SherbyNet - /page/wikibook/entry.mu

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<title>Frame-dragging</title>

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Frame-dragging

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the effect on spacetime caused by a rotating mass. For video frame editing, see <a href="Frame_rate" title="Frame rate">Frame rate</a>.</div>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="General_relativity" title="General relativity">General relativity</a></th></tr><tr><td class="sidebar-image"><div class="sidebar-caption" style="padding:0.5em 0.2em 0.6em;border-bottom:1px solid #aaa; display:block;margin-bottom:0.1em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}">

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<annotation encoding="application/x-tex">{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}</annotation>

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</math><img src="./124ab80fcb17e2733cc17ff6f93da5e52f355c77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.468ex; height:2.843ex;" alt="{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}" loading="lazy"></div></td></tr><tr><td class="sidebar-content" style="padding-bottom:0.75em;">

<ul><li><a href="Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li>

<li><div class="hlist"><ul><li><a href="History_of_general_relativity" title="History of general relativity">History</a></li><li><a href="Timeline_of_gravitational_physics_and_relativity" title="Timeline of gravitational physics and relativity">Timeline</a></li><li><a href="Tests_of_general_relativity" title="Tests of general relativity">Tests</a></li></ul></div></li>

<li><a href="Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Fundamental concepts</div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li>

<li><a href="Special_relativity" title="Special relativity">Special relativity</a></li>

<li><a href="World_line" title="World line">World line</a></li>

<li><a href="Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo-Riemannian manifold</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Phenomena</div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content">

<ul><li><a href="Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Kepler problem</a></li>

<li><a href="Gravitational_lens" title="Gravitational lens">Gravitational lensing</a></li>

<li><a href="Gravitational_redshift" title="Gravitational redshift">Gravitational redshift</a></li>

<li><a href="Gravitational_time_dilation" title="Gravitational time dilation">Gravitational time dilation</a></li>

<li><a href="Gravitational_wave" title="Gravitational wave">Gravitational waves</a></li>

<li><a href="Geodetic_effect" title="Geodetic effect">Geodetic effect</a></li>

<li><a href="Event_horizon" title="Event horizon">Event horizon</a></li>

<li><a href="Gravitational_singularity" title="Gravitational singularity">Singularity</a></li>

<li><a href="Black_hole" title="Black hole">Black hole</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="background:#ececff; font-style:italic;font-weight:normal;">

<a href="Spacetime" title="Spacetime">Spacetime</a></th></tr><tr><td class="sidebar-content">

<ul><li><a href="Spacetime_diagram" title="Spacetime diagram">Spacetime diagrams</a></li>

<li><a href="Minkowski_space" title="Minkowski space">Minkowski spacetime</a></li>

<li><a href="Wormhole" title="Wormhole">Einstein–Rosen bridge</a></li></ul></td>

</tr></tbody></table></div></div></td>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Equations</li><li>Formalisms</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;padding-bottom:0;margin-bottom:0;"><tbody><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;">

Equations</th></tr><tr><td class="sidebar-content" style="padding-top:0;">

<ul><li><a href="Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li>

<li><a href="Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li>

<li><a href="Friedmann_equations" title="Friedmann equations">Friedmann</a></li>

<li><a href="Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li>

<li><a href="Mathisson%E2%80%93Papapetrou%E2%80%93Dixon_equations" title="Mathisson–Papapetrou–Dixon equations">Mathisson–Papapetrou–Dixon</a></li>

<li><a href="Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;">

Formalisms</th></tr><tr><td class="sidebar-content" style="padding-top:0;">

<li><a href="Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;">

Advanced theory</th></tr><tr><td class="sidebar-content" style="padding-top:0;">

<ul><li><a href="Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li>

<li><a href="Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a>)</li>

<li><a href="Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li>

<li><a href="Einstein%E2%80%93Rosen_metric" title="Einstein–Rosen metric">Einstein–Rosen waves</a></li>

<li><a href="Wormhole" title="Wormhole">Wormhole</a></li>

<li><a href="G%C3%B6del_metric" title="Gödel metric">Gödel</a></li>

<li><a href="Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a></li>

<li><a href="Kerr%E2%80%93Newman%E2%80%93de%E2%80%93Sitter_metric" title="Kerr–Newman–de–Sitter metric">Kerr–Newman–de Sitter</a></li>

<li><a href="Kasner_metric" title="Kasner metric">Kasner</a></li>

<li><a href="Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li>

<li><a href="Milne_model" title="Milne model">Milne</a></li>

<li><a href="Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Robertson–Walker</a></li>

<li><a href="Oppenheimer%E2%80%93Snyder_model" title="Oppenheimer–Snyder model">Oppenheimer–Snyder</a></li>

<li><a href="Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li>

<li><a href="Hartle%E2%80%93Thorne_metric" title="Hartle–Thorne metric">Hartle–Thorne</a></li>

<li><a href="Vaidya_metric" title="Vaidya metric">Vaidya</a></li>

<li><a href="Peres_metric" title="Peres metric">Peres</a></li>

<li><a href="De_Sitter%E2%80%93Schwarzschild_metric" title="De Sitter–Schwarzschild metric">De Sitter-Schwarzschild</a></li>

<li><a href="McVittie_metric" title="McVittie metric">McVittie</a></li>

<li><a href="Kerr%E2%80%93Newman%E2%80%93de%E2%80%93Sitter_metric" title="Kerr–Newman–de–Sitter metric">Kerr-Newman-de-Sitter</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Albert_Einstein" title="Albert Einstein">Einstein</a></li>

<li><a href="Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li>

<li><a href="David_Hilbert" title="David Hilbert">Hilbert</a></li>

<li><a href="Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li>

<li><a href="Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li>

<li><a href="Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li>

<li><a href="Hans_Reissner" title="Hans Reissner">Reissner</a></li>

<li><a href="Gunnar_Nordstr%C3%B6m" title="Gunnar Nordström">Nordström</a></li>

<li><a href="Arthur_Eddington" title="Arthur Eddington">Eddington</a></li>

<li><a href="Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li>

<li><a href="Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li>

<li><a href="Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li>

<li><a href="Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li>

<li><a href="J._Robert_Oppenheimer" title="J. Robert Oppenheimer">Oppenheimer</a></li>

<li><a href="Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a></li>

<li><a href="John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li>

<li><a href="Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li>

<li><a href="James_M._Bardeen" title="James M. Bardeen">Bardeen</a></li>

<li><a href="Arthur_Geoffrey_Walker" title="Arthur Geoffrey Walker">Walker</a></li>

<li><a href="Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li>

<li><a href="J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li>

<li><a href="Roger_Penrose" title="Roger Penrose">Penrose</a></li>

<li><a href="Stephen_Hawking" title="Stephen Hawking">Hawking</a></li>

<li><a href="Amal_Kumar_Raychaudhuri" title="Amal Kumar Raychaudhuri">Raychaudhuri</a></li>

<li><a href="Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li>

<li><a href="Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li>

<li><a href="Willem_Jacob_van_Stockum" title="Willem Jacob van Stockum">van Stockum</a></li>

<li><a href="Ezra_T._Newman" title="Ezra T. Newman">Newman</a></li>

<li><a href="Kip_Thorne" title="Kip Thorne">Thorne</a></li>

<li><a href="List_of_contributors_to_general_relativity" title="List of contributors to general relativity">others</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-below hlist" style="background-color: transparent; border-color: #A2B8BF">

<ul><li> <a href="Portal%3APhysics" title="Portal:Physics">Physics portal</a></li>

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Frame-dragging is an effect on <a href="Spacetime" title="Spacetime">spacetime</a>, predicted by <a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s <a href="General_relativity" title="General relativity">general theory of relativity</a>, that is due to non-static stationary distributions of <a href="Mass%E2%80%93energy" class="mw-redirect" title="Mass–energy">mass–energy</a>. A stationary <a href="Field_(physics)" title="Field (physics)">field</a> is one that is in a steady state, but the masses causing that field may be non-static ⁠— rotating, for instance. More generally, the subject that deals with the effects caused by mass–energy currents is known as <a href="Gravitoelectromagnetism" title="Gravitoelectromagnetism">gravitoelectromagnetism</a>, which is analogous to the magnetism of <a href="Classical_electromagnetism" title="Classical electromagnetism">classical electromagnetism</a>.

The first frame-dragging effect was derived in 1918, in the framework of general relativity, by the Austrian physicists <a href="Josef_Lense" title="Josef Lense">Josef Lense</a> and <a href="Hans_Thirring" title="Hans Thirring">Hans Thirring</a>, and is also known as the <a href="Lense%E2%80%93Thirring_effect" class="mw-redirect" title="Lense–Thirring effect">Lense–Thirring effect</a>.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a> They predicted that the rotation of a massive object would distort the <a href="Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">spacetime metric</a>, making the orbit of a nearby test particle <a href="Precess" class="mw-redirect" title="Precess">precess</a>. This does not happen in <a href="Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a> for which the <a href="Gravitational_field" title="Gravitational field">gravitational field</a> 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.

<div class="mw-heading mw-heading2"><h2 id="Effects">Effects</h2></div>

Rotational frame-dragging (the <a href="Lense%E2%80%93Thirring_effect" class="mw-redirect" title="Lense–Thirring effect">Lense–Thirring effect</a>) appears in the <a href="General_principle_of_relativity" class="mw-redirect" title="General principle of relativity">general principle of relativity</a> 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 <a href="Gravity_Probe_B" title="Gravity Probe B">Gravity Probe B</a> experiment. Qualitatively, frame-dragging can be viewed as the gravitational analog of <a href="Electromagnetic_induction" title="Electromagnetic induction">electromagnetic induction</a>.

Also, an inner region is dragged more than an outer region. This produces locally rotating frames. For example, imagine that a north–south-oriented ice skater, in orbit over the equator of a rotating black hole and rotationally at rest with respect to the stars, extends her arms. The arm extended toward the black hole will be "torqued" spinward due to gravitomagnetic induction ("torqued" is in quotes because gravitational effects are not considered "forces" under <a href="General_relativity" title="General relativity">GR</a>). 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 <a href="Equivalence_principle" title="Equivalence principle">equivalence principle</a>, 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 <a href="Hyperfine_structure" title="Hyperfine structure">hyperfine structure</a> in atomic spectra due to nuclear spin. A useful metaphor is a <a href="Planetary_gear" class="mw-redirect" title="Planetary gear">planetary gear</a> 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 <a href="Mach's_principle" title="Mach's principle">Mach's principle</a>.)

Another consequence is that, for an object constrained in an equatorial orbit, but not in freefall, it weighs more if orbiting anti-spinward, and less if orbiting spinward. For example, in a suspended equatorial bowling alley, a bowling ball rolled anti-spinward would weigh more than the same ball rolled in a spinward direction. Note, frame dragging will neither accelerate nor slow down the bowling ball in either direction. It is not a "viscosity". Similarly, a stationary <a href="Plumb-bob" class="mw-redirect" title="Plumb-bob">plumb-bob</a> 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. However, if a "yoyo" plumb-bob (with axis perpendicular to the equatorial plane) is slowly lowered, over the equator, toward the static limit, the yoyo will spin up in a counter rotating direction. However, any denizens inside the yoyo will not feel any torque and will not experience any felt change in angular momentum.

Linear frame dragging is the similarly inevitable result of the general principle of relativity, applied to <a href="Linear_momentum" class="mw-redirect" title="Linear momentum">linear momentum</a>. 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).<a href="#cite_note-4">[4]</a>

Static mass increase is a third effect noted by Einstein in the same paper.<a href="#cite_note-5">[5]</a> The effect is an increase in <a href="Inertia" title="Inertia">inertia</a> 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.

<div class="mw-heading mw-heading2"><h2 id="Experimental_tests">Experimental tests</h2></div>

In 1976 Van Patten and Everitt<a href="#cite_note-6">[6]</a><a href="#cite_note-7">[7]</a> 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<a href="#cite_note-8">[8]</a> who proposed to launch a passive, geodetic satellite in an orbit identical to that of the <a href="LAGEOS" title="LAGEOS">LAGEOS</a> 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, <a href="LARES_(satellite)" title="LARES (satellite)">LARES</a>, 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 (<a href="Satellite_laser_ranging" title="Satellite laser ranging">SLR</a>) technique to measure the Lense–Thirring effect dates to 1977–1978.<a href="#cite_note-9">[9]</a> Tests started to be effectively performed by using the LAGEOS and LAGEOS II satellites in 1996,<a href="#cite_note-10">[10]</a> according to a strategy<a href="#cite_note-11">[11]</a> 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<a href="#cite_note-12">[12]</a><a href="#cite_note-13">[13]</a> by discarding the perigee of LAGEOS II and using a linear combination.<a href="#cite_note-14">[14]</a> Recently, a comprehensive overview of the attempts to measure the Lense-Thirring effect with artificial satellites was published in the literature.<a href="#cite_note-15">[15]</a> The overall accuracy reached in the tests with the LAGEOS satellites is subject to some controversy.<a href="#cite_note-16">[16]</a><a href="#cite_note-17">[17]</a><a href="#cite_note-18">[18]</a>

The <a href="Gravity_Probe_B" title="Gravity Probe B">Gravity Probe B</a> experiment<a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a> was a satellite-based mission by a Stanford group and NASA, used to experimentally measure another gravitomagnetic effect, the <a href="Schiff_precession" class="mw-redirect" title="Schiff precession">Schiff precession</a> of a gyroscope,<a href="#cite_note-21">[21]</a><a href="#cite_note-22">[22]</a><a href="#cite_note-23">[23]</a> 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<a href="#cite_note-24">[24]</a> 256–128%, with the hope of reaching about 13% in December 2007.<a href="#cite_note-25">[25]</a>

In 2008 the Senior Review Report of the NASA Astrophysics Division Operating Missions stated that it was unlikely that the Gravity Probe B team will be able to reduce the errors to the level necessary to produce a convincing test of currently untested aspects of General Relativity (including frame-dragging).<a href="#cite_note-26">[26]</a><a href="#cite_note-27">[27]</a>

On May 4, 2011, the Stanford-based analysis group and NASA announced the final report,<a href="#cite_note-28">[28]</a> 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.<a href="#cite_note-29">[29]</a><a href="#cite_note-PRL-30">[30]</a>

NASA published claims of success in verification of frame dragging for the <a href="GRACE_(satellite)" class="mw-redirect" title="GRACE (satellite)">GRACE twin satellites</a><a href="#cite_note-31">[31]</a> and Gravity Probe B,<a href="#cite_note-32">[32]</a> both of which claims are still in public view. A research group in Italy,<a href="#cite_note-33">[33]</a> USA, and UK also claimed success in verification of frame dragging with the Grace gravity model, published in a peer reviewed journal. All the claims include recommendations for further research at greater accuracy and other gravity models.

In the case of stars orbiting close to a spinning, supermassive black hole, frame dragging should cause the star's orbital plane to <a href="Lense%E2%80%93Thirring_precession" title="Lense–Thirring precession">precess</a> about the black hole spin axis. This effect should be detectable within the next few years via <a href="Astrometry" title="Astrometry">astrometric</a> monitoring of stars at the center of the <a href="Milky_Way" title="Milky Way">Milky Way</a> galaxy.<a href="#cite_note-34">[34]</a>

By comparing the rate of orbital precession of two stars on different orbits, it is possible in principle to test the <a href="No-hair_theorem" title="No-hair theorem">no-hair theorems</a> of general relativity, in addition to measuring the spin of the black hole.<a href="#cite_note-35">[35]</a>

<div class="mw-heading mw-heading2"><h2 id="Astronomical_evidence">Astronomical evidence</h2></div>

<a href="Relativistic_jet" class="mw-redirect" title="Relativistic jet">Relativistic jets</a> may provide evidence for the reality of frame-dragging. <a href="Gravitoelectromagnetism" title="Gravitoelectromagnetism">Gravitomagnetic</a> forces produced by the <a href="Lense%E2%80%93Thirring_precession" title="Lense–Thirring precession">Lense–Thirring effect</a> (frame dragging) within the <a href="Ergosphere" title="Ergosphere">ergosphere</a> of <a href="Rotating_black_hole" title="Rotating black hole">rotating black holes</a><a href="#cite_note-36">[36]</a><a href="#cite_note-37">[37]</a> combined with the energy extraction mechanism by <a href="Roger_Penrose" title="Roger Penrose">Penrose</a><a href="#cite_note-38">[38]</a> have been used to explain the observed properties of <a href="Relativistic_jet" class="mw-redirect" title="Relativistic jet">relativistic jets</a>. The gravitomagnetic model developed by <a href="Reva_Kay_Williams" class="mw-redirect" title="Reva Kay Williams">Reva Kay Williams</a> predicts the observed high energy particles (~GeV) emitted by <a href="Quasars" class="mw-redirect" title="Quasars">quasars</a> and <a href="Active_galactic_nuclei" class="mw-redirect" title="Active galactic nuclei">active galactic nuclei</a>; the extraction of X-rays, γ-rays, and relativistic e−–e+ pairs; the collimated jets about the polar axis; and the asymmetrical formation of jets (relative to the orbital plane).

The Lense–Thirring effect has been observed in a binary system that consists of a massive <a href="White_dwarf" title="White dwarf">white dwarf</a> and a <a href="Pulsar" title="Pulsar">pulsar</a>.<a href="#cite_note-39">[39]</a>

<div class="mw-heading mw-heading2"><h2 id="Mathematical_derivation">Mathematical derivation</h2></div>

Frame-dragging may be illustrated most readily using the <a href="Kerr_metric" title="Kerr metric">Kerr metric</a>,<a href="#cite_note-kerr_1963-40">[40]</a><a href="#cite_note-41">[41]</a> which describes the geometry of <a href="Spacetime" title="Spacetime">spacetime</a> in the vicinity of a mass M rotating with <a href="Angular_momentum" title="Angular momentum">angular momentum</a> J, and <a href="Boyer%E2%80%93Lindquist_coordinates" title="Boyer–Lindquist coordinates">Boyer–Lindquist coordinates</a> (see the link for the transformation):

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}c^{2}d\tau ^{2}=&\left(1-{\frac {r_{\text{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_{\text{s}}r\alpha ^{2}}{\rho ^{2}}}\sin ^{2}\theta \right)\sin ^{2}\theta \ d\phi ^{2}+{\frac {2r_{\text{s}}r\alpha c\sin ^{2}\theta }{\rho ^{2}}}d\phi dt\end{aligned}}}</annotation>

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</math><img src="./0ef9cb97aecf9b2bb42b3a5469716e1b9d248a1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:68.264ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}c^{2}d\tau ^{2}=&\left(1-{\frac {r_{\text{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_{\text{s}}r\alpha ^{2}}{\rho ^{2}}}\sin ^{2}\theta \right)\sin ^{2}\theta \ d\phi ^{2}+{\frac {2r_{\text{s}}r\alpha c\sin ^{2}\theta }{\rho ^{2}}}d\phi dt\end{aligned}}}" loading="lazy"></dd></dl>

where rs is the <a href="Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild radius</a>

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<annotation encoding="application/x-tex">{\displaystyle r_{\text{s}}={\frac {2GM}{c^{2}}}}</annotation>

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</math><img src="./e2769d1ec3f716c2035d0956f746243102a501b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.295ex; height:5.676ex;" alt="{\displaystyle r_{\text{s}}={\frac {2GM}{c^{2}}}}" loading="lazy"></dd></dl>

and where the following shorthand variables have been introduced for brevity

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<annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {J}{Mc}}}</annotation>

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</math><img src="./b5bf0e22506f5f60c23eeb195f2c65c246b367ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.871ex; height:5.176ex;" alt="{\displaystyle \alpha ={\frac {J}{Mc}}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \rho ^{2}=r^{2}+\alpha ^{2}\cos ^{2}\theta }</annotation>

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</math><img src="./1f67c3bd726469ef1ea5d341e3bee4e102ee669a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.87ex; height:3.176ex;" alt="{\displaystyle \rho ^{2}=r^{2}+\alpha ^{2}\cos ^{2}\theta }" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \Lambda ^{2}=r^{2}-r_{\text{s}}r+\alpha ^{2}}</annotation>

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</math><img src="./4bb44a5095611fc758cd3e1f039fe34e7c831e09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.069ex; height:3.009ex;" alt="{\displaystyle \Lambda ^{2}=r^{2}-r_{\text{s}}r+\alpha ^{2}}" loading="lazy"></dd></dl>

In the non-relativistic limit where M (or, equivalently, rs) goes to zero, the Kerr metric becomes the orthogonal metric for the <a href="Oblate_spheroidal_coordinates" title="Oblate spheroidal coordinates">oblate spheroidal coordinates</a>

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<annotation encoding="application/x-tex">{\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}}</annotation>

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</math><img src="./6c2aea2eecef375e06b6b27107fbc103d3c4a8ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:58.681ex; height:6.176ex;" alt="{\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}}" loading="lazy"></dd></dl>

We may rewrite the Kerr metric in the following form

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<annotation encoding="application/x-tex">{\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}}</annotation>

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</math><img src="./40485c6a2b931eef8504c1736e148e5cd65dfd8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:65.69ex; height:7.676ex;" alt="{\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}}" loading="lazy"></dd></dl>

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 <a href="Colatitude" title="Colatitude">colatitude</a> θ

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<annotation encoding="application/x-tex">{\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 }}}</annotation>

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</math><img src="./da3d7bae5d84bdf0971f83b8734ccb7c0920f8f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:40.851ex; height:6.009ex;" alt="{\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 }}}" loading="lazy"></dd></dl>

In the plane of the equator this simplifies to:<a href="#cite_note-42">[42]</a>

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<annotation encoding="application/x-tex">{\displaystyle \Omega ={\frac {r_{s}\alpha c}{r^{3}+\alpha ^{2}r+r_{s}\alpha ^{2}}}}</annotation>

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</math><img src="./acfe3f132a550e0a841600f924dc2f4df3844fbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.581ex; height:5.343ex;" alt="{\displaystyle \Omega ={\frac {r_{s}\alpha c}{r^{3}+\alpha ^{2}r+r_{s}\alpha ^{2}}}}" loading="lazy"></dd></dl>

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 <a href="Ergosphere" title="Ergosphere">ergosphere</a> of a rotating <a href="Black_hole" title="Black hole">black hole</a>. The Kerr metric has two surfaces on which it appears to be singular. The inner surface corresponds to a spherical <a href="Event_horizon" title="Event horizon">event horizon</a> similar to that observed in the <a href="Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild metric</a>; this occurs at

<msub>

<mtext>inner</mtext>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msqrt>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{\text{inner}}={\frac {r_{\text{s}}+{\sqrt {r_{\text{s}}^{2}-4\alpha ^{2}}}}{2}}}</annotation>

</semantics>

</math><img src="./ff42e954cd368dd8cc30a8aaca806d364fa24a93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.616ex; height:6.176ex;" alt="{\displaystyle r_{\text{inner}}={\frac {r_{\text{s}}+{\sqrt {r_{\text{s}}^{2}-4\alpha ^{2}}}}{2}}}" loading="lazy"></dd></dl>

where the purely radial component grr of the metric goes to infinity. The outer surface can be approximated by an <a href="Oblate_spheroid" class="mw-redirect" title="Oblate spheroid">oblate spheroid</a> with lower spin parameters, and resembles a pumpkin-shape<a href="#cite_note-visser-43">[43]</a><a href="#cite_note-blundell-44">[44]</a> with higher spin parameters. It touches the inner surface at the poles of the rotation axis, where the colatitude θ equals 0 or π; its radius in Boyer-Lindquist coordinates is defined by the formula

<msub>

<mtext>outer</mtext>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msqrt>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{\text{outer}}={\frac {r_{\text{s}}+{\sqrt {r_{\text{s}}^{2}-4\alpha ^{2}\cos ^{2}\theta }}}{2}}}</annotation>

</semantics>

</math><img src="./b7c2f6074e3cd823a489d0a1782dace2e0528867.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.736ex; height:6.176ex;" alt="{\displaystyle r_{\text{outer}}={\frac {r_{\text{s}}+{\sqrt {r_{\text{s}}^{2}-4\alpha ^{2}\cos ^{2}\theta }}}{2}}}" loading="lazy"></dd></dl>

where the purely temporal component gtt of the metric changes sign from positive to negative. The space between these two surfaces is called the <a href="Ergosphere" title="Ergosphere">ergosphere</a>. A moving particle experiences a positive <a href="Proper_time" title="Proper time">proper time</a> along its <a href="Worldline" class="mw-redirect" title="Worldline">worldline</a>, its path through <a href="Spacetime" title="Spacetime">spacetime</a>. 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.

<div class="mw-heading mw-heading3"><h3 id="Lense–Thirring_effect_inside_a_rotating_shell">Lense–Thirring effect inside a rotating shell</h3></div>

The <a href="Lense%E2%80%93Thirring_effect" class="mw-redirect" title="Lense–Thirring effect">Lense–Thirring effect</a> inside a rotating shell was taken by <a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a> as not just support for, but a vindication of <a href="Mach's_principle" title="Mach's principle">Mach's principle</a>, in a letter he wrote to <a href="Ernst_Mach" title="Ernst Mach">Ernst Mach</a> in 1913 (five years before Lense and Thirring's work, and two years before he had attained the final form of <a href="General_relativity" title="General relativity">general relativity</a>). A reproduction of the letter can be found in <a href="Gravitation_(book)" title="Gravitation (book)">Misner, Thorne, Wheeler</a>.<a href="#cite_note-mtw-45">[45]</a> The general effect scaled up to cosmological distances, is still used as a support for Mach's principle.<a href="#cite_note-mtw-45">[45]</a>

Inside a rotating spherical shell the acceleration due to the Lense–Thirring effect would be<a href="#cite_note-phister-46">[46]</a>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

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</mrow>

</msub>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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]}</annotation>

</semantics>

</math><img src="./746015edea3a36294f39471033272aa670bde435.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.499ex; height:2.843ex;" alt="{\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]}" loading="lazy"></dd></dl>

where the coefficients are

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}d_{1}&={\frac {4MG}{3Rc^{2}}}\\d_{2}&={\frac {4MG}{15Rc^{2}}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./2ac15042c3354e9c2f74ea61d8f2f4ad52104894.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:13.099ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}d_{1}&={\frac {4MG}{3Rc^{2}}}\\d_{2}&={\frac {4MG}{15Rc^{2}}}\end{aligned}}}" loading="lazy"></dd></dl>

for MG ≪ Rc2 or more precisely,

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

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<msup>

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{1}={\frac {4\alpha (2-\alpha )}{(1+\alpha )(3-\alpha )}},\qquad \alpha ={\frac {MG}{2Rc^{2}}}}</annotation>

</semantics>

</math><img src="./efc1ce55001a974810b865a8848362c5ffe92244.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.886ex; height:6.509ex;" alt="{\displaystyle d_{1}={\frac {4\alpha (2-\alpha )}{(1+\alpha )(3-\alpha )}},\qquad \alpha ={\frac {MG}{2Rc^{2}}}}" loading="lazy"></dd></dl>

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.<a href="#cite_note-47">[47]</a>

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<ul><li><a href="Geodetic_effect" title="Geodetic effect">Geodetic effect</a></li>

<li><a href="Gravity_Recovery_and_Climate_Experiment" class="mw-redirect" title="Gravity Recovery and Climate Experiment">Gravity Recovery and Climate Experiment</a></li>

<li><a href="Gravitomagnetism" class="mw-redirect" title="Gravitomagnetism">Gravitomagnetism</a></li>

<li><a href="Mach's_principle" title="Mach's principle">Mach's principle</a></li>

<li><a href="Broad_iron_K_line" class="mw-redirect" title="Broad iron K line">Broad iron K line</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-mtw-45">^ <a href="#cite_ref-mtw_45-0">a</a> <a href="#cite_ref-mtw_45-1">b</a> Misner, Thorne, Wheeler, Gravitation, Figure 21.5, page 544

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<li id="cite_note-phister-46"><a href="#cite_ref-phister_46-0">^</a> <cite id="CITEREFPfister2005" class="citation journal cs1">Pfister, Herbert (2005). <a rel="nofollow" class="external text" href="http://philsci-archive.pitt.edu/archive/00002681/">"On the history of the so-called Lense–Thirring effect"</a>. General Relativity and Gravitation. 39 (11): 1735–1748. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007GReGr..39.1735P">2007GReGr..39.1735P</a>. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.693.4061">10.1.1.693.4061</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10714-007-0521-4">10.1007/s10714-007-0521-4</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:22593373">22593373</a>.</cite>

</li>

<li id="cite_note-47"><a href="#cite_ref-47">^</a> <cite id="CITEREFPfister1985" class="citation journal cs1">Pfister, H.; et al. (1985). "Induction of correct centrifugal force in a rotating mass shell". Classical and Quantum Gravity. 2 (6): 909–918. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1985CQGra...2..909P">1985CQGra...2..909P</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0264-9381%2F2%2F6%2F015">10.1088/0264-9381/2/6/015</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:250883114">250883114</a>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFRenzetti2013" class="citation journal cs1">Renzetti, G. (May 2013). <a rel="nofollow" class="external text" href="https://doi.org/10.2478%2Fs11534-013-0189-1">"History of the attempts to measure orbital frame-dragging with artificial satellites"</a>. <a href="Central_European_Journal_of_Physics" class="mw-redirect" title="Central European Journal of Physics">Central European Journal of Physics</a>. 11 (5): 531–544. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013CEJPh..11..531R">2013CEJPh..11..531R</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2478%2Fs11534-013-0189-1">10.2478/s11534-013-0189-1</a>.</cite></li>

<li><cite id="CITEREFGinzburg1959" class="citation journal cs1">Ginzburg, V. L. (May 1959). "Artificial Satellites and the Theory of Relativity". <a href="Scientific_American" title="Scientific American">Scientific American</a>. 200 (5): 149–160. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1959SciAm.200e.149G">1959SciAm.200e.149G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican0559-149">10.1038/scientificamerican0559-149</a>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://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</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080619060938/http://www.nasa.gov/home/hqnews/2004/oct/HQ_04351_time_drags.html">Archived</a> 2008-06-19 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Relativity254" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align:center;"><div id="Relativity254" style="font-size:114%;margin:0 4em"><a href="Theory_of_relativity" title="Theory of relativity">Relativity</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="Special_relativity" title="Special relativity">Special relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Principle_of_relativity" title="Principle of relativity">Principle of relativity</a> (<a href="Galilean_invariance" title="Galilean invariance">Galilean relativity</a></li>

<li><a href="Galilean_transformation" title="Galilean transformation">Galilean transformation</a>)</li>

<li><a href="Special_relativity" title="Special relativity">Special relativity</a></li>

<li><a href="Doubly_special_relativity" title="Doubly special relativity">Doubly special relativity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Frame_of_reference" title="Frame of reference">Frame of reference</a></li>

<li><a href="Speed_of_light" title="Speed of light">Speed of light</a></li>

<li><a href="Hyperbolic_orthogonality" title="Hyperbolic orthogonality">Hyperbolic orthogonality</a></li>

<li><a href="Rapidity" title="Rapidity">Rapidity</a></li>

<li><a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a></li>

<li><a href="Proper_length" title="Proper length">Proper length</a></li>

<li><a href="Proper_time" title="Proper time">Proper time</a></li>

<li><a href="Proper_acceleration" title="Proper acceleration">Proper acceleration</a></li>

<li><a href="Mass_in_special_relativity" title="Mass in special relativity">Relativistic mass</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li>

<li><a href="List_of_textbooks_on_relativity" title="List of textbooks on relativity">Textbooks</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Time_dilation" title="Time dilation">Time dilation</a></li>

<li><a href="Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence (E=mc2)</a></li>

<li><a href="Length_contraction" title="Length contraction">Length contraction</a></li>

<li><a href="Relativity_of_simultaneity" title="Relativity of simultaneity">Relativity of simultaneity</a></li>

<li><a href="Relativistic_Doppler_effect" title="Relativistic Doppler effect">Relativistic Doppler effect</a></li>

<li><a href="Thomas_precession" title="Thomas precession">Thomas precession</a></li>

<li><a href="Ladder_paradox" title="Ladder paradox">Ladder paradox</a></li>

<li><a href="Twin_paradox" title="Twin paradox">Twin paradox</a></li>

<li><a href="Terrell_rotation" title="Terrell rotation">Terrell rotation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="Spacetime" title="Spacetime">Spacetime</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Light_cone" title="Light cone">Light cone</a></li>

<li><a href="World_line" title="World line">World line</a></li>

<li><a href="Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagram</a></li>

<li><a href="Biquaternion" title="Biquaternion">Biquaternions</a></li>

<li><a href="Minkowski_space" title="Minkowski space">Minkowski space</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="General_relativity" title="General relativity">General relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li>

<li><a href="Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li>

<li><a href="Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a></li>

<li><a href="Penrose_diagram" title="Penrose diagram">Penrose diagram</a></li>

<li><a href="Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li>

<li><a href="Mach's_principle" title="Mach's principle">Mach's principle</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="ADM_formalism" title="ADM formalism">ADM formalism</a></li>

<li><a href="BSSN_formalism" title="BSSN formalism">BSSN formalism</a></li>

<li><a href="Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li>

<li><a href="Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li>

<li><a href="Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian formalism</a></li>

<li><a href="Raychaudhuri_equation" title="Raychaudhuri equation">Raychaudhuri equation</a></li>

<li><a href="Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein equation</a></li>

<li><a href="Ernst_equation" title="Ernst equation">Ernst equation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Black_hole" title="Black hole">Black hole</a></li>

<li><a href="Event_horizon" title="Event horizon">Event horizon</a></li>

<li><a href="Gravitational_singularity" title="Gravitational singularity">Singularity</a></li>

<li><a href="Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Two-body problem</a></li></ul>

<ul><li><a href="Gravitational_wave" title="Gravitational wave">Gravitational waves</a>: <a href="Gravitational-wave_astronomy" title="Gravitational-wave astronomy">astronomy</a></li>

<li><a href="Gravitational-wave_observatory" title="Gravitational-wave observatory">detectors</a> (<a href="LIGO" title="LIGO">LIGO</a> and <a href="LIGO_Scientific_Collaboration" title="LIGO Scientific Collaboration">collaboration</a></li>

<li><a href="Virgo_interferometer" title="Virgo interferometer">Virgo</a></li>

<li><a href="LISA_Pathfinder" title="LISA Pathfinder">LISA Pathfinder</a></li>

<li><a href="Hulse%E2%80%93Taylor_binary" class="mw-redirect" title="Hulse–Taylor binary">Hulse–Taylor binary</a></li></ul>

<ul><li><a href="Tests_of_general_relativity" title="Tests of general relativity">Other tests</a>: <a href="Apsidal_precession" title="Apsidal precession">precession</a> of Mercury</li>

<li><a href="Gravitational_lens" title="Gravitational lens">lensing</a> (together with <a href="Einstein_cross" class="mw-redirect" title="Einstein cross">Einstein cross</a> and <a href="Einstein_rings" class="mw-redirect" title="Einstein rings">Einstein rings</a>)</li>

<li><a href="Gravitational_redshift" title="Gravitational redshift">redshift</a></li>

<li><a href="Shapiro_time_delay" title="Shapiro time delay">Shapiro delay</a></li>

<li> / <a href="Geodetic_effect" title="Geodetic effect">geodetic effect</a> (<a href="Lense%E2%80%93Thirring_precession" title="Lense–Thirring precession">Lense–Thirring precession</a>)</li>

<li><a href="Pulsar_timing_array" title="Pulsar timing array">pulsar timing arrays</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Advanced theories</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke theory</a></li>

<li><a href="Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein</a></li>

<li><a href="Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li>Cosmological: <a href="Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Friedmann–Lemaître–Robertson–Walker</a> (<a href="Friedmann_equations" title="Friedmann equations">Friedmann equations</a>)</li>

<li><a href="Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li>

<li><a href="Kasner_metric" title="Kasner metric">Kasner</a></li>

<li><a href="BKL_singularity" title="BKL singularity">BKL singularity</a></li>

<li><a href="G%C3%B6del_metric" title="Gödel metric">Gödel</a></li>

<li><a href="Milne_model" title="Milne model">Milne</a></li></ul>

<ul><li>Spherical: <a href="Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a></li>

<li><a href="Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation" title="Tolman–Oppenheimer–Volkoff equation">Tolman–Oppenheimer–Volkoff equation</a>)</li>

<li><a href="Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li></ul>

<ul><li>Axisymmetric: <a href="Kerr_metric" title="Kerr metric">Kerr</a> (<a href="Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a>)</li>

<li><a href="Weyl%E2%88%92Lewis%E2%88%92Papapetrou_coordinates" class="mw-redirect" title="Weyl−Lewis−Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li>

<li><a href="Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li>

<li><a href="Relativistic_disk" title="Relativistic disk">discs</a></li></ul>

<ul><li>Others: <a href="Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li>

<li><a href="Ozsv%C3%A1th%E2%80%93Sch%C3%BCcking_metric" title="Ozsváth–Schücking metric">Ozsváth–Schücking</a></li>

<li><a href="Alcubierre_drive" title="Alcubierre drive">Alcubierre</a></li>

<li><a href="Ellis_wormhole" title="Ellis wormhole">Ellis</a></li></ul>

<ul><li>In computational physics: <a href="Numerical_relativity" title="Numerical relativity">Numerical relativity</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Scientists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li>

<li><a href="Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li>

<li><a href="Albert_Einstein" title="Albert Einstein">Einstein</a></li>