Gravitoelectromagnetism, abbreviated GEM, refers to a set of
formal analogies between the equations for
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
and
relativistic gravitation; specifically: between
Maxwell's field equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
Th ...
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
Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
effects of moving electric charge. 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.
Background
This approximate reformulation of
gravitation as described by
general relativity in the
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, non-axisymmetric, rotating object will experience acceleration not predicted by a purely Newtonian (gravitoelectric) gravity 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
An astrophysical jet is an astronomical phenomenon where outflows of ionised matter are emitted as an extended beam along the axis of rotation. When this greatly accelerated matter in the beam approaches the speed of light, astrophysical jets beco ...
s.
Roger Penrose had proposed a mechanism that relies on
frame-dragging-related effects for extracting energy and momentum from rotating
black hole
A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can defo ...
s.
Reva Kay Williams
Reva Kay Williams is an American astrophysicist. Williams is the first Black American woman to receive a Ph.D. in theoretical astrophysics and the first person to successfully work out the Penrose process using Einstein's Theory of Relativity to ...
, 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
quasars and
active galactic nuclei
An active galactic nucleus (AGN) is a compact region at the center of a galaxy that has a much-higher-than-normal luminosity over at least some portion of the electromagnetic spectrum with characteristics indicating that the luminosity is not prod ...
; 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 is currently analyzing data from the first direct test of GEM, the
Gravity Probe B satellite experiment, to see whether they are consistent with gravitomagnetism. The
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 experiments, which use retroreflectors on the ...
also plans to observe gravitomagnetism effects.
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
Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fie ...
. 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
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
for
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
, called the "GEM equations", can be derived. GEM equations compared to Maxwell's equations are:
where:
* E
g is the gravitoelectric field (conventional
gravitational field), with SI unit m⋅s
−2
* E is the
electric field
* B
g is the gravitomagnetic field, with SI unit s
−1
* B is the
magnetic field
* ''ρ''
g is
mass density with, SI unit kg⋅m
−3
* ''ρ'' is
charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in co ...
* J
g is mass current density or
mass flux (J
g = ''ρ''
gv
ρ, where v
ρ is the
velocity of the mass flow), with SI unit kg⋅m
−2⋅s
−1
* J is electric
current density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional are ...
* ''G'' is the
gravitational constant
* ''ε''
0 is the
vacuum permittivity
* ''c'' is both the
speed of propagation of gravity and the
speed of light.
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
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
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:
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
g in the GEM equations must be multiplied by − and E
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 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. (See ''
Relativistic wave equations
In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the con ...
'' 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. This can be expressed as an attractive or repulsive gravitomagnetic component.
Gravitomagnetic arguments also predict that a flexible or fluid
toroidal mass undergoing
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-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 a
chiral
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from i ...
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.
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 B
g near a rotating body can be derived from the GEM equations. It is exactly half of the
Lense–Thirring precession rate, and is given by:
:
where L is the
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syste ...
of the body. At the equatorial plane, r and L are perpendicular, so their
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...
vanishes, and this formula reduces to:
:
The magnitude of angular momentum of a homogeneous ball-shaped body is:
:
where:
*
is the
moment of inertia of a ball-shaped body (see:
list of moments of inertia
Moment of inertia, denoted by , measures the extent to which an object resists rotational acceleration about a particular axis, it is the rotational analogue to mass (which determines an object's resistance to ''linear'' acceleration). The momen ...
);
*
is the
angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
;
*''m'' is the
mass;
*''r'' is the
radius;
*''T'' is the rotational period.
Gravitational waves have equal gravitomagnetic and gravitoelectric components.
Earth
Therefore, the magnitude of
Earth's gravitomagnetic field at its
equator is:
:
where
is
Earth's gravity. The field direction coincides with the angular moment direction, i.e. north.
From this calculation it follows that Earth's equatorial gravitomagnetic field is about , or . 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 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. 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 transformation
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
s, the GEM equations are not. The fact that ''ρ''
g and ''j''
g do not form a
four-vector (instead they are merely a part of the
stress–energy tensor) is the basis of this difference.
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
*
Anti-gravity
Anti-gravity (also known as non-gravitational field) is a hypothetical phenomenon 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 ...
*
Artificial gravity
Artificial gravity is the creation of an inertial force that mimics the effects of a gravitational force, usually by rotation.
Artificial gravity, or rotational gravity, is thus the appearance of a centrifugal force in a rotating frame of ref ...
*
Frame-dragging
*
Geodetic effect
*
Gravitational radiation
*
Gravity Probe B
*
Kaluza–Klein theory
*
Linearized gravity
In the theory of general relativity, linearized gravity is the application of perturbation theory to the metric tensor that describes the geometry of spacetime. As a consequence, linearized gravity is an effective method for modeling the effec ...
*
Speed of gravity § Electrodynamical analogies
*
Stationary spacetime
*
Non-Relativistic Gravitational Fields
References
Further reading
Books
*
*
*
*
*
*
*
*
*
Papers
*
*
*
*
* in
*
*
*
External links
Gravity Probe B: Testing Einstein's Universenews on tentative result of European Space Agency (
esa
, owners =
, headquarters = Paris, Île-de-France, France
, coordinates =
, spaceport = Guiana Space Centre
, seal = File:ESA emblem seal.png
, seal_size = 130px
, image = Views in the Main Control Room (120 ...
) research
In Search of Gravitomagnetism NASA, 20 April 2004.
*
ttp://www.arxiv.org/abs/gr-qc/0610015 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.
{{theories of gravitation
General relativity
Effects of gravitation
Tests of general relativity