Gravitoelectromagnetism
Gravitoelectromagnetism, abbreviated GEM, refers to a set of
The analogy and equations differing only by some small factors were first published in 1893, before general relativity, by
Background
This approximate reformulation of
Indirect validations of gravitomagnetic effects have been derived from analyses of
A group at Stanford University is currently[when?] analyzing data from the first direct test of GEM, the Gravity Probe B satellite experiment, to see whether they are consistent with gravitomagnetism.[9] The Apache Point Observatory Lunar Laser-ranging Operation also plans to observe gravitomagnetism effects.[citation needed]
-
... or, equivalently, current I, same field profile, and field generation due to rotation.
-
Fluid mechanics – rotational 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
GEM equations | Maxwell's equations |
---|---|
where:
- Eg is the gravitoelectric field (conventional gravitational field), with SI unit m⋅s−2
- E is the electric field
- Bg is the gravitomagnetic field, with SI unit s−1
- B is the magnetic field
- ρg is mass density with, SI unit kg⋅m−3
- ρ is charge density
- Jg is mass current density or mass flux (Jg = ρgvρ, where vρ is the velocity of the mass flow), with SI unit kg⋅m−2⋅s−1
- J is electric current density
- G is the gravitational constant
- ε0 is the vacuum permittivity
- c is both the 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 and the vector potential according to:
and:
Inserting this four potentials into the Gaussian law for the gravitaion field (first line of the table) and Ampère's circuital law (fourth line of the table) and applying the
For a stationary situation () the
The wave equation for the gravitomagnetic potential can also be solved for a rotating spherical body (which is actually 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:
GEM equation | EM equation |
---|---|
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:[13]
GEM equation | EM equation |
---|---|
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 Bg in the GEM equations must be multiplied by −1/2c and Eg 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 order
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]. This can be expressed as an attractive or repulsive gravitomagnetic component.
Gravitomagnetic arguments also predict that a flexible or fluid
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 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]
Modelling this complex behaviour as a curved spacetime problem has yet to be done and is believed to be very difficult.[citation needed]
Gravitomagnetic fields of astronomical objects
This section's factual accuracy is disputed. (May 2013) |
This section needs additional citations for verification. (February 2024) |
A rotating spherical body with a homogeneous density distribution produces a stationary gravitomagnetic potential which is described by:
Due to the body's angular velocity the velocity inside the body can be described as . Therefore
has to be solved to obtain the gravitomagnetic potential . The analytical solution outside of the body is (see for example [16]):
where:
- is the angular momentum vector;
- is the moment of inertia of a ball-shaped body (see: list of moments of inertia);
- is the angular velocity;
- m is the mass;
- R is the radius;
- T is the rotational period.
The formula for the gravitomagnetic field Bg can now be obtained by:
It is exactly half of the Lense–Thirring precession rate. At the equatorial plane, r and L are perpendicular, so their dot product vanishes, and this formula reduces to:
Gravitational waves have equal gravitomagnetic and gravitoelectric components.[17]
Earth
Therefore, the magnitude of Earth's gravitomagnetic field at its equator is:
where is
From this calculation it follows that Earth's equatorial gravitomagnetic field is about 1.012×10−14 Hz,[18] or 3.1×10−7 g/c. 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
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. 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 under Lorentz transformations, the GEM equations are not. The fact that ρg and jg 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]
Although GEM may hold approximately in two different reference frames connected by a
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
References
- S2CID 118596433.
- ^ O. Heaviside (1893). Electromagnetic Theory: A Gravitational and Electromagnetic Analogy. Vol. 1. The Electrician. pp. 455–464.
- ^
Bibcode:1969NCimR...1..252P.
- ^
R.K. Williams (1995). "Extracting x rays, Ύ rays, and relativistic e−e+ pairs from supermassive Kerr black holes using the Penrose mechanism". Physical Review. 51 (10): 5387–5427. 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. S2CID 1350543.
- .
- ^ 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. .
- ^ Gravitomagnetism in Quantum Mechanics, 2014 https://www.slac.stanford.edu/pubs/slacpubs/14750/slac-pub-14775.pdf
- ISBN 0-691-03323-4
- ^ B. Mashhoon; F. Gronwald; H.I.M. Lichtenegger (2001). "Gravitomagnetism and the Clock Effect". Gyros, Clocks, Interferometers...: Testing Relativistic Graviy in Space. Lecture Notes in Physics. Vol. 562. pp. 83–108. )
- ^
S.J. Clark; R.W. Tucker (2000). "Gauge symmetry and gravito-electromagnetism". Classical and Quantum Gravity. 17 (19): 4125–4157. S2CID 15724290.
- arXiv:gr-qc/0311030.
- ^ B. Mashhoon (2000). "Gravitoelectromagnetism". Reference Frames and Gravitomagnetism – Proceedings of the XXIII Spanish Relativity Meeting. pp. 121–132. )
- ^ R.L. Forward (1963). "Guidelines to Antigravity". American Journal of Physics. 31 (3): 166–170. .
- ^
A. Malcherek (2023). Elektromagnetismus und Gravitation (2. ed.). Springer-Vieweg. ISBN 978-3-658-42701-6.
- OCLC 904397831.
- ^ "2*pi*radius of Earth*earth gravity/(5*c^2*day) – Google Search". google.com.
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. ISBN 9780521829519.
- L. H. Ryder (2009). Introduction to General Relativity. Cambridge University Press. pp. 200–207. ISBN 9780521845632.
- J. B. Hartle (2002). Gravity: An Introduction to Einstein's General Relativity. Addison-Wesley. pp. 296, 303. ISBN 9780805386622.
- S. Carroll (2003). Spacetime and Geometry: An Introduction to General Relativity. Addison-Wesley. p. 281. ISBN 9780805387322.
- ISBN 978-0-7167-5016-1.
- L. Iorio, ed. (2007). Measuring Gravitomagnetism: A Challenging Enterprise. Nova. ISBN 978-1-60021-002-0.
- ISBN 978-0-917406-09-6.
- ISBN 978-0-917406-15-7.
- Antoine Acke (2018). Gravitation explained by Gravitoelectromagnetism. LAP. ISBN 978-613-9-93065-4.
Papers
- S.J. Clark; R.W. Tucker (2000). "Gauge symmetry and gravito-electromagnetism". Classical and Quantum Gravity. 17 (19): 4125–4157. S2CID 15724290.
- R.L. Forward (1963). "Guidelines to Antigravity". American Journal of Physics. 31 (3): 166–170. .
- R.T. Jantzen; P. Carini; D. Bini (1992). "The Many Faces of Gravitoelectromagnetism". Annals of Physics. 215 (1): 1–50. S2CID 6691986.
- B. Mashhoon (2000). "Gravitoelectromagnetism". Reference Frames and Gravitomagnetism – Proceedings of the XXIII Spanish Relativity Meeting. pp. 121–132. )
- B. Mashhoon (2003). "Gravitoelectromagnetism: a Brief Review". ISBN 978-1-60021-002-0.
- M. Tajmar; C.J. de Matos (2001). "Gravitomagnetic Barnett Effect". Indian Journal of Physics B. 75: 459–461. 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. S2CID 14846902.
- A. Bakopoulos; P. Kanti (2016). "Novel Ansatzes and Scalar Quantities in Gravito-Electromagnetism". General Relativity and Gravitation. 49 (3): 44. S2CID 119232668.
External links
- 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 the Wayback 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.