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Gravitation:Theories and Experiments
Part I: Clifford M. Will, WUGRAV, Washington U., St. Louis, USA
Phenomenological approach
Part II: Gilles Esposito-Farese, GRεCO / IAP, Paris, France, [email protected]
Field-theoretical approach
• A: Scalar-tensor gravity (October 10th & 11th)
• B: Binary-pulsar tests (October 11th)
• C: Modified Newtonian dynamics (October 12th)
Gravitation: Theories and Experiments
Part II: Field-theoretical approach(Gilles Esposito-Farese)
A. Scalar-tensor gravity------------------------1. General relativistic action2. Higher-order gravity3. Einstein and Jordan frames4. Scalar-tensor theories5. Nordström, Brans-Dicke and generalizations6. Weak-field predictions7. Strong-field predictions8. Gravitational waves
B. Binary-pulsar tests-----------------------1. Pulsars2. Post-Keplerian formalism3. PSRs B1913+16 and B1534+124. The dissymmetric PSR J1141-65455. The double pulsar J0737-30396. Constraints on scalar-tensor theories7. Comparison with LIGO/VIRGO and LISA8. Null tests of symmetry principles
C. Modified Newtonian dynamics--------------------------------1. Dark matter2. Milgrom's MOND phenomenology3. Various theoretical attempts4. Aquadratic (k-essence) models5. Light deflection6. Disformal and vector-tensor theories7. Experimental issues8. Pioneer anomaly
Smatter [ , gmn ]matterMATTER–GRAVITY COUPLING
Metric coupling chosen to satisfy the (weak) equivalence principle
acceleration
gravitation
Impossible to determinefrom a local experimentif there is acceleration
or gravitation(Einstein 1907)
€
Smatter [ , gmn ]matterMATTER–GRAVITY COUPLING
Metric coupling chosen to satisfy the (weak) equivalence principle
€
(special relativity)
freely fallingelevator
Earth
Smatter [ , gµν ]mattergµν =
11
11
λµν = 0
MATTER–GRAVITY COUPLING
Metric coupling:
Freely falling elevator (= Fermi coordinate system)
⇒1 Constancy of the constants 2 Local Lorentz invariance
Space & time independence of coupling constants Local non-gravitational experiments areand mass scales of the Standard Model Lorentz invariant
Oklo natural fission reactor Isotropy of space verified at the 10–27 level|α/α| < 7×10–17 yr–1 << 10–10 yr–1 (cosmo) [Prestage et al. 85, Lamoreaux et al. 86,[Shlyakhter 76, Damour & Dyson 96] Chupp et al. 89]
3 Universality of free fall 4 Universality of gravitational redshift
Non self-gravitating bodies fall with the same In a static Newtonian potentialacceleration in an external gravitational field g00 = –1 + 2 U(x)/c2 + O(1/c4) the time measured by two clocks isLaboratory: 4×10–13 level [Baessler et al. 99] τ1/τ2 = 1 + [U(x1)–U(x2)]/c2 + O(1/c4) Flying hydrogen maser clock: 2×10–4 level : 2×10–13 level [Williams et al. 04] [Vessot et al. 79–80, Pharao/Aces will give 5×10–6]
.
acceleration
gravitation
⇒ Whatever their composition,lower clocks are slower
Doppler effect(cf. fire-truck siren)
(⇒ impossible tosynchronize
even static clocks)
4 Universality of gravitational redshift (time dilation)
Conclusion of experimental tests in the Parametrized Post-Newtonian formalism
0 0.5 1 1.5 2
0.5
1
1.5
2
GeneralRelativity
Lunar Laser Ranging
Mercury perihelion shift
Mars radar ranging&
Very Long Baseline Interferometry&
Time delay for Cassini spacecraft
βPPN
γPPN
ξα1,2,3ζ1,2,3,4
0.996 0.998 1 1.002 1.004
0.996
0.998
1
1.002
1.004
generalrelativity
βPPN
γPPNLLR
VLBICassini
GENERAL RELATIVITYis essentially the onlytheory consistent with
weak-field experiments
This is because $ also a deformation of space:
Light deflection and the equivalence principle
acceleration
gravitation
fi Modificationof the stars’
apparent position
Sun
Earth
In 1911–14, Einstein predictshalf the correct value
Sun
EarthNordström’stheory 1913
SunEarth
Einstein’sgeneral relativity 1915
[Eddington 1919]
ln A(j)
j
b0
a0
curvature
slope
j0
ln A(j) = a0 (j–j0) + 1 b0 (j–j0)2 + …2
jmatter
jj
j
j
j
...
Geff = G ( 1 + a02 )
gPPN– 1 a02
bPPN– 1 a02 b0 a0 a0
b0
a0 a0
scalargraviton
-6 -4 -2 0 2 4 6b0
General Relativity
|a0|
0.025
0.030
0.035
0.010
0.015
0.020
0.005
LLR
perihelionshift
VLBI
LLR
jmatter
j
matterj
Cassini
S = 16 p G Ú -g {R - 2 ( mj)2 } + Smatter[matter , gmn A2(j) gmn]
1
Tensor-scalar theories
spin 2 spin 0 physical metric
* * *
Vertical axis (b0 = 0) : Jordan–Fierz–Brans–Dicke theory a0 = 2 wBD + 3 Horizontal axis (a0 = 0) : perturbatively equivalent to G.R.
2 1
Deviations from general relativity due to the scalar field
• At any order in 1 , the deviations involve at least two a0 factors:cn
scalar…
a0
a0
a0 a0
a0
a0
graviton
= small deviations!
• But nonperturbative strong-field effects may occur:
a0 + a1 Gm + a2
Gm 2 + …
Rc2 Rc2
[ ]deviations = a0 ¥
< 10-5
2
LARGE for Gm ª 0.2 ?Rc2
α0 = 0 ϕc
ϕc(at the center of the star)
Energy
“spontaneous scalarization”
ln A(ϕ)
ϕϕ0
β0 < 0 large slope ~ αA⇒ large deviations from General Relativity for neutron stars
small m
/R (Sun)
critical m/R
large m/R
(neutron star)
E ≈ ∫ [ (∇ϕ)2 + ρ eβ0ϕ2/ 2 ]12—
R ϕc21
2— + m eβ0ϕc
2/ 2
parabola Gaussianif β0< 0
ϕ0
matter-scalarcoupling functionNo deviation from
General Relativityin weak-field conditions
0.6
0.4
0.2
00.5 1 1.5 2 2.5 3
mA/m—
|αA|
criticalmass
maximummass
maximummass in GR
scalar charge
baryonic mass
ϕneutron star
[T. Damour & G.E-F 1993]