Lense–Thirring precession

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,^[1] the effect should be renamed as Einstein-Thirring-Lense 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.

The status of experimental verification of the Lense-Thirring effect, and of frame dragging in general, is given in the frame dragging article.

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]

$ds^2=\left(1-\frac{2GM}{rc^2}\right)c^2 dt^2 - \left(1+\frac{2GM}{rc^2}\right)d\sigma^2 +4G\epsilon_{ijk}S^k \frac{x^i}{c^3r^3} \; cdt \;dx^j$

The symbols are:

• $ds^2$ the metric
• $d\sigma^2= dx^2 +dy^2 + dz^2 = dr^2+r^2 d \theta\ ^2+ r^2 \sin^2 \theta\ d \varphi\ ^2$ the flat-space line element in three dimensions
• $r=\sqrt{x^2+y^2+z^2}$ the "radial" position of the observer
• $c$ the speed of light
• $G$ the gravitational constant
• $\epsilon_{ijk}$ the completely antisymmetric Levi-Civita symbol
• $M=\int T^{00} d^3x$ the mass of the rotating body
• $S_k=\int \epsilon_{klm}x^lT^{m0}d^3x$ the angular momentum of the rotating body.
• $T^{\mu\nu}$ 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]

$0 = r\frac{d^2 \varphi}{dt^2}+2\frac{GJ}{c^2r^3} \frac{dr}{dt}$

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

$0=r\frac{d^2 \varphi}{dt^2}+2\omega\frac{dr}{dt}$

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:^[3]

$\Omega^k = \frac{G}{c^2 r^3}\left[S^k - 3\frac{(S \cdot x) x^k}{r^2}\right]$

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

$\frac{dL^i}{dt} = \epsilon_{ijk} \Omega^j L^k$

The rate of precession is given by

$\epsilon_{ijk} \Omega^k = \Gamma_{ij0}$

Where $\Gamma_{ij0}$ is the Christoffel symbol for the above metric. Misner, Thorne, Wheeler, op. cit.^[3] provide hints on how to most easily calculate this.

Gravitomagnetic analysis

It is popular in some circles to use the gravitomagnetic 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 $h_{\mu\nu}=g_{\mu\nu}-\eta_{\mu\nu}$ can be read off from the Lense-Thirring metric given above, where $ds^2=g_{\mu\nu}dx^\mu dx^\nu$ and $\eta_{\mu\nu}dx^\mu dx^\nu=c^2dt^2-dx^2-dy^2-dz^2$. In this approach, one writes the linearized metric, given in terms of the gravitomagneitc potentials $\phi$ and $\vec{A}$ is

$h_{00} = \frac{-2\phi}{c^2}$

and

$h_{0i} = \frac{2A_i}{c^2}$

where

$\phi = \frac{-GM}{r}$

is the gravito-electric potential, and

$\vec{A} = \frac {G}{r^3c} \vec{S} \times \vec{r}$

is the gravitomagnetic potential. Here, $\vec{r}$ is the 3D spatial coordinate of the observer, and $\vec{S}$ is the angular momentum of the rotating body, exactly as defined above. The corresponding fields are

$\vec{E} = -\nabla\phi - \frac{1}{2c} \frac{\partial \vec{A}}{\partial t}$

for the gravito-electric field, and

$\vec{B} = \frac{1}{2} \vec{\nabla} \times \vec{A}$

is the gravitomagnetic field. It is then a matter of plugging and chugging to obtain

$\vec{B} = -\frac{G}{2cr^3} \left[ \vec{S} - 3 \frac{(\vec{S}\cdot\vec{r}) \vec{r}}{r^2}\right]$

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 gyromagnetic ratio is (curiously!) two.

The gravitomagnetic analog of the Lorentz force is given by

$\vec{F} = m \vec{E} + 4m \vec{v}\times\vec{B}$

where $m$ is the velocity of a test particle moving with velocity $\vec{v}$. 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 $\vec{v}=-\hat{r}dr/dt$; 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 pendulm is perpendicular to the Earth's surface, in the radial direction. Thus, we may take $\hat{z} \cdot \hat{r} = \cos\theta$ where $\theta$ is the latitude. Similarly, the location of the observer $r$ is at the Earth's surface $R$. This leaves rate of precession is as

$\Omega_\text{LT} = \frac{2}{5}\frac{G M \omega}{c^2 R}\cos\theta.$

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 of:

$\Omega_\text{LT}=2.2 \cdot 10^{-4} \text{ arcseconds}/\text{day}.$

At this rate a Foucault pendulum would have to oscillate for more than 16000 years to precess 1 degree. Despite being quite small, notice that it is still two orders of magnitude large 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.

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^[4]

$\frac{d\Omega}{dt} = \frac{2GS}{c^2a^3(1-e^2)^{3/2}} = \frac{2G^2M^2\chi}{c^3a^3(1-e^2)^{3/2}}$

where

• a and e are the semimajor axis and eccentricity of the orbit
• M is the mass of the black hole
• χ is the dimensionless spin parameter (0<χ<1).

Lense–Thirring precession of stars near the Milky Way supermassive black hole is expected to be measurable within the next few years.^[5]

The precessing stars also exert a torque back on the black hole, causing its spin axis to precess, at a rate^[6]

$\frac{d\boldsymbol{S}}{dt} = \frac{2G}{c^2}\sum_j \frac{\boldsymbol{L}_j\times\boldsymbol{S}}{a_j^3(1-e_j^2)^{3/2}}$

where

• L_j is the angular momentum of the j'th star
• (a_j,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 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 (the "Bardeen-Petterson effect").^[7]

References

1. Pfister, H. (November 2007). "On the history of the so-called Lense–Thirring effect". General Relativity and Gravitation 39 (11): 1735–1748. Bibcode:2007GReGr..39.1735P. doi:10.1007/s10714-007-0521-4.
2. Ronald Adler, Maurice Bazin, Menahem Schiffer, Introduction to General Relativity (1965) McGraw-Hill Book Company ISBN 0-07-000423-4 (See section 7.7)
3. Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation (1973) W. H,. Freeman ISBN 0-7167-0334-3 (See chapter 19)
4. Merritt, David (2013). Dynamics and Evolution of Galactic Nuclei. Princeton, NJ: Princeton University Press. p. 169. ISBN 9781400846122.
5. Eisenhauer, Frank; et al. (March 2011). "GRAVITY: Observing the Universe in Motion". The Messenger 143: 16–24. Bibcode:2011Msngr.143...16E.
6. Merritt, David; Vasiliev, Eugene (November 2012). "Spin evolution of supermassive black holes and galactic nuclei". Physical Review D 86 (10): 102002. arXiv:1205.2739. Bibcode:2012PhRvD..86b2002A. doi:10.1103/PhysRevD.86.022002.
7. Bardeen, James M.; Petterson, Jacobus A. (January 1975). "The Lense-Thirring Effect and Accretion Disks around Kerr Black Holes". The Astrophysical Journal Letters 195: L65. Bibcode:1975ApJ...195L..65B. doi:10.1086/181711.

External links

• (German) explanation of Thirring-Lense effect Has pictures for the satellite example.