# Lense–Thirring precession

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In general relativity, Lense–Thirring precession or the Lense–Thirring effect (named after Josef Lense and Hans Thirring) is a relativistic correction to the precession of a gyroscope near a large rotating mass such as the Earth. It is a gravitomagnetic frame-dragging effect. According to a recent historical analysis by Pfister,[1] the effect should be renamed as Einstein-Thirring-Lense effect. It is a prediction of general relativity consisting of secular precessions of the longitude of the ascending node and the argument of pericenter of a test particle freely orbiting a central spinning mass endowed with angular momentum $S$.

The difference between de Sitter precession and the Lense–Thirring effect is that the de Sitter effect is due simply to the presence of a central mass, whereas the Lense–Thirring effect 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.

## Derivation

Before we can calculate this we want to find the gravitomagnetic field. The gravitomagnetic field in the equatorial plane of a rotating star:

$\boldsymbol{B}=\frac{3}{5}R^{2}q\Big(\boldsymbol{\omega}\cdot\boldsymbol{r}\frac{\boldsymbol{r}}{r^5}-\frac{1}{3}\frac{\boldsymbol{\omega}}{r^{3}}\Big).$

If we use then:

$\boldsymbol{\omega}=-4\int\frac{\rho\boldsymbol{u}\,dV}{r}.$

We get:

$\boldsymbol{B}=\frac{12}{5}R^2 q\Big(\boldsymbol{\omega}\cdot\boldsymbol{r}\frac{\boldsymbol{r}}{r^5}-\frac{1}{3}\frac {\boldsymbol{\omega}}{r^3}\Big).$

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 $\theta$ is the latitude:

$\boldsymbol{B} = - \left(\frac{1}{3}\frac {\boldsymbol{\omega}}{r^3}\cos\theta\right).$

The absolute value of this would then be:

$\boldsymbol{B}=-\frac{4}{5}\frac {\boldsymbol{\omega} m R^2}{r^3}\cos\theta.$

This is the gravitomagnetic field. We know there is a strong relation between the angular velocity in the local inertial system, $\boldsymbol{\Omega}_{\text{LIF}}$, and the gravitomagnetic field:

Therefore 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:

$\Omega_\text{LT} = -\frac{2}{5}\frac{G m \omega}{c^2 R}\cos\theta.$

As an example the latitude of the city of Nijmegen in the Netherlands is used for reference. This latitude gives a value for the Lense–Thirring precession of:

$\Omega_\text{LT}=-2.2 \cdot 10^{-4} \text{ arcseconds}/\text{day}.$

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:

$\Omega_\text{rel} = \frac {3\pi G m}{c^2 r}.$

At this rate a Foucault pendulum would have to oscillate for more than 16000 years to precess 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 criticized 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 became 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). 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 precesses, at a very small rate, in the same direction as the rotation of the Earth.

## Astrophysical importance

A star orbiting a spinning supermassive black hole experiences Lense–Thirring precession, causing its orbital line of nodes to precess at a rate[2]

$\frac{d\Omega}{dt} = \frac{2GS}{c^2a^3(1-e^2)^{3/2}} = \frac{2G^2M^2\chi}{c^3a^3(1-e^2)^{3/2}}$

where

• a and e are the semimajor axis and eccentricity of the orbit
• M is the mass of the black hole
• χ is the dimensionless spin parameter (0<χ<1).

Lense–Thirring precession of stars near the Milky Way supermassive black hole is expected to be measurable within the next few years.[3]

The precessing stars also exert a torque back on the black hole, causing its spin axis to precess, at a rate[4]

$\frac{d\boldsymbol{S}}{dt} = \frac{2G}{c^2}\sum_j \frac{\boldsymbol{L}_j\times\boldsymbol{S}}{a_j^3(1-e_j^2)^{3/2}}$

where

• Lj is the angular momentum of the j'th star
• (aj,ej) are its semimajor axis and eccentricity.

A gaseous accretion disk that is tilted with respect to a spinning black hole will experience Lense–Thirring precession, at a rate given by the above equation, after setting e=0 and identifying a with the disk radius. Because the precession rate varies with distance from the black hole, the disk will "wrap up", until viscosity forces the gas into a new plane, aligned with the black hole's spin axis (the "Bardeen-Petterson effect").[5]

## References

1. ^ Pfister, H. (November 2007). "On the history of the so-called Lense–Thirring effect". General Relativity and Gravitation 39 (11): 1735–1748. Bibcode:2007GReGr..39.1735P. doi:10.1007/s10714-007-0521-4.
2. ^ Merritt, David (2013). Dynamics and Evolution of Galactic Nuclei. Princeton, NJ: Princeton University Press. p. 169. ISBN 9781400846122.
3. ^ Eisenhauer, Frank; et al. (March 2011). "GRAVITY: Observing the Universe in Motion". The Messenger 143: 16–24. Bibcode:2011Msngr.143...16E.
4. ^ Merritt, David; Vasiliev, Eugene (November 2012). "Spin evolution of supermassive black holes and galactic nuclei". Physical Review D 86 (10): 102002. arXiv:1205.2739. Bibcode:2012PhRvD..86b2002A. doi:10.1103/PhysRevD.86.022002.
5. ^ Bardeen, James M.; Petterson, Jacobus A. (January 1975). "The Lense-Thirring Effect and Accretion Disks around Kerr Black Holes". The Astrophysical Journal Letters 195: L65. Bibcode:1975ApJ...195L..65B. doi:10.1086/181711.