Frame-dragging

This article is about the effect on spacetime caused by a rotating mass. For video frame editing, seeFrame rate.

Frame-draggingis an effect onspacetime,predicted byAlbert Einstein'sgeneral theory of relativity,that is due to non-static stationary distributions ofmass–energy.A stationaryfieldis one that is in a steady state, but the masses causing that field may be non-static ⁠— rotating, for instance. More generally, the subject that deals with the effects caused by mass–energy currents is known asgravitoelectromagnetism,which is analogous to the magnetism ofclassical electromagnetism.

The first frame-dragging effect was derived in 1918, in the framework of general relativity, by the Austrian physicistsJosef LenseandHans Thirring,and is also known as theLense–Thirring effect.They predicted that the rotation of a massive object would distort thespacetime metric,making the orbit of a nearby test particleprecess.This does not happen inNewtonian mechanicsfor which thegravitational fieldof 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.

Effects

Rotational frame-dragging(theLense–Thirring effect) appears in thegeneral principle of relativityand 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 theGravity Probe Bexperiment. Qualitatively, frame-dragging can be viewed as the gravitational analog ofelectromagnetic induction.

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" underGR). 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 theequivalence 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 thehyperfine structurein atomic spectra due to nuclear spin. A useful metaphor is aplanetary gearsystem with the black hole being the sun gear, the ice skater being a planetary gear and the outside universe being the ring gear. SeeMach's principle.

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 stationaryplumb-bobsuspended over the rotating object will not list. It will hang vertically. If it starts to fall, induction will push it in the spinward direction. However, if a "yoyo" plumb-bob (with axis perpendicular to the equatorial plane) is slowly lowered, over the equator, toward the static limit, the yoyo will spin up in a counter rotating direction. Curiously, any denizens inside the yoyo will not feel any torque and will not experience any felt change in angular momentum.

Linear frame draggingis the similarly inevitable result of the general principle of relativity, applied tolinear 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 increaseis a third effect noted by Einstein in the same paper.The effect is an increase ininertiaof 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.

Experimental tests

In 1976 Van Patten and Everittproposed 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 Ciufoliniwho proposed to launch a passive, geodetic satellite in an orbit identical to that of theLAGEOSsatellite, 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 andLAGEOS IIsatellites in 1996,according to a strategyinvolving 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 2004–2006by 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.

TheGravity Probe Bexperimentwas a satellite-based mission by a Stanford group and NASA, used to experimentally measure another gravitomagnetic effect, theSchiff precessionof 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 of256–128%, 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 theGRACE twin satellitesand 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 toprecessabout the black hole spin axis. This effect should be detectable within the next few years viaastrometricmonitoring of stars at the center of theMilky Waygalaxy.

By comparing the rate of orbital precession of two stars on different orbits, it is possible in principle to test theno-hair theoremsof general relativity, in addition to measuring the spin of the black hole.

Astronomical evidence

Relativistic jetsmay provide evidence for the reality of frame-dragging.Gravitomagneticforces produced by theLense–Thirring effect(frame dragging) within theergosphereofrotating black holescombined with the energy extraction mechanism byPenrosehave been used to explain the observed properties ofrelativistic jets.The gravitomagnetic model developed byReva Kay Williamspredicts the observed high energy particles (~GeV) emitted byquasarsandactive 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 massivewhite dwarfand apulsar.

Mathematical derivation

Frame-dragging may be illustrated most readily using theKerr metric,which describes the geometry ofspacetimein the vicinity of a massMrotating withangular momentumJ,andBoyer–Lindquist coordinates(see the link for the transformation):

${\begin{aligned}c^{2}d\tau ^{2}=&\left(1-{\frac {r_{s}r}{\rho ^{2}}}\right)c^{2}dt^{2}-{\frac {\rho ^{2}}{\Lambda ^{2}}}dr^{2}-\rho ^{2}d\theta ^{2}\\&{}-\left(r^{2}+\alpha ^{2}+{\frac {r_{s}r\alpha ^{2}}{\rho ^{2}}}\sin ^{2}\theta \right)\sin ^{2}\theta \ d\phi ^{2}+{\frac {2r_{s}r\alpha c\sin ^{2}\theta }{\rho ^{2}}}d\phi dt\end{aligned}}$

wherer_sis theSchwarzschild radius

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

and where the following shorthand variables have been introduced for brevity

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

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

$\Lambda ^{2}=r^{2}-r_{s}r+\alpha ^{2}\,\!$

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

$c^{2}d\tau ^{2}=c^{2}dt^{2}-{\frac {\rho ^{2}}{r^{2}+\alpha ^{2}}}dr^{2}-\rho ^{2}d\theta ^{2}-\left(r^{2}+\alpha ^{2}\right)\sin ^{2}\theta d\phi ^{2}$

We may rewrite the Kerr metric in the following form

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

This metric is equivalent to a co-rotating reference frame that is rotating with angular speed Ω that depends on both the radiusrand thecolatitudeθ

$\Omega =-{\frac {g_{t\phi }}{g_{\phi \phi }}}={\frac {r_{s}\alpha rc}{\rho ^{2}\left(r^{2}+\alpha ^{2}\right)+r_{s}\alpha ^{2}r\sin ^{2}\theta }}$

In the plane of the equator this simplifies to:

$\Omega ={\frac {r_{s}\alpha c}{r^{3}+\alpha ^{2}r+r_{s}\alpha ^{2}}}$

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

The two surfaces on which theKerr metricappears to have singularities; the inner surface is theoblate spheroid-shapedevent horizon,whereas the outer surface is pumpkin-shaped.Theergospherelies between these two surfaces; within this volume, the purely temporal componentg_ttis 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 theergosphereof a rotatingblack hole.The Kerr metric has two surfaces on which it appears to be singular. The inner surface corresponds to a sphericalevent horizonsimilar to that observed in theSchwarzschild metric;this occurs at

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

where the purely radial componentg_rrof the metric goes to infinity. The outer surface can be approximated by anoblate spheroidwith lower spin parameters, and resembles a pumpkin-shapewith higher spin parameters. It touches the inner surface at the poles of the rotation axis, where the colatitudeθequals 0 or π; its radius in Boyer-Lindquist coordinates is defined by the formula

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

where the purely temporal componentg_ttof the metric changes sign from positive to negative. The space between these two surfaces is called theergosphere.A moving particle experiences a positiveproper timealong itsworldline,its path throughspacetime.However, this is impossible within the ergosphere, whereg_ttis negative, unless the particle is co-rotating with the interior massMwith an angular speed at least of Ω. However, as seen above, frame-dragging occurs about every rotating mass and at every radiusrand colatitudeθ,not only within the ergosphere.

Lense–Thirring effect inside a rotating shell

TheLense–Thirring effectinside a rotating shell was taken byAlbert Einsteinas not just support for, but a vindication ofMach's principle,in a letter he wrote toErnst Machin 1913 (five years before Lense and Thirring's work, and two years before he had attained the final form ofgeneral relativity). A reproduction of the letter can be found inMisner, 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

${\bar {a}}=-2d_{1}\left({\bar {\omega }}\times {\bar {v}}\right)-d_{2}\left[{\bar {\omega }}\times \left({\bar {\omega }}\times {\bar {r}}\right)+2\left({\bar {\omega }}{\bar {r}}\right){\bar {\omega }}\right]$

where the coefficients are

${\begin{aligned}d_{1}&={\frac {4MG}{3Rc^{2}}}\\d_{2}&={\frac {4MG}{15Rc^{2}}}\end{aligned}}$

forMG≪Rc²or more precisely,

$d_{1}={\frac {4\alpha (2-\alpha )}{(1+\alpha )(3-\alpha )}},\qquad \alpha ={\frac {MG}{2Rc^{2}}}$

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.

See also

• Geodetic effect
• Gravity Recovery and Climate Experiment
• Gravitomagnetism
• Mach's principle
• Broad iron K line

Further reading

• Renzetti, G. (May 2013)."History of the attempts to measure orbital frame-dragging with artificial satellites".Central European Journal of Physics.11(5): 531–544.Bibcode:2013CEJPh..11..531R.doi:10.2478/s11534-013-0189-1.
• Ginzburg, V. L. (May 1959). "Artificial Satellites and the Theory of Relativity".Scientific American.200(5): 149–160.Bibcode:1959SciAm.200e.149G.doi:10.1038/scientificamerican0559-149.

External links

• NASA RELEASE: 04-351 As The World Turns, It Drags Space And TimeArchived2008-06-19 at theWayback Machine