In
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, 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. 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 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.
According to a 2007 historical analysis by Herbert Pfister, the effect should be renamed the
Einstein–Thirring–Lense effect.
Lense–Thirring metric
The gravitational field of a spinning spherical body of constant density was studied by Lense and Thirring in 1918, in the
weak-field approximation. They obtained the metric
where the symbols represent:
*
the
metric,
*
the flat-space
line element in three dimensions,
*
the "radial" position of the observer,
*
the
speed of light,
*
the
gravitational constant,
*
the completely antisymmetric
Levi-Civita symbol,
*
the mass of the rotating body,
*
the angular momentum of the rotating body,
*
the
energy–momentum tensor.
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.
Coriolis term
The frame-dragging effect can be demonstrated in several ways. 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. So, for example, an (instantaneously) radially infalling geodesic at the equator will satisfy the equation
[
where
* is the time,
* is the azimuthal angle (longitudinal angle),
* is the magnitude of the angular momentum of the spinning massive body.
The above can be compared to the standard equation for motion subject to the Coriolis force:
where is the angular velocity of the rotating coordinate system. Note that, in either case, if the observer is not in radial motion, i.e. if , there is no effect on the observer.
]
Precession
The frame-dragging effect will cause a gyroscope to precess. The rate of precession is given by[
where:
* is the angular velocity of the precession, a vector, and one of its components,
* the angular momentum of the spinning body, as before,
* the ordinary flat-metric inner product of the position and the angular momentum.
That is, if the gyroscope's angular momentum relative to the fixed stars is , then it precesses as
The rate of precession is given by
where is the Christoffel symbol for the above metric. '' Gravitation'' by Misner, Thorne, and Wheeler][ provides hints on how to most easily calculate this.
]
Gravitoelectromagnetic analysis
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. The linearized metric can be read off from the Lense–Thirring metric given above, where , and . In this approach, one writes the linearized metric, given in terms of the gravitomagnetic potentials and is
and
where
is the gravito-electric potential, and
is the gravitomagnetic potential. Here is the 3D spatial coordinate of the observer, and is the angular momentum of the rotating body, exactly as defined above. The corresponding fields are
for the gravitoelectric field, and
is the gravitomagnetic field. It is then a matter of substitution and rearranging to obtain