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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 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, 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 \mathrm ds^2 =\left(1-\frac\right)c^2\,\mathrm dt^2 -\left(1+\frac\right)\,\mathrm d\sigma^2 +4G\epsilon_S^k \frac c \,\mathrm dt\,\mathrm dx^j, where the symbols represent: * \mathrm ds^2 the metric, * \mathrm d\sigma^2 = \mathrm dx^2 + \mathrm dy^2 + \mathrm dz^2 = \mathrm dr^2 + r^2\mathrm d\theta^2 + r^2\sin^2\theta\,\mathrm d\varphi^2 the flat-space line element in three dimensions, * r = \sqrt the "radial" position of the observer, * c the speed of light, * G the gravitational constant, * \epsilon_ the completely antisymmetric Levi-Civita symbol, * M = \int T^ \,\mathrm d^3x the mass of the rotating body, * S_k = \int \epsilon_x^l T^ \,\mathrm d^3x the angular momentum of the rotating body, * T^ 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 r\frac +2\frac\frac =0, where * t is the time, * \varphi is the azimuthal angle (longitudinal angle), * 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: r\frac +2\omega\frac = 0, where \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 dr/dt = 0, 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 \Omega^k = \frac \left ^k - 3 \frac\right where: * \Omega is the angular velocity of the precession, a vector, and \Omega_k one of its components, * S_k the angular momentum of the spinning body, as before, * S \cdot x 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 L^i, then it precesses as \frac = \epsilon_ \Omega^j L^k. The rate of precession is given by \epsilon_ \Omega^k = \Gamma_, where \Gamma_ 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 h_ = g_ - \eta_ can be read off from the Lense–Thirring metric given above, where ds^2 = g_ \,dx^\mu \,dx^\nu, and \eta_ \,dx^\mu \,dx^\nu = c^2 \,dt^2 - dx^2 - dy^2 - dz^2. In this approach, one writes the linearized metric, given in terms of the gravitomagnetic potentials \phi and \vec is h_ = \frac and h_ = \frac, where \phi = \frac is the gravito-electric potential, and \vec = \frac \vec \times \vec is the gravitomagnetic potential. Here \vec is the 3D spatial coordinate of the observer, and \vec is the angular momentum of the rotating body, exactly as defined above. The corresponding fields are \vec = -\nabla\phi - \frac for the gravitoelectric field, and \vec = \vec \times \vec is the gravitomagnetic field. It is then a matter of substitution and rearranging to obtain \vec = -\frac \left \vec - 3 \frac\right/math> as the gravitomagnetic field. Note that it is half the Lense–Thirring precession frequency. In this context, Lense–Thirring precession can essentially be viewed as a form of Larmor precession. 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 \vec = m \vec + m \frac \times \vec, where m is the mass of a test particle moving with velocity \vec. This can be used in a straightforward way to compute the classical motion of bodies in the gravitomagnetic field. For example, a radially infalling body will have a velocity \vec = -\hat \,dr/dt; direct substitution yields the Coriolis term given in a previous section.


Example: Foucault's pendulum

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 R, the moment of inertia is given by 2MR^2/5, so that the absolute value of the angular momentum S is \Vert S\Vert = 2MR^2\omega/5, with \omega the angular speed of the spinning ball. The direction of the spin of the Earth may be taken as the ''z'' axis, whereas the axis of the pendulum is perpendicular to the Earth's surface, in the radial direction. Thus, we may take \hat \cdot \hat = \cos\theta, where \theta is the
latitude In geography, latitude is a geographic coordinate system, geographic coordinate that specifies the north-south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from −90° at t ...
. Similarly, the location of the observer r is at the Earth's surface R. This leaves rate of precession is as \Omega_\text = \frac \frac \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 \Omega_\text = 2.2 \cdot 10^ \text/\text. 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. The above does not include the de Sitter precession; it would need to be added to get the total relativistic precessions on Earth.


Experimental verification

The Lense–Thirring effect, and the effect of frame dragging in general, continues to be studied experimentally. 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 hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
s and
neutron star A neutron star is the gravitationally collapsed Stellar core, core of a massive supergiant star. It results from the supernova explosion of a stellar evolution#Massive star, massive star—combined with gravitational collapse—that compresses ...
s, 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 In astronomy and planetary science, a magnetosphere is a region of space surrounding an astronomical object in which charged particles are affected by that object's magnetic field. It is created by a celestial body with an active interior Dynamo ...
, auroras and mass composition. 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. Results from astrophysical settings are presented after the following section.


Astrophysical setting

A star orbiting a spinning supermassive black hole experiences Lense–Thirring precession, causing its orbital line of nodes to precess at a rate \frac = \frac = \frac, where * ''a'' and ''e'' are the
semimajor axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the long ...
and eccentricity of the orbit, * ''M'' is the mass of the black hole, * ''χ'' is the dimensionless spin parameter (0 < ''χ'' < 1). The precessing stars also exert a torque back on the black hole, causing its spin axis to precess, at a rate \frac = \frac\sum_j \frac, where * L''j'' is the angular momentum of the ''j''th star, * ''a''''j'' and ''e''''j'' 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 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 Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
forces the gas into a new plane, aligned with the black hole's spin axis.


Astrophysical tests

The orientation of an astrophysical jet can be used as evidence to deduce the orientation of an accretion disk; a rapidly changing jet orientation suggests a reorientation of the accretion disk, as described above. Exactly such a change was observed in 2019 with the black hole X-ray binary in V404 Cygni.
Pulsar A pulsar (''pulsating star, on the model of quasar'') is a highly magnetized rotating neutron star that emits beams of electromagnetic radiation out of its Poles of astronomical bodies#Magnetic poles, magnetic poles. This radiation can be obse ...
s 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 A white dwarf is a Compact star, stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very density, dense: in an Earth sized volume, it packs a mass that is comparable to the Sun. No nuclear fusion takes place i ...
, 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. 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 The Milky Way or Milky Way Galaxy is the galaxy that includes the Solar System, with the name describing the #Appearance, galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars in other arms of the galax ...
, using the GRAVITY instrument of the Very Large Telescope. The star orbits with a period of 16 years, and it should be possible to constrain the angular momentum of the black hole by observing the star over two to three periods (32 to 48 years).


See also

* Gravity Probe B


References


External links


(German) explanation of Thirring–Lense effect
has pictures for the satellite example. {{DEFAULTSORT:Lense-Thirring Precession Precession General relativity