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 ~!T = −(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 f0g-vector is a time-like vector, Minkowski-orthogonal to the three space-like vectors f1; 2; 3g. 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 ^ = j~vj , then a boost has the form ~vx^ t^0 = γ 1 − c2 x^0 = γ(^x − ~vt) y^0 =y ^ z^0 =z ^ 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 2 3 γ −γβx −γβy −γβz 2 6 βx βxβy βxβz 7 6−γβx 1 + (γ − 1) β2 (γ − 1) β2 (γ − 1) β2 7 L = 6 2 7 : 6 βy βx βy βy βz 7 6−γβy (γ − 1) β2 1 + (γ − 1) β2 (γ − 1) β2 7 4 2 5 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 2 3 2 3 γ −γβ 0 0 γ0 0 −γ0β0 0 6 7 6 7 6−γβ γ 0 07 6 0 1 0 07 6 7 0 6 7 L = 6 7 ;L = 6 7 6 0 0 1 07 6−γ0β0 0 γ0 07 4 5 4 5 0 0 0 1 0 0 0 1 1 have the products 2 3 γγ0 −γβ −γγ0β0 0 6 7 6−γγ0β γ γγ0ββ0 07 0 6 7 LL = 6 7 (2) 6 −γ0β0 0 γ0 07 4 5 0 0 0 1 2 3 γγ0 −γγ0β −γ0β0 0 6 7 6 −γβ γ 0 07 0 6 7 L L = 6 7 ; (3) 6−γγ0β0 γγ0ββ0 γ0 07 4 5 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 ^l. When applying boosts first to the e^ frame and then frome ^ to thee ^0 frame, we will find that even though the coordinate axes of ^l ande ^ are parallel to each other, and the coordinate axes ofe ^ ande ^0 are also parallel to each other, the axes of ^l and e^0 are not in general parallel. 0 e^0 e^0 1 0 e^3 0 e^2 00 β β0 L0 00 Rθ L ^l0 e^0 β L ^l1 e^1 ^l3 e^3 ^l2 e^2 Figure 1: Three inertial reference frames with individual relative velocities. The Lorentz transformation from ^l toe ^0 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 ^ ^ y^ plane. 2 3 1 0 0 0 6 7 60 cos(θ) sin(θ) 07 6 7 Rθ = 6 7 60 − sin(θ) cos(θ) 07 4 5 0 0 0 1 2 3 γγ0 −γγ0β −γ0β0 0 6 7 6 γ(γ0β0 sin(θ) − β cos(θ)) −γ(γ0ββ0 sin(θ) − cos(θ)) −γ0 sin(θ) 07 00 −1 0 6 7 L = Rθ L L = 6 7 (4) 6−γ(β sin(θ) + γ0β0 cos(θ)) γ(sin(θ) + γ0ββ0 cos(θ)) γ0 cos(θ) 07 4 5 0 0 0 1 By requiring that the matrix L00, as a boost, be symmetric, the elements of the rotational matrix can be determined 8 −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 2 3 γγ0 −γγ0β −γ0β0 0 6 0 γ2γ02β2 γγ02ββ0 7 00 6−γγ β 1 + γγ0+1 γγ0+1 07 L = 6 0 0 7 : (6) 6 0 0 γγ02ββ0 γ (γ+γ ) 7 6 −γ β 0 0 07 4 γγ +1 γγ +1 5 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 e^0 0 e^3 0 e^0 e^1 2 e^0 e^1 e^3 e^2 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 = ke^ dτ 0 where τ is the proper time and k is some proportonality factor. Further, ~q · e^0 = 0 and d d~q d^e (~q · e^ ) = · e^ + ~q · 0 = k + ~q ·~b = 0 dτ 0 dτ 0 dτ ~ d^e0 where b = dτ is the world acceleration ande ^0 · e^0 = 1. Therefore the Fermi-Walker transport equation for the vector ~q is d~q = −(~q ·~b)^e dτ 0 ~ ~ By differentiating two space-like vectors, ~q = −(~q · b)^e0 and ~p = −(~p · b)^e0 , we find that the scalar product is preserved by Fermi-Walker transport. d (~q · ~p) = −(~q ·~b)(~p · e^ ) − (~q · e^ )(~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.