# Thomas Precession

1. Sep 1, 2011

### utesfan100

I have been reading about Thomas precession and have encountered many papers that appear to give conflicting accounts of its magnitude and direction.

In particular I am considering the portion of the DeSitter precession that is accounted for by the Thomas precession. I have found values ranging from -1/6 to 1/6 to 1/2. The value of 1/2 appears to also require applying the equivalence principle to the energy of the satellite, rather than to rest mass.

I have considered the Laplace-Runge-Lenz vector, and the angle this makes with the radial vector. This angle rotates and depends only on orbital parameters at the satellite. This angular velocity should experience time dilation relative to the orbital velocity. This produced a precession of $\gamma-1$ times the orbital period, in the direction of rotation. This is 1/6 the DeSitter precession.

Often this effect is discussed in terms of quantum mechanics, specifically the orbit of electrons around a hydrogen nucleus. Here a negative value is derived. Electrons are thought to be in orbits where the wave form resonates. This would require the waveform to conform to the actual circumference, requiring a precession with the inverse of what I found above from the electrons perspective. To a first order approximation this changes the sign of the highest order term, causing a pseudo force adjustment of -1/6 the DeSitter precession for classical approximations.

Is this a correct way to understand this effect?

2. Sep 3, 2011

### Bill_K

utesfan100, Unfortunately your confusion is well-founded - many of the available descriptions of Thomas precession are naive, unclear, contradictory and even incorrect. Almost any rotation you encounter in relativity has been at one time labeled as due to a Thomas precession, or partly so. You seem to have touched all the bases on the subject, but maybe I can make a few remarks:

Thomas precession - a kinematic effect in special relativity resulting from the transport of a local inertial frame along a nongeodesic curve in flat space.

DeSitter precession - a dynamic effect in general relativity resulting from the transport of a local inertial frame along a geodesic in curved space.

These two are unrelated. The correct value of the Thomas precession for a particle in a circular path of radius r and angular velocity ω is given in MTW: ωThomas = (γ-1)ω ≈ ½ v2ω in a retrograde sense.

This statement is widespread and especially unfortunate. The picture of an electron traveling in an orbit whose circumference is an integral number of wavelengths was a hypothesis of the pre-quantum Bohr model of the atom. The Bohr model predicted an L·S term in the fine structure of the energy levels that was twice the observed magnitude. Thomas claimed his precession effect could get the right magnitude. His argument was no longer relevant after the development of the Dirac equation.

3. Sep 4, 2011

### utesfan100

Is this from the frame of a distant observer or the electron?

Suppose we have an electron orbiting a positively charged sphere in a large vacuum chamber/Faraday cage. Further, let's suppose the gravitational DeSitter precession is much smaller than the electrical Thomas precession.

One could define an electrical Laplace-Runge-Lenz vector for this system. If the Thomas Precession is from the electron's perspective, the results are equivalent to my time dilation estimate.

...

I suppose it would be more instructive to actually use the square approximation and play with it myself to see the motions :)

If I am right, the angels measured between one edge frame and either of its adjacent frames should be less than a right angle, adding up to less than a full circle over a circuit. Taken to the continuous limit, once this gap is overcome by the moving frame, it would have precessed in the prograde direction by this gap angle.

Either way, finding the speed where one polygon has the apparent angle of another polygon would provide an elegant example of this effect. Can the square frame have the local angle of a triangle or a pentagon?

Doesn't the Dirac equation formulate the idea that a wave needs to be coherent over its motion within special relativity in a formal yet elegant tensor notation?

I think that is what I meant by "Electrons are thought to be in orbits where the wave form resonates." I don't want to think about this too hard though, as my head might enter an entangled state with equal + and - half spin.

4. Sep 5, 2011

### utesfan100

So I considered a collection of particles moving in a square relative to some frame, reflecting off of particle mirrors at the corners. It occurred to me that, from the particle's perspective, these mirrors should be length contracted.

At $v=\sqrt{\frac{2}{3}} c$ the 45 degree angle is length contracted to a 30 degree angle. This shrinks the 90 degree internal angle to a 60 degree internal angle. The particle observes a triangle of reflections.

Thus the particles observe a complete cycle every three reflections, while only traversing 270 degrees in the inertial frame.

Considering this in the continuous limit, it appears this effect is similar to the length contraction of the particles from the rest frame, tied to a fixed circumference. This is the inverse of the time dilation and thus opposite in the sign of the first order perturbations.

It is interesting to note that the time-dilation/length contraction effects produce different results in this model. This is also an intuitive example of how accelerating reference frames can break this symmetry.

5. Sep 6, 2011

### Bill_K

Thomas precession is a continuous process and requires a continuous description. I hope you would not try to describe the orbit of a planet around the sun by approximating it with a triangle.

6. Sep 6, 2011

### utesfan100

My square and triangle results were for particles traveling in a square reflected by mirrors. This is more tangible than the limiting polygon, as the maximum edge length goes to 0, I would use to approximate an orbit.

In particular, it allows the sign of the effect to be seen clearly.

Thank you for clarifying my thoughts to where I now see that my further thoughts do not fit under the topic of Thomas Precession.

7. Sep 6, 2011