Gravitomagnetism The Myth and the Legend

1. GravitomagnetismThe Myth and the Legend Eric L. Michelsen

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. Topics • Einstein’s Theory of Gravity • Metric Theories of Gravity • The Gravitomagnetic Term • Precessing Gyroscopes • Gravity Probe B • Lunar Orbital Perturbations • The Controversy Prerequisite: some familiarity with General Relativity helps • Papers: • [1] Gravitomagnetic Effects, M. L. Ruggiero et. al., arXiv:gr-qc/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 • [2] The Role of Gravitomagnetism on Gyroscopes and the Moon, Tom Murphy, UCSD, as yet unpublished.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.

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 dependson the test-body velocity. • It’s not like being dragged in a stream

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

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

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.

8. A Perturbing Thought • Nonlinear equations are hard to solve • Use perturbation theory: • h just makes the equations simpler

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.

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

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)

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.

13. Gravitomagnetically Precessing Gyroscopes z y x • 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 BG BG

14. Gravity Probe B z y x • 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 L polar orbit precession L

15. Lunar Orbital Effects y x • 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 magnified view

16. Lunar Orbit Perturbations y x • 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 V BG a

17. Lunar Orbit Perturbations: Part Deux y x • 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 offset orbit u BG a

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]