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In general relativity, Lense–Thirring precession or the Lense–Thirring effect (Austrian German: [ˈlɛnsə 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 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 ${\displaystyle S}$.

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.

According to a 2007 historical analysis by Herbert Pfister,^[1] the effect should be renamed the Einstein–Thirring–Lense effect.

The 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^[2][3]

$${\displaystyle \mathrm {d} s^{2}=\left(1-{\frac {2GM}{rc^{2}}}\right)c^{2}\,\mathrm {d} t^{2}-\left(1+{\frac {2GM}{rc^{2}}}\right)\,\mathrm {d} \sigma ^{2}+4G\epsilon _{ijk}S^{k}{\frac {x^{i}}{c^{3}r^{3}}}c\,\mathrm {d} t\,\mathrm {d} x^{j},}$$

• ${\displaystyle \mathrm {d} s^{2}}$ the metric,
• ${\displaystyle \mathrm {d} \sigma ^{2}=\mathrm {d} x^{2}+\mathrm {d} y^{2}+\mathrm {d} z^{2}=\mathrm {d} r^{2}+r^{2}\mathrm {d} \theta ^{2}+r^{2}\sin ^{2}\theta \,\mathrm {d} \varphi ^{2}}$ the flat-space line element in three dimensions,
• ${\textstyle r={\sqrt {x^{2}+y^{2}+z^{2}}}}$ the "radial" position of the observer,
• ${\displaystyle c}$ the speed of light,
• ${\displaystyle G}$ the gravitational constant,
• ${\displaystyle \epsilon _{ijk}}$ the completely antisymmetric Levi-Civita symbol,
• ${\textstyle M=\int T^{00}\,\mathrm {d} ^{3}x}$ the mass of the rotating body,
• ${\textstyle S_{k}=\int \epsilon _{klm}x^{l}T^{m0}\,\mathrm {d} ^{3}x}$ the angular momentum of the rotating body,
• ${\displaystyle T^{\mu \nu }}$ 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.

The 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^[2]

$${\displaystyle r{\frac {\mathrm {d} ^{2}\varphi }{\mathrm {d} t^{2}}}+2{\frac {GJ}{c^{2}r^{3}}}{\frac {\mathrm {d} r}{\mathrm {d} t}}=0,}$$

• ${\displaystyle t}$ is the time,
• ${\displaystyle \varphi }$ is the azimuthal angle (longitudinal angle),
• ${\displaystyle J=\Vert S\Vert }$ 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:

$${\displaystyle r{\frac {\mathrm {d} ^{2}\varphi }{\mathrm {d} t^{2}}}+2\omega {\frac {\mathrm {d} r}{\mathrm {d} t}}=0,}$$

where ${\displaystyle \omega }$ 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 ${\displaystyle dr/dt=0}$Zdroj:https://en.wikipedia.org?pojem=Lense–Thirring_precession
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