1. Gravitomagnetism The Myth and the Legend
Eric L. Michelsen
3/30/2005, Revised 1/2010

2. 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

3. Topics Einstein’s Theory of Gravity Metric Theories of Gravity
Prerequisite: some familiarity with General Relativity helps
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:gr-qc/ v2, 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[2] The Role of Gravitomagnetism on Gyroscopes and the Moon, Tom Murphy, UCSD, as yet unpublished.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/ , 1/7/2003.
3/30/2005, Revised 1/2010

4. 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

5. GR: A Metric Theory of Gravity
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)
Field Equation
sources of field
spacetime of all physics
metric tensor field
mass/ energy
(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

6. 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 squared-distance between two events, minus the squared-distance light travels in the time between the events:
(c dt)2
dx2
−ds2
In general, the metric tensor field defines the dot product of any two vectors
3/30/2005, Revised 1/2010

7. 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
Field Equations
sources of fields
Fields:
metric tensor field
scalar field
other fields
spacetime of all physics
mass/ energy
Field equations relate all the fields, to define the all-important metric tensor field.
3/30/2005, Revised 1/2010

8. A Perturbing Thought Nonlinear equations are hard to solve
Use perturbation theory:
h just makes the equations simpler
3/30/2005, Revised 1/2010

9. The Gravitomagnetic Term
Use perturbation theory to compute the weak-field, non-relativistic perturbation to the metric:
Compare to E&M (tensor vs. vector):
Can jump right to gravity waves; but let’s not.
3/30/2005, Revised 1/2010

10. The Gravitomagnetic Field
Use the perturbed metric to compute the equations of motion. (Solve the geodesic equation.) Gravitomagnetic term: ai x y vi
rij
source of field vj
Left hand rule
Compare to Biot-Savart:
No standard convention for factors of 2, signs, etc.
3/30/2005, Revised 1/2010

11. Where Did the Tensor Go?
To order (1/c2), only the first row and column of h are significant:
Reduces equations to vectors (rank-1 tensors)
3/30/2005, Revised 1/2010

12. Gravitational “Maxwell’s Equations”
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

13. Gravitomagnetically Precessing Gyroscopes
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 ai vi L 
precession BG L vi ai x y z BG BG
3/30/2005, Revised 1/2010

14. Gravity Probe B Equatorial precession opposite direction of earth spin
Partially cancels GPB signal: total precession = ¼ polar precession
Dipole approximation no good: altitude 640 km = 0.1 R
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 x y z
3/30/2005, Revised 1/2010

15. 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 x y
magnified view
3/30/2005, Revised 1/2010

16. Lunar Orbit Perturbations
Velocities: both objects orbit the sun at ~30 km /s
Lorentz contraction: should contract tangential size, but not elongate?? a
to sun BG V
elongated orbit D V
BG = 0
BG = 0
orbital elongation ~ cos 2D
≈ 5 meters x y V BG a
3/30/2005, Revised 1/2010

17. Lunar Orbit Perturbations: Part Deux
Velocity: moon orbits earth at ~1 km/s
to sun u BG a D V
BG = 0
BG = 0
orbital offset ~ cos D
≈ 5 meters x y
offset orbit u BG a
3/30/2005, Revised 1/2010

18. The Controversy
Lunar Laser Ranging (LLR) confirms the gravitomagnetic term to 0.1%
Gravity Probe B will confirm it with a different method to only 1%
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 N-body equations of motion – motional, gravitomagnetic, non-linear, inductive, etc. – contribute to the measured details of the lunar orbit, so LLR achieves near-completeness 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