FYGB08 Project Thomas precession

Author: SSN: Daniel Kronberg 850810-5999

Abstract This project is a derivation of Thomas precession, a relativistic correction to a vector traveling along a curvilinear trajectory. We consider the precession as a consequence of consecutive restricted Lorentz transformations between instantly inertial reference frames along the trajectory. These tranformations ~v×~v˙ result in an observed retrograde precession of the vector, with the angular velocity ~ωT = −(γ − 1) v2 . 1 Introduction This project is an attempt at a straightforward explanation of Thomas precession by discussing Thomas- Wigner rotation algebraically as a consequence of consecutive restricted Lorentz transformations. Fermi- Walker transport is approached in a fairly conceptual manner as a motivation of how results based on a set of inertial reference frames can be applied to accelerated systems. It is possible to derive the Thomas angular velocity directly from the equations given in the Fermi-Walker section, but here we have chosen to derive it from Thomas-Wigner rotation. Illustrated reference frames are shown as a tetrad of vectors, where the {0}-vector is a time-like vector, Minkowski-orthogonal to the three space-like vectors {1, 2, 3}.

2 Lorentz transformations Lorentz transformations are used to connect observations between inertial reference frames. In particular they give the correct transformation rules at relativistic velocities. A single Lorentz transformation along a given direction is called a restricted Lorentz transformation, or boost, and affects the time component and ~v the spatial component parallel to the velocity. If the basis vectors are chosen so thatx ˆ = |~v| , then a boost has the form  ~vxˆ tˆ0 = γ 1 − c2 xˆ0 = γ(ˆx − ~vt) yˆ0 =y ˆ zˆ0 =z ˆ

1 ~v γ = , β = , p1 − β2 c

where ~v is the boost velocity, and c is the speed of light. A transformation from the primed frame into some other frame with a collinear velocity has the same form, and is equivalent to a single boost from the first frame with a total velocity given by

β + β0 β00 = . (1) 1 + ββ0 Any boost can be represented by a symmetric matrix, and the general boost form is   γ −γβx −γβy −γβz 2  βx βxβy βxβz  −γβx 1 + (γ − 1) β2 (γ − 1) β2 (γ − 1) β2  L =  2  .  βy βx βy βy βz  −γβy (γ − 1) β2 1 + (γ − 1) β2 (γ − 1) β2   2  βz βx βz βy βz −γβz (γ − 1) β2 (γ − 1) β2 1 + (γ − 1) β2

However, the product of two non-collinear boosts is not equivalent to any single boost, but instead to some boost and some rotation. For example, two perpendicular boosts     γ −γβ 0 0 γ0 0 −γ0β0 0     −γβ γ 0 0  0 1 0 0   0   L =   ,L =    0 0 1 0 −γ0β0 0 γ0 0     0 0 0 1 0 0 0 1

1 have the products   γγ0 −γβ −γγ0β0 0   −γγ0β γ γγ0ββ0 0 0   LL =   (2)  −γ0β0 0 γ0 0   0 0 0 1   γγ0 −γγ0β −γ0β0 0    −γβ γ 0 0 0   L L =   , (3) −γγ0β0 γγ0ββ0 γ0 0   0 0 0 1

which are not symmetric; notice further that the product is non-commutative. As a result, the coordinate axes of the final reference frame are not parallel with those of the first, but are instead rotated by some angle θ about an axis orthogonal to the boosts. We can hence express the product of two boosts as a product of a boost and a rotation. This rotation is called Thomas rotation, Wigner rotation, or Thomas-Wigner rotation, after Llewellyn Thomas and Eugene Wigner. We will in the next section derive the explicit expression of the rotation matrix represnting the Thomas-Wigner rotation.

3 Thomas-Wigner rotation

Consider two inertial reference frames, observed from the lab frame ˆl. When applying boosts first to the eˆ frame and then frome ˆ to thee ˆ0 frame, we will find that even though the coordinate axes of ˆl ande ˆ are parallel to each other, and the coordinate axes ofe ˆ ande ˆ0 are also parallel to each other, the axes of ˆl and eˆ0 are not in general parallel.

0 eˆ0

eˆ0 1 0 eˆ3 0 eˆ2

00 β β0 L0 00 Rθ L

ˆl0 eˆ0

β L ˆl1 eˆ1 ˆl3 eˆ3 ˆl2 eˆ2

Figure 1: Three inertial reference frames with individual relative velocities.

The Lorentz transformation from ˆl toe ˆ0 is given by the matri L0L. As previously discussed, this can be 0 00 written as the product of a rotation and a boost, i.e. L L = RθL . In order to find the rotation angle θ between the two frames, we follow the method outlined in [1] by introducing an arbitrary rotation matrix and using it to find an expression for the boost L00. Since the coordinate axes can always be chosen parallel

2 to the boosts, it is sufficient to consider rotation in thex ˆ ∧ yˆ plane.   1 0 0 0   0 cos(θ) sin(θ) 0   Rθ =   0 − sin(θ) cos(θ) 0   0 0 0 1   γγ0 −γγ0β −γ0β0 0    γ(γ0β0 sin(θ) − β cos(θ)) −γ(γ0ββ0 sin(θ) − cos(θ)) −γ0 sin(θ) 0 00 −1 0   L = Rθ L L =   (4) −γ(β sin(θ) + γ0β0 cos(θ)) γ(sin(θ) + γ0ββ0 cos(θ)) γ0 cos(θ) 0   0 0 0 1

By requiring that the matrix L00, as a boost, be symmetric, the elements of the rotational matrix can be determined  −γ0 sin(θ) = γ(sin(θ) + γ0ββ0 cos(θ))  −γγ0β = γ(γ0β0 sin(θ) − β cos(θ)) −γ0β0 = −γ(β sin(θ) + γ0β0 cos(θ)) =⇒

γ + γ0 γγ0ββ0 cos(θ) = , sin(θ) = − . (5) γγ0 + 1 γγ0 + 1

Substituting the resulting expressions into L00 gives us   γγ0 −γγ0β −γ0β0 0  0 γ2γ02β2 γγ02ββ0  00 −γγ β 1 + γγ0+1 γγ0+1 0 L =  0 0  . (6)  0 0 γγ02ββ0 γ (γ+γ )   −γ β 0 0 0  γγ +1 γγ +1  0 0 0 1

This method can be applied to any two boosts by splitting them up into collinear and perpendicular components and adding their velocities according to equation (1). For three or more arbitrary boosts, the expressions are more complex, but in the situations that are interesting when discussing Thomas precession, we need only deal with motion in a plane.

4 Fermi-Walker transport The results from the previous section can be applied easily when dealing only with inertial frames, however, many interesting systems are accelerated, and it is not immediately evident how Thomas-Wigner rotation would apply to such systems. In order to describe accelerated systems, we will consider Fermi-Walker transport, a form of parallel-transport used to move a vector along a curvilinear trajectory with the minimum possible amount of rotation. In this model, we treat the trajectory of the accelerated system as consisting of an infinite series of instantaneously inertial frames, each related to its neighbors by an infinitesimal boost in the direction of acceleration. This is a commonly used approximation in relativity theory [2].

3 0 eˆ0

0 eˆ3

0 eˆ0 eˆ1 2 eˆ0

eˆ1 eˆ3 eˆ2

Figure 2: Two instantaneously inertial frames on a curved world line, representing an accelerated system.

Let us describe Fermi-Walker transport in some more detail, following [3]. Consider an accelerated particle and its world-line, observed from the Thomas conventional frame; an inertial frame related to the lab frame by a single boost to some initial frame of the particle. Then consider the space-like hyperplane that is orthogonal to the time-axis. Any purely space-like vector ~q in the accelerated system must be continuously projected into the hyperplane of the next inertial frame. Each infinitesimal projection will be in the direction of the time-axis, and so the evolution of the vector ~q will be d~q = keˆ dτ 0 where τ is the proper time and k is some proportonality factor. Further,

~q · eˆ0 = 0

and d d~q dˆe (~q · eˆ ) = · eˆ + ~q · 0 = k + ~q ·~b = 0 dτ 0 dτ 0 dτ

~ dˆe0 where b = dτ is the world acceleration ande ˆ0 · eˆ0 = 1. Therefore the Fermi-Walker transport equation for the vector ~q is d~q = −(~q ·~b)ˆe dτ 0 ~ ~ By differentiating two space-like vectors, ~q = −(~q · b)ˆe0 and ~p = −(~p · b)ˆe0 , we find that the scalar product is preserved by Fermi-Walker transport. d (~q · ~p) = −(~q ·~b)(~p · eˆ ) − (~q · eˆ )(~p ·~b) = 0 dτ 0 0 Therefore the change in any space-like vector parallel-transported along this curved world-line is a rotation only. The length of the vector remains constant. With the basic concepts of Fermi-Walker transport in mind, we can now apply the results to the description of Thomas-Wigner rotation.

5 Thomas precession As a vector is Fermi-Walker transported over a curvilinear trajectory, it will experience countless infinitesimal Thomas-Wigner rotations. Since the rotation angle is dependent on the relative velocities of the boosts, the percieved rotation angle will depend on the reference system from which the parallel-transported frame is observed. However, if the motion of a parallel-transported vector, such as a spin-vector, is periodic, and obervations are made at integer multiples of the period, the rotations will add

4 up to produce a net retrograde precession that is independent of the observer’s frame of reference. The angular velocity of this precession can be found by considering each individual infinitesimal set of boosts to have the rotational angle given by (5). If we then assume that the second boost, representing the change in velocity from one instant inertial frame to the next, is much smaller than the total relative velocity as seen from the observer frame, β >> β0, and also small enough compared to the speed of light that γ0 ≈ 1, we can make the approximation

γββ0 cos(θ) ≈ 1 , sin(θ) ≈ . γ + 1

As the rotational angle is infinitesimal, we can also make the approximation sin(θ) ≈ θ. We then subsitute in the components of the velocity vectors as transformed by the associated boost L00, to the first order, as done in [2], i.e.

β0 β00 ≈ β , β00 = , β002 ≈ β2 , γ00 ≈ γ . x y γ We can now find the expression for the Thomas angular velocity.

γ2 ~v˙ × ~v ~v × ~v˙ ~ω = = −(γ − 1) T γ + 1 c2 v2

Where ~v and ~v˙ are measured in the same inertial frame. The Thomas angular velocity is observed, for example, when considering a spinning gyroscope traveling in an orbit around a central force. In this case the spin vector of the gyroscope will precess as described. Another common example is the spin-orbit interaction of electrons. In that case, however, Thomas precession by itself can not account for the motion of the spin vector. Larmor precession must also be taken into account.

6 Discussion The greatest difficulty with this project, despite the rather paradoxical nature of Thomas-Wigner rotation 00 β0 itself, was understanding the approximation βy ≈ γ . It’s still not quite clear to me why this is a good approximation, and I have been unable to find a satisfactory explanation. Because of this, obtaining the correct result for the Thomas angular velocity was initially troublesome. Fortunately, accepting that the acceleration in a curvilinear orbit has this value when seen from some inertial lab-frame allowed me to calculate the angular velocity correctly based on both methods presented in [1] and [2], though only the first is shown here. The more geometrically focused method demonstrated in [3] was more difficult to apply to the end result, but I find the approach very interesting and I hope to be able to revisit it in the future, in order to properly understand it.

5 References [1] Fererro, R. & Thibeault, M. (2002) Generic composition of boosts: an elementary derivation of the Wigner rotation arxiv.org/abs/physics/0211022v1 [physics.ed-ph] [2] Goldstein, H. & Poole, C.P. & Safko, J.L. (2002) Classical Mechanics (3rd ed.). Pearson Education Limited, Harlow.

[3] Notes 5, Thomas Precession. (2002). Retrieved January 15, 2015, from http://bohr.physics.berkeley.edu/classes/221/0708/notes/thomprec.pdf