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Basic Levels of Experiments

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

EFT Wokshop, Pittsburg, July 2007

Laboratory Tests:theoretical motivations

The Bullet Cluster

- 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

EFT Wokshop, Pittsburg, July 2007

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

EFT Wokshop, Pittsburg, July 2007

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 ]

EFT Wokshop, Pittsburg, July 2007

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. Nobili’s lecture)

EFT Wokshop, Pittsburg, July 2007

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 force+1/r² law

EFT Wokshop, Pittsburg, July 2007

Local Lorentz Invariance

[Credit: Clifford M. Will]

The limitsassume a speed of Earth of 370 km/srelative to the mean rest frame of the universe.

EFT Wokshop, Pittsburg, July 2007

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

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 Earth’s oblateness, tidal effects, the Shapiro time delay, and other post Newtonian effects.

EFT Wokshop, Pittsburg, July 2007

Gravitational Red Shift

[Credit: Clifford M. Will ]

Selected tests of local position

invariance via gravitational redshift

experiments, showingbounds on

which measures degree of deviation

of redshift from the Einstein formula.

In null redshift experiments, the bound is on the difference inbetween different kinds of clocks.

EFT Wokshop, Pittsburg, July 2007

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)

EFT Wokshop, Pittsburg, July 2007

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)

EFT Wokshop, Pittsburg, July 2007

Solar System Tests: Classic

- Advance of Perihelion
- Bending of Light
- Shapiro Time Delay

EFT Wokshop, Pittsburg, July 2007

Advance of Perihelion

p

Q: To what extent does the orbital

motion of the Sun contribute to ?

EFT Wokshop, Pittsburg, July 2007

Bending of Light

Traditionally the bending of

light is computed in a static-field

approximation.

Q: What physics is behind the

static approximation?

EFT Wokshop, Pittsburg, July 2007

The Shapiro Time Delay

Eikonal Equation:

A plane-wave eikonal

(static gravity field):

Solar System Tests: Advanced

- Gravimagnetic Field Measurement
- LAGEOS
- Gravity Probe B
- Cassini

- The Speed of Gravity
- The Pioneer Anomaly

EFT Wokshop, Pittsburg, July 2007

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

EFT Wokshop, Pittsburg, July 2007

Gravity Probe B

Residual noise: GP-B Gyro #1 Polhode Motion (torque-free Euler-Poinsot precession)

=>

Mission

begins

=>

Mission

ends

EFT Wokshop, Pittsburg, July 2007

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)

EFT Wokshop, Pittsburg, July 2007

Propagation of light in time-dependent gravitational field: light and gravity null cones

Observer

Future gravity null cone

Star’s 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

Observer’s world line

Planet’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).

EFT Wokshop, Pittsburg, July 2007

The speed-of-gravity VLBI experiment with Jupiter light in general relativity(Fomalont & Kopeikin, Astrophys. J., 598, 704, 2003)

Position of Jupiter taken from

the JPL ephemerides (radio/optics)

undeflected position of the quasar

5

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.

The Pioneer Anomaly light in general relativity

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: light in general relativityRetroreflector’s Positions on the Moon

Credit: T. Murphy (UCSD) light in general relativity

Lunar Laser Ranging: TechnologyEFT Wokshop, Pittsburg, July 2007

LLR and the Strong Principle of Equivalence light in general relativity

Inertial mass

Gravitational mass

The Nordtvedt effect: 4(-1)-(-1)=-0.0007±0.0010

Moon

Earth

Moon

Earth

To the Sun

To the Sun

Gauge Freedom in the Earth-Moon-Sun System light in general relativity

Sun

Moon

Earth

Boundary of the local

Earth-Moon reference

frame

Example of the gauge modes: light in general relativity

- 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)

Effect of the Lorentz and Einstein contractions light in general relativity

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 light in general relativityof 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

EFT Wokshop, Pittsburg, July 2007

Are the gauge modes observable? light in general relativity

- 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 light in general relativity

- 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

EFT Wokshop, Pittsburg, July 2007

Deriving the Equations of Motion light in general relativity

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

Coordinate transformations

between the global and local frames

External multipole moments in terms of

external gravitational potentials

Laws of motion: external

Laws of transformation of the

internal and external moments

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

Lorentz-Droste, 1917 light in general relativity

Einstein-Infeld-Hoffman, 1938

Petrova, 1940

Fock, 1955

(see Havas, 1989, 1993 for

interesting historic details)

Equations of Motionin a binary systemCarmeli, 1964

Ohta, Okamura, Kiida, Kimura,

1974

Damour-Deruelle, 1982

Kopeikin, 1985

Schaefer, 1985

…

Grishchuk-Kopeikin, 1983

Damour, 1983

Kopeikin, PhD 1986

EFT Wokshop, Pittsburg, July 2007

Orbital Parameterization light in general relativity(Klioner & Kopeikin, ApJ, 427, 951, 1994)

f

To observer

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

EFT Wokshop, Pittsburg, July 2007

Timing Model light in general relativity

Pulsar’s

rotational

frequency

derivative

Pulsar’s

rotational

frequency

Pulse’s

number

Emission

time

Roemer

delay

Time of

arrival

Proper

motion

delay

Parallax

delay

Einstein

delay

Shapiro

delay

Bending

Delay

Plasma

delay

Atomic

(proper)

time

EFT Wokshop, Pittsburg, July 2007

Keplerian Parameters light in general relativity

- Projected semi-major axis:
- Eccentricity:
- Orbital Period:
- Longitude of periastron:
- Julian date of periastron:
- Keplerian parameters => Mass function:

EFT Wokshop, Pittsburg, July 2007

Credit: Esposito-Farese light in general relativity

Four binary pulsars testsEFT Wokshop, Pittsburg, July 2007

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.

EFT Wokshop, Pittsburg, July 2007

Geodetic precession in PSR 1913+16 orbital geometry of a binary pulsar

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 orbital geometry of a binary pulsar(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 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

EFT Wokshop, Pittsburg, July 2007

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

EFT Wokshop, Pittsburg, July 2007

Matching of Local and Global Frames gravitational wave detector science

(u, w)

Global coordinates (t, x)

Matching Domain

Coordinate Transformations between Local and Global Frames gravitational wave detector science

EFT Wokshop, Pittsburg, July 2007

The Law of Motion of the Origin of the Local Frame in the Global Frame

External Grav. Potentials

Inertial Forces

EFT Wokshop, Pittsburg, July 2007

Fixing the Origin of the Local Frame Global Frame

Definition of Spherical Symmetry Global Frame

- Definition in terms of internal multipole moments
- Definition in terms of internal distributions of density, energy, stresses, etc.

EFT Wokshop, Pittsburg, July 2007

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 multipoles?

Internal Multipole Moments in the Global Frame multipoles?

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

Multipolar Expansion of the post-Newtonian Potentials

Multipolar Expansion of the post-Newtonian Potentials [ ]

These terms

are absorbed

to the Tolman

(relativistic)

mass

Translational Equations of Motion ]

gravitational mass

inertial mass

Newtonian force

the Nordtvedt parameter

the effective mass

B

Einstein-Infeld-Hoffmann Force ]

What masses in 2 PNA?

Post-Newtonian Spin-Orbit Coupling Force ]

These terms are not spins.

Magnitude of the post-Newtonian Forces ]

- = ( ) - structure-dependent ellipticity of the body (Love’s number)

For ordinary stars:

For black holes:

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

Magnitude of the post-Newtonian Forces ]

For black hole:

EFT Wokshop, Pittsburg, July 2007

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