Gravitomagnetism refers to a set of
formal analogiesAnalogy is a cognitive process of transferring information or meaning from a particular subject to another particular subject , and a linguistic expression corresponding to such a process...
between Maxwell's field equations and an approximation, valid under certain conditions, to the
Einstein field equationsThe Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
for
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. The most common version of GEM is valid only far from isolated sources, and for slowly moving test particles. The equations were first published in 1893, before general relativity, by
Oliver HeavisideOliver Heaviside was a selftaught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations , reformulated Maxwell's field equations in terms of electric and...
as a separate theory expanding Newton's law.
Background
This approximate reformulation of
gravitationGravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
as described by
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
makes an apparent force appear in a
frame of referenceA frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
different from a freely moving, gravitating body. By analogy with electromagnetism, this apparent force is called the
gravitomagnetic force, since it arises in the same way that a moving electric charge creates a magnetic field, the analogous force in
special relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
. The main consequence of the
gravitomagnetic force, or acceleration, is that a freefalling 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 jetRelativistic jets are extremely powerful jets of plasma which emerge from presumed massive objects at the centers of some active galaxies, notably radio galaxies and quasars. Their lengths can reach several thousand or even hundreds of thousands of light years...
s.
Roger PenroseSir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...
had proposed a frame dragging mechanism for extracting energy and momentum from rotating black holes. Reva Kay Williams, University of Florida, developed a rigorous proof that validated Penrose's mechanism. Her model showed how the LenseThirring effect could account for the observed high energies and luminosities of
quasarA quasistellar radio source is a very energetic and distant active galactic nucleus. Quasars are extremely luminous and were first identified as being high redshift sources of electromagnetic energy, including radio waves and visible light, that were pointlike, similar to stars, rather than...
s and
active galactic nucleiAn active galactic nucleus is a compact region at the centre of a galaxy that has a much higher than normal luminosity over at least some portion, and possibly all, of the electromagnetic spectrum. Such excess emission has been observed in the radio, infrared, optical, ultraviolet, Xray and...
; the collimated jets about their polar axis; and the asymmetrical jets (relative to the orbital plane). All of those observed properties could be explained in terms of gravitomagnetic effects. Williams’ application of Penrose's mechanism can be applied to black holes of any size. Relativistic jets can serve as the largest and brightest form of validations for gravitomagnetism.
A group at
Stanford UniversityThe Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is a private research university on an campus located near Palo Alto, California. It is situated in the northwestern Santa Clara Valley on the San Francisco Peninsula, approximately northwest of San...
is currently analyzing data from the first direct test of GEM, the
Gravity Probe BGravity Probe B is a satellitebased mission which launched on 20 April 2004 on a Delta II rocket. The spaceflight phase lasted until 2005; its aim was to measure spacetime curvature near Earth, and thereby the stress–energy tensor in and near Earth...
satellite experiment, to see if they are consistent with gravitomagnetism. The
Apache Point Observatory Lunar Laserranging OperationThe Apache Point Observatory Lunar Laserranging Operation, or APOLLO, is a project at the Apache Point Observatory in New Mexico. It is an extension and advancement of previous Lunar Laser Ranging Experiment, which uses retroreflectors on the Moon to track changes in lunar orbital distance and...
also plans to observe gravitomagnetism effects.
Equations
According to
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, the
gravitational fieldThe gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
produced by a rotating object (or any rotating massenergy) can, in a particular limiting case, be described by equations that have the same form as the
magnetic fieldA magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
in
classical electromagnetismClassical electromagnetism is a branch of theoretical physics that studies consequences of the electromagnetic forces between electric charges and currents...
. Starting from the basic equation of general relativity, the Einstein field equation, and assuming a weak
gravitational fieldThe gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
or reasonably flat spacetime, the gravitational analogs to
Maxwell's equationsMaxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
for
electromagnetismElectromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
, called the "GEM equations", can be derived. GEM equations compared to Maxwell's equations in
SISi, si, or SI may refer to : Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
units are:
GEM equations 
Maxwell's equations 








where:
 E_{g} is the static gravitational field
The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
(conventional gravity, also called gravitoelectric in analogous usage);
 E is the electric field
In physics, an electric field surrounds electrically charged particles and timevarying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
;
 B_{g} is the gravitomagnetic field;
 B is the magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
;
 ρ_{g} is mass density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
;
 ρ is charge density
The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...
:
 J_{g} is mass current density
Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...
(J_{g} = ρ_{g} v_{ρ}, where v_{ρ} is the velocityIn physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
of the mass flow generating the gravitomagnetic field);
 J is electric current density
Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...
;
 G is the gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
;
 ε_{0} is the vacuum permittivity;
 c is the speed of propagation of gravity
In the context of classical theories of gravitation, the speed of gravity is the speed at which changes in a gravitational field propagate. This is the speed at which a change in the distribution of energy and momentum of matter results in subsequent alteration, at a distance, of the gravitational...
(which is equal to the speed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
according to general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
).
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 forceIn physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...
equation:
.
where:
 m is the mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
of the test particleIn physical theories, a test particle is an idealized model of an object whose physical properties are assumed to be negligible except for the property being studied, which is considered to be insufficient to alter the behavior of the rest of the system...
;
 v is the instantaneous velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
of the test particle.
The 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 forceIn physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...
equation is multiplied by 2, as shown above. The factors 2 or 1/2 arise because the gravitational field is caused by the
stressenergy tensorThe stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and nongravitational force fields...
which is a second rank tensor, as opposed to the electromagnetic field which is caused by the
fourcurrentIn special and general relativity, the fourcurrent is the Lorentz covariant fourvector that replaces the electromagnetic current density, or indeed any conventional charge current density...
, a first rank tensor. This difference becomes intuitively clear when one compares noninvariance of relativistic mass to electric
charge invarianceCharge invariance refers to the fixed electrostatic potential of a particle, regardless of speed. For example, an electron has a specific rest charge. Accelerate that electron, and the charge remains the same . The key word here is relativistic...
. This is often referred to as gravity being a spin2 field and electromagnetism being a spin1 field.
In Planck units
From comparison of GEM equations and Maxwell's equations it is obvious that −1/(4π
G) is the gravitational analog of vacuum permittivity
ε_{0}. Adopting
Planck unitsIn physics, Planck units are physical units of measurement defined exclusively in terms of five universal physical constants listed below, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units. Planck units elegantly simplify...
normalizes
G,
c and 1/(4π
ε_{0}) 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
electromagnetismElectromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
: electrostatic charges of identical sign repel each other, while masses attract each other. Hence the GEM equations are simply
Maxwell's equationsMaxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
with
massMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
(or mass density) substituting for
chargeIn physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges are associated with conserved quantum numbers.Formal definition:...
(or
charge densityThe linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...
), and −
G replacing the Coulomb force constant 1/(4π
ε_{0}).
The following Table summarizes the results thus far:
4π appears in both the GEM and Maxwell equations, because
Planck unitsIn physics, Planck units are physical units of measurement defined exclusively in terms of five universal physical constants listed below, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units. Planck units elegantly simplify...
normalize
G and 1/(4π
ε_{0}) to 1, and not 4π
G and 1/
ε_{0}.
Higherorder effects
Some higherorder 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 will 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
toroidToroid may refer to*Toroid , a doughnutlike solid whose surface is a torus.*Toroidal inductors and transformers which have wire windings on circular ring shaped magnetic cores.*Vortex ring, a toroidal flow in fluid mechanics....
al mass undergoing
minor axisIn geometry, the semiminor axis is a line segment associated with most conic sections . One end of the segment is the center of the conic section, and it is at right angles with the semimajor axis...
rotational acceleration (accelerating "smoke ring" rotation) will tend to pull matter through the throat (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
gforceThe gforce associated with an object is its acceleration relative to freefall. This acceleration experienced by an object is due to the vector sum of nongravitational forces acting on an object free to move. The accelerations that are not produced by gravity are termed proper accelerations, and...
s.
Consider a toroidal mass with two degrees of rotation (both major axis and minoraxis spin, both turning inside out and revolving). This represents a "special case" in which gravitomagnetic effects generate a
chiralA chiral phenomenon is one that is not identical to its mirror image . The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity...
corkscrewlike 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 minoraxis 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 fieldIn theoretical physics a Coriolis field is one the apparent gravitational fields felt by a rotating or forciblyaccelerated body, together with the centrifugal field and the Euler field.Mathematical expression:...
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 to be very difficult.
Gravitomagnetic fields of astronomical objects
The formula for the gravitomagnetic field
B_{g} near a rotating body can be derived from the GEM equations. It is given by:
where
L is the
angular momentumIn physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
of the body. At the equatorial plane,
r and
L are perpendicular, so their
dot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equallength sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
vanishes, and this formula reduces to:
The magnitude of angular momentum of a homogeneous ballshaped body is:
where:
 is the moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...
of a ballshaped body (see: list of moments of inertia);
 is the angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
;
 m is the mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
;
 r is the radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
;
 T is the rotational period.
Earth
Therefore, the magnitude of
EarthEarth is the third planet from the Sun, and the densest and fifthlargest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
's gravitomagnetic field at its
equatorAn equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....
is:
where
is
Earth's gravityThe gravity of Earth, denoted g, refers to the acceleration that the Earth imparts to objects on or near its surface. In SI units this acceleration is measured in metres per second per second or equivalently in newtons per kilogram...
. The force direction coincides with the angular moment direction, i.e. north.
From this calculation it follows that Earth's equatorial gravitomagnetic field is about Hz, or in units of
standard gravityStandard gravity, or standard acceleration due to free fall, usually denoted by g0 or gn, is the nominal acceleration of an object in a vacuum near the surface of the Earth. It is defined as precisely , or about...
(9.81 m/s
^{2}) divided by the speed of light. Such a field is extremely weak and requires extremely sensitive measurements to be detected. One experiment to measure such field was the
Gravity Probe BGravity Probe B is a satellitebased mission which launched on 20 April 2004 on a Delta II rocket. The spaceflight phase lasted until 2005; its aim was to measure spacetime curvature near Earth, and thereby the stress–energy tensor in and near Earth...
mission.
Pulsar
If the preceding formula is used with the second fastestspinning pulsar known,
PSR J17482446ad (which rotates 716 times per second), assuming a radius of 16 km, and two solar masses, then
equals about 166 Hz. This would be easy to notice. However, the pulsar is spinning at a quarter of the speed of light at the equator, and its radius is only three times more than its
Schwarzschild radiusThe Schwarzschild radius is the distance from the center of an object such that, if all the mass of the object were compressed within that sphere, the escape speed from the surface would equal the speed of light...
. 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.
Fringe physics
Incomplete understanding of the meaning of the similarity of the gravitomagnetic formulas, above, and
Maxwell's equationsMaxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.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 equationsThe Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
, on the other hand, is firmly part of
General RelativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. It is an approximation to the
current standard theory of gravitationGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, and has testable predictions, which are in the final stages of being directly tested by the
Gravity Probe BGravity Probe B is a satellitebased mission which launched on 20 April 2004 on a Delta II rocket. The spaceflight phase lasted until 2005; its aim was to measure spacetime curvature near Earth, and thereby the stress–energy tensor in and near Earth...
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 antigravity
Antigravity is the idea of creating a place or object that is free from the force of gravity. It does not refer to the lack of weight under gravity experienced in free fall or orbit, or to balancing the force of gravity with some other force, such as electromagnetism or aerodynamic lift...
devices;
 Eugene Podkletnov
Dr Yevgeny Podkletnov is a Russian engineer, formerly affiliated with the Materials Science Department at the Tampere University of Technology, Finland, who is best known for his controversial work on a socalled gravity shielding device...
's claims to have constructed gravityshielding devices and gravitational reflection beams.
 Any proposal to produce gravitation using electrical circuits.
Though it should be noted that the similarity of gravitomagnetic force laws to electromagnetic force laws offers some evidence of similar or shared foundational mechanisms. See also
KaluzaKlein theoryIn physics, Kaluza–Klein theory is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism. The theory was first published in 1921. It was proposed by the mathematician Theodor Kaluza who extended general relativity to a fivedimensional spacetime...
and
Heim theoryHeim theory is a physics theory, initially proposed by a German physicist, the late Burkhard Heim, that attempts to develop a theory of everything. Heim theory's six dimensional model was later extended to eight and twelve dimensions, in collaboration with W. Dröscher...
.
See also
 Linearized gravity
Linearized gravity is an approximation scheme in general relativity in which the nonlinear contributions from the spacetime metric are ignored, simplifying the study of many problems while still producing useful approximate results.The method:...
 Geodetic effect
The geodetic effect represents the effect of the curvature of spacetime, predicted by general relativity, on a vector carried along with an orbiting body...
 Gravitational radiation
 Gravity Probe B
Gravity Probe B is a satellitebased mission which launched on 20 April 2004 on a Delta II rocket. The spaceflight phase lasted until 2005; its aim was to measure spacetime curvature near Earth, and thereby the stress–energy tensor in and near Earth...
 Framedragging
Einstein's general theory of relativity predicts that nonstatic, stationary massenergy distributions affect spacetime in a peculiar way giving rise to a phenomenon usually known as framedragging...
Further reading
in
External links
 Gravity Probe B: Testing Einstein's Universe
 Gyroscopic Superconducting Gravitomagnetic Effects news on tentative result of European Space Agency (esa) research
 In Search of gravitomagnetism, NASA, 20 April 2004.
 Gravitomagnetic London MomentNew test of General Relativity?
 Measurement of Gravitomagnetic and Acceleration Fields Around Rotating Superconductors M. Tajmar, et al., 17 October 2006.
 Test of the LenseThirring effect with the MGS Mars probe, New Scientist
New Scientist is a weekly nonpeerreviewed Englishlanguage international science magazine, which since 1996 has also run a website, covering recent developments in science and technology for a general audience. Founded in 1956, it is published by Reed Business Information Ltd, a subsidiary of...
, January 2007.