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Topic 4: Special Relativityand Thomas Precession

Dr. Bill PezzagliaCSUEB Physics

Updated Nov 2, 2010

1Lecture Series: The Spin of the Matter, Physics 4 250, Fall 2010

If you can't explain it simply, you don't understand it well enough.

Albert Einstein (1879-1955)

1905 Special Relativity

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Index: Rough Draft

A. Lorentz Transformations

B. Velocity Addition

C. Acceleration & Thomas Precession

D. References

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A. Lorentz Transformations

1. Special Relativity Effects

2. Lorentz Transformation

3. Clifford Algebra form of LT

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A1. Lorentz-FitzGerald Contraction 5

1889 FitzGerald, 1892 Lorentz

•Propose a moving meter stick will appear to shrink in length

L’ = L [1-v 2/c2]1/2

•1905 Einstein deduces this from his postulates of relativity.

A2. Time Dilation 6

Proposed 1897 by Larmor1904 by Lorentz

• A moving clock will appear torun slower

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2

1c

v

tt

−

=′

Paradoxically, two observers each moving with a clock, sees the OTHER clock running slowly

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A.3 Special Relativity7

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)

• 1904 Lorentz, 1905 Poincare & Einstein• Motion is equivalent to a hyperbolic rotation in sp acetime

• Angle ββββ is the “rapidity”: tanh ββββ=v/c

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=

y

x

ct

y

x

ct

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0coshsinh

0sinhcosh

'

'

'

ββββ

22 /1

1cosh

cv−== γβ

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Clifford Algebra Form of Rotation

• Rotation of any quantity “Q” in “xy” plane by angle φcan be expressed in exponential “half angle” form:

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22122/ˆˆ sinˆˆcos21 φφφ eeeR ee +==

1)ˆˆ( 221 −=ee

1' −= RQRQ

Note “spacelike” bivectors square negative:

Clifford Algebra Form of LT

• Hyperbolic Rotation in “TX” plane by rapidityangle β has same exponential “half angle” form:

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21422/ˆˆ sinhˆˆcosh14 βββ eeeL ee +==

1' −= LQLQ

1)ˆˆ( 214 +=eeNote “Timelike” bivectors square positive:because

1)ˆ(

1)ˆ(2

1

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+=

−=

e

e

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Hyperbolic Geometry Review

• Spacetime is Hyperbolic, not trigonometric

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( )( )

1sinhcosh

cosh

sinhtanh

cosh

sinh

22

21

21

−=−

=

+=

−=−

−

ββ

ββ

ββ

ee

ee

B. Velocity Addition Formula

• Velocities don’t add:

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21 VVV +≠

• Rapidities DO add: 21 βββ +=

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B.1 Addition of Velocities 13

• Speeds add: V=

• Adding anything to speed of light gives speed of light

Parallel Velocity Addition (Einstein 1905)

• Rapidity angles add, not velocities

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( )

221

21

21

2121

1

tanh

tanhtanh1

tanhtanhtanh

c

VVVV

V

c

V

+

+=

=

++=+

β

ββββββ

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Velocity Composition

• The general addition of velocities in two different directions is neither commutative or associative:

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( )( )

2

2

2

1

1

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1 1

2

cu

cvuuvu

c

vuvu

−=

××

+++•+

=⊕

γ

λγ

Thomas-Wigner Rotation 1939

• 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?

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( )( )( )( )( )( ) 232113

121332

313221

eeeeee

eeeeee

eeeeee

===

Spacelike bivectors are closed under multiplication

( )( )( )( )( )( ) 131434

323424

212414

eeeeee

eeeeee

eeeeee

===

Timelike bivectors are notclosed under multiplication, they yield rotations

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Derivation using Half Angles

• I have not seen this done anywhere, but possibly Abraham A. Ungar has it in his book somewhere.

• Want to show:

• where {αβγ} are timelike bivectors for Lorentz

• θ is spacelike bivector of rotation

• Expand:

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)sinˆ)(cossinhˆ(cosh)sinhˆ)(coshsinhˆ(cosh 22222222θθγγββαα θγβα ++=++

2/ˆ2/ˆ2/ˆ2/ˆ θθγγββαα eeee =

1ˆ

1ˆˆˆ2

222

−=

+===

θγβα

Solution of Wigner Rotation

• Solution is analogous to the Rodrigues formula!

• Net Velocityis symmetric!

• Rotation

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22

222

tanhtanhˆˆ1

tanhˆtanhˆtanhˆ

βα

βαγ

βαβαγ

•++=

22

222

tanhtanhˆˆ1

tanhtanhˆˆtanˆ

βα

βαθ

βαβαθ•+

⊗=

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C. Thomas Precession19

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

• Acceleration perpendicular to velocity causes an extra rotation of the reference frame, with angular velocity:

• The change in any vector in lab frame is related to change in rotating (due to accelerating) body frame by:

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vacth

rr×≈ 221ω

Vd

Vd

d

Vdth

bodylab

rrrr

×+= ωττ

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Thomas Precession in Orbits

• 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.

• 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:

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2

2

2c

vTth πωθ =≈∆

dayarcdayrads sec/103/105.1 611 −− ×=×≈∆θ

1. Fermi-Walker Transport

• Basis vectors ek fixed to an accelerating NON-rotating rigid body will change according to:

– 4 velocity “u”– 4 acceleration:

• The change in any vector in lab frame is related to change in (accelerating) body frame by:

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( )( )auua

eaue

kkc

kck

−=

•∧=

2

2

1

1&

( ) Vaud

Vd

d

Vdc

bodylab

rrr

•∧+= 21ττ

ua &=

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Clifford Algebra derivation

• Lorentz Transformation of basis vector (at time “t”)

• Operator for boost in direction of velocity described by rapidity β

• I think you can show that the Fermi-Walker transport bivector is given:

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1)0()( −= LeLte jk

111 222−− −=≡∧ LLRLau

c&&

2ˆ ββeL =

Incomplete derivation

• Differentiating exponential operator I get:

• The last term is the Thomas Precession

• I haven’t yet shown this gives exactly

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( )τββββ

τβββ

d

d

d

dLL

ˆˆcosh1sinh

ˆˆ2 1 ⊗−++=− &&

111 222−− −=≡∧ LLRLau

c&&

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Thomas Term from Fermi-Walker

• Expanding the 4-vectors one gets

• The last term is simply the linear acceleration (timelikebivector)

• The first term is a spacelike bivector, i.e. a rotational angular velocity

• Ouch, not quite right. Should be:

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kkaeceavaurrr

433 γγ +∧≈∧

avcth

rrr ×= 312 γω

vacth

rrr ×+

=1

212 γ

γω

Larmor Precession in Atomic Orbits

• An electron, with magnetic moment µµµµ which is moving with constantvelocity in electromagnetic field will have a torque on it:

• In rest frame of atom, there is no magnetic field, only electric field. This is proportional to the acceleration of the electron. Substitute this for “E” in above

• The magnetic moment is proportional to the spin. Substitute this for µµµµ

• We get precession rate:

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( )EvBSc

×−×= rrr&

21µ

Sm

e rr ≈µ

amEerr =

SS

c

av

Larmor

Larmor

×=

×+=

ω

ω

&

rr

2

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Larmor plus Thomas

• Note the Larmor precession is just double the Thomas, and with opposite sign.

• The total precession is hence the algebraic sum

• Thus, observed precession is half the expected Larmorprecession (“The Thomas half”)

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thLarmor c

av ωω 22

−=×+=rr

221

c

avthLarmornet

rr×=+= ωωω

References

• 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.

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References

• 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

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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)

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Things to do• Obviously need to clean up the derivation of Thomas precession term.

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