Math 2.0

Relativity and gravitational theory Definition

Gravitoelectromagnetism

Builds on Linearized gravityGeodesics in general relativityPoisson's equationMaxwell's equations

Definition

Work in linearized gravity on ℝ4\R^4 with coordinates (ct,x)(ct, x), x∈ℝ3x \in \R^3, in the harmonic gauge. Let the source be stationary matter with mass density ρ(x)\rho(x) and velocity field u(x)u(x) with ∣u∣≪c\abs{u} \ll c, and neglect its stresses. Define the gravitoelectric potential Φ\Phi and the gravitomagnetic vector potential AA as the decaying solutions of ∇2Φ=4πGρ,∇2A=4πGcρu.\nabla^2 \Phi = 4\pi G \rho, \qquad \nabla^2 A = \frac{4\pi G}{c} \rho u. Then the metric is, to first order, ds2=−(1+2Φc2)c2dt2+8cA⋅dx dt+(1−2Φc2)∣dx∣2.ds^2 = -\Big(1 + \frac{2\Phi}{c^2}\Big) c^2 dt^2 + \frac{8}{c} A \cdot dx \, dt + \Big(1 - \frac{2\Phi}{c^2}\Big) \abs{dx}^2. The gravitoelectric field is Eg=−∇ΦE_g = -\nabla \Phi and the gravitomagnetic field is Bg=∇×AB_g = \nabla \times A. Gravitoelectromagnetism is the description of weak stationary gravity by the pair (Eg,Bg)(E_g, B_g).

Remarks

In this normalization the fields satisfy the Maxwell-like equations ∇⋅Eg=−4πGρ,∇×Eg=0,∇⋅Bg=0,∇×Bg=−4πGcρu,\nabla \cdot E_g = -4\pi G \rho, \quad \nabla \times E_g = 0, \quad \nabla \cdot B_g = 0, \quad \nabla \times B_g = -\frac{4\pi G}{c} \rho u, where the last one uses the gauge condition ∇⋅A=0\nabla \cdot A = 0. A test particle with velocity vv moving along a geodesic has, to first order in v/cv/c, d2xdt2=Eg+4c v×Bg.\frac{d^2 x}{dt^2} = E_g + \frac{4}{c}\, v \times B_g. The factor 44 and the minus signs differ from electromagnetism (Maxwell's equations); other authors rescale AA and BgB_g, so the numerical factors in the literature vary.

Example: a mass MM at rest at the origin gives Φ=−GM/r\Phi = -GM/r, Eg=−GMx/r3E_g = -GM x/r^3, and A=0A = 0, so Bg=0B_g = 0; this is Newtonian gravity.

Non-example: the strong field near a neutron star is outside the definition, because the linear approximation fails. So is a source moving close to the speed of light, because the slow-motion approximation fails.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph
Geodesics in general relativity, concept, no evidence levelGeodesics in general …conceptLinearized gravity, concept, no evidence levelLinearized gravityconceptMaxwell's equations, concept, no evidence levelMaxwell's equationsconceptPoisson's equation, concept, no evidence levelPoisson's equationconceptGravitoelectromagnetism, definition, no evidence levelGravitoelectromagneti…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Geodesics in general relativity concept
  2. Linearized gravity concept
  3. Maxwell's equations concept
  4. Poisson's equation concept
  5. Gravitoelectromagnetism definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Gravitoelectromagnetism
About this page
Tag85B3EF1DS5
URIm2://85B3EF1DS5
Kinddefinition
WritingW0 stub
MSC83C25, 83C10
DefinesGravitoelectromagnetism
UsesLinearized gravity, Geodesics in general relativity, Poisson's equation, Maxwell's equations
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.