GRAVITYGRAVITY PROBE-BPROBE-B

(What shall it measure and what (What shall it measure and what for)for)

Bartolome Alles, INFN Pisa

High-Energy Astrophysics Journal Club, Pisa, 16 January 2008

1. Excerpt of Einsteinian Gravity (1)1. Excerpt of Einsteinian Gravity (1)

The presence of The presence of matter influences the matter influences the spacetime fabric. The spacetime fabric. The Minkowski flat metric Minkowski flat metric is converted into the is converted into the metric tensor gmetric tensor g for a for a curved spacetimecurved spacetime..

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

Matter contents is codified in the Energy-Momentum tensor T. This is a symmetric, divergenceless tensor.

1. Excerpt of Einsteinian Gravity (2)1. Excerpt of Einsteinian Gravity (2)

gRRG2

1

The only tensors with two indices (which therefore can be equated to Tby covariance), divergenceless and depending on g and its first two derivatives are the Einstein-Hilbert

and the metric tensor gitself (D. Lovelock ‘71, ‘72). R is the Ricci 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

2

Tc

GG N

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)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),

dctdrc

JGrd

rc

MGctd

rc

MGds NNN 2

3

2

22

22 sin

421)(

21

where J is the angular momentum of the sphere (rotating aroundits 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)2. A layman’s excursion in Astrophysics (1)

A false colour image of the A false colour image of the two lobes of the Radio two lobes of the Radio Source NGC 6251 taken Source NGC 6251 taken from WSRT Radio- from WSRT Radio- Telescope.Telescope.

2. A layman’sexcursion inAstrophysics(2)

A montage showingsuccessive enlarge-ments of the aboveimage by usingRadio TelescopesArrays (M.C. Begel-man et al. ‘84).

2. A layman’s excursion in Astrophysics (3)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)2. A layman’s excursion in Astrophysics (4) AGN possibly harbour very compact giant objects which attract AGN possibly harbour very compact giant objects which attract

material that form an accretion disk orbiting on or close to its material that form an accretion disk orbiting on or close to its equatorial plane. Models predict a disk orthogonal to the angular equatorial plane. Models predict a disk orthogonal to the angular momentum of the object (at least within 100 pc) that ejects momentum of the object (at least within 100 pc) that ejects material in the two opposite directions perpendicular to the disk material in the two opposite directions perpendicular to the disk giving rise to the jet and counterjet.giving rise to the jet and counterjet.

The jets stay well collimated from 10 to 1000 Kpc until they end The jets stay well collimated from 10 to 1000 Kpc until they end their trip colliding against the intergalactic medium and forming their trip colliding against the intergalactic medium and forming the two radiating heads that advance by ram pressure balance at the two radiating heads that advance by ram pressure balance at a velocity much smaller than the original jet speed (M.J. Rees ‘85).a velocity much smaller than the original jet speed (M.J. Rees ‘85).

It is generally accepted that the lasting good collimation of the It is generally accepted that the lasting good collimation of the jets is due to the “frame dragging”, one of the consequences of jets is due to the “frame dragging”, one of the consequences of the t-r terms in the Kerr metric.the t-r terms in the Kerr metric.

Better resolution has allowed to view the accreting disks of several Better resolution has allowed to view the accreting disks of several AGN (see next page). They display a warping (like that of the brim 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 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-still another conse- quence of the t-r term in the metric: the so-called Lense-Thirring (L-T) precession.called Lense-Thirring (L-T) precession.

Some high frequency QPO in neutron stars could be explained by Some high frequency QPO in neutron stars could be explained by the periodic passing of the tilted accreting disk (L. Stella, M. Vietri the periodic passing of the tilted accreting disk (L. Stella, M. Vietri ‘97). The tilt would be produced again by L-T precession.‘97). The tilt would be produced again by L-T precession.

2. A layman’s excursion in Astrophysics (5)2. A layman’s excursion in Astrophysics (5)

NGC 6251 3C 449

Bogart’s hat

NGC 6251 has been downloadedfrom Hubble web site and 3C 449from G.R. Tremblay et al. ‘06.

2. A layman’s excursion in Astrophysics (6)2. A layman’s excursion in Astrophysics (6)

Another kind of precession, the so-called de Sitter or geodetic Another kind of precession, the so-called de Sitter or geodetic precession, can be responsible for similar effects. This precession precession, can be responsible for similar effects. This precession is due to the sole presence of a mass (it needs not rotating). When 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 a companion star is present, it is generally dominant over the L-T effect.effect.

The de Sitter precession has been already detected in the Moon-The de Sitter precession has been already detected in the Moon-Earth system, regarded as a gyroscope orbiting in the Sun’s 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) gravitational field (I.I. Shapiro et al. ‘88; J.G. Williams et al., ‘96) finding agreement with the theoretical prediction within 1finding 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 precessions3. de Sitter and Lense-Thirring precessions

TLdeSitterSdt

Sd ,

The spin of the gyroscope in its local frame makes a precession under the law

where the angular speeds are

3232

ˆˆ3,

2

3

r

JrrJ

c

Grv

rc

MG NTL

NdeSitter

to lowest order in the Newton constant GN/c .

These two are the General Relativistic precession effects on gyroscopes. We shall calculate both expressions by using qualitative arguments.

2

4. Lense-Thirring precession (1)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, g=+h

where for all ,|h|1 (this condition is surely satisfied in the terres- trial gravitational field). Then the field equations become

0,2

1,

16,2

22

hhhhT

c

Gh N

ct

where h is the trace of hevaluated with . The last condition derives fromthe (gauge) freedom to choose the coordinate set. Famous retarded solutionsto the above equation are graviational waves, Lense-Thirring metric, N celes-tial bodies metric, etc.

4. Lense-Thirring precession (2)4. Lense-Thirring precession (2)

.1

,2

200

200i

i Ac

hAc

h

.,1 0 ABAAc

E t

Let us consider the following definitions

Now define the functions

They satisfy the equations

.4

16,1

,4,0

0

00

Ec

cTGBBc

E

TGEB

ti

Nt

N

In the Faraday’s law a higher order term was included for clarity.

4. Lense-Thirring precession (3)4. Lense-Thirring precession (3)

0,04 0 JAAc tt

The gauge fixing condition and the divergence of the Energy-Momentum tensor yield

where 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

B

c

vE

dt

rdxxx

2

2

0

which imply that the role of a charge q is played by the negative of themass, q=-m.

4. Lense-Thirring precession (4)4. Lense-Thirring precession (4)

.

ˆˆ322

33 r

rJrJ

c

G

cr

JrGB N

N

.ˆsin2 eJ

c

GvBv N

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

For particles with radial velocity the Lorentz force turns out to be propor-tional to

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

TLBmc

q

2since q=-m.

5. de Sitter precession (1)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,

cr

MGrvE

c

vB N

systemproper 344

which by the same mechanism as in the Lense-Thirring case, inducesa precession in the gyroscope. This produces a 4 of the total deSitter 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)5. de Sitter precession (2)

A straightforward calculation A straightforward calculation of the Thomas term leads to a of the Thomas term leads to a con- tribution which is -1/3 of con- tribution which is -1/3 of the total de Sitter precession. the total de Sitter precession. The preces-sion angle is The preces-sion angle is . This angle is seen in the . This angle is seen in the figure. The spin vector (black figure. The spin vector (black arrow) will get inclined after a arrow) will get inclined after a complete orbit. The angular complete orbit. The angular precession velocity is precession velocity is divided by the period of an divided by the period of an orbit.orbit.

The vector product vxr can be The vector product vxr can be deduced by studying the deduced by studying the several orientations of the several orientations of the black arrow (vertical, black arrow (vertical, horizontal or pointing to the horizontal or pointing to the reader).reader).

BA

2

6. GPB Experiment (1)6. GPB Experiment (1)

L.I. Schiff

C.W.F. Everitt

6. GPB Experiment (2)6. GPB Experiment (2)

A satellite provided with a telescope A satellite provided with a telescope and four gyroscopes was put in a and four gyroscopes was put in a polar orbit around the Earth at 642 polar orbit around the Earth at 642 Km alti-tude from April 2004 to Km alti-tude from April 2004 to August 2005.August 2005.

The telescope constantly pointed to-The telescope constantly pointed to-wards the star IM Pegasi (HR 8703).wards the star IM Pegasi (HR 8703).

One of the gyros was used as a One of the gyros was used as a drag-free mass in order to correct drag-free mass in order to correct the orbit of the GPB satellite from the orbit of the GPB satellite from small disturb-ances (solar wind, small disturb-ances (solar wind, ripples in the outer atmospheric ripples in the outer atmospheric layers, etc.). The pre- cession of the layers, etc.). The pre- cession of the other three (for redun- dancy) gyros other three (for redun- dancy) gyros are measured.are measured.

de Sitter and L-T precessions were de Sitter and L-T precessions were ex-pected to be measured with ex-pected to be measured with precisions of 0.01% and 1% precisions of 0.01% and 1% respectively.respectively.

6. GPB Experiment (3)6. GPB Experiment (3)

beam splitter prism_X

optical sensor prism_Y The guide star was chosen following the requirements: (i) never hided by the Sun, (ii) bright enough for the telescope onii) bright enough for the telescope on board to detect it, (iii) it must be a radioboard to detect it, (iii) it must be a radio source too and (iv) be close to a quasar. source too and (iv) be close to a quasar.

The gyros are spun up at the beginning of the mission with their angular momen- 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 about their orientations.

6. GPB Experiment (4)6. GPB Experiment (4) The Earth magnetic field (which could greatly disturb the SQUID’s) is The Earth magnetic field (which could greatly disturb the SQUID’s) is

shielded by covering the experimental payload with lead bags.shielded by covering the experimental payload with lead bags. The fourth gyro is used as a proof mass to correct the satellite The fourth gyro is used as a proof mass to correct the satellite

trajectory from tiny (order milli-Newton) external disturbances. The trajectory from tiny (order milli-Newton) external disturbances. The instruments are dipped into liquid He at 1.8 °K to keep instruments are dipped into liquid He at 1.8 °K to keep superconducting properties. As He vaporizes, liquid and vapour are superconducting properties. As He vaporizes, liquid and vapour are being separated by a porous plug and the vapour used as a propellent 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 for 16 micro-thrusters (sort of pores) that correct the spacecraft orbit from little deviations.from little deviations.

Apart from the proper motion of the guide star, data analysis must Apart from the proper motion of the guide star, data analysis must also take into account and subtract effects from light aberration, also take into account and subtract effects from light aberration, similar relativ-istic precessions due to Sun, Moon and other planets, similar relativ-istic precessions due to Sun, Moon and other planets, Sun oblateness, etc.Sun oblateness, etc.

NO RESULTS YET! NO RESULTS YET! Unexpected torques acting on the gyroscopes have Unexpected torques acting on the gyroscopes have been detected. They are likely due to a non-uniform Nb coating which been detected. They are likely due to a non-uniform Nb coating which induces static charges on the gyros and create torques with the induces static charges on the gyros and create torques with the electrodes. GPB team claims to have been able to model such torques electrodes. GPB team claims to have been able to model such torques and the announce of final results is expected by May 2008…and the announce of final results is expected by May 2008…

4 4

6. GPB Experiment (5)

7. LAGEOS Experiment (1)7. LAGEOS Experiment (1)

LAGEOS satellite

I. Ciufolini

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 =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.

7. LAGEOS Experiment (2)7. LAGEOS Experiment (2)

Orbits of LAGEOSand 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 times larger than the L-T effect.7

7. LAGEOS Experiment (3)7. LAGEOS Experiment (3)

The Newtonian and L-T contributions are

./311

2

,''''14

4

,/1261

2/314sin7

8

5

1

cos

2

3

2/3232

32

22

22

22

2

4222

2

yearmaseDc

JG

rdrYrl

I

yeare

e

D

RII

eD

R

NTLI

ll

l

orbitNewtonI

The multipoles Il are poorly known and this fact does not allow to separatethe 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)7. LAGEOS Experiment (4)

yearmascc IIIII /2.6021 Indeed the linear combination

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 ofthe non-spherical shape of Earth (by use of the recently launched satellitesCHAMP and GRACE) in order to have a good control of I2l (l>2), makespossible 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. Conclusions8. Conclusions

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

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

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