Lense-Thirring effect

The Lense-Thirring effect arises from the off-diagonal terms in the metric tensor that appear when a massive object rotates. In the weak-field limit, the gravitational field can be decomposed into a gravitoelectric part (the usual Newtonian gravity) and a gravitomagnetic part, which is sourced by mass currents. This gravitomagnetic field exerts a torque on a gyroscope or a test particle's orbital plane, causing a precession known as frame-dragging.

For a satellite in a circular orbit around a rotating body, the orbital plane precesses at a rate given by the Lense-Thirring formula. The effect is extremely small: for Earth, the predicted precession for a satellite like LAGEOS is about 31 milliarcseconds per year, which is tiny compared to the Newtonian precession due to Earth's oblateness (which is thousands of times larger).¹

Josef Lense and Hans Thirring published their derivation in 1918, just three years after Einstein's final formulation of general relativity. They considered the weak-field limit of the Einstein field equations and showed that a rotating mass would induce a precession of a gyroscope's axis. Their work was largely theoretical and remained untested for decades due to the smallness of the effect.

In the 1960s, the concept of gravitomagnetism was further developed by Robert L. Forward and others, who drew analogies with electromagnetism. The first serious experimental proposals to measure the effect came in the 1970s, leading to the Gravity Probe B mission and the use of laser-ranged satellites like LAGEOS.²

The first unambiguous measurement of the Lense-Thirring effect was reported in 2004 by Ignazio Ciufolini and colleagues, using the LAGEOS and LAGEOS II satellites. They measured a precession of about 39 milliarcseconds per year, with an accuracy of about 10%, consistent with the prediction of general relativity.³

In 2011, the Gravity Probe B mission, which used ultra-precise gyroscopes in a polar orbit, measured the frame-dragging effect to be 37.2 milliarcseconds per year with an error of about 19%, also confirming the prediction. More recent analyses of the LAGEOS satellites and the LARES satellite have improved the accuracy to about 5%.⁴

The Lense-Thirring effect plays a crucial role in astrophysics, particularly around black holes and neutron stars. In the vicinity of a rotating black hole, frame-dragging causes the accretion disk to precess, which can modulate the X-ray emission from the system. This phenomenon is invoked to explain quasi-periodic oscillations in X-ray binaries and the behavior of active galactic nuclei.⁵

Frame-dragging also affects the orbits of pulsars in binary systems and the spin-orbit coupling in coalescing binary black holes, which is relevant for gravitational wave astronomy. The detection of gravitational waves by LIGO and Virgo has provided indirect evidence for the effect, as the waveforms include the spin-induced precession of the orbital plane.⁶

One lesser-known aspect is the Lense-Thirring effect on the Moon's orbit. The Moon's orbit is also predicted to precess due to Earth's rotation, but the effect is only about 1 part in 10 billion of the Newtonian precession, making it extremely difficult to detect. However, lunar laser ranging experiments have placed upper bounds on the effect, which are consistent with general relativity.

Another subtlety is the distinction between the Lense-Thirring precession of a gyroscope and the precession of an orbital plane. The former is often called the Schiff precession, after Leonard Schiff who calculated it in 1960. The two effects are related but distinct, and both were measured by Gravity Probe B.²

In the strong-field regime, the Lense-Thirring effect can lead to the Bardeen-Petterson effect, where a misaligned accretion disk around a rotating black hole is forced into alignment with the black hole's equatorial plane. This effect is important for understanding the geometry of accretion flows in active galactic nuclei and X-ray binaries.⁵

Frame-dragging

The dragging of spacetime around a rotating massive body, as predicted by general relativity.

Gravitomagnetic field

The gravitational analog of a magnetic field, produced by mass currents.

Gyroscope

A device used to measure orientation, whose axis precesses due to frame-dragging.

LAGEOS

Laser Geodynamics Satellite, a series of satellites used for precise measurements of Earth's gravitational field.