Lense-Thirring precession

Lense-Thirring precession is a general relativistic correction to the precession of a gyroscope near a large rotating mass such as the Earth. Before we can calculate this we want to find the Magnetic Field. The Magnetic Field in the equatorial plane of a rotating star:

:

If we use then:

:

We get:

:

When we look at Foucault's pendulum we only have to take the perpendicular-component to the Earth's surface. This means the first part of the equation cancels, where the radius $r$ equals $R$ and $heta$ is the latitude:

:

The absolute value of this would then be:

:

This is the gravimetric field. We know there is a strong relation between the angular velocity in the local inertial system, , and the gravimetric field:

We know the Earth introduces a precession on all gyroscopes in a stationary system surrounding the Earth. This precession is called the Lense-Thirring precession with a magnitude:

:$Omega\_\{LT\}=-frac\{2\}\{5\}frac\{G\; m\; omega\}\{c^\{2\}\; R\}cos\; heta$

We used the latitude of the city of Nijmegen in the Netherlands for reference. Here this effect gives us the value for the Lense-Thirring precession:

:$Omega\_\{LT\}=-2.2\; cdot\; 10^\{-4\}\; arcseconds/\; day$

The total relativistic precessions on Earth is given by the sum of the De sitter precession and the Lense-Thirring precession. This can be calculated by:

:$Omega\_\{rel\}=frac\; \{3pi\; G\; m\}\{c^\{2\}r\}$

This means Foucault's Pendulum should oscillate for more than 16000 years to precede 1 degree.

Intuitive explanation:

According to Newtonian Mechanics, a body rotates or does not rotate relative to an absolute space. This absolute space is fixed. Ernst Mach criticised this idea, and proposed that the absolute space does not exist, it should be defined by the bodies that exist in the universe. So when we see a body rotating it would be rotating relative to the rest of the bodies in the universe. This idea that the bodies define in some way the reference frames got incarnated in the relativistic theory of gravitation, proposed by Albert Einstein in 1915. As a consequence, the rotation of nearby objects affects the rotation of other objects. This is the Lense-Thirring effect.

As an example of the Lense-Thirring effect consider the following:

Think of a satellite rotating around the earth. According to Newtonian Mechanics, if there are no external forces applied to the satellite but the gravitation force exerted by the earth, it will keep rotating in the same plane forever. This will be the case whether the earth rotates around its axis or not. But with General Relativity, we find that the rotation of the earth exerts a force to the satellite, so that the rotation plane of the satellite rotates, by a very small amount, in the same direction as the rotation of the earth.

See also

*Frame-dragging

External links

* [http://homepage.univie.ac.at/Franz.Embacher/Rel/Thirring-Lense/ThirringLense1.pdf (German) explanation of Thirring-Lense effect] Has pictures for the satellite example.