GetWiki
geodetic effect
ARTICLE SUBJECTS
being →
database →
ethics →
fiction →
history →
internet →
language →
linux →
logic →
method →
news →
policy →
purpose →
religion →
science →
software →
truth →
unix →
wiki →
ARTICLE TYPES
essay →
feed →
help →
system →
wiki →
ARTICLE ORIGINS
critical →
forked →
imported →
original →
geodetic effect
please note:
- the content below is remote from Wikipedia
- it has been imported raw for GetWiki
{{Short description|Precession of satellite orbits due to a celestial body's presence affecting spacetime}}{{about|precession of orbiting bodies|observing binary stars|de Sitter double star experiment}}File:Gravity_Probe_turning_axis.gif|thumb|236px|A representation of the geodetic effect, with values for Gravity Probe BGravity Probe BThe geodetic effect (also known as geodetic precession, de Sitter precession or de Sitter effect) represents the effect of the curvature of spacetime, predicted by general relativity, on a vector carried along with an orbiting body. For example, the vector could be the angular momentum of a gyroscope orbiting the Earth, as carried out by the Gravity Probe B experiment. The geodetic effect was first predicted by Willem de Sitter in 1916, who provided relativistic corrections to the EarthâMoon system's motion. De Sitter's work was extended in 1918 by Jan Schouten and in 1920 by Adriaan Fokker.BOOK, Studies in the History of General Relativity, Jean Eisenstaedt, Anne J. Kox,weblink 42, 0-8176-3479-7, Birkhäuser, 1988, It can also be applied to a particular secular precession of astronomical orbits, equivalent to the rotation of the LaplaceâRungeâLenz vector.JOURNAL, de Sitter, W, 1916, On Einstein's Theory of Gravitation and its Astronomical Consequences, Mon. Not. R. Astron. Soc., 77, 155â184, 1916MNRAS..77..155D, 10.1093/mnras/77.2.155, free, The term geodetic effect has two slightly different meanings as the moving body may be spinning or non-spinning. Non-spinning bodies move in geodesics, whereas spinning bodies move in slightly different orbits.Rindler, p. 254.The difference between de Sitter precession and LenseâThirring precession (frame dragging) is that the de Sitter effect is due simply to the presence of a central mass, whereas LenseâThirring precession is due to the rotation of the central mass. The total precession is calculated by combining the de Sitter precession with the LenseâThirring precession.- the content below is remote from Wikipedia
- it has been imported raw for GetWiki
Experimental confirmation
The geodetic effect was verified to a precision of better than 0.5% percent by Gravity Probe B, an experiment which measures the tilting of the spin axis of gyroscopes in orbit about the Earth.WEB, Everitt, C.W.F., Parkinson, B.W.,weblink Gravity Probe B Science ResultsâNASA Final Report, 2009, 2009-05-02, The first results were announced on April 14, 2007 at the meeting of the American Physical Society.NEWS, Kahn, Bob, April 14, 2007, Was Einstein right? Scientists provide first public peek at Gravity Probe B results, Stanford News,weblink January 3, 2023,Formulae
{{General relativity sidebar |phenomena}}To derive the precession, assume the system is in a rotating Schwarzschild metric. The nonrotating metric is
ds^2 = dt^2 left(1-frac{2m}{r}right) - dr^2 left(1 - frac{2m}{r}right)^{-1} - r^2 (dtheta^2 + sin^2 theta , dphi'^2) ,
where c = G = 1.We introduce a rotating coordinate system, with an angular velocity omega, such that a satellite in a circular orbit in the θ = Ï/2 plane remains at rest. This gives us
dphi = dphi' - omega , dt.
In this coordinate system, an observer at radial position r sees a vector positioned at r as rotating with angular frequency Ï. This observer, however, sees a vector positioned at some other value of r as rotating at a different rate, due to relativistic time dilation. Transforming the Schwarzschild metric into the rotating frame, and assuming that theta is a constant, we find
& - dr^2 left(1-frac{2m}{r}right)^{-1} - frac{r^2 beta - 2mrbeta}{1-2m/r - r^2 betaomega^2} , dphi^2,
end{align}with beta = sin^2(theta). For a body orbiting in the θ = Ï/2 plane, we will have β = 1, and the body's world-line will maintain constant spatial coordinates for all time. Now, the metric is in the canonical form
ds^2 = e^{2Phi}left(dt - w_i , dx^i right)^2 - k_{ij} , dx^i , dx^j.
From this canonical form, we can easily determine the rotational rate of a gyroscope in proper time
& = frac{ sqrt{beta} omega (r -3 m) }{ r- 2 m - beta omega^2 r^3 } = sqrt{beta}omega.
end{align}where the last equality is true only for free falling observers for whichthere is no acceleration, and thus Phi,_{i} = 0. This leads to
Thomas precession
One can attempt to break down the de Sitter precession into a kinematic effect called Thomas precession combined with a geometric effect caused by gravitationally curved spacetime. At least one authorRindler, Page 234 does describe it this way, but others state that "The Thomas precession comes into play for a gyroscope on the surface of the Earth ..., but not for a gyroscope in a freely moving satellite."Misner, Thorne, and Wheeler, Gravitation, p. 1118 An objection to the former interpretation is that the Thomas precession required has the wrong sign. The Fermi-Walker transport equationMisner, Thorne, and Wheeler, Gravitation, p. 165, pp. 175-176, pp. 1117-1121 gives both the geodetic effect and Thomas precession and describes the transport of the spin 4-vector for accelerated motion in curved spacetime. The spin 4-vector is orthogonal to the velocity 4-vector. Fermi-Walker transport preserves this relation. If there is no acceleration, Fermi-Walker transport is just parallel transport along a geodesic and gives the spin precession due to the geodetic effect. For the acceleration due to uniform circular motion in flat Minkowski spacetime, Fermi Walker transport gives the Thomas precession.See also
- Frame-dragging
- Geodesics in general relativity
- Gravity well
- Timeline of gravitational physics and relativity
Notes
{{reflist}}References
- Wolfgang Rindler (2006) Relativity: special, general, and cosmological (2nd Ed.), Oxford University Press, {{ISBN|978-0-19-856731-8}}
External links
- Gravity Probe B websites at weblink" title="web.archive.org/web/20090922110819weblink">NASA and Stanford University
- weblink" title="web.archive.org/web/20100319230735weblink">Precession in Curved Space "The Geodetic Effect"
- Geodetic Effect
- content above as imported from Wikipedia
- "geodetic effect" does not exist on GetWiki (yet)
- time: 5:09am EDT - Sat, May 18 2024
- "geodetic effect" does not exist on GetWiki (yet)
- time: 5:09am EDT - Sat, May 18 2024
[ this remote article is provided by Wikipedia ]
LATEST EDITS [ see all ]
GETWIKI 23 MAY 2022
The Illusion of Choice
Culture
Culture
GETWIKI 09 JUL 2019
Eastern Philosophy
History of Philosophy
History of Philosophy
GETWIKI 09 MAY 2016
GetMeta:About
GetWiki
GetWiki
GETWIKI 18 OCT 2015
M.R.M. Parrott
Biographies
Biographies
GETWIKI 20 AUG 2014
GetMeta:News
GetWiki
GetWiki
© 2024 M.R.M. PARROTT | ALL RIGHTS RESERVED