Lense-Thirring Frame Dragging

Lense–Thirring Frame Dragging

Audience

Difficulty	Intermediate

Notation on this page

The Lense–Thirring effect (also called frame-dragging) is the prediction of General Relativity that a rotating mass drags spacetime around with itself. Discovered in 1918 by Josef Lense and Hans Thirring, the effect was confirmed experimentally in 2011 by the Gravity_Probe_B satellite mission.

Frame-dragging is the gravitational analogue of a magnetic field surrounding a moving electric current — a key prediction of GEM.

The classical prediction

For a stationary, axisymmetric mass M rotating with angular momentum J, the gravitomagnetic field at distance r from the centre is:

 Bg(r) = (G / c2 r3) [ 3 (r̂ · J)r̂ − J ]

This is the gravitational analogue of a magnetic dipole field. A test gyroscope at distance r experiences a precession of its spin axis:

 ΩLT = (G / c2 r3) [ 3 (r̂ · J)r̂ − J ] · (1/2)

For Earth (J ≈ 5.86 × 10³³ kg m²/s) at the orbital altitude of Gravity_Probe_B (~ 642 km), the predicted Lense–Thirring precession is approximately 39.2 milliarcseconds per year.

The physical picture

Imagine spacetime as a viscous fluid surrounding a spinning ball. As the ball rotates, it drags the surrounding fluid along with it. In GR the "fluid" is the inertial-reference-frame structure of spacetime itself — and a freely-falling gyroscope (which would, in flat spacetime, keep its axis fixed relative to the distant stars) is instead carried along by the rotating frame.

The effect is tiny in everyday situations (Earth's rotation drags inertial frames by less than a millionth of a degree per year) but becomes dramatic near rotating black holes, where frame-dragging is so extreme it produces the "ergosphere" — a region where no inertial observer can remain stationary relative to the distant stars.

Experimental confirmation

Gravity Probe B (2011)

The dedicated Gravity_Probe_B mission flew four superconducting gyroscopes in polar orbit around Earth from 2004 to 2005, with data analysis completed in 2011. The mission measured both:

• Geodetic effect (de Sitter precession): 6601.8 ± 18.3 mas/yr (GR prediction: 6606 mas/yr — confirmed to 0.3 %).
• Frame-dragging (Lense–Thirring): 37.2 ± 7.2 mas/yr (GR prediction: 39.2 mas/yr — confirmed to ~19 %).

This is the cleanest direct measurement of frame-dragging to date. Published as Physical Review Letters 106: 221101 (2011).

LAGEOS satellites

Two passive laser-ranging satellites (LAGEOS, LAGEOS 2) have been used since 1976 to measure frame-dragging via the precession of their orbital plane. Ciufolini and collaborators (2004, 2010) report agreement with GR's frame-dragging prediction at the 10 % level.

LARES (2012–present)

The LARES satellite — designed specifically to improve frame-dragging measurements — has confirmed the LT effect to about 5 % accuracy by 2016.

Frame-dragging near astrophysical objects

The effect becomes important near:

• Rotating black holes — the Kerr metric describes spacetime around a spinning black hole; frame-dragging is responsible for the ergosphere and for the Penrose process (extracting rotational energy).
• Neutron stars — typical frame-dragging frequencies ~ 10⁻² Hz near the surface of a millisecond pulsar.
• The galactic centre — Sgr A* exhibits frame-dragging effects on stellar orbits at the milli-arcsecond level.

Coupling to ψ in the present framework

In the psionic framework, frame-dragging is unmodified at the sanity-check limit 11 where ψ → 0. The standard prediction is recovered exactly — and Gravity_Probe_B confirms it.

In regions where ψ is dynamically significant, additional gravitomagnetic source terms appear:

$ \mathbf {B} _{g}(\mathbf {r} )=\mathbf {B} _{g}^{\text{standard}}(\mathbf {r} )+{\bigl (}{\text{extra }}\psi {\text{-contribution from }}\!\int \mathbf {j} _{\psi }\,d^{3}r'{\bigr )} $

This is the rigorous mathematical basis for the prediction that rotating superconductors and other strong-ψ-coupling systems should show anomalously large frame-dragging. The Tajmar 2007 anomaly (28 orders of magnitude larger than GR predicts in a rotating superconductor) is consistent with this picture.

Why the factor of 4

A famous puzzle in GEM: the factor of 4 in the gravitomagnetic Lorentz analog F = m(E_g + 4v × B_g/c) compared with the EM Lorentz force F = q(E + v × B). The factor comes from the spin-2 nature of gravity: a gravitational "dipole moment" sourced by a mass current couples 4× as strongly as the corresponding electromagnetic dipole would. This is not an arbitrary normalisation but a consequence of the tensor structure of GR.

Sanity checks

• Non-rotating mass (J = 0) → B_g = 0; no frame-dragging. ✓
• v → 0 → only the Newtonian (gravitoelectric) force; the LT force vanishes. ✓
• ψ → 0 → standard Kerr-like frame-dragging. ✓ (Sanity_Check_Limits §11.)
• Weak-field limit → linearised GEM equations. ✓ (Gravitoelectromagnetism.)

See Also

• What_is_Frame_Dragging
• Gravity_Probe_B
• Gravitoelectromagnetism
• Gravitomagnetic_London_Moment
• Modified_Einstein_Equations_with_Psi
• Famous_Experiments

References

• Lense, J., Thirring, H. (1918). "Über den Einfluß der Eigenrotation der Zentralkörper auf die Bewegung der Planeten und Monde nach der Einsteinschen Gravitationstheorie." Physikalische Zeitschrift 19: 156–163.
• Everitt, C. W. F., et al. (2011). "Gravity Probe B: Final Results of a Space Experiment to Test General Relativity." Physical Review Letters 106: 221101.
• Ciufolini, I., et al. (2004). "Confirmation of the General Relativistic Prediction of the Lense-Thirring Effect." Nature 431: 958–960.
• Ciufolini, I., et al. (2016). "A test of general relativity using the LARES and LAGEOS satellites and a GRACE Earth gravity model." European Physical Journal C 76: 120.