing the anomaly which would be observed.

When expressed in terms of the anomalous potentials deﬁning the extended met-

ric (61), the anomalous acceleration contains several contributions due to different

sources of perturbation, affecting in particular the propagation of lightlike signals

and the trajectories of massive bodies. We shall consider the simpliﬁed case of a re-

mote probe which is moving on an escape trajectory in the ecliptic plane and which

is reaching the outer part of the solar system. Several contributions to the anomalous

524 M.-T. Jaekel and S. Reynaud

acceleration may then be neglected and the latter takes a simpliﬁed expression [63]

ıa ' ıa

sec

C ıa

ann

;

ıa

sec

'

.ıg

/ CŒRr



ı.g

 ıg





Œg



ır;

ıa

ann



ı



: (67)

.r; / represents the position of the probe with respect to the Sun, in the ecliptic

plane (r is the radial distance and  the difference of angular positions between

the probe and the Earth) and  denotes the impact parameter (with respect to the

Sun) of the lightlike signal propagating between the probe and the Earth. The accel-

eration anomaly divides into two parts, a secular anomaly ıa

sec

which varies over

large times only, typically for variations of the distance between the probe and the

Earth of the order of several A.U., and a modulated anomaly ıa

ann

which describes

variations with annual or semiannual periodicity. One notes that, due to several sim-

pliﬁcations in the previous representation (neglecting in particular the motions of

the stations, effects of the Earth atmosphere, deviations from the ecliptic plane, etc.)

additional modulations, in particular with daily periodicity, have been ignored. Both

parts of the anomalous acceleration are generated by anomalies in the two gravita-

tional potentials (ıg

;ıg

) and in the probe trajectory (ır). In particular, the latter

cannot be ignored in the case of Pioneer 10/11 probes as no range capabilities (al-

lowing one to directly determine the radial position of the probe) were available for

them. Anomalous metrics (67) can be seen to lead to anomalous accelerations which

exhibit the same qualitative features as the Pioneer anomaly [5, 71, 73, 79].

To obtain from the observed Pioneer anomaly quantitative constraints on the two

gravitational potentials, one needs to enter a detailed analysis of navigation data,

performing this analysis in the enlarged phenomenological framework provided by

metric extensions of GR. As already shown in the above-simpliﬁed model (67), the

acceleration anomaly exhibits a secular part and modulations which both depend on

anomalies in the gravitational potentials and in the trajectory. Correlated anomalies

should then play an essential role when confronting models pertaining to the ex-

tended phenomenological framework with a detailed analysis of Pioneer data [63].

Although a precise confrontation is not yet available, some consequences can

nonetheless be drawn from the general form of the secular anomaly observed on the

Pioneer probes. As shown by Eqs. 67 (anomaly modulations are used to eliminate

the trajectory anomaly [63]), a constant anomalous acceleration ıa Da



over the distances covered by the Pioneer probes may follow from different forms of

the anomalous gravitational potentials (61). On one hand, if only a perturbation of

Newton potential is assumed, the latter should behave linearly with the heliocentric

distance, that is, ı˚

' r=l

, to produce a constant acceleration. This form how-

ever conﬂicts with tests which have been performed between the Earth and Mars

[5, 61, 85], so that an anomaly limited to the Newtonian sector is only allowed at

large heliocentric distances [75], where it must still remain compatible with the

Mass, Inertia, and Gravitation 525

ephemeris of outer planets. On the other hand, a constant acceleration is also ob-

tained for an anomaly in the second sector which behaves quadratically with the

heliocentric distance, that is, ı˚

. A combination of anomalies in the

two sectors may also lead to the form taken by Pioneer data [63]. One notes that

these properties cannot be obtained from PPN metrics (63) and that the form just

given for the anomalous potential in the second sector ı˚

corresponds to a non

vanishing constant curvature in the outer part of the solar system.

A remarkable feature of the phenomenological framework derived from metric

extensions of GR (58, 61) is that, besides producing Pioneer-like anomalies, it also

allows one to preserve the agreement with classical gravity tests. An important part

of gravity tests is provided by ephemerides of inner planets, and in particular by per-

ihelion precession anomalies [42]. Anomalies in perihelion precessions of planets

(with respect to Newtonian gravitation) are well accounted for by the nonlinear de-

pendence of GR metric on Newton potential (62). Metric extensions (58, 61) induce

modiﬁcations which generalize the corrections obtained in the PPN framework (63).

The anomalous gravitational potentials ı˚

and ı˚

may be seen as promoting the

parameters ˇ and  to functions which depend on the heliocentric distance [62]. Ob-

served anomalies in perihelion precessions constrain a particular combination of the

two gravitational potentials for ranges around a few A.U. where planet ephemerides

are known with precision. Provided the latter combination remains small within

these ranges, compatibility of extended metrics with gravity tests in ensured.

Precise gravity tests are obtained by measurements of the deﬂection induced on

a lightlike signal by the Sun gravitational ﬁeld, as for instance those performed

with Cassini probe [9]. Light deﬂection measurements can be seen to explore the

behavior of gravitational potentials in the Sun vicinity, that is, at small heliocen-

tric distances. The deﬂection angle is determined by the impact parameter of the

lightlike signal and, within a PPN framework, is directly related to the value of the

parameter . In the framework of metric extensions (58, 61), the effect of anomalous

gravitational potentials on the deﬂection angle may be interpreted as due to a gen-

eralized parameter  , with a functional dependence on the impact parameter [61].

Deviations of this generalized parameter from its constant value corresponding to

GR are given by a combination of the two anomalous gravitational potentials (61),

so that light deﬂection measurements provide constraints on the behavior of gravita-

tional potentials in the range of a few solar radii. A precise analysis shows that these

constraints still allow different behaviors of the two gravitational potentials with re-

spect to the radial distance [60]. When compared with the value predicted by GR

or even a PPN metric, the deﬂection angle might then show an anomalous behavior

which becomes observable for large values of the impact parameter. Although light

deﬂection is more important, within the PPN framework, for small impact param-

eters and is thus usually measured with lightlike signals grazing the Sun surface, a

global mapping of light deﬂection over the sky, including large impact parameters,

is programmed in the future mission GAIA [37]. A detected anomalous behavior of

deﬂection angles could point at the necessity to use an enlarged phenomenological

framework like that provided by metric extensions of GR.

526 M.-T. Jaekel and S. Reynaud

5Conclusion

Despite difﬁculties that emerge at their interface, quantum theory and relativity the-

ory cannot be considered as independent frameworks, with quantum applications

excluding motion and gravitation and relativity being limited to a representation

of classical systems. Quantum ﬁelds possess ﬂuctuations which result in mechan-

ical effects and affect inertial responses. This entails a revision of our notions of

mass and motion which becomes necessary when dealing with microscopic systems.

Quantum ﬁeld ﬂuctuations also modify gravitation with the possibility of leading to

observable effects at macroscopic scales. The effects of quantum ﬂuctuations on rel-

ativistic systems cannot be ignored and it appears necessary to have a representation

which is compatible with both frameworks.

We have shown that quantum ﬁeld ﬂuctuations, even in vacuum, must be taken

into account when analyzing mechanical effects such as inertia. As exempliﬁed on

Casimir energy and contrarily to a common opinion, vacuum ﬁeld ﬂuctuations do

contribute to inertia and the induced inertial mass complies both with quantum and

with relativistic requirements. These properties are more easily put into evidence

using the linear response formalism, which thus allows one to connect in a con-

sistent way the quantum and relativistic frameworks. Inﬁnities that are responsible

for difﬁculties when interfacing quantum and relativity theories can be treated by

renormalization techniques. The linear response formalism not only recovers the

fundamental connection between interacting and free quantum ﬁelds lying at the

basis of QFT, but also allows one to implement in a general way the fundamen-

tal relation made by relativity between motions and the symmetries of space-time.

A treatment of motion in quantum vacuum may thus be given which remains con-

sistent with the relativistic conception of space-time and which brings new light on

the role played by quantum vacuum with respect to inertia [45,59].

The effects of quantum ﬂuctuations on inertia have important consequences for

the notion of mass. There results that the mass parameter used for characteriz-

ing motion is a quasistatic approximation. Mass possesses quantum ﬂuctuations

which cannot be neglected in high frequency regimes, or equivalently at very

short timescales, and these ﬂuctuations should be described by the correlations of

a quantum operator representing the mass observable [53]. This modiﬁcation of

the representation of mass is accompanied by similar extensions for positions and

motions in space-time in order to comply with the requirements imposed on observ-

ables by quantum and relativity theories. Remarkably, these extensions are ensured

in a consistent way by the existence of a single algebra where quantum observables

and relativistic transformations are simultaneously represented. This unique alge-

bra, built on space-time symmetries, provides relations between mass and motions,

including uniformly accelerated motions, which hold in the quantum framework.

This allows in particular to write a quantum version of Einstein effect and shows

that the equivalence principle can be extended to the quantum framework [58].

According to the principles of relativity, quantum ﬁeld ﬂuctuations affect not

only inertial but also gravitational masses, with the consequence that classical grav-

itation must be modiﬁed to account for radiative corrections. Presently, theoretical

Mass, Inertia, and Gravitation 527

analysis and experimental observations both support the equivalence principle as

a faithful and well-tested basis for a gravitation theory. These arguments strongly

favor metric theories but do not constrain the gravitational coupling. If classical

tests performed up to now in the solar system tend to conﬁrm Einstein–Hilbert

Lagrangian, the corresponding coupling looses its simple form when corrections

induced by quantum ﬂuctuations are taken into account. Although applying renor-

malization techniques to gravitation theory appears as a formidable task, theoretical

arguments and observations both point at the possibility of corrections to gravitation

occurring at the classical level, that is, at large length scales. Radiative corrections

induce modiﬁcations which amount to replace Newton gravitation constant by two

gravitation running coupling constants corresponding to two sectors with different

conformal weights. These may equivalently be represented by metric extensions of

GR which can be characterized either by two nonvanishing Ricci curvatures or by

two gravitational potentials. Besides modiﬁcations of Newton potential, a gravita-

tional potential in a second sector opens new phenomenological possibilities. Metric

extensions of GR appear as an efﬁcient way to parametrize gravitation theories lying

in the neighborhood of GR and offer larger possibilities than the usual PPN phe-

nomenological framework. Having also the ability to remain compatible with clas-

sical gravity tests, they may account for Pioneer-like anomalies and may further lead

to other correlated anomalies which could be observed in future experiments [64].

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