Bimetric gravity

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https://en.wikipedia.org/wiki/Bimetric_gravity 

Bimetric gravity or bigravity refers to two different classes of theories. The first class of theories relies on modified mathematical theories of gravity (or gravitation) in which two metric tensors are used instead of one. The second metric may be introduced at high energies, with the implication that the speed of light could be energy-dependent, enabling models with a variable speed of light.

If the two metrics are dynamical and interact, a first possibility implies two graviton modes, one massive and one massless; such bimetric theories are then closely related to massive gravity. Several bimetric theories with massive gravitons exist, such as those attributed to Nathan Rosen (1909–1995) or Mordehai Milgrom with relativistic extensions of Modified Newtonian Dynamics (MOND). More recently, developments in massive gravity have also led to new consistent theories of bimetric gravity. Though none has been shown to account for physical observations more accurately or more consistently than the theory of general relativity, Rosen's theory has been shown to be inconsistent with observations of the Hulse–Taylor binary pulsar. Some of these theories lead to cosmic acceleration at late times and are therefore alternatives to dark energy. Bimetric gravity is also at odds with measurements of gravitational waves emitted by the neutron-star merger GW170817.

On the contrary, the second class of bimetric gravity theories does not rely on massive gravitons and does not modify Newton's law, but instead describes the universe as a manifold having two coupled Riemannian metrics, where matter populating the two sectors interact through gravitation (and antigravitation if the topology and the Newtonian approximation considered introduce negative mass and negative energy states in cosmology as an alternative to dark matter and dark energy). Some of these cosmological models also use a variable speed of light in the high energy density state of the radiation-dominated era of the universe, challenging the inflation hypothesis.

Rosen's bigravity (1940 to 1989)

In general relativity (GR), it is assumed that the distance between two points in spacetime is given by the metric tensor. Einstein's field equation is then used to calculate the form of the metric based on the distribution of energy and momentum.

In 1940, Rosen proposed that at each point of space-time, there is a Euclidean metric tensor ${\displaystyle {\gamma }_{ij}}$ in addition to the Riemannian metric tensor ${\displaystyle g_{ij}}$. Thus at each point of space-time there are two metrics:

1. ${\displaystyle d{s}^2=g_{ij}d{x}^{i}d{x}^j}$
2. ${\displaystyle d{\sigma }^2={\gamma }_{ij}d{x}^{i}d{x}^j}$

The first metric tensor, ${\displaystyle g_{ij}}$, describes the geometry of space-time and thus the gravitational field. The second metric tensor, ${\displaystyle {\gamma }_{ij}}$, refers to the flat space-time and describes the inertial forces. The Christoffel symbols formed from ${\displaystyle g_{ij}}$ and ${\displaystyle {\gamma }_{ij}}$ are denoted by ${\displaystyle {\{}_{jk}^i\}}$ and ${\displaystyle {\mathrm{\Gamma }}_{jk}^i}$ respectively.

Since the difference of two connections is a tensor, one can define the tensor field ${\displaystyle {\mathrm{\Delta }}_{jk}^i}$ given by:

${\displaystyle {\mathrm{\Delta }}_{jk}^i={\{}_{jk}^i\}-{\mathrm{\Gamma }}_{jk}^i}$

(1)

Two kinds of covariant differentiation then arise: ${\displaystyle g}$-differentiation based on ${\displaystyle g_{ij}}$ (denoted by a semicolon, e.g. ${\displaystyle X_{;a}}$), and covariant differentiation based on ${\displaystyle {\gamma }_{ij}}$ (denoted by a slash, e.g. ${\displaystyle X_{/a}}$). Ordinary partial derivatives are represented by a comma (e.g. ${\displaystyle X_{,a}}$). Let ${\displaystyle R_{ijk}^h}$ and ${\displaystyle P_{ijk}^h}$ be the Riemann curvature tensors calculated from ${\displaystyle g_{ij}}$ and ${\displaystyle {\gamma }_{ij}}$, respectively. In the above approach the curvature tensor ${\displaystyle P_{ijk}^h}$ is zero, since ${\displaystyle {\gamma }_{ij}}$ is the flat space-time metric.

A straightforward calculation yields the Riemann curvature tensor

${\displaystyle \begin{array}{rl}{R}_{ijk}^h& =P_{ijk}^h-{\mathrm{\Delta }}_{ij/k}^h+{\mathrm{\Delta }}_{ik/j}^h+{\mathrm{\Delta }}_{mj}^h{\mathrm{\Delta }}_{ik}^m-{\mathrm{\Delta }}_{mk}^h{\mathrm{\Delta }}_{ij}^m\\ & =-{\mathrm{\Delta }}_{ij/k}^h+{\mathrm{\Delta }}_{ik/j}^h+{\mathrm{\Delta }}_{mj}^h{\mathrm{\Delta }}_{ik}^m-{\mathrm{\Delta }}_{mk}^h{\mathrm{\Delta }}_{ij}^m\end{array}}$

Each term on the right hand side is a tensor. It is seen that from GR one can go to the new formulation just by replacing {:} by ${\displaystyle \mathrm{\Delta }}$ and ordinary differentiation by covariant ${\displaystyle \gamma }$-differentiation, ${\displaystyle \sqrt{-g}}$ by ${\displaystyle \sqrt{\frac{g}{\gamma }}}$, integration measure ${\displaystyle d^{4}x}$ by ${\displaystyle \sqrt{-\gamma }{d}^{4}x}$, where ${\displaystyle g=det(g_{ij})}$, ${\displaystyle \gamma =det({\gamma }_{ij})}$ and ${\displaystyle d^{4}x=d{x}^{1}d{x}^{2}d{x}^{3}d{x}^4}$. Having once introduced ${\displaystyle {\gamma }_{ij}}$ into the theory, one has a great number of new tensors and scalars at one's disposal. One can set up other field equations other than Einstein's. It is possible that some of these will be more satisfactory for the description of nature.

The geodesic equation in bimetric relativity (BR) takes the form

${\displaystyle \frac{d^2{x}^i}{d{s}^2}+{\mathrm{\Gamma }}_{jk}^i\frac{d{x}^j}{ds}\frac{d{x}^k}{ds}+{\mathrm{\Delta }}_{jk}^i\frac{d{x}^j}{ds}\frac{d{x}^k}{ds}=0}$

(2)

It is seen from equations (1) and (2) that ${\displaystyle \mathrm{\Gamma }}$ can be regarded as describing the inertial field because it vanishes by a suitable coordinate transformation.

Being the quantity ${\displaystyle \mathrm{\Delta }}$ a tensor, it is independent of any coordinate system and hence may be regarded as describing the permanent gravitational field.

Rosen (1973) has found BR satisfying the covariance and equivalence principle. In 1966, Rosen showed that the introduction of the space metric into the framework of general relativity not only enables one to get the energy momentum density tensor of the gravitational field, but also enables one to obtain this tensor from a variational principle. The field equations of BR derived from the variational principle are

${\displaystyle K_j^i=N_j^i-\frac{1}{2}{\delta }_j^{i}N=-8\pi \kappa T_j^i}$

(3)

where

${\displaystyle N_j^i=\frac{1}{2}{\gamma }^{\alpha \beta }(g^{hi}{g}_{hj/\alpha }{)}_{/\beta }}$

or

${\displaystyle \begin{array}{rl}{N}_j^i& =\frac{1}{2}{\gamma }^{\alpha \beta }\{{\left(g^{hi}{g}_{hj,\alpha }\right)}_{,\beta }-{\left(g^{hi}{g}_{mj}{\mathrm{\Gamma }}_{h\alpha }^m\right)}_{,\beta }-{\gamma }^{\alpha \beta }{\left({\mathrm{\Gamma }}_{j\alpha }^i\right)}_{,\beta }+{\mathrm{\Gamma }}_{\lambda \beta }^i\left[g^{h\lambda }{g}_{hj,\alpha }-g^{h\lambda }{g}_{mj}{\mathrm{\Gamma }}_{h\alpha }^m-{\mathrm{\Gamma }}_{j\alpha }^{\lambda }\right]-\\ & {\mathrm{\Gamma }}_{j\beta }^{\lambda }\left[g^{hi}{g}_{h\lambda ,\alpha }-g^{hi}{g}_{m\lambda }{\mathrm{\Gamma }}_{h\alpha }^m-{\mathrm{\Gamma }}_{\lambda \alpha }^i\right]+{\mathrm{\Gamma }}_{\alpha \beta }^{\lambda }\left[g^{hi}{g}_{hj,\lambda }-g^{hi}{g}_{mj}{\mathrm{\Gamma }}_{h\lambda }^m-{\mathrm{\Gamma }}_{j\lambda }^i\right]\}\end{array}}$

with

${\displaystyle N=g^{ij}{N}_{ij}}$, ${\displaystyle \kappa =\sqrt{\frac{g}{\gamma }}}$

and ${\displaystyle T_j^i}$ is the energy-momentum tensor.

The variational principle also leads to the relation

${\displaystyle T_{j;i}^i=0}$.

Hence from (3)

${\displaystyle K_{j;i}^i=0}$,

which implies that in a BR, a test particle in a gravitational field moves on a geodesic with respect to ${\displaystyle g_{ij}.}$

Rosen continued improving his bimetric gravity theory with additional publications in 1978 and 1980, in which he made an attempt "to remove singularities arising in general relativity by modifying it so as to take into account the existence of a fundamental rest frame in the universe." In 1985 Rosen tried again to remove singularities and pseudo-tensors from General Relativity. Twice in 1989 with publications in March and November Rosen further developed his concept of elementary particles in a bimetric field of General Relativity.

It is found that the BR and GR theories differ in the following cases:

• propagation of electromagnetic waves
• the external field of a high density star
• the behaviour of intense gravitational waves propagating through a strong static gravitational field.

The predictions of gravitational radiation in Rosen's theory have been shown since 1992 to be in conflict with observations of the Hulse–Taylor binary pulsar.

Massive bigravity

Main article: Massive gravity

Since 2010 there has been renewed interest in bigravity after the development by Claudia de Rham, Gregory Gabadadze, and Andrew Tolley (dRGT) of a healthy theory of massive gravity. Massive gravity is a bimetric theory in the sense that nontrivial interaction terms for the metric ${\displaystyle g_{\mu \nu }}$ can only be written down with the help of a second metric, as the only nonderivative term that can be written using one metric is a cosmological constant. In the dRGT theory, a nondynamical "reference metric" ${\displaystyle f_{\mu \nu }}$ is introduced, and the interaction terms are built out of the matrix square root of ${\displaystyle g^{-1}f}$.

In dRGT massive gravity, the reference metric must be specified by hand. One can give the reference metric an Einstein–Hilbert term, in which case ${\displaystyle f_{\mu \nu }}$ is not chosen but instead evolves dynamically in response to ${\displaystyle g_{\mu \nu }}$ and possibly matter. This massive bigravity was introduced by Fawad Hassan and Rachel Rosen as an extension of dRGT massive gravity.

The dRGT theory is crucial to developing a theory with two dynamical metrics because general bimetric theories are plagued by the Boulware–Deser ghost, a possible sixth polarization for a massive graviton. The dRGT potential is constructed specifically to render this ghost nondynamical, and as long as the kinetic term for the second metric is of the Einstein–Hilbert form, the resulting theory remains ghost-free.

The action for the ghost-free massive bigravity is given by

${\displaystyle S=-\frac{M_g^2}{2}\int d^{4}x\sqrt{-g}R(g)-\frac{M_f^2}{2}\int d^{4}x\sqrt{-f}R(f)+m^2{M}_g^2\int d^{4}x\sqrt{-g}{\displaystyle \sum _{n=0}^4{\beta }_n{e}_n(\mathbb{X})+\int d^{4}x\sqrt{-g}{\mathcal{L}}_{\mathrm{m}}(g,{\mathrm{\Phi }}_i).}}$

As in standard general relativity, the metric ${\displaystyle g_{\mu \nu }}$ has an Einstein–Hilbert kinetic term proportional to the Ricci scalar ${\displaystyle R(g)}$ and a minimal coupling to the matter Lagrangian ${\displaystyle {\mathcal{L}}_{\mathrm{m}}}$, with ${\displaystyle {\mathrm{\Phi }}_i}$ representing all of the matter fields, such as those of the Standard Model. An Einstein–Hilbert term is also given for ${\displaystyle f_{\mu \nu }}$. Each metric has its own Planck mass, denoted ${\displaystyle M_g}$ and ${\displaystyle M_f}$ respectively. The interaction potential is the same as in dRGT massive gravity. The ${\displaystyle {\beta }_i}$ are dimensionless coupling constants and ${\displaystyle m}$ (or specifically ${\displaystyle {\beta }_i^{1/2}m}$) is related to the mass of the massive graviton. This theory propagates seven degrees of freedom, corresponding to a massless graviton and a massive graviton (although the massive and massless states do not align with either of the metrics).

The interaction potential is built out of the elementary symmetric polynomials ${\displaystyle e_n}$ of the eigenvalues of the matrices ${\displaystyle \mathbb{K}=\mathbb{I}-\sqrt{g^{-1}f}}$ or ${\displaystyle \mathbb{X}=\sqrt{g^{-1}f}}$, parametrized by dimensionless coupling constants ${\displaystyle {\alpha }_i}$ or ${\displaystyle {\beta }_i}$, respectively. Here ${\displaystyle \sqrt{g^{-1}f}}$ is the matrix square root of the matrix ${\displaystyle g^{-1}f}$. Written in index notation, ${\displaystyle \mathbb{X}}$ is defined by the relation

${\displaystyle X^{\mu }{}_{\alpha }{X}^{\alpha }{}_{\nu }=g^{\mu \alpha }{f}_{\nu \alpha }.}$

The ${\displaystyle e_n}$ can be written directly in terms of ${\displaystyle \mathbb{X}}$ as

${\displaystyle \begin{array}{rl}{e}_0(\mathbb{X})& =1,\\ e_1(\mathbb{X})& =[\mathbb{X}],\\ e_2(\mathbb{X})& =\frac{1}{2}\left([\mathbb{X}{]}^2-[{\mathbb{X}}^2]\right),\\ e_3(\mathbb{X})& =\frac{1}{6}\left([\mathbb{X}{]}^3-3[\mathbb{X}][{\mathbb{X}}^2]+2[{\mathbb{X}}^3]\right),\\ e_4(\mathbb{X})& =\det \mathbb{X},\end{array}}$

where brackets indicate a trace, ${\displaystyle [\mathbb{X}]\equiv X^{\mu }{}_{\mu }}$. It is the particular antisymmetric combination of terms in each of the ${\displaystyle e_n}$ which is responsible for rendering the Boulware–Deser ghost nondynamical.

Gravitoelectromagnetism

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Gravitoelectromagnetism

 

Diagram regarding the confirmation of gravitomagnetism by Gravity Probe B

Gravitoelectromagnetism, abbreviated GEM, refers to a set of formal analogies between the equations for electromagnetism and 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 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 jets. Roger Penrose had proposed a mechanism that relies on frame-dragging-related effects for extracting energy and momentum from rotating black holes. Reva Kay Williams, 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; 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 also plans to observe gravitomagnetism effects.

Physical analogues of fields

• 

  Gravitomagnetism – gravitomagnetic field H due to (total) angular momentum J.

• 

  Electromagnetism – magnetic field B due to a dipole moment m...

• 

  ... 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 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:

GEM equations	Maxwell's equations
${\displaystyle \mathrm{\nabla }\cdot {\mathbf{E}}_{\text{g}}=-4\pi G{\rho }_{\text{g}}}$	${\displaystyle \mathrm{\nabla }\cdot \mathbf{E}=\frac{\rho }{{ϵ}_0}}$
${\displaystyle \mathrm{\nabla }\cdot {\mathbf{B}}_{\text{g}}=0}$	${\displaystyle \mathrm{\nabla }\cdot \mathbf{B}=0}$
${\displaystyle \mathrm{\nabla }\times {\mathbf{E}}_{\text{g}}=-\frac{\mathrm{\partial }{\mathbf{B}}_{\text{g}}}{\mathrm{\partial }t}}$	${\displaystyle \mathrm{\nabla }\times \mathbf{E}=-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}}$
${\displaystyle \mathrm{\nabla }\times {\mathbf{B}}_{\text{g}}=-\frac{4\pi G}{c^2}{\mathbf{J}}_{\text{g}}+\frac{1}{c^2}\frac{\mathrm{\partial }{\mathbf{E}}_{\text{g}}}{\mathrm{\partial }t}}$	${\displaystyle \mathrm{\nabla }\times \mathbf{B}=\frac{1}{{ϵ}_0{c}^2}\mathbf{J}+\frac{1}{c^2}\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}}$

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
• 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
• G is the gravitational constant
• ε₀ 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 equation:

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

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:

GEM equation	EM equation
${\displaystyle {\mathcal{S}}_{\text{g}}=-\frac{c^2}{4\pi G}{\mathbf{E}}_{\text{g}}\times 4{\mathbf{B}}_{\text{g}}}$	${\displaystyle \mathcal{S}=c^2{\epsilon }_0\mathbf{E}\times \mathbf{B}}$

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 of B_g in the GEM equations must be multiplied by −1/2c 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 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 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:

${\displaystyle {\mathbf{B}}_{\text{g}}=\frac{G}{2c^2}\frac{\mathbf{L}-3(\mathbf{L}\cdot \mathbf{r}/r)\mathbf{r}/r}{r^3},}$

where L is the angular momentum of the body. At the equatorial plane, r and L are perpendicular, so their dot product vanishes, and this formula reduces to:

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

The magnitude of angular momentum of a homogeneous ball-shaped body is:

${\displaystyle L=I_{\text{ball}}\omega =\frac{2m{r}^2}{5}\frac{2\pi }{T}}$

where:

• ${\displaystyle I_{\text{ball}}=\frac{2m{r}^2}{5}}$ is the moment of inertia of a ball-shaped body (see: list of moments of inertia);
• ${\displaystyle \omega }$ is the angular velocity;
• 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:

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

where ${\displaystyle g=G\frac{m}{r^2}}$ 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 1.012×10⁻¹⁴ Hz, 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 PSR J1748-2446ad (which rotates 716 times per second), assuming a radius of 16 km, and two solar masses, then

${\displaystyle B_{\text{g}}=\frac{2\pi Gm}{5r{c}^{2}T}}$

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 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 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.