Colloquium at the University of Mississippi, Oxford, USA

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Spacetime astrometry and

gravitational experiments in the solar system

Sergei KopeikinUniversity of Missouri

October 14, 2014

Colloquium at the University of Mississippi, Oxford, USA

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AbstractAstrometry is the branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. The main goal of spacetime astrometry is to build the inertial coordinate system in the sky and to test general theory of relativity as well as other fundamental theories. Modern astrometry uses the sophisticated technologies and techniques including the satellites in deep space, ultra-precise atomic clocks, very long baseline interferometry (VLBI) and Doppler tracking. We overview the current astrometric space missions and discuss the theoretical principles of the gravitational experiments utilizing the light propagation through the gravitational field of the massive bodies in the solar system. We pay a special attention to the goals and results of the light-propagation experiments in time-dependent gravitational field of planets and Sun which were conducted in the last decade. We will also touch upon a possibility of the local measurement of the Hubble constant with spacecraft’s Doppler tracking without making a direct observation of cosmological objects (quasars, supernova).

October 14, 2014

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Contents1. Astrometric Experiments2. Gravitational Field Model3. Light-ray Propagation4. Light-ray Deflection Angle5. Gravitomagnetism and the speed of gravity6. Gravitational Time Delay7. The idea of the speed-of-gravity experiment8. Jovian 2002 and Cronian 2009 experiments9. Cassini gravitomagnetic experiment10. “Pioneer anomaly” - Local measurement of the

Hubble constant?

October 14, 2014

Colloquium at the University of Mississippi, Oxford, USA

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Astrometry in Space

October 14, 2014

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SIM

SIM PlanetQuest has been designed as a space-based 9-m baseline optical Michelson interferometer operating in the visible waveband. This mission might open up many areas of astrophysics, via astrometry with unprecedented accuracy. Over a narrow field of view (1°), SIM aimed to achieve an accuracy of 1 µas in a single measurement!

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GAIA Gaia: was launched in 2013. It scans the sky continuously according to a pre-defined pattern. The satellite rotates around its spin axis at a rate of 60 arcsec/s, equivalent to a spin period of 6 hours. The spin axis itself precesses at a fixed angle of 45 degrees to the Sun. The line of sight of the two astrometric instruments are separated by the 'basic angle', which is 106.5 degrees. Astrometric precision 10 μas.

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JASMINE = Japan Astrometry Satellite Mission for INfrared Exploration. It will survey the Milky Way and its bulge in the infrared band around 1 milli-micron, measure positions, distances, and proper motion of several hundred million stars at high accuracy approaching 10 μas. Launch date: 2020÷24.

October 14, 2014

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Square Kilometer Array (SKA)

The SKA will be an interferometric array of individual antenna stations, synthesizing an aperture with a diameter of up to several thousand kilometers. The SKA is a new generation radio telescope that will be 100 times as sensitive as the best present-day instruments. It will unlock information from the very early Universe and, using novel capabilities, be able to undertake entirely new classes of observation including VLBI with a micro-arcsecond resolution.

October 14, 2014

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Mauna Kea

Hawaii Owens Valley

California Brewster

Washington North Liberty

Iowa Hancock

New Hampshire

Kitt Peak Arizona

Pie Town New Mexico

Fort Davis Texas

Los Alamos New Mexico

St. Croix Virgin Islands

October 14, 2014

Colloquium at the University of Mississippi, Oxford, USA

Colloquium at the University of Mississippi, Oxford, USA

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VERA VLBI Exploration of

Radio Astrometry is the first VLBI array dedicated to phase-referencing micro-arcsecond astrometry.

S269 (Sharpless 269) is a massive star forming region toward constellation Orion. VERA has successfully measured its trigonometric parallax of 189 +/- 8 micro-arcsecond. This is the smallest parallax ever measured, corresponding to a source distance to 17,250 light year (~ 5.3 Kpc).

October 14, 2014

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Gravitational Field Model

October 14, 2014

Colloquium at the University of Mississippi, Oxford, USA

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Existing and incoming astrometric facilities demand new approach in theoretical understanding of light propagation through the variable gravitational fields generated by moving, oscillating, and rotating massive bodies as well as the field of gravitational waves.

October 14, 2014

Colloquium at the University of Mississippi, Oxford, USA

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1. Linearized general relativity

2. The harmonic gauge

3. The gravity field equation (c = 1)

hg

02

1 hh

02

2

2

ht

October 14, 2014

Colloquium at the University of Mississippi, Oxford, USA

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Retarded gravitational potentials2

00

0

00

2 2 ( ) ( )...

4 ( ) 2 ( )...

2 ( )...

i ij

i i j

i ij

i j

ij

ij ij

s s

s

M I Ih

r x r x x

s

r

I Ih

r x r

Ih h

r

t

s

s r

( ) ( ) ( ) ( ) ( ) ( )i i ij i j ijP P PI Mx I Mxs s s s x s sJ

the retarded time:

October 14, 2014

Colloquium at the University of Mississippi, Oxford, USA

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Light-ray Propagation

October 14, 2014

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The light-ray perturbation

perturbationunpertrurbednull vector

0

1

2

0

1

2

dKK K

d

dxK k

d

h h h

x x x

dk

d

dh k h k k

d x

The unperturbed equationof light ray

The perturbed equation of light ray

The Christoffel symbols

The wave vector decomposition

The light-ray geodesic

October 14, 2014

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The unperturbed light-ray trajectory

22

)(

dr

kx iiiN

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Light-ray Deflection Angle

October 14, 2014

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The light-ray deflection angle

jpipjipjipjijp

iQ

ijj

jijij

iD

iiM

iQ

iD

iM

iSun

i

ij

jii

ji

jppjii

ii

i

nmmnmmmmnnnnd

sI

nd

sIkmmnn

d

sI

nd

M

d

dxkk

hkkkhkhkd

dhkk

d

xd

3

2

002

2

)(4

)(4)(4

14

2

1

2

1

2

1

Time argument is the retarded time: s = t - r

Gravitational field of a moving planet is localized on null cone and interacts with light with retardation.

October 14, 2014

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The deflection equations and the central inverse mapping

R

Mvk

d

R

mmznznmznzd

L

mmsnsnmsnsd

RJ

mmznnzd

L

n

P

Q

D

M

14

))((2)()(

))((2)()(

)()(

cos1

limb

limb

222

2

222

2

2

October 14, 2014

Colloquium at the University of Mississippi, Oxford, USA

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Snapshot deflection patterns

Monopole

Dipole

Quadrupole

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Dynamic deflection patterns

Circle Cardioid Cayley’s sextic

0

2

cos2

X

Mr

r

0

2cos1

X

Lrp

p

2

02

cos33cos

X

Lrq

q

L

L

March 21, 1988Treuhaft & LoweDSN JPL NASA

September 8, 2002Fomalont & KopeikinVLBA+MPfRA

Not measured yet(SIM, SKA, Gaia,JASMINE, VERA?)

October 14, 2014

Colloquium at the University of Mississippi, Oxford, USA

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Gravitomagnetism and the speed of gravity

October 14, 2014

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Gravitomagnetism

GRAVITOMAGNETIC FIELD arises from moving masses just as a magnetic field arises from moving electric charges.

The gravitoelectric potentialThe leading term is U=GM/r.

The gravitomagnetic potentialThe leading term is (v/c)U.

00

2

2h

c

ii hc

A 0

2

4

g h The metric tensor

October 14, 2014

Colloquium at the University of Mississippi, Oxford, USA

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Two types of gravitomagnetic field

Extrinsic (Lorentz-Einstein): caused by translational currents of matter induced by motion of massive bodies in space with respect to observer

October 14, 2014

Intrinsic (Lense-Thirring): caused by rotating currents of matter induced by angular momentum of the massive body

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Post-Newtonian parameter labels time-dependent gravitational effects and characterizes the speed of the respond of the gravitational field to the positional changes of a massive body. We call it the “speed of gravity” parameter

Hence,

The speed of gravity is “the speed of light” entering the gravity sector of the fundamental interactions.

/ εgc c

Speed-of-gravity Parameterization of Gravitomagnetism

Gravity Fields

Gauge condition

Einstein’s Field Equations

October 14, 2014

εg

c

c

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Gravitational Time Delay

October 14, 2014

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Gravitational Time Delay

October 14, 2014

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Extrinsic gravitomagnetic force on a test particle

October 14, 2014

2extrinsicgm noise2 2

extrinsicgm

2 2

2 2 2

these terms vanish in the field of a ro

41

4

4 1 1 3 4

2

d

dt c c

c

c t c t c c cc c t

v vv v F F

F v A

A v v v vA

tating mass being at rest

Massive body must move wrt observer to generate the extrinsic GM. How to measure it?

USE PHOTONS ! For photons that amplifies the PN termcv k

extrinsicgm noise

"Newtonian" force

extrinsicgm 2

post-Newtonian force of the order of V/

s depending on v/c = O(1)

2 4

2 4 14 4

2c

dcdt

c t c t c

kk k F F

AF k A k k k A

2 2

2

2

post-Newtonian force of the order of V /c

t

Colloquium at the University of Mississippi, Oxford, USA

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Parameterized Time Delay Equation

Kopeikin S. (2004) Class. Quant. Grav., 21, 3251 Kopeikin S. (2006) Int. J. Mod. Phys. D, 15, 305 Kopeikin S. & Fomalont E. (2006) Found. Phys., No. 1, pp. 1 - 42Kopeikin & Makarov (2007) Phys. Rev. D, 75, 062002

1

0 0

1 0 1 0 1 0 0 0

1 0

( )

1| | ( , ) ( ) ( )

( , )1( , ) ( , ( )) 1

2N

N

t t

Nt t

t t t t t c t tc

ht t dt k k h t t d k k

x x

x x x x k

xx

October 14, 2014

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Gravitational Time Delay by a moving body

October 14, 2014

00 0

0 0 0 0

1 11 0 3

22 4

| ( ) | | ( ) | | ( ) |

photon: ( ) ( ) massive body: ( ) ( )

| ( )1( , ) 2 1 ln

ijij i

g

N

g

GMGM GMh h h

t t t c

t c t t t t t

sGMt t

c c

v

x z x z x z

x x x k z z v

x zk v

1 1

0 0 0 0

2 2

1 1 1 1 0 0 0 02 2

1 1 1 1

| ( )

| ( ) | ( )

( ) ( ) | ( ) | ( ) ( ) | ( ) |

1| ( ) |

g g g g

g

s

s s

v vs t t O s t t O

c c c c

s t tc

k x z

x z k x z

v vz z x z z z x z

x z 0 0 0 0

1 | ( ) |

g

s t tc

x z

Look like a retarded time

Colloquium at the University of Mississippi, Oxford, USA

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The idea of the speed-of-gravity experiment

October 14, 2014

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The Minkowski diagram of the light-gravity field interaction

Leonid observes.

Leonid’s world line

Kip’s world line

Planet’s world line

Future gravity null cone

Future gravity null cone

Future gravity null cone

Future gravity null cone

Future gravity null coneLight n

ull co

ne

Light n

ull co

ne

Kip emits light

October 14, 2014

Colloquium at the University of Mississippi, Oxford, USA

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The null cones for gravitational field and light

Observer and planet are at rest Planet moves uniformly relative to observer

October 14, 2014

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Jovian 2002 and Cronian 2009 experiments

October 14, 2014

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The Jovian 2002 experiment

Position of Jupiter taken fromthe JPL ephemerides

Position of Jupiterdetermined from thegravitational deflectionof light by Jupiter

The retardation effect was measured with 20% of accuracy, thus, proving that the null cone for gravity and light coincides (Fomalont & Kopeikin 2003)

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

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Edward B. Fomalont(observation, data processing)

Sergei M. Kopeikin(theory, interpretation)

The speed-of-gravity experiment (2002)

VLBA support: NRAO and MPIfR (Bonn)

October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA

Albuquerque 2002

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Basic Interferometry

(in one minute)

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Limitations to Positional Accuracy• Location of Radio Telescope Position on earth (1 cm) Earth Rotation and orientation (5 cm)• Time synchronization (50 psec)• Array stability (5 cm)• Propagation in troposphere and ionosphere Very variable in time and space (5 cm in 10 min) CONVERSION FACTORS for astrometry: 1 cm = 30 psec = 300 microarcsec 0.03cm = 1 psec = 10 microarcsec

Phase-referencing VLBI technique can achieve 10 microarcsec!

October 14, 2014

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Interpreting the speed-of-gravity experiment

October 14, 2014

Kopeikin & Fomalont - gravity sector of GR is compatible with SR

speed of gravity = speed of light [ = 1 ]

gravitomagnetic (velocity-induced) field of moving Jupiter

1. Will – aberration of light (radiowaves) from the quasar

2. Asada, Carlip – speed of light (radiowaves) from the quasar

3. Nordtvedt – retardation of radio waves from the quasar in Jovian’s magnetosphere

4. Pascual-Sanchez – the Römer delay of light (already known since 1676)

5. Samuel – retardation of radio waves emitted by Jupiter itself

6. Van Flandern – the quantity measured was already known to propagate at the speed of light

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Light Deflection Experiment with Saturn and Cassini spacecraft as a calibrator

(Proc. IAU Symp. 261, 2009)

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Cassini Gravitomagnetic Experiment

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Gravitomagnetic Field in the Cassini Experiment(Kopeikin et al., Phys. Lett. A, 2007)

Gravitomagnetic Doppler shiftdue to the orbitalmotion of the Sun

Bertotti-Iess-Tortora, Nature, 2004

However, the gravitomagnetic contribution was not analyzed

51 (2.1 2.3) 10

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Gravitational time delay in the ODP code

October 14, 2014

1 2 12Cassini-Earth 3

1 2 12

1 1 1 2 2 2

The linearized w.r.t. v/c time delay equation can be

re-formulated as follows ( arXiv:0809.3433)

12 1 ln

( ) ( )

Kopeikin

R R RGM

c R R Rc

t t

k v

R x z R x z 12 1 2

1 0 1 0 2 0 2 0

=| |

( ) v( ) ( ) ( )

Notice that velocity of the light-ray deflecting body

enters the argument of the logarithm in the time delay.

R

t t t t t t

R R

z z z z v

v

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1. Cassini solar conjunction experiment has a potential to detect the gravitomagnetic field of the moving Sun directly!

2. It requires re-processing of the data 3. The announced value for is based on the implicit assumption

that the gravitomagnetic deflection of light agrees with GR, but this assumption was not tested.

Numerical Estimates for Cassini Doppler Shift

• The peak value of the Doppler shift is caused by orbital motion of Earth and reaches .

• R.M.S. error of the measurements is • Doppler shift due to the orbital motion of Sun is • The value of (-1) would be affected by the solar

motion by the amount if the gravitomagnetic deflection of light were not in accordance with GR

10106 14101

13109.2

51 (2.1 2.3) 10

4102.1

October 14, 2014

Conclusions

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PROGRESS IN MEASUREMENTS OF THE GRAVITATIONAL BENDING OF RADIO WAVES

USING THE VLBA

October 14, 2014

E. Fomalont, S. Kopeikin, G. Lanyi, and J. Benson The Astrophysical Journal, 699, 1395 (2009)

γ = 0.9998 ± 0.0003

October 2005

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Pioneer Anomaly: Local measurement of the Hubble constant?

October 14, 2014

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Heat recoilexplanationof the Pioneeranomaly

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Background metric

October 14, 2014

Standard assumption of gravitational experimental physics is that spacetime is asymptotically flat

where t is the proper time measured by static observers.In fact, we live in the expanding universe described on all scales by the Robertson-Walker metric

where t is the proper time measured by the Hubble observers.

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Local Diffeomorphism

October 14, 2014

We introduce the conformal time:

where .It reduces the RW metric to the conformally-flat form:

Now, we look for a local diffeomorphism reducing the RW metric to the Minkowski metric:

which means

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Special Conformal Transformation

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Expand the scale factor,

and substitute it to the local diffeomorphism . Compare with the Taylor expansion of the special conformal transformation w.r.t. vector . It yields

Local Minkowski coordinates are defined by the special conformal transformation

Local Minkowski Coordinates

October 14, 2014

where t is the proper time measured by the Hubble observer.

The Minkowski time coordinate is not the proper time except for the time-like world line

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The Christoffel symbols are nil in the local Minkowski coordinates. According to EEP any test particle moves along a geodesic which are straight lines

One can prove that on photon’s worldline (but remember that is not a proper time of observer).We want to parameterize the geodesic with the proper time t measured by the observer along her/his worldline:

Einstein’s principle of equivalence

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Motion of light in local coordinates

October 14, 2014

EEP, applied to a conformal manifold, tells us that a freely-moving particle experiences a geometric (Finsler-type) force because for a particle moving with the velocity v

In particular, equation of motion of photons in the local coordinates in cosmology

Light (in local coordinates) moves non-uniformly!

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Doppler shift

October 14, 2014

𝑃1

𝑃2Emitter’s world line

Receiver’s world line𝜔1

𝜔2

�⃗�

𝑃0

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Doppler shift

October 14, 2014

Frequency of radio waves:

Doppler shift:

Light-ray trajectory:

Observer’s proper time:

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Time derivatives

October 14, 2014

Relation of the proper time of moving clocks to the cosmic time:

Light-ray path:

Relation of the cosmic time at the point of emission to that at the point of observation

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Doppler tracking experiment

October 14, 2014

Doppler shift equation:

predicts gravitational blue shift of frequency for static observers in cosmology:

Pioneer anomaly may have a cosmological explanation!

+ _

Doppler shift for distant quasarsDoppler shift for local (static) observers

Δ𝜔𝜔1

=∑𝑖=1

𝑁 𝛿𝜔𝑖

𝜔1

=𝐻 (𝑡𝑁−𝑡 1 )Integrated Doppler shift: has the same sign and

magnitude as the Pioneer anomaly.

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Thank you!

October 14, 2014