Gravitomagnetism (sometimes
Gravitoelectromagnetism, abbreviated
GEM), refers to a set of
formal analogiesAnalogy is a cognitive process of transferring information 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 Einstein's theory of general 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. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
. The most common version of GEM is valid only far from isolated sources, and for slowly moving test particles.
Background
This approximate reformulation of
gravitationGravitation is a natural phenomenon by which objects with mass attract one another. In everyday life, gravitation is most commonly thought of as the agency which lends weight to objects with mass. Gravitation causes dispersed matter to coalesce, thus accounting for the existence of the Earth, the...
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. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
makes a "
fictitious forceA fictitious force, also called a pseudo force, d'Alembert force or inertial force, is an apparent force that acts on all masses in a non-inertial frame of reference, such as a rotating reference frame...
" 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 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 forceA fictitious force, also called a pseudo force, d'Alembert force or inertial force, is an apparent force that acts on all masses in a non-inertial frame of reference, such as a rotating reference frame...
" in
special relativitySpecial relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies"...
. 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 jetRelativistic jets are extremely powerful jets of plasma which emerge from 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 Lense-Thirring effect could account for the observed high energies and luminosities of
quasarA quasi-stellar radio source is a powerfully energetic and distant galaxy with an active galactic nucleus. Quasars were first identified as being high redshift sources of electromagnetic energy, including radio waves and visible light, that were point-like, similar to stars, rather than extended...
s and
active galactic nucleiAn active galactic nucleus is a compact region at the centre of a galaxy which has a much higher than normal luminosity over some or all of the electromagnetic spectrum . A galaxy hosting an AGN is called an active galaxy...
; 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 located in Stanford, California, United States...
is currently analyzing data from the first direct test of GEM, the
Gravity Probe BGravity Probe B is a satellite-based mission which launched on 20 April 2004. The spaceflight phase lasted until 2005, and data analysis is expected to continue through 2010. Its aim is 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.
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. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
, the
gravitational fieldA gravitational field is a model used within physics to explain how gravity exists in the universe. In its original concept, gravity was a force between point masses...
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 fieldMagnetic fields surround magnetic materials and electric currents and are detected by the force they exert on other magnetic materials and moving electric charges...
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 fieldA gravitational field is a model used within physics to explain how gravity exists in the universe. 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 four partial differential equations that relate the electric and magnetic fields to their sources, charge density and current density. These equations can be combined to show that light is an electromagnetic wave...
for
electromagnetismElectromagnetism is the physics of the electromagnetic field, a field that exerts a force on particles with the property of electric charge and is reciprocally affected by the presence and motion of such particles....
, called the "GEM equations", can be derived. GEM equations compared to Maxwell's equations in
SISi, si, or SI may refer to :- Places :* Mount Si, a mountain in state of Washington* Si County, county in Anhui, China* Si River, a river in China* Slovenia, a European nation Si, si, or SI may refer to (all SI unless otherwise stated):- Places :* Mount Si, a mountain in state of Washington* Si...
are:
| GEM equations |
Maxwell's equations |
|
|
|
|
|
|
|
|
where:
- Eg is the static gravitational field
A gravitational field is a model used within physics to explain how gravity exists in the universe. In its original concept, gravity was a force between point masses...
(conventional gravity, also called gravitoelectric for the sake of analogy);
- E is the electric field
In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. This electric field exerts a force on other electrically charged objects...
;
- Bg is the gravitomagnetic field;
- B is the magnetic field
Magnetic fields surround magnetic materials and electric currents and are detected by the force they exert on other magnetic materials and moving electric charges...
;
- ρ is mass density
The density of a material is defined as its mass per unit volume. The symbol of density is ρ .- Formula :Mathematically:where: is the density, is the mass, is the volume....
;
- ρem is charge density
The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume. It is measured in coulombs per metre , square metre , or cubic metre , respectively. Since there are positive as well as negative charges, the charge density can take on negative values....
:
- J 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 = ρ vρ, where vρ is the velocityIn physics, velocity is the rate of change of position. It is a vector physical quantity; both speed and direction are required to define it. In the SI system, it is measured in meters per second: or ms-1. The scalar absolute value of velocity is speed...
of the mass flow generating the gravitomagnetic field);
- Jem 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...
;
- ε0 is the vacuum permittivity;
- c is the speed of propagation of gravity
In the context of classical theories of gravitation, the speed of gravity refers to the speed at which a gravitational field propagates. This is the speed at which changes in the distribution of energy and momentum result in noticeable changes in the gravitational field which they produce.The...
(equal to, 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. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
, the speed of lightIn physics, the speed of light is a physical constant, the speed at which electromagnetic radiation, such as light, travels in free space . Its value is 299,792,458 metres per second...
).
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
In physics, mass commonly refers to any of three properties of matter, which have been shown experimentally to be equivalent: inertial mass, active gravitational mass and passive gravitational mass...
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 the rate of change of position. It is a vector physical quantity; both speed and direction are required to define it. In the SI system, it is measured in meters per second: or ms-1. The scalar absolute value of velocity is speed...
of the test particle.
Acceleration of any test particle is simply:.
In some literature, all instances of
Bg in the GEM equations are multiplied by 1/2, a factor absent from Maxwell's equations. This factor vanishes if
Bg 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 and 1/2 arise because gravitational field is caused by
stress-energy 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 non-gravitational force fields...
which is second rank tensor, as opposed to electromagnetic field which is caused by
four-currentIn special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density, or indeed any conventional charge current density. Its four components are given by:where...
which is first rank tensor. That difference becomes intuitively clear when non-invariance of relativistic mass is compared 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
spinIn particle physics and quantum mechanics, spin is a fundamental characteristic property of elementary particles including the force carriers , composite particles , and atomic nuclei....
-2 field and electromagnetism being spin-1 field.
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 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 one when expressed in terms of these units. Planck units elegantly simplify...
normalizes
G,
c and 1/(4πε
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 the physics of the electromagnetic field, a field that exerts a force on particles with the property of electric charge and is reciprocally affected by the presence and motion of such particles....
: 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 four partial differential equations that relate the electric and magnetic fields to their sources, charge density and current density. These equations can be combined to show that light is an electromagnetic wave...
with
massIn physics, mass commonly refers to any of three properties of matter, which have been shown experimentally to be equivalent: inertial mass, active gravitational mass and passive gravitational mass...
(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. It is measured in coulombs per metre , square metre , or cubic metre , respectively. Since there are positive as well as negative charges, the charge density can take on negative values....
), and -
G replacing the Coulomb force constant 1/(4πε
0).
The following Table summarizes the results thus far:
Common Structure of the Maxwell and
GEM Equations Given 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 one when expressed in terms of these units. Planck units elegantly simplify...
. |
ι = 1 (Maxwell) or -1 (GEM). |
4
π 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 one when expressed in terms of these units. Planck units elegantly simplify...
normalize
G and 1/(4πε
0) to 1, and not 4π
G and ε
0.
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
toroidToroid may refer to*Toroid , a doughnut-like solid whose surface is a torus.*Toroidal inductors and transformers which have wire windings on circular ring shaped magnetic cores.*Vortex ring, a toroid-shaped flow in fluid mechanics....
al mass undergoing
minor axisIn geometry, the semi-minor 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 semi-major 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-forceThe g-force experienced by an object is its acceleration relative to free-fall. The term g-force is considered a misnomer, as g-force is not a force but an acceleration....
s.
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
chiralA phenomenon is said to be chiral if it 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...
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 fieldIn theoretical physics a Coriolis field is the apparent gravitational field felt by a rotating or forcibly-accelerated body.In a theory that conforms to some versions of Mach's principle, this “apparent”, “fictitious” or “pseudo-gravitational” field effect can be treated as genuine.As an example,...
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.
Gravitomagnetic field of Earth
Formula for gravitomagnetic field
Bg near a rotating body can be derived from the GEM equations and is:
where
L is the
angular momentumAngular momentum is a quantity that is useful in describing the rotational state of a physical system. For a rigid body rotating around an axis of symmetry , the angular momentum can be expressed as the product of the body's moment of inertia and its angular velocity...
of the body. At the equatorial plane,
r and
L are perpendicular, so their
dot productIn mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. It is the standard inner product of the orthonormal Euclidean space...
vanishes, and this formula reduces to:
Magnitude of angular momentum of a homogeneous ball-shaped body is:
where:
- is the moment of inertia
Moment of inertia, also called mass moment of inertia or the angular mass, is a measure of an object's resistance to changes in its rotation rate. It is the rotational analog of mass. That is, it is the inertia of a rigid rotating body with respect to its rotation...
of a ball-shaped 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
In physics, mass commonly refers to any of three properties of matter, which have been shown experimentally to be equivalent: inertial mass, active gravitational mass and passive gravitational mass...
;
- 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....
;
- T is the rotational period.
Therefore, magnitude of
EarthEarth is the third planet from the Sun. It is the fifth largest of the eight planets in the solar system, and the largest of the terrestrial planets in the Solar System in terms of diameter, mass and density...
's gravitomagnetic field at it's
equatorThe equator is the intersection of the Earth's surface with the plane perpendicular to the Earth's axis of rotation and containing the Earth's center of mass. In simpler language, it is an imaginary line on the Earth's surface equidistant from the North Pole and South Pole that divides the Earth...
is:
where is the
Earth's gravityEarth's gravity, 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 m/s
2...
.
From this calculation it follows that Earth's equatorial gravitomagnetic field is about
Bg, Earth =
HzThe hertz is a unit of frequency. It is defined as the number of complete cycles per second. It is the basic unit of frequency in the International System of Units , and is used worldwide in both general-purpose and scientific contexts...
. Such an 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 satellite-based mission which launched on 20 April 2004. The spaceflight phase lasted until 2005, and data analysis is expected to continue through 2010. Its aim is to measure spacetime curvature near Earth, and thereby the stress-energy tensor in and near Earth...
experiment.
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 four partial differential equations that relate the electric and magnetic fields to their sources, charge density and current density. These equations can be combined to show that light is an electromagnetic wave...
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 Einstein's theory of general 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 unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
. 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. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
, and has testable predictions, which are in the final stages of being directly tested by the
Gravity Probe BGravity Probe B is a satellite-based mission which launched on 20 April 2004. The spaceflight phase lasted until 2005, and data analysis is expected to continue through 2010. Its aim is 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 forceIn physics, the electromagnetic force is the force that the electromagnetic field exerts on electrically charged particles. It is the electromagnetic force that holds electrons and protons together in atoms, and which hold atoms together to make molecules. The electromagnetic force operates via...
law, gravitomagnetism should not be confused with any of the following:
- Claims to have constructed anti-gravity
In physical cosmology, astronomy and celestial mechanics, anti-gravity 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 experienced in free fall or orbit, nor to balancing the force of gravity with some other force, such as...
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 so-called gravity shielding device...
's claims to have constructed gravity-shielding devices and gravitational reflection beams.
- Any proposal to produce gravitation using electrical circuits.
See also
- Anti-gravity
In physical cosmology, astronomy and celestial mechanics, anti-gravity 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 experienced in free fall or orbit, nor to balancing the force of gravity with some other force, such as...
- 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 satellite-based mission which launched on 20 April 2004. The spaceflight phase lasted until 2005, and data analysis is expected to continue through 2010. Its aim is to measure spacetime curvature near Earth, and thereby the stress-energy tensor in and near Earth...
- Frame-dragging
Albert Einstein's theory of general relativity predicts that rotating bodies drag spacetime around themselves in a phenomenon referred to as frame-dragging. The rotational frame-dragging effect was first derived from the theory of general relativity in 1918 by the Austrian physicists Josef Lense...
Further reading
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External links