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Topic 4: Special Relativity and Thomas Precession

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Topic 4: Special Relativity and Thomas Precession Lecture Series: The Spin of the Matter, Physics 4250, Fall 2010 1 Topic 4: Special Relativity and Thomas Precession Dr. Bill Pezzaglia CSUEB Physics Updated Nov 2, 2010 2 If you can't explain it simply, you don't understand it well enough. Albert Einstein (1879-1955) 1905 Special Relativity 1 Index: Rough Draft 3 A. Lorentz Transformations B. Velocity Addition C. Acceleration & Thomas Precession D. References 4 A. Lorentz Transformations 1. Special Relativity Effects 2. Lorentz Transformation 3. Clifford Algebra form of LT 2 A1. Lorentz-FitzGerald Contraction 5 1889 FitzGerald, 1892 Lorentz •Propose a moving meter stick will appear to shrink in length L’ = L [1-v2/c 2]1/2 •1905 Einstein deduces this from his postulates of relativity. A2. Time Dilation 6 Proposed 1897 by Larmor 1904 by Lorentz • A moving clock will appear to run slower t t ′ = v 2 1 − c 2 Paradoxically, two observers each moving with a clock, sees the OTHER clock running slowly 3 7 A.3 Special Relativity 1905 Einstein postulates: • Motion is relative • Speed of light is same for all observers From these postulates he recovers Lorentz’s transformations Lorentz Transformation (1904) 8 • 1904 Lorentz, 1905 Poincare & Einstein • Motion is equivalent to a hyperbolic rotation in spacetime • Angle βββ is the “rapidity”: tanh βββ=v/c ct ' cosh β sinh β 0 ct x' = sinh β cosh β 0 x y' 0 0 1 y 1 cosh β = γ = 1 − v 2 / c 2 4 Clifford Algebra Form of Rotation 9 • Rotation of any quantity “Q” in “xy ” plane by angle φ can be expressed in exponential “half angle” form: Q'= R Q R−1 ˆ ˆ φ φ φ = e1e2 2/ = + R e cos 2 e1eˆˆ 2 sin 2 Note “spacelike” bivectors square negative: 2 = − (e1eˆˆ 2 ) 1 Clifford Algebra Form of LT 10 • Hyperbolic Rotation in “TX ” plane by rapidity angle β has same exponential “half angle” form: Q'= L Q L−1 ˆ ˆ β β β = e4e1 2/ = + L e cosh 2 eˆ4eˆ1 sinh 2 Note “Timelike” bivectors square positive: (eˆ eˆ )2 = +1 because 4 1 2 = − (eˆ4 ) 1 2 = + (eˆ1) 1 5 Hyperbolic Geometry Review 11 • Spacetime is Hyperbolic, not trigonometric β = 1 ( β − −β ) sinh 2 e e β = 1 ()β + −β cosh 2 e e sinh tanh = cosh cosh 2 −sinh 2 = −1 B. Velocity Addition Formula 12 ≠ + • Velocities don’t add: V V1 V2 β = β + β • Rapidities DO add: 1 2 6 B.1 Addition of Velocities 13 • Speeds add: V= • Adding anything to speed of light gives speed of light Parallel Velocity Addition (Einstein 1905) 14 • Rapidity angles add, not velocities tanh β + tanh β tanh ()β + β = 1 2 1 2 + β β 1 tanh 1 tanh 2 V tanh β = c V +V V = 1 2 VV 1+ 1 2 c2 7 Velocity Composition 15 • The general addition of velocities in two different directions is neither commutative or associative: 1 γ ⊕ = + + 1 ()×()× u v u v 2 u u v u •v c + λ 1+ 1 c2 γ = 1 1− u2 c2 Thomas-Wigner Rotation 1939 16 • 1775 Euler: The composition of two rotations is a single rotation. • Wigner shows that in general the composition of two Lorentz transformations is a Lorentz transformation plus a rotation • WHY isn’t it just a Lorentz transformation? Spacelike bivectors are Timelike bivectors are not closed under multiplication closed under multiplication, they yield rotations ( )( ) = (e e )(e e ) = e e e1e2 e2e3 e1e3 4 1 4 2 1 2 ( )( ) = (e e )(e e ) = e e e2e3 e3e1 e2e1 4 2 4 3 2 3 ( )( ) = (e e )(e e ) = e e e3e1 e1e2 e3e2 4 3 4 1 3 1 8 Derivation using Half Angles 17 • I have not seen this done anywhere, but possibly Abraham A. Ungar has it in his book somewhere. αˆα 2/ βˆβ 2/ ˆγγ 2/ θˆθ 2/ • Want to show: e e = e e • where { αβγ } are timelike bivectors for Lorentz αˆ 2 = βˆ 2 = γˆ2 = +1 • θ is spacelike bivector of rotation θˆ2 = −1 • Expand: α +α α β + βˆ β = γ +γ γ θ +θˆ θ (cosh 2 ˆ sinh 2 )(cosh 2 sinh 2 ) (cosh 2 ˆsinh 2 )(cos 2 sin 2 ) Solution of Wigner Rotation 18 • Solution is analogous to the Rodrigues formula! α β • Net Velocity γ αˆ tanh + βˆ tanh γˆ tanh = 2 2 is symmetric! 2 + α • βˆ α β 1 ˆ tanh 2 tanh 2 αˆ ⊗ βˆ tanh α tanh β • Rotation θˆ tan θ = 2 2 2 + α • βˆ α β 1 ˆ tanh 2 tanh 2 9 19 C. Thomas Precession Llewellyn Hilleth Thomas (1903-1992) • 1926 proposes “Thomas Precession ” to explain the difference between predictions made by spin- orbit coupling theory and experimental observations. Thomas Precession in Brief 20 • Acceleration perpendicular r r to velocity causes an extra ω ≈ 1 a ×v rotation of the reference th 2c2 frame, with angular velocity: • The change in any vector r r dV dV r r in lab frame is related to = +ω ×V dτ dτ th change in rotating (due to lab body accelerating) body frame by: 10 Thomas Precession in Orbits 21 • Atomic: After electron completes one closed orbit (in period “T”), there will be an “anomalous” net rotation of the (spin axis of the) electron due to the acceleration being perpendicular to the velocity. v2 ∆θ ≈ ω T = 2π th c2 • This is also true for any circular orbit, for example a Foucault pendulum, which precesses at rate of 222 ° a day at Hayward, would have an additional Thomas precession due to rotation of earth on the order of: ∆θ ≈ 5.1 ×10 −11 rads / day = 3×10 −6 arc sec/ day 1. Fermi-Walker Transport 22 • Basis vectors ek fixed to an e& = 1 (u ∧ a)• e accelerating NON-rotating k c2 k rigid body will change = 1 ()a u − u a according to: c2 k k – 4 velocity “u” – 4 acceleration: a = u& • The change in any vector r r dV dV r in lab frame is related to = + 1 ()u∧a •V change in (accelerating) dτ dτ c2 body frame by: lab body 11 Clifford Algebra derivation 23 • Lorentz Transformation of e (t) = L e )0( L−1 basis vector (at time “t”) k j • Operator for boost in βˆ β direction of velocity = 2 described by rapidity β L e • I think you can show that the Fermi-Walker transport bivector is given: 1 u ∧ a ≡ 2L&R−1 = −2LL&−1 c2 Incomplete derivation 24 • Differentiating exponential operator I get: − dβˆ dβˆ 2L&L 1 = β& βˆ + sinh β + ()1− cosh β βˆ ⊗ dτ dτ • The last term is the Thomas Precession • I haven’t yet shown this gives exactly 1 u ∧ a ≡ 2L&R−1 = −2LL&−1 c2 12 Thomas Term from Fermi-Walker 25 • Expanding the 4-vectors one gets r r r ∧ ≈ γ 3 ∧ + γ 3 k u a v a ce 4ek a • The last term is simply the linear acceleration (timelike bivector) • The first term is a spacelike bivector,r i.e. a rotationalr r angular velocity ω = 1 γ 3v × a th c2 • Ouch, not quite right. Should be: r γ 2 r r ω = 1 a ×v th c2 γ +1 Larmor Precession in Atomic Orbits 26 • An electron, with magnetic moment µµµ r r r which is moving with constant & = µ ×( − 1 × ) velocity in electromagnetic field will S B 2 v E have a torque on it: c • In rest frame of atom, there is no r r magnetic field, only electric field. = This is proportional to the eE ma acceleration of the electron. Substitute this for “E” in above r e r • The magnetic moment is µ ≈ S proportional to the spin. Substitute this for µµµ m r r • We get precession rate: × ω = + v a Larmor c2 & = ω × S Larmor S 13 Larmor plus Thomas 27 r r • Note the Larmor precession is v × a ω = + = −2ω just double the Thomas, and Larmor c2 th with opposite sign. • The total precession is hence the algebraic sum r r v × a ω = ω +ω = 1 net Larmor th 2 c2 • Thus, observed precession is half the expected Larmor precession (“The Thomas half”) References 28 • Baylis, Electrodyanmics, has a coherent description of Thomas Precession, using 3D Clifford algebra (i.e. Pauli Algebra). The Wigner rotation is derived in equation (4.29). Thomas Precession in section 4.6. • L. H. Thomas, "Motion of the spinning electron", Nature 117, 514, (1926) • E. P. Wigner, "On unitary representations of the inhomogeneous Lorentz group", Ann. Math. 40, 149–204 (1939). • Tomonaga, The Story of Spin, Lecture 2 and 11. 14 References 29 • Foucault http://en.wikipedia.org/wiki/Foucault_pendulum • Thomas: http://www.columbia.edu/acis/history/thomas.html • Abraham A. Ungar, Beyond the Einstein Addition Law and Its Gyroscopic Thomas Precession 30 Hmk Ideas • Calculate the Thomas precession rate for the moon. How many years would it take to have the side of the moon facing us change by 180 degrees? • Calculate the Thomas precession rate for a point on the equator of the earth • Does the Kimbal experiment measure Thomas precession due to earth’s rotation? (this is separate from the Foucault precession) 15 31 Things to do • Obviously need to clean up the derivation of Thomas precession term. 16.
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