Gravitation: Theories & Experiments
Clifford M. Will and Gilles Esposito-Farèse
Part 1
Clifford Will
James S. McDonnell Professor of Physics
McDonnell Center for the Space Sciences
Department of Physics
Washington University, St. Louis USA
http://wugrav.wustl.edu/people/CMW
cmw@wuphys.wustl.edu
Outline of the Lectures
Lecture 1: The Einstein Equivalence Principle
Lecture 2: Post-Newtonian Limit of GR
Lecture 3: The Parametrized Post-Newtonian Framework
Lecture 4: Tests of the PPN Parameters
Outline 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 WEP
o Tests of local Lorentz invariance
o Tests of local position invariance
 Metric theories of gravity
 Non metric theories of gravity
 Physics in curved spacetime
Lecture 2: Post-Newtonian Limit of GR
Lecture 3: The Parametrized Post-Newtonian Framework
Lecture 4: Tests of the PPN Parameters
Special Relativistic Electrodynamics
I   m0a c 
a
1

16

F  A ,  A ,
ea
 u u d  
c
a
 


  F F d x
4

A dx 
The Weak Equivalence Principle (WEP)
400 CE Ioannes Philiponus: “…let fall from the same height
1553
1586
two weights of which one is many times as heavy as the
other …. the difference in time is a very small one”
Giambattista Benedetti
proposed equality
Simon Stevin
experiments
1589-92 Galileo Galilei
Leaning Tower of Pisa?
1670-87 Newton
pendulum experiments
1889, 1908 Baron R. von Eötvös
1990s
torsion balance experiments (10-9)
UW (Eöt-Wash)
10-13
Bodies fall in a gravitational field with an acceleration
that is independent of mass, composition or internal structure
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this pi cture.
The Einstein Equivalence Principle (EEP)
 Test bodies fall with the same acceleration
Weak Equivalence Principle (WEP)
 In a local freely falling frame, physics (nongravitational) is independent of frame’s velocity
Local Lorentz Invariance (LLI)
In a local freely falling frame, physics (nongravitational) is independent of frame’s location
Local Position Invariance (LPI)
Tests of the Weak Equivalence Principle
APOLLO (LLR) 10-13
Microscope 10-15(2008)
STEP 10-18 (?)

Lorentz non-invariant EM action
I   m0a

1 v a dt  ea  (  A  v a )dt
2
a
1

8
a
2
2 2
3
(E

c
B
)d
x dt

Under a Lorentz transformation, eg

t   (t  vx)
  1/ 1 v 2
x   (x  vt)
E 2  c 2B2  E 2  c 2B2
2
(1 c ) {2v  (E  B)  v 2 (ET2  BT2 )}
2
Tests of Local Lorentz Invariance
Light falling down a tower
v  gt
 gh

Tests of Local Position Invariance
ACES(2010) 10-6
Tests of Local Position Invariance
Constant Limit (yr-1)

W
me/mp
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
<1 X 10-11
0.15 Oklo reactor
<5 X 10-12
109
BBN
<3 X 10-15
2-3
Quasar spectra
Metric 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   m0a c 
a
1

16

ea
 u u d 
a c
 
  F F d 4 x

A dx 
Metric theories, nonmetric theories and electrodynamics
I   m0a c 
a
1

16

ea
g u u d 
a c
 
gg  g F F d 4 x

A dx 
Metric theories, nonmetric theories and electrodynamics
I   m0a c 
a
1

16

ea
g u u d 
a c
 
hh  h  F F d 4 x

A dx 
The Th Framework
I   m0a

T  Hv a dt  ea  (  A  v a )dt
2
a
a
1

8
2
1 2
3
(

E


B
)d
x dt

T, H, ,  are functions of an external static spherical potential U(r)

Metric theory action iff
with
    (H /T)1/ 2
g00  T(U)
gij  H(U)ij

Metric theories, nonmetric theories and electrodynamics
I   m0a c 
a
1

16

ea
g u u d 
a c
 
gg  g F F d 4 x

A dx 
TH Framework: Violation of WEP
TH Framework: Violation of LLI
I   m0a

1 v a dt  ea  (  A  v a )dt
2
a

a
1
8
 (E
2
 c 2 B 2 )d 3 x dt

J=3/2
BL  0, c 1 c 1, BLV c 1, BL ||V
Standard Model Extension (SME)
Kostelecky et al
L =    (k )  (D  )† D   m 2 †
1  
    (k F )  F F
4
D      ieA 
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
D. Mattingly, Living Reviews in Relativity 8, 2005-5
Electrodynamics in curved spacetime
I   m0a c 
a
1

16

ea
g u u d 
a c
 
gg  g F F d 4 x

A dx 
Outline of the Lectures
Lecture 1: The Einstein Equivalence Principle
Lecture 2: Post-Newtonian Limit of GR
Lecture 3: The Parametrized Post-Newtonian Framework
Lecture 4: Tests of the PPN Parameters