Lecture Series: The Spin of the Matter, Physics 4250, Fall Topic 8: Geometrodynamics of spinning particles First Draft.

Transcription

1 Lectue Seies: The Spin of the Matte, Physics 450, Fall 00 Topic 8: Geometodynamics of spinning paticles Fist Daft D. Bill Pezzaglia CSUEB Physics Updated Nov 8, 00 Outline A. Maxwellian Gavity B. Gavitomagnetic Moment C. Pecession D. Papapetou Equations E. Refeences

2 A. Maxwellian Gavity Gavity Analog of Electomagnetism Maxwell poposes gavitoelectic theoy 894 Heaviside (in 3 weeks) extends ideas to include gavitomagnetism 95 Einstein s Theoy (gavity is a tenso field, not a vecto field). GavitoElectic Field 4 (a) Electic Field 8 Faaday Fist poposes ideas of Lines of Foce Foce on Chage: F = qe ρ Gauss s Law: E = ε 0

3 (b) GavitoElectic Field is g 5 Gavitational Analogy (to electic foce): Foce Law: F = mg Field is acceleation of gavity g Chage is (gavitational) mass m (c) Gauss s Law fo g 6 83 Gauss s Law yields field aound chage: E = E = ρ ε 0 Q 4πε 0 ˆ Newton s Law implies field aound mass: g = GM ˆ 864 Maxwell wites: (764 Lagange) g = 4π Gρ 3

4 . GavitoMagnetic Field 7 (a) Magnetic Foce (86 Maxwell) F = q E + v B ( ) (b). Gavitational Analogy to Magnetic Heaviside poposes gavitomagnetic field g B F = m ( g + v ) E g B g B has not yet been diectly veified! [moe on Gavity Pobe B late] 4

5 (c) Ampee s Law fo g B Maxwell wites that moving chage ρ ceates magnetic field E B = μ 0 ρ v + c t 894 Heaviside poposes that moving mass ρ ceates gavito-magnetic field (effect vey small) [I don t know if he had the displacement tem] 4πG g E g B = ρ v + c c t B. GavitoMagnetic Moment 0. Magnetic Dipoles (a) Magnets ceate fields Magnets have poles No magnetic monopoles (why? Nobody knows!) B = 0 5

6 (b). Electomagnets 83 William Stugeon shows Cuent Loop is equivalent to a magnet Magnetic Moment (N tuns, Cuent I, aea A) μ = NIA (c). Gyomagnetic Ratio Rotating Ball of Chage (e.g. electon) should have magnetic moment popotional to spin: Gyomagnetic Ratio: μ = γ γ = ge m Anomalous g facto g= fo classical objects g= howeve fo pointlike electon (why?) S 6

7 . Gavitomagnetic Moment is Spin 3 Rotating Ball of mass should have gavitomagnetic moment popotional to spin angula momentum μ = S g G Anomalous g facto value is disputed g= fo classical objects??? g= fo pointlike electon???? 3. Dipole Fields 4 (a) Magnetic Field of Dipole At the equato, field is paallel to suface of eath with value B( ) = μ 0 4π μ 3 7

8 (b). Gavitomagnetic Field of Eath 5 In analogy, the spin angula momentum L of the eath ceates gavitomagnetic field. At equato it is paallel to suface of eath with value g B G = c L 3 (c). Balls fall sideways! 6 Acceleation on falling ball (equato of eath) F G v L a = = g E + v g B = g + 3 m c Einstein Equivalence Pinciple is valid! Big and small balls will tavel same path. Homewok #: Afte falling distance d how fa sideways will a solid spinning ball go? 8

9 C. Pecession 7. Lamo Pecession (a) Toque on Magnet τ = μ B (b) Pecession (c) Lamo Fequency ds dt = γ S B S ω = & = γ B S. Schiff Pecession (960) (a) Gavitomagnetic Toque on Spin τ = S g B 8 (b) Pecession equals gavitomagnetic field (c) Fame Dagging pecession due to angula momentum L of Eath S & ω = = S ω = g B GL c R 3 Homewok #: Calculate pecession ate at equato of eath! 9

10 3. de Sitte Pecession (96) (a) Toque on Moving Magnetic Dipole τ μ B c ( v E) = 9 (b) Gavitational Analogy τ S ( g v g ) = B c E (c) Geodetic pecession due to gavitoelectic field ds dt = c S ( v g) Homewok #3: Calculate pecession ate fo gyoscope in obit aound eath 4a. Gavity Pobe B (Launched 004) 0 Confimed Geodetic to % still analyzing Fame-dagging (gavitomagnetic( gavitomagnetic) τ = S g B τ = S ( v ) g E 0

11 4b. The Kimball Expeiment CSUEB Attempt to measue nuclea spin pecession due to gavity inteaction 5-laye magnetic shield constucted: magnetic field educed 0,000,000 times! D. Spin and Inetial Mass Thee ae 3 ways to think about mass. Inetial Mass F=ma. Passive Gavitational Mass F=mg 3. Active Gavitational Mass GM g = The Weak Equivalence pinciple says that inetial mass equals passive gavitational mass Conventional view is that spin has no contibution to inetial mass (?)

12 . Gadient Foces Violate EEP 3 Gadient Foces due to spin (a) Enegy of Moving Magnetic Dipole U μ = ( B v E) c (b) Gavity Analogy U S ( g v g ) = B c E (c) Sten-Gelach Foce will depend upon spin! F = U. Gavitational Spin Obit Coupling 4 Moving Spin in gavitoelectic field violates EEP Gadient Foce: Will not modify falling motion But, Gyoscope with spin S in obit with angula momentum L aound static mass will expeience spin dependent foce F L ˆ g m ± m S c = F = c ( S v g ) v g g = E GM

13 3. Gavitational Spin-Spin Coupling Gadient Foce: F = ( S ) g B 5 Eath with angula momentum L ceates gavitomagnetic field Foce between spinning eath and Gyoscope of spin S is analogous to foce between two magnets F = 3G L S ˆ 4 c Spinning bodies will fall slowe/faste! Homewok #4: How fast must a ball spin to be weightless? 4. Will mass be changed by Spin? 6 In EM, gadient foces make mass dynamic. Has anyone yet done the extension to Maxwellian Gavity? The analogous equation would be: m m ± S g v g ' 0 c ( ) = 0 Hence spin-field inteaction would change the inetial est mass of paticle B Would it change the gavitational mass as well? Will it cause spinning balls to fall slowe o faste? c E 3

14 D. Papapetou Equations In cuved space, paticles follow paths of least distance in 4D spacetime, called geodesics 7 But this doesn t give the ight answe fo spinning paticles (o if thee is tosion). Spinning Paticles Violate (Weak) EEP (95) Papapetou Equations: Shows spinning paticles will deviate fom geodesics 8 Geodesic Pat EEP violating Theefoe, a spinning body will not fall the same as a non-spinning one. Violates Weak EEP. (but not Stong EEP?) 4

15 . Spin Papapetou Equation Thee is moe disageement on pope fom My studies suggest: 9 S& μν = x& α Γ ω αμ S ων Geodesic Pat m S αβ R ω αβμ EEP violating? S ων 3. Fenkel Equations (96) 30 Spinning Paticles don t behave the way you expect As it acceleates, the speed inceases on the ight, but deceases on the left Highe speed inceases the mass on the ight side Causes sideways contibution to momentum (will this save the Stong EEP fo falling gyoscope?) Newton s Momentum Fomula must be modified: 5

16 E. Refeences 3 Slides of all my talks available at: McDonald, Am. J. Phys. 65, 7 (997) 59- (on Heaviside Gavity) Olive Heaviside, "A Gavitational and Electomagnetic Analogy",Pat I, The Electician, 3, 8-8 (893); Pat II, The Electician, 3, 359 (893). O. Heaviside. Electomagnetic Theoy, Vol, pp , ( The Electician Pinting and Publishing Co., London, 894). P. Nahin, Olive Heaviside (John Hopkins Univ Pess, 00), does not mention much on his gavity theoy howeve. R.L. Fowad, "Geneal Relativity fo the Expeimentalist", Poceedings of the Institute fo Radio Enginees 49, (96). This supposedly was the fist evival of Heaviside's theoy. Tomonaga, The Stoy of Spin, Univ Chicago Pess (997) Bassett & Edney, Intoducing Relativity, Icon Books (00), p. 56 Hatle, Gavity, Addison Wesley (003) D Inveno, Intoducing Einstein s Relativity, Claendon Pess (99) Misne, Thone & Wheele, Gavitation, Feeman & Company (973) Baut, Electodynamics and Classical Theoy of Fields and Paticles Wald, R, "Gavitational Spin Inteaction," Physical Review D, 6(), , (97). Final Exam Talks 0 min? In finals week at final exam slot fo this time (Dec 0?) Ideas fo topics to follow 6

17 Topic Ideas Discuss the Sagnac effect, ecent expeiments and usage in technology Discuss one of the paadoxes (Ehenfest, Banett, Schiff), ecent eview of papes. Quatenions: eview usage in animation, and/o demonstate that otational calculations ae indeed faste/bette than standad methods. Fenkel equations: I have a shot animation poject witten up that someone could do, demonstate that thee is a sideways shift in a moving spinning paticle Thomas Pecession: discuss effect in diffeent contexts, fo example due to otation of eath, o obits of planets. Photon Spin Paadox: discuss expeiments which show that ciculaly polaized light does, o does not cay spin. Papapetou Equations: Discuss whethe these equations have been veified (stoage ings?) 7