Definitions

# Frame-dragging

Albert Einstein's theory of general relativity predicts that rotating bodies drag spacetime around themselves in a phenomenon referred to as frame-dragging. The rotational frame-dragging effect was first derived from the theory of general relativity in 1918 by the Austrian physicists Josef Lense and Hans Thirring, and is also known as the Lense-Thirring effect. Lense and Thirring predicted that the rotation of an object would alter space and time, dragging a nearby object out of position compared to the predictions of Newtonian physics. The predicted effect is incredibly small — about one part in a few trillion. In order to detect it, it is necessary to look at a very massive object, or build an instrument that is incredibly sensitive. More generally, the subject of field effects caused by moving matter is known as gravitomagnetism.

## Frame dragging effects

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 rotating 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 around the object faster than light moving against the rotation as seen by a distant observer. It is now the best-known effect, partly thanks to the Gravity Probe B experiment.

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 to derive from the same equation of general relativity. It is also a tiny effect that is difficult to confirm experimentally.

## Experimental tests of frame-dragging

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 and endowed 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, LARES, WEBER-SAT. Although extensively studied by various groups, such an idea has not yet been implemented. The butterfly configuration would allow, in principle, to measure not only the sum of the nodes but also the difference of the perigees, although such Keplerian orbital elements are more affected by the non-gravitational perturbations like the direct solar radiation pressure: the use of the active, drag-free technology would be required. Other proposed approaches involved the use of a single satellite to be placed in near polar orbit of low altitude, but such a strategy has been shown to be unfeasible. In order to enhance the possibilities of being implemented, it has been recently claimed that LARES/WEBER-SAT would be able to measure the effects induced by the multidimensional braneworld model by Dvali, Gabadaze and Porrati and to improve by two orders of magnitude the present-day level of accuracy of the equivalence principle. Such claims have been shown to be highly unrealistic.

Limiting ourselves 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 involving only the nodes of both the spacecraft. Although the predictions of general relativity are compatible with the experimental results, the realistic evaluation of the total error raised a debate. Another test of the Lense-Thirring effect in the gravitational field of Mars, performed by suitably interpreting the data of the Mars Global Surveyor (MGS) spacecraft, has been recently reported. Also such a test raised a debate. Attempts to detect the Lense-Thirring effect induced by the Sun's rotation on the orbits of the inner planets of the Solar System have been reported as well: the predictions of general relativity are compatible with the estimated corrections to the perihelia precessions, although the errors are still large. However, the inclusion of the radiometric data from the Magellan orbiter recently allowed Pitjeva to greatly improve the determination of the unmodelled precession of the perihelion of Venus. It amounts to -0.0004 +/- 0.0001 arcseconds/century, while the Lense-Thirring effect for the Venus' periehlion is just -0.0003 arcseconds/century. The system of the Galilean satellites of Jupiter was investigated as well, following the original suggestion by Lense and Thirring.

The Gravity Probe B experiment is currently under way to experimentally measure another gravitomagnetic effect, i.e. the Schiff precession of a gyroscope, to an expected 1% accuracy or better. Unfortunately, it seems that such an ambitious goal will not be achieved: indeed, first preliminary results released in April 2007 point toward a so far obtained accuracy of 256-128%, with the hope of reaching about 13% in December 2007. A 1% measurement of the Lense-Thirring effect in the gravitational field of the Earth could be obtained by launching at least two entirely new satellites, preferably endowed with active mechanisms of compensation of the non-gravitational forces, in rather eccentric orbits, as stated in 2005 by Iorio. Recently, the Italian Space Agency (ASI) has announced that the LARES satellite will be launched with a VEGA rocket at the end of 2008 The goal of LARES is to measure the Lense-Thirring effect to 1%, but there are doubts that this can be achieved, mainly due to the relatively low-orbit which LARES should be inserted into bringing into play more mismodelled even zonal harmonics. That is, spherical harmonics of the Earth's gravitational field caused by mass concentrations (like mountains) can drag a satellite in a way which may be difficult to distinguish from frame-dragging. Recently, an indirect test of the gravitomagnetic interaction accurate to 0.1% has been reported by Murphy et al with the Lunar Laser Ranging (LLR) technique, but Kopeikin questioned the ability of LLR to be sensible to gravitomagnetism.

## 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 Sir Roger Penrose -->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-ray and γ-ray photons; the collimated jets about the polar axis; and the asymmetrical formation of jets (relative to the orbital plane).

## Mathematical derivation of frame-dragging

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


c^{2} dtau^{2} = left(1 - frac{r_{s} r}{rho^{2}} right) c^{2} dt^{2} - frac{rho^{2}}{Lambda^{2}} dr^{2} - rho^{2} dtheta^{2}

- left(r^{2} + alpha^{2} + frac{r_{s} r alpha^{2}}{rho^{2}} sin^{2} theta right) sin^{2} theta dphi^{2} + frac{2r_{s} ralpha c sin^{2} theta }{rho^{2}} dphi dt

where rs is the Schwarzschild 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 where M (or, equivalently, rs) goes to zero, the Kerr metric becomes the orthogonal metric for the oblate spheroidal coordinates


c^{2} dtau^{2} = c^{2} dt^{2} - frac{rho^{2}}{r^{2} + alpha^{2}} dr^{2} - rho^{2} dtheta^{2} - left(r^{2} + alpha^{2} right) sin^{2}theta dphi^{2}

We may re-write the Kerr metric in the following form


c^{2} dtau^{2} = left(g_{tt} - frac{g_{tphi}^{2}}{g_{phiphi}} right) dt^{2} + g_{rr} dr^{2} + g_{thetatheta} dtheta^{2} + g_{phiphi} left(dphi + frac{g_{tphi}}{g_{phiphi}} 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


Omega = -frac{g_{tphi}}{g_{phiphi}} = frac{r_{s} alpha r c}{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.

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


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

where the purely radial component grr of the metric goes to infinity. The outer surface is not a sphere, but an oblate spheroid that touches the inner surface at the poles of the rotation axis, where the colatitude θ equals 0 or π; its radius is defined by the formula


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

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.

### Lense-Thirring effect inside a rotating shell

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

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

where the coefficients are

$d_1 = frac\left\{4MG\right\}\left\{3Rc^2\right\}$

$d_2 = frac\left\{4MG\right\}\left\{15Rc^2\right\}$

for MG<$d_1= frac\left\{4 alpha\left(2- alpha\right)\right\}\left\{\left(1+ alpha\right)\left(3- alpha\right)\right\}, alpha=frac\left\{MG\right\}\left\{2Rc^2\right\}$

The space-time inside the rotating spherical shell will not be flat. To have flat space-time inside, the rotating sphere should have non-spherical shape