Frame-dragging

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Frame-dragging is an effect on

Static mass increase is a third effect noted by Einstein in the same paper.[5] 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^[6][7] 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^[8] 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.^[9] Tests started to be effectively performed by using the LAGEOS and LAGEOS II satellites in 1996,^[10] according to a strategy^[11] 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^[12][13] by discarding the perigee of LAGEOS II and using a linear combination.^[14] Recently, a comprehensive overview of the attempts to measure the Lense-Thirring effect with artificial satellites was published in the literature.^[15] The overall accuracy reached in the tests with the LAGEOS satellites is subject to some controversy.^[16][17][18]

The

NASA published claims of success in verification of frame dragging for the

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.^[34]

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.^[35]

The Lense–Thirring effect has been observed in a binary system that consists of a massive white dwarf and a pulsar.^[39]

Frame-dragging may be illustrated most readily using the Kerr metric,^[40][41] 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):

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

where r_s is the Schwarzschild radius

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

and where the following shorthand variables have been introduced for brevity

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

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

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

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

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

We may rewrite the Kerr metric in the following form

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

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

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

In the plane of the equator this simplifies to:^[42]

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

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

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

where the purely radial component g_rr of the metric goes to infinity. The outer surface can be approximated by an

Inside a rotating spherical shell the acceleration due to the Lense–Thirring effect would be[46]

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

where the coefficients are

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

for MG ≪ Rc² or more precisely,

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

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.^[47]

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