Gravitoelectromagnetism
[1] The most common version of GEM is valid only far from isolated sources, and for slowly moving test particles.
The analogy and equations differing only by some small factors were first published in 1893, before general relativity, by Oliver Heaviside as a separate theory expanding Newton's law of universal gravitation.
The main consequence of the gravitomagnetic field, or velocity-dependent acceleration, is that a moving object near a massive, rotating object will experience acceleration that deviates from that predicted by a purely Newtonian gravity (gravitoelectric) field.
Roger Penrose had proposed a mechanism that relies on frame-dragging-related effects for extracting energy and momentum from rotating black holes.
[3] Reva Kay Williams, University of Florida, developed a rigorous proof that validated Penrose's mechanism.
analyzing data from the first direct test of GEM, the Gravity Probe B satellite experiment, to see whether they are consistent with gravitomagnetism.
[9] The Apache Point Observatory Lunar Laser-ranging Operation also plans to observe gravitomagnetism effects.
GEM equations compared to Maxwell's equations are:[11][12] where: Faraday's law of induction (third line of the table) and the Gaussian law for the gravitomagnetic field (second line of the table) can be solved by the definition of a gravitation potential
can also be solved for a rotating spherical body (which is a stationary case) leading to gravitomagnetic moments.
For a test particle whose mass m is "small", in a stationary system, the net (Lorentz) force acting on it due to a GEM field is described by the following GEM analog to the Lorentz force equation: where: The GEM Poynting vector compared to the electromagnetic Poynting vector is given by:[13] The literature does not adopt a consistent scaling for the gravitoelectric and gravitomagnetic fields, making comparison tricky.
For example, to obtain agreement with Mashhoon's writings, all instances of Bg in the GEM equations must be multiplied by −1/2c and Eg by −1.
This difference becomes clearer when one compares non-invariance of relativistic mass to electric charge invariance.
[15] Consider a toroidal mass with two degrees of rotation (both major axis and minor-axis spin, both turning inside out and revolving).
This represents a "special case" in which gravitomagnetic effects generate a chiral corkscrew-like gravitational field around the object.
The reaction forces to dragging at the inner and outer equators would normally be expected to be equal and opposite in magnitude and direction respectively in the simpler case involving only minor-axis spin.
[citation needed] Modelling this complex behaviour as a curved spacetime problem has yet to be done and is believed to be very difficult.
The analytical solution outside of the body is (see for example[16]): where: The formula for the gravitomagnetic field Bg can now be obtained by: It is exactly half of the Lense–Thirring precession rate.
This factor of two can be explained completely analogous to the electron's g-factor by taking into account relativistic calculations.
At the equatorial plane, r and L are perpendicular, so their dot product vanishes, and this formula reduces to: Gravitational waves have equal gravitomagnetic and gravitoelectric components.
When such fast motion and such strong gravitational fields exist in a system, the simplified approach of separating gravitomagnetic and gravitoelectric forces can be applied only as a very rough approximation.
The fact that ρg and jg do not form a four-vector (instead they are merely a part of the stress–energy tensor) is the basis of this difference.
Note that the GEM equations are invariant under translations and spatial rotations, just not under boosts and more general curvilinear transformations.
