Gravitomagnetism
The Myth and the Legend
Eric L. Michelsen
3/30/2005, Revised 1/2010
1
My One Sentence
Gravity includes a velocity-independent force (Newton) and a
velocity-dependent force (gravitomagnetic), closely
analogous to the electric and magnetic fields in E&M
•
Purely relativistic effect, not in Newtonian gravity
3/30/2005, Revised 1/2010
2
Topics
Prerequisite: some familiarity with General Relativity helps
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•
•
•
•
•
•
Einstein’s Theory of Gravity
Metric Theories of Gravity
The Gravitomagnetic Term
Precessing Gyroscopes
Gravity Probe B
Lunar Orbital Perturbations
The Controversy
•
Papers:



[1] Gravitomagnetic Effects, M. L. Ruggiero et. al., arXiv:grqc/0207065v2, 7/9/2004.
[2] The Role of Gravitomagnetism on Gyroscopes and the Moon, Tom
Murphy, UCSD, as yet unpublished.
[3] Lunar Laser Ranging – A Comprehensive Probe of Post-Newtonian
Gravity, K. Nordtvedt, arXiv:gr-qc/0301024, 1/7/2003.
3/30/2005, Revised 1/2010
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Gravitomagnetism
•
•
•
The orphan child of gravitational physics
Renewed interest with launch of Gravity Probe B
Lorentz invariance implies a gravitomagnetic field [1, p3]




•
“Any theory that combines Newtonian gravity together with
Lorentz invariance in a consistent way, must include a
gravitomagnetic field, which is generated by mass current.” [1]
Dr. O’Neil says that a 1/r2 force law is not Lorentz invariant
Since Lorentz transformation includes velocity, any 1/r2 force must
be accompanied by a source-velocity-dependent field
Given the structure of the Lorentz transformation, the velocity
dependent field must be a Biot-Savart-like magnetic field (to within
a constant factor).
Aka “frame-dragging” – Bad Name


Bad name because the direction of force depends
on the test-body velocity.
It’s not like being dragged in a stream
3/30/2005, Revised 1/2010
4
GR: A Metric Theory of Gravity
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•
•
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A metric theory of gravity defines a metric tensor field throughout all space
The metric tensor field describes the “shape” (curvature) of space
All physics, gravitational and otherwise, occurs in the physical spacetime
described by the metric tensor field.
The only dynamic field is the metric tensor field (loosely analogous to the EM
field in electromagnetics)
sources of
field
mass/
energy


Field Equation
R
1
 g  R  T
2
metric tensor field
spacetime
of all
physics
(smoothly curved manifold)
R and R are nonlinear functions of g
Just about the simplest metric theory of gravity there is
3/30/2005, Revised 1/2010
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The Metric Tensor Field
•
•
The metric tensor field quantifies intervals, frame independent
measures of the separation between two events.
In an inertial frame (flat space), the squared-interval is the squareddistance between two events, minus the squared-distance light travels
in the time between the events:
ds    cdt   dx  dy  dz 
2
2
2
2
2
3

 
g dx dx 
, 0
flat space :
g 
 1

0

0

0


where x 0 , x1 , x 2 , x 3   t , x, y , z 
0 0 0

1 0 0
 
0 1 0

0 0 1
dx2
(c dt)2
−ds2
•
In general, the metric tensor field defines the dot product of any two
vectors
3/30/2005, Revised 1/2010
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Metric Theories of Gravity
•
By definition [Will, 1993], a metric theory of gravity defines a metric
tensor field throughout all space



But other unobservable fields may be defined
Their only purpose is to define the metric tensor through field equations
In the end, only the metric tensor field affects observable physics
sources of
fields
mass/
energy
Field
Equations
Fields:
• metric tensor field
• scalar field
• other fields
spacetime of
all physics
Field equations relate all the fields, to define the all-important metric
tensor field.
3/30/2005, Revised 1/2010
7
A Perturbing Thought
•
•
Nonlinear equations are hard to solve
Use perturbation theory:
g 
 1

0

0

0
0
1
0
0
0
0
1
0
0

0
 h ,
0

1
where
h  1
 
1
Define h  h   h  
2
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h just makes the equations simpler
3/30/2005, Revised 1/2010
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•
The Gravitomagnetic Term
Use perturbation theory to compute the weak-field, nonrelativistic perturbation to the metric:
 2
2




 h  16 GT
2
 t



 2
2




 h  0
2
 t

Compare to E&M (tensor vs. vector):
 2
2 

  2    A  4 j
 t

•
vacuum
vacuum

 2
2 
  2    A  0
 t

Can jump right to gravity waves; but let’s not.
3/30/2005, Revised 1/2010
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The Gravitomagnetic Field
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Use the perturbed metric to compute the equations of
motion. (Solve the geodesic equation.)
Gravitomagnetic term:
ai  kM j
v j  rˆij 

v 
i
r2

BG (r )  kM
 v  rˆ 
r2
ai
y
x
vi
rij
source of
field



vj
Left hand rule
q  v  rˆ 
B(r) 
Compare to Biot-Savart:
c r2
No standard convention for factors of 2, signs, etc.
3/30/2005, Revised 1/2010
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Where Did the Tensor Go?
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To order (1/c2), only the first row and column of h are
significant:
  1 
 1 
 1 
 1 
O
O
O
O






 2 

2
2
2
r
c
r
c
r
c






 r c 

  1 

O
~
0
~
0
~
0
  2 

r
c



h ~
  1 

~0
~0
~0 
O 2 
 r c

  1 

~0
~0
~ 0 
 O  2 
 r c

•
Reduces equations to vectors (rank-1 tensors)
3/30/2005, Revised 1/2010
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Gravitational “Maxwell’s Equations”
EG   
1 A
,
2c t
  EG  4 G  ,
  EG  
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1 BG
,
2c t
BG    A.
Lorenz gauge :
1  1
 A  0
c t 2
  BG  0
1
1
4 G
  B G  EG 
j
2
c
c
Valid for weak field, non-relativistic speeds
Imply propagating waves: gravity waves
Factors of 2 are remnants of rank-2 tensor wave equation,
and spin 2 gravitons
No standard conventions for factors of 2, signs, etc.
3/30/2005, Revised 1/2010
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Gravitomagnetically Precessing Gyroscopes
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Use the solar system barycentric frame
Source of gravitomagnetic field is earth’s spin
Precession at poles is same direction as earth spin
This is not geodetic precession; gravitomagnetism is much
smaller
mass element
vi
L
ai

BG
z
y
BG
BG
3/30/2005, Revised 1/2010
L
vi
ai
precession
x
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Gravity Probe B
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Equatorial precession opposite direction of earth spin

Dipole approximation no good: altitude 640 km = 0.1 R

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Partially cancels GPB signal: total precession = ¼ polar precession
Dipole approximation is never much good: if far enough for dipole,
effect is too small to see
Do the integral: 42 mas/y is the published number
precession
polar orbit
L
precession
L
z
y
x
3/30/2005, Revised 1/2010
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•
Lunar Orbital Effects
Solar system barycentric frame: Source of gravitomagnetic
field is earth’s orbit around sun

Spin of the earth is negligible
We decompose the lunar velocity into two components


V: Lunar motion around sun = earth’s motion around sun
u: Lunar motion around earth
u
vmoon = V + u
Sun
V
Earth
V
3/30/2005, Revised 1/2010
y
x
magnified view
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Lunar Orbit Perturbations
Velocities: both objects orbit the sun at ~30 km /s
Lorentz contraction: should contract tangential size, but not elongate??
to sun
a
BG
V
elongated
orbit
D
V
BG =
0
BG =
0
orbital elongation
~ cos 2D
≈ 5 meters
BG
3/30/2005, Revised 1/2010
a
V
y
x
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Lunar Orbit Perturbations: Part Deux
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Velocity: moon orbits earth at ~1 km/s
u
to sun
BG
a
D
V
BG =
0
BG =
0
orbital offset
~ cos D
≈ 5 meters
BG
3/30/2005, Revised 1/2010
a
u
y
x
offset
orbit
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The Controversy
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Lunar Laser Ranging (LLR) confirms the gravitomagnetic
term to 0.1%
Gravity Probe B will confirm it with a different method to
only 1%

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
Word-of-mouth claims say there is more to GPB than just the
gravitomagnetic effect
But [2] did the math, and recovers the published value of 42 mas/y
Is GPB new physics?
“Most all of the 1/c2 order, post-Newtonian terms in the Nbody equations of motion – motional, gravitomagnetic,
non-linear, inductive, etc. – contribute to the measured
details of the lunar orbit, so LLR achieves nearcompleteness as a gravity experiment and probe.” [3, p1]
Possible confusion due to Sun’s quadrupolar tidal field,
which produces cos 2D term (but 90o out of phase). [3, p3]
3/30/2005, Revised 1/2010
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