Geodetic effect

This article is about precession of orbiting bodies. For observing binary stars, see de Sitter double star experiment.

A representation of the geodetic effect.

The 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 Earth–Moon system's motion. De Sitter's work was extended in 1918 by Jan Schouten and in 1920 by Adriaan Fokker.^[1]It can also be applied to a particular secular precession of astronomical orbits, equivalent to the rotation of the Laplace–Runge–Lenz vector.^[2]

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.^[3]

The difference between de Sitter precession and Lense–Thirring precession (frame dragging) is that the de Sitter effect is due simply to the presence of a central mass, whereas Lense–Thirring precession is due to the rotation of the central mass. The total precession is calculated by combining the de Sitter precession with the Lense–Thirring precession.

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.^[4] The first results were announced on April 14, 2007 at the meeting of the American Physical Society.^[5]

Formulae

General relativity

G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}

Fundamental concepts

Phenomena

Spacetime
Kepler problem Gravitational lensing Gravitational waves Frame-dragging Geodetic effect Event horizon Singularity Black hole
Spacetime diagrams Minkowski spacetime Einstein–Rosen bridge

Equations
Formalisms

Equations
Linearized gravity Einstein field equations Friedmann Geodesics Mathisson–Papapetrou–Dixon Hamilton–Jacobi–Einstein
Formalisms
ADM BSSN Post-Newtonian
Advanced theory
Kaluza–Klein theory Quantum gravity

Solutions

Scientists

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 (d\theta^2 + \sin^2 \theta \, d\phi'^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 = π/2 plane remains at rest. This gives us

d\phi = d\phi' - \omega \, dt.

In this coordinate system, an observer at radial position r sees a vector positioned at r as rotating with angular frequency ω. 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

ds^2 = \left(1-\frac{2m}{r}-r^2 \beta\omega^2 \right)\left(dt-\frac{r^2 \beta\omega}{1-2m/r-r^2 \beta\omega^2} \, d\phi\right)^2 - dr^2 \left(1-\frac{2m}{r}\right)^{-1} - \frac{r^2 \beta - 2mr\beta}{1-2m/r - r^2 \beta\omega^2} \, d\phi^2

with $\beta = \sin^2(\theta)$ . For a body orbiting in the θ = π/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^{2\Phi}\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

\Omega = \frac{\sqrt{2}}{4} e^\Phi [k^{ik}k^{jl}(\omega_{i,j}-\omega_{j,i})(\omega_{k,l} - \omega_{l,k})]^{1/2} = \frac{ \sqrt{\beta} \omega (r -3 m) }{ r- 2 m - \beta \omega^2 r^3 } =\sqrt{\beta}\omega.

where the last equality is true only for free falling observers for which there is no acceleration, and thus $\Phi,_{i} = 0$ . This leads to

\Phi,_i = \frac{2m/r^2 - 2r\beta\omega^2}{2(1-2m/r-r^2 \beta\omega^2)} = 0.

Solving this equation for ω yields

\omega^2 = \frac{m}{r^3 \beta}.

This is essentially Kepler's law of periods, which happens to be relativistically exact when expressed in terms of the time coordinate t of this particular rotating coordinate system. In the rotating frame, the satellite remains at rest, but an observer aboard the satellite sees the gyroscope's angular momentum vector precessing at the rate ω. This observer also sees the distant stars as rotating, but they rotate at a slightly different rate due to time dilation. Let τ be the gyroscope's proper time. Then

\Delta \tau = \left(1-\frac{2m}{r} - r^2 \beta\omega^2 \right)^{1/2} \, dt = \left(1-\frac{3m}{r}\right)^{1/2} \, dt.

The −2m/r term is interpreted as the gravitational time dilation, while the additional −m/r is due to the rotation of this frame of reference. Let α' be the accumulated precession in the rotating frame. Since $\alpha' = \Omega \Delta \tau$ , the precession over the course of one orbit, relative to the distant stars, is given by:

\alpha = \alpha' + 2\pi = -2 \pi \sqrt{\beta}\Bigg( \left(1-\frac{3m}{r} \right)^{1/2} - 1 \Bigg).

With a first-order Taylor series we find

\alpha \approx \frac{3\pi m}{r}\sqrt{\beta} = \frac{3\pi m}{r}\sin(\theta).

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 author^[6] 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."^[7] An objection to the former interpretation is that the Thomas precession required has the wrong sign.

Notes

↑ Jean Eisenstaedt, Anne J. Kox (1988). Studies in the History of General Relativity. Birkhäuser. p. 42. ISBN 0-8176-3479-7.
↑ de Sitter, W (1916). "On Einstein's Theory of Gravitation and its Astronomical Consequences". Mon. Not. Roy. Astron. Soc. 77: 155–184. Bibcode:1916MNRAS..77..155D.
↑ Rindler, p. 254.
↑ Everitt, C.W.F.; Parkinson, B.W. (2009). "Gravity Probe B Science Results—NASA Final Report" (PDF). Retrieved 2009-05-02.
↑ http://einstein.stanford.edu/content/press_releases/SU/pr-aps-041807.pdf
↑ Rindler, Page 234
↑ Misner, Thorne, and Wheeler, Gravitation, p. 1118

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 NASA and Stanford University
Precession in Curved Space "The Geodetic Effect"
Geodetic Effect

Relativity

Special
relativity

Background	Principle of relativity Introduction to special relativity Special relativity

Foundations	Relative motion Frame of reference Speed of light Maxwell's equations

Formulation	Galilean relativity Galilean transformation Lorentz transformation

Consequences	Time dilation Relativistic mass Mass–energy equivalence Length contraction Relativity of simultaneity Relativistic Doppler effect Thomas precession Relativistic disks

Spacetime	Minkowski spacetime World line Spacetime diagrams Light cone

General
relativity

Background	Introduction Mathematical formulation Resources

Fundamental concepts	Special relativity Equivalence principle World line Riemannian geometry Minkowski diagram

Phenomena	Two-body problem Lenses Waves Frame-dragging Geodetic effect Event horizon Singularity Black hole

Equations	Linearized gravity Post-Newtonian formalism Einstein field equations Geodesic equation Friedmann equations ADM formalism BSSN formalism Hamilton–Jacobi–Einstein equation

Advanced theories	Kaluza–Klein Quantum gravity Brans–Dicke theory

Solutions	Schwarzschild Reissner–Nordström Gödel Kerr Kerr–Newman Kasner Taub–NUT Milne Robertson–Walker pp-wave van Stockum dust