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Gravitoelectromagnetism

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Analogies between Maxwell's and Einstein's field equations

This article is about the gravitational analog of electromagnetism as a whole. For the specific gravitational analog of magnetism, seeframe-dragging.

Diagram regarding the confirmation of gravitomagnetism byGravity Probe B

Gravitoelectromagnetism, abbreviatedGEM, is a set offormal analogies between the equations forelectromagnetism andrelativistic gravitation. More specifically, it is an analogy betweenMaxwell's field equations and an approximation, valid under certain conditions, to theEinstein field equations forgeneral relativity.Gravitomagnetism is thekinetic effects of gravity, in analogy to themagnetic effects of movingelectric charge.^[1] The most common version of GEM is valid only far from isolated sources, and for slowly movingtest particles.

The analogy and equations differing only by some small factors were first published in 1893, before general relativity, byOliver Heaviside as a separate theory expandingNewton's law of universal gravitation.^[2]^{[better source needed]}

Background

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This approximate reformulation ofgravitation as described bygeneral relativity in theweak field limit makes an apparent field appear in aframe 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 thegravitoelectric andgravitomagnetic 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 thegravitomagnetic 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 ofrelativistic jets.Roger Penrose had proposed a mechanism that relies onframe-dragging-related effects for extracting energy and momentum from rotatingblack holes.^[3]Reva Kay Williams developed a rigorous proof that validatedPenrose's mechanism.^[4] Her model showed how theLense–Thirring effect could account for the observed high energies and luminosities ofquasars andactive galactic nuclei; the collimated jets about their polar axis; and the asymmetrical jets (relative to the orbital plane).^[5]^[6] All of those observed properties could be explained in terms of gravitomagnetic effects.^[7] Williams's application of Penrose's mechanism can be applied to black holes of any size.^[8] Relativistic jets can serve as the largest and brightest form of validations for gravitomagnetism.

TheGravity Probe B satellite experiment, using four cryogenic gyroscopes in Earth orbit, provided direct evidence of frame-dragging consistent with theoretical predictions of general relativity.^[9]^[10]

Physical analogues of fields^[11]

Gravitomagnetism – gravitomagnetic fieldH due to (total)angular momentumJ.
Electromagnetism –magnetic fieldB due to adipole momentm ...
... or, equivalently,currentI, same field profile, and field generation due to rotation.
Fluid mechanics – rotationalfluid 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

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According togeneral relativity, thegravitational 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 inclassical electromagnetism. Starting from the basic equation of general relativity, theEinstein field equation, and assuming a weak gravitational field or reasonablyflat spacetime, the gravitational analogs toMaxwell's equations forelectromagnetism, called the "GEM equations", can be derived. GEM equations compared to Maxwell's equations are:^[12]^[13]

Comparison of GEM and Maxwell's equations
Law	GEM equations	Maxwell's equations
Gauss's law	$\nabla \cdot \mathbf {E} _{\text{g}}=-4\pi G\rho _{\text{g}}\$	$\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}$
Gauss's law for magnetism	$\nabla \cdot \mathbf {B} _{\text{g}}=0\$	$\nabla \cdot \mathbf {B} =0\$
Faraday's law of induction	$\nabla \times \mathbf {E} _{\text{g}}+{\frac {\partial \mathbf {B} _{\text{g}}}{\partial t}}=0$	$\nabla \times \mathbf {E} +{\frac {\partial \mathbf {B} }{\partial t}}=0$
Ampère–Maxwell law	$\nabla \times \mathbf {B} _{\text{g}}-{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} _{\text{g}}}{\partial t}}=-{\frac {4\pi G}{c^{2}}}\mathbf {J} _{\text{g}}$	$\nabla \times \mathbf {B} -{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}={\frac {1}{\varepsilon _{0}c^{2}}}\mathbf {J}$

where:

E_g is the gravitoelectric field (conventionalgravitational field), with SI unit m⋅s⁻²
E is theelectric field, with SI unit kg⋅m⋅s⁻³⋅A⁻¹
B_g is the gravitomagnetic field, with SI unit s⁻¹
B is themagnetic field, with SI unit kg⋅s⁻²⋅A⁻¹
ρ_g ismass density, with SI unit kg⋅m⁻³
ρ ischarge density, with SI unit A⋅s⋅m⁻³
J_g is mass current density ormass flux, with SI unit kg⋅m⁻²⋅s⁻¹
J is electriccurrent density, with SI unit A⋅m⁻²
G is thegravitational constant, with SI unit m³⋅kg⁻¹⋅s⁻²
ε₀ is thevacuum permittivity, with SI unit kg⁻¹⋅m⁻³⋅s⁴⋅A²
c is both thespeed of propagation of gravity and thespeed of light, with SI unit m⋅s⁻¹.

Potentials

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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 $\phi _{\text{g}}$ and the vector potential $\mathbf {A} _{\text{g}}$ according to: $\mathbf {E} _{\text{g}}:=-{\mbox{grad}}~\phi _{\text{g}}-{\frac {\partial \mathbf {A} _{\text{g}}}{\partial t}}=-\nabla ~\phi _{\text{g}}-{\frac {\partial \mathbf {A} _{\text{g}}}{\partial t}}$ and $\mathbf {B} _{\text{g}}:={\mbox{rot}}~\mathbf {A} _{\text{g}}=\nabla \times \mathbf {A} _{\text{g}}$

Inserting this four potentials $\left(\phi _{\text{g}},\mathbf {A} _{\text{g}}\right)$ 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 theLorenz gauge the following inhomogeneous wave-equations are obtained: $-{\frac {1}{c^{2}}}{\frac {\partial ^{2}\phi _{\text{g}}}{\partial t^{2}}}+\nabla ^{2}~\phi _{\text{g}}=4\pi G\rho _{\text{g}}$ $-{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} _{\text{g}}}{\partial t^{2}}}+\nabla ^{2}~\mathbf {A} _{\text{g}}={\frac {4\pi G}{c^{2}}}\rho _{\text{g}}\mathbf {v}$

For a stationary situation ( $\partial \phi _{\text{g}}/\partial t=0$ ) thePoisson equation of the classical gravitation theory is obtained. In a vacuum ( $\rho _{\text{g}}=0$ ) awave equation is obtained under non-stationary conditions. GEM therefore predicts the existence ofgravitational waves. In this way GEM can be regarded as a generalization of Newton's gravitation theory.

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

Lorentz force

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For a test particle whose massm 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 theLorentz force equation:

Lorentz force equations in GEM and electromagnetism
GEM equation	EM equation
$\mathbf {F} _{\text{g}}=m\left(\mathbf {E} _{\text{g}}\ +\ \mathbf {v} \times 4\mathbf {B} _{\text{g}}\right)$	$\mathbf {F} =q\left(\mathbf {E} \ +\ \mathbf {v} \times \mathbf {B} \right)$

where:

v is thevelocity of thetest particle
m is themass of the test particle
q is theelectric charge of the test particle.

Poynting vector

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The GEM Poynting vector compared to the electromagneticPoynting vector is given by:^[14]

Poynting vector equations in GEM and electromagnetism
GEM equation	EM equation
${\mathcal {S}}_{\text{g}}=-{\frac {c^{2}}{4\pi G}}\mathbf {E} _{\text{g}}\times 4\mathbf {B} _{\text{g}}$	${\mathcal {S}}=c^{2}\varepsilon _{0}\mathbf {E} \times \mathbf {B}$

Scaling of fields

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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 ofB_g in the GEM equations must be multiplied by⁠−+1/2c⁠ andE_g 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 orderstress–energy tensor, as opposed to the source of the electromagnetic field being the first orderfour-current tensor. This difference becomes clearer when one compares non-invariance ofrelativistic mass to electriccharge 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.^[15] (SeeRelativistic wave equations for more on "spin-1" and "spin-2" fields.)

Higher-order effects

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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]}. This can be expressed as an attractive or repulsive gravitomagnetic component.

Gravitomagnetic arguments also predict that a flexible or fluidtoroidal mass undergoingminor 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 anyg-forces.^[16]

Consider a toroidal mass with two degrees of rotation: major-axis spin (rotation around the throat) and minor-axis spin ("smoke ring" rotation). This represents a case in which gravitomagnetic effects generate achiral 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. Whenboth rotations are applied simultaneously, these two sets of reaction forces can be said to occur at different depths in a radialCoriolis field that extends across the rotating torus, making it more difficult to establish whether cancellation is complete.^{[citation needed]} Modelling this complex behaviour as a curved spacetime problem has yet to be done.^{[citation needed]}

Gravitomagnetic fields of astronomical objects

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

Due to the body's angular velocity ${\boldsymbol {\omega }}$ the velocity inside the body can be described as $\mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r}$ . Therefore $\nabla ^{2}\mathbf {A} _{\text{g}}=-{\frac {4\pi G}{c^{2}}}\rho _{\text{g}}{\boldsymbol {\omega }}\times \mathbf {r}$ has to be solved to obtain the gravitomagnetic potential $\mathbf {A} _{\text{g}}$ . The analytical solution outside of the body is (see for example^[17]): $\mathbf {A} _{\text{g}}=-{\frac {G}{2c^{2}}}{\frac {\mathbf {L} \times \mathbf {r} }{r^{3}}}$ with $\mathbf {L} =I_{\text{ball}}{\boldsymbol {\omega }}={\frac {2mR^{2}}{5}}{\frac {2\pi }{T}}$ where:

$\mathbf {L}$ is theangular momentum vector;
${\textstyle I_{\text{ball}}={\frac {2mR^{2}}{5}}}$ is themoment of inertia of a ball-shaped body (see:List of moments of inertia);
${\textstyle {\boldsymbol {\omega }}={\frac {2\pi }{T}}}$ is theangular velocity;
m is themass;
R is theradius;
T is the rotational period.

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

It is exactly half of theLense–Thirring precession rate. This suggests that the gravitomagnetic analog of theg-factor is two. This factor of two can be explained completely analogous to the electron'sg-factor by taking into account relativistic calculations. At the equatorial plane,r andL are perpendicular, so theirdot product vanishes, and this formula reduces to: $\mathbf {B} _{\text{g}}={\frac {G}{2c^{2}}}{\frac {\mathbf {L} }{r^{3}}},$

Gravitational waves have equal gravitomagnetic and gravitoelectric components.^[18]

Earth

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Therefore, the magnitude ofEarth's gravitomagnetic field at itsequator is: $\mathbf {B} _{\text{g,Earth}}={\frac {G}{5c^{2}}}{\frac {m}{r}}{\frac {2\pi }{T}}={\frac {2\pi rg_{0}}{5c^{2}T}},$ where $g_{0}=G{\frac {m}{r^{2}}}$ isEarth'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 about1.012×10⁻¹⁴ Hz.^[a] Such a field is extremely weak and requires extremely sensitive measurements to be detected. One experiment to measure such a field was theGravity Probe B mission.

Pulsar

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If the preceding formula is used with the pulsarPSR J1748-2446ad (which rotates 716 times per second), assuming a radius of16 km and a mass of two solar masses, then $\mathbf {B} _{\text{g}}={\frac {2\pi Gm}{5rc^{2}T}}$ equals about166 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 itsSchwarzschild 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

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While Maxwell's equations are invariant underLorentz transformations, the GEM equations are not. The fact thatρ_g andj_g do not form afour-vector is the basis of this difference.^{[citation needed]} Instead they are merely a part of thestress–energy tensor, which is what acts as the source of the gravitational field ingeneral relativity.

Although GEM may hold approximately in two different reference frames connected by aLorentz 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.

Notes

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^2πR_🜨g₀ / (5c² × 1 day)

References

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^David Delphenich (2015)."Pre-metric electromagnetism as a path to unification".Unified Field Mechanics: Natural Science Beyond the Veil of Spacetime, Morgan State University, USA, 16–19 November 2014:215–220.arXiv:1512.05183.doi:10.1142/9789814719063_0023.ISBN 978-981-4719-05-6.S2CID 118596433.
^O. Heaviside (1893).Electromagnetic Theory: A Gravitational and Electromagnetic Analogy. Vol. 1. The Electrician. pp. 455–464.
^R. Penrose (1969). "Gravitational collapse: The role of general relativity".Rivista del Nuovo Cimento. Numero Speciale 1:252–276.Bibcode:1969NCimR...1..252P.
^R. K. Williams (1995). "Extracting x rays, γ rays, and relativistic e⁻e⁺ pairs from supermassive Kerr black holes using the Penrose mechanism".Physical Review.51 (10):5387–5427.Bibcode:1995PhRvD..51.5387W.doi:10.1103/PhysRevD.51.5387.PMID 10018300.
^R. K. Williams (2004). "Collimated escaping vortical polar e⁻e⁺ jets intrinsically produced by rotating black holes and Penrose processes".The Astrophysical Journal.611 (2):952–963.arXiv:astro-ph/0404135.Bibcode:2004ApJ...611..952W.doi:10.1086/422304.S2CID 1350543.
^A. Danehkar (2020)."Gravitational fields of the magnetic-type".International Journal of Modern Physics D.29 (14): 2043001.arXiv:2006.13287.Bibcode:2020IJMPD..2943001D.doi:10.1142/S0218271820430014.
^R. K. Williams (2005). "Gravitomagnetic field and Penrose scattering processes".Annals of the New York Academy of Sciences. Vol. 1045. pp. 232–245.
^R. K. Williams (2001). "Collimated energy–momentum extraction from rotating black holes in quasars and microquasars using the Penrose mechanism".AIP Conference Proceedings. Vol. 586. pp. 448–453.arXiv:astro-ph/0111161.Bibcode:2001AIPC..586..448W.doi:10.1063/1.1419591.
^R. J. Adler; P. Chen (2011).Gravitomagnetism in Quantum Mechanics(PDF) (Report).SLAC National Accelerator Laboratory.
^Everitt, C. W. F.; DeBra, D. B.; Parkinson, B. W.; Turneaure, J. P.; Conklin, J. W.; Heifetz, M. I.; Keiser, G. M.; Silbergleit, A. S.; Holmes, T.; Kolodziejczak, J.; Al-Meshari, M.; Mester, J. C.; Muhlfelder, B.; Solomonik, V. G.; Stahl, K. (31 May 2011)."Gravity Probe B: Final Results of a Space Experiment to Test General Relativity".Physical Review Letters.106 (22) 221101.arXiv:1105.3456.Bibcode:2011PhRvL.106v1101E.doi:10.1103/PhysRevLett.106.221101.ISSN 0031-9007.PMID 21702590.
^I. Ciufolini; J. A. Wheeler (1995).Gravitation and Inertia. Princeton Physics Series.ISBN 0-691-03323-4.
^B. Mashhoon; F. Gronwald; H. I. M. Lichtenegger (2001), "Gravitomagnetism and the Clock Effect",Gyros, Clocks, Interferometers ...: Testing Relativistic Gravity in Space, Lecture Notes in Physics, vol. 562, pp. 83–108,arXiv:gr-qc/9912027,Bibcode:2001LNP...562...83M,CiteSeerX 10.1.1.340.8408,doi:10.1007/3-540-40988-2_5,ISBN 978-3-540-41236-6,S2CID 32411999{{citation}}: CS1 maint: work parameter with ISBN (link)
^S. J. Clark; R. W. Tucker (2000). "Gauge symmetry and gravito-electromagnetism".Classical and Quantum Gravity.17 (19):4125–4157.arXiv:gr-qc/0003115.Bibcode:2000CQGra..17.4125C.doi:10.1088/0264-9381/17/19/311.S2CID 15724290.
^B. Mashhoon (2007). "Gravitoelectromagnetism: A Brief Review". In Iorio (ed.).The Measurement of Gravitomagnetism: A Challenging Enterprise. Nova Science Publishers.arXiv:gr-qc/0311030.Bibcode:2003gr.qc....11030M.
^B. Mashhoon (2000). "Gravitoelectromagnetism".Reference Frames and Gravitomagnetism – Proceedings of the XXIII Spanish Relativity Meeting. pp. 121–132.arXiv:gr-qc/0011014.Bibcode:2001rfg..conf..121M.CiteSeerX 10.1.1.339.476.doi:10.1142/9789812810021_0009.ISBN 978-981-02-4631-0.S2CID 263798773.
^R. L. Forward (1963). "Guidelines to Antigravity".American Journal of Physics.31 (3):166–170.Bibcode:1963AmJPh..31..166F.doi:10.1119/1.1969340.
^A. Malcherek (2023).Elektromagnetismus und Gravitation: Vereinheitlichung und Erweiterung der Klassischen Physik [Electromagnetism and Gravity: Unification and Expansion of Classical Physics] (in German) (2nd ed.). Springer-Vieweg.ISBN 978-3-658-42701-6.
^H. Pfister; M. King (24 February 2015).Inertia and Gravitation: the Fundamental Nature and Structure of Space–time. Lecture Notes in Physics. Springer. p. 147.ISBN 978-3-319-15036-9.OCLC 904397831.

External links

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Gravity Probe B: Testing Einstein's Universe
Gyroscopic Superconducting Gravitomagnetic Effects news on tentative result of European Space Agency (esa) research
In Search of Gravitomagnetism Archived 9 October 2006 at theWayback Machine, NASA, 20 April 2004.
Gravitomagnetic London Moment – New test of General Relativity?
Measurement of Gravitomagnetic and Acceleration Fields Around Rotating Superconductors M. Tajmar, et al., 17 October 2006.
Test of the Lense–Thirring effect with the MGS Mars probe,New Scientist, January 2007.

Theories of gravitation

Standard

Newtonian gravity (NG)	Newton's law of universal gravitation Gauss's law for gravity Poisson's equation for gravity History of gravitational theory
General relativity (GR)	Introduction History Mathematics Exact solutions Resources Tests Post-Newtonian formalism Linearized gravity ADM formalism Gibbons–Hawking–York boundary term

Alternatives to
general relativity

Paradigms	Classical theories of gravitation Quantum gravity Theory of everything
Classical	Poincaré gauge theory Einstein–Cartan Teleparallelism Bimetric theories Gauge theory gravity Composite gravity f(R) gravity Infinite derivative gravity Massive gravity Modified Newtonian dynamics, MOND AQUAL Tensor–vector–scalar Nonsymmetric gravitation Scalar–tensor theories Brans–Dicke Scalar–tensor–vector Conformal gravity Scalar theories Nordström Whitehead Geometrodynamics Induced gravity Degenerate Higher-Order Scalar-Tensor theories
Quantum-mechanical	Euclidean quantum gravity Canonical quantum gravity Wheeler–DeWitt equation Loop quantum gravity Spin foam Causal dynamical triangulation Asymptotic safety in quantum gravity Causal sets DGP model Rainbow gravity theory
Unified-field-theoric	Kaluza–Klein theory Supergravity
Unified-field-theoric and quantum-mechanical	Noncommutative geometry Semiclassical gravity Superfluid vacuum theory Logarithmic BEC vacuum String theory M-theory F-theory Heterotic string theory Type I string theory Type 0 string theory Bosonic string theory Type II string theory Little string theory Twistor theory Twistor string theory
Generalisations / extensions of GR	Liouville gravity Lovelock theory (2+1)-dimensional topological gravity Gauss–Bonnet gravity Jackiw–Teitelboim gravity

Pre-Newtonian
theories and
toy models

Movatterモバイル変換

Gravitoelectromagnetism

Background

Equations

Potentials

Lorentz force

Poynting vector

Scaling of fields

Higher-order effects

Gravitomagnetic fields of astronomical objects

Earth

Pulsar

Lack of invariance

See also

Notes

References

Further reading

Books

Papers

External links