Lense-Thirring precession
Encyclopedia
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, the effect should be renamed as Einstein-Thirring-Lense effect. It is a prediction of general relativity
consisting of secular precession
s 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
.
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.
field. The gravitomagnetic 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
equals 
and 
is the latitude:
The absolute value of this would then be:
This is the gravitomagnetic field. We know there is a strong relation between the angular velocity in the local inertial system,
, 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:
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:
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:
At this rate a Foucault pendulum
would have to oscillate for more than 16000 years to precede 1 degree.
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.
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, Lense–Thirring precession or the Lense–Thirring effect (named after Josef Lense
Josef Lense
Josef Lense was an Austrian physicistIn 1914 Lense got his doctorate under Samuel Oppenheim. From 1927-28 he was Professor ordinarius and from 1928-1946 Professor extraordinarius for applied mathematics at the Technical University of Munich...
and Hans Thirring
Hans Thirring
Hans Thirring was an Austrian theoretical physicist, professor, and father of the physicist Walter Thirring....
) is a relativistic
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
correction to the precession
Precession
Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...
of a gyroscope
Gyroscope
A gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum. In essence, a mechanical gyroscope is a spinning wheel or disk whose axle is free to take any orientation...
near a large rotating mass such as the Earth. It is a gravitomagnetic
Gravitomagnetism
Gravitomagnetism , refers to a set of formal analogies between Maxwell's field equations and an approximation, valid under certain conditions, to the Einstein field equations for general relativity. The most common version of GEM is valid only far from isolated sources, and for slowly moving test...
frame-dragging
Frame-dragging
Einstein's general theory of relativity predicts that non-static, stationary mass-energy distributions affect spacetime in a peculiar way giving rise to a phenomenon usually known as frame-dragging...
effect. According to a recent historical analysis by Pfister, the effect should be renamed as Einstein-Thirring-Lense effect. It is a prediction of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
consisting of secular precession
Precession
Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...
s 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
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
.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 gravitomagneticGravitomagnetism
Gravitomagnetism , refers to a set of formal analogies between Maxwell's field equations and an approximation, valid under certain conditions, to the Einstein field equations for general relativity. The most common version of GEM is valid only far from isolated sources, and for slowly moving test...
field. The gravitomagnetic 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
equals and is the latitude:The absolute value of this would then be:
This is the gravitomagnetic field. We know there is a strong relation between the angular velocity in the local inertial system,
, 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:
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:
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:
At this rate a Foucault pendulum
Foucault pendulum
The Foucault pendulum , or Foucault's pendulum, named after the French physicist Léon Foucault, is a simple device conceived as an experiment to demonstrate the rotation of the Earth. While it had long been known that the Earth rotated, the introduction of the Foucault pendulum in 1851 was the...
would have to 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 MachErnst Mach
Ernst Mach was an Austrian physicist and philosopher, noted for his contributions to physics such as the Mach number and the study of shock waves...
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.
External links
- (German) explanation of Thirring-Lense effect Has pictures for the satellite example.