# GRAVITY PROBE-B by wanghonghx

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```									       GRAVITY PROBE-B
(What shall it measure and what for)

Bartolome Alles, INFN Pisa

High-Energy Astrophysics Journal Club, Pisa, 16 January 2008
1. Excerpt of Einsteinian Gravity (1)

   The presence of matter
influences the spacetime
fabric. The Minkowski flat
metric hmn is converted
into the metric tensor gmn
for a curved spacetime.

   Newtonian gravity now becomes free fall motion in the
bosom of a curved manifold.

   Matter contents is codified in the Energy-Momentum
tensor Tmn . This is a symmetric, divergenceless tensor.
1. Excerpt of Einsteinian Gravity (2)

 The only tensors with two indices (which therefore can be equated
to Tmn by covariance), divergenceless and depending on gmn and
its
first two derivatives are the Einstein-Hilbert
1
G  R  R g 
2
and the metric tensor gmn itself (D. Lovelock ‘71, ‘72). Rmn is the Ricc
tensor and R its trace.
 The field equations for the metric tensor are thus (the proportionality
constant is chosen by comparing the NR limit with the Poisson equa-
tion)
8 GN
G   2 T .
c
 The Kerr metric is valid for the vacuum around a rotating spherical
mass (R.P. Kerr ‘63; M.M. Schiffer et al. ‘73; R.J. Finkelstein ‘75).
1. Excerpt of Einsteinian Gravity (3)

 For small mass M and slow rotation the above Kerr metric takes
the form of Lense-Thirring (J. Lense, H. Thirring ‘18),

 2GN M          2GN M  2 4GN J 2
ds  1  2 d (ct )  1      d r  3 sin  d dct
2                  2

   c r             c r 
2
cr
where J is the angular momentum of the sphere (rotating around
its Z axis).

 When the sphere stops rotating, the above metric turns into the
Schwarzschild solution (for small M and in isotropic coordinates).

 The t-t and r-r terms in the metric have been experimentally tested,
(perihelia advances, light rays bending, gravitational redshift, etc.).
However the t-r term, typical of a rotating system, has never been
verified.
2. A layman’s excursion in Astrophysics (1)

A false colour image of the two
lobes of the Radio Source NGC
Telescope.
2. A layman’s
excursion in
Astrophysics
(2)

A montage showing
successive enlarge-
ments of the above
image by using
Arrays (M.C. Begel-
man et al. ‘84).
2. A layman’s excursion in Astrophysics (3)

The main jet from the galaxy nucleus of NGC 6251.
2. A layman’s excursion in Astrophysics (4)
   AGN possibly harbour very compact giant objects which attract material
that form an accretion disk orbiting on or close to its equatorial plane.
Models predict a disk orthogonal to the angular momentum of the object
(at least within 100 pc) that ejects material in the two opposite directions
perpendicular to the disk giving rise to the jet and counterjet.
   The jets stay well collimated from 10 to 1000 Kpc until they end their trip
colliding against the intergalactic medium and forming the two radiating
heads that advance by ram pressure balance at a velocity much smaller
than the original jet speed (M.J. Rees ‘85).
   It is generally accepted that the lasting good collimation of the jets is due
to the “frame dragging”, one of the consequences of the t-r terms in the
Kerr metric.
   Better resolution has allowed to view the accreting disks of several AGN
(see next page). They display a warping (like that of the brim of Humphrey
Bogart’s hat). It has been advanced that this effect is still another conse-
quence of the t-r term in the metric: the so-called Lense-Thirring (L-T)
precession.
   Some high frequency QPO in neutron stars could be explained by the
periodic passing of the tilted accreting disk (L. Stella, M. Vietri ‘97). The tilt
would be produced again by L-T precession.
2. A layman’s excursion in Astrophysics (5)

NGC 6251                             3C 449

from Hubble web site and 3C 449
from G.R. Tremblay et al. ‘06.

Bogart’s hat
2. A layman’s excursion in Astrophysics (6)
   Another kind of precession, the so-called de Sitter or geodetic precession,
can be responsible for similar effects. This precession is due to the sole
presence of a mass (it needs not rotating). When a companion star is
present, it is generally dominant over the L-T effect.
   The de Sitter precession has been already detected in the Moon-Earth
system, regarded as a gyroscope orbiting in the Sun’s gravitational field
(I.I. Shapiro et al. ‘88; J.G. Williams et al., ‘96) finding agreement with the
theoretical prediction within 1%.

 GPB is aimed at measuring with unprecedent
accuracy the above two precession effects on
gyroscopes orbiting the Earth.

 In the present context a gyroscope is defined
as an object spinning with angular momentum S
around one of its symmetry axes and with no
torques (all forces, if any, act on its center of
mass).
3. de Sitter and Lense-Thirring precessions

 The spin of the gyroscope in its local frame makes a precession under
the law
dS
 S   ,    deSitter   L T
dt
where the angular speeds are

 deSitter   
3 GN M                                
GN J  3r r  J
v  r ,  L T  2
ˆˆ
2 3
2 c r                  c       r3

to lowest order in the Newton constant GN/c 2 .

 These two are the General Relativistic precession effects on gyroscopes.
We shall calculate both expressions by using qualitative arguments.
4. Lense-Thirring precession (1)
 Since we are working at the lowest order, terms proportional to M are
neglected during the calculation. In this case the t-r sector is crucial. There-
upon the metric is no longer static.

 Let us consider a small deviation to the flat Minkowski metric, gmn=hmn+hmn
where for all m,n         |hmn|<<1 (this condition is surely satisfied in the terres-
trial gravitational field). Then the field equations become
  2   2  h    16 GN T , h   h  1 h , h  ,  0
 ct                                        
                       c2                  2
where h is the trace of hmn evaluated with hmn. The last condition derives from
the (gauge) freedom to choose the coordinate set. Famous retarded solutions
to the above equation are graviational waves, Lense-Thirring metric, N celes-
tial bodies metric, etc.
4. Lense-Thirring precession (2)

 Let us consider the following definitions
2 0         1 i
h00   2 A , h0i  2 A .
c           c

 Now define the functions
1
E    t A  A0 , B    A .
c
They satisfy the equations

  B  0 ,   E  4 GN T00 ,
1                              4
  E    t B ,   B  16 GN cT 0i   t E .
c                              c
In the Faraday’s law a higher order term was included for clarity.
4. Lense-Thirring precession (3)
 The gauge fixing condition and the divergence of the Energy-Momentum
tensor yield
4
 t A0    A  0 ,  t     J  0
c
where r and J are naturally defined from the above equations.

 With the adopted notation, Einstein field equations to lowest order look
like Maxwell classical electrodynamics. To understand what plays the role
of electric charge, it is enough to write the equations for the free fall

d2r              
   x x   0
x      
                       E  v  B
dt 2        c    
          
which imply that the role of a charge q is played by the negative of the
mass, q=-m.
4. Lense-Thirring precession (4)

 By use of the above definitions the gravitomagnetic field can be computed
from the Lense-Thirring metric,

B  
                             
 2 GN r  J   2GN J  3 r  J r .
ˆ     ˆ
       cr 3     c       r3
            
For particles with radial velocity the Lorentz force turns out to be propor-
tional to
GN
v  B  v 2 J sin  e .
ˆ
c
This is the frame-dragging effect.
 In the present context a gyroscope is equivalent to a magnetic dipole and,
following Larmor’s theorem, a magnetic field will make it carry out a pre-
cession according to the angular speed
q
B   L T
2mc
since q=-m.
5. de Sitter precession (1)
 The de Sitter precession has two origins. On the one hand it is due to the
very same mechanism by which spin-orbit interactions raise in Atomic
Physics. Move to the proper reference system of the gyroscope and the
gravitoelectric field will become a gravitomagnetic field,

v              GN M
B proper system  4  E  4 v  r 3
c               r c

which by the same mechanism as in the Lense-Thirring case, induces
a precession in the gyroscope. This produces a 4/3 of the total de
Sitter precession.
 Notice that this term has been calculated in the proper frame of the
gyroscope and the additional Thomas correction must also be considered
since gravitational interactions have been treated as usual accelerations,
(the r.h.s. of the Newton equation) instead of spacetime curvature (the
l.h.s).
5. de Sitter precession (2)
   A straightforward calculation of
the Thomas term leads to a con-
tribution which is -1/3 of the total
de Sitter precession. The preces-             A
2
sion angle is dj=-pb             .                B
This angle is seen in the figure.
The spin vector (black arrow) will
get inclined after a complete orbit.
The angular precession velocity is
dj divided by the period of an
orbit.
   The vector product vxr can be
deduced by studying the several
orientations of the black arrow
(vertical, horizontal or pointing to
6. GPB Experiment (1)

L.I. Schiff

C.W.F. Everitt
6. GPB Experiment (2)

   A satellite provided with a telescope
and four gyroscopes was put in a polar
orbit around the Earth at 642 Km alti-
tude from April 2004 to August 2005.
   The telescope constantly pointed to-
wards the star IM Pegasi (HR 8703).
   One of the gyros was used as a drag-
free mass in order to correct the orbit
of the GPB satellite from small disturb-
ances (solar wind, ripples in the outer
atmospheric layers, etc.). The pre-
cession of the other three (for redun-
dancy) gyros are measured.
   de Sitter and L-T precessions were ex-
pected to be measured with precisions
of 0.01% and 1% respectively.
6. GPB Experiment (3)

prism_Y   optical sensor
 The guide star was chosen following the
requirements: (i) never hided by the Sun,
(ii) bright enough for the telescope on
board to detect it, (iii) it must be a radio
source too and (iv) be close to a quasar.
 The gyros are spun up at the beginning
of the mission with their angular momen-
beam splitter                prism_X     tum made to point at the guide star.
 Each gyroscope is a quartz ball coated
with superconducting Nb. Electric charges
applied to three pairs of electrodes keep
the gyros spinning without mechanical
contacts. London magnetic fields created
by the rotating superconducting Nb are
revealed by SQUID’s yielding information
6. GPB Experiment (4)
   The Earth magnetic field (which could greatly disturb the SQUID’s) is
   The fourth gyro is used as a proof mass to correct the satellite trajectory
from tiny (order milli-Newton) external disturbances. The instruments are
dipped into liquid He 4 at 1.8 °K to keep superconducting properties. As He 4
vaporizes, liquid and vapour are being separated by a porous plug and the
vapour used as a propellent for 16 micro-thrusters (sort of pores) that
correct the spacecraft orbit from little deviations.
   Apart from the proper motion of the guide star, data analysis must also
take into account and subtract effects from light aberration, similar relativ-
istic precessions due to Sun, Moon and other planets, Sun oblateness, etc.
   NO RESULTS YET! Unexpected torques acting on the gyroscopes have been
detected. They are likely due to a non-uniform Nb coating which induces
static charges on the gyros and create torques with the electrodes. GPB
team claims to have been able to model such torques and the announce of
final results is expected by May 2008…
6. GPB Experiment (5)
7. LAGEOS Experiment (1)
 LAGEOS (Laser Geodynamics Satellite) was launched in ’76 to study crustal
movement, continental drift, Earth shape (geoid), etc. Its orbit lies at 5900
Km over the Earth surface with an inclination a=109.94° and an excentricity
e=0.004. In ‘92 a second similar satellite (LAGEOS II) was put in orbit with
a different orbital inclination.

 Nodes and perigees are also affected by the L-T dragging. The idea was to
make use of data from these satellites to measure such effects.

LAGEOS satellite

I. Ciufolini
7. LAGEOS Experiment (2)

Orbits of LAGEOS
and LAGEOS II.

 In principle the idea was to compare the shift of the nodes. However the New-
tonian contribution to this shift is about 10 7 times larger than the L-T effect.
7. LAGEOS Experiment (3)

 The Newtonian and L-T contributions are

 R  cos
2
                  2
1  3e 2 / 2    

   Newton      3
  orbit       
5  R 
I 2  I 4       
 7 sin   4
2
                126 / year ,
I
2                 
D  1  e2 2   
         8 D                   
1 e  
2 2


I 2l 
4
4l  1

 r ' Y2l  '  r ' d r ' ,
2l                 3

                2GN J
 L T                         31 mas / year .
I

c D 1 e
2 3

2 3/ 2

The multipoles I2l are poorly known and this fact does not allow to separate
the extremelly small relativistic effect from the dominant Newtonian part.
 One possible solution is to combine data from LAGEOS with data from an-
other (planned) satellite, called LAGEOS III, which would follow a supple-
mentary orbit in such a way that the two Newtonian terms cancel out.

 A third solution consists in combining data from LAGEOS and LAGEOS II in
order to single out the relativistic term (I. Ciufolini et al. ‘04).
7. LAGEOS Experiment (4)
 Indeed the linear combination
 I  c1 II  c2II  60 .2 mas / year
               
does not depend on I2 or I4. c1=0.295 and c2=-0.35 are calculable coeffi-
cients. This, together with the inclusion of a more accurate knowledge of
the non-spherical shape of Earth (by use of the recently launched satellites
CHAMP and GRACE) in order to have a good control of I2l (l>2), makes
possible to extract the L-T effect with much smaller classical uncertainties.

 The reported error on the result of I. Ciufolini has been criticized on the
basis of a misuse of the errors in the CHAMP and GRACE determinations
of I2l (L. Iorio ‘04).

 Moreover one has to subtract other sources of uncertainty: influences from
Moon, Sun and other planets, oceanic and crustal tides, Sun radiance, Earth
albedo, solar wind, atmospheric drag, interplanetary dust, Yarkovsky effect,
etc. There are claims that these errors have been underestimated.
8. Conclusions

1)   Relativistic frame dragging is a very attractive mechan-
ism to explain several observations in high-energy
astrophysics.

2)   However it has never been experimentally verified.
The difficulty lying in the fact that such effects on
Earth are extremely faint.

3)   We have described two experiments devised to test
the frame dragging in the terrestrial gravitational field:
Gravity Probe-B and LAGEOS I & II. To date, system-
atic errors have prevented both experiments from pro-
viding a confident answer to the quest.

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