Transcript Radiointerferometric Measurement of the Speed of

Radiointerferometric Measurement
of the Speed of Gravity
Sergei Kopeikin
Department of Physics and Astronomy
University of Missouri, Columbia, USA
Colloquium at the Lawrence
Berkeley National Laboratory
(California, Berkeley, 05/05/05)
359th WE-Heraeus-Seminar
Lasers,Clocks, and Drag-Free
(ZARM, Bremen, 05/30/05)
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Agenda
•
•
•
•
General relativity and linearized gravity
The Lienard-Wiechert retarded potentials
The Shapiro time delay
The speed and aberration of gravity and
light
• The Jovian deflection experiment on the
speed of gravity
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General Relativity
Created by Albert
Einstein in 1915.
Widely accepted
after triumphal
experimental
confirmation by
Sir A. Eddington’s
expedition which
observed the
gravitational
deflection of starlight
in complete (~50%)
agreement
with general relativity.
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D. Hilbert
H.A. Lorentz
W. De Sitter
A. Eddington
M. Planck
P.A.M. Dirac
A.A. Friedmann
A.F. Ioffe
L.D. Landau
V.A. Fock
H. Weyl
T. Levi-Civita
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Basic Geometric Variables:

g (t , x )
• The Metric Tensor
• The Affine Connection

  (t , x )

• The Curvature Tensor


R
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


x







x



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








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Einstein’s Theory of Gravity
• The metric tensor  gravitational potential(s)
• The affine connection  the gravity force
• The Principle of Equivalence  (the covariant derivative )



1   g  g  g  
 g       
2
x
x 
 x
•The Gravity Field Equations

R

1 
8 G 
  R  4 T 
2
c



G

0


T

 0


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Matter tells space-time
how to curve.
Space-time tells matter
how to move.
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Linearized General Relativity


g    h

1   h h h 
        
2
x
x 
 x
The Harmonic (Lorentz) gauge
h 
x
1 h 

0

2 x
Linearized Einstein’s Gravity Field Equations
 1 2
16 G  
1  
2 
  2 2    h    4 T    T 
c 
2

 c t

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The origin of the fundamental constants
VLBI,
LIGO
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Bradley,
MichelsonMorley
Super-colliders,
Early universe
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What is the “speed of light”?
gravity
weak
G, c
𝛼𝑤 , c
c
𝛼𝑠 , c
electromagnetic
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strong
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The Liénard-Wiechert Retarded
Gravitational Potentials
T

t, x    m 
N
a 1

h


u
 t  ua  t   (3)  x  x a (t ) 
4Gm u u  1 2 
t, x   2

c
u r
 

r  x  xa ( s )
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a
1 
a



1
s  t  x  xa  s 
c
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The Liénard-Wiechert
Gravitational Potentials
The Liénard-Wiechert
Electromagnetic Potentials
Observer
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Observer
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Question:
how to measure the speed of gravity?
Proposal 1: observe propagation of electromagnetic
versus gravitational waves and compare their
speeds of propagation
Proposal 2: observe propagation of electromagnetic waves
in time-dependent gravitational field and compare
the retardation of gravity effect against that for light
Experimental Technique:
1.
Gravitational Wave Detectors
2.
Very Long Baseline Interferometry
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Propagation of Light: Static Gravitational Field and Static
Observer
1915 - optics
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1964 - radio
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The Shapiro Time Delay in the Static Field
Eikonal Equation:
g

 
0


x x
Electromagnetic Phase:
(1   )Gm  xE xP 
   0  k x 
ln  2 
2
c
 D 

PPN-parameter
I.I. Shapiro et al.
PRL, 26, No 18, 1132-35 (1971)
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Berotti-Iess (2004): -1 = 0.00002
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Measuring Position of Gravitational Lens without Its Direct
Observation in Optics/Radio
of the light deflection
Errors in measuring the
light deflection
Limit the precision of
measuring the gravitational
position of the lens
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Gravity displaces the true position
of each star in a radial direction
out of a point-like gravitational
lens (Sun, Jupiter). This property
of gravity can be used in order to
determine position of the
“gravitational image” of the lens
in the sky without observing the
lens in optics/radio! (Kopeikin
2001).
Vector field of the gravitational
force deflecting the light
converges to the point where the
lens’ center-of-gravity is located.
Thus, measuring the deflection of
light (time delay) one can
determine gravitational position
of the lens in the sky without its
direct observation in optics/radio
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The Aberration of Light and Gravity
A stargazer moving in flat space-time
with speed v sees positions of all stars
shifted at the angle of aberration v/c.
This is because light propagates with
fundamental speed of c and the
electromagnetic field is Lorentzinvariant.
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The angle of the aberration
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How does the gravity
force aberrate ?
Is the speed of the Lorentz
transformation for gravity
the same as for light ?
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The Lorentz Invariance of Gravity and Light
Optical position of the Sun
Gravitational position of the Sun
measued from the VLBI time delay
A picture of the solar eclipse in a geocentric frame with the optical positions of stars, and the
optical and gravitational positions of the Sun – both shifted due to the aberration (the Lorentz
transformation). The aberration is the same for both light and gravity in Einstein’s general
relativity theory because their fundamental speeds are the same BUT we need to prove it
experimentally !
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The Aberration of Gravity and its Fundamental Speed.
Gravitational deflection of light to determine position of grav. lens.
Position of the quasar deflected by the gravity force and shifted by the
aberration of light. The speed of gravity is equal to the speed of light.
Gravitational position
of the Sun in the
plane of the sky
measured in the
geocentric reference
frame in the case
when the speed of
gravity is infinite.
The aberration of gravity
angle viewed from the Earth
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Position of the quasar deflected by the
gravity force and shifted by the aberration
of light. The speed of gravity is infinite.
Position of the quasar shifted by
the aberration of light only. Gravitational
deflection of light is switched off.
Gravitational position of the Sun in the plane
of the sky measured in the geocentric
reference
frame in the case when the speed of gravity
equals the speed of light.
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The Aberration of Gravity and its Fundamental Speed
The aberration of gravity effect is obtained
after solving the triangles shown in the picture

Gravitational
deflection of
light  = 4Gm
  g




𝛿𝛼 =
2
c d

4𝐺𝑚 𝑟
𝑐2𝑑 𝑑
1−
𝑣
The impact
𝑐 parameter
𝑐
𝑐𝑔
of the light ray  = d/r
r – the distance ``lens-observer”
d – the impact distance of the
light ray
The gravity aberration angle  g  v / c g
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x J ( s)  x J (t )  v J (t )( s  t ) 18 ...
The Aberration versus Retardation of Gravity
dK
Gm  K  u


 
 F    K K   1    2
2
c
r
d
R

r  x  xJ ( s )
rR  u r 
The retarded time of the gravity field


k
u
Gm     


 r 
K  k  1    2
k   k u  
 
c  k r  
rR 
(1   )Gm


   0  k x 
(
k
u
)
ln

k
r




2
c

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
 2
 r  
 u 
 rR

1
s  t  | x  x J (s) |
c
Lorentz invariant equation
for the bending of light
Lorentz invariant equation
for the electromagnetic phase
delay
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Measuring the Fundamental Speed of Gravity with VLBI
The basic idea:
Observer (VLBI antenna)
at time t and position x
1.
2.
3.
Measure the gravitational
deflection of quasar’s light by
moving lens (Jupiter, Sun);
Use this information to find
out the (retarded) position of
the lens from which it deflects
the light ;
Compare the retarded position
of the lens obtained from the
gravitational time delay with
that calculated from the JPL
ephemeris with light timedelay taken into account.
Jupiter’s world line
Null cone
Jupiter’s ephemeris position
st
1
x  x J ( s)
c
Jupiter’s gravitational position
measured from VLBI time delay
x J (s)
x J (s)
Quasar
st
1
x  x J ( s)
cg
The Jovian
Experiment
Edward Fomalont
(observation +
data processing)
Sergei Kopeikin
(theory + proposal)
(with a gracious support of NRAO, MPIfR,
and the correlator technicians)
The Experiment
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Motion of Jupiter
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The Basic Concept of VLBI Measurement
• The radio signals from
observed source are digitized
and recorded on magnetic
tape at the antennas.
• The recorded tapes are
packed and shipped to the
correlator.
• The tapes are mounted on
the playback drives, are
synchronized, and then played
back in unison.
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•The correlator mathematically
recombines all of the data
from each antenna to simulate
a single antenna up to 8000
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Conferences, Long Island, Bahamasmiles in diameter.
Basic Interferometry
(in one minute)
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The Very Long Baseline Array
Mauna Kea
Hawaii
Owens Valley
California
Brewster
Washington
North Liberty
Iowa
Hancock
New Hampshire
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Kitt Peak
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Pie Town
Fort Davis
Los Alamos
St. Croix
The Correlator Side
The photograph at left is the correlator. It is a
special purpose supercomputer designed to
perform the one special task of recombining all of
the astronomical data fed to it by the playback
drives. It is capable of performing about 36 billion
floating point computations per second. Most of
the work of the correlator takes place in specially
designed micro-chips that are very good at
performing mathematical functions called fast
Fourier transforms. The correlator is full of them.
The photograph below is the correlator control
console.
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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:
1 cm = 30 psec = 300 microarcsec
0.03cm = 1 psec = 10 microarcsec
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Source Structure Stability Over Experiment
Source Stability
2 mas
ticks
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Effect of Troposphere
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Two calibrators – phase-referencing technique.
Factor
of 3 increase
in accuracy
using
calibrators
Reconstruction
of the
wave front
with2 accuracy
10 microarcseconds = a human hair seen from a distance 500 miles !
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Measured Delays for Each Source
r
e
s
i
d
u
a
l
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Residual Delays for J0842 Compared on Several Days for a
Few Baselines
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Jupiter’s retarded position from the gravitational time delay (green
points) versus Jupiter’s retarded position from JPL ephemeris
(magenta solid line)
magnetosphere
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Results of Experiment
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_____________________________
Future Experiments
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 Testing higher-order effects beyond Shapiro delay
-- light bending at 1 microarcsec at Jupiter’s limb
 Testing the aberration of gravity effect with the Sun
(October 2005). The effect = 37 mas, accuracy 1%
 Testing the bending of light by Jupiter’s quadrupolar field
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