Gravitoelectromagnetism

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,refers to a set offormal analogiesbetween the equations forelectromagnetismandrelativisticgravitation;specifically: betweenMaxwell's field equationsand an approximation, valid under certain conditions, to theEinstein field equationsforgeneral relativity.Gravitomagnetismis a widely used term referring specifically to the kinetic effects of gravity, in analogy to themagneticeffects of moving electric charge.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 Heavisideas a separate theory expandingNewton's law of universal gravitation.

Background

This approximate reformulation ofgravitationas described bygeneral relativityin theweak field limitmakes an apparent field appear in aframe of referencedifferent 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 thegravitoelectricandgravitomagneticfields, 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 thegravitomagneticfield, 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 Penrosehad proposed a mechanism that relies onframe-dragging-related effects for extracting energy and momentum from rotatingblack holes.Reva Kay Williams,University of Florida, developed a rigorous proof that validatedPenrose's mechanism.Her model showed how theLense–Thirring effectcould account for the observed high energies and luminosities ofquasarsandactive galactic nuclei;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's 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 atStanford Universityis currentlyanalyzing data from the first direct test of GEM, theGravity Probe Bsatellite experiment, to see whether they are consistent with gravitomagnetism.TheApache Point Observatory Lunar Laser-ranging Operationalso plans to observe gravitomagnetism effects.

Physical analogues of fields

• 
  Gravitomagnetism– gravitomagnetic fieldHdue to (total)angular momentumJ.
• 
  Electromagnetism–magnetic fieldBdue to adipole momentm...
• 
  ... or, equivalently,currentI,same field profile, and field generation due to rotation.
• 
  Fluid mechanics– rotationalfluid dragof 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 togeneral relativity,thegravitational fieldproduced 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 weakgravitational fieldor reasonablyflat spacetime,the gravitational analogs toMaxwell's equationsforelectromagnetism,called the "GEM equations", can be derived. GEM equations compared to Maxwell's equations are:

GEM equations	Maxwell's equations
$\nabla \cdot \mathbf {E} _{\text{g}}=-4\pi G\rho _{\text{g}}\$	$\nabla \cdot \mathbf {E} ={\frac {\rho }{\epsilon _{0}}}$
$\nabla \cdot \mathbf {B} _{\text{g}}=0\$	$\nabla \cdot \mathbf {B} =0\$
$\nabla \times \mathbf {E} _{\text{g}}+{\frac {\partial \mathbf {B} _{\text{g}}}{\partial t}}=0$	$\nabla \times \mathbf {E} +{\frac {\partial \mathbf {B} }{\partial t}}=0$
$\nabla \times \mathbf {B} _{\text{g}}-{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} _{\text{g}}}{\partial t}}=-{\frac {4\pi G}{c^{2}}}\mathbf {J} _{\text{g}}$	$\nabla \times \mathbf {B} -{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}={\frac {1}{\epsilon _{0}c^{2}}}\mathbf {J}$

where:

• E_gis the gravitoelectric field (conventionalgravitational field), with SI unit m⋅s⁻²
• Eis theelectric field
• B_gis the gravitomagnetic field, with SI unit s⁻¹
• Bis themagnetic field
• ρ_gismass density,with SI unit kg⋅m⁻³
• ρischarge density
• J_gis mass current density ormass flux(J_g=ρ_gv_ρ,wherev_ρis thevelocityof the mass flow), with SI unit kg⋅m⁻²⋅s⁻¹
• Jis electriccurrent density
• Gis thegravitational constant
• ε₀is thevacuum permittivity
• cis both thespeed of propagation of gravityand thespeed 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$\phi _{\text{g}}$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 gaugethe following inhomogeneous wave-equations are obtained:

$-{\frac {1}{c^{2}}}{\frac {\partial ^{2}\phi _{\text{g}}}{\partial t^{2}}}+\nabla ^{2}~\phi _{\text{g}}=4\pi G\rho _{g}$

$-{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} _{\text{g}}}{\partial t^{2}}}+\nabla ^{2}~\mathbf {A} _{\text{g}}={\frac {4\pi G}{c^{2}}}\rho _{g}\mathbf {v}$

For a stationary situation ($\partial \phi _{\text{g}}/\partial t=0$) thePoisson equationof the classical gravitation theory is obtained. In a vacuum ($\rho _{\text{g}}=0$) awave equationis 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

For a test particle whose massmis "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 forceequation:

GEM equation	EM equation
$\mathbf {F} _{\text{g}}=m\left(\mathbf {E} _{\text{g}}\ +\mathbf {v} \times 4\mathbf {B} _{\text{g}}\right)$	$\mathbf {F} _{\text{e}}=q\left(\mathbf {E} \ +\ \mathbf {v} \times \mathbf {B} \right)$

where:

• vis thevelocityof thetest particle
• mis themassof the test particle
• qis theelectric chargeof the test particle.

Poynting vector

The GEM Poynting vector compared to the electromagneticPoynting vectoris given by:

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

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_gin the GEM equations must be multiplied by −⁠1/2c⁠andE_gby −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-currenttensor. This difference becomes clearer when one compares non-invariance ofrelativistic massto 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.(SeeRelativistic wave equationsfor more on "spin-1" and "spin-2" fields).

Higher-order effects

Some higher-order gravitomagnetic effects can reproduce effects reminiscent of the interactions of more conventional polarized charges. For instance, if two wheels are spun on a common axis, the mutual gravitational attraction between the two wheels will be greater if they spin in opposite directions than in the same direction.This can be expressed as an attractive or repulsive gravitomagnetic component.

Gravitomagnetic arguments also predict that a flexible or fluidtoroidalmass undergoingminor axisrotational 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.

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 achiralcorkscrew-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. Whenbothrotations are applied simultaneously, these two sets of reaction forces can be said to occur at different depths in a radialCoriolis fieldthat 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

A rotating spherical body with a homogeneous density distribution produces a stationary gravitomagnetic potential, which is described by:

$\nabla ^{2}\mathbf {A} _{\text{g}}=-{\frac {4\pi G}{c^{2}}}\rho _{g}\mathbf {v}$

Due to the body's angular velocity$\mathbf {\omega }$the velocity inside the body can be described as$\mathbf {v} =\mathbf {\omega } \times \mathbf {r}$.Therefore

$\nabla ^{2}\mathbf {A} _{\text{g}}=-{\frac {4\pi G}{c^{2}}}\rho _{g}\mathbf {\omega } \times \mathbf {r}$

has to be solved to obtain the gravitomagnetic potential$\mathbf {A} _{\text{g}}$.The analytical solution outside of the body is (see for example):

$\mathbf {A} _{\text{g}}=-{\frac {G}{2c^{2}}}{\frac {\mathbf {L} \times \mathbf {r} }{r^{3}}}\quad {\text{with}}\quad \mathbf {L} ={\frac {2}{5}}mR^{2}\mathbf {\omega } \quad {\text{or}}\quad L=I_{\text{ball}}\omega ={\frac {2mR^{2}}{5}}{\frac {2\pi }{T}}\quad$

where:

• $\mathbf {L}$is theangular momentumvector;
• $I_{\text{ball}}={\frac {2mR^{2}}{5}}$is themoment of inertiaof a ball-shaped body (see:list of moments of inertia);
• $\omega \$is theangular velocity;
• mis themass;
• Ris theradius;
• Tis the rotational period.

The formula for the gravitomagnetic fieldB_gcan 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 precessionrate. This suggests that the gravitomagnetic analog of theg-factoris 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,randLare perpendicular, so theirdot productvanishes, and this formula reduces to:

$\mathbf {B} _{\text{g}}={\frac {G}{2c^{2}}}{\frac {\mathbf {L} }{r^{3}}},$

Gravitational waveshave equal gravitomagnetic and gravitoelectric components.

Earth

Therefore, the magnitude ofEarth's gravitomagnetic field at itsequatoris:

$B_{\text{g,Earth}}={\frac {G}{5c^{2}}}{\frac {m}{r}}{\frac {2\pi }{T}}={\frac {2\pi rg}{5c^{2}T}},$

where$g=G{\frac {m}{r^{2}}}$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 about1.012×10⁻¹⁴Hz.Such a field is extremely weak and requires extremely sensitive measurements to be detected. One experiment to measure such field was theGravity Probe Bmission.

Pulsar

If the preceding formula is used with the pulsarPSR J1748-2446ad(which rotates 716 times per second), assuming a radius of 16 km and a mass of two solar masses, then

$B_{\text{g}}={\frac {2\pi Gm}{5rc^{2}T}}$

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 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

While Maxwell's equations are invariant underLorentz transformations,the GEM equations are not. The fact thatρ_gandj_gdo not form afour-vector(instead they are merely a part of thestress–energy tensor) is the basis of this difference.

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.

See also

• 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
• Stationary spacetime

Further reading

Books

• M. P. Hobson; G. P. Efstathiou; A. N. Lasenby (2006).General Relativity: An Introduction for Physicists.Cambridge University Press. pp. 490–491.ISBN9780521829519.
• L. H. Ryder (2009).Introduction to General Relativity.Cambridge University Press. pp. 200–207.ISBN9780521845632.
• J. B. Hartle (2002).Gravity: An Introduction to Einstein's General Relativity.Addison-Wesley. pp. 296, 303.ISBN9780805386622.
• S. Carroll (2003).Spacetime and Geometry: An Introduction to General Relativity.Addison-Wesley. p. 281.ISBN9780805387322.
• J.A. Wheeler(1990). "Gravity's next prize: Gravitomagnetism".A journey into gravity and spacetime.Scientific American Library. pp. 232–233.ISBN978-0-7167-5016-1.
• L. Iorio, ed. (2007).Measuring Gravitomagnetism: A Challenging Enterprise.Nova.ISBN978-1-60021-002-0.
• O.D. Jefimenko(1992).Causality, electromagnetic induction, and gravitation: a different approach to the theory of electromagnetic and gravitational fields.Electret Scientific.ISBN978-0-917406-09-6.
• O.D. Jefimenko(2006).Gravitation and Cogravitation.Electret Scientific.ISBN978-0-917406-15-7.

Papers

• 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.S2CID15724290.
• R.L. Forward (1963). "Guidelines to Antigravity".American Journal of Physics.31(3): 166–170.Bibcode:1963AmJPh..31..166F.doi:10.1119/1.1969340.
• R.T. Jantzen; P. Carini; D. Bini (1992). "The Many Faces of Gravitoelectromagnetism".Annals of Physics.215(1): 1–50.arXiv:gr-qc/0106043.Bibcode:1992AnPhy.215....1J.doi:10.1016/0003-4916(92)90297-Y.S2CID6691986.
• 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.CiteSeerX10.1.1.339.476.doi:10.1142/9789812810021_0009.ISBN978-981-02-4631-0.S2CID263798773.
• B. Mashhoon (2003). "Gravitoelectromagnetism: a Brief Review".arXiv:gr-qc/0311030.in
• L. Iorio, ed. (2007).Measuring Gravitomagnetism: A Challenging Enterprise.Nova. pp. 29–39.ISBN978-1-60021-002-0.
• M. Tajmar; C.J. de Matos (2001). "Gravitomagnetic Barnett Effect".Indian Journal of Physics B.75:459–461.arXiv:gr-qc/0012091.Bibcode:2000gr.qc....12091D.
• L. Filipe Costa; Carlos A. R. Herdeiro (2008). "A gravito-electromagnetic analogy based on tidal tensors".Physical Review D.78(2): 024021.arXiv:gr-qc/0612140.Bibcode:2008PhRvD..78b4021C.doi:10.1103/PhysRevD.78.024021.S2CID14846902.
• A. Bakopoulos; P. Kanti (2016). "Novel Ansatzes and Scalar Quantities in Gravito-Electromagnetism".General Relativity and Gravitation.49(3): 44.arXiv:1610.09819.Bibcode:2017GReGr..49...44B.doi:10.1007/s10714-017-2207-x.S2CID119232668.

External links

• Gravity Probe B: Testing Einstein's Universe
• Gyroscopic Superconducting Gravitomagnetic Effectsnews on tentative result of European Space Agency (esa) research
• In Search of GravitomagnetismArchived9 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 SuperconductorsM. Tajmar, et al., 17 October 2006.
• Test of the Lense–Thirring effect with the MGS Mars probe,New Scientist,January 2007.