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1 Topic 8: Geometrodynamics of spinning particles First Draft Dr. Bill Pezzaglia CSUEB Physics Updated Nov 28, 2010 Lecture Series: The Spin of the Matter, Physics 4250, Fall 2010 Outline A. Maxwellian Gravity B. Gravitomagnetic Moment C. Precession D. Papapetrou Equations E. References 2

Outline 2 - clifford.org · Larmor Precession (a)Torque on Magnet (b)Precession (c) Larmor Frequency C. Precession 17 B ... • Quaternions: review usage in animation, and/or demonstrate

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Topic 8: Geometrodynamicsof spinning particles

First Draft

Dr. Bill PezzagliaCSUEB Physics

Updated Nov 28, 2010

1Lecture Series: The Spin of the Matter, Physics 4250, Fall 2010

Outline

A. Maxwellian GravityB. Gravitomagnetic MomentC. PrecessionD. Papapetrou EquationsE. References

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Gravity Analog of Electromagnetism

• 1864 Maxwell proposesgravitoelectric theory

• 1894 Heaviside (in 3 weeks)extends ideas to includegravitomagnetism

• 1915 Einstein’s Theory(gravity is a tensor field,not a vector field)

A. Maxwellian Gravity 3

(a) Electric Field

• 1821 Faraday First proposes ideas of “Lines of Force”

• Force on Charge:

• Gauss’s Law:

1. GravitoElectric Field 4

0ερ

=•∇ Er

EqFrr

=

3

Gravitational Analogy (to electric force):

• Force Law:

• “Field” is acceleration of gravity “g”

• “Charge” is (gravitational) mass “m”

(b) GravitoElectric Field is “g” 5

gmF rr=

• 1813 Gauss’s Law

• yields field around charge:

• Newton’s Law implies“field” around mass:

• 1864 Maxwell writes:(1764 Lagrange)

(c) Gauss’s Law for “g” 6

rr

QE ˆ4 2

0πε=

r0ερ

=•∇ Er

rrGMg ˆ

2

−=

r

ρπ Gg 4−=•∇r

4

(a) Magnetic Force (1861 Maxwell)2. GravitoMagnetic Field 7

( )BvEqFrrrr

×+=

• 1889 Heaviside proposes “gravitomagnetic field” gB

(b). Gravitational Analogy to Magnetic 8

( )BE gvgmF rrrr×+=

• gB has not yet been directly verified![more on Gravity Probe B later]

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• 1856 Maxwell writes that movingcharge ρ creates magnetic field

• 1894 Heaviside proposes that moving mass ρcreates gravito-magnetic field (effect very small) [I don’t know if he had the “displacement” term]

(c) Ampere’s Law for “gB” 9

tE

cvB

∂∂

+=×∇r

rr20

1ρμ

tg

cv

cGg E

B ∂∂

+−=×∇r

rr2

14 ρπ

1. Magnetic Dipoles

(a) Magnets create fields• Magnets have 2 poles• No magnetic monopoles

(why? Nobody knows!)

B. GravitoMagnetic Moment 10

0=•∇ Br

6

• 1823 William Sturgeon shows Current Loop is equivalent to a magnet

• Magnetic Moment(N turns, Current I, area A)

(b). Electromagnets 11

NIA=μ

• Rotating Ball of Charge (e.g. electron) should have magnetic moment proportional to spin:

• Gyromagnetic Ratio:

• Anomalous “g” factor– g=1 for classical objects– g=2 however for pointlike electron (why?)

(c). Gyromagnetic Ratio 12

Srr γμ =

mge2

7

• Rotating Ball of mass should have gravitomagnetic moment proportional to spin angular momentum

• Anomalous “g” factor value is disputed– g=1 for classical objects???– g=2 for pointlike electron????

2. Gravitomagnetic Moment is Spin 13

SgG

rr=μ

(a) Magnetic Field of Dipole• At the equator, field is

parallel to surface of earth with value

3. Dipole Fields 14

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4)(

rrB μ

πμ

⎟⎠⎞

⎜⎝⎛=

8

• In analogy, the spin angular momentum “L” of the earth creates gravitomagnetic field.

• At equator it is parallel to surface of earth with value

(b). Gravitomagnetic Field of Earth 15

32 rL

cGgB ⎟⎠⎞

⎜⎝⎛=

• Acceleration on falling ball (equator of earth)

• Einstein Equivalence Principle is valid!Big and small balls will travel same path.

• Homework #1: After falling distance “d” how far sideways will a solid spinning ball go?

(c). Balls fall sideways! 16

32 rLv

cGggvg

mFa BE

rrrrrr

rr ×

⎟⎠⎞

⎜⎝⎛+=×+==

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1. Larmor Precession(a) Torque

on Magnet

(b) Precession

(c) LarmorFrequency

C. Precession 17

Brrr

×= μτ

BSdtSd rrr

×= γ

BSS γω ==&

(a) GravitomagneticTorque on Spin

(b) Precession equalsgravitomagnetic field

(c) “Frame Dragging”precession due to angular momentum “L” of Earth

2. Schiff Precession (1960) 18

BgS rrr×=τ

32 RcGL

BgSS==

Homework #2: Calculate precession rate at equator of earth!

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(a) Torque on MovingMagnetic Dipole

(b) GravitationalAnalogy

(c) “Geodetic” precessiondue to gravitoelectric field

3. de Sitter Precession (1916) 19

( )EvBc

rrrrr×−×= 2

1μτ

( )gvSdtSd

c

rrrr

××−= 21

Homework #3: Calculate precession rate for gyroscope in orbit around earth

( )EcB gvgS rrrrr×−×= 2

•• Confirmed Geodetic to 1%Confirmed Geodetic to 1%•• still analyzing Framestill analyzing Frame--dragging (dragging (gravitomagneticgravitomagnetic))

4a. Gravity Probe B (Launched 2004) 20

BgS rrr×=τ

( )EgvS rrrr××=τ

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•• Attempt to measure nuclear spin precession due Attempt to measure nuclear spin precession due to gravity interactionto gravity interaction

4b. The Kimball Experiment CSUEB 21

55--layer magnetic shieldlayer magnetic shieldconstructed:constructed:

magnetic field reducedmagnetic field reduced10,000,000 times!10,000,000 times!

There are 3 ways to think about mass

D. Spin and Inertial Mass 22

1. Inertial Mass F=ma

2. Passive Gravitational Mass F=mg

3. Active Gravitational Mass2r

GMg =

The “Weak Equivalence principle” says that inertial mass equals passive gravitational mass

Conventional view is that spin has no contribution to inertial mass (?)

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Gradient Forces due to spin

1. Gradient Forces Violate EEP 23

(a) Energy of MovingMagnetic Dipole

(b) Gravity Analogy

(c) Stern-Gerlach Forcewill depend upon spin!

( )EvBUc

rrrr×−•−= 2

UF −∇=r

( )EcB gvgSU rrrr×−•−= 2

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2. Gravitational Spin Orbit Coupling 24

• Gradient Force:

• Will not modify falling motion

• But, Gyroscope with spin “S” in orbit with angular momentum “L” around static mass will experience spin dependent force

⎥⎦

⎤⎢⎣

⎡ •±−= 22

2ˆcmrSLmgrFrr

r

( )EcgvSF rrrr

ו∇−= 21

Moving Spin in gravitoelectric field violates EEP

gv rr ||

2rGMg =

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3. Gravitational Spin-Spin Coupling 25

• Gradient Force:

• Earth with angular momentum “L” creates gravitomagnetic field

• Force between spinning earthand Gyroscope of spin “S” isanalogous to force betweentwo magnets

rrc

SLGF ˆ342

rrr •−=

( )BgSF rrr•−∇=

Spinning bodies will fall slower/faster !

Homework #4: How fast must a ball spin to be “weightless”?

4. Will mass be changed by Spin? 26

In EM, gradient forces make mass dynamic. Has anyone yet done the extension to MaxwellianGravity? The analogous equation would be:

• Hence spin-field interaction would change the inertial rest mass of particle

• Would it change the gravitational mass as well?

• Will it cause spinning balls to fall slower or faster?

( )EcBcgvgSmm rrrr

×−•±= 2211

00'

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D. Papapetrou Equations

In curved space, particles follow paths of least distance in 4D spacetime, called geodesics

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But this doesn’t give the right answer for spinning particles (or if there is torsion)

1. Spinning Particles Violate (Weak) EEP

• (1951) Papapetrou Equations:Shows spinning particles will deviate from “geodesics”

• Therefore, a spinning bodywill not fall the same as anon-spinning one.

• Violates Weak EEP.(but not Strong EEP?)

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Geodesic Part EEP violating

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2. Spin Papapetrou Equation

• There is more disagreement on proper formMy studies suggest:

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Geodesic Part EEP violating?

ωνω

αβμαβ

ωνωαμ

αμν SRSSxS m2

1−Γ= &&

3. Frenkel Equations (1926)

Spinning Particles don’tbehave the way you expect

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As it accelerates, the speed increases on the right, but decreases on the left

Higher speed increases the mass on the right side

Causes sideways contribution to momentum(will this save the Strong EEP for falling gyroscope?)

Newton’s Momentum Formula must be modified:

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E. References

• McDonald, Am. J. Phys. 65, 7 (1997) 591-2 (on Heaviside Gravity)• Oliver Heaviside, "A Gravitational and Electromagnetic Analogy",Part I, The

Electrician, 31, 281-282 (1893); Part II, The Electrician, 31, 359 (1893).• O. Heaviside. Electromagnetic Theory, Vol 1, pp. 455-465, (“The Electrician” Printing

and Publishing Co., London, 1894).• P. Nahin, “Oliver Heaviside” (John Hopkins Univ Press, 2002), does not mention much

on his gravity theory however.• R.L. Forward, "General Relativity for the Experimentalist", Proceedings of the Institute

for Radio Engineers 49, 892-904 (1961). This supposedly was the first revival of Heaviside's theory.

• Tomonaga, “The Story of Spin”, Univ Chicago Press (1997)• Bassett & Edney, “Introducing Relativity”, Icon Books (2002), p. 56• Hartle, “Gravity”, Addison Wesley (2003)• D’Inverno, “Introducing Einstein’s Relativity”, Clarendon Press (1992)• Misner, Thorne & Wheeler, “Gravitation”, Freeman & Company (1973)• Barut, Electrodynamics and Classical Theory of Fields and Particles• Wald, R, "Gravitational Spin Interaction," Physical Review D, 6(2), 406-413, (1972).

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Slides of all my talks available at:http://www.clifford.org/wpezzag/talks.html

Final Exam Talks

• 20 min?• In finals week at final exam slot for this

time (Dec 10?)

• Ideas for topics to follow

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Topic Ideas• Discuss the Sagnac effect, recent experiments and usage in

technology• Discuss one of the paradoxes (Ehrenfest, Barnett, Schiff), recent

review of papers.• Quaternions: review usage in animation, and/or demonstrate that

rotational calculations are indeed faster/better than standard methods.

• Frenkel equations: I have a short animation project written up that someone could do, demonstrate that there is a sideways shift in a moving spinning particle

• Thomas Precession: discuss effect in different contexts, for example due to rotation of earth, or orbits of planets.

• Photon Spin Paradox: discuss experiments which show that circularly polarized light does, or does not carry “spin”.

• Papapetrou Equations: Discuss whether these equations have been verified (storage rings?)