Gravitoelectromagnetism: A Brief Review
Bahram Mashhoon
Abstract and summary · read the original at the source
In one page
Bahram Mashhoon, at the University of Missouri, sets out the exact sense in which gravity looks like electromagnetism. Take Einstein’s equations, keep the first-order terms, and they rearrange into Maxwell’s: a gravitoelectric field produced by mass, a gravitomagnetic field produced by mass in motion, an induction law joining them the way Faraday’s does, and a Lorentz force law for a test particle. Mashhoon gives two routes to it. The first is that linearised expansion about flat spacetime. The second builds the same fields out of the curvature tensor directly, in a freely falling frame, and works in any curved spacetime at all. The joint holding both together is what he calls the gravitational Larmor theorem: locally, a gravitomagnetic field is a rotation, exactly as a gravitoelectric field is an acceleration — Einstein’s elevator, except the elevator has to spin as well as accelerate. Two results come with it: the precession rate of a gyroscope near a rotating mass, and a gravitational Poynting vector.
Why it matters hereChapter 11 is gravity control, and every serious attempt at it — Podkletnov’s disks, the Li and Torr prediction, Tajmar’s spinning rings — is written in this language, because gravitoelectromagnetism is the one place where general relativity hands you a field you can in principle drive with a current. This is the review those papers cite for the equations. Chapter 3 needs the deeper point Mashhoon makes: the gravitomagnetic field and a rotation are locally the same thing, which is inertia and gravity being two readings of one quantity. Martin Tajmar’s gravitomagnetic London moment measurements are at /library/stm-21102decd7 and /library/stm-1cb16dd1bc, the LARES and LAGEOS satellite test of frame dragging at /library/stm-e79ea933f1, and the Defense Intelligence Agency’s own survey of superconductors in gravity research, which opens from these same equations, at /library/stm-2aafe96904.
What it claims
01Linearise Einstein’s field equations about flat spacetime and they take Maxwell’s form. Writing the mass density and the mass current as source terms gives a gravitoelectric potential and a gravitomagnetic vector potential, and the resulting fields obey four equations that match electromagnetism one for one: the divergence of the gravitoelectric field is set by mass density, the gravitomagnetic field is divergence-free, a changing gravitomagnetic field induces a gravitoelectric one, and the curl of the gravitomagnetic field is set by the mass current plus a displacement term. The continuity equation for mass falls out of them, as it should.Section 1.2, Equations 1.4 to 1.9
Settled physics02The force law comes with it: a test particle of mass m feels minus m times the gravitoelectric field, plus a velocity-cross-gravitomagnetic-field term carrying a factor of two. That factor is not an accident of bookkeeping — the ratio of gravitomagnetic to gravitoelectric charge is always exactly two because linearised gravity is a spin-two field, where for a spin-one field such as Maxwell’s the ratio is always one.Section 1.2, Equation 1.11; Section 1.3
Settled physics03The gravitational Larmor theorem is Einstein’s equivalence principle written in this language. In a small enough neighbourhood the GEM fields can be replaced exactly by an accelerated frame in flat spacetime, with the gravitoelectric field appearing as a translational acceleration and the gravitomagnetic field as a frequency of rotation. Einstein’s elevator therefore has to spin as well as accelerate: rotation of the frame is what stands in for the gravitomagnetic field of the source.Section 1.3, Equations 1.19 to 1.23
Settled physics04A gyroscope carries a gravitomagnetic dipole moment equal to minus its spin over the speed of light, and precesses in the exterior field of a rotating mass at a rate set by the source’s angular momentum divided by the square of the speed of light times the cube of the distance, in the standard dipole pattern. Evidence for the Earth’s gravitomagnetic field had already been offered from laser ranging of the LAGEOS satellites, and measuring this precession directly with superconducting gyroscopes in a polar orbit was the aim of NASA’s Gravity Probe B.Section 1.3, Equation 1.24; Section 1.1
Settled physics05Gravitational energy circulates around a rotating body. The GEM Poynting vector, built from the cross product of the two fields exactly as in electromagnetism, gives a flow that goes round a stationary source of mass M and angular momentum J in the same sense as the rotation, with a flow velocity proportional to the angular momentum per unit mass divided by the radius and by the sine of the polar angle. The flow is divergence-free, and its circulation does not depend on radial distance at all.Section 1.2, Equations 1.17 and 1.18
Published and peer-reviewed06Intrinsic quantum spin couples to rotation and, through the Larmor theorem, to gravitomagnetic fields — so a spin-half particle behaves essentially like a tiny gyroscope, with an energy shift given by minus the rotation frequency dotted into the spin. The rotation half is confirmed: a frequency offset between mercury nuclear spins and the Earth’s rotation, and helicity-rotation coupling measured to high accuracy with rotating GPS receivers. The gravitational half is the open measurement. At the Earth’s surface the up-versus-down splitting is about ten to the minus nineteen electron volts from the planet’s rotation and about ten to the minus twenty-nine from its gravitomagnetic field; near Jupiter the gravitomagnetic term rises to about ten to the minus twenty-seven, which Mashhoon expects to come within reach of an orbiting magnetometer. The same coupling implies a gravitomagnetic Stern-Gerlach force that does not depend on mass — so the weight of a body depends on its spin.Section 1.5, Spin-Rotation-Gravity Coupling
What to watch
The way in
https://arxiv.org/abs/gr-qc/0311030LICENCE. Posted to arXiv as gr-qc/0311030 on 8 November 2003 and revised on 17 April 2008; the deposit carries arXiv’s assumed licence for legacy submissions rather than a Creative Commons statement, so this page holds the summary, the claims and the author’s own abstract and sends the reader to the source. The author’s note on the record says the revised version was published as the third chapter of The Measurement of Gravitomagnetism: A Challenging Enterprise, edited by Lorenzo Iorio (Nova Science, New York, 2007), pages 29 to 39. Fifteen pages, no figures, thirty-five reference groups, from the Department of Physics and Astronomy at the University of Missouri-Columbia.
How to cite it
Bahram Mashhoon (2003) Gravitoelectromagnetism: A Brief Review. arXiv:gr-qc/0311030
Where it sits in the curriculum
Gravity control and superconductorsInertia and gravity from the vacuum