Space-Time Electrodynamics, and Magnetic Monopoles

Transcription

1 Gauge Insiue Journal Space-Time lecrodnamics and Magneic Monopoles June 203 Absrac Mawell s lecrodnamics quaions for he 3- Space Vecor Fields disallow magneic monopoles. Those equaions could no be wrien for he Space-Time Vecor Fields because i is believed ha he curl of a four dimensional Vecor Field is a si dimensional Vecor Field.

2 Gauge Insiue Journal Recenl we showed ha he 4-space curl is a 4-vecor and supplied he correc formula for i. Thus we obain he Curls of he Space-Time and of he Space-ime. We show ha he Curl of he space-ime wih 0 differs from he Curl of he Spaial. Similarl he Curl of he space-ime wih 0 differs from he Curl of he Spaial. On he oher hand he divergence of he space-ime wih 0 equals he divergence of he spaial and he divergence of he space-ime wih 0 equals he divergence of he spaial. Also he Power ensi of he Space-ime wih 0 and wih 0 equals he Power densi of he Spaial and. The ivergence of he space-ime allows Magneic Monopoles. The space-ime and can be derived from space-ime poenials ϕ and Α and remain unchanged under gauge ransformaions of hese poenials. 2

3 Gauge Insiue Journal Kewords: Infiniesimal Infinie-per-real per-real Cardinal Infini. Non-Archimedean Calculus Limi Coninui erivaive Inegral Gradien ivergence Curl Mawell quaions lecrodnamics lecromagneic Fields Space-ime Vecor Fields Magneic Monopoles 2000 Mahemaics Subjec Classificaion 2635; 2630; 265; 2620; 26A06; 26A2; 030; 0355; 037; 035; 46S20; 97I40; 97I30. 3

4 Gauge Insiue Journal Conens Inroducion. 4-Space Curl 2. 4-Vecors Cross Produc 3. Space-ime lecric and Magneic Fields 4. Curls of Space-ime Fields 5. ivergences of Spaceime Fields and Magneic Monopoles 6. Space-ime Poenials and heir Gauge Transformaions References. 4

5 Gauge Insiue Journal Inroducion 0. lecro-magneic Fields In lecrodnamics we denoe () () () isplacemen (Coulomb/(meer) 2 ) () () lecric. (Vols/meer) () The isplacemen and he lecric Fields are relaed b where in an isoropic medium 2 ε ε ε 0 n ε 0 ( + χ) Permiivi n Refracive Inde χ Suscepibili. In nonlinear isoropic medium χ ma depend on powers of he lecric field. In Opical Fibers we assume () (3) 3 χ χ + χ. 5

6 Gauge Insiue Journal Then assuming a harmonic plane elecric field oscillaing a angular speed ω propagaing along he Fiber he refracive inde depends on ω oo and n n( ω ) In a non-isoropic medium ε ε ij is a 3 3 mari. If non-linear each ε depends on ω and ij he elecric field. Similarl we denoe () () () () () () Inducion (Weber/(meer) Magneic (Ampere/meer) The Inducion and he Magneic Fields are relaed b where in an isoropic medium μ 2 ) μ μ μ Permeabili 0 r μ r Relaive Permeabili. 0.2 lecrodnamics quaions in 3-space 6

7 Gauge Insiue Journal Mawell s lecrodnamics quaions for he 3-Space Vecor Fields are (I) iv + + ρ (Coulomb/(meer) 3 ) where ρ elecric charge densi. iv 0 (II) (Weber/(meer) 3 ) + + where he magneic charge densi is ero. Thus quaion (II) means no magneic monopoles. (III) Curl (Vol/(meer) 2 ) (IV) Curl J + (Ampere/(meer) 2 ) J J J where J conducion Curren ensi. 7

8 Gauge Insiue Journal 0.3 rroneous Curl in Four imensions Mawell s equaions could no be wrien for he Space-Time Vecor Fields because i is believed ha he curl of a four dimensional Vecor Field is a si dimensional Vecor Field. The definiion of he Curl of a Vecor Funcion is based on he area densi of circulaion and requires he concep of circulaion and he use of infiniesimals. Since infiniesimals were avoided and limis are vague he Curl in man es is defined b he resul ha follows from is definiion. Namel in Caresian Coordinaes he Curl is defined b P () R Q Q () P R R ( Q P + + Q R R Q P Q 8

9 Gauge Insiue Journal P Q R. Clearl he 3-Space Curl is a 3-dimensional vecor funcion wih componens in he direcion of he uni vecors and. I is no self eviden how his resul ha became definiion ma be generalied o 4-Space wih is base of four uni vecors and. For insance we can add a column o obain P Q R S???? bu wha will be a fourh raw of ha 4 4 deerminan? I is less eviden how he fac ha here are si erms of he form i j j i ha deermine he 4-space curl lead o he belief ha he 4- Space curl is si dimensional. 9

10 Gauge Insiue Journal To obain he formula for he 4-Space curl we need o define i hrough he circulaions in 4-space. 0.4 Correc 4-space Curl In [an4] we showed ha he 4-space curl is a 4-vecor and supplied he correc formula for i. Appling his formula o he Space-Time and we obain he lecrodnamics quaions for he Space-Time lecro-magneic Fields. Those equaions allow Magneic Monopoles. 0

11 Gauge Insiue Journal. Curl of 4-Vecor Le P () Q () R () and S () be hperreal differeniable funcions defined on an infiniesimal area elemen ds. he uni vecors ds projecs ono si 2-planes generaed b and. - projecion wih area dd and normal - projecion wih area dd and normal - projecion wih area d d and normal - projecion wih area d d and normal

12 Gauge Insiue Journal - projecion wih area dd and normal ( ) ( ) ( ) 0 - projecion wih area dd and normal ( ) ( ) ( ) 0 The projeced areas are ds nds dd ds nds dd + dd ds nds dd + dd ds ds dd n The projecions areas are walls of a bo wih vere a ( ) and sides d d d and d. 2

13 Gauge Insiue Journal Given posiive orienaion of a righ hand ssem P () P ( ) dl+ Q ( ) dl Q () l l dd ( ds ) P () Q () Q () Q ( ) dl+ R ( ) dl R () l l dd ( ds ) Q () R () R () R ( ) dl+ S ( ) dl S () l l dd ( ds ) R () S () S () S ( ) dl+ P ( ) dl P () l l dd ( ds ) S () P () S () S ( ) dl+ Q ( ) dl Q () l l dd ( ds ) 3

14 Gauge Insiue Journal S () Q () P () P ( ) dl+ R ( ) dl R () l l dd ( ds ) P () R () The 4-space Curl is he sum of he si area curls. Tha is P () Q () R () S () R S P P S Q S P R Q Q R R S S P P R P Q S Q Q R S R P S R P +. Q P Q S + R Q 4

15 Gauge Insiue Journal 2. Cross-Produc of 4-Vecors The Cross-produc of 4-vecors is he sum of si crossproducs. Tha is P P 2 Q Q 2 R R 2 S S 2 P P 2 R R 2 S S + 2 P P + 2 R R 2 S S 2 R S S P P R R S S P P R P P 2 S S 2 Q Q Q Q 2 Q Q 2 R R 2 P Q S Q Q R P Q S Q Q R 5

16 Gauge Insiue Journal R S R2 S 2 S P P R + S2 P2 P2 R 2 P Q S Q + P2 Q2 S2 Q2 Q R Q2 R 2. 6

17 Gauge Insiue Journal 3. Space-ime lecric and Magneic Fields We shall assume ha Space-ime elecric and magneic fields have four componens along he aes. 3. Space-Time lecric-flu ( isplacemen) Field () () () () (Coulomb/(meer) 2 ). 3.2 Space-Time Poenial-Gradien (lecric) Field () () () () (Vols/meer). 7

18 Gauge Insiue Journal 3.3 Space-ime Magneic-Flu (Inducion) Field () () () () (Weber/(meer) 2 ) 3.4 Space-ime Curren-Gradien (Magneic) Field () () () () (Ampere/meer) We shall assume ha he Space-ime Magneic and he lecric Fields are perpendicular. 3.5 Assuming Spaial Spaial Proof:

19 Gauge Insiue Journal We shall assume ha he Space-ime Magneic and he lecric Fields propagae in direcion perpendicular o boh 3.7 Assuming Propagaion in direcion Perpendicular o Space-ime and o Space-ime Propagaion is in direcion perpendicular o he spaial and o he spaial 3.9 Space-ime Power ensi + + Proof: 2. 9

20 Gauge Insiue Journal Power ensi of Space-ime Power ensi of Spaial 20

21 Gauge Insiue Journal 4. Curls of Space-Time Fields 4. Curl of Space-ime () () () + () 0 Proof: Appling he Formula for he 4-dimensional Curl () + () () + ()

22 Gauge Insiue Journal 4.2 () () () Curl of Space-ime Curl of Spaial 4.4 Curl of Space-ime () 0 () J () J + () 0 J Proof: Appling he Formula for he 4-dimensional Curl () + J () () + J+ () J + 22

23 Gauge Insiue Journal 0 J J. + 0 J 4.5 () 0 () J () J + 0 J Curl of Space-ime Curl of Spaial 23

24 Gauge Insiue Journal 5. ivergences of Space-ime Fields and Magneic Monopoles 5. ivergence of Space-ime Proof: () () ρ () + () () () () ρ () lecric Monopoles even wih ρ 0 24

25 Gauge Insiue Journal 5.3 () () () 0 ρ ivergence of Space-ime ivergence of Spaial 5.5 () () () () () () Proof:. () () 25

26 Gauge Insiue Journal Magneic Monopoles 5.7 () () () ivergence of Space-ime ivergence of Spaial 26

27 Gauge Insiue Journal 6. Space-ime Poenials and heir Gauge Transformaions enoe a 4-vecor b (4) V (3) and a 3-vecor b V We define a space-ime scalar elecric poenial ϕ () and a space-ime 4-vecor Magneic Poenial We define A (4) () A () A () A (). A () 6. Space-ime lecric Field (4) (4) (4) ϕ A. 6.2 (4) (3) ϕ A (4) (3) ϕ A (4) (3) ϕ A. (4) ϕ A ϕ A 27

28 Gauge Insiue Journal We define 6.3 Space-ime Magneic Inducion (4) (4) (4) A. 6.4 A A (4) A A A A + (3) (4) (4) A A + A A (3) (4) A A (3) A A (3) A A. (3) A + A (3) 6.5 ψ (4) (4) (4) 0 Proof: ψ ψ ψ ψ ψ ψ ψ ψ + ψ ψ ψ ψ ψ + ψ ψ ψ (4) (4) 0. 28

29 Gauge Insiue Journal 6.6 Gauge Transformaion of ϕ and A (4) For an scalar Poenial Λ() he Poenials Φ ϕ Λ (4) (4) (4) A A + Λ are called Gauge Transformaion of ϕ and A (4) 6.7 For an scalar Poenial Λ() (4) (4) (4) Φ A (4) (4) (4) A. Proof: (4) (4) (4) ϕ A (4) (4) (4) ( ϕ Λ ) ( A + Λ). Φ A (4) (4) (4) ( A + Λ). (4) (4) (4) A (4) A (4) 29

30 Gauge Insiue Journal References [Cohen]. Richard Cohen The Phsics Quick Reference Guide American Insiue of Phsics 996. [an] Infiniesimals in Gauge Insiue Journal Vol.6 No 4 November 200; [an2] Infiniesimal Calculus in Gauge Insiue Journal Vol.7 No 4 November 20; [an3] Infiniesimal Vecor Calculus posed o ecember 20; [an4] 4-space Curl is a 4-Vecor posed o Januar 202; [Fischer] A C Fischer-Cripps The Phsics Companion Insiue of Phsics [ughes/galord] William ughes and ber Galord asic quaions of ngineering Science Schaum