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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 • • • • • • • 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 [2] The Role of Gravitomagnetism on Gyroscopes and the Moon, Tom Murphy, UCSD, as yet unpublished.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 3 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 • • • • 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 5 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 6 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 • h just makes the equations simpler 3/30/2005, Revised 1/2010 8 • 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 9 The Gravitomagnetic Field • 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 10 Where Did the Tensor Go? • 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 11 Gravitational “Maxwell’s Equations” EG 1 A , 2c t EG 4 G , EG • • • • 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 12 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 vi L ai BG z y BG BG 3/30/2005, Revised 1/2010 L vi ai precession x 13 Gravity Probe B • • Equatorial precession opposite direction of earth spin Dipole approximation no good: altitude 640 km = 0.1 R • 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 14 • • 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 15 • • 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 16 Lunar Orbit Perturbations: Part Deux • 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 17 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 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 18