Metric extensions of General Relativityand gravitation in the solar system
Marc-Thierry Jaekel
Laboratoire de Physique TheoriqueEcole Normale Superieure, CNRS, UPMC
24, rue Lhomond, F-75231 Paris Cedex 05, France
GPhysLes Houches, October 20th, 2009
[email protected]@spectro.jussieu.fr
Outline
• Radiative corrections and metric extensions of General Relativity• Tests of GR and anomalies.• Phenomenology and gravitation in the solar system.• Conclusion
Gravitation and GRGravitation is geometry:GR identifies gravitation with the metric field in a Riemannian space-time
ds ≡ gµνdxµdxν
• ideal clocks measure the proper time along their trajectories: τ ≡Rds
• freely falling probes (masses and light) follow geodesics: δ(Rds) =
Coupling to the metric field leads to the universality of free fall.The equivalence principle is the best tested property of nature.
Gravitation is one of the four fundamental interactions:sources couple to curvature through their energy-momentum tensor
• one curvature tensor has a null divergence (Bianchi identities)Eµν ≡ Rµν −
gµνR, ∇νEµν =
like the energy-momentum tensor (conservation laws)∇νTµν =
• in GR, the two tensors are simply proportional to each other
Eµν =πGNc
Tµν Einstein− Hilbert equationsNewton gravitation constant GN is the less well known fundamental constant.
GR as a Quantum Field TheoryThe four interactions (Electro-weak, QCD, GR) call for a unified treatment
Eµν = Rµν − gµνR
Fµν = ∇µAν −∇νAµ
Eµν =πGNc
Tµν ∇νFµν = gJµ
GN must be a scale dependent coupling constant.Quantum fluctuations of gravitation are perturbations of metric fields
gµν = ηµν + hµν , ηµν = diag(,−,−,−) , |hµν | <<
hµν(x) ≡Z
dk
(π)e−ikxhµν [k]
The geometrical invariants become gauge invariant fields:(Riemann, scalar, Einstein) curvatures
Rλµνρ =
{kλkνhµρ − kλkρhµν − kµkνhλρ + kµkρhλν}
Rµν = Rλµλν , R = Rµµ , Eµν = Rµν − ηµνR
Einstein-Hilbert equations:metric fluctuations and energy-momentum tensor fluctuations are coupled.
Metric extensions of GRQuantum fluctuations of metric fields and stress tensors modify the gravitonpropagator i.e. the effective coupling between metric fields and sources
GN GN
GN
GN
Tµν Tλρ
hµν hλρ
Radiative corrections introduce a coupling to squares of curvatures:
• GR is embedded in renormalizable theories.• GN becomes scale dependent (a running coupling constant).• Renormalization introduces additional gravitational coupling constants.
GR is extended to a theory which preserves its geometric basis:
• gravitation is still described by a metric theory.• it may remain close to GR within a large range of scales.• the corrections to GR differ in two sectors.
Gravitation couplingsThe graviton couples differently to massive and massless fields (trace andtraceless energy-momentum tensors):the two sectors of Weyl (traceless) and scalar (trace) curvatures correspondto two different running coupling constants.
In the linearized approximation, the two sectors can be separated withprojectors. For a stationary pointlike source:
Tµν = δµδνT, T = Mcδ(k)
Eµν = E()µν + E()
µν , πµν ≡ ηµν −kµkνk
E()µν = {πµπν −
πµνπ
} πG
()
cT, E()
µν =πµνπ
πG()
cT
G() = GN + δG(), G() = GN + δG()
The two couplings are equivalent to two gravitation potentials
g = + (ΦN + δΦN ), gij = −(− (ΦN + δΦN − δΦP ))δij
In general, the gravitation equations may be written as response equations
Eµν [k] = χλρµν [k] Tλρ[k] = {πGNc
δλµδρν + δχλρµν [k]}Tλρ[k]
M.-T. Jaekel, S. Reynaud, Ann. Physik (1995) 68
Anomalous curvaturesThe gravitation equations have solutions which remain in the vicinity of GRmetric: (for a static point-like source, in Schwartzschild coordinates)ˆ
Eνµ˜
st=πGNM
cδµδ
νδ
()(x), [g]st = − GNM
cr= −
[grr]st
General metric solutions are characterized by perturbations of Einsteincurvature (which does not vanish in empty space)
Eµν ≡ [Eµν ]st + δEµν , δEµν (x) ≡Zdx′ δχµρνλ(x, x′)Tλρ (x′)
The two gravitation running coupling constants are equivalent to twoindependent components of Einstein curvature.The two anomalous curvatures are seen as anomalous parts in the twometric components describing isotropic solutions
g = [g]st + δg, grr = [grr]st + δgrr
δg[g]st
=
Zdu
[g]st
Z u δEu
du′ +
ZδErru
du
[g]st
δgrr[grr]st
= − u
[g]st
ZδEu
du, u ≡
r
Gravitation potentials
GR metric is determined by its vanishing Einstein curvature and is describedby a single Newtonian gravitation potential ΦN .
The two independent anomalous curvatures are equivalent to two gravitationpotentials ΦN + δΦN and δΦP which replace ΦN
δE ≡ u(δΦN − δΦP )′′, δErr ≡ uδΦ′P ()′ ≡ ∂u
The two potentials describe the anomalous parts of the metric components inthe isotropic case
δgrr =u
(− κu)(δΦN − δΦP )′, κ ≡ GNM
c
δg = δΦN + κ(− κu)
Zu(δΦN − δΦP )′ − δΦN
(− κu)du
The two sectors and the non linearity of gravitation are taken into account.The two potentials provide a gauge-independent parametrization of metrictheories in the vicinity of GR.
M.-T. Jaekel, S. Reynaud, Class. Quantum Grav. 23 (2006) 777
Gravitation in the solar system
Tests in the solar system are presently performed by comparing observationswith the predictions obtained from a family of parametrized metrics.In the approximation of a pointlike gravitational source (and ignoring effectsdue to its rotation) PPN metrics may be written (in isotropic coordinates)
g = + φ+ βφ + . . . , φ = −GNMcr
grr = −+ γφ+ . . .
Eddington parameters γ and β describe effects on light deflection and on thetrajectories of massive bodies.PPN metrics are particular cases of metric extensions of GR
δΦN = (β − )φ +O(φ), δΦP = −(γ − )φ+O(φ)
δE =
rO(φ), δErr =
r((γ − )φ+O(φ)) [PPN]
Tests of the metric nature of gravitation
• Eotvos type experiments• Tests of universality of free fall• Earth-Moon distance measurements
Relative accelerationbetween test bodiesof different compositions
η ≡ a − aa + a
The equivalence principleis presently tested at −
to be improved to −
(Microscope).C.F. Will Living Reviews in Relativity, 9 (2006) 3
Tests of Newton potential
The Newtonian dependence of the potential in the first sectoris well tested within a large range of scales.
log 1
0α
log10λ (m)
Geophysical
Laboratory
Satellites
PlanetaryLLR
Search for aYukawa correctionδΦN (r) = αe−
rλ φ(r)
Experiments in thesubmillimeter range:fifth force tests.
At long ranges:best fits performedwith observations onartificial probesand planets (ephemerids).J. Coy, E. Fischbach, R. Hellings, C. Talmadge, E.M. Standish (2003)
Significant deviations remain possible at very short and very long ranges
Tests of PPN parameters β, γTests in the solar system compare observations with PPN predictions
δΦN (r) = (β − )φ(r), δΦP (r) = −(γ − )φ(r)
• Ranging on planets• Astrometry and VLBI• Lunar laser ranging• Doppler velocimetry on probes• Light deflection
Tests of β, γ are consistent with GR and bound allowed deviations
|γ − | < × −, |β − | < × −
Measurement of the two-wayrelativistic frequency shiftdue to the Sun gravitation(Cassini)B. Bertotti, L. Iess and P. Tortora,Nature 425 (2003) 374
An extra acceleration was detected in the fit of Cassini data (3 nm/s2)and interpreted by the Cassini Team as RTG radiation acceleration.
Anomalies in the solar system
Extension of Pioneer 10/11missions by NASA, after theirplanetary objectives were met:the best long-range testof gravity to date.Pioneer 10/11 probes have shownanomalies after their last flyby.
A radio signal is sent from the Earth,transponded back by the probe,and received by a station on Earth.J. Anderson et al.,Phys. Rev. D 65 (2002) 082004
Doppler residuals show a nearly linear dependence in time
vobs − vmodel ' −aP (t− tin), aP ' . nm s−2
No conventional explanation has been totally successful up to now.
Post-Einsteinian phenomenology
Metric extensions of GR provide the appropriate frameworkfor analysing gravitation tests performed in the solar system.Light-like propagation is characterized by the time delay function
cT (r1, r2, φ) ≡Z r2
r1
− grrg00
(r)drq− grr
g00(r)− ρ2
r2
, φ =
Z r2
r1
ρdr/r 2q− grr
g00(r)− ρ2
r2
The time delay function is determined by the potentials in the two sectors.The second time derivative (or time derivative of the Doppler signal)gives a difference with GR which is interpreted as an anomalous acceleration
δa ' δasec + δaann
δasec ' −c2
2∂r (δg00) + [r2]st
δ(g00grr )
2− δg00
ff− c2
2∂2
r [g00]st δr2
δaann 'ddt
nhφi
stδρo
The Pioneer-like anomaly has a secular part δasec and a modulated part δaann.The secular and modulated anomalies are correlated.
M.-T. Jaekel, S. Reynaud, Class. Quantum Grav. 23 (2006) 7561
Planetary ephemerides
Metric extensions of GR give a time delay function and equations forgeodesics which are parametrized by two potentials ΦN and ΦP .Models for the two potentials may be designed to provide functions to beused in place of the usual expressions obtained from GR or PPN metrics.
Expressions for the perihelion precessions of planets depend on the twopotentials, and generalize those obtained from PPN metrics
δ∆$
2π' u (uδΦP)′′ − c2u
2GNMδΦ′′N , (u =
1r
)
+e2u2
8
„“u2δΦ′′P + uδΦ′P
”′′− c2u
2GNMδΦ′′′′N
«+ . . .
M.-T. Jaekel, S. Reynaud, Class. Quantum Grav. 23 (2006) 777
The same parametrized functions must be used• when analysing ranging and Doppler data obtained from probe tracking• when realizing the fit of the parameters determining planet ephemerides.
Planet ephemerides provide sensitive probes of gravitation at the A.U scale.
Light deflection
Light deflection depends on a (conformal) combination of the two potentials
2δΦN(r)− δΦP(r) ≡ −G0Mc2r
+Mc2 rζ0(r)
In the usual PPN framework, the potential in the second sector results in anEddington parameter γ which depends on the impact parameter ρ
δγ(ρ) =2(G0 −GN)
GN− ζ0(ρ)ρ2
GN
With respect to GR, the deflection angle shows an anomalous part
δ∆θ ' −GNMc2
∂
∂ρ
„δγ(ρ)ln
4r1r2
ρ2
«GR deflection angles increase with smaller impact parametersbut anomalies may increase with larger impact parameters.
Precise light deflection tests at large angles (GAIA) provide sensitive probesof gravitation at the solar radius scale.
M.-T. Jaekel, S. Reynaud, Class. Quantum Grav. 22 (2005) 2135
Conclusion
• For gravitation to be like the other fundamental interations, GN mustdepend on scale and GR undergo scale dependent modifications.Indeed, observations point at potential modifications of GR at largelength scales which may already occur at the solar system scale.
• Radiative corrections preserve the geometric nature of gravitation: thisproperty is also verified at an extremely good level.From symmetry arguments, two different gravitational running couplingconstants must emerge in the sectors of Weyl and scalar curvatures.Two functions, two components of Einstein curvature or two potentials,parametrize the metric theories to be confronted to tests.
• Such metric extensions of GR, which generalize the usual PPN metrics,can pass the existing tests while exhibiting Pioneer-like anomalies.They also predict further correlated anomalies, like modulated Doppleranomalies, anomalous light deflection, ....These anomalies could be exhibited by further analyses of available dataor by experiments in future space missions.
