Gravitation: Theories & ExperimentsGravitation: Theories & Experiments

Clifford WillJames S. McDonnell Professor of PhysicsMcDonnell Center for the Space SciencesDepartment of PhysicsWashington University, St. Louis USA

http://wugrav.wustl.edu/people/CMWcmw@wuphys.wustl.edu

Clifford M. Will and Gilles Esposito-Farèse

Part 1

Outline of the LecturesOutline of the Lectures

Lecture 1: The Einstein Equivalence PrincipleLecture 2: Post-Newtonian Limit of GRLecture 3: The Parametrized Post-Newtonian FrameworkLecture 4: Tests of the PPN Parameters

Outline of the LecturesOutline of the Lectures

Lecture 1: The Einstein Equivalence Principle Review of dynamics in special relativity The weak equivalence principle The Einstein equivalence principle Tests of EEP

o Tests of WEPo Tests of local Lorentz invarianceo Tests of local position invariance

Metric theories of gravity Non metric theories of gravity Physics in curved spacetime

Lecture 2: Post-Newtonian Limit of GRLecture 3: The Parametrized Post-Newtonian FrameworkLecture 4: Tests of the PPN Parameters

Special Relativistic Electrodynamics

€

I = − m0ac −η μν uμ uν dτ +∫a

∑ ea

ca

∑ Aμ dx μ∫

−1

16π−ηη μαη νβ Fμν Fαβ d4 x∫

Fμν = Aν ,μ − Aμ ,ν

400 CE Ioannes Philiponus: “…let fall from the same heighttwo weights of which one is many times as heavy as theother …. the difference in time is a very small one”

1553 Giambattista Benedettiproposed equality

1586 Simon Stevinexperiments

1589-92 Galileo GalileiLeaning Tower of Pisa?

1670-87 Newtonpendulum experiments

1889, 1908 Baron R. von Eötvöstorsion balance experiments (10-9)

1990s UW (Eöt-Wash) 10-13

The Weak Equivalence Principle (WEP)The Weak Equivalence Principle (WEP)

Bodies fall in a gravitational field with an accelerationthat is independent of mass, composition or internal structure

QuickTime™ and aPhoto - JPEG decompressor

are needed to see this picture.

The Einstein Equivalence Principle (EEP)The Einstein Equivalence Principle (EEP)

Test bodies fall with the same accelerationWeak Equivalence Principle (WEP)

In a local freely falling frame, physics (non-gravitational) is independent of frame’s velocity

Local Lorentz Invariance (LLI)In a local freely falling frame, physics (non-gravitational) is independent of frame’s location

Local Position Invariance (LPI)

Tests of the Weak Equivalence PrincipleTests of the Weak Equivalence Principle

APOLLO (LLR) 10-13

Microscope 10-15(2008)

STEP 10-18 (?)

€

I = − m0a 1− va2 dt +∫

a

∑ ea

a

∑ (−Φ + A ⋅va )dt∫

−1

8π(E 2 − c 2B2)d3x∫ dt

€

E 2 − c 2B2 → E 2 − c 2B2

+(1− c 2)γ 2{2v ⋅(E × B) + v 2(ET2 + BT

2 )}

Lorentz non-invariant EM actionLorentz non-invariant EM action

Under a Lorentz transformation, eg

€

′ t = γ(t − vx)

x = γ(x − vt)γ =1/ 1− v 2

Tests of Local Lorentz InvarianceTests of Local Lorentz Invariance

€

v = gt

= gh

Light falling down a tower

Tests of Local Position InvarianceTests of Local Position Invariance

ACES(2010) 10-6

Constant

Limit (yr-1) Z Method

<30 X 10-16 0 Clock comparisons

<0.5 X 10-16 0.15 Oklo reactor

<3.4 X 10-16 0.45 187Re decay

(6.4±1.4) X 10-

16

3.7 Quasar spectra

<1.2 X 10-16 2.3 Quasar spectra

W

<1 X 10-11 0.15 Oklo reactor

<5 X 10-12 109 BBN

me/mp <3 X 10-15 2-3 Quasar spectra

Tests of Local Position InvarianceTests of Local Position Invariance

Metric Theories of GravityMetric Theories of Gravity

Spacetime is endowed with a metric g

The world lines of test bodies are geodesics of that metric

In a local freely falling frame (local Lorentz, or inertial frame), the non-gravitational laws of physics are those from special relativity

“universal coupling principle”

Metric theories, nonmetric theories and electrodynamics

€

I = − m0ac −η μν uμ uν dτ +∫a

∑ ea

ca

∑ Aμ dx μ∫

−1

16π−ηη μαη νβ Fμν Fαβ d4 x∫

Metric theories, nonmetric theories and electrodynamics

€

I = − m0ac −gμν uμ uν dτ +∫a

∑ ea

ca

∑ Aμ dx μ∫

−1

16π−ggμα gνβ Fμν Fαβ d4 x∫

Metric theories, nonmetric theories and electrodynamics

€

I = − m0ac −gμν uμ uν dτ +∫a

∑ ea

ca

∑ Aμ dx μ∫

−1

16π−hhμα hνβ Fμν Fαβ d4 x∫

€

I = − m0a T − Hva2 dt +∫

a

∑ ea

a

∑ (−Φ + A ⋅va )dt∫

−1

8π(εE 2 − μ−1B2)d3x∫ dt

The ThThe Th Framework Framework

T, H, , are functions of an external static spherical potential U(r)

Metric theory action iff

€

= =(H /T)1/ 2

€

g00 = −T(U)

gij = H(U)δ ij

with

Metric theories, nonmetric theories and electrodynamics

€

I = − m0ac −gμν uμ uν dτ +∫a

∑ ea

ca

∑ Aμ dx μ∫

−1

16π−ggμα gνβ Fμν Fαβ d4 x∫

THTH Framework: Violation of WEP Framework: Violation of WEP

THTH Framework: Violation of LLI Framework: Violation of LLI

€

I = − m0a 1− va2 dt +∫

a

∑ ea

a

∑ (−Φ + A ⋅va )dt∫

−1

8π(E 2 − c 2B2)d3x∫ dt

€

BL ≠ 0, c =1

€

c ≠1, BL⊥V

€

c ≠1, BL ||V

Standard Model Extension (SME)Standard Model Extension (SME)

If the universe is fundamentally isotropic

•Clock comparisons•Clocks vs cavities•Time of flight of

high energy photons•Birefringence in

vacuum•Neutrino

oscillations•Threshold effects in

particle physics

€

L = η + (kφ )μν[ ](Dμφ)† Dν φ − m2φ†φ

−1

4η μαη νβ + (kF )μναβ

[ ]Fμν Fαβ

Dμφ = ∂μφ + ieAμφ

Kostelecky et al

D. Mattingly, Living Reviews in Relativity 8, 2005-5

Electrodynamics in curved spacetime

€

I = − m0ac −gμν uμ uν dτ +∫a

∑ ea

ca

∑ Aμ dx μ∫

−1

16π−ggμα gνβ Fμν Fαβ d4 x∫

Outline of the LecturesOutline of the Lectures

Lecture 1: The Einstein Equivalence PrincipleLecture 2: Post-Newtonian Limit of GRLecture 3: The Parametrized Post-Newtonian FrameworkLecture 4: Tests of the PPN Parameters