Gravitomagnetism

# Gravitomagnetism

Overview
Gravitomagnetism refers to a set of formal analogies
Analogy
Analogy 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 equations
Einstein field equations
The 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 relativity
General relativity
General 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 Heaviside
Oliver Heaviside
Oliver Heaviside was a self-taught 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.

This approximate reformulation of gravitation
Gravitation
Gravitation, 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 relativity
General relativity
General 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 reference
Frame of reference
A 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.
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Encyclopedia
Gravitomagnetism refers to a set of formal analogies
Analogy
Analogy 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 equations
Einstein field equations
The 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 relativity
General relativity
General 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 Heaviside
Oliver Heaviside
Oliver Heaviside was a self-taught 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 gravitation
Gravitation
Gravitation, 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 relativity
General relativity
General 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 reference
Frame of reference
A 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 relativity
Special relativity
Special 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 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 jet
Relativistic jet
Relativistic 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 Penrose
Roger Penrose
Sir 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 quasar
Quasar
A quasi-stellar 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 point-like, similar to stars, rather than...

s and active galactic nuclei
Active galactic nucleus
An 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, ultra-violet, X-ray 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 University
Stanford University
The 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 B
Gravity Probe B
Gravity Probe B is a satellite-based 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 Laser-ranging Operation
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 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 relativity
General relativity
General 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 field
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...

produced by a rotating object (or any rotating mass-energy) can, in a particular limiting case, be described by equations that have the same form as the magnetic field
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;...

in classical electromagnetism
Classical electromagnetism
Classical 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 field
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...

or reasonably flat spacetime, the gravitational analogs to Maxwell's equations
Maxwell's equations
Maxwell'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 electromagnetism
Electromagnetism
Electromagnetism 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 SI
Si
Si, 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:
• Eg is the static gravitational field
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
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

;
• Bg is the gravitomagnetic field;
• B is the magnetic field
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;...

;
• ρg is mass density
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
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...

:
• Jg is mass current density
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...

(Jg = ρg vρ, where vρ is the velocity
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 mass flow generating the gravitomagnetic field);
• J is electric current density
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
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
Speed 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 light
Speed of light
The 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 relativity
General relativity
General 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 force
Lorentz force
In 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
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 particle
Test particle
In 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
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 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 force
Lorentz force
In 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 stress-energy tensor
Stress-energy tensor
The 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 a second rank tensor, as opposed to the electromagnetic field which is caused by the four-current
Four-current
In 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...

, a first rank tensor. This difference becomes intuitively clear when one compares non-invariance of relativistic mass to electric charge invariance
Charge invariance
Charge 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 spin-2 field and electromagnetism being a 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 ε0. Adopting Planck units
Planck units
In 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/(4πε0) to 1, thereby eliminating these constants from both sets of equations. The two sets of equations then become identical but for the minus sign preceding 4π in the GEM equations. These minus signs stem from an essential difference between gravity and electromagnetism
Electromagnetism
Electromagnetism 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 equations
Maxwell's equations
Maxwell'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 mass
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:...

(or mass density) substituting for charge
Charge (physics)
In 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 density
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...

), and −G replacing the Coulomb force constant 1/(4πε0).

The following Table summarizes the results thus far:
Common Structure of the Maxwell and

GEM Equations Given Planck units
Planck units
In 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...

.

ι = +1 (Maxwell) or −1 (GEM).

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

## 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
Toroid
Toroid 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 toroidal flow in fluid mechanics....

al mass undergoing minor axis
Semi-minor axis
In 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...

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-force
G-force
The g-force associated with an object is its acceleration relative to free-fall. This acceleration experienced by an object is due to the vector sum of non-gravitational 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 minor-axis spin, both turning inside out and revolving). This represents a "special case" in which gravitomagnetic effects generate a chiral
Chirality (physics)
A 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...

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
Coriolis field
In theoretical physics a Coriolis field is one the apparent gravitational fields felt by a rotating or forcibly-accelerated 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 Bg near a rotating body can be derived from the GEM equations. It is given by:

where L is the angular momentum
Angular momentum
In 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 product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length 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 ball-shaped body is:

where:
• is the moment of inertia
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 ball-shaped body (see: list of moments of inertia);
• is the angular velocity
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
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 Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest 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 equator
Equator
An 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 gravity
Earth's gravity
The 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 gravity
Standard gravity
Standard 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/s2) 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 B
Gravity Probe B
Gravity Probe B is a satellite-based 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 fastest-spinning pulsar known, PSR J1748-2446ad (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 radius
The 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 equations
Maxwell's equations
Maxwell'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 equations
Einstein field equations
The 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 Relativity
General relativity
General 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 gravitation
General relativity
General 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 B
Gravity Probe B
Gravity Probe B is a satellite-based 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 anti-gravity
Anti-gravity
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 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
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.

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 Kaluza-Klein theory
Kaluza–Klein theory
In 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 five-dimensional spacetime...

and Heim theory
Heim theory
Heim 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...

.

• Linearized gravity
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
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...

• Gravity Probe B
Gravity Probe B
Gravity Probe B is a satellite-based 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...

• Frame-dragging
Frame-dragging
Einstein's general theory of relativity predicts that non-static, stationary mass-energy distributions affect spacetime in a peculiar way giving rise to a phenomenon usually known as frame-dragging...