The first frame-dragging effect was derived in 1918, in the framework of general relativity, by the Austrian physicists
Josef Lense and
Hans Thirring, and is also known as the
Lense–Thirring effect. They predicted that the rotation of a massive object would distort the
spacetime metric, making the orbit of a nearby test particle
precess. This does not happen in
Newtonian mechanics for which the
gravitational field of a body depends only on its mass, not on its rotation. The Lense–Thirring effect is very small – about one part in a few trillion. To detect it, it is necessary to examine a very massive object, or build an instrument that is very sensitive.
In 2015, new general-relativistic extensions of Newtonian rotation laws were formulated to describe geometric dragging of frames which incorporates a newly discovered antidragging effect.
Rotational frame-dragging (the
Lense–Thirring effect) appears in the
general principle of relativity and similar theories in the vicinity of rotating massive objects. Under the Lense–Thirring effect, the frame of reference in which a clock ticks the fastest is one which is revolving around the object as viewed by a distant observer. This also means that light traveling in the direction of rotation of the object will move past the massive object faster than light moving against the rotation, as seen by a distant observer. It is now the best known frame-dragging effect, partly thanks to the
Gravity Probe B experiment. Qualitatively, frame-dragging can be viewed as the gravitational analog of
Also, an inner region is dragged more than an outer region. This produces interesting locally rotating frames. For example, imagine that a north–south-oriented ice skater, in orbit over the equator of a rotating black hole and rotationally at rest with respect to the stars, extends her arms. The arm extended toward the black hole will be "torqued" spinward due to gravitomagnetic induction ("torqued" is in quotes because gravitational effects are not considered "forces" under
GR). Likewise the arm extended away from the black hole will be torqued anti-spinward. She will therefore be rotationally sped up, in a counter-rotating sense to the black hole. This is the opposite of what happens in everyday experience. There exists a particular rotation rate that, should she be initially rotating at that rate when she extends her arms, inertial effects and frame-dragging effects will balance and her rate of rotation will not change. Due to the
equivalence principle, gravitational effects are locally indistinguishable from inertial effects, so this rotation rate, at which when she extends her arms nothing happens, is her local reference for non-rotation. This frame is rotating with respect to the fixed stars and counter-rotating with respect to the black hole. This effect is analogous to the
hyperfine structure in atomic spectra due to nuclear spin. A useful metaphor is a
planetary gear system with the black hole being the sun gear, the ice skater being a planetary gear and the outside universe being the ring gear. See
Another interesting consequence is that, for an object constrained in an equatorial orbit, but not in freefall, it weighs more if orbiting anti-spinward, and less if orbiting spinward. For example, in a suspended equatorial bowling alley, a bowling ball rolled anti-spinward would weigh more than the same ball rolled in a spinward direction. Note, frame dragging will neither accelerate nor slow down the bowling ball in either direction. It is not a "viscosity". Similarly, a stationary
plumb-bob suspended over the rotating object will not list. It will hang vertically. If it starts to fall, induction will push it in the spinward direction.
Linear frame dragging is the similarly inevitable result of the general principle of relativity, applied to
linear momentum. Although it arguably has equal theoretical legitimacy to the "rotational" effect, the difficulty of obtaining an experimental verification of the effect means that it receives much less discussion and is often omitted from articles on frame-dragging (but see Einstein, 1921).
Static mass increase is a third effect noted by Einstein in the same paper. The effect is an increase in
inertia of a body when other masses are placed nearby. While not strictly a frame dragging effect (the term frame dragging is not used by Einstein), it is demonstrated by Einstein that it derives from the same equation of general relativity. It is also a tiny effect that is difficult to confirm experimentally.
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, cheaper 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 deg 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,
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 back to 1977–1978. Tests have 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. The latest tests with the LAGEOS satellites have been 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.
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 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 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 should be detectable within the next few years via
astrometric monitoring of stars at the center of the
Milky Way galaxy.
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.
and where the following shorthand variables have been introduced for brevity
In the non-relativistic limit where M (or, equivalently, rs) goes to zero, the Kerr metric becomes the orthogonal metric for the
oblate spheroidal coordinates
We may rewrite the Kerr metric in the following form
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
In the plane of the equator this simplifies to:
Thus, an inertial reference frame is entrained by the rotating central mass to participate in the latter's rotation; this is frame-dragging.
The two surfaces on which the
Kerr metric appears to have singularities; the inner surface is the
event horizon, whereas the outer surface is pumpkin-shaped. The
ergosphere lies between these two surfaces; within this volume, the purely temporal component gtt is negative, i.e., acts like a purely spatial metric component. Consequently, particles within this ergosphere must co-rotate with the inner mass, if they are to retain their time-like character.
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
where the purely radial component grr 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
where the purely temporal component gtt 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 gtt 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.
Inside a rotating spherical shell the acceleration due to the Lense–Thirring effect would be
where the coefficients are
for MG ≪ Rc2 or more precisely,
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.
^Thirring, H. (1918). "Über die Wirkung rotierender ferner Massen in der Einsteinschen Gravitationstheorie". Physikalische Zeitschrift. 19: 33.
1918PhyZ...19...33T. [On the Effect of Rotating Distant Masses in Einstein's Theory of Gravitation]
^Thirring, H. (1921). "Berichtigung zu meiner Arbeit: 'Über die Wirkung rotierender Massen in der Einsteinschen Gravitationstheorie'". Physikalische Zeitschrift. 22: 29.
1921PhyZ...22...29T. [Correction to my paper "On the Effect of Rotating Distant Masses in Einstein's Theory of Gravitation"]
^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.
1918PhyZ...19..156L. [On the Influence of the Proper Rotation of Central Bodies on the Motions of Planets and Moons According to Einstein's Theory of Gravitation]
^Van Patten, R. A.; Everitt, C. W. F. (1976). "A possible experiment with two counter-rotating drag-free satellites to obtain a new test of Einstein's general theory of relativity and improved measurements in geodesy". Celestial Mechanics. 13 (4): 429–447.
^Everitt, C. W. F, The Gyroscope Experiment I. General Description and Analysis of Gyroscope Performance. In: Bertotti, B. (Ed.), Proc. Int. School Phys. "Enrico Fermi" Course LVI. New Academic Press, New York, pp. 331–360, 1974. Reprinted in: Ruffini, R. J., Sigismondi, C. (Eds.), Nonlinear Gravitodynamics. The Lense–Thirring Effect. World Scientific, Singapore, pp. 439–468, 2003.
^Everitt, C. W. F., et al., Gravity Probe B: Countdown to Launch. In: Laemmerzahl, C., Everitt, C. W. F., Hehl, F. W. (Eds.), Gyros, Clocks, Interferometers...: Testing Relativistic Gravity in Space. Springer, Berlin, pp. 52–82, 2001.
^Pugh, G. E., Proposal for a Satellite Test of the Coriolis Prediction of General Relativity, WSEG, Research Memorandum No. 11, 1959. Reprinted in: Ruffini, R. J., Sigismondi, C. (Eds.), Nonlinear Gravitodynamics. The Lense–Thirring Effect. World Scientific, Singapore, pp. 414–426, 2003.
^Schiff, L., On Experimental Tests of the General Theory of Relativity, Am. J. Phys., 28, 340–343, 1960.
^Muhlfelder, B., Mac Keiser, G., and Turneaure, J., Gravity Probe B Experiment Error, poster L1.00027 presented at the American Physical Society (APS) meeting in Jacksonville, Florida, on 14–17 April 2007, 2007.