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, gef@iap.fr

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]