Math 2.0

Relativity and gravitational theory Classical tests and observational relativity Proposition

Lense–Thirring precession

Builds on Frame-draggingGravitoelectromagnetismLinearized gravityKepler orbitPrecessionAngular momentumGeodetic effect

Statement

Let GG be the gravitational constant and cc the speed of light. Let a body of angular momentum JJ create a weak, slowly rotating gravitational field, described by linearized gravity. Let a test body move on a bound orbit close to a Kepler orbit with semi-major axis aa and eccentricity ee around it. Then, to first order in G/c2G/c^2, the line of nodes of the orbit precesses in the sense of the rotation, with average angular velocity Ω˙LT=2GJc2a3(1−e2)3/2.\dot\Omega_{LT} = \frac{2 G J}{c^2 a^3 (1 - e^2)^{3/2}}.

Remarks

In the weak field a rotating body adds off-diagonal terms g0ig_{0i} to the metric. They form a gravitomagnetic potential shaped like a magnetic dipole field, in the language of gravitoelectromagnetism. The resulting gravitomagnetic force acts on the orbiting body like a magnetic force on a moving charge, and averaging it over one Kepler orbit gives the precession above. The same field makes a gyroscope at position 𝐫\mathbf{r} precess relative to the distant stars with angular velocity Gc2r3[3(𝐉⋅𝐫^)𝐫^−𝐉]\frac{G}{c^2 r^3}\left[3(\mathbf{J}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}} - \mathbf{J}\right], where 𝐫^=𝐫/r\hat{\mathbf r} = \mathbf r/r. Lense and Thirring derived the effect in 1918. For Earth satellites it is a few tens of milliarcseconds per year, and it was measured with the LAGEOS satellites. It is one form of frame-dragging.

Going deeper

For a student

The Lense–Thirring precession is the gravitational analogue of the precession of a charged particle's orbit around a magnetic dipole. It must be separated from much larger Newtonian precessions caused by the oblateness of the Earth, which is why satellite tests combine several orbits. For a gyroscope, the analogous effect is the frame-dragging precession measured by Gravity Probe B, which sits on top of the larger geodetic effect.

Reading path

To understand this page from the ground up, read these 25 pages in order. The foundations come first.

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Boyer–Lindquist coordinates, concept, no evidence levelBoyer–Lindquist coord…conceptCross product, concept, no evidence levelCross productconceptDerivative, concept, no evidence levelDerivativeconceptEuclidean space, object, no evidence levelEuclidean spaceobjectEuclidean vector, concept, no evidence levelEuclidean vectorconceptFermi–Walker transport, concept, no evidence levelFermi–Walker transportconceptGeneral relativity, concept, no evidence levelGeneral relativityconceptGeodesics in general relativity, concept, no evidence levelGeodesics in general …conceptGravitoelectromagnetism, concept, no evidence levelGravitoelectromagneti…conceptKepler orbit, concept, no evidence levelKepler orbitconceptKilling vector field, concept, no evidence levelKilling vector fieldconceptLinearized gravity, concept, no evidence levelLinearized gravityconceptOrdinary differential equation, concept, no evidence levelOrdinary differential…conceptParameterized post-Newtonian formalism, concept, no evidence levelParameterized post-Ne…conceptPrecession, concept, no evidence levelPrecessionconceptRicci curvature, concept, no evidence levelRicci curvatureconceptVelocity, concept, no evidence levelVelocityconceptClassical mechanics, concept, no evidence levelClassical mechanicsconceptSchwarzschild metric, example, no evidence levelSchwarzschild metricexampleMomentum, concept, no evidence levelMomentumconceptKerr metric, example, no evidence levelKerr metricexampleAngular momentum, concept, no evidence levelAngular momentumconceptFrame-dragging, statement, no evidence levelFrame-draggingstatementGeodetic effect, statement, no evidence levelGeodetic effectstatementLense–Thirring precession, statement, no evidence levelLense–Thirring preces…statement
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. Boyer–Lindquist coordinates concept
  2. Cross product concept
  3. Derivative concept
  4. Euclidean space object
  5. Euclidean vector concept
  6. Fermi–Walker transport concept
  7. General relativity concept
  8. Geodesics in general relativity concept
  9. Gravitoelectromagnetism concept
  10. Kepler orbit concept
  11. Killing vector field concept
  12. Linearized gravity concept
  13. Ordinary differential equation concept
  14. Parameterized post-Newtonian formalism concept
  15. Precession concept
  16. Ricci curvature concept
  17. Velocity concept
  18. Classical mechanics concept
  19. Schwarzschild metric example
  20. Momentum concept
  21. Kerr metric example
  22. Angular momentum concept
  23. Frame-dragging statement
  24. Geodetic effect statement
  25. Lense–Thirring precession statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Lense%E2%80%93Thirring_precession
About this page
Tag801B900M5K
URIm2://801B900M5K
Kindstatement, proposition
WritingW0 stub
MSC83C10, 83C25
StatusThis copy of the site has no build data. Run m2 status 801B900M5K.
AboutFrame-dragging
UsesFrame-dragging, Gravitoelectromagnetism, Linearized gravity, Kepler orbit, Precession, Angular momentum, Geodetic effect
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Changed3 November 2026

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