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

Background

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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, it is called the gravetomagnetic force, since it arises the same way that a moving electric charge creates a magnetic field, which is also a "fictitious force" in Special Relativity. The main consequence of the gravetomagnetic force, or acceleration, is that a free-falling object near a massive rotating object, will begin to rotate.

The effects of such a gravitational field, often loosely referred to as gravitomagnetic effects, are among the last basic predictions of general relativity not yet directly tested. A group at Stanford University is currently analyzing data from the first direct test of GEM, the Gravity B satellite experiment. Frame-dragging is often mentioned as a gravitomagnetic effect, but the Lense-Thirring effect (precession) may be a more appropriate example.

Equations

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According to general relativity, the gravitational field produced by a rotating object (or any rotating mass-energy) is formally analogous to 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, Mashhoon, Gronwald, and Lichtenegger, and Clark and Tucker have derived the following gravitational analogs to Maxwell's equations for electromagnetism. They are called the "GEM equations":

$\backslash nabla\; \backslash cdot\; \backslash mathbf\{E\}\; =\; -4\; \backslash pi\; G\; \backslash rho\; \backslash$

$\backslash nabla\; \backslash cdot\; \backslash mathbf\{B\}\; =\; 0\; \backslash$

$\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; -\backslash frac\{1\}\{c\}\; \backslash frac\{\backslash partial\; \backslash mathbf\{B\}\; \}\; \{\backslash partial\; t\}\; \backslash$

$\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash frac\{1\}\{c\}\; \backslash left(\; -4\; \backslash pi\; G\; \backslash mathbf\{J\}\; +\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\; \{\backslash partial\; t\}\; \backslash right)\; =\; \backslash frac\{1\}\{c\}\; \backslash left(\; -4\; \backslash pi\; G\; \backslash rho\; \backslash mathbf\{v\}\_\{\backslash rho\}\; +\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\; \{\backslash partial\; t\}\; \backslash right)\; \backslash$

where:

• E is the static gravitational field (conventional gravity, also called gravitoelectric for the sake of analogy);
• B is the gravitomagnetic field;
• ρ is mass density (instead of electric charge density);
• v_ρ is the velocity of the mass flow generating the gravitomagnetic field;
• J is mass current density (J = ρ v_ρ);
• G is the gravitational constant;
• c is the speed of propagation of gravity (according to General Relativity, equal to the speed of light).

For a test particle of small mass m, the net (Lorentz) force acting on it due to GEM fields is described by the following GEM analog to the Lorentz force equation:

$\backslash mathbf\{F\}\_\{m\}\; =\; m\; \backslash left(\; \backslash mathbf\{E\}\; +\; \backslash frac\{\backslash mathbf\{v\}\_\{m\}\}\; \{c\}\; \backslash times\; 2\; \backslash mathbf\{B\}\; \backslash right)$.

where:

• m is the mass of the test particle;
• v_m is the instantaneous velocity of the test particle;

In the literature, all instances of B in the GEM equations are multiplied by 1/2, a factor absent from Maxwell's equations. This factor is unnecessary if B in the GEM version of the Lorentz force equation is multiplied by 2, as shown above. The factors 2 and 1/2 arise because the effective gravitomagnetic charge is twice the static gravitational (gravitoelectric) charge, a remnant of the spin-2 character of the gravitational field. For a pure spin-1 field such as the genuine electromagnetic field, the magnetic charge equals the electric charge.

Comparison with electromagnetism

The above GEM equations are very similar to Maxwell's equations in free space, expressed using CGS units.

$\backslash nabla\; \backslash cdot\; \backslash mathbf\{E\}\; =\; 4\backslash pi\backslash rho$

$\backslash nabla\; \backslash cdot\; \backslash mathbf\{B\}\; =\; 0$

$\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; -\backslash frac\{1\}\{c\}\; \backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\; \{\backslash partial\; t\}$

$\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash frac\{1\}\{c\}\; \backslash left(\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\; \{\backslash partial\; t\}\; +\; 4\backslash pi\; \backslash mathbf\{J\}\; \backslash right)$

Adopting Planck units eliminates G and c from both sets of equations by normalizing these constants to 1. The two sets of equations are now identical but for the minus sign preceding 4π in the GEM equations. These two minus signs stem from an essential difference between gravity and electromagnetism: electrostatic and magnetic forces of the same type repel, while gravity assures that two positive masses attract. Hence the GEM equations are simply Maxwell's equations with mass (or mass density) substituting for charge (or charge density), and -Gρ replacing ρ. The following Table summarizes the results thus far:

GEM Equations Given Planck units.

$\backslash nabla\; \backslash cdot\; \backslash mathbf\{B\}\; =\; 0$

$\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; -\backslash partial\; \backslash mathbf\{B\}/\; \backslash partial\; t$

$\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash iota\; 4\backslash pi\backslash mathbf\{J\}\; +\; \backslash partial\; \backslash mathbf\{E\}/\; \backslash partial\; t$

$\backslash iota$ = 1 (Maxwell) or -1 (GEM).

Common Structure of the Maxwell and

$\backslash nabla\; \backslash cdot\; \backslash mathbf\{E\}\; =\; \backslash iota\; 4\backslash pi\backslash rho$

The factor of 4π that recurs in these equations equals, by definition, the mantissa of the permeability of free space μ₀. The value of μ₀ is 10^-7 for SI units, and 1 for Planck and CGS units. The unnormalized (SI units) version of the fourth Maxwell equation is:

$\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash mu\_0\; \backslash mathbf\{J\}\; +\; \backslash mu\_0\; \backslash epsilon\_0\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\; \{\backslash partial\; t\}\; =\; \backslash frac\{1\}\{c^2\}\; \backslash left(\; \backslash frac\; \{\backslash mathbf\{J\}\}\; \{\backslash epsilon\_0\}\; +\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\; \{\backslash partial\; t\}\; \backslash right)$

where ε₀ is the permittivity of free space, and μ₀ε₀ = c^-2. CGS and Planck units both normalize the Coulomb force constant to 1. Hence such units effectively normalize ε₀ to (4π)^-1, whence the factor of 4π in the CGS and Planck versions of Maxwell's equations. Normalizing G to (4π)^-1 and ε₀ to 1, so that G and the Coulomb force constant become identical, eliminates the factor 4π from the table above.

Higher-order effects

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Some of the higher-order gravitomagnetic effects can begin to 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 them feeling conventional g-forces, although whether this is a practical suggestion is not clear.

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 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 is very difficult.

Gravitomagnetic field of Earth

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Bg,Earth = 10^-14 rad.s^-1 ("Experimental Detection of the Gravitomagnetic London Moment" by Martin Tajmar, Florin Plesescu, Klaus Marhold & Clovis J. de Matos)

See the Gravity Probe B experiment.

Superconductor gravitomagnetic field

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A paper Experimental Detection of the Gravitomagnetic London Moment claims to have measured a gravitomagnetic field created by an accelerating superconducting disc, giving, for such a small and light object a very strong field of 10^-4g.

Fringe physics

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Incomplete understanding of the meaning of the similarity of the gravetomagnetic formulas, above, and Maxwell's equations for (real) electricity and magnetism have given rise to fringe physics which mainstream physicists find both tedious and alarming.

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 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, which are rejected by mainstream scientists:

• the so-called Electric Universe "theory", which claims to identify gravity as a form of electromagnetism;
• claims to have constructed anti-gravity devices;
• Eugene Podkletnov's claims to have constructed gravity-shielding devices and gravitational reflection beams.

See also

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• Anti-gravity
• Gravitational radiation
• Gravity Probe B

External links

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• In Search of gravitomagnetism, NASA, 20 April 2004.
• Towards a new test of general relativity?, ESA General Studies Programme, 23 March 2006.
• Gravitomagnetic London Moment-New test of General Relativity?

References

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• Forward, Robert (1963) "Guidelines to antigravity," American Journal of Physics 31: 166-70 (Members only access).
• eprint version
• A recent introduction to GEM by a leading expert.
• John Archibald Wheeler (1990) A journey into gravity and spacetime. See pp.232-233 ("Gravity's next prize: Gravitomagnetism").

General relativity | Effects of gravitation

Гравитомагнетизъм | Gravitomagnetisme | Гравитомагнетизм | 重力磁性