Tests of Gravity Sergei Kopeikin Sternberg Astronomical Institute, Moscow 1986 GrishchukZeldovich

Tests of Gravity

Sergei Kopeikin Sternberg Astronomical Institute, Moscow 1986

Grishchuk Zeldovich

EFT Wokshop, Pittsburg, July 2007 2

Basic Levels of Experiments

• Laboratory

• Earth/Moon

• Solar System

• Binary Pulsars

• Cosmology

• Gravitational Detectors


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


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


Principle of Equivalence:torsion balance tests

2 /

12 1 24

r

b

g eV q q

r

m c

2- limits on the strength of a Yukawa-type PE-violation coupled to baryon number. [Credit: Jens H Gundlach ]


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)15m/m 10 18m/m 10

17m/m 10

http://eotvos.dm.unipi.it/grex2003/index_file/v3_document.htm




Newtonian 1/r² Law 2- limits on 1/r² violations.[Credit: Jens H Gundlach 2005 New J. Phys. 7 205 ]

/1 212

21 22

1

1 ...2

rGm mV e

rGm m

r rr

Eöt-Wash 1/r² test data with therotating pendulum

=1; =250 m

Casimir force+1/r² law


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.

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

http://blog.wired.com/geekdad/2007/03/proving_the_the.html

http://relativity.livingreviews.org/Articles/lrr-2003-1


Global Positioning System1. 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.


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.


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)

( , )x ct x

1 2 3, ,

1 2 3 4, , ,

( , | , , ,...)g ct x


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)


Solar System Tests: Classic

• Advance of Perihelion

• Bending of Light

• Shapiro Time Delay


Advance of Perihelion

1 21 2

1 2

; =

m mm m m

m m

p

32 1 3 10 Q: To what extent does the orbital

motion of the Sun contribute to ?


Bending of Light

Traditionally the bending of

light is computed in a static-field

approximation.

Q: What physics is behind the

static approximation?

The Shapiro Time Delay

0gx x

0 2 2

(1 )ln E Px xGm

k xc D

Eikonal Equation:

A plane-wave eikonal (static gravity field):

(PRL, 26, 1132, 1971)


Limits on the parameter [Credit: Clifford M. Will ]


Solar System Tests: Advanced

• Gravimagnetic Field Measurement – LAGEOS

– Gravity Probe B

– Cassini

• The Speed of Gravity

• The Pioneer Anomaly


LAGEOS (Ciufolini, PRL, 56, 278, 1986)

3 2 3/ 2

2

(1 )L T

S

a e

-131 mas yrL T

Measured with 15%

error budget by

Ciufolini & Pavlis, Nature 2004

J2 perturbation is

totally suppressed

with k = 0.545

http://www.nature.com/nature/journal/v431/n7011/full/nature03007.html





Gravity Probe B

2 3

1 2 3

1

2

31 11

2 4

S LT T

S

LT

T

dSS

d

GM r v

c r

s n n sGS

c r

v A

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

=> =>Mission

beginsMission

ends


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)

http://arxiv.org/ps/gr-qc/0604060



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

Observer

Observer

Observer’s world line

Star’s world line

Planet’s world line

Future gravity null cone




Future gravity null coneLight n

ull co

ne

Light n

ull co

ne


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).

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

Position of Jupiter taken fromthe JPL ephemerides (radio/optics)

Position of Jupiter asdetermined from thegravitational deflectionof light from the quasar

1

2

3

54

10 microarcseconds = the width of a typical strand of a human hair from a distance of 650 miles.

Measured with 20% of accuracy, thus, proving that the null cone is a bi-characteristic hypersurface (speed of gravity = speed of light)

undeflected position of the quasar

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.

http://en.wikipedia.org/wiki/Doppler_effect

http://en.wikipedia.org/wiki/Sun

Lunar Laser Ranging: Retroreflector’s Positions on the Moon


Lunar Laser Ranging: TechnologyCredit: T. Murphy (UCSD)

LLR and the Strong Principle of Equivalence

Inertial mass

Gravitational mass

To the Sun To the Sun

EarthEarth

Moon Moon

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

Earth-Moon Sun-planets gauge modes

Earth-Moon Earth-Moon

Sun-planets

, ,gauge modes

16

0

0

T

Gauge Freedom in the Earth-Moon-Sun System

Moon EarthSun

Boundary of the localEarth-Moon reference frame ( , )w u w

'

'

'

x x

g gx x

R R

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)

2 SunIAU

constant+secular+periodic terms

1

2

GMdBv Q

du r

1( )

2i j

ijD u v v

[ ] [ ] [ ] [ ]Sun SunSun IAU3 3

(1 2 ) 2(1 )ij i j i j i j ijdF GM GMv w v w v Q R

du r r

SunIAU( )

GME u Y

r

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 variationof about 2 millimeters.

Earth

The Lorentzcontraction

The Einsteincontraction


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

( )i i iB

u t

w x x t

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

Deriving the Equations of MotionLagrangian-based theory of gravity

Laws of transformation of theinternal and external moments

Boundary and initial conditions:External problem - global frame

Field equations: tensor, vector, scalar

Laws of motion: external

External multipole moments in terms of external gravitational potentials

Matching of external and internal solutions

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

Coordinate transformations between the global and local frames

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


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


Orbital Parameterization(Klioner & Kopeikin, ApJ, 427, 951, 1994)

– Osculating Elements

– Blandford-Teukolsky– Epstein-Haugan– Brumberg– Damour-Deruelle

To observer

f


Timing Model

Pulse’s

number

Pulsar’s

rotational

frequency

Pulsar’s

rotational

frequency

derivative

Emission

time

Time of

arrival

Roemer

delayProper

motion

delay

Parallax

delay

Einstein

delay

Shapiro

delay

Bending

Delay

Plasma

delay

Atomic

(proper)

time


Keplerian Parameters

• Projected semi-major axis:

• Eccentricity:

• Orbital Period:

• Longitude of periastron:

• Julian date of periastron:

– Keplerian parameters => Mass function:

0

e

0T

bP

( , ,sin )p cf m m i


Post-Keplerian Parameters

s

Two more "radiation" parameters: and x e


Four binary pulsars testsCredit: Esposito-Farese


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)

14

21

(7.88 0.01) 10

1.6 10

obs

GR

x

x

PSR J0437-4715

(0.236 0.017) M

(1.58 0.18) Mc

p

m

m

Shapiro delay in the pulsar PSRJ 1909-3744 timingsignal due to the gravitational field of its companion.

Geodetic precession in PSR 1913+16

1.21 deg yr -1

Pulsar’s Spin

Axis

Orbital Spin Axis

Credit: M. Kramer & D. Lorimer

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 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


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

Reference Frames in N-body Problem: global and local frames

LR

gr

1e

3e

2e

0e

Matching of Local and Global Frames

x

w

x

wwug

x

w

x

uwug

cx

u

x

uwug

cxtg

ji

ij

i

i

),(),(2

),(1

),( 0002

Matching Domain

(u, w) Global coordinates (t, x)


Coordinate Transformations between Local and Global Frames


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

External Grav. Potentials Inertial Forces

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.

Definition of Spherical Symmetry in terms of intrinsic multipoles?

Active mass multipolemoment

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 tothe 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

0 0

Multipolar Expansion of the post-Newtonian Potentials [ ]

LL

L STF)(KSTFSTF

These termsare absorbedto the Tolman(relativistic) mass

The Inertial Forces

Translational Equations of Motion

the Nordtvedt parameter

gravitational mass

Newtonian force

the effective mass

tidal

inertial mass

B

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

Ntidal R

LFF

2

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

6232

c

v

R

L

v

v

sound

Kepler

sound

Kepler

Lvsound ,

For ordinary stars:

For black holes:

Ngsound

KeplerN

sound

Keplertidal c

v

r

L

v

v

R

L

v

vFFF

105252

Ntidal c

vFF

10


NEIH c

vFF

2

NNS R

L

c

v

R

L

c

v

c

vFFF

22

Spin-dependent terms 4th-order moment-of-inertia terms

For maximal Kerr black hole:

NNS c

v

c

vFFF

43

Spin-dependent terms 4th-order moment-of-inertia terms


tidalNIGR R

L

c

vFFF

42

For black hole:

NIGR R

L

c

vFF

22

)1(

tidalNIGR c

vFFF

10

NIGR c

vFF

6

)1(


Documents

Tests of Gravity Sergei Kopeikin Sternberg Astronomical Institute, Moscow 1986 GrishchukZeldovich