Tests%20of%20Gravity

1
Tests of Gravity

• Sergei Kopeikin

Sternberg Astronomical Institute, Moscow 1986
Grishchuk
Zeldovich
2
Basic Levels of Experiments

• Laboratory
• Earth/Moon
• Solar System
• Binary Pulsars
• Cosmology
• Gravitational Detectors

3
Laboratory Tests theoretical motivations

• Alternative (classic) theories of gravity with
  short-range forces
• Scalar-tensor
• Vector-tensor TeVeS
• Tensor-tensor (Milgrom, Bekenstein)
• Non-symmetric connection (torsion)
• Super-gravity, M-theory
• Strings, p-branes
• Loop quantum gravity
• Extra dimensions, the hierarchy problem
• Cosmological acceleration

The Bullet Cluster
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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 Equivalencetorsion balance tests
2-? limits on the strength of a Yukawa-type
PE-violation coupled to baryon number.
Credit Jens H Gundlach
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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 A.
  Nobilis lecture)

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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
Casimir force1/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
• 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)

Mach-Zender Interferometer
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Global Positioning System

• 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.
• 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.
• The Sagnac effect for a receiver at rest on the
  equator is 133 ns, it may be larger for moving
  receivers.
• At the sub-nanosecond level additional
  corrections apply, including the contribution
  from Earths 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
• A metric tensor 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)
• - preferred frame effects (0 in GR)
• - 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
Q To what extent does the orbital motion
of the Sun contribute to ???
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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
A plane-wave eikonal (static gravity field)
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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)
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
Residual noise GP-B Gyro 1 Polhode Motion
(torque-free Euler-Poinsot precession)
gt
gt
Mission begins
Mission ends
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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(2.12.3)
?
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Propagation of light in time-dependent
gravitational field light and gravity null cones
Observer
Future gravity null cone
Stars world line
Observer
Future gravity null cone
Future gravity null cone
Future gravity null cone
Light null cone
Future gravity null cone
Light null cone
Observers world line
Planets world line
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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 planets 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
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1
Position of Jupiter as determined from
the gravitational deflection of light from the
quasar
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.
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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.
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Lunar Laser Ranging Retroreflectors Positions
on the Moon
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Lunar Laser Ranging Technology
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)-0.00070.0010
Moon
Earth
Moon
Earth
To the Sun
To the Sun
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Gauge Freedom in the Earth-Moon-Sun System
Sun
Moon
Earth
Boundary of the local Earth-Moon reference frame
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Example of the gauge modes

• TT-TCB transformation of time scales
• Lorentz contraction of the local coordinates
• Einstein contraction of the local coordinates
• Relativistic Precession (de Sitter,
  Lense-Thirring, Thomas)

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Effect of the Lorentz and Einstein contractions
Magnitude of the contractions is about 1 meter!
Ellipticity of the Earths orbit leads to its
annual variation of about 2 millimeters.
The Lorentz contraction
Earth
The Einstein contraction
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The gauge modes in EIH equations of a three-body
problem

• Newtonian-like transformation of the
  Einstein-Infeld-Hoffman (EIH) force
• 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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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.

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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
Boundary and initial conditions Internal problem
- local frame(s)
External solution of the field equations metric
tensor other fields in entire space
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
External multipole moments in terms of external
gravitational potentials
Coordinate transformations between the global
and local frames
Laws of transformation of the internal and
external moments
Laws of motion external
Laws of motion internal Fixing the origin of
the local frame
Equations of motion external
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
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Equations of Motionin 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)
f

• Osculating Elements
• Blandford-Teukolsky
• Epstein-Haugan
• Brumberg
• Damour-Deruelle

To observer
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Timing Model
Pulsars rotational frequency derivative
Pulsars rotational frequency
Pulses number
Emission time
Roemer delay
Time of arrival
Proper motion delay
Parallax delay
Einstein delay
Shapiro delay
Bending Delay
Plasma delay
Atomic (proper) time
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Keplerian Parameters

• Projected semi-major axis
• Eccentricity
• Orbital Period
• Longitude of periastron
• Julian date of periastron
• Keplerian parameters gt Mass function

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Post-Keplerian Parameters
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
Shapiro delay in the pulsar PSRJ 1909-3744
timing signal due to the gravitational field of
its companion.
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Geodetic precession in PSR 191316
1.21 deg yr
-1
Credit M. Kramer D. Lorimer
Pulsars Spin Axis
Orbital Spin Axis
To observer
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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
  pulsars companion
• Since these perturbations have different
  orbital-phase 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 bodys spherical symmetry
• The effacing principle

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Reference Frames in N-body Problem global and
local frames
R
L
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Matching of Local and Global Frames
(u, w)
Global coordinates (t, x)
Matching Domain
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Coordinate Transformations between Local and
Global Frames
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The Law of Motion of the Origin of the Local
Frame in the Global Frame
External Grav. Potentials
Inertial Forces
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Fixing the Origin of the Local Frame
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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
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Intrinsic Definition of Spherical Symmetry
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Definition of Spherical Symmetry Gravitational
Potential
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Integrals from the Spherical Distribution of
Matter
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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)
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Multipolar Expansion of the Newtonian Potential
in the Global Frame
Multipolar Expansion of the post-Newtonian
Potentials
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Multipolar Expansion of the post-Newtonian
Potentials
These terms are absorbed to the
Tolman (relativistic) mass
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The Inertial Forces
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Translational Equations of Motion
gravitational mass
inertial mass
Newtonian force
the Nordtvedt parameter
the effective mass
B
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Einstein-Infeld-Hoffmann Force
What masses in 2 PNA?
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Post-Newtonian Spin-Orbit Coupling Force
These terms are not spins.
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Post-Newtonian Brumbergs Force
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The Effacing-Principle-Violating Forces
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Magnitude of the post-Newtonian Forces

• ?( ) - structure-dependent
  ellipticity of the body (Loves number)

For ordinary stars For black holes
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Magnitude of the post-Newtonian Forces
Spin-dependent terms
4th-order moment-of-inertia terms
For maximal Kerr black hole
Spin-dependent terms
4th-order moment-of-inertia terms
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Magnitude of the post-Newtonian Forces
For black hole
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• Thank You!