Gravitomagnetism
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Gravitoelectromagnetism, abbreviated GEM, refers to a set of formal analogies between the equations for
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
and relativistic gravitation; specifically: between Maxwell's field equations and an approximation, valid under certain conditions, to the
Einstein field equations In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...
for
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. Gravitomagnetism is a widely used term referring specifically to the kinetic effects of gravity, in analogy to the magnetic effects of moving electric charge. The most common version of GEM is valid only far from isolated sources, and for slowly moving
test particles In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insuf ...
. The analogy and equations differing only by some small factors were first published in 1893, before general relativity, by
Oliver Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed ...
as a separate theory expanding Newton's law.


Background

This approximate reformulation of gravitation as described by
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
in the weak field limit makes an apparent field appear in a frame of reference different from that of a freely moving inertial body. This apparent field may be described by two components that act respectively like the electric and magnetic fields of electromagnetism, and by analogy these are called the ''gravitoelectric'' and ''gravitomagnetic'' fields, since these arise in the same way around a mass that a moving electric charge is the source of electric and magnetic fields. The main consequence of the ''gravitomagnetic'' field, or velocity-dependent acceleration, is that a moving object near a massive, non-axisymmetric, rotating object will experience acceleration not predicted by a purely Newtonian (gravitoelectric) gravity field. More subtle predictions, such as induced rotation of a falling object and precession of a spinning object are among the last basic predictions of general relativity to be directly tested. Indirect validations of gravitomagnetic effects have been derived from analyses of relativistic jets. Roger Penrose had proposed a mechanism that relies on
frame-dragging Frame-dragging is an effect on spacetime, predicted by Albert Einstein's general theory of relativity, that is due to non-static stationary distributions of mass–energy. A stationary field is one that is in a steady state, but the masses ca ...
-related effects 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 Lense–Thirring effect could account for the observed high energies and luminosities of quasars and
active galactic nuclei An active galactic nucleus (AGN) is a compact region at the center of a galaxy that has a much-higher-than-normal luminosity over at least some portion of the electromagnetic spectrum with characteristics indicating that the luminosity is not prod ...
; 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 University is currently analyzing data from the first direct test of GEM, the
Gravity Probe B Gravity Probe B (GP-B) was a satellite-based experiment to test two unverified predictions of general relativity: the geodetic effect and frame-dragging. This was to be accomplished by measuring, very precisely, tiny changes in the direction of ...
satellite experiment, to see whether they are consistent with gravitomagnetism. The
Apache Point Observatory Lunar Laser-ranging Operation The Apache Point Observatory Lunar Laser-ranging 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 experiments, which use retroreflectors on the ...
also plans to observe gravitomagnetism effects.


Equations

According to
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, 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 in
classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fie ...
. Starting from the basic equation of general relativity, the
Einstein field equation In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...
, and assuming a weak gravitational field or reasonably
flat spacetime In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
, the gravitational analogs to
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
for
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
, called the "GEM equations", can be derived. GEM equations compared to Maxwell's equations are: where: * Eg is the gravitoelectric field (conventional gravitational field), with SI unit m⋅s−2 * E is the electric field * Bg is the gravitomagnetic field, with SI unit s−1 * B is the magnetic field * ''ρ''g is
mass density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
with, SI unit kg⋅m−3 * ''ρ'' is
charge density In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in ...
* Jg is mass current density or
mass flux In physics and engineering, mass flux is the rate of mass flow. Its SI units are kg m−2 s−1. The common symbols are ''j'', ''J'', ''q'', ''Q'', ''φ'', or Φ (Greek lower or capital Phi), sometimes with subscript ''m'' to indicate mass is th ...
(Jg = ''ρ''gvρ, where vρ is the
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
of the mass flow), with SI unit kg⋅m−2⋅s−1 * J is electric current density * ''G'' is the gravitational constant * ''ε''0 is the vacuum permittivity * ''c'' is both the speed of propagation of gravity and the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
.


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: * v is the
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
of the
test particle In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insuf ...
* ''m'' is the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
of the test particle * ''q'' is the
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
of the test particle.


Poynting vector

The GEM Poynting vector compared to the electromagnetic Poynting vector is given by:


Scaling of fields

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 − and Eg by −1. These factors variously modify the analogues of the equations for the Lorentz force. There is no scaling choice that allows all the GEM and EM equations to be perfectly analogous. The discrepancy in the factors arises because the source of the gravitational field is the second order stress–energy tensor, as opposed to the source of the electromagnetic field being the first order
four-current In special and general relativity, the four-current (technically the four-current density) is the four-dimensional analogue of the electric current density. Also known as vector current, it is used in the geometric context of ''four-dimensional spa ...
tensor. This difference becomes clearer when one compares non-invariance of
relativistic mass The word "mass" has two meanings in special relativity: '' invariant mass'' (also called rest mass) is an invariant quantity which is the same for all observers in all reference frames, while the relativistic mass is dependent on the velocity o ...
to electric
charge invariance Charge invariance refers to the fixed value of the electric charge of a particle regardless of its motion. Like mass, total spin and magnetic moment, particle's charge quantum number remains unchanged between two reference frames in relative mot ...
. This can be traced back to the spin-2 character of the gravitational field, in contrast to the electromagnetism being a spin-1 field. (See ''
Relativistic wave equations In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the con ...
'' for more on "spin-1" and "spin-2" fields).


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 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
toroid In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its ...
al mass undergoing
minor axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lon ...
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 g-forces. 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 Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...
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 to be very difficult.


Gravitomagnetic fields of astronomical objects

The formula for the gravitomagnetic field Bg near a rotating body can be derived from the GEM equations. It is exactly half of the
Lense–Thirring precession In general relativity, Lense–Thirring precession or the Lense–Thirring effect (; named after Josef Lense and Hans Thirring) is a relativistic correction to the precession of a gyroscope near a large rotating mass such as the Earth. It is a ...
rate, and is given by: :\mathbf_\text = \frac \frac, where L is the
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
of the body. At the equatorial plane, r and L are perpendicular, so their
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
vanishes, and this formula reduces to: :\mathbf_\text = \frac \frac, The magnitude of angular momentum of a homogeneous ball-shaped body is: : L=I_\text \omega= \frac \frac where: *I_\text = \frac is the moment of inertia of a ball-shaped body (see:
list of moments of inertia Moment of inertia, denoted by , measures the extent to which an object resists rotational acceleration about a particular axis, it is the rotational analogue to mass (which determines an object's resistance to ''linear'' acceleration). The momen ...
); * \omega \ is the angular velocity; *''m'' is the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
; *''r'' is the
radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
; *''T'' is the rotational period. Gravitational waves have equal gravitomagnetic and gravitoelectric components.


Earth

Therefore, the magnitude of
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
's gravitomagnetic field at its equator is: :B_\text = \frac \frac \frac = \frac, where g = G \frac is
Earth's gravity The gravity of Earth, denoted by , is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation). It is a vector quanti ...
. The field direction coincides with the angular moment direction, i.e. north. From this calculation it follows that Earth's equatorial gravitomagnetic field is about , or . 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 Gravity Probe B (GP-B) was a satellite-based experiment to test two unverified predictions of general relativity: the geodetic effect and frame-dragging. This was to be accomplished by measuring, very precisely, tiny changes in the direction of ...
mission.


Pulsar

If the preceding formula is used with the pulsar PSR J1748-2446ad (which rotates 716 times per second), assuming a radius of 16 km, and two solar masses, then :B_\text = \frac 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 radius The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteris ...
. 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.


Lack of invariance

While Maxwell's equations are invariant under
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
s, the GEM equations are not. The fact that ''ρ''g and ''j''g do not form a
four-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
(instead they are merely a part of the stress–energy tensor) is the basis of this difference. Although GEM may hold approximately in two different reference frames connected by a
Lorentz boost In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
, there is no way to calculate the GEM variables of one such frame from the GEM variables of the other, unlike the situation with the variables of electromagnetism. Indeed, their predictions (about what motion is free fall) will probably conflict with each other. Note that the GEM equations are invariant under translations and spatial rotations, just not under boosts and more general curvilinear transformations. Maxwell's equations can be formulated in a way that makes them invariant under all of these coordinate transformations.


See also

*
Anti-gravity Anti-gravity (also known as non-gravitational field) is a hypothetical phenomenon 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 t ...
*
Artificial gravity Artificial gravity is the creation of an inertial force that mimics the effects of a gravitational force, usually by rotation. Artificial gravity, or rotational gravity, is thus the appearance of a centrifugal force in a rotating frame of re ...
*
Frame-dragging Frame-dragging is an effect on spacetime, predicted by Albert Einstein's general theory of relativity, that is due to non-static stationary distributions of mass–energy. A stationary field is one that is in a steady state, but the masses ca ...
* Geodetic effect * Gravitational radiation *
Gravity Probe B Gravity Probe B (GP-B) was a satellite-based experiment to test two unverified predictions of general relativity: the geodetic effect and frame-dragging. This was to be accomplished by measuring, very precisely, tiny changes in the direction of ...
* Kaluza–Klein theory *
Linearized gravity In the theory of general relativity, linearized gravity is the application of perturbation theory to the metric tensor that describes the geometry of spacetime. As a consequence, linearized gravity is an effective method for modeling the effects ...
* Speed of gravity § Electrodynamical analogies * Stationary spacetime * Non-Relativistic Gravitational Fields


References


Further reading


Books

* * * * * * * * *


Papers

* * * * * in * * *


External links


Gravity Probe B: Testing Einstein's Universe


news on tentative result of European Space Agency (
esa , owners = , headquarters = Paris, Île-de-France, France , coordinates = , spaceport = Guiana Space Centre , seal = File:ESA emblem seal.png , seal_size = 130px , image = Views in the Main Control Room (120 ...
) research
In Search of Gravitomagnetism
NASA, 20 April 2004.

* ttp://www.arxiv.org/abs/gr-qc/0610015 Measurement of Gravitomagnetic and Acceleration Fields Around Rotating SuperconductorsM. Tajmar, et al., 17 October 2006.
Test of the Lense–Thirring effect with the MGS Mars probe
''
New Scientist ''New Scientist'' is a magazine covering all aspects of science and technology. Based in London, it publishes weekly English-language editions in the United Kingdom, the United States and Australia. An editorially separate organisation publish ...
'', January 2007. {{theories of gravitation General relativity Effects of gravitation Tests of general relativity