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General relativity
G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}\,
Einstein field equations
Introduction to...
Mathematical formulation of...
Phenomena
Kepler problem · Lenses · Waves
Frame-dragging · Geodetic effect
Event horizon · Singularity
Black hole
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Albert Einstein's theory of general relativity predicts that rotating bodies drag spacetime around themselves in a phenomenon referred to as frame-dragging. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the General relativity (GR is a Theory of Gravitation that was developed by Albert Einstein between 1907 and 1915 The mathematics of general relativity refers to various mathematical structures and techniques that are used in studying Albert Einstein 's theory of General The Kepler problem in general relativity involves solving for the motion of two spherical bodies interacting with one another by Gravitation, as described by the theory of A gravitational lens is formed when the light from a very distant bright source (such as a Quasar) is "bent" around a massive object (such as a cluster of In Physics, a gravitational wave is a Fluctuation in the Curvature of Spacetime which propagates as a wave, traveling outward from The geodetic effect represents the effect of the curvature of Spacetime, predicted by General relativity, on a spinning moving body In General relativity, an event horizon is a boundary in Spacetime, an area surrounding a Black hole or a Wormhole, inside which events cannot A gravitational singularity (sometimes spacetime singularity) is approximately a place where quantities which are used to measure the Gravitational field become A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS The rotational frame-dragging effect was first derived from the theory of general relativity in 1918 by the Austrian physicists Joseph Lense and Hans Thirring, and is also known as the Lense-Thirring effect. Year 1918 ( MCMXVIII) was a Common year starting on Tuesday (link will display the full calendar of the Gregorian calendar (or a Common Hans Thirring ( March 23, 1888 - March 22, 1976) was an Austrian theoretical physicist, Professor, and father [1][2][3] 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. Gravitomagnetism (sometimes Gravitoelectromagnetism, abbreviated GEM) refers to a set of formal analogies between Maxwell's field

Contents

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. A principle of relativity is a criterion for judging physical theories, stating that they are inadequate if they do not prescribe the exact same laws of physics in 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. Gravity Probe B ( GP-B) is a Satellite -based mission which launched in 2004

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). [4]

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 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[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 and endowed with drag-free apparatus. A somewhat equivalent, cheaper 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 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,[9][10] 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,[11][12][13] 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,[14][15] but such a strategy has been shown to be unfeasible. [16][17][18] 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[19] induced by the multidimensional braneworld model by Dvali, Gabadaze and Porrati[20] and to improve by two orders of magnitude the present-day level of accuracy of the equivalence principle. [21] Such claims have been shown to be highly unrealistic. [22][23]

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. In satellite laser ranging ( SLR) a global network of observation stations measure the round trip time of flight of ultrashort pulses of Light to Satellites [24][25] Tests have started to be effectively performed by using the LAGEOS and LAGEOS II satellites in 1996,[26] according to a strategy[27] involving the use of a suitable combination of the nodes of both satellites and the perigee of LAGEOS II. LAGEOS, or Laser Geodynamics Satellites are a series of scientific research Satellites designed to provide an orbiting laser ranging benchmark for geodynamical The latest tests with the LAGEOS satellites have been performed in 2004-2006[28][29] by discarding the perigee of LAGEOS II and using a linear combination[30][31][32][33][34][35] involving only the nodes of both the spacecraft. LAGEOS, or Laser Geodynamics Satellites are a series of scientific research Satellites designed to provide an orbiting laser ranging benchmark for geodynamical Although the predictions of general relativity are compatible with the experimental results, the realistic evaluation of the total error raised a debate. [36][37][38][39][40][41] 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. [42] Also such a test raised a debate. [43][44][45] 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:[46] the predictions of general relativity are compatible with the estimated corrections to the perihelia precessions,[47] although the errors are still large. The system of the Galilean satellites of Jupiter was investigated as well,[48] following the original suggestion by Lense and Thirring.

The Gravity Probe B experiment[49][50] is currently under way to experimentally measure another gravitomagnetic effect, i. Gravity Probe B ( GP-B) is a Satellite -based mission which launched in 2004 e. the Schiff precession of a gyroscope,[51][52] 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[53] 256-128%, with the hope of reaching about 13% in December 2007. [54] 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. [55] Recently, the Italian Space Agency (ASI) has announced that the LARES satellite will be launched with a VEGA rocket at the end of 2008 [1]. The goal of LARES is to measure the Lense-Thirring effect to 1%, but there are doubts that this can be achieved[56][57]. Recently, an indirect test of the gravitomagnetic interaction accurate to 0. 1% has been reported by Murphy et al[58] with the Lunar Laser Ranging (LLR) technique, but Kopeikin[59] questioned the ability of LLR to be sensible to gravitomagnetism.

Astronomical evidence

Relativistic Jet. The environment around the AGN where the relativistic plasma is collimated into jets which escape along the pole of the supermassive black hole
Relativistic Jet. The environment around the AGN where the relativistic plasma is collimated into jets which escape along the pole of the supermassive black hole

Relativistic jets may provide evidence for the reality of frame-dragging. An active galactic nucleus ( AGN) is a compact region at the centre of a Galaxy which has a much higher than normal luminosity over some or all of the Electromagnetic Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In Physics and Chemistry, plasma is an Ionized Gas, in which a certain proportion of Electrons are free rather than being bound A supermassive black hole is a Black hole with a Mass of an order of magnitude between 105 and 1 The lower-energy non-relativistic version of this phenomenon is described at Polar jet. Gravitomagnetic forces produced by the Lense-Thirring effect (frame dragging) within the ergosphere of rotating black holes[60][61] combined with the energy extraction mechanism by Sir Roger Penrose[62] have been used to explain the observed properties of relativistic jets. Gravitomagnetism (sometimes Gravitoelectromagnetism, abbreviated GEM) refers to a set of formal analogies between Maxwell's field The ergosphere is a region located outside a Rotating black hole. Black hole#Major features of rotating black holes A rotating black hole is a Black hole that possesses Angular momentum. Sir Roger Penrose, PhD, OM, FRS (born 8 August 1931) is an English Mathematical physicist and Emeritus The lower-energy non-relativistic version of this phenomenon is described at Polar jet. 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). A quasar (contraction of QUASi-stellAR radio source) is an extremely powerful and distant Active galactic nucleus. An active galactic nucleus ( AGN) is a compact region at the centre of a Galaxy which has a much higher than normal luminosity over some or all of the Electromagnetic

Mathematical derivation of frame-dragging

Frame-dragging may be illustrated most readily using the Kerr metric,[63][64] which describes the geometry of spacetime in the vicinity of a mass M rotating with angular momentum J

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} + a^{2} + \frac{r_{s} r \alpha^{2}}{\rho^{2}} \sin^{2} \theta \right) \sin^{2} \theta \ d\phi^{2} + \frac{2r_{s} r\alpha}{\rho^{2}} d\phi 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} 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 re-write 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 radius r and the colatitude θ

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

Thus, an inertial reference frame is entrained by the rotating central mass to participate in the latter's rotation; this is frame-dragging. In General relativity, the Kerr metric (or Kerr vacuum) describes the geometry of Spacetime around a rotating massive body SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside Oblate spheroidal coordinates are a three-dimensional orthogonal Coordinate system that results from rotating the two-dimensional elliptic coordinate system In Spherical coordinates, colatitude is the Complementary angle of the Latitude, i

The two surfaces on which the Kerr metric appears to have singularities; the inner surface is the spherical event horizon, whereas the outer surface is an oblate spheroid. 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.
The two surfaces on which the Kerr metric appears to have singularities; the inner surface is the spherical event horizon, whereas the outer surface is an oblate spheroid. In General relativity, the Kerr metric (or Kerr vacuum) describes the geometry of Spacetime around a rotating massive body In General relativity, an event horizon is a boundary in Spacetime, an area surrounding a Black hole or a Wormhole, inside which events cannot An oblate Spheroid is a rotationally symmetric Ellipsoid having a polar axis shorter than the diameter of the equatorial circle whose plane The ergosphere lies between these two surfaces; within this volume, the purely temporal component gtt is negative, i. The ergosphere is a region located outside a Rotating black hole. 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 ergosphere is a region located outside a Rotating black hole. A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e 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} - 4\alpha^{2}}}{2}

where the purely radial component grr of the metric goes to infinity. In General relativity, an event horizon is a boundary in Spacetime, an area surrounding a Black hole or a Wormhole, inside which events cannot In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside 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} - 4\alpha^{2} \cos^{2}\theta}}{2}

where the purely temporal component gtt of the metric changes sign from positive to negative. An oblate Spheroid is a rotationally symmetric Ellipsoid having a polar axis shorter than the diameter of the equatorial circle whose plane The space between these two surfaces is called the ergosphere. The ergosphere is a region located outside a Rotating black hole. A moving particle experiences a positive proper time along its worldline, its path through spacetime. In relativity, proper time is Time measured by a single Clock between events that occur at the same place as the clock In physics the world line of an object is the unique path of that object as it travels through 4- Dimensional Spacetime. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS 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.

See also

References

  1. ^ Thirring, H. In General relativity, the Kerr metric (or Kerr vacuum) describes the geometry of Spacetime around a rotating massive body The geodetic effect represents the effect of the curvature of Spacetime, predicted by General relativity, on a spinning moving body Gravitomagnetism (sometimes Gravitoelectromagnetism, abbreviated GEM) refers to a set of formal analogies between Maxwell's field In Theoretical physics, particularly in discussions of gravitation theories, Mach's principle (or Mach's conjecture is the name given by Einstein to a In astronomy the broad iron K line is a Spectral line that is an accurate measure of a Black hole 's immense gravitational force The lower-energy non-relativistic version of this phenomenon is described at Polar jet. Lense-Thirring precession is a general relativistic correction to the Precession of a Gyroscope near a large rotating mass such as the Earth Über die Wirkung rotierender ferner Massen in der Einsteinschen Gravitationstheorie. Physikalische Zeitschrift 19, 33 (1918). [On the Effect of Rotating Distant Masses in Einstein's Theory of Gravitation]
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