View source for Gravitoelectromagnetism

{{Short description|Analogies between Maxwell's and Einstein's field equations}}
{{About|the gravitational analog of electromagnetism as a whole|the specific gravitational analog of magnetism|frame-dragging}}
{{Use dmy dates|date=May 2022}}
[[File:Gravity Probe B Confirms the Existence of Gravitomagnetism-en.jpg|thumb|400 px|Diagram regarding the confirmation of gravitomagnetism by [[Gravity&nbsp;Probe&nbsp;B]]]]

'''Gravitoelectromagnetism''', abbreviated '''GEM''', refers to a set of [[Analogy|formal analogies]] between the equations for [[electromagnetism]] and [[General relativity|relativistic]] [[gravitation]]; specifically: between [[Maxwell's field equations]] and an approximation, valid under certain conditions, to the [[Einstein field equations]] for [[general relativity]]. '''Gravitomagnetism''' is a widely used term referring specifically to the kinetic effects of gravity, in analogy to the [[Magnetic field|magnetic]] effects of moving electric charge.<ref>{{cite journal|author=David Delphenich|year=2015|title=Pre-metric electromagnetism as a path to unification|url=https://www.worldscientific.com/doi/abs/10.1142/9789814719063_0023|journal=Unified Field Mechanics: Natural Science Beyond the Veil of Spacetime, Morgan State University, USA, 16–19 November 2014|pages=215–220|doi=10.1142/9789814719063_0023|arxiv=1512.05183|isbn=978-981-4719-05-6|s2cid=118596433}}</ref> The most common version of GEM is valid only far from isolated sources, and for slowly moving [[test particles]].

The analogy and equations differing only by some small factors were first published in 1893, before general relativity, by [[Oliver Heaviside]] as a separate theory expanding [[Newton's law of universal gravitation]].<ref>
{{cite book
 |author=O. Heaviside
 |author-link=Oliver Heaviside
 |year=1893
 |publisher=The Electrician
 |title=Electromagnetic Theory: A Gravitational and Electromagnetic Analogy
 |url=https://archive.org/details/electromagnetic00heavgoog/page/n481/mode/2up
 |volume=1
 |pages=455–464
}}</ref>{{better source needed|reason=one that mentions the connection between Heaviside and modern GEM|date=June 2014}}

== Background ==
This approximate reformulation of [[gravitation]] as described by [[general relativity]] in the [[Weak field approximation|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, rotating object will experience acceleration that deviates from that predicted by a purely Newtonian gravity (gravitoelectric) 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 jet]]s. [[Roger Penrose]] had proposed a mechanism that relies on [[frame-dragging]]-related effects for extracting energy and momentum from rotating [[black hole]]s.<ref>
{{cite journal
 |author=R. Penrose |author-link=Roger Penrose
 |title=Gravitational collapse: The role of general relativity
 |journal=Rivista del Nuovo Cimento
 |volume=Numero Speciale 1 |pages=252–276
 |date=1969
 |bibcode=1969NCimR...1..252P
}}</ref> [[Reva Kay Williams]], University of Florida, developed a rigorous proof that validated [[Penrose process|Penrose's mechanism]].<ref>
{{cite journal
 |author=R.K. Williams
 |title=Extracting x rays, γ rays, and relativistic e<sup>−</sup>e<sup>+</sup> pairs from supermassive Kerr black holes using the Penrose mechanism
 |journal=Physical Review
 |volume=51 |issue=10 |pages=5387–5427
 |date=1995
 |doi=10.1103/PhysRevD.51.5387
 |bibcode=1995PhRvD..51.5387W
 |pmid=10018300
}}</ref> Her model showed how the [[Lense–Thirring precession|Lense–Thirring effect]] could account for the observed high energies and luminosities of [[quasar]]s and [[Active galactic nucleus|active galactic nuclei]]; the collimated jets about their polar axis; and the asymmetrical jets (relative to the orbital plane).<ref>
{{cite journal
 |author=R.K. Williams
 |title=Collimated escaping vortical polar e<sup>−</sup>e<sup>+</sup> jets intrinsically produced by rotating black holes and Penrose processes
 |journal=The Astrophysical Journal
 |volume=611 |issue= 2|pages=952–963
 |date=2004
 |doi=10.1086/422304
 |bibcode=2004ApJ...611..952W
 |arxiv=astro-ph/0404135 |s2cid=1350543
}}</ref><ref name="Danehkar">{{cite journal |last=Danehkar|first=A.|date=2020|title=Gravitational fields of the magnetic-type|journal=[[International Journal of Modern Physics D]]|volume=29|issue=14|pages=2043001|bibcode=2020IJMPD..2943001D|arxiv=2006.13287|doi=10.1142/S0218271820430014|doi-access=free}}</ref> All of those observed properties could be explained in terms of gravitomagnetic effects.<ref>
{{cite conference
 |author=R.K. Williams
 |title=Gravitomagnetic field and Penrose scattering processes
 |book-title=Annals of the New York Academy of Sciences
 |volume=1045 |pages=232–245
 |date=2005
}}</ref> Williams's application of Penrose's mechanism can be applied to black holes of any size.<ref>
{{cite conference
 |author=R.K. Williams
 |date=2001
 |title=Collimated energy–momentum extraction from rotating black holes in quasars and microquasars using the Penrose mechanism
 |book-title=AIP Conference Proceedings
 |volume=586 |pages=448–453
 |arxiv=astro-ph/0111161
 |doi=10.1063/1.1419591
 |bibcode=2001AIPC..586..448W
}}</ref> Relativistic jets can serve as the largest and brightest form of validations for gravitomagnetism.

A group at [[Stanford University]] is currently{{when|date=May 2023}} analyzing data from the first direct test of GEM, the [[Gravity Probe B]] satellite experiment, to see whether they are consistent with gravitomagnetism.<ref>Gravitomagnetism in Quantum Mechanics, 2014 https://www.slac.stanford.edu/pubs/slacpubs/14750/slac-pub-14775.pdf</ref> The [[Apache Point Observatory Lunar Laser-ranging Operation]] also plans to observe gravitomagnetism effects.{{Citation needed|date=June 2014}}

{{gallery
|title=Physical analogues of fields<ref>Gravitation and Inertia, I. Ciufolini and J.A. Wheeler, Princeton Physics Series, 1995, {{ISBN|0-691-03323-4}}</ref>
|File:Gravitomagnetic field due to angular momentum.svg|'''Gravitomagnetism''' – gravitomagnetic field {{math|'''H'''}} due to (total) [[angular momentum]] {{math|'''J'''}}.
|File:Magnetic field due to dipole moment.svg|'''[[Electromagnetism]]''' – [[magnetic field]] {{math|'''B'''}} due to a [[Magnetic moment|dipole moment]] {{math|'''m'''}}...
|File:Magnetic field due to current.svg|... or, equivalently, [[Electric current|current]] {{mvar|I}}, same field profile, and field generation due to rotation.
|File:Rotational fluid drag of a solid sphere immersed in fluid.svg|'''[[Fluid mechanics]]''' – rotational [[Drag (physics)|fluid drag]] of a solid sphere immersed in fluid, analogous directions and senses of rotation as magnetism, analogous interaction to frame dragging for the gravitomagnetic interaction.
}}

== Equations ==

According to [[general relativity]], 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]]. Starting from the basic equation of general relativity, the [[Einstein field equation]], and assuming a weak gravitational field or reasonably [[flat spacetime]], the gravitational analogs to [[Maxwell's equations]] for [[electromagnetism]], called the "GEM equations", can be derived. GEM equations compared to Maxwell's equations are:<ref>
{{citation
 |author1=B. Mashhoon |author2=F. Gronwald |author3=H.I.M. Lichtenegger
 |title=Gravitomagnetism and the Clock Effect
 |work=Gyros, Clocks, Interferometers ...: Testing Relativistic Gravity in Space
 |year=2001
 |pages=83–108
 |volume=562
 |series=Lecture Notes in Physics
 |arxiv=gr-qc/9912027
 |bibcode = 2001LNP...562...83M |doi=10.1007/3-540-40988-2_5 |isbn=978-3-540-41236-6 |citeseerx=10.1.1.340.8408 |s2cid=32411999
}}</ref><ref>
{{cite journal
 |author1=S.J. Clark |author2=R.W. Tucker |title=Gauge symmetry and gravito-electromagnetism
 |journal=Classical and Quantum Gravity
 |volume=17 |issue= 19|pages=4125–4157
 |date=2000
 |doi=10.1088/0264-9381/17/19/311
 |arxiv = gr-qc/0003115 |bibcode = 2000CQGra..17.4125C |s2cid=15724290
}}</ref>

{| class="wikitable"
|-
 ! GEM equations
 ! Maxwell's equations
|-
 |<math> \nabla \cdot \mathbf{E}_\text{g} = -4 \pi G \rho_\text{g} \ </math>
 |<math> \nabla \cdot \mathbf{E} =  \frac{\rho}{\epsilon_0} </math>
|-
 |<math> \nabla \cdot \mathbf{B}_\text{g} = 0 \ </math>
 |<math> \nabla \cdot \mathbf{B} = 0 \ </math>
|-
 |<math> \nabla \times \mathbf{E}_\text{g} + \frac{\partial \mathbf{B}_\text{g} } {\partial t} = 0 </math>
 |<math> \nabla \times \mathbf{E} + \frac{\partial \mathbf{B} } {\partial t} = 0 </math>
|-
 |<math> \nabla \times \mathbf{B}_\text{g} - \frac{1}{c^2} \frac{\partial \mathbf{E}_\text{g}} {\partial t} = -\frac{4 \pi G}{c^2} \mathbf{J}_\text{g} </math>
 |<math> \nabla \times \mathbf{B} - \frac{1}{c^2} \frac{\partial \mathbf{E}} {\partial t} = \frac{1}{\epsilon_0 c^2} \mathbf{J} </math>
|-
|}
where:
* '''E'''<sub>g</sub> is the gravitoelectric field (conventional [[gravitational field]]), with SI unit m⋅s<sup>−2</sup>
* '''E''' is the [[electric field]]
* '''B'''<sub>g</sub> is the gravitomagnetic field, with SI unit s<sup>−1</sup>
* '''B''' is the [[magnetic field]]
* ''ρ''<sub>g</sub> is [[density|mass density]], with SI unit kg⋅m<sup>−3</sup>
* ''ρ'' is [[charge density]]
* '''J'''<sub>g</sub> is mass current density or [[mass flux]], with SI unit kg⋅m<sup>−2</sup>⋅s<sup>−1</sup>
* '''J''' is electric [[current density]]
* ''G'' is the [[gravitational constant]]
* ''ε''<sub>0</sub> is the [[vacuum permittivity]]
* ''c'' is both the [[Speed of gravity|speed of propagation of gravity]] and the [[speed of light]].

=== Potentials ===

Faraday's law of induction (third line of the table) and the Gaussian law for the gravitomagnetic field (second line of the table) can be solved by the definition of a gravitation potential <math>\phi_\text{g}</math> and the vector potential <math>\mathbf{A}_\text{g}</math> according to:
: <math>\mathbf{E}_\text{g} := - \mbox{grad}~\phi_\text{g} - \frac{\partial \mathbf{A}_\text{g}}{\partial t} = - \nabla~\phi_\text{g} - \frac{\partial \mathbf{A}_\text{g}}{\partial t}</math>
and
: <math>\mathbf{B}_\text{g}  := \mbox{rot}~\mathbf{A}_\text{g} = \nabla \times \mathbf{A}_\text{g}</math>

Inserting this four potentials <math>\left( \phi_\text{g}, \mathbf{A}_\text{g} \right)</math> into the Gaussian law for the gravitation field (first line of the table) and Ampère's circuital law (fourth line of the table) and applying the [[Lorenz gauge]] the following inhomogeneous wave-equations are obtained:
: <math> - \frac{1}{c^2} \frac{\partial^2 \phi_\text{g}}{\partial t^2} + \nabla^2~\phi_\text{g}  = 4 \pi G \rho_\text{g} </math>
: <math>- \frac{1}{c^2} \frac{\partial^2 \mathbf{A}_\text{g}}{\partial t^2} + \nabla^2~\mathbf{A}_\text{g} = \frac{4 \pi G}{c^2}  \rho_\text{g} \mathbf{v} </math>

For a stationary situation (<math> \partial \phi_\text{g}/\partial t = 0</math>) the [[Poisson equation]] of the classical gravitation theory is obtained. In a vacuum (<math>\rho_\text{g} = 0</math>) a [[wave equation]] is obtained under non-stationary conditions. GEM therefore predicts the existence of [[gravitational waves]]. In this way GEM can be regarded as a generalization of Newton's gravitation theory.

The wave equation for the gravitomagnetic potential <math>\mathbf{A}_\text{g}</math> can also be solved for a rotating spherical body (which is a stationary case) leading to gravitomagnetic moments.

=== 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:

{| class="wikitable"
|-
! GEM equation
! EM equation
|-
| <math>\mathbf{F}_\text{g} = m \left( \mathbf{E}_\text{g} \ + \ \mathbf{v} \times 4 \mathbf{B}_\text{g} \right) </math>
| <math>\mathbf{F} = q \left( \mathbf{E} \ + \ \mathbf{v} \times \mathbf{B} \right) </math>
|-
|}
where:
* '''v''' is the [[velocity]] of the [[test particle]]
* ''m'' is the [[mass]] of the test particle
* ''q'' is the [[electric charge]] of the test particle.

=== Poynting vector ===
The GEM Poynting vector compared to the electromagnetic [[Poynting vector]] is given by:<ref>
{{Cite book |last=B. Mashhoon |title=The Measurement of Gravitomagnetism: A Challenging Enterprise. |date=2007 |publisher=Nova Science Publishers |editor-last=Iorio |chapter=Gravitoelectromagnetism: A Brief Review |bibcode=2003gr.qc....11030M|eprint=gr-qc/0311030 }}</ref>
{| class="wikitable"
|-
! GEM equation
! EM equation
|-
| <math>\mathcal{S}_\text{g} = -\frac{c^2}{4 \pi G} \mathbf{E}_\text{g} \times 4 \mathbf{B}_\text{g} </math>
| <math>\mathcal{S} = c^2 \varepsilon_0 \mathbf{E} \times \mathbf{B} </math>
|-
|}

=== 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 '''B'''<sub>g</sub> in the GEM equations must be multiplied by −{{sfrac|1|2''c''}} and '''E'''<sub>g</sub> by −1.  These factors variously modify the analogues of the equations for the Lorentz force<!-- and the Poynting vector-->.  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]] tensor.  This difference becomes clearer when one compares non-invariance of [[relativistic mass]] to electric [[charge invariance]]. This can be traced back to the spin-2 character of the gravitational field, in contrast to the electromagnetism being a spin-1 field.<ref>
{{cite book
 |author=B. Mashhoon
 |title=Reference Frames and Gravitomagnetism – Proceedings of the XXIII Spanish Relativity Meeting |chapter=Gravitoelectromagnetism
 |date=2000
 |arxiv=gr-qc/0011014 |bibcode = 2001rfg..conf..121M |doi = 10.1142/9789812810021_0009
 |pages=121–132
 |isbn=978-981-02-4631-0
 |citeseerx=10.1.1.339.476
 |s2cid=263798773
}}</ref> (See ''[[Relativistic wave equations]]'' 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{{Citation needed|date=November 2023}}. This can be expressed as an attractive or repulsive gravitomagnetic component.

Gravitomagnetic arguments also predict that a flexible or fluid [[toroid]]al mass undergoing [[semi-minor axis|minor 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]]s.<ref>
{{cite journal
 |author=R.L. Forward
 |title=Guidelines to Antigravity
 |journal=American Journal of Physics
 |volume=31 |issue=3 |pages=166–170
 |date=1963
 |doi=10.1119/1.1969340
 |bibcode = 1963AmJPh..31..166F
}}</ref>

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 [[Chirality (physics)|chiral]] corkscrew-like gravitational field around the object. The reaction forces to dragging at the inner and outer equators would normally be expected to be equal and opposite in magnitude and direction 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.{{Citation needed|date=May 2012}}

Modelling this complex behaviour as a curved spacetime problem has yet to be done and is believed to be very difficult.{{Citation needed|date=May 2012}}

== Gravitomagnetic fields of astronomical objects ==
{{disputed section|date=May 2013}}
{{refimprove section | date=February 2024}}

A rotating spherical body with a homogeneous density distribution produces a stationary gravitomagnetic potential, which is described by:
: <math> \nabla^2 \mathbf{A}_\text{g} = - \frac{4 \pi G}{c^2}  \rho_g \mathbf{v} </math>

Due to the body's angular velocity <math>\mathbf{\omega}</math> the velocity inside the body can be described as <math>\mathbf{v} = \mathbf{\omega} \times \mathbf{r}</math>. Therefore 
: <math> \nabla^2 \mathbf{A}_\text{g} = - \frac{4 \pi G}{c^2}  \rho_g \mathbf{\omega} \times \mathbf{r} </math>
has to be solved to obtain the gravitomagnetic potential <math>\mathbf{A}_\text{g}</math>. The analytical solution outside of the body is (see for example<ref>
{{cite book
 |author=A. Malcherek
 |title=Elektromagnetismus und Gravitation
 |edition=2.
 |publisher = Springer-Vieweg
 |date=2023
 |isbn=978-3-658-42701-6
}}</ref>):
: <math>\mathbf{A}_{\text{g}} = - \frac{G}{2 c^2} \frac{\mathbf{L}  \times \mathbf{r}}{r^3} \quad \text{with} \quad \mathbf{L} = \frac{2}{5} m R^2 \mathbf{\omega} \quad \text{or} \quad L=I_\text{ball} \omega= \frac{2 m R^2}{5} \frac{2 \pi}{T}\quad</math>
where:
* <math>\mathbf{L}</math> is the [[angular momentum]] vector;
* <math>I_\text{ball} = \frac{2 m R^2}{5}</math> is the [[moment of inertia]] of a ball-shaped body (see: ''[[List of moments of inertia]]'');
* <math> \omega \ </math> is the [[angular velocity]];
* ''m'' is the [[mass]];
* ''R'' is the [[radius]];
* ''T'' is the rotational period.

The formula for the gravitomagnetic field '''B'''<sub>g</sub> can now be obtained by:
: <math>\mathbf{B}_\text{g} = \nabla \times \mathbf{A}_\text{g} = \frac{G }{2 c^2} \frac{\mathbf{L} - 3(\mathbf{L} \cdot \mathbf{r}/r) \mathbf{r}/r}{r^3},</math>

It is exactly half of the [[Lense–Thirring precession]] rate. This suggests that the gravitomagnetic analog of the [[G-factor (physics)|''g''-factor]] is two. This factor of two can be explained completely analogous to the electron's ''g''-factor by taking into account relativistic calculations. At the equatorial plane, '''r''' and '''L''' are perpendicular, so their [[dot product]] vanishes, and this formula reduces to:
: <math>\mathbf{B}_\text{g} = \frac{G }{2 c^2} \frac{\mathbf{L}}{r^3},</math>

[[Gravitational wave]]s have equal gravitomagnetic and gravitoelectric components.<ref>{{Cite book |author1=((Pfister, Herbert, 1936-))|url=https://www.worldcat.org/oclc/904397831 |title=Inertia and gravitation : the fundamental nature and structure of space–time |author2=King, Markus |date=24 February 2015 |page=147 |isbn=978-3-319-15036-9 |publisher=Springer |location=Cham |oclc=904397831}}</ref>

=== Earth ===
Therefore, the magnitude of [[Earth]]'s gravitomagnetic field at its [[equator]] is:
: <math>B_\text{g,Earth} = \frac{G }{5 c^2} \frac{m}{r} \frac{2 \pi}{T} = \frac{2 \pi r g}{5c^2 T},</math>
where <math> g = G \frac{m}{r^2} </math> is [[Earth's gravity]]. The field direction coincides with the angular moment direction, i.e. north.

From this calculation it follows that the strength of the Earth's equatorial gravitomagnetic field is about {{val|1.012|e=-14|u=Hz}}.<ref>2''π''[[Earth radius|{{Earth radius}}]][[Gravity of Earth|''g''<sub>0</sub>]]/(5''c''<sup>2</sup> × 1&nbsp;[[day]])</ref> 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]] 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&nbsp;km and a mass of two solar masses, then
: <math>B_\text{g}  = \frac{2 \pi G m}{5rc^2 T}</math>
equals about 166&nbsp;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 its [[Schwarzschild radius]]. 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 ==
{{Unreferenced section|date=July 2024}}
<!-- Commented out until the reference has been reviewed and approved so that it becomes a reliable source. Also this version would need to be described in the article. 
In 2013 there appeared an article <ref>  Piotr Ogonowski, Piotr Skindzier, Maxwell-like picture of General Relativity and its Planck limit, arXiv:1301.2758, http://arxiv.org/abs/1301.2758</ref> that shows, there exists Lorentz invariant version of GEM equations that reproduce General Relativity formula, what opens new areas for research.{{dubious|date=August 2013}}
-->
While Maxwell's equations are invariant under [[Lorentz transformation#Transformation of the electromagnetic field|Lorentz transformation]]s, the GEM equations are not.  The fact that ''&rho;''<sub>g</sub> and ''j''<sub>g</sub> do not form a [[four-vector]] (instead they are merely a part of the [[stress–energy tensor]]) is the basis of this difference.{{Citation needed|reason=Explanation needed why these exclude each other, other wiki page should be enough.|date=June 2017}}

Although GEM may hold approximately in two different reference frames connected by a [[Lorentz boost]], 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 ==
{{cols|colwidth=27em}}
* [[Anti-gravity]]
* [[Artificial gravity]]
* [[Frame-dragging]]
* [[Geodetic effect]]
* [[Gravitational radiation]]
* [[Gravity Probe B]]
* [[Kaluza–Klein theory]]
* [[Linearized gravity]]
* [[Modified Newtonian dynamics]]
* [[Non-Relativistic Gravitational Fields]]
* [[Speed of gravity#Electrodynamical analogies|Speed of gravity § Electrodynamical analogies]]
* [[Stationary spacetime]]
{{colend}}

== References ==
{{reflist}}

== Further reading ==

{{refbegin|30em}}
; Books :
* {{cite book
 |author1=M. P. Hobson |author2=G. P. Efstathiou |author3=A. N. Lasenby |title=General Relativity: An Introduction for Physicists
 |pages = 490–491
 |isbn = 9780521829519
 |year=2006
 |publisher = Cambridge University Press 
 |url=https://books.google.com/books?id=xma1QuTJphYC&q=hobson+general+relativity+gravitomagnetic+field&pg=PA496
}}
* {{cite book
 |author=L. H. Ryder
 |title=Introduction to General Relativity
 |pages = 200–207
 |isbn = 9780521845632
 |year=2009
 |publisher = Cambridge University Press 
 |url=https://books.google.com/books?id=JhzNo9twSXgC&q=ryder+general+relativity+gravitomagnetic+field&pg=PA203
}}
* {{cite book
 |author=J. B. Hartle
 |title=Gravity: An Introduction to Einstein's General Relativity
 |pages = 296, 303
 |isbn = 9780805386622
 |year=2002
 |publisher = Addison-Wesley
}}
* {{cite book
 |author=S. Carroll
 |title=Spacetime and Geometry: An Introduction to General Relativity
 |page = 281
 |isbn = 9780805387322
 |year=2003
 |publisher = Addison-Wesley
}}
* {{cite book
 |author=J.A. Wheeler
 |author-link=John Archibald Wheeler
 |chapter=Gravity's next prize: Gravitomagnetism
 |title=A journey into gravity and spacetime
 |pages=232–233
 |publisher=Scientific American Library
 |date=1990
 |isbn=978-0-7167-5016-1
}}
* {{cite book
 |editor=L. Iorio 
 |title=Measuring Gravitomagnetism: A Challenging Enterprise
 |publisher=Nova
 |isbn=978-1-60021-002-0
 |date=2007
}}
* {{cite book
 |last=O.D. Jefimenko
 |author-link=Oleg D. Jefimenko
 |title=Causality, electromagnetic induction, and gravitation : a different approach to the theory of electromagnetic and gravitational fields
 |publisher=Electret Scientific
 |date=1992
 |isbn=978-0-917406-09-6
}}
* {{cite book
 |last=O.D. Jefimenko
 |author-link=Oleg D. Jefimenko
 |title=Gravitation and Cogravitation
 |publisher=Electret Scientific
 |date=2006
 |isbn=978-0-917406-15-7
}}

; Papers :
* {{cite arxiv
 |arxiv=0802.3346
 |title=The gravitomagnetic field of a sphere, Gravity Probe B and the LAGEOS satellites
 |author=J. Biemond
 |date=2008
}}
* {{cite journal
 |author1=S.J. Clark |author2=R.W. Tucker |title=Gauge symmetry and gravito-electromagnetism
 |journal=Classical and Quantum Gravity
 |volume=17 |issue= 19|pages=4125–4157
 |date=2000
 |doi=10.1088/0264-9381/17/19/311
 |arxiv = gr-qc/0003115 |bibcode = 2000CQGra..17.4125C |s2cid=15724290
}}
* {{cite journal
 |author=R.L. Forward
 |title=Guidelines to Antigravity
 |journal=American Journal of Physics
 |volume=31 |issue=3 |pages=166–170
 |date=1963
 |doi=10.1119/1.1969340
 |bibcode = 1963AmJPh..31..166F
}}
* {{cite journal
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== External links ==
* [http://einstein.stanford.edu/ Gravity Probe B: Testing Einstein's Universe]
* [http://www.esa.int/SPECIALS/GSP/SEM0L6OVGJE_0.html Gyroscopic Superconducting Gravitomagnetic Effects] news on tentative result of European Space Agency ([[esa]]) research
* [http://www.nasa.gov/vision/universe/solarsystem/19apr_gravitomagnetism.html In Search of Gravitomagnetism] {{Webarchive|url=https://web.archive.org/web/20061009003037/http://www.nasa.gov/vision/universe/solarsystem/19apr_gravitomagnetism.html |date=9 October 2006 }}, NASA, 20 April 2004.
* [http://www.physorg.com/news12054.html Gravitomagnetic London Moment – New test of General Relativity?]
* [http://www.arxiv.org/abs/gr-qc/0610015 Measurement of Gravitomagnetic and Acceleration Fields Around Rotating Superconductors] M. Tajmar, et al., 17 October 2006.
* [https://www.newscientist.com/article/mg19325874.800-loner-stakes-claim-to-gravity-prize.html Test of the Lense–Thirring effect with the MGS Mars probe], ''[[New Scientist]]'', January 2007.

{{theories of gravitation}}

[[Category:General relativity]]
[[Category:Effects of gravity]]
[[Category:Tests of general relativity]]