formulasearchengine - User contributions [en]

Nonnegative rank (linear algebra)

2015-01-06T15:56:24Z

14.139.128.12: /* Formal Definition */

The writer is called Wilber Pegues. Invoicing is my occupation. Mississippi is where his home is. It's not a common thing but what I like performing is to climb but I don't have the time recently. Here is my homepage ... [http://bigpolis.com/blogs/post/6503 free psychic reading]

Strain energy

2014-01-09T08:21:39Z

14.139.128.12:

{{more footnotes|date=April 2011}}
The '''Walker circulation''', also known as the '''Walker cell''', is a conceptual model of the air flow in the [[tropics]] in the lower atmosphere ([[troposphere]]). According to this model parcels of air follow a closed circulation in the [[zonal]] and vertical directions. This circulation, which is roughly consistent with observations, is caused by differences in heat distribution between ocean and land. It was discovered by [[Gilbert Walker]]. In addition to motions in the zonal and vertical direction the tropical atmosphere also has considerable motion in the [[meridional]] direction as part of, for example, the [[Hadley Circulation]].

==Walker's methodology==
Walker determined that the time scale of a year (used by many studying the atmosphere) was unsuitable because geospatial relationships could be entirely different depending on the season. Thus, Walker broke his temporal analysis into December–February, March–May, June–August, and September–November.

Walker then selected a number of "centers of action", which included areas such as the Indian Peninsula. The centers were in the hearts of regions with either permanent or seasonal high and low pressures. He also added points for regions where rainfall, wind or temperature was an important control.

He examined the relationships of the summer and winter values of pressure and rainfall, first focusing on summer and winter values, and later extending his work to the spring and autumn.

He concludes that variations in temperature are generally governed by variations in pressure and rainfall. It had previously been suggested that sunspots could be the cause of the temperature variations, but Walker argued against this conclusion by showing monthly correlations of sunspots with temperature, winds, cloud cover, and rain that were inconsistent.

Walker made it a point to publish all of his correlation findings, both of relationships found to be important as well as relationships that were found to be unimportant. He did this for the purpose of dissuading researchers from focusing on correlations that did not exist.

==Mathematical basis==
The statistical model involved in the analysis of atmospheric data that led to the discovery of the Walker circulation is called an autoregressive (AR) process.

===Autocorrelation function===
As background, first consider the autocorrelation function. An autocorrelation function in a measure of the dependence of time series values at one time on the values at another time. Given the time series <math>x(n), n=1, 2, ...N</math>, the autocorrelation function at lag <math>k</math> is defined as:

<math> R_{xx}(k) = \frac{1}{(N-k)} \sum_{i=1}^{N-k} x(i) x(i+k).\,</math>

The value of the autocorrelation function at lag <math>0</math> is the power of <math>x(n)</math>, or its variance if the mean value of <math>x(n)</math> is zero:

<math> R_{xx}(0) = \frac{1}{N} \sum_{i=1}^{N} x(i)^2.\,</math>

Moreover, <math> \sqrt{R_{xx}(\infty)},</math> is the mean value for random processes.

The autocorrelation function may be used to detect deterministic components masked in a random background because autocorrelation functions of deterministic data (like sine wave) persist over all time displacements, while autocorrelation functions of stochastic processes tend to zero for large time displacement (for 0-mean time series).
<ref>[http://www.cbi.dongnocchi.it/glossary/Autocorrelation.html AUTOCORRELATION FUNCTION]</ref>

===Autoregressive model===
{{Main|Autoregressive model}}
Next, consider the autoregressive model proposed by Walker. [[Autoregressive model|Autoregressive Modeling]] is mathematical modeling of a time series based on the assumption that each value of the series depends only on a weighted sum of the previous values of the same series plus "noise".

If <math>x(j)</math> is the <math>j</math>-th value of the time series, the AR model of order <math>p</math> is given by:

<math> x(j) = \sum_{i=1}^p a_i x(j-i) + n(j).\,</math>

where <math>n(j)</math> is the noise. The order, <math>p</math>, can be considered as an index of the lag within the time series of which data will be considered for the analysis. The larger the lag, the larger the system of equations to be solved.

The AR coefficients can be estimated from the autocorrelation sequence by solving the [[#Yule-Walker equations|Yule-Walker equations]].<ref>[http://www.cbi.dongnocchi.it/glossary/AR.html AUTOREGRESSIVE MODELING]</ref>

The generalized matrix version of the AR(''p'') model is given by the equation

<math> X_t = \sum_{i=1}^p \varphi_i X_{t-i}+ \varepsilon_t.\,</math>

Gidon Eshel provides a useful breakdown of the Yule-Walker equations that discusses their relation to between the least squares approach for fitting an AR(p) model.<ref>[http://www-stat.wharton.upenn.edu/~steele/Courses/956/ResourceDetails/YWSourceFiles/YW-Eshel.pdf Yule-Walker equations]</ref>

===Yule-Walker equations===
The AR(''p'') model is given by the equation

<math> X_t = \sum_{i=1}^p \varphi_i X_{t-i}+ \varepsilon_t.\,</math>

It is based on parameters <math>\varphi_i</math> where <math>i = 1, \ldots, p</math>. There is a direct correspondence between these parameters and the covariance function of the process, and this correspondence can be inverted to determine the parameters from the autocorrelation function (which is itself obtained from the covariances). This is done using the '''Yule-Walker equations''':

<math> \gamma_m = \sum_{i=1}^p \varphi_i \gamma_{m-i} + \sigma_\varepsilon^2\delta_m </math>

where <math>m = 0, \ldots, p</math>, yielding <math>p+1</math> equations. <math>\gamma_m</math> is the [[autocovariance function]] of <math>X</math>, <math>\sigma_\varepsilon</math> is the [[standard deviation]] of the input noise process, and <math>\delta_m</math> is the [[Kronecker delta function]].

Because the last part of the equation is non-zero only if <math>m=0</math>, the equation is usually solved by representing it as a matrix for <math>m>0</math>, thus getting equation

<math>\begin{bmatrix}\gamma_1 \\ \gamma_2 \\ \gamma_3 \\ \vdots \\ \end{bmatrix} =
\begin{bmatrix}
\gamma_0 & \gamma_{-1} & \gamma_{-2} & \dots \\
\gamma_1 & \gamma_0 & \gamma_{-1} & \dots \\
\gamma_2 & \gamma_{1} & \gamma_{0} & \dots \\
\vdots & \vdots & \vdots & \ddots \\
\end{bmatrix}
\begin{bmatrix}
\varphi_{1} \\
\varphi_{2} \\
\varphi_{3} \\
\vdots \\
\end{bmatrix}</math>

solving all <math>\varphi</math>. For <math>m=0</math> have

<math> \gamma_0 = \sum_{i=1}^p \varphi_i \gamma_{-i} + \sigma_\varepsilon^2 </math>

which allows us to solve <math>\sigma_\varepsilon^2</math>.

The above equations (the Yule-Walker equations) provide one route to estimating the parameters of an AR(p) model, by replacing the theoretical covariances with estimated values. One way of specifying the estimated covariances is equivalent to a calculation using [[least squares regression]] of values <math>X_t</math> on the <math>p</math> previous values of the same series.

==Oceanic effects==
The Walker Circulations of the tropical Indian, Pacific, and Atlantic basins result in westerly surface winds in Northern Summer in the first basin and easterly winds in the second and third basins. As a result the temperature structure of the three oceans display dramatic asymmetries. The equatorial Pacific and Atlantic both have cool surface temperatures in Northern Summer in the east, while cooler surface temperatures prevail only in the western Indian Ocean. And these changes in surface temperature reflect changes in the depth of the thermocline.

Changes in the Walker Circulation with time occur in conjunction with changes in surface temperature. Some of these changes are forced externally, such as the seasonal shift of the Sun into the Northern Hemisphere in summer. Other changes appear to be the result of coupled ocean-atmosphere feedback in which, for example, easterly winds cause the sea surface temperature to fall in the east, enhancing the zonal heat contrast and hence intensifying easterly winds across the basin. These anomalous easterlies induce more equatorial upwelling and raise the thermocline in the east, amplifying the initial cooling by the southerlies. This coupled ocean-atmosphere feedback was originally proposed by Bjerknes. From an oceanographic point of view, the equatorial cold tongue is caused by easterly winds. Were the earth climate symmetric about the equator, cross-equatorial wind would vanish, and the cold tongue would be much weaker and have a very different zonal structure than is observed today.<ref>[http://cat.inist.fr/?aModele=afficheN&cpsidt=2154325 Ocean-atmosphere interaction in the making of the Walker circulation and equatorial cold tongue]</ref>
The Walker cell is indirectly related to [[upwelling]] off the coasts of [[Peru]] and [[Ecuador]]. This brings [[nutrient]]-rich cold water to the surface, increasing fishing stocks.

==El Niño==
{{Main|El Niño–Southern Oscillation}}

[[Image:LaNina.png|thumb|A schematic diagram of the quasi-equilibrium and [[El Niño|La Niña]] phase of the southern oscillation. The '''Walker circulation''' is seen at the surface as easterly trade winds which move water and air warmed by the sun towards the west. The western side of the equatorial Pacific is characterized by warm, wet low pressure weather as the collected moisture is dumped in the form of typhoons and thunderstorms. The ocean is some 60 cm higher in the western Pacific as the result of this motion. The water and air are returned to the east. Both are now much cooler, and the air is much drier. An El Niño episode is characterised by a breakdown of this water and air cycle, resulting in relatively warm water and moist air in the eastern Pacific.]]

The Walker circulation is caused by the [[pressure gradient force]] that results from a [[High pressure area|high pressure system]] over the eastern Pacific ocean, and a [[low pressure system]] over [[Indonesia]]. When the Walker circulation weakens or reverses, an [[El Niño]] results, causing the ocean surface to be warmer than average, as upwelling of cold water occurs less or not at all. An especially strong Walker circulation causes a [[La Niña]], resulting in cooler ocean temperatures due to increased upwelling.

A scientific study published in May 2006 in the journal [[Nature (journal)|Nature]] indicates that the Walker circulation has been slowing since the mid-19th Century. The authors argue that [[global warming]] is a likely causative factor in the weakening of the wind pattern.<ref>[http://www.gfdl.noaa.gov/research/climate/highlights/index.html#walker A tropical atmospheric circulation slow-down]</ref> However, a new study from The Twentieth Century Reanalysis Project shows that the Walker circulation has not been slowing (or increasing) from 1871-2008.<ref>The Twentieth Century Reanalysis Project. Quarterly Journal of the Royal Meteorological Society, 137: 1–28. {{doi|10.1002/qj.776}}, http://onlinelibrary.wiley.com/doi/10.1002/qj.776/abstract</ref>

==See also==
*[[Atmospheric circulation]]
*[[Earth's atmosphere]]

==References==
* Walker Institute, University of Reading, UK. http://www.walker-institute.ac.uk/about/sir_gilbert.htm
* Walker, JM. Pen Portrait of Sir Gilbert Walker, CSI, MA, ScD, FRS. Weather 1997 (Volume 52, No.7, pages 217-220)
* Walker, G.T. and Bliss, E.W., 1930. World Weather IV, Memoirs of the Royal Meteorological Society, 3, (24), 81-95.
* Walker, G.T. and Bliss, E.W., 1937. World Weather VI, Memoirs of the Royal Meteorological Society, 4, (39), 119-139.
* Walker, G.T., 1923. Correlation in seasonal variations of weather, VIII. A preliminary study of world weather. Memoirs of the India Meteorological Department, 24, (4), 75-131.
* Walker, G.T., 1924. Correlation in seasonal variations of weather, IX. A further study of world weather. Memoirs of the India Meteorological Department, 24, (9),275-333. http://www.rmets.org/about/history/classics.php
* Katz, R.W. Sir Gilbert Walker and a Connection between El Nino and Statistics. Statistical Science, 17 (2002), 97-117. http://amath.colorado.edu/courses/4540/2004Spr/walkerss.pdf
* Climate research summary - [http://www.gfdl.noaa.gov/research/climate/highlights/index.html#walker Walker Circulation: a tropical atmospheric circulation slow-down] Text and graphics from [[NOAA]] / [[Geophysical Fluid Dynamics Laboratory]]
* [http://www.eurekalert.org/pub_releases/2006-05/ncfa-sit050106.php Slowdown in tropical Pacific flow pinned on climate change] - press release from [[University Corporation for Atmospheric Research]].
* [http://www.nature.com/nature/journal/v441/n7089/abs/nature04744.html Weakening of tropical Pacific atmospheric circulation due to anthropogenic forcing ] 4 May 2006 in [[Nature (journal)|Nature]].
* [http://www.breitbart.com/news/2006/05/03/D8HCH8VO2.html Associated Press news story, 3 May 2006: "Global Warming Cited in Wind Shift"]
* [http://www.atmos-chem-phys.net/12/9791/2012/acp-12-9791-2012.html Tropical convective transport and the Walker circulation], 29 October 2012 in [[Atmospheric Chemistry and Physics]]
'''In-line citations'''
{{Reflist}}

{{Use dmy dates|date=September 2010}}

{{DEFAULTSORT:Walker Circulation}}
[[Category:Tropical meteorology]]

[[fr:Circulation atmosphérique#Circulation de Walker]]

Least trimmed squares

2013-12-09T06:35:43Z

14.139.128.12: /* Description of method */

{{General relativity|cTopic=Phenomena}}
[[Einstein]]'s [[general theory of relativity]] predicts that non-static, stationary [[mass–energy]] distributions affect [[spacetime]] in a peculiar way giving rise to a phenomenon usually known as '''frame-dragging'''. 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]].<ref>{{Cite journal|last=Thirring |first=H. |authorlink= |coauthors= |year=1918 |month= |title=Über die Wirkung rotierender ferner Massen in der Einsteinschen Gravitationstheorie |journal=Physikalische Zeitschrift |volume=19 |issue= |page=33 |bibcode=1918PhyZ...19...33T |url= |accessdate= |quote= }} [On the Effect of Rotating Distant Masses in Einstein's Theory of Gravitation]</ref><ref>{{Cite journal|last=Thirring |first=H. |authorlink= |coauthors= |year=1921 |month= |title=Berichtigung zu meiner Arbeit: ‘Über die Wirkung rotierender Massen in der Einsteinschen Gravitationstheorie’ |journal=Physikalische Zeitschrift |volume=22 |issue= |page=29 |bibcode=1921PhyZ...22...29T |url= |accessdate= |quote= }} [Correction to my paper "On the Effect of Rotating Distant Masses in Einstein's Theory of Gravitation"]</ref><ref>{{Cite journal|last=Lense |first=J. |authorlink= |coauthors=Thirring, H. |year=1918 |month= |title=Über den Einfluss der Eigenrotation der Zentralkörper auf die Bewegung der Planeten und Monde nach der Einsteinschen Gravitationstheorie |journal=Physikalische Zeitschrift |volume=19 |issue= |pages=156–163 |id= |url= |accessdate= |quote= |bibcode = 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]</ref> They predicted that the rotation of a massive object would distort the [[Metric tensor (general relativity)|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. More generally, the subject of effects caused by mass–energy currents is known as [[gravitomagnetism]], in analogy with [[classical electromagnetism]].

==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 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 electromagnetic induction.

Also, an inner region is dragged more than an outer region. This produces interesting locally rotating frames. For example, imagine that an ice skater, in orbit over the equator of a 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|Principle of Equivalence]] 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 [[Mach'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 or 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).<ref>Einstein, A ''The Meaning of Relativity'' (contains transcripts of his 1921 Princeton lectures).</ref>

'''Static mass increase''' is a third effect noted by Einstein in the same paper.<ref>{{Cite book|title=The Meaning of Relativity |last=Einstein |first=A. |authorlink= |coauthors= |year=1987 |publisher=Chapman and Hall |location=London |isbn= |page= |pages=95–96 |url= }}</ref> 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.

==Experimental tests of frame-dragging==

===Proposals===
In 1976 Van Patten and Everitt<ref>{{Cite journal|last=Van Patten |first=R. A. |authorlink= |coauthors=Everitt, C. W. F. |year=1976 |month= |title=Possible Experiment with Two Counter-Orbiting Drag-Free Satellites to Obtain a New Test of Einsteins's General Theory of Relativity and Improved Measurements in Geodesy |journal=Phys. Rev. Lett. |volume=36 |issue=12 |pages=629–632 |doi=10.1103/PhysRevLett.36.629 |url= |accessdate= |quote= |bibcode=1976PhRvL..36..629V}}</ref><ref>{{Cite journal|last=Van Patten |first=R. A. |authorlink= |coauthors=Everitt, C. W. F. |year=1976 |month= |title=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 |journal=Celest. Mech. Dyn. Astron. |volume=13 |issue=4 |pages=429–447 |doi=10.1007/BF01229096 |url= |accessdate= |quote= |bibcode = 1976CeMec..13..429V }}</ref> 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<ref>{{Cite journal|last=Ciufolini |first=I. |authorlink= |coauthors= |year=1986 |month= |title=Measurement of Lense–Thirring Drag on High-Altitude Laser-Ranged Artificial Satellites |journal=Phys. Rev. Lett. |volume=56 |issue=4 |pages=278–281 |doi=10.1103/PhysRevLett.56.278 |url= |accessdate= |quote= |pmid=10033146 |bibcode=1986PhRvL..56..278C}}</ref> 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 (satellite)|LARES]], WEBER-SAT. Although extensively studied by various groups,<ref>Ries, J.C., Eanes, R.J., Watkins, M.M., Tapley, B., Joint NASA/ASI Study on Measuring the Lense–Thirring Precession Using a Second LAGEOS Satellite, ''CSR-89-3'', Center for Space Research, The University of Texas at Austin, 1989.</ref><ref>{{Cite journal|last=Iorio |first=L. |authorlink= |coauthors=Lucchesi, D. M.; Ciufolini, I. |year=2002 |month= |title=The LARES Mission Revisited: An Alternative Scenario |journal=Class. Quantum Grav. |volume=19 |issue= 16|pages=4311–4325 |doi=10.1088/0264-9381/19/16/307 |url= |accessdate= |quote= |arxiv = gr-qc/0203099 |bibcode = 2002CQGra..19.4311I }}</ref> 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,<ref>{{Cite journal|last=Iorio |first=L. |authorlink= |coauthors= |year=2003 |month= |title=A new proposal for measuring the Lense–Thirring effect with a pair of supplementary satellites in the gravitational field of the Earth |journal=Phys. Lett. A |volume=308 |issue=2–3 |pages=81–84 |doi=10.1016/S0375-9601(02)01800-5 |url= |accessdate= |quote= |arxiv = gr-qc/0206073 |bibcode = 2003PhLA..308...81I }}</ref><ref>{{Cite journal|last=Iorio |first=L. |authorlink= |coauthors= |year=2003 |month= |title=On a new observable for measuring the Lense–Thirring effect with Satellite Laser Ranging |journal=Gen. Relativ. Gravit. |volume=35 |issue= 9|pages=1583–1595 |doi=10.1023/A:1025727001141 |url= |accessdate= |quote= |arxiv = gr-qc/0206074 |bibcode = 2003GReGr..35.1583I }}</ref><ref>{{Cite journal|last=Iorio |first=L. |authorlink= |coauthors=Lucchesi, D. M. |year=2003 |month= |title=LAGEOS-type Satellites in Critical Supplementary Orbital Configuration and the Lense—Thirring Effect Detection |journal=Class. Quantum Grav. |volume=20 |issue= 13|pages=2477–2490 |doi=10.1088/0264-9381/20/13/302 |url= |accessdate= |quote= |arxiv = gr-qc/0209027 |bibcode = 2003CQGra..20.2477I }}</ref> 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,<ref>Lucchesi, D.M., Paolozzi, A., A cost effective approach for LARES satellite, ''paper presented at XVI Congresso Nazionale AIDAA (24–28 September 2001, Palermo)'', 2001.</ref><ref>Ciufolini, I., On the orbit of the LARES satellite, (Preprint http://www.arxiv.org/abs/gr-qc/0609081), 2006.</ref> but such a strategy has been shown to be unfeasible.<ref>Peterson, G.E., Estimation of the Lense–Thirring precession using laser-ranged satellites, ''CSR-97-1'', Center for Space Research, The University of Texas at Austin, 1997.</ref><ref>Iorio, L., A critical approach to the concept of a polar, low-altitude LARES satellite, ''Class. Quantum Grav.'', '''19''', L175–L183, 2002.</ref><ref>Iorio, L., A comment on the paper "On the orbit of the LARES satellite", by I. Ciufolini, ''Planet. Space Sci.'',''' 55''', 1198–1200, 2007.</ref> 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<ref>Ciufolini, I., LARES/WEBER-SAT, frame-dragging and fundamental physics, (Preprint http://arxiv.org/abs/gr-qc/0412001), 2004.</ref> induced by the [[DGP model|multidimensional braneworld model]] by Dvali, Gabadaze and Porrati<ref>Dvali, G., Gabadadze, G., Porrati, M., 4D Gravity on a Brane in 5D Minkowski Space, ''Phys. lett. B'', '''485''', 208–214, 2000.</ref> and to improve by two orders of magnitude the present-day level of accuracy of the equivalence principle.<ref>Ciufolini, I., Frame Dragging and Lense–Thirring Effect, ''Gen. Relativ. Gravit.'', '''36''', 2257–2270, 2004.</ref> Iorio claimed these improvements were unrealistic.<ref>Iorio, L., On the possibility of testing the Brane-World scenario with orbital motions in the Solar System, ''J. Cosmol. Astrpart. Phys.'', '''7''', 8, 2005.</ref><ref>Iorio, L., LARES/WEBER-SAT and the equivalence principle, ''Europhys. Lett.'', '''80''', 40007, 2007. See also [http://arxiv.org/abs/0706.1930 this] preprint</ref>

===Analysis of experimental data===
[[Image:LAGEOS-NASA.jpg|right|thumb|150px|The LAGEOS-1 satellite. ([[Diameter|D]]=60 cm)]]
Limiting the scope to the scenarios involving existing orbiting bodies, the first proposal to use the [[LAGEOS]] satellite and the Satellite Laser Ranging ([[Satellite laser ranging|SLR]]) technique to measure the Lense–Thirring effect dates back to 1977–1978.<ref>Cugusi, L., Proverbio E. Relativistic effects on the Motion of the Earth's. Satellites, paper presented at the International Symposium on Satellite Geodesy in Budapest from June 28 to July 1, 1977, ''J. of Geodesy'', '''51''', 249–252, 1977.</ref><ref>Cugusi, L., Proverbio, E., Relativistic Effects on the Motion of Earth's Artificial Satellites, ''Astron. Astrophys'', '''69''', 321–325, 1978.</ref> Tests have started to be effectively performed by using the LAGEOS and LAGEOS II satellites in 1996,<ref>Ciufolini, I., Lucchesi, D.M., Vespe, F., Mandiello, A., Measurement of Dragging of Inertial Frames and Gravitomagnetic Field Using Laser-Ranged Satellites, ''Il Nuovo Cimento A'', '''109''', 575–590, 1996.</ref> according to a strategy<ref>Ciufolini, I., On a new method to measure the gravitomagnetic field using two orbiting satellites., ''Il Nuovo Cimento A'', '''109''', 1709–1720, 1996.</ref> 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<ref>Ciufolini, I., and Pavlis, E.C., A confirmation of the general relativistic prediction of the Lense–Thirring effect, ''Nature'', '''431''', 958–960, 2004</ref><ref>Ciufolini, I., Pavlis, E.C., and Peron, R., Determination of frame-dragging using Earth gravity models from CHAMP and GRACE, ''New Astron.'', '''11''', 527–550, 2006.</ref> by discarding the perigee of LAGEOS II and using a linear combination<ref>Pavlis, E.C., Geodetic contributions to gravitational experiments in space. In: Cianci, R., Collina, R., Francaviglia, M., Fré, P. (Eds.), ''Recent Developments in General Relativity. 14th SIGRAV Conference on General Relativity and Gravitational Physics, Genova, Italy, September 18–22, 2000''. Springer, Milano, pp. 217–233, 2002.</ref><ref>Iorio, L., Morea, A., The impact of the new Earth gravity models on the measurement of the Lense–Thirring effect, ''Gen. Relativ. Gravit.'', '''36''', 1321–1333, 2004. (Preprint http://www.arxiv.org/abs/gr-qc/0304011).</ref><ref>Iorio, L., The new Earth gravity models and the measurement of the Lense–Thirring effect. In: Novello, M., Bergliaffa, S.P., Ruffini, R. (Eds.), ''On Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories'', World Scientific, Singapore, pp. 1011–1020, 2003. (Preprint http://www.arxiv.org/abs/gr-qc/0308022).</ref><ref>Iorio, L., The impact of the new CHAMP and GRACE Earth gravity models on the measurement of the general relativistic Lense–Thirring effect with the LAGEOS and LAGEOS II satellites. In: Reigber, Ch., Luehr, H., Schwintzer, P., Wickert, J. (Eds.), '' '', Earth Observation with CHAMP. Results from Three Years in Orbit, Springer-Verlag, Berlin, pp. 187–192, 2003. (Preprint http://arxiv.org/abs/gr-qc/0309092)</ref><ref>Ries, J.C., Eanes, R.J., Tapley, B.D., Lense–Thirring Precession Determination from Laser Ranging to Artificial Satellites. In: Ruffini, R., Sigismondi, C. (Eds.), ''Nonlinear Gravitodynamics. The Lense–Thirring Effect'', World Scientific, Singapore, pp. 201–211, 2003a.</ref><ref>Ries, J.C., Eanes, R.J., Tapley, B.D., Peterson, G.E., Prospects for an Improved Lense–Thirring Test with SLR and the GRACE Gravity Mission. In: Noomen, R., Klosko, S., Noll, C., Pearlman, M. (Eds.), ''Proceedings of the 13th International Laser Ranging Workshop, NASA CP 2003–212248'', NASA Goddard, Greenbelt, 2003b. (Preprint http://cddisa.gsfc.nasa.gov/lw13/lw$\_${proceedings}.html$\#$science).</ref> involving only the nodes of both the spacecraft.
Although the predictions of general relativity are compatible with the experimental results, realistic evaluation of the total error raised a debate.<ref>Iorio, L., On the reliability of the so far performed tests for measuring the Lense–Thirring effect with the LAGEOS satellites, ''New Astron.'', '''10''', 603–615, 2005.</ref><ref>Ciufolini, I., and Pavlis, E.C., On the Measurement of the Lense–Thirring effect Using the Nodes of the LAGEOS Satellites in reply to "On the reliability of the so-far performed tests for measuring the Lense–Thirring effect with the LAGEOS satellites" by L. Iorio, ''New Astron.'', '''10''', 636–651, 2005.</ref><ref>Lucchesi, D.M., The impact of the even zonal harmonics secular variations on the Lense–Thirring effect measurement with the two Lageos satellites, ''Int. J. of Mod. Phys. D'', '''14''', 1989–2023, 2005.</ref><ref>Iorio, L., A critical analysis of a recent test of the Lense–Thirring effect with the LAGEOS satellites, ''J. of Geodesy'', '''80''', 128–136, 2006.</ref><ref>Iorio, L., An assessment of the measurement of the Lense–Thirring effect in the Earth gravity field, in reply to: ``On the measurement of the Lense–Thirring effect using the nodes of the LAGEOS satellites, in reply to ``On the reliability of the so far performed tests for measuring the Lense–Thirring effect with the LAGEOS satellites" by L. Iorio," by I. Ciufolini and E. Pavlis, ''Planet. Space Sci.'', '''55''', 503–511, 2007.</ref><ref>{{Cite journal|last=Iorio |first=L. |authorlink= |coauthors= |date=February 2010|title= Conservative evaluation of the uncertainty in the LAGEOS–LAGEOS II Lense–Thirring test |journal=Centr. Eur. J. Phys. |volume= 8|issue= 1|pages= 25–32|doi = 10.2478/s11534-009-0060-6 |url= |accessdate= |quote=|bibcode = 2010CEJPh...8...25I
|arxiv = 0710.1022 }}</ref><ref>{{Cite journal|last=Iorio |first=L. |authorlink= |coauthors=Ruggiero, M.L.; Corda, C. |date=June 2013|title= Novel considerations about the error budget of the LAGEOS-based tests of frame-dragging with GRACE geopotential models |journal=Acta Astronaut. |volume= 91|issue= 10-11|pages= 141–148|doi = 10.1016/j.actaastro.2013.06.002 |url= |accessdate= |quote=|arxiv = 1307.0753 |bibcode = 2013AcAau..91..141I }}</ref>

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.<ref>Iorio, L., A note on the evidence of the gravitomagnetic field of Mars, ''Class. Quantum Grav.'', '''23''', 5451–5454, 2006.</ref> There is also debate about this test.<ref>Iorio, L., Testing frame-dragging with the Mars Global Surveyor spacecraft in the gravitational field of Mars. In: Iorio, L. (Ed.), ''The Measurement of Gravitomagnetism: A Challenging Enterprise'', Nova publishers, Hauppauge (NY), pp. 177–187, 2007. (Preprint http://www.arxiv.org/abs/gr-qc/0701042), 2007.</ref><ref>{{Cite journal|last=Krogh |first=K. |authorlink= |coauthors= |date=November 2007 |title=Comment on 'Evidence of the gravitomagnetic field of Mars' |journal=Class. Quantum Grav. |volume=24 |issue= 22|pages=5709–5715 |id= |url= |accessdate= |quote= |doi=10.1088/0264-9381/24/22/N01 |bibcode = 2007CQGra..24.5709K }} (Preprint http://arxiv.org/abs/astro-ph/0701653)</ref><ref>{{Cite journal|last=Iorio |first=L. |authorlink= |coauthors= |date=June 2010|title= On the Lense–Thirring test with the Mars Global Surveyor in the gravitational field of Mars |journal=Centr. Eur. J. Phys. |volume= 8|issue= 3|pages= 509–513 |doi = 10.2478/s11534-009-0117-6 |url= |accessdate= |quote= |arxiv = gr-qc/0701146 |bibcode = 2010CEJPh...8..509I }}</ref> 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:<ref>{{Cite journal|last=Iorio |first=L. |authorlink= |coauthors= |year=2007 |month= |title=First preliminary tests of the general relativistic gravitomagnetic field of the Sun and new constraints on a Yukawa-like fifth force from planetary data |journal=Planet. Space Sci. |volume=55 |issue=10 |pages=1290–1298 |doi=10.1016/j.pss.2007.04.001 |url= |accessdate= |quote= |bibcode=2007P&SS...55.1290I|arxiv = gr-qc/0507041 }}</ref> the predictions of general relativity are compatible with the estimated corrections to the perihelia precessions,<ref>{{Cite journal|last=Pitjeva |first=E. V. |authorlink=Elena V. Pitjeva |coauthors= |year=2005 |month= |title=Relativistic Effects and Solar Oblateness from Radar Observations of Planets and Spacecraft |journal=Astron. Lett. |volume=31 |issue=5 |pages=340–349 |doi=10.1134/1.1922533 |url= |accessdate= |quote= |bibcode = 2005AstL...31..340P }}</ref> although the errors are still large. However, the inclusion of the radiometric data from the Magellan orbiter recently allowed [[Elena V. Pitjeva|Pitjeva]] to greatly improve the determination of the unmodelled precession of the perihelion of Venus. It amounts to −0.0004±0.0001 [[arcsecond]]s/century, while the Lense–Thirring effect for the Venus' perihelion is just −0.0003 arcseconds/century.<ref>{{Cite journal|last =Iorio |first= L. |year=2008|title= Advances in the measurement of the Lense–Thirring effect with planetary motions in the field of the Sun |journal=Schol. Res. Exchange|volume=2008|id=105235}}</ref> The system of the Galilean satellites of Jupiter was investigated as well,<ref>{{Cite journal|last=Iorio |first=L. |authorlink= |coauthors=Lainey, V. |year=2005 |month= |title=The Lense–Thirring effect in the Jovian system of the Galilean satellites and its measurability |journal=Int. J. Mod. Phys. D |volume=14 |issue=12 |pages=2039–2050 |doi=10.1142/S0218271805008133 |url= |accessdate= |quote= |arxiv = gr-qc/0508112 |bibcode = 2005IJMPD..14.2039I }}</ref> following the original suggestion by Lense and Thirring.

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,<ref>{{Cite journal|last=Murphy |first=T. W. |authorlink= |coauthors=Nordtvedt, K.; Turyshev, S. G. |year=2007 |month= |title=The Gravitomagnetic Influence on Gyroscopes and on the Lunar Orbit |journal=Phys. Rev. Lett. |volume=98 |issue=7 |page=071102 |doi=10.1103/PhysRevLett.98.071102 |id= |url= |accessdate= |quote= |pmid=17359012 |arxiv=gr-qc/0702028 |bibcode=2007PhRvL..98g1102M}}</ref> but Kopeikin questioned the ability of LLR to be sensitive to gravitomagnetism.<ref>{{Cite journal|last=Kopeikin |first= |authorlink= |coauthors= |year=2007 |month= |title=Comment on ‘The gravitomagnetic influence on gyroscopes and on the lunar orbit’ |journal=Phys. Rev. Lett. |volume=98 |issue= 22|page=229001 |doi=10.1103/PhysRevLett.98.229001 |url= |accessdate= |quote= |pmid=17677884 |first1=SM |bibcode=2007PhRvL..98v9001K|arxiv = gr-qc/0702120 }}</ref>

The [[Gravity Probe B]] experiment<ref>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.</ref><ref>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.</ref> was a satellite-based mission by a Stanford group and NASA, used to experimentally measure another gravitomagnetic effect, the [[Schiff precession]] of a gyroscope,<ref>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.</ref><ref>[[Leonard I. Schiff|Schiff, L.]], On Experimental Tests of the General Theory of Relativity, ''Am. J. of Phys.'', '''28''', 340–343, 1960.</ref> 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<ref>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.</ref> 256–128%, with the hope of reaching about 13% in December 2007.<ref>StanfordNews 4/14/07, downloadable at http://einstein.stanford.edu/</ref>
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).<ref>[http://nasascience.nasa.gov/astrophysics/about-us/science-strategy/senior-reviews/AstroSR08_Report.pdf] ''Report of the 2008 Senior Review of the
Astrophysics Division Operating Missions'', NASA</ref><ref>[http://www.newscientist.com/article/dn13938-gravity-probe-b-scores-f-in-nasa-review.html ''Gravity Probe B scores 'F' in NASA review''], Jeff Hecht, New Scientist – Space, May 20, 2008</ref>
On May 4, 2011, the Stanford-based analysis group and NASA announced the final report,<ref>http://einstein.stanford.edu/highlights/status1.html</ref> 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.<ref>/http://www.sciencenews.org/view/generic/id/73870/title/Gravity_Probe_B_finally_pays_off_</ref> The findings were accepted for publication in the journal [[Physical Review Letters]].<ref name=PRL>{{cite news
| url=http://prl.aps.org/accepted/L/ea070Y8dQ491d22a28828c95f660a57ac82e7d8c0
| journal = Physical Review Letters
| title = Gravity Probe B: Final results of a space experiment to test general relativity
| work=
| author=
| date=2011-05-01
| accessdate=2011-05-06 }}
</ref>

===Possible future tests===
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 with active mechanisms of compensation of the non-gravitational forces, in eccentric orbits, as stated in 2005 by [[Lorenzo Iorio]].<ref>Iorio, L., The impact of the new Earth gravity models on the measurement of the Lense–Thirring effect with a new satellite, ''New Astron.'', '''10 ''', 616–635, 2005.</ref> On 13 February 2012 the [[Italian Space Agency]] (ASI) launched the [[LARES (satellite)|LARES]] satellite with a [[Vega (rocket)|Vega]] rocket.<ref>
{{cite web
|url=http://www.esa.int/esaCP/SEMK3RGXTWG_index_0.html
|title=Overview of ESA activities in 2012 of interest to media.
}}</ref> The goal of LARES is to measure the Lense–Thirring effect to 1%, but L. Iorio and other researchers raised doubts that this can be achieved,<ref>{{Cite journal|last=Iorio|first= L.|year=2009|title= Towards a 1% measurement of the Lense–Thirring effect with LARES?|journal=Adv. Space Res.|volume=43|issue=7|pages=1148–1157|doi=10.1016/j.asr.2008.10.016|bibcode=2009AdSpR..43.1148I|arxiv = 0802.2031 }}</ref><ref>{{Cite journal|last=Iorio|first= L.|year=2009|title= Will the recently approved LARES mission be able to measure the Lense–Thirring effect at 1%? |journal= Gen. Relativ. Gravit.|volume=41|issue=8|pages=1717–1724|doi=10.1007/s10714-008-0742-1|bibcode = 2009GReGr..41.1717I |arxiv = 0803.3278 }}</ref><ref>{{Cite journal|last=Iorio|first= L.|date=December 2009|title= An Assessment of the Systematic Uncertainty in Present and Future Tests of the Lense–Thirring Effect with Satellite Laser Ranging |journal= Space Sci. Rev.|volume=148|issue=1-4|pages=363–381 |doi=10.1007/s11214-008-9478-1|bibcode = 2009SSRv..148..363I |arxiv = 0809.1373 }}</ref><ref>Iorio, L., Advances in the measurement of the Lense–Thirring effect with Satellite Laser Ranging in the gravitational field of the Earth, in V.V. Koslovskiy (ed.), ''Progress in Laser and Electro-Optics Research'', NOVA publishers, Hauppauge, N.Y., 2008. ISBN 978-1-60456-558-4</ref><ref>{{citation|journal = Acta Phys. Polonica B |last=Iorio|first= L.|year=2010|title= On the impact of the atmospheric drag on the LARES mission |month = April |volume=41|issue=4|pages=753–765 }} [http://th-www.if.uj.edu.pl/acta/vol41/pdf/v41p0753.pdf PDF of the paper]</ref><ref>{{cite journal
|last1=Renzetti |first1=G.
|year=2012
|title=Are higher degree even zonals really harmful for the LARES/LAGEOS frame-dragging experiment?
|journal=[[Canadian Journal of Physics]]
|volume=90 |issue=9 |pages=883–888
|bibcode=2012CaJPh..90..883R
|doi=10.1139/p2012-081
}}</ref><ref>{{cite journal
|last1=Renzetti |first1=G.
|year=2013
|title=First results from LARES: An analysis
|journal=[[New Astronomy]]
|volume=23-24
|bibcode=2013NewA...23...63R
|pages=63–66
|doi=10.1016/j.newast.2013.03.001
}}</ref> 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{{Citation needed|date=July 2013}}; I. Ciufolini and coworkers <ref>{{cite web|title=The LARES Scientific Team - Official LARES Web Site|url=http://www.lares-mission.com/sciteam.html}}</ref> offered replies.<ref>{{cite journal|last=Ciufolini|first=I.|coauthors=A. Paolozzi, E. C. Pavlis, J. C. Ries, R. Koenig, R. A. Matzner, G. Sindoni, H. Neumayer|title=Towards a One Percent Measurement of Frame Dragging by Spin with Satellite Laser Ranging to LAGEOS, LAGEOS 2 and LARES and GRACE Gravity Models|journal=Space Science Reviews|year=2009|volume=148|issue=1-4|pages=71–104|doi=10.1007/s11214-009-9585-7|bibcode = 2009SSRv..148...71C }}</ref>

In the case of stars orbiting close to a spinning, supermassive black hole, frame dragging should cause the star's orbital plane to [[Lense–Thirring precession|precess]] about the black hole spin axis. This effect should be detectable within the next few years via [[astrometry|astrometric]] monitoring of stars at the center of the [[Milky Way]] galaxy.<ref>{{Cite journal
| last = Merritt
| first = D.
| last2 = Alexander
| first2 = T.
| last3 = Mikkola
| first3 = S.
| last4 = Will
| first4 = C.
| author-link = David Merritt
| title = Testing Properties of the Galactic Center Black Hole Using Stellar Orbits
| journal = Physical Review D
| volume = 81
| issue = 6
| page = 062002
| date =
| year = 2010
| bibcode = 2010PhRvD..81f2002M
| doi = 10.1103/PhysRevD.81.062002
| postscript =  |arxiv = 0911.4718 }}
</ref>
By comparing the rate of orbital precession of two stars on different orbits, it is possible in principle to test the [[no-hair theorem]]s of general relativity, in addition to measuring the spin of the black hole.<ref>{{Cite journal
| last = Will
| first = C.
| author-link = Clifford Will
| title = Testing the General Relativistic "No-Hair" Theorems Using the Galactic Center Black Hole Sagittarius A*
| journal = Astrophysical Journal Letters
| volume = 674
| issue = 1
| pages = L25–L28
| date =
| year = 2008
| doi = 10.1086/528847
| postscript = 
| bibcode=2008ApJ...674L..25W|arxiv = 0711.1677 }}
</ref>

==Astronomical evidence==
[[File:Galaxies AGN Inner-Structure-of.jpg|right|thumb|Relativistic Jet. The environment around the [[Active galactic nucleus|AGN]] where the [[special relativity|relativistic]] [[Plasma (physics)|plasma]] is collimated into jets which escape along the pole of the [[supermassive black hole]]]]

[[Relativistic jet]]s 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 hole]]s<ref>{{Cite journal|last=Williams |first=R. K. |authorlink= |coauthors= |year=1995 |month= |title=Extracting X rays, Ύ rays, and relativistic e−–e+ pairs from supermassive Kerr black holes using the Penrose mechanism |journal=Physical Review D |volume=51 |issue=10 |pages=5387–5427 |doi=10.1103/PhysRevD.51.5387 |url= |accessdate= |quote= |bibcode = 1995PhRvD..51.5387W }}</ref><ref>{{Cite journal|last=Williams |first=R. K. |authorlink= |coauthors= |year=2004 |month= |title=Collimated escaping vortical polar e−–e+ jets intrinsically produced by rotating black holes and Penrose processes |journal=The Astrophysical Journal |volume=611 |issue= 2|pages=952–963 |doi=10.1086/422304 |url= |accessdate= |quote= |bibcode=2004ApJ...611..952W|arxiv = astro-ph/0404135 }}</ref> combined with the energy extraction mechanism by [[Roger Penrose|Penrose]]<ref>{{Cite journal|last=Penrose |first=R. |authorlink= |coauthors= |year=1969 |month= |title=Gravitational collapse: The role of general relativity |journal=Nuovo Cimento Rivista |volume=1 |issue=Numero Speciale |pages=252–276 |bibcode=1969NCimR...1..252P |url= |accessdate= |quote= }}</ref> have been used to explain the observed properties of [[relativistic jet]]s. 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-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).

==Mathematical derivation of frame-dragging==
Frame-dragging may be illustrated most readily using the [[Kerr metric]],<ref name="kerr_1963">{{Cite journal| last = Kerr | first = R. P. | authorlink = Roy Kerr | year = 1963 | title = Gravitational field of a spinning mass as an example of algebraically special metrics | journal = Physical Review Letters | volume = 11| issue = 5 | pages = 237–238 | doi = 10.1103/PhysRevLett.11.237 | bibcode=1963PhRvL..11..237K}}</ref><ref>{{Cite book| last = Landau | first = LD | authorlink = Lev Landau | coauthors = Lifshitz, EM | year = 1975 | title = The Classical Theory of Fields (Course of Theoretical Physics, Vol. 2) | edition = revised 4th English | publisher = Pergamon Press | location = New York | isbn = 978-0-08-018176-9 |pages = 321–330}}</ref> which describes the geometry of [[spacetime]] in the vicinity of a mass ''M'' rotating with [[angular momentum]] ''J''

:<math>
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}
</math>
::::<math>
- \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
</math>

where ''r''''s'' is the [[Schwarzschild metric|Schwarzschild radius]]

:<math>
r_{s} = \frac{2GM}{c^{2}}
</math>

and where the following shorthand variables have been introduced for brevity

:<math>
\alpha = \frac{J}{Mc}
</math>

:<math>
\rho^{2} = r^{2} + \alpha^{2} \cos^{2} \theta\,\!
</math>

:<math>
\Lambda^{2} = r^{2} - r_{s} r + \alpha^{2}\,\!
</math>

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]]

:<math>
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}
</math>

We may rewrite the Kerr metric in the following form

:<math>
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}
</math>

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

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

In the plane of the equator this simplifies to:<ref>{{Cite journal|last=Tartaglia |first=A. |authorlink= |coauthors= |year=2008 |month= |title=Detection of the gravitometric clock effect |journal= |volume= |issue= |pages= |id= |url= |accessdate= |quote= |arxiv=gr-qc/9909006v2|bibcode = 2000CQGra..17..783T |doi = 10.1088/0264-9381/17/4/304 }}</ref>

:<math>
\Omega = \frac{r_{s} \alpha c}{r^{3} + \alpha^{2} r + r_{s} \alpha^{2}}
</math>

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

[[File:Ergosphere.svg|thumb|300px|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.]]

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

:<math>
r_{inner} = \frac{r_{s} + \sqrt{r_{s}^{2} - 4\alpha^{2}}}{2}
</math>

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

:<math>
r_{outer} = \frac{r_{s} + \sqrt{r_{s}^{2} - 4\alpha^{2} \cos^{2}\theta}}{2}
</math>

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 to the Lense–Thirring effect would be<ref name=phister>{{Cite journal|last=Pfister |first=Herbert |authorlink= |coauthors= |year=2005 |month= |title=On the history of the so-called Lense–Thirring effect |journal=General Relativity and Gravitation |volume=39 |issue=11 |pages=1735–1748 |doi=10.1007/s10714-007-0521-4 |url=http://philsci-archive.pitt.edu/archive/00002681/ |accessdate= |quote= |bibcode = 2007GReGr..39.1735P }}</ref>

: <math>
\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]
</math>

where the coefficients are

: <math>
d_1 = \frac{4MG}{3Rc^2}
</math>

: <math>
d_2 = \frac{4MG}{15Rc^2}
</math>

for ''MG'' ≪ ''Rc''2 or more precisely,

: <math>
d_1 = \frac{4 \alpha(2 - \alpha)}{(1 + \alpha)(3- \alpha)}, \qquad \alpha=\frac{MG}{2Rc^2}
</math>

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.<ref>{{Cite journal|last=Pfister |first=H. |authorlink= |coauthors=''et al.'' |year=1985 |month= |title=Induction of correct centrifugal force in a rotating mass shell |journal=Class. Quantum Grav. |volume=2 |issue=6 |pages=909–918 |doi=10.1088/0264-9381/2/6/015 |url= |accessdate= |quote= |bibcode = 1985CQGra...2..909P }}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Kerr metric]]
* [[Geodetic effect]]
* [[Gravity Recovery and Climate Experiment]]
{{Col-2-of-3}}
* [[Gravitomagnetism]]
* [[Mach's principle]]
* [[Broad iron K line]]
{{Col-3-of-3}}
* [[Relativistic jet]]
* [[Lense–Thirring precession]]
* [[Woodward effect]]
{{col-end}}

==References==
{{Reflist|2}}

==Further reading==
*{{Cite journal|last=Renzetti |first=G. |date=May 2013 |title=History of the attempts to measure orbital frame-dragging with artificial satellites |journal=Centr. Eur. J. Phys. |volume = 11 |issue = 5 |pages = 531–544|doi=10.2478/s11534-013-0189-1|bibcode = 2013CEJPh.tmp...67R }}

==External links==
* [http://www.nasa.gov/home/hqnews/2004/oct/HQ_04351_time_drags.html NASA RELEASE: 04-351 As The World Turns, It Drags Space And Time]
* [http://space.newscientist.com/article/mg19325874.800-loner-stakes-claim-to-gravity-prize.html ''New Scientist'' press release of the MGS test by Iorio in the gravitational field of Mars]
* [http://arxiv.org/abs/gr-qc/0701141 Paper by Giampiero Sindoni, Claudio Paris and Paolo Ialongo about the Mars-MGS test (''unpublished'')]
* [http://arxiv.org/abs/gr-qc/0703020 Paper by G. Felici about the Mars-MGS test (''unpublished'')]
* [http://arxiv.org/abs/astro-ph/0701653 Paper by Kris Krogh about the Mars-MGS test]
* [http://arxiv.org/abs/gr-qc/0601015 Reply by Ignazio Ciufolini and Erricos Pavlis about some criticisms by Iorio]
* [http://arxiv.org/abs/astro-ph/0210139v4 Frame dragging applied to relativistic jets]
* [http://www.phy.duke.edu/~kolena/framedrag.html Duke University press release: General Relativistic Frame Dragging]
* [http://msnbc.msn.com/id/3077887/ MSNBC report on X-ray observations]
* [http://xxx.lanl.gov/abs/gr-qc/9704065 Ciufolini et al. LAGEOS paper 1997 – 25% error]
* [http://arxiv.org/abs/gr-qc/0209109 Ciufolini update Sep 2002 – 20% error]
* [http://www.phy.duke.edu/~kolena/framedrag.html Press release regarding LAGEOS study]
* [http://cddisa.gsfc.nasa.gov/lw13/lw_proceedings.html#science Preprint by Ries et al.]
* [http://www.nature.com/news/2004/041018/full/041018-11.html Ciufolini and Pavlis ''Nature'' new article on 2004 re-analysis of the LAGEOS data]
* [http://www.arxiv.org/abs/gr-qc/0411024 Iorio '' New Astronomy'' general paper with full references]
* [http://www.arxiv.org/abs/gr-qc/0412057 Iorio ''J. of Geodesy'' paper on the impact of the secular variations of the even zonal harmonics of the geopotential]
* [http://www.arxiv.org/abs/gr-qc/0608119 Iorio ''Planetary Space Science'' paper]
* [http://dx.doi.org/10.1007/s10714-008-0742-1 Iorio ''General Relativity and Gravitation'' paper on LARES]
* [http://dx.doi.org/10.1016/j.asr.2008.10.016 Iorio ''Advances in Space Research'' paper on LARES]
* [http://arxiv.org/abs/0809.1373 An Assessment of the Systematic Uncertainty in Present and Future Tests of the Lense–Thirring Effect with Satellite Laser Ranging Iorio''Space Science Reviews'' paper on LARES]
* [http://arxiv.org/abs/0808.0658 Advances in the measurement of the Lense–Thirring effect with Satellite Laser Ranging in the gravitational field of the Earth Iorio invited book chapter on LARES]
''An early version of this article was adapted from public domain material from http://science.msfc.nasa.gov/newhome/headlines/ast06nov97_1.htm ''
* [http://www.asi.it/it/attivita/cosmologia/lares ''ASI (Italian Space Agency) announces the launch of LARES Mission'']
* [http://lares.diaa.uniroma1.it/ ''Official Site of LARES Mission'']

{{Relativity}}

{{DEFAULTSORT:Frame-Dragging}}
[[Category:Tests of general relativity]]
[[Category:Effects of gravitation]]
[[Category:Frames of reference]]
[[Category:Concepts in physics]]

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