Thomas precession

In physics the Thomas precession, named after Llewellyn Thomas, is a special relativistic correction to the precession of a gyroscope in a rotating non-inertial frame. It can be understood as a consequence of the fact that the space of velocities in relativity is hyperbolic, and so parallel transport of a vector (the gyroscope's angular velocity) around a circle (its linear velocity) leaves it pointing in a different direction.

History

Thomas precession in relativity was already known to Silberstein [L. Silberstein, The Theory of Relativity (MacMillan London 1914)] in 1914. But the only knowledge Thomas had of relativistic precession came from de Sitter's paper on the relativistic precession of the moon, first published in a book by Eddington [A.S. Eddington, The Mathematical Theory of Relativity (Cambridge 1924)] .

In 1925 Thomas relativistically recomputed the precessional frequency of the doublet separation in the fine structure of the atom. He thus found the missing factor 1/2 which came to be known as the Thomas half.

This discovery of the relativistic precession of the electron spin led to the understanding of the significance of the relativistic effect. The effect was therefore named Thomas precession

Physics

To find the Thomas precession we assume the system is in a rotating Minkowski space. The metric in cylindrical coordinates $r,\; phi\text{'},\; z$ is given by:

:

The frame rotates with a constant angular velocity $omega$ so $phi\; =\; phi\text{'}-omega\; t$ and the real metric is given by:

:

This is the so called canonical form of the metric.

:

From this we can read of the relativistic vector potential:

:

We can now calculate the rotational rate of the gyroscope using:

::$=\; frac\{omega\}\{(1\; -\; r^\{2\}omega^\{2\})\}$

In this equation $k$ is the spatial component of the metric. And using the convention $romega\; =\; v$ for the velocity we find:

:$Omega\; =\; frac\{v\}\{r(1\; -\; v^\{2\})\}$

So after one revolution the correction to the precession is given by:

:$alpha\; =\; Omega\; Delta\; au\; -\; 2pi\; =\; 2pi\; left\; [frac\{1\}\{sqrt\{1-v^\{2\}\; -\; 1\; ight]\; approx\; pi\; v^\{2\}$

Applications

In Quantum Mechanics

In quantum mechanics Thomas precession is a correction to the spin-orbit interaction, which takes into account the relativistic time dilation between the electron and the nucleus in hydrogenic atoms.

Basically, it states that spinning objects precess when they accelerate in special relativity because Lorentz boosts do not commute with each other.

In a Foucault pendulum

Thomas precession gives a correction to the precession of a Foucault pendulum. For a Foucault pendulum located in the city of Nijmegen in the Netherlands the correction is:

$omega\; approx\; 9.5\; cdot\; 10^\{-7\},\; mathrm\{arcseconds\}\; /\; mathrm\{day\}$

References

Textbooks

*

External links

* [http://www.mathpages.com/rr/s2-11/2-11.htm Mathpages article on Thomas Precession]
* [http://bohr.physics.berkeley.edu/classes/221/0708/notes/thomprec.pdf Alternate, detailed derivation of Thomas Precession] (by Robert Littlejohn)