Spacetime astrometry
and
gravitational experiments
in the solar system
Sergei Kopeikin
University of Missouri
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
1
Abstract
Astrometry 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, ultraprecise 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
Colloquium at the University of Mississippi,
Oxford, USA
2
Contents
1. Astrometric Experiments
2. Gravitational Field Model
3. Light-ray Propagation
4. Light-ray Deflection Angle
5. Gravitomagnetism and the speed of gravity
6. Gravitational Time Delay
7. The idea of the speed-of-gravity experiment
8. Jovian 2002 and Cronian 2009 experiments
9. Cassini gravitomagnetic experiment
10. “Pioneer anomaly” - Local measurement of
the Hubble constant?
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
3
Astrometry in Space
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
4
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!
October 14, 2014
Colloquium at the University of
Mississippi, Oxford, USA
5
GAIA
October 14, 2014
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.
Colloquium at the University of
Mississippi, Oxford, USA
6
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
Colloquium at the University of
Mississippi, Oxford, USA
7
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
8
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.
Colloquium at the University of
October 14, 2014
Mississippi, Oxford, USA
9
Mauna Kea
Hawaii
at the University of Mississippi,
Oxford, USA
October 14, 2014
Kitt Peak
Arizona
Owens Valley
California
Brewster
Washington
North Liberty
Iowa
Hancock
New Hampshire
Pie Town
New Mexico
Fort Davis
Texas
Los Alamos
New Mexico
St. Croix
Virgin Islands
10
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
Colloquium at the University of
Mississippi, Oxford, USA
11
Gravitational Field Model
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
12
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
13
1. Linearized general relativity
g      h
2. The harmonic gauge
h


1
2


h  0
3. The gravity field equation (c = 1)
2 
 2
  2    h  0
 t



October 14, 2014
Colloquium at the University of
Mississippi, Oxford, USA
14
Retarded gravitational potentials
h00 
i
2
 I  ij  ( s ) 
  2 I (s) 



  ...
i 
i
j 
x  r  x x 
r

2M
r

   ij 

4 I (s)

2I
(s)

  ...


j

r
x 
r


i
h0 i

2I
hij   ij h00 
 ij 
I (s)  M xP (s)
October 14, 2014
 ...
r
the retarded time:
i
(s)
i
str
I
 ij 
i
j
( s)  M xP ( s) xP ( s)  J
Colloquium at the University of
Mississippi, Oxford, USA
 ij 
(s)
15
Light-ray Propagation
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
16
The light-ray perturbation
dK
The light-ray geodesic
The unperturbed equation
of light ray
The perturbed equation
of light ray
October 14, 2014

K




dk


dx
0


d
1


k


2

p ertu rb atio n
u n p ertru rb ed
n u ll vecto r


 h 
 h  
  h 





 

x

x

x



d
d

   K K
d
The wave vector decomposition
The Christoffel symbols


d
0

 h
Colloquium at the University of
Mississippi, Oxford, USA

k


1
2



x

h


k k

17

The unperturbed light-ray trajectory
x N ( )  k   
i
r 
October 14, 2014
Colloquium at the University of
Mississippi, Oxford, USA
i
 d
2
i
2
18
Light-ray Deflection Angle
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
19
The light-ray deflection angle
2
d x
d
i
2
1


 h

k k

2
     k k j 
i
i
j
i
d  
1 i
1 i j p

 k h i   k h 00  k k k h jp 
d 
2
2


i
j
dx
d
 
i
   Sun   M   D   Q
i
i
M 
i
i
4M
i
i
1   n i
d

j

i
D

d
Q 
i
4 I (s)
4I
2
 jp 
d
3
n n
(s)
i
j
m m
n n
October 14, 2014
i
i
j
j

j
j
4k I (s)
n
Time argument is the retarded time: s = t - r
i
d
n n m m m m n m m n
p
i
j
p
i
j
p
i
p
j

Gravitational field of a moving planet is
localized on null cone and interacts with
light with retardation.
Colloquium at the University of
Mississippi, Oxford, USA
20
The deflection equations and the central inverse mapping

M
  1  cos  n

D

L
d
 Q  J 2
 
( z  n ) n  ( z  m ) m 
R
2
d
2
L
2
d
2
( s  n )
( z  n )
2
2

 ( s  m ) n  2 ( s  n )( s  m ) m
2

 ( z  m ) n  2 ( z  n )( z  m ) m
2


R
   limb
d

 limb  4 1  k  v P
October 14, 2014
 MR
Colloquium at the University of
Mississippi, Oxford, USA
21
Snapshot deflection patterns
Monopole
Dipole
Quadrupole
October 14, 2014
Colloquium at the University of
Mississippi, Oxford, USA
22
Dynamic deflection patterns
Circle
  2 r cos 
r 
2M
X0
March 21, 1988
Treuhaft & Lowe
DSN JPL NASA
October 14, 2014
Cardioid
  p 1  cos 2 
pr
L
X0
September 8, 2002
Fomalont & Kopeikin
VLBA+MPfRA
Colloquium at the University of
Mississippi, Oxford, USA
Cayley’s sextic
  q L cos 3  3 cos  
r L 

q L  
2  X 0 
2
Not measured yet
(SIM, SKA, Gaia,
JASMINE, VERA?)
23
Gravitomagnetism and the
speed of gravity
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
24
Gravitomagnetism
GRAVITOMAGNETIC FIELD arises from
moving masses just as a magnetic field
arises from moving electric charges.
g        h 
 
c
2
h 00
2
Ai  
October 14, 2014
c
2
4
h0 i
The metric tensor
The gravitoelectric potential
The leading term is U=GM/r.
The gravitomagnetic potential
The leading term is (v/c)U.
Colloquium at the University of Mississippi,
Oxford, USA
25
Two types of gravitomagnetic
field
Intrinsic (Lense-Thirring):
caused by rotating
currents of matter
induced by angular
momentum
of the massive body
October 14, 2014
Extrinsic (Lorentz-Einstein):
caused by translational
currents of matter induced
by motion of massive
bodies in space with
respect to observer
Colloquium at the University of Mississippi,
Oxford, USA
26
Speed-of-gravity Parameterization of Gravitomagnetism
Post-Newtonian
parameter  labels timedependent 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 c g  c / ε
Gravity Fields
Gauge condition
Hence,
c
cg
The speed of gravity is
“the speed of light”
entering the gravity sector
of the fundamental
interactions.
Einstein’s Field Equations
October 14, 2014
ε
Colloquium at the University of Mississippi,
Oxford, USA
27
Gravitational Time Delay
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
28
Gravitational Time Delay
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
29
Extrinsic gravitomagnetic force on a test
particle
2

v 
4
 1  2     2 v  v   
dt
c 
c

dv
ex trin sic
Fg m

4
c

ex trin sic
Fg m
 F n o ise
v    A 
2
2

v   1  
1 
 v
 v
 v

  3  2  
 
 A 


  4
2
2
c t
c

t
c
c
c
c
2
c

t






4 A
th ese term s van ish in th e field o f a ro tatin g m ass b ein g at rest
M assive b o d y m u st m o ve w rt o b server to g en erate th e ex trin sic G M . H o w to m easu re it?
USE PHOTONS !
c
dk
dt
F o r p h o to n s v  c k th at am p lifies th e P N term s d ep en d in g o n v/c = O (1 )
 2   4k  k   

ex trin sic
Fg m
 F n o ise
"N ew to n ian " fo rce
F
ex trin sic
gm
 4k     A 
 2 
k 
 4 k  
 c t
p o st-N ew to n ian fo rce o f th e o rd er o f V / c
October 14, 2014

 k  A 


4 A
c t


1
2c
2
2
t
2

p o st-N ew to n ian fo rce o f th e o rd er o f V
Colloquium at the University of Mississippi,
Oxford, USA
2
/c
2
30
Parameterized Time Delay Equation
t1  t 0 
1
c
 ( t1 , t 0 ) 
| x 1  x 0 |   ( t1 , t 0 )
1
2

t1
t0
x N (t )  x 0  c k (t  t0 )



h
(

,
x
)
t




 
 
dt  k k h  ( t , x N ( t ))     1   d  k k 


t0


 x  x N (  ) 

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 - 42
Kopeikin & Makarov (2007) Phys. Rev. D, 75, 062002
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
31
Gravitational Time Delay
by a moving body
h00 
2G M
hij 
| x  z (t ) |
photon: x
2 G M  ij
| x  z (t ) |
x N (t )  x 0  ck (t  t0 )

GM 
1
 ( t1 , t 0 )  2 3  1 
k  v  ln


c 
cg

m assive bo dy: z ( t )  z 0  v ( t  t 0 )
 | x 1  z ( s1 ) |  k   x 1  z ( s1 )  


|
x

z
(
s
)
|

k

x

z
(
s
)


 0

0
0
0
 v2 
z ( s1 )  z ( t1 ) 
| x 1  z ( t1 ) |  O  2 
c 
cg
 g 
v
h0 i
 v 




| x  z ( t ) |  c g 
4G M
 v2 
z ( s0 )  z (t0 ) 
| x 0  z (t0 ) |  O  2 
c 
cg
 g 
v
Look like a retarded time
s1  t1 
1
cg
October 14, 2014
| x 1  z ( t1 ) |
s0  t0 
Colloquium at the University of Mississippi,
Oxford, USA
1
cg
| x 0  z (t0 ) |
32
The idea of the speed-of-gravity
experiment
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
33
The Minkowski diagram of the light-gravity field interaction
Leonid observes.
Kip’s world line
Future gravity null cone
Future gravity null cone
Future gravity null cone
Future gravity null cone
Future gravity null cone
Kip emits light
October 14, 2014
Planet’s world line
Colloquium at the University of
Mississippi, Oxford, USA
Leonid’s
world line
34
The null cones for gravitational field and light
Observer and planet are at rest
October 14, 2014
Planet moves uniformly relative to observer
Colloquium at the University of
Mississippi, Oxford, USA
35
Jovian 2002 and Cronian 2009
experiments
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
36
The Jovian 2002 experiment
Position of Jupiter taken from
the JPL ephemerides
Position of Jupiter
determined from the
gravitational deflection
of light by Jupiter
10 microarcseconds = the width of a typical
strand of a human hair from a distance of
650 miles!!!
October 14, 2014
The retardation effect was measured with 20% of
accuracy, thus, proving that the null cone for gravity
and light coincides (Fomalont & Kopeikin 2003)
Colloquium at the University of
Mississippi, Oxford, USA
37
The speed-of-gravity
experiment (2002)
Edward B. Fomalont
(observation, data processing)
Sergei M. Kopeikin
(theory, interpretation)
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
Albuquerque 2002
VLBA support: NRAO and MPIfR (Bonn)
38
Basic Interferometry
(in one minute)
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
39
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
Colloquium at the University of Mississippi,
Oxford, USA
40
Interpreting the speed-of-gravity experiment
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
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
41
Light Deflection Experiment with Saturn
and Cassini spacecraft as a calibrator
(Proc. IAU Symp. 261, 2009)
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
42
Cassini Gravitomagnetic Experiment
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
43
Gravitomagnetic Field in the Cassini Experiment
(Kopeikin et al., Phys. Lett. A, 2007)
Gravitomagnetic
Doppler shift
due to the orbital
motion of the Sun
Bertotti-Iess-Tortora, Nature, 2004
  1  (2.1  2.3)  10
5
However, the gravitomagnetic
contribution was not analyzed
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
44
Gravitational time delay in the ODP code
T h e lin earized w .r.t. v/c tim e d elay eq u atio n can b e
re-fo rm u lated as fo llo w s ( K o p eikin arX iv:0 8 0 9 .3 4 3 3)
 C assin i-E arth
 R1  R 2  R1 2 
GM 
1

 2

 1  k  v  ln 
3
c
R

R

R
c 

2
12 
 1
R 1  x 1  z ( t1 )
R 2  x 2  z (t2 )
z ( t1 )  z 0  v ( t1  t 0 )
R1 2 = | R 1  R 2 |
z ( t2 )  z 0  v ( t2  t0 )
N o tice th at velo city v o f th e lig h t-ray d eflectin g b o d y
en ters th e arg u m en t o f th e lo g arith m in th e tim e d elay.
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
45
•
Numerical Estimates for Cassini Doppler Shift
The peak value of the Doppler shift is caused by
orbital motion of Earth and reaches 6  10 .
R.M.S. error of the measurements is  1  10
Doppler shift due to the orbital motion of Sun is 2 .9  10
The value of (-1) would be affected by the solar
motion by the amount  1 . 2  10 if the
gravitomagnetic deflection of light were not in
accordance with GR
 10
•
•
•
 14
4
Conclusions
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   1  (2.1  2.3)  10 is based on the implicit
assumption that the gravitomagnetic deflection of light agrees with GR,
but this assumption was not tested.
5
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
46
 13
PROGRESS IN MEASUREMENTS OF THE
GRAVITATIONAL BENDING OF RADIO WAVES
USING THE VLBA
E. Fomalont, S. Kopeikin, G. Lanyi, and J. Benson
The Astrophysical Journal, 699, 1395 (2009)
γ = 0.9998 ± 0.0003
October 2005
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
47
Pioneer Anomaly:
Local measurement of the Hubble
constant?
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
48
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
49
Heat recoil
explanation
of the Pioneer
anomaly
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
50
Background metric
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.
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
51
Local Diffeomorphism
We introduce the conformal time:
where 𝑎 𝜂 ≡ 𝑅 𝑡 𝜂 .
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
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
52
Special Conformal Transformation
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
53
Local Minkowski Coordinates
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 𝑏𝛼 . It yields
Local Minkowski coordinates are defined by the special conformal transformation
where t is the proper time measured by the Hubble observer.
The Minkowski time coordinate 𝒙𝟎 is not the proper time except for the timelike world line 𝒚𝒊 = 𝟎 𝐨𝐫 𝒙𝒊 = 𝟎 .
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
54
Einstein’s principle of equivalence
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 𝜎 = 𝑥 0 on photon’s worldline (but
remember that 𝑥 0 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:
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
55
Motion of light in local coordinates
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
1
0
𝑥 = 𝑡 + 𝐻𝑣 2 𝑡 2
2
In particular, equation of motion of photons in the
local coordinates in cosmology
Light (in local coordinates) moves non-uniformly!
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
56
Doppler shift
𝜔2
Emitter’s world line
𝑃2
𝑘
𝜔1
𝑃1
Receiver’s world line
𝑃0
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
57
Doppler shift
Frequency of radio waves:
Doppler shift:
Light-ray trajectory:
Observer’s proper time:
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
58
Time derivatives
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
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
59
Doppler tracking experiment
Doppler shift equation:
predicts gravitational blue shift of frequency
for static observers in cosmology:
_
+
Doppler shift for local (static) observers
Integrated Doppler shift:
Δ𝜔
=
𝜔1
Doppler shift for distant quasars
𝑁
𝛿𝜔𝑖
= 𝐻 𝑡𝑁 − 𝑡1
𝑖=1 𝜔1
has the same sign and
magnitude as the Pioneer anomaly.
Pioneer anomaly may have a cosmological explanation!
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
60
Thank you!
October 14, 2014
Colloquium at the University of Mississippi,
Oxford, USA
61