1. Gravity Probe B and Relativistic Precession
Presentation by Reece Boston
Oct 19, 2012
2. Introduction to Experiment
The Gravity Probe B Relativity Mission, fifty-year-long effort of
Stanford University, collaboration with NASA and Marshall
Space Flight Center
Experimental preparation started in1960 with Schiff and
Fairbank, spacecraft launched in Aug, 2004, data collection
finished Aug, 2005, data published 2011
Gyroscopes in drag-free satellite pointed at distant star HR8703
Tests two predictions of GR:
Geodetic Precession : Related to curvature of spacetime due
only to mass of earth. Two sources, “Gravitoelectric” just from
potential, and another due to space curvature.
Frame-Dragging : Related to the curvature due to mass-current
of spinning earth. Called “Gravitomagnetic”
Both effects very minute, even in conditions of experiment; order
of arcsecond/year.
This necessitates very accurate measuring devices.
3. Whence Precession?
Result of “fictitious torques” due to curvature of spacetime.
Consider familiar frustration of derivatives in 2D polar
coordinates.
Divergence in Cartesian components is
· F(r) =
∂
∂x
,
∂
∂y
· (Fx , Fy ) =
∂Fx
∂x
+
∂Fy
∂y
.
EASY!
So in polar components, with = ( ∂
∂r , 1
r
∂
∂φ )
· F(r) =
∂
∂r
,
1
r
∂
∂φ
· (Fr , Fφ) =
∂Fr
∂r
+
1
r
∂Fφ
∂φ
.
EASY! but WRONG!
Vectors can only be compared at the same point. Derivative
compares vectors at different points.
4. Whence Precession?
In taking divergence, we have terms of form
lim
→0
ˆn(r + u ) · F(r + u) − ˆn(r) · F(r)
= lim
→0
ˆn(r + u) − ˆn(r)
· F + ˆn ·
F(r + u ) − F(r)
.
To do the subtraction, we need to properly move the vector
F(r + u ) from its home at r + u , and put it at r.
This is called Parallel Transport: vector maintains same
orientation relative to u.
Because unit vectors will change, the components will change.
To first order in , change is linear
F(r + u )|| transport to r = F(r + u )at r + ˆeαΓα
βγFβ
uγ
The Γα
βγ are given by
Γα
βγ =
1
2
gαδ ∂gβδ
∂xγ
+
∂gγδ
∂xβ
−
∂gβγ
∂xδ
.
5. Whence Precession?
Consider particle on trajectory r = w(λ), with velocity u = ˙w.
How does v(r) change?
Directional derivative uv(r) =
v(r+udλ)||−v(r)
dλ .
Parallel transported v(r + udλ)|| = v(r + λu)at r + ˆeαΓα
βγvβuγ .
Hence, change in vector along trajectory is
uν
νvµ
=
∂vµ
∂xν
uν
+ Γµ
αβvα
uβ
.
If v = u, then uu = a and Γα
βγuβuγ is effective force.
Since Γα
βγ = 1
2gαδ ∂gβδ
∂xγ +
∂gγδ
∂xβ −
∂gβγ
∂xδ , metric coefficients are
effective potentials.
If uv = 0, then v is parallel-transported along trajectory w(λ).
A “geodesic” defines local inertial frame, follows trajectory
uu = 0.
6. Whence Precession?
So if a gyro is spinning with initial spin four-vector s = (0, s),
and moving on geodesic, uu = 0, then in its own rest frame s
remains space-like, u · s = 0.
Hence, s is parallel transported along trajectory defined by
four-velocity u, us = 0.
Gives gyroscopic equation
dsα
dτ
+ Γα
βγuγ
sβ
= 0.
Changed by effective “torques” from Christoffel symbols, in
analogy to ds
dt = Ω × s from classical mechanics.
These “torques” come from Γα
βγ, hence from gµν, hence related
to the curvature of spacetime.
Let’s find the precession terms!
7. Gravitoelectromagnetism
We now split metric tensor gαβ in to three parts
gravitoelectric scalar potential Φ = −c2
2 (g00 + 1)
gravitomagnetic vector potential γ = g0j ˆej
space curvature tensor gjk
We can define gravitoelectric field g = − Φ
We can define gravitomagnetic field H = × γ.
We can even define “Maxwell Equations” for these [c.f.
Braginsky, Caves, Thorne 1977]
· g = −4πGρ × g = 0
· H = 0 × H = 4[−4πGρv/c + (1/c)(∂g/∂t)]
From analogy to EM then, for spinning, spherical earth
g = −
GM
r2
ˆr and H =
2G
c
L − 3(L · ˆr)ˆr
r3
H will cause frame-dragging, g and gjk will cause geodesic
effect.
8. Frame-Dragging Precession
Gryoscope has spin s; this is a spinning clump of mass, or a
gravitomagnetic dipole, moment µ = 1
2s.
Magnetic dipole-dipole interaction experiences torque µ × B.
Earth-Gyroscope interaction is dipole-dipole, so s experiences
torque ds
dt = s
2 × H
c . [Thorne, 1988]
So precession due to gravitomagnetic effect is
ΩGM = −
1
2c
H =
G
c2
3(ˆr · L)ˆr − L
r3
.
9. Gravitoelectric Precession
Gravitomagnetic dipole experiences no torque from a stationary
gravitoelectric field , but an orbiting dipole sees an orbiting
Earth.
In electrodynamics, this produces induced magnetic field
Binduced = −v
c × E.
So we have induced gravitomagnetic field Hinduced = −v
c × g.
As above, this results in torque s × Hinduced
2c and precession
ΩGE = −
1
2c
Hinduced =
GM
2c2r2
ˆr × v.
10. Space Curvature Precession
So far all these effects have direct parallels in EM; however, the
3-tensor term gjk has no parallel.
gjk contains information on how space is curved due to the mass
monopole M – not spacetime curvature, but space curvature.
Can be visualized with example of vector on cone [c.f. Thorne]
Turns out to be ΩSC = 2ΩGE .
So total precession Ω = 3GM
2c2r2 ˆr × v + G
c2
3(ˆr·L)ˆr−L
r3 .
11. Experimental Set-Up
William Fairbank said:
“No mission could be simpler than GP-B: It’s just a star, a
telescope, and a spinning sphere.”
The Star:
Named HR8703, or IM Pegasi, optical and radio-star
Chosen for its brightness, its location near equator, highly
determinable proper motion [Buchman, et al., 2000]
Uncertainties in proper motion of star directly related to
experimental uncertainty [Mester, et al.,2004]
Motion of HR8703 is determinable because it is radio-star, able to
use Very-Long-Baseline Interferometry [Shapiro, et al., 2012]
Harvard Smithsonian Astronomical Observatory in separate
experiment established 0.1 marcsec/yr proper motion
uncertainties.
The Telescope:
The Spinning Sphere:
12. Experimental Set-Up
William Fairbank said:
“No mission could be simpler than GP-B: It’s just a star, a
telescope, and a spinning sphere.”
The Star:
The Telescope:
Set to track HR8703; provides “inertial” direction for comparison
to spin direction of gyroscope
Housed in atmosphere-drag-free satellite system; essentially an
exterior case that absorbs all shocks from collisions and
rebalances with very precise thrusters.
Makes realistic approximate local inertial frame. [c.f. Hartle,
2003]
[Hartle, Gravity, 2003 pg. 181]
The Spinning Sphere:
13. Experimental Set-Up
William Fairbank said:
“No mission could be simpler than GP-B: It’s just a star, a
telescope, and a spinning sphere.”
The Star:
The Telescope:
[Hartle, Gravity, 2003 pg. 181]
The Spinning Sphere:
14. Experimental Set-Up
William Fairbank said:
“No mission could be simpler than GP-B: It’s just a star, a
telescope, and a spinning sphere.”
The Star:
The Telescope:
The Spinning Sphere:
Have won Guinness World Record for most-perfectly spherical
objects ever.[Blau 2011]
Peak-to-valley asphericity of 25 nm, compared to 1.9 cm radius;
about 1 part per million asphericity.[Mester 2004]
Spin axes of gyros needs to be perfectly aligned with principal
axes of rotation
Made of fused quartz, coated in niobium for superconduction,
kept at 1.8K
Electrostatically suspended and positioned inside quartz housing
Coupled to SQUID magnetometer, which reads magnetic moment
of sphere
15. Experimental Set-Up
William Fairbank said:
“No mission could be simpler than GP-B: It’s just a star, a
telescope, and a spinning sphere.”
The Star:
The Telescope:
The Spinning Sphere:
17. How’d They Do?
Einstein predicts Geodetic Precession 6.606 arcec/year,
Framedragging Precession 0.0392 arcsec/year. [Everitt 2011]
Results show Geodetic Precession 6.602 ± 0.018 and
Frame-dragging Precession 0.0372 ± 0.0072 [Blau 2011]
Experiment confirms Geodetic to 0.2% and Frame-dragging to
18.4%; were hoping for 0.01% and 1%.
Gyros were not perfect spheres, so spin axis and principal axis
did not align perfectly. Caused more error than anticipated.
Tests still confirm Einstein’s predictions.
