General Lorentz Transformation Consider a Lorentz Transformation with

General Lorentz Transformation • Consider a Lorentz Transformation, with arbitrary v, : ct´ = γ(ct - β r) r´ = r + β-2(β r)(γ -1)β - γctβ • Transformation matrix in 4 d spacetime: x´ L x • Can write L as: L = RL 0 = L 0´R´. Here R rotation matrix, as in Ch. 4. L 0 Pure “boost” or pure velocity transformation. For example: (v || x): γ -βγ 0 0 L 0 = -γβ γ 0 0 0 0 1

• For any Lorentz transformation x´ L x, there exists an Inverse Transformation L-1: L-1 L = LL-1 = 1, x = L-1 x´ • Existence of L-1 puts 4 constraints on diagonal elements of L & 6 constraints on the offdiagonal elements of L. 10 constraints on 16 elements of L Only 6 independent elements of L PHYSICS: 3 elements of L correspond to 3 components of relative velocity & 3 correspond to the Euler angles of rotation

Sect. 7. 3: Velocity Addition & Thomas Precession • Some of the following is from the E&M book by Jackson! • Velocity Addition: Recall the Galilean Transformation: See figure: Moving point P with velocity u´ in frame S´. S´is moving with velocity v in positive x 1 direction with respect to frame S: Velocity u of point P with respect to frame S is obviously: u = u´ + v

• Velocity Addition: Lorentz Transformation (Method 1) Figure: Moving point P, velocity u´ in S´. S´ moves with velocity v in the +x 1 direction with respect to S: Find the components of velocity u of P with respect to S. Inverse Lorentz transform (differential form) is: (ILT) dx 0 = γ(dx 0´ + βdx 1´), dx 2 = dx 2´ dx 1 = γ(dx 1´ + βdx 0´), dx 3 = dx 3´ β = (v/c), γ = [1 - β 2]-½ u´ components: ui´ (dxi´/dt´) = c (dxi´/dx 0´) (i =1, 2, 3) u components: ui (dxi/dt) = c (dxi/dx 0) (i =1, 2, 3) To find relations between these, use them with above ILT equations. Divide dx 1, dx 2 & dx 3 by dx 0 & manipulate.

• Velocity Addition: Lorentz Transformation (Method 1) (ILT): dx 0 = γ(dx 0´ + βdx 1´), dx 2 = dx 2´ dx 1 = γ(dx 1´ + βdx 0´), dx 3 = dx 3´ ui´ (dxi´/dt´) = c (dxi´/dx 0´) (i =1, 2, 3) ui (dxi/dt) = c (dxi/dx 0) (i =1, 2, 3) For example: u 1 = c(dx 1/dx 0) = c[γ(dx 1´ + βdx 0´)][γ(dx 0´ + βdx 1´)]-1 = c[(dx 1´/dx 0´) + β][1 + β(dx 1´/dx 0´)]-1 Or: u 1 = [u 1´ + v]/[1 + (vu 1´)/c 2] Similarly: u 2 = c(dx 2/dx 0) = γ-1(u 2´)[1 + (vu 1´)/c 2]-1 u 3 = c(dx 3/dx 0) = γ-1(u 3´)[1 + (vu 1´)/c 2]-1

• Velocity Addition: Lorentz Transformation (Method 1) More generally: u|| component of u || v u component of u v Find: u|| = [u||´ + v][1 + (v u´)/c 2]-1 u = γ-1(u ´)[1 + (v u´)/c 2]-1 Einstein Velocity Addition Formula • For u´ & v parallel, get the Goldstein result: u = [u´ + v][1 + (vu´)/c 2]-1 (or βu = [βu´ + β][ 1+ βu´β]-1) • Get Galilean result: u = u´ + v for v << c (γ 1, β 0) • Can also get relations between spherical angles in S´ & S (see figure). Can show (student exercise): = ´ and tanθ = γ-1(u´sinθ´)(u´cosθ´ + v)-1 Also: u = [(u´)2 + v 2 + 2 u´vcosθ´ - (u´vsinθ´)2/c 2]½[1 + (u´v/c 2)cosθ´]-1

• Velocity Addition: Lorentz Transformation (Method 2 Goldstein): 3 inertial systems: S 1, S 2, S 3, with aligned x axes. S 2 moves with v with respect to S 1. S 3 moves with v´ with respect to S 2. GOAL: Find the velocity v´´ of S 3 with respect to S 1. • Transforming from S 1 to S 3 Equivalent to applying 2 successive Lorentz boosts: L 1 -2, to transform from S 1 to S 2 followed by L 2 -3, to transform from S 2 to S 3. That is, can write (matrix multiplication): L 1 -3 L 2 -3 L 1 -2

• Velocity Addition: Lorentz Transformation (Method 2 Goldstein): Transforming from S 1 to S 3 L 1 -3 L 2 -3 L 1 -2 Or: (β = v/c, γ = [1 - β 2]-½ , β´ = v´/c, γ´ = [1 - (β´)2]-½) γ´´ -β´´γ´´ 0 -β´´ γ´´ 0 0 0 1 (β´´ = v´´/c, γ ´´= [1 -(β´´)2]-½) The 2 forms must be the same: β´´ = (β + β´)/(1+ ββ´) Or: v´´ = (v + v´)/(1+ vv´/c 2) Einstein Velocity Addition Formula (parallel velocities)

Sect. 7. 3: Thomas Precession • Product of 2 Lorentz Transformations (LT) is a LT. If L& L´ are LT’s: L´´ L´ L is also a LT. NOTE! They do not commute! Order matters! • If L´ & L are pure boosts, but their velocities are not parallel, L´´ will involve a boost PLUS a rotation. This rotation Thomas Precession. • Consider this under the ASSUMPTION (Approximation): Speed v´ associated with L´ is small compared to speed v associated with L & also v´ << c

• Consider 3 inertial systems: S 1, S 2, S. S 2 moves with v = cβ with respect to S 1. S 3 moves with v´ = cβ´ with respect to S 2. – No loss of generality to have the axes of S 1 so that β || xaxis of S 1 & so that β´ is in the x´-y´ plane of S 2. • Transform from S 1 to S 3 Apply 2 Lorentz boosts: L, to transform from S 1 to S 2 followed by L´, to transform from S 2 to S 3. Can write (matrix multiplication): L´´ L´ L.

• L´´ is not symmetric It is a rotation + a velocity boost. – Let S 3 move with v´´ = cβ´´ with respect to S 1. • Off diagonal elements in L´´ corresponding to z motion are zero The rotation is about an axis xy plane. • Approximation: v´ = cβ´ is small compared to v = cβ. Also v´ << c β´ << 1, γ´ 1. Keep only terms to 1 st order in β´. This implies:

• Using this approximation in the result for L´´ gives: • This is a combination of a boost & a rotation. Now, to find (separate out) the rotation part: Consider (in the same approximation) the inverse Lorentz Transformation BACK from S 3 to S 1. Can show (see text) this is:

• Can get rotation matrix (under same assumption) by: • Compare with the rotation matrices in Ch. 4 (generalized to include the 4 th dimension) & get: 1 0 0 cos(ΔΩ) sin(ΔΩ) 0 R 0 -sin(ΔΩ) cos(ΔΩ) 0 0 1 R corresponds to S 3 being rotated with respect to S 1 about the z axis through an angle ΔΩ

• When comparing the 2 expressions, recall that the approximations we’ve made mean that the off diagonal element (γ - 1)(βy´´)β -1 is very small so that: sin(ΔΩ) (γ - 1)(βy´´)β -1 is very small and thus, sin(ΔΩ) ΔΩ = (γ - 1)(βy´´)β -1 & cos(ΔΩ) 1 • Summary: Rotation part of L´´ L´ L corresponds to S 3 rotating with respect to S 1 about z axis through (very small, in this approx) angle: ΔΩ = (γ - 1)(βy´´)β-1 Thomas Precession Angle Very important & very well verified experimentally in atomic physics!

• Briefly consider a particle moving in the lab frame with velocity v which is NOT CONSTANT. Since the particle rest frame is accelerated with respect to the lab frame, it is NOT an inertial system & thus, strictly speaking we cannot connect the 2 frames with a Lorentz Transformation (LT)! • However, we can circumvent this by the following device: Imagine an number of inertial systems, each moving with uniform velocity relative to the lab system. As the velocity of the particle changes, for each infinitesimal time change dt, we can match the particle velocity to that of one of these inertial frames. At each instant, we can make a LT from the lab to the particle frame. That is, in some sense, we can describe the transformation by a time dependent LT! • We mention this now because a rotating frame (as in Thomas Precession) is not an inertial frame!

Thomas Precession: ΔΩ = (γ - 1)(βy´´)β-1 • Application: S 1 = lab system. S 2, S 3 = 2 of these instantaneous rest frames for a particle undergoing Thomas Precession just described. A small time Δt apart. • In Δt, a lab observer will see a change Δv in the particle velocity. Our discussion shows that this will have only a y component: Δv = cβy´´. Initially, choose velocity v = cβ along the x axis. The vector velocity has rotated through the infinitesimal angle ΔΩ given above. Using all of this, we can write the vector angle as: ΔΩ = - (γ - 1)(v Δv)/v 2 (1) Dividing (1) by Δt gives an instantaneous angular velocity of the particle: ω (ΔΩ/Δt) or: ω = - (γ - 1)(v a)v-2 (2) Here: a (Δv/Δt) = particle acceleration as seen from the lab system S 1

The Thomas Precession frequency: ω = - (γ - 1)(v a)/v 2 (2) • v is usually small enough that γ = [1 - β 2]-½ 1 + (½)β 2 The Thomas Precession frequency becomes: ω = (½)(a v)c-2 (3) • This was originally proposed to explain the anomalous magnetic moment of the electron. It led to the hypothesis that electrons have an internal quantum number called spin angular momentum. See the discussion in the Relativity chapter in J. D. Jackson’s E&M book for more details!