Geodetic effect

The geodetic effect or the geodetic precession , also called de Sitter effect or de Sitter precession after Willem de Sitter , is an effect of the general theory of relativity on the axis of rotation of a top on an orbit in the gravitational field of a central mass. One observes the precession of a freely falling top in the gravitational field, there is no classical equivalent. Since this effect is not predicted by Newton's theory , it is a test of general relativity .

The somewhat smaller Lense-Thirring-Effect also acts on the axis of rotation of a top, but is caused by the rotation of the central mass. The geodetic effect, on the other hand, is solely explained by the presence of the central mass. Both effects together result in the general relativistic overall effect.

Theoretical background

The external gravitational field of a central mass is described in the general theory of relativity by the Schwarzschild metric , that is, the metric is in 4-dimensional spherical coordinates ${\displaystyle (ct,r,\theta ,\phi )}$

${\displaystyle (g_{\mu ,\nu })={\begin{pmatrix}1-{\frac {r_{\mathrm {S} }}{r}}&0&0&0\\0&-(1-{\frac {r_{\mathrm {S} }}{r}})^{-1}&0&0\\0&0&-r^{2}&0\\0&0&0&-r^{2}\sin ^{2}\theta \end{pmatrix}}}$

It stands for the speed of light and for the Schwarzschild radius of the central mass, i.e. , where is the gravitational constant and the mass of the field-generating central body. In some representations so-called natural units are used, i.e. That is, it is and often is set, which we do not want to do here. ${\displaystyle c}$ ${\displaystyle r_{\mathrm {S} }}$ ${\displaystyle \textstyle r_{\mathrm {S} }={\frac {2GM}{c^{2}}}}$ ${\displaystyle G}$ ${\displaystyle M}$ ${\displaystyle c=1}$ ${\displaystyle G=1}$

For the relativistic angular momentum vector , the equation of motion applies to the exclusive effect of gravity ${\displaystyle (s^{\mu })}$

${\displaystyle {\frac {\mathrm {D} s^{\mu }}{\mathrm {d} \tau }}=0}$      or      , ${\displaystyle {\frac {\mathrm {d} s^{\mu }}{\mathrm {d} \tau }}=-\Gamma _{\kappa \nu }^{\mu }s^{\kappa }u^{\nu }}$

where stands for the covariant derivative according to the proper time , the setting of which to zero expresses the absence of other forces. When translating back to ordinary derivatives, the Christoffel symbols appear as usual because of the space-time curvature , which are composed of the metric and its derivatives (see article Christoffel symbol ). after all, it is the four speed of the free falling top. ${\displaystyle \mathrm {D} }$ ${\displaystyle \tau }$ ${\displaystyle \Gamma _{\kappa \nu }^{\mu }}$ ${\displaystyle g_{\mu \nu }}$ ${\displaystyle (u^{\nu })}$

If one looks at a circular path in the plane , then with a constant . Note that due to the choice of 4-dimensional spherical coordinates, the last component is the change in time , i.e. the angular velocity of the body. Furthermore, many of the Christoffel symbols disappear for an orbit in this plane, which considerably simplifies the equations of motion. Ultimately, the solution of these equations leads to a vector, which changes periodically with an angular frequency , which in a first approximation with respect to powers of to ${\displaystyle \textstyle \theta ={\frac {\pi }{2}}}$ ${\displaystyle (u^{\nu })=(u^{0},0,0,\omega _{0})}$ ${\displaystyle \omega _{0}}$ ${\displaystyle \phi }$ ${\displaystyle (s^{\mu })}$ ${\displaystyle \omega }$ ${\displaystyle \textstyle {\frac {r_{\mathrm {S} }}{r}}}$

${\displaystyle \omega =\omega _{0}{\sqrt {1-{\frac {3r_{\mathrm {S} }}{2r}}}}}$

results. After the full period of rotation , the spin vector according to the above formula has not reached the starting position, but rather a phase difference of ${\displaystyle \textstyle {\frac {2\pi }{\omega _{0}}}}$ ${\displaystyle \omega }$

${\displaystyle (\omega _{0}-\omega ){\frac {2\pi }{\omega _{0}}}=2\pi -2\pi {\frac {\omega }{\omega _{0}}}=2\pi -2\pi {\sqrt {1-{\frac {3r_{\mathrm {S} }}{2r}}}}\cong \pi {\frac {3r_{\mathrm {S} }}{2r}}}$,

in the last step the Taylor series was broken off after the linear term. This is the geodetic precession per revolution.

Experimental confirmation

The geodetic precession of the moon

If the earth-moon system is viewed as a top that revolves around the sun, then the geodetic effect must have an influence on the plane of rotation of this top, i.e. on the plane of the lunar orbit. De Sitter recognized this as early as 1916 and predicted an effect of around 2 arc seconds per century. If one uses the Schwarzschild radius of the sun in the above formula and that is the distance between the earth and the sun, i.e. the distance between the top and the central mass, and multiplying this by 100, then, as de Sitter, gets about 2 arc seconds . Although this is many orders of magnitude less than the influence of the other planets (a full precession after 18.6 years ), this effect was proven with an accuracy of 0.6% using LLR data , which allow the lunar orbit to be measured with centimeter accuracy become. ${\displaystyle r_{\mathrm {S} }\cong 3\,\mathrm {km} }$ ${\displaystyle r\cong 150\cdot 10^{6}\,\mathrm {km} }$

Gravity Probe B Experiment

With the help of a satellite that was put into orbit and equipped with high-precision gyroscopes , the geodetic effect could be confirmed with an accuracy of within the framework of the Gravity Probe B experiment . ${\displaystyle 3\cdot 10^{-3}}$

PSR B1913 + 16

The PSR B1913 + 16 system is a binary star system that consists of two neutron stars . Because the axes of rotation are not perpendicular to the plane of the orbit, a geodetic precession also occurs here. This is reflected in the change in the pulsar spin and thus in the received radiation profile. The observations are compatible with the geodetic precession. Conversely, the variables observed allow conclusions to be drawn about a geometric model of PSR B1913 + 16. One of the predictions of the model is that this system will no longer be observable from 2025, as the radiation cone will then no longer hit the earth.

Individual evidence

1. ↑ Torsten Fließbach : General Theory of Relativity . Springer, 2016, ISBN 978-3-662-53105-1 , Chapter 29: Geodetic precession .
2. ↑ Wolfgang Rindler : Theory of Relativity: Special, General and Cosmological . Wiley VHC, 2006, ISBN 978-3-527-41173-3 , Chapter 11.13: de-sitter precession using rotating coordinates .
3. ↑ Matthias Bartelmann, Björn Feuerbacher, Timm Krüger, Dieter Lüst, Anton Rebhan, Andreas Wipf: Theoretical Physics . Springer, 2015, ISBN 978-3-642-54617-4 , p. 345: Geodetic precession and Lense-Thirring effect
4. ↑ Torsten Fließbach: General Theory of Relativity . Springer, 2016, ISBN 978-3-662-53105-1 , Chapter 29: Geodetic precession (The calculations in this textbook are a bit more general, the article only deals with the Schwarzschild metric).
5. ^ ^{A b} Clifford M. Will : The Confrontation between General Relativity and Experiment . Chapter 4.4.2. Geodetic Precession , arxiv : 1403.7377 .
6. ↑ Jürgen Müller, Liliane Biskupek, Franz Hofmann, Enrico Mai: Lunar Laser Ranging and Relativity . In Frontiers in Relativistic Celestial Mechanics . de Gruyter, 2014, ISBN 978-3-11-034545-2 , chapter 4.4. Geodesic Precession .
7. ^ Michael Kramer : Determination of the Geometry of the PSR B1913 + 16 System by Geodetic Precession . In: The Astrophysical Journal . tape 509 , no. 2 , December 20, 1998, pp. 856-860 , doi : 10.1086 / 306535 .