In general relativity, Lense–Thirring precession or the Lense–Thirring effect (Austrian German: [ˈlɛnsɛ ˈtɪrɪŋ]; 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 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
The gravitational field of a spinning spherical body of constant density was studied by Lense and Thirring in 1918, in the weak-field approximation.
where the symbols represent: The above is the weak-field approximation of the full solution of the Einstein equations for a rotating body, known as the Kerr metric, which, due to the difficulty of its solution, was not obtained until 1965.
One way is to solve for geodesics; these will then exhibit a Coriolis force-like term, except that, in this case (unlike the standard Coriolis force), the force is not fictional, but is due to frame dragging induced by the rotating body.
where The above can be compared to the standard equation for motion subject to the Coriolis force:
where: That is, if the gyroscope's angular momentum relative to the fixed stars is
Gravitation by Misner, Thorne, and Wheeler[3] provides hints on how to most easily calculate this.
It is popular in some circles to use the gravitoelectromagnetic approach to the linearized field equations.
The reason for this popularity should be immediately evident below, by contrasting it to the difficulties of working with the equations above.
In this approach, one writes the linearized metric, given in terms of the gravitomagnetic potentials
The factor of 1/2 suggests that the correct gravitomagnetic analog of the g-factor is two.
This factor of two can be explained completely analogous to the electron's g-factor by taking into account relativistic calculations.
The gravitomagnetic analog of the Lorentz force in the non-relativistic limit is given by
This can be used in a straightforward way to compute the classical motion of bodies in the gravitomagnetic field.
; direct substitution yields the Coriolis term given in a previous section.
To get a sense of the magnitude of the effect, the above can be used to compute the rate of precession of Foucault's pendulum, located at the surface of the Earth.
For a solid ball of uniform density, such as the Earth, of radius
At this rate a Foucault pendulum would have to oscillate for more than 16000 years to precess 1 degree.
Despite being quite small, it is still two orders of magnitude larger than Thomas precession for such a pendulum.
There are two basic settings for experimental tests: direct observation via satellites and spacecraft orbiting Earth, Mars or Jupiter, and indirect observation by measuring astrophysical phenomena, such as accretion disks surrounding black holes and neutron stars, or astrophysical jets from the same.
The Juno spacecraft's suite of science instruments will primarily characterize and explore the three-dimensional structure of Jupiter's polar magnetosphere, auroras and mass composition.
[4] As Juno is a polar-orbit mission, it will be possible to measure the orbital frame-dragging, known also as Lense–Thirring precession, caused by the angular momentum of Jupiter.
where 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.
Exactly such a change was observed in 2019 with the black hole X-ray binary in V404 Cygni.
[9] Pulsars emit rapidly repeating radio pulses with extremely high regularity, which can be measured with microsecond precision over time spans of years and even decades.
A 2020 study reports the observation of a pulsar in a tight orbit with a white dwarf, to sub-millisecond precision over two decades.
The precise determination allows the change of orbital parameters to be studied; these confirm the operation of the Lense–Thirring effect in this astrophysical setting.
[10] It may be possible to detect the Lense–Thirring effect by long-term measurement of the orbit of the S2 star around the supermassive black hole in the center of the Milky Way, using the GRAVITY instrument of the Very Large Telescope.