Gravitomagnetism (sometimes Gravitoelectromagnetism, abbreviated GEM), refers to a set of formal analogies between Maxwell's field equations and an approximation, valid under certain conditions, to the Einstein field equations for general relativity. The most common version of GEM is valid only far from isolated sources, and for slowly moving test particles.

[edit] Background

This approximate reformulation of gravitation as described by general relativity makes a "fictitious force" appear in a frame of reference different from a moving, gravitating body. By analogy with electromagnetism, this fictitious force is called the gravitomagnetic force, since it arises in the same way that a moving electric charge creates a magnetic field, the analogous "fictitious force" in special relativity. The main consequence of the gravitomagnetic force, or acceleration, is that a free-falling object near a massive rotating object will itself rotate. This prediction, often loosely referred to as a gravitomagnetic effect, is among the last basic predictions of general relativity yet to be directly tested.

Indirect validations of gravitomagnetic effects have been derived from analyses of relativistic jets. Roger Penrose had proposed a frame dragging mechanism for extracting energy and momentum from rotating black holes.^[1] Reva Kay Williams, University of Florida, developed a rigorous proof that validated Penrose's mechanism.^[2] Her model showed how the Lense-Thirring effect could account for the observed high energies and luminosities of quasars and active galactic nuclei; the collimated jets about their polar axis; and the asymmetrical jets (relative to the orbital plane).^[3] All of those observed properties could be explained in terms of gravitomagnetic effects.^[4] Williams’ application of Penrose's mechanism can be applied to black holes of any size.^[5] Relativistic jets can serve as the largest and brightest form of validations for gravitomagnetism.

A group at Stanford University is currently analyzing data from the first direct test of GEM, the Gravity Probe B satellite experiment, to see if they are consistent with gravitomagnetism.

[edit] Equations

According to general relativity, the gravitational field produced by a rotating object (or any rotating mass-energy) can, in a particular limiting case, be described by equations that have the same form as the magnetic field in classical electromagnetism. Starting from the basic equation of general relativity, the Einstein field equation, and assuming a weak gravitational field or reasonably flat spacetime, the gravitational analogs to Maxwell's equations for electromagnetism, called the "GEM equations", can be derived. GEM equations compared to Maxwell's equations in SI are:^{[6][7][8][9][10]}

GEM equations	Maxwell's equations

where:

• E_g is the static gravitational field (conventional gravity, also called gravitoelectric for the sake of analogy);
• E is the electric field;
• B_g is the gravitomagnetic field;
• B is the magnetic field;
• ρ is mass density;
• ρ_em is charge density:
• J is mass current density (J = ρ v_ρ, where v_ρ is the velocity of the mass flow generating the gravitomagnetic field);
• J_em is electric current density;
• G is the gravitational constant;
• ε₀ is the vacuum permittivity;
• c is the speed of propagation of gravity (equal to, by general relativity, the speed of light).

[edit] Lorentz force

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:

• m is the mass of the test particle;
• v is the instantaneous velocity of the test particle.

Acceleration of any test particle is simply:

.

In some literature, all instances of B_g in the GEM equations are multiplied by 1/2, a factor absent from Maxwell's equations. This factor vanishes if B_g in the GEM version of the Lorentz force equation is multiplied by 2, as shown above. The factors 2 and 1/2 arise because gravitational field is caused by stress-energy tensor which is second rank tensor, as opposed to electromagnetic field which is caused by four-current which is first rank tensor. That difference becomes intuitively clear when non-invariance of relativistic mass is compared to electric charge invariance. This is often referred to as gravity being spin-2 field and electromagnetism being spin-1 field.

[edit] In Planck units

From comparison of GEM equations and Maxwell's equations it is obvious that −1/(4πG) is the gravitational analog of vacuum permittivity ε₀. Adopting Planck units normalizes G, c and 1/(4πε₀) to 1, thereby eliminating these constants from both sets of equations. The two sets of equations then become identical but for the minus sign preceding 4π in the GEM equations. These minus signs stem from an essential difference between gravity and electromagnetism: electrostatic charges of identical sign repel each other, while masses attract each other. Hence the GEM equations are simply Maxwell's equations with mass (or mass density) substituting for charge (or charge density), and −G replacing the Coulomb force constant 1/(4πε₀).

The following Table summarizes the results thus far:

Common Structure of the Maxwell and GEM Equations Given Planck units.

ι = +1 (Maxwell) or −1 (GEM).

4π appears in both the GEM and Maxwell equations, because Planck units normalize G and 1/(4πε₀) to 1, and not 4πG and ε₀.

[edit] Higher-order effects

Some higher-order gravitomagnetic effects can reproduce effects reminiscent of the interactions of more conventional polarized charges. For instance, if two wheels are spun on a common axis, the mutual gravitational attraction between the two wheels arguably ought to be greater if they spin in opposite directions than in the same direction. This can be expressed as an attractive or repulsive gravitomagnetic component.

Gravitomagnetic arguments also predict that a flexible or fluid toroidal mass undergoing minor axis rotation ("smoke ring" rotation) will tend to pull matter preferentially in through one throat and expel it from the other (a case of rotational frame dragging, acting through the throat). In theory, this configuration might be used for accelerating objects (through the throat) without such objects experiencing any g-forces.^[11]

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. When both rotations are applied simultaneously, these two sets of reaction forces can be said to occur at different depths in a radial Coriolis field that extends across the rotating torus, making it more difficult to establish that cancellation is complete.

Modelling this complex behaviour as a curved spacetime problem has yet to be done and is believed very difficult.

[edit] Gravitomagnetic field of Earth

Formula for gravitomagnetic field B_g near a rotating body can be derived from the GEM equations and is: ^[7]

where L is the angular momentum of the body. At the equatorial plane, r and L are perpendicular, so their dot product vanishes, and this formula reduces to:

Magnitude of angular momentum of a homogeneous ball-shaped body is:

where:

• is the moment of inertia of a ball-shaped body (see: list of moments of inertia);
• is the angular velocity;
• m is the mass;
• r is the radius;
• T is the rotational period.

Therefore, magnitude of Earth's gravitomagnetic field at it's equator is:

where is the Earth's gravity.

From this calculation it follows that Earth's equatorial gravitomagnetic field is about B_{g, Earth} = 1.012×10⁻¹⁴ Hz.^[12] Such a field is extremely weak and requires extremely sensitive measurements to be detected. One experiment to measure such field was the Gravity Probe B experiment.

[edit] Fringe physics

Incomplete understanding of the meaning of the similarity of the gravitomagnetic formulas, above, and Maxwell's equations for (real) electricity and magnetism have given rise to fringe physics. Use of the gravitomagnetic analogy for a simplified form of the Einstein field equations, on the other hand, is firmly part of General Relativity. It is an approximation to the current standard theory of gravitation, and has testable predictions, which are in the final stages of being directly tested by the Gravity Probe B experiment. Despite the use of the word magnetism in gravitomagnetism, and despite the similarity of the GEM force laws to the (real) electromagnetic force law, gravitomagnetism should not be confused with any of the following:

• Claims to have constructed anti-gravity devices;
• Eugene Podkletnov's claims to have constructed gravity-shielding devices and gravitational reflection beams.
• Any proposal to produce gravitation using electrical circuits.

[edit] See also

• Anti-gravity
• Geodetic effect
• Gravitational radiation
• Gravitational induction
• Gravity Probe B
• Frame-dragging

[edit] References

[edit] Further reading

• S.J. Clark, R.W. Tucker (2000). "Gauge symmetry and gravito-electromagnetism". Classical and Quantum Gravity 17: 4125–4157. doi:10.1088/0264-9381/17/19/311. 
• R.L. Forward (1963). "Guidelines to Antigravity". American Journal of Physics 31 (3): 166–170. doi:10.1119/1.1969340. 
• O. Heaviside (1893). "A gravitational and electromagnetic analogy". The Electrician 31: 81–82. http://www.as.wvu.edu/phys/OJ/Heavisid.htm. 
• L. Iorio (ed.) (2007). Measuring Gravitomagnetism: A Challenging Enterprise. Nova. ISBN 1600210023. 
• R.T. Jantzen, P. Carini, D. Bini (1992). "The Many Faces of Gravitoelectromagnetism". Annals of Physics 215: 1–50. doi:10.1016/0003-4916(92)90297-Y. arΧiv:gr-qc/0106043. 
• O.D. Jefimenko (1992). Causality, electromagnetic induction, and gravitation : a different approach to the theory of electromagnetic and gravitational fields. Electret Scientific. ISBN 0917406095. 
• B. Mashhoon (2000). "Gravitoelectromagnetism". arΧiv:gr-qc/0011014 [gr-qc]. 
• B. Mashhoon (2003). "Gravitoelectromagnetism: a Brief Review". arΧiv:gr-qc/0311030 [gr-qc].  in L. Iorio (ed.) (2007). Measuring Gravitomagnetism: A Challenging Enterprise. Nova. p. 29–39. ISBN 1600210023. 
• M. Tajmar, C.J. de Matos (2001). "Gravitomagnetic Barnett Effect". Indian Journal of Physics B 75: 459–461. arΧiv:gr-qc/0012091. 
• J.A. Wheeler (1990). "Gravity's next prize: Gravitomagnetism". A journey into gravity and spacetime. Scientific American Library. pp. 232–233. ISBN 0716750163. 
• L. Filipe Costa, Carlos A. R. Herdeiro (2007). "A gravito-electromagnetic analogy based on tidal tensors". arΧiv:gr-qc/0612140 [gr-qc]. 

[edit] External links

• Gyroscopic Superconducting Gravitomagnetic Effectsnews on tentative result of European Space Agency (esa) research
• In Search of gravitomagnetism, NASA, 20 April 2004.
• Gravitomagnetic London Moment-New test of General Relativity?
• Measurement of Gravitomagnetic and Acceleration Fields Around Rotating Superconductors M. Tajmar, et al., 17 October 2006.
• Test of the Lense-Thirring effect with the MGS Mars probe, New Scientist, January 2007.