Frame-dragging

Experimental tests

In 1976 Van Patten and Everitt proposed to implement a dedicated mission aimed to measure the Lense–Thirring node precession of a pair of counter-orbiting spacecraft to be placed in terrestrial polar orbits with drag-free apparatus. A somewhat equivalent, less expensive version of such an idea was put forth in 1986 by Ciufolini who proposed to launch a passive, geodetic satellite in an orbit identical to that of the LAGEOS satellite, launched in 1976, apart from the orbital planes which should have been displaced by 180 degrees apart: the so-called butterfly configuration. The measurable quantity was, in this case, the sum of the nodes of LAGEOS and of the new spacecraft, later named LAGEOS III, LARES, WEBER-SAT.

Limiting the scope to the scenarios involving existing orbiting bodies, the first proposal to use the LAGEOS satellite and the Satellite Laser Ranging (SLR) technique to measure the Lense–Thirring effect dates to 1977–1978. Tests started to be effectively performed by using the LAGEOS and LAGEOS II satellites in 1996, according to a strategy involving the use of a suitable combination of the nodes of both satellites and the perigee of LAGEOS II. Further tests with the LAGEOS satellites were performed in 2004–2006 by discarding the perigee of LAGEOS II and using a linear combination. Recently, a comprehensive overview of the attempts to measure the Lense-Thirring effect with artificial satellites was published in the literature. The overall accuracy reached in the tests with the LAGEOS satellites is subject to some controversy.

The Gravity Probe B experiment was a satellite-based mission by a Stanford group and NASA, used to experimentally measure another gravitomagnetic effect, the Schiff precession of a gyroscope, to an expected 1% accuracy or better. Unfortunately such accuracy was not achieved. The first preliminary results released in April 2007 pointed towards an accuracy of 256–128%, with the hope of reaching about 13% in December 2007. In 2008 the Senior Review Report of the NASA Astrophysics Division Operating Missions stated that it was unlikely that the Gravity Probe B team will be able to reduce the errors to the level necessary to produce a convincing test of currently untested aspects of General Relativity (including frame-dragging). On May 4, 2011, the Stanford-based analysis group and NASA announced the final report, and in it the data from GP-B demonstrated the frame-dragging effect with an error of about 19 percent, and Einstein's predicted value was at the center of the confidence interval.

NASA published claims of success in verification of frame dragging for the GRACE twin satellites and Gravity Probe B, both of which claims are still in public view. A research group in Italy, USA, and UK also claimed success in verification of frame dragging with the Grace gravity model, published in a peer reviewed journal. All the claims include recommendations for further research at greater accuracy and other gravity models.

In 2026, an analysis combining LARES-2 and LAGEOS laser-ranging data with GRACE Earth-gravity models reported a measurement of terrestrial frame-dragging with a relative uncertainty at the one-part-in-a-thousand level, an order-of-magnitude improvement over previous Solar System determinations.

In the case of stars orbiting close to a spinning, supermassive black hole, frame dragging should cause the star's orbital plane to precess about the black hole spin axis. This effect was predicted to be detectable within the next few years via astrometric monitoring of stars at the center of the Milky Way galaxy. It was confirmed by the Max Planck Institute 16 April 2020.

By comparing the rate of orbital precession of two stars on different orbits, it is possible in principle to test the no-hair theorems of general relativity, in addition to measuring the spin of the black hole.

Astronomical evidence

Relativistic jets may provide evidence for the reality of frame-dragging. Gravitomagnetic forces produced by the Lense–Thirring effect (frame dragging) within the ergosphere of rotating black holes combined with the energy extraction mechanism by Penrose have been used to explain the observed properties of relativistic jets. The gravitomagnetic model developed by Reva Kay Williams predicts the observed high energy particles (~GeV) emitted by quasars and active galactic nuclei; the extraction of X-rays, γ-rays, and relativistic e⁻–e⁺ pairs; the collimated jets about the polar axis; and the asymmetrical formation of jets (relative to the orbital plane).

The Lense–Thirring effect has been observed in a binary system that consists of a massive white dwarf and a pulsar.

Mathematical derivation

Frame-dragging may be illustrated most readily using the Kerr metric, which describes the geometry of spacetime in the vicinity of a mass M rotating with angular momentum J, and Boyer–Lindquist coordinates (see the link for the transformation):

${\displaystyle \begin{array}{cc}{c}^{2}d{\tau }^2=& \left(1-\frac{r_{\text{s}}r}{{\rho }^2}\right)c^{2}d{t}^2-\frac{{\rho }^2}{{\mathrm{\Lambda }}^2}d{r}^2-{\rho }^{2}d{\theta }^2\\ & -\left(r^2+{\alpha }^2+\frac{r_{\text{s}}r{\alpha }^2}{{\rho }^2}{\sin }^2\theta \right){\sin }^2\theta d{\varphi }^2+\frac{2r_{\text{s}}r\alpha c{\sin }^2\theta }{{\rho }^2}d\varphi dt\end{array}}$

where r_s is the Schwarzschild radius

${\displaystyle r_{\text{s}}=\frac{2GM}{c^2}}$

and where the following shorthand variables have been introduced for brevity

${\displaystyle \alpha =\frac{J}{Mc}}$

${\displaystyle {\rho }^2=r^2+{\alpha }^2{\cos }^2\theta }$

${\displaystyle {\mathrm{\Lambda }}^2=r^2-r_{\text{s}}r+{\alpha }^2}$

In the non-relativistic limit where M (or, equivalently, r_s) goes to zero, the Kerr metric becomes the orthogonal metric for the oblate spheroidal coordinates

${\displaystyle c^{2}d{\tau }^2=c^{2}d{t}^2-\frac{{\rho }^2}{r^2+{\alpha }^2}d{r}^2-{\rho }^{2}d{\theta }^2-\left(r^2+{\alpha }^2\right){\sin }^2\theta d{\varphi }^2}$

We may rewrite the Kerr metric in the following form

${\displaystyle c^{2}d{\tau }^2=\left(g_{tt}-\frac{g_{t\varphi }^2}{g_{\varphi \varphi }}\right)d{t}^2+g_{rr}d{r}^2+g_{\theta \theta }d{\theta }^2+g_{\varphi \varphi }{\left(d\varphi +\frac{g_{t\varphi }}{g_{\varphi \varphi }}dt\right)}^2}$

This metric is equivalent to a co-rotating reference frame that is rotating with angular speed Ω that depends on both the radius r and the colatitude θ

${\displaystyle \mathrm{\Omega }=-\frac{g_{t\varphi }}{g_{\varphi \varphi }}=\frac{r_s\alpha rc}{{\rho }^2\left(r^2+{\alpha }^2\right)+r_s{\alpha }^{2}r{\sin }^2\theta }}$

In the plane of the equator this simplifies to:

${\displaystyle \mathrm{\Omega }=\frac{r_s\alpha c}{r^3+{\alpha }^{2}r+r_s{\alpha }^2}}$

Thus, an inertial reference frame is entrained by the rotating central mass to participate in the latter's rotation; this is frame-dragging.

An extreme version of frame dragging occurs within the ergosphere of a rotating black hole. The Kerr metric has two surfaces on which it appears to be singular. The inner surface corresponds to a spherical event horizon similar to that observed in the Schwarzschild metric; this occurs at

${\displaystyle r_{\text{inner}}=\frac{r_{\text{s}}+\sqrt{r_{\text{s}}^2-4{\alpha }^2}}{2}}$

where the purely radial component g_rr of the metric goes to infinity. The outer surface can be approximated by an oblate spheroid with lower spin parameters, and resembles a pumpkin-shape with higher spin parameters. It touches the inner surface at the poles of the rotation axis, where the colatitude θ equals 0 or π; its radius in Boyer-Lindquist coordinates is defined by the formula

${\displaystyle r_{\text{outer}}=\frac{r_{\text{s}}+\sqrt{r_{\text{s}}^2-4{\alpha }^2{\cos }^2\theta }}{2}}$

where the purely temporal component g_tt of the metric changes sign from positive to negative. The space between these two surfaces is called the ergosphere. A moving particle experiences a positive proper time along its worldline, its path through spacetime. However, this is impossible within the ergosphere, where g_tt is negative, unless the particle is co-rotating with the interior mass M with an angular speed at least of Ω. However, as seen above, frame-dragging occurs about every rotating mass and at every radius r and colatitude θ, not only within the ergosphere.

Lense–Thirring effect inside a rotating shell

The Lense–Thirring effect inside a rotating shell was taken by Albert Einstein as not just support for, but a vindication of Mach's principle, in a letter he wrote to Ernst Mach in 1913 (five years before Lense and Thirring's work, and two years before he had attained the final form of general relativity). A reproduction of the letter can be found in Misner, Thorne, Wheeler. The general effect scaled up to cosmological distances, is still used as a support for Mach's principle.

Inside a rotating spherical shell the acceleration due to the Lense–Thirring effect would be

${\displaystyle \overline{a}=-2d_1\left(\overline{\omega }\times \overline{v}\right)-d_2\left[\overline{\omega }\times \left(\overline{\omega }\times \overline{r}\right)+2\left(\overline{\omega }\overline{r}\right)\overline{\omega }\right]}$

where the coefficients are

${\displaystyle \begin{array}{cc}{d}_1& =\frac{4MG}{3R{c}^2}\\ d_2& =\frac{4MG}{15R{c}^2}\end{array}}$

for MG ≪ Rc² or more precisely,

${\displaystyle d_1=\frac{4\alpha (2-\alpha )}{(1+\alpha )(3-\alpha )},\alpha =\frac{MG}{2R{c}^2}}$

The spacetime inside the rotating spherical shell will not be flat. A flat spacetime inside a rotating mass shell is possible if the shell is allowed to deviate from a precisely spherical shape and the mass density inside the shell is allowed to vary.