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/[email protected]
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
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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
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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