COMPLETE SOLUTIONS OF THE ECE2 FIELD EQUATIONS. ( \V\VW.et3m.net. W'vvw.archive.org.

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1 COMPLETE SOLUTIONS OF THE ECE2 FIELD EQUATIONS. by M. W. Evans and H. Eckardt Civil List and AlAS I UPITEC ( \V\VW.et3m.net. W'vvw.archive.org. ABSTRACT Complete solutions of the ECE2 field equations are given for an electromagnetic and gravitational free space plane wave, a static magnetic flux density, B, a static gravitomagnetic field, a static electric field strength E and the gravitostatic acceleration due to gravity g. In each case the complete solutions include the spin connection four vector components. Keywords, ECE2. complete solutions for the free space plane wave and static fields.

2 INTRODUCTION In the immediately preceding pap:r (UFT380) work was initiated towards a complete solution of the ECE2 covariant field equations { 1 12} of electromagnetism and gravitation. These complete solutions are also solutions ofthe ECE2 hydrodynamic field equations. The general solution requires consideration of a set of seven nonlinear partial differential equations in seven unknowns, the three components of the vector potential and the four components of the spin cmmection four vector. In Section 2, the solutions are written out in full for a free space plane wave of electromagnetism and gravitation, and for static fields in electromagnetism and gravitation. This paper is a brief synopsis of detailed calculations contained in the notes accompanying UFT381 on and (referred to as "combined sites"). Note 381 ( 1) gives the complete solution for plane waves of electromagnetism and gravitation in free space, solutions which include the relevant spin connections and which obey the antisymmetry laws ofece2. Note 381(2) gives particular solutions of the antisymmetry laws. Note 381(3) uses these particular solutions to give the complete solution for the static magnetic flux density (B) and static gravitomagnetic field (.!l ). Notes 381(4) to 381(8) gives the complete solution for the static electric field strength E. Section 3 gives a numerical and graphical analysis of selected solutions. 2. SOME COMPLETE SOLUTIONS The ECE2 covariant field equations of electrodynamics { 1 12} are: ~ (0 ~. ~ ~ ;. E ~ ( I 66 ( ~) 'r~y..~ ~ (KoG!.\\'\)<.~) (>}

3 in the notation of UFT316 and UFT317. Here B is the magnetic flux density. E the electric field strength, { is the electric charge density, E 0 is the vacuum permittivity ins. I. Units, J is the electric current density, and t 6 components of the kappa four vector: is the vacuum permeability. The are defined by: VCo ~;}_ ~~ 0, {&) ( and ~ \< l... {o) ( The spin cmmection four vector is defined by: c:j ~ ~ ( ~ ' ~) (\o) and the potential four vector is defined by: where f is as follows: ( j_ A\ (_ J ) tl1e scalar potential and~ is the vector potential. The relevant S. I. Units are

4 _, Yr.. The components of the kappa four vector are: and A (t~) in which the ECE hypothesis has been used: Ab =. F1 ( o) ~ 0 A.,. A c ) "\J In general the field equations allow for the existence of a magnetic charge I current density. i.e. a magnetic monopole (or charge density) and a magnetic current density. The magnetic charge density is zero if and only if: V< ~ _ 0 _ (r1) vc:c: ~ \ ~ x ~ ~ Q.. ( \~) In free space: 0 (\"') (do)

5 " ~)(..~ _..L~~Q_ c'l dt and ~. ~ c =( = 6 :1.3) It0 ~~ + ~ ')<_,0 ~= Q_. b4) c. where: Eqs. ( \1 ) to ( :l) ) are satisfied by the plane waves:, r ( ) r ( ) f ' ' ) o d..~ \C ~ ~ \..:\~ ' Ji... 'f ( 0 (o) ( ' ' ') ~~) ~ ~ D \~+~.e.... JJ 1":_ wt'vl~ (:n) ' and where G.> is the angular frequency of the wave at a point Z and instant t. The wave vector is defined by ~") \1( o ::: 0 I ~ \.{ "::. 2.. l '(); Eqs. ( d.3> ) to ( )\.r ) are satisfied by:, l \

6 i I< so: \Io (o) :::.. '(. CJ 0 C 0') and ~ (o") ;:... ( w. c~:l) The electric field strenath o IS. given. by: and the maanetic ::o flu x d ensity. by: ~ ::. d_fi 0 fl...::;: " I\ at When used with: Eqs. ( 3~ ) i and ( ~l.r) imply: (~~~) + ~ (wo~) ~ Y _Q_. {lj)

7 JA>< f Jl = CJ1. A"' = i vcj~1)(. (~.+0 the antisymmetry laws red uce to: d A f I Jl "" (.)l At " i ltlz. Ai (4d.) w"1'.a 1.,. (.) 1 A~ (~ It follows that: c~). and by inspection. the spin coimecti on ls. t 11e plane wave: (0 ~ ~; ( ii_ +~)e~if c~) w ~ A ~!=\ ( o) ( ( o) From Eq. ( 3~ ): where A 1 ls t 1e complex conjugate of A: f (\ 1 :... (\ {o) (, f ( ~ ~ ~~~ _;i...+_i)z. *) Finally, Eq. ( ~b ) implies that: ( 4Y,)

8 i I< The complete solution for the free space plane waves in the absence of a magnetic. VCo D J \X: C>o ::...OJ This is a simple example solution. In general as described in UFT380, the homogeneous field equations: 0 together with Eq. ( 3l ) to Eq. ( 3 q ) give seven equations in seven unknowns. These equations are given in UFT380. The antisymmetry laws ( ~\ ) to ( 3>'\ ) are fundamental to physics, as discussed in UFT131 UFT134 and in UFT350. They are a rigorous constraint and allow only certain types of solution. Note 381 (2) discusses some particular solutions such as: J~( (s~)

9 Another set of particular solutions is: CJz Ai { ss) 2>A~ )/ )A~ o ",(., Az (5b) ;;~} c.>, A~ (s~ These particular sol~~s are useful for finding the static magnetic flux density from the ECE2 field equations as follows. which has self consistent particular solutions: c) At ~X _w"a 7 = l _w 1 \ A~ )i ) { S"i) _ {bo) (t;g) Consider the well known { 1 12} magnetic planar vector potential: It follows that: which is Eq. ( 5~ ), Q.E.D. Using:

10 Eq. ( bd ) gives: Fm1hermore: From Eqs ( Sl ),( Lo ), c (~)and ( ~s ): C.,1::: _, \ w ')C... "::. ( ) )<.. ({,~ and the spin connection vector is: ~ Q.E.D. It follows as in Note 381(3) that: (. i) (,1) \ j_ I 6 \ + and: which is the required static magnetic flux density in the Z axis, Q. E. D. The electric field strength is zero in magnetostatics, so: 0 There is no time dependence in magnetostatics, so:

11 da ()t ~: 0 ' (10 It follows that: wo 0 _(!J) and that Eq. ( This is true from Eq. ( ) reduces to: The ECE2 Ampere ' Law: d :5t b ~ ), Q. E. D. ~ l (~)<.~) at. means that J vanishes for the magnetic flux density ( ~ '\ ). A net current density of zero is consistent with the fact that the electric charge density is zero because there is no electric field strength present. The complete solution for the static magnetic flux density B is therefore: ~ Ā = )~( )~ '~ ~...e l ~ ~ (.) (/! t Xi) ;t \ r.... \ ~~ ~ I The ECE2 field equations for the static electric field strength E are:

12 Eq. ( ll ) is the anti symmetry law of ECE2 electrostatics. Eq. ( ~ \ ) in component form gives three equations: " JAz _ J_Ac = w tfh_ wz A1 }/ )l 0 A)< )fh. = CJ~ A")(. C> '><. A l )l ;)"A da1 _ JAx "" )f w.,._ A1 Wy A~_ Eqs. ( As for magnetostatics: JA _ o (u) so ()"t ~ Y ( o. ~) "' (.>. ~ ><8_ + ~ )<_ 3_ w. ~ 0 (t1) which gives three components equations:

13 The Coulomb law: g1ves: J q. A t (J. '3... ~ t ~. 'S!. (.). dt in seven unknowns:,..., w w 7,wz. vvd...,.._.. ) J given that f If, is known experimentally.. So the static electric flux density can be found in general by solving these equations. An example solution can be found by assuming the Coulomb field, which is one of the

14 most accurately tested laws in physics: This has the solution :..,Z";l ~ rrf,\ J q )? =. e._:l r T ~'fifo(~ so the electric field strength is proportional to A: From Eq. ( 'b6 ): _, ~ l+ir E C> ( =.e:l ~ (~~ ~Eo (b J \2 woa. t::l \ (~ A J~o df :::. 0 ( '\q) so the scalar spin c01mection is time independent: From Eq. ( \<b ): Using: :o.w.,'!xa tbx3_w. =2. _(lot) (\o~)

15 it follows that: so 0 A possible solution is: so A '\1 (,..)e;, c ( ( \6~) <~ and Qo c... (to') ' The complete solution for the static electric field strength is \: = wo f\ ~0 :.C ;, ) _~~.(to~),3. ~'" Eo < j ~ _ ( \ o ~) C) ::. \ (' ). c A =~e._") ~) (''") ':) 4'\\ fo ( C (\\t) As shown in detail in Note 381 (5) a solution of identical structure exists for the gravitostatic field equations of ECE2:

16 where '[',h. is the gravitational scalar potential Q is the gravitational vector potential Q is the gravitational rour vector C(.t.. = ( ;; > ~ 1 l In) g is the gravitostatic field, i2_ is the gravitomagnetic field, G is Newton s constant and f~ is the source mass density. The relevant S. I. Units are as follows: ~ ~s. ~ Cf ) _, We s Vhls. l rhs (( I (\\~ Finally. Notes 381(6) and 381(7) check that the antisymmetry laws are obeyed. In ECE2 electrostatics for example: so:

17 et cyclicum. Therefore the antisymmetry law.r A s 1 or are obeyed: )A"" )A 1 ~ w.,._ n;_, w 7 1i x = o ( t0 ~f )~ et cyclicum.

18 i I< '.. ACKNOWLEDGMENTS The British Government is thanked for a Civil List Pension and the staff of AlAS and others for many interesting discussions. Dave Burleigh. CEO of Annexa Inc.. is thanked for hosting \V\V\v.aias.us. site maintenance and feedback software and hardware maintenance. Alex Hill is thanked for translation and broadcasting. and Robert Cheshire for broadcasting. REFERENCES ( 1} M. Vv'. Evans. H. Eckardt, D. W. Lindstrom and S. J. Crothers. ECE2: The Second Paradigm Shift'' (open access on combined sites \vww.aias.us and as UFT366 and epubli in prep.. translation by Alex Hill) {2} M. W. Evans. H. Eckardt. D. W. Lindstrom and S. J. Crothers. "The Principles ofece" (open access as UFT350 and Spanish section. epubli. Berlin hardback. New Generation. London. softback. translation by Alex Hill. Spanish section). {3} M. W. Evans. S. J. Crothers, H. Eckardt and K. Pendergast. "Criticisms of the Einstein Field Equation" (open access as UFT301. Cambridge International. 2010). {4} M. W. Evans. H. Eckardt and D. W. Lindstrom. "Generally Covariant Unified Field Theory" (Abramis in seven volumes softback. open access in relevant UFT papers. combined sites). {5} L. Felker, "The Evans Equations of Unified Field Theory" (Abramis open access as UFT302. Spanish translation by Alex Hill). ( 6} H. Eckardt. "The ECE Engineering Model" (Open access as UFT303. collected equations). [ 7} M. W. Evans. ' Collected Scientometrics (Open access as UFT307. New Generation 2015).

19 ( 8} M. W. Evans and L. B Crowell, 'Classical and Quantum Electrodynamics and the B(3) Field'" (World Scientific Open Access Omnia Opera Section ohvv.rw.aias.us). [9} M. W. Evans and S. Kielich (eds.), 'Modern Nonlinear Optics" (Wiley Interscience. New York ! ) in two editions and six volumes. {I 0] M. W. Evans and J. P. Vigier. ''The Enigmatic Photon'. (Kluwer to in five volumes hardback and softback. open access Omnia Opera Section of W\\W.aias.us ). [11} M. W. Evans. Ed.. 'Definitive Refutations of the Einsteinian General Relativity'' (Cambridge International open access on combined sites). [12} M.W. Evans and A. A. Hasanein. "The Photomagneton in Quantum Field Theory'' (World Scientific, 1994 ).