Download Tests of GR - High Energy Experiment

Tests of Gravity
Sergei Kopeikin
Sternberg Astronomical Institute, Moscow 1986
Grishchuk
Zeldovich
Basic Levels of Experiments
• Laboratory
• Earth/Moon
• Solar System
• Binary Pulsars
• Cosmology
• Gravitational Detectors
EFT Wokshop, Pittsburg, July 2007
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Laboratory Tests: theoretical motivations
• Alternative (“classic”) theories of gravity with shortrange forces
The Bullet Cluster
– Scalar-tensor
– Vector-tensor
– Tensor-tensor
TeVeS
(Milgrom, Bekenstein)
– Non-symmetric connection (torsion)
• Super-gravity, M-theory
• Strings, p-branes
• Loop quantum gravity
• Extra dimensions, the hierarchy problem
• Cosmological acceleration
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Laboratory Tests: experimental techniques
• Principle of Equivalence
– Torsion balance (Eötvös-type experiment)
– Rotating torsion balance
– Rotating source
– Free-fall in lab
– Free-fall in space
• Newtonian 1/r² Law (a fifth force)
– Torsion balance
– Rotating pendulum
– Torsion parallel-plate oscillator
– “Spring board” resonance oscillator
– Ultra-cold neutrons
• Extra dimensions and the compactification scale
– Large Hadron Collider
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Principle of Equivalence:
torsion balance tests
2- limits on the strength of a Yukawa-type PE-violation
coupled to baryon number. [Credit: Jens H Gundlach ]
g2
e r / 
V12 
q1q2
4
r

mb c
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Principle of Equivalence:
• Free-fall in Lab
– Galileo Galilei
– NIST Boulder
– ZARM Bremen
– Stratospheric balloons
– Lunar feather-hammer test (David Scott – Apollo 15)
• Free-fall in Space
– SCOPE (French mission
– STEP (NASA/ESA mission
– GG (Italian mission
 m/m 10
15
 m/m 1018
 m/m 1017
EFT Wokshop, Pittsburg, July 2007
)
)
A. Nobili’s lecture)
6
Newtonian 1/r² Law
2- limits on 1/r² violations.
[Credit: Jens H Gundlach 2005 New J. Phys. 7 205 ]
Eöt-Wash 1/r² test data with the
rotating pendulum
=1; =250 m


Gm1m2
1   e r / 
r
Gm1m2 

 2


1    r  2 r  ... 
r 

2

V12 
Casimir force+1/r²
law
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Local Lorentz Invariance
[Credit: Clifford M. Will]
The limits assume a speed
of Earth of 370 km/s
relative to the mean rest
frame of the universe.
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Gravitational Red Shift
• Ground
– Mössbauer effect (Pound-Rebka 1959)
– Neutron interferometry
(Colella-Overhauser-Werner 1975)
– Atom interferometry
– Clock metrology
Mach-Zender Interferometer
– Proving the Theory of Relativity in Your Minivan
• Air
– Häfele & Keating (1972)
– Alley (1979)
• Space
– Gravity Probe A (Vessot-Levine 1976)
– GPS (Relativity in the Global Positioning System)
Global Positioning System
1.
The combined effect of second order Doppler shift
(equivalent to time dilation) and gravitational red shift
phenomena cause the clock to run fast by 38 s per day.
2.
The residual orbital eccentricity causes a sinusoidal variation
over one revolution between the time readings of the
satellite clock and the time registered by a similar clock on
the ground. This effect has typically a peak-to-peak
amplitude of 60 - 90 ns.
3.
The Sagnac effect – for a receiver at rest on the equator is
133 ns, it may be larger for moving receivers.
4.
At the sub-nanosecond level additional corrections apply,
including the contribution from Earth’s oblateness, tidal
effects, the Shapiro time delay, and other post Newtonian
effects.
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Gravitational Red Shift
[Credit: Clifford M. Will ]
Selected tests of local position
invariance via gravitational redshift
experiments, showing bounds on 
which measures degree of deviation
of redshift from the Einstein formula.
In null redshift experiments, the
bound is on the difference in 
between different kinds of clocks.
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The PPN Formalism: the postulates
•
A global coordinate frame x  (ct , x )
•
A metric tensor g  (ct , x |  ,  ,  ,...) with 10 potentials and 10
parameters
 - curvature of space (= 1 in GR)

- non-linearity of gravity (=1 in GR)

- preferred location effects (=0 in GR)
1 ,  2 , 3 - preferred frame effects (=0 in GR)
 1 ,  2 ,  3 ,  4 - violation of the linear momentum conservation (=0 in GR)
•
Stress-energy tensor: a perfect fluid
•
Stress-energy tensor is conserved (“comma goes to semicolon” rule)
•
Test particles move along geodesics
•
Maxwell equations are derived under assumption that the principle of
equivalence is valid (“comma goes to semicolon” rule)
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The PPN Formalism: the difficulties
• The structure of the metric tensor in arbitrary coordinates is known
only in one (global) coordinate system
• Gauge-invariance is not preserved
• Oservables and gravitational variables are disentangled
• PPN parameters are gauge-dependent
• PPN formalism derives equations of motion of test point particles
under assumption that the weak principle of equivalence is valid but
it does not comply with the existence of the Nordtvedt effect
• PPN is limited to the first post-Newtonian approximation
• Remedy:
– Damour & Esposito-Farese, Class. Quant. Grav., 9, 2093 (1992)
– Kopeikin & Vlasov, Phys. Rep., 400, 209-318 (2004)
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Solar System Tests: Classic
• Advance of Perihelion
• Bending of Light
• Shapiro Time Delay
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Advance of Perihelion
p
m  m1  m2 ;
2    1  3 103
=
m1m2
m1  m2
Q: To what extent does the orbital
motion of the Sun contribute to ?
EFT Wokshop, Pittsburg, July 2007
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Bending of Light

Traditionally the bending of
light is computed in a static-field
approximation.
Q: What physics is behind the
static approximation?
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The Shapiro Time Delay (PRL, 26, 1132, 1971)
Eikonal Equation:
  
g
0


x x
A plane-wave eikonal
(static gravity field):
(1   )Gm  xE xP 
   0  k x 
ln  2 
2
c
 D 

Limits on the parameter 
[Credit: Clifford M. Will ]
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Solar System Tests: Advanced
• Gravimagnetic Field Measurement
– LAGEOS
– Gravity Probe B
– Cassini
• The Speed of Gravity
• The Pioneer Anomaly
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LAGEOS (Ciufolini, PRL, 56, 278, 1986)

 L T 
2 S
a3 (1  e2 )3/ 2

LT  31 mas yr -1
Measured with 15%
error budget by
Ciufolini & Pavlis, Nature 2004
J2 perturbation is
totally suppressed
with k = 0.545
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Gravity Probe B
dS
  S
d
   S   LT  T
1  GM r  v

S      2  3
2 c
r

1
1  GS s  3n  n  s 
 LT   1    1  2
2
4  c
r3
T  v  A
Residual noise: GP-B Gyro #1 Polhode Motion (torque-free Euler-Poinsot
precession)
Mission
begins
=>
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=>
Mission
ends
21
Cassini Measurement of Gravimagnetic Field
(Kopeikin et al., Phys. Lett. A 2007)
Mass current
due to the orbital
motion of the Sun
Bertotti-Iess-Tortora, Nature, 2004
-1=(2.1±2.3)
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Propagation of light in time-dependent gravitational field:
light and gravity null cones
Observer
Star’s world line
Future gravity null cone
Observer
Future gravity null cone
Future gravity null cone
Future gravity null cone
Future gravity null cone
Planet’s world line
Observer’s
world line
The null-cone bi-characteristic interaction of
gravity and light in general relativity
Any of the Petrov-type gravity field obeys the principle of causality, so that
even the slowly evolving "Coulomb component" of planet’s gravity field can
not transfer information about the planetary position with the speed faster than
the speed of light (Kopeikin, ApJ Lett., 556, 1, 2001).
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The speed-of-gravity VLBI experiment with Jupiter
(Fomalont & Kopeikin, Astrophys. J., 598, 704, 2003)
Position of Jupiter taken from
the JPL ephemerides (radio/optics)
undeflected position of the quasar
Position of Jupiter as
determined from the
gravitational deflection
of light from the quasar
5
1
4
2
3
Measured with 20% of accuracy, thus, proving
that the null cone is a bi-characteristic
hypersurface (speed of gravity = speed of light)
10 microarcseconds = the width of a typical strand
of a human hair from a distance of 650 miles.
The Pioneer Anomaly
The anomaly is seen in radio Doppler and ranging data, yielding information on the
velocity and distance of the spacecraft. When all known forces acting on the
spacecraft are taken into consideration, a very small but unexplained force remains.
It causes a constant sunward acceleration of (8.74 ± 1.33) × 10−10 m/s2 for both
Pioneer spacecrafts.
Lunar Laser Ranging:
Retroreflector’s Positions on the Moon
Lunar Laser Ranging: Technology
EFT Wokshop, Pittsburg, July 2007
Credit: T. Murphy (UCSD)
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LLR and the Strong Principle of
Equivalence
Inertial mass
Gravitational mass
The Nordtvedt effect: 4(-1)-(-1)=-0.0007±0.0010
Earth
Moon
Moon
Earth
To the Sun
To the Sun
Gauge Freedom in the Earth-Moon-Sun System
x '  x   
g '  g  



    Earth-Moon
  Sun-planets
  gauge
modes
 
x


x 
R '  R


 Earth-Moon
 16 TEarth-Moon
Sun
Moon
Earth

 Sun-planets
0

 ,
 ,
 gauge





modes
 0
Boundary of the local
Earth-Moon reference
frame w  (u, w)
Example of the gauge modes:
– TT-TCB transformation of time scales
1 2 GM Sun
v 
2
r
dB

du
– Lorentz contraction of the local coordinates
Dij (u ) 
– Einstein contraction of the local coordinates
constant+secular+periodic terms
1 i j
vv
2
GM Sun
 YIAU
r
– Relativistic Precession (de Sitter, Lense-Thirring, Thomas)
E (u )  
dFij
du
 (1  2 )
GM Sun [i j ]
GM Sun [i j ] [i j ]
[ ij ]
v
w

2(1


)
v
w

v
Q

R
Sun
IAU
r3
r3
 QIAU
Effect of the Lorentz and Einstein contractions
Magnitude of the contractions is about 1 meter!
Ellipticity of the Earth’s orbit leads to its annual variation
of about 2 millimeters.
The Lorentz
contraction
Earth
The Einstein
contraction
The gauge modes in EIH equations
of a three-body problem:
• “Newtonian-like” transformation of the EinsteinInfeld-Hoffman (EIH) force
u t
w  x  x (t )
i
i
i
B
• This suppresses all gauge modes in the
coordinate transformation from the global to
local frame but they all appear in the geocentric
EIH equations as spurious relativistic forces
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33
Are the gauge modes
observable?
• Einstein: no – they do not present in observational data
• LLR team (Murphy, Nordtvedt, Turyshev, PRL 2007)
– yes – the “gravitomagnetic” modes are observable
• Kopeikin, S., PRL., 98, 229001 (2007)
The LLR technique involves processing data with two
sets of mathematical equations, one related to the
motion of the moon around the earth, and the other
related to the propagation of the laser beam from earth
to the moon. These equations can be written in different
ways based on "gauge freedom“, the idea that arbitrary
coordinates can be used to describe gravitational
physics. The gauge freedom of the LLR technique shows
that the manipulation of the mathematical equations is
causing JPL scientists to derive results that are not
apparent in the data itself.
Binary Pulsar Tests
• Equations of Motion
• Orbital Parametrization
• Timing Formula
• Post-Keplerian Formalism
– Gravitational Radiation
– Geodetic Precession
– Three-dimensional test of gravity
• Extreme Gravity: probing black hole physics
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Deriving the Equations of Motion
Lagrangian-based theory of gravity
Field equations: tensor, vector, scalar
Boundary and initial conditions:
External problem - global frame
External solution of the field equations:
metric tensor + other fields in entire space
Boundary and initial conditions:
Internal problem - local frame(s)
Internal solution of the field equations:
metric tensor + other fields in a local domain;
external and internal multipole moments
Matching of external and internal solutions
Coordinate transformations
between the global and local frames
Laws of motion: external
External multipole moments in terms of
external gravitational potentials
Laws of transformation of the
internal and external moments
Equations of motion: external
Laws of motion: internal;
Fixing the origin of the local frame
Equations of motion: internal
Effacing principle: equations of motion of spherical and non-rotating bodies depend only on
their relativistic masses – bodies’ moments of inertia does not affect the equations
Equations of Motion
in a binary system
Lorentz-Droste, 1917
Einstein-Infeld-Hoffman, 1938
Petrova, 1940
Fock, 1955
(see Havas, 1989, 1993 for
interesting historic details)
Carmeli, 1964
Ohta, Okamura, Kiida, Kimura,
1974
Damour-Deruelle, 1982
Kopeikin, 1985
Schaefer, 1985
…
Grishchuk-Kopeikin, 1983
Damour, 1983
Kopeikin, PhD 1986
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Orbital Parameterization
(Klioner & Kopeikin, ApJ, 427, 951, 1994)
– Osculating Elements
f
r (t )  a (t ) 1  e(t ) cos E  ,
n(t  T0 )  E  e(t ) sin E  l (t ),
  f   (t ),
tan
–
–
–
–
f
1  e(t )
E

tan
2
1  e(t )
2
To observer
Blandford-Teukolsky
Epstein-Haugan
Brumberg
Damour-Deruelle
r (t )  a p 1  er cos U  ,
n(t  T0 )  U  e sin U ,



Pb
2 
n
1
(t  T0 )  ,

Pb  2 Pb



   0  (1  k ) Ae ,
tan
Ae
2

k

n
,
1  e
U
tan ,
1  e
2
er  e 1   r  ,
e  e 1    .
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Timing Model
1
N (T )  N 0  pT   pT 2   intrinsic (T )
2
noise
Pulsar’s
rotational
frequency
Pulse’s
number
Pulsar’s
rotational
frequency
derivative
Emission
time
DM
t  D T   R        E   S   B   2   grav. wave (t )
f
noise
Time of
arrival
Roemer
delay
 t
R
Proper
motion
delay


Parallax Einstein Shapiro Bending
delay
delay
delay
Delay



E

S
Plasma
delay
  clock (t )
noise
Atomic
(proper)
time
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Keplerian Parameters
• Projected semi-major axis:
• Eccentricity:
• Orbital Period:
e
• Longitude of periastron:
Pb
• Julian date of periastron:
0
T0
– Keplerian parameters => Mass function: f (m p , mc ,sin i )
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Post-Keplerian Parameters


Two more "radiation" parameters: x and e
s
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Four binary pulsars tests
Credit: Esposito-Farese
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A test of general relativity from the three-dimensional
orbital geometry of a binary pulsar
(van Straten, Bailes, Britton, Kulkarni, et al. Nature 412, 158, 2001)
PSR J0437-4715
xobs  (7.88  0.01) 1014
xGR  1.6 1021
Shapiro delay in the pulsar PSRJ 1909-3744 timing
signal due to the gravitational field of its companion.
EFT Wokshop, Pittsburg, July 2007
mc  (0.236  0.017) M
mp  (1.58  0.18) M
43
Geodetic precession in PSR 1913+16
1.21 deg yr
-1
Credit: M. Kramer & D. Lorimer
Pulsar’s Spin
Axis
Orbital Spin Axis
To observer
Extreme Gravity: detecting black hole
with pulsar timing (Wex & Kopeikin, ApJ, 1999)
– Timing of a binary pulsar allows us to measure the
quadrupolar-field and spin-orbit-coupling
perturbations caused by the presence of the pulsar’s
companion
– Since these perturbations have different orbitalphase dependence, one can measure the
quadrupole and the spin of the companion
– Black hole physics predicts a unique relationship
between the spin and the quadrupole because of the
“no-hair theorem”
– Comparision of the mesured value of spin against the
quadrupole allows us to see if the companion is a
black hole and explore the black hole physics
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Finite Size Effects in the PN Equations of
Motion: gravitational wave detector science
• Reference frames in N-body problem
• Definition of body’s spherical symmetry
• The effacing principle
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46
Reference Frames in N-body Problem:
global and local frames
R
L  rg
Matching of Local and Global Frames

e3

e0

e1
(u, w)
Global coordinates (t, x)

e2
Matching Domain
1
u u 2
u wi
wi w j
g  (t , x)  2 g 00 (u, w)    g 0i (u, w)    g ij (u, w)  
c
x x
c
x x
x x
Coordinate Transformations between Local and
Global Frames
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49
The Law of Motion of the Origin of the Local Frame in the Global Frame
External Grav. Potentials
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Inertial Forces
50
Fixing the Origin of the Local Frame
Definition of Spherical Symmetry
• Definition in terms of internal multipole
moments
• Definition in terms of internal distributions
of density, energy, stresses, etc.
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Definition of Spherical Symmetry in terms of intrinsic multipoles?
Active mass
multipole
moment
Mass density
Scalar mass
multipole moments
Conformal mass
multipole moments
Scalar mass
multipole moments
Intrinsic Definition of Spherical Symmetry
Definition of Spherical Symmetry: Gravitational Potential
Integrals from the Spherical Distribution of Matter
Internal Multipole Moments in the Global Frame
Dipole is not zero
Quadrupole is not zero,
but proportional to
the moment of inertia of the
second order:
The assumption of spherical symmetry in the global coordinates
leads to 1PN force first calculated by Brumberg (1972)
Multipolar Expansion of the Newtonian Potential in the Global Frame
0
0
Multipolar Expansion of the post-Newtonian Potentials
Multipolar Expansion of the post-Newtonian
Potentials [ STF  STF   K L (STF) L ]
L
These terms
are absorbed
to the Tolman
(relativistic)
mass
The Inertial Forces
Translational Equations of Motion
tidal
inertial mass
gravitational mass
Newtonian force
the Nordtvedt parameter
B
the effective mass
Einstein-Infeld-Hoffmann Force
What masses in 2 PNA?
Post-Newtonian Spin-Orbit Coupling Force
These terms are not spins.
Post-Newtonian Brumberg’s Force
The Effacing-Principle-Violating Forces
Magnitude of the post-Newtonian Forces
2
L
Ftidal     FN
R
= (vsound , L ) - structure-dependent ellipticity of the body (Love’s number)
3
6
 vKepler   L 
  Kepler   v 
    
  
  
 vsound   R 
 sound   c 
2
2
2
 vKepler   L 
 vKepler 
   FN  

For ordinary stars: Ftidal  
 vsound   R 
 vsound 
10
v
For black holes:  Ftidal    FN
c
5
2
L
 
r 
 g
5
10
v
  FN
c
Magnitude of the post-Newtonian Forces
2
v
FEIH    FN
c
2
2
 v  v  L 
 v   L 
FS     FN      FN
 c  c  R 
 c  R
Spin-dependent terms
4th-order moment-of-inertia terms
For maximal Kerr black hole:
3
4
v
v
FS    FN    FN
c
c
Spin-dependent terms
4th-order moment-of-inertia terms
Magnitude of the post-Newtonian Forces
2
4
v  L
FIGR      FN  Ftidal
c R
2
FIGR
For black hole:
10
FIGR
v
   FN  Ftidal
c
6
FIGR
2
v  L
 (   1)    FN
c R
v
 (   1)  FN
c
EFT Wokshop, Pittsburg, July 2007
69