Development of Spin Connection Resonance in the Coulomb Law

Transcription

1 22 Development of Spin Connection Resonance in the Coulomb Law by Myon W. Evans, Alpha Institute fo Advanced Study, Civil List Scientist. and and H. Eckadt, Alpha Institute fo Advanced Study, Deputy Diecto and Fellow. Abstact It is shown that the Coulomb law if developed in the context of a geneally covaiant unified field theoy poduces seveal classes of esonant phenomena due to the fact that the electomagnetic field is the Catan tosion of Einstein Catan Evans (ECE) field theoy. The electomagnetic potential is always defined with the spin connection, so the possibility of esonance is always pesent, in the sense that the potential of the Coulomb law can be amplified by damped o undamped esonance. In suitable mateials this esonance poduces fee electons which may be used fo powe geneation as fist demonstated by Tesla. Keywods: ECE theoy, spin connection esonance in the Coulomb law, new souces of electic powe Intoduction Recently a geneally covaiant unified field theoy has been developed based on standad Catan geomety [1 8] - Einstein Catan Evans (ECE) field theoy. One of the many consequences of the ECE theoy is that the fundamental

2 Development of Spin Connection Resonance in the Coulomb Law laws of classical electodynamics ae augmented. The ECE theoy educes to these well known laws in well defined limits, but also gives moe infomation based on the fact that the electomagnetic field tenso is the Catan tosion within a popotionality ca () in volts. The electomagnetic potential is always defined as the Catan tetad, so that the electomagnetic field always contains the spin connection. In the absence of the spin connection the ECE theoy educes staightfowadly to the standad Maxwell Heaviside (MH) theoy [9], because without a spin connection, space-time educes to the flat Minkowski space-time of MH theoy. In Section 22.2 the most geneal ECE equation of the Coulomb law is developed to show that thee exists a class of esonant solutions which can be demonstated staightfowadly. The Coulomb limit is defined and conditions fo damped and undamped esonance discussed. In Section 22.3, anothe novel class of esonant solutions is obtained by consideing a heteodyne type diving foce with a simple spin connection. Thee is feedom of choice of spin connection as long as the eduction to the Coulomb law is well defined. In the vast majoity of cases the Coulomb law is obseved to be vey accuate, but Tesla [1] was the fist to demonstate expeimentally that esonant powe can be obtained fom space-time. Theefoe behind the Coulomb law is hidden a new wold of possibilities fo obtaining esonant electic powe fom space-time Simple Resonant Solutions The basic spin connection esonance (SCR) equation of the Coulomb law [1 8] is witten in tems of the adial coodinate as: 2 ( ) φ 2 φ ω + φ ( 2 2ω + 2 ω ) = ρ (22.1) ɛ whee φ is the potential of the Coulomb law, is the adial coodinate, ω the spin connection, ρ the chage density and ɛ the vacuum pemittivity in S.I. units. Eq. (22.1) can be educed staightfowadly to the basic stuctue of the damped esonato equation, which was discoveed in the eighteenth centuy [11]: d 2 x dx +2β d2 d + κ2 x = A cos(κ). (22.2) In Eq. (22.2) β takes the ole of the fiction coefficient, and κ is a Hooke s law type wave-numbe. The ight hand side tem in Eq. (22.2) is a cosinal diving tem with a chaacteistic wave-numbe κ, and A is a popotionality constant.

3 Eq. (22.1) educes to Eq. (22.2) when: 22.2 Simple Resonant Solutions 325 ( ω =2 β 1 ), (22.3) κ 2 = 4 ( β 1 ) + ω. (22.4) Solving these equations defines the condition unde which the spin connection gives the simple esonance equation (22.2): ω = κ 2 4β log e 4. (22.5) Unde this condition, Eq. (22.1) becomes: 2 φ φ +2β 2 + κ2 φ = ρ, (22.6) ɛ an equation which gives well known esonant solutions and thei equivalent cicuits, so that the cicuits used fo example by Tesla can be designed and etched on to foundy mateial. Reduction to the Coulomb law occus when β = 1. (22.7) It is seen fom Eqns. (22.3) and (22.4) that unde the condition (22.7): so that the Coulomb law is obtained: ω =,κ 2 =, (22.8) 2 φ φ = ρ. (22.9) ɛ In geneal howeve thee is no eason to assume that condition (22.7) must always hold. The Coulomb law holds expeimentally in the vast majoity of applications, but in geneal elativity it is geatly eniched by the spin connection. The taditional stuctue (22.9) is egained if and only if the fiction coefficient is defined by Eq. (22.7). Reduction to the undamped esonato occus when: β = (22.1)

4 Development of Spin Connection Resonance in the Coulomb Law which implies: ω = 2, ω = 2 2,κ2 = 2. (22.11) If thee is dispesion [9] in the wave-numbe κ it becomes complex valued: The conjugate poduct is: and is positive valued, but the squae is : κ = κ + iκ. (22.12) κ κ = κ 2 + κ 2 (22.13) κ 2 = κ 2 κ 2 +2iκ κ. (22.14) Theefoe an undamped esonato equation of the type: 2 φ 2 + κ κ φ = ρ ɛ (22.15) can exist. At esonance it is well known that the solutions of the undamped esonato become infinite, signifying the elease of fee electons into a powe cicuit fom well chosen mateials [1 8]. 1 β = β = 1 β = 2 5 ω Fig Gaph of ω () foκ = 2 and thee β values.

5 22.3 Diect Solution of the SCR Equation fo the Coulomb Law 327 In Fig the function ω () (Eq. (22.5)) is plotted fo thee β values. Fo β =,ω, takes a fom of a shifted 1 function. Because we chose κ =2= const., we get a shift compaed to the theoetical esult of Eq. (22.11) fo ω. Fo all values of β thee is a common adius value ( = l) at which omega is zeo Diect Solution of the SCR Equation fo the Coulomb Law In [1 8] it was shown that the adial diffeential equation d d Φ Φ 2 + d2 d 2 Φ= ρ() ɛ (22.16) has to be solved in ode to get esonances of the Coulomb law. Φ() denotes the adial dependence of the potential and ρ() is a chage density seving as a diving foce. In Eq. (22.16) a special fom of the spin connection has been assumed. Accoding to the theoy of odinay diffeential equations, the most geneal solution consists of the geneal solution of the homogeneous equation (ρ = ) to which one paticula solution of the inhomogeneous equation has to be added. The solution of the homogeneous equation is d d Φ Φ 2 + d2 d 2 Φ = (22.17) Φ=k 2 k 1 2 (22.18) with abitay constants k 1 and k 2. By choosing k 1 = q (2πɛ,k ) 2 = we obtain the solution Φ C = q (22.19) 4πɛ which is exactly the Coulomb potential of a point chage q. Thus Eq. (22.17) is fully compatible to the Poisson equation fo a vanishing diving foce. The paticula solution of the full equation (22.16) is given by Φ p = 2 ρ()d 2 ρ()d 2ɛ (22.2) (we used the compute algeba system Maxima [12] to obtain this). Fom this solution it can be seen that the key fo the occuence of esonances

6 Development of Spin Connection Resonance in the Coulomb Law ae the two integals. If an oscillating function is inseted fo ρ(), at least one of the integals has to go to inifity fo cetain paametes contained in ρ(). In the following we will investigate this behaviou fo seveal pedefined functions ρ(). Ou fist choice is ρ() :=A cos(κ) (22.21) which is oscillating with an amplitude A and spatial fequency κ. The fist integal of (22.21) then esults to 2 cos(κ)da = ((κ2 2 2) sin(κ)+2κ cos(κ))a κ 3 (22.22) and the second to 2 cos(κ)da = 2 sin(κ)a. (22.23) κ Combining the tems of (22.21) leads to the paticula solution (sin(κ) κ cos(κ))a Φ p1 = ɛ κ 3 (22.24) This is an oscillating function without esonances (all paametes set to unity), see Fig Φ p1 () Fig Paticula solution Φ p1 ().

7 22.3 Diect Solution of the SCR Equation fo the Coulomb Law ρ() Fig Heteodyne function (intefeence of two nea-by fequencies). Next we use the diving foce ρ() :=A cos((κ + δ)) cos((κ δ)) (22.25) This is a heteodyne beat which is an intefeence of two nea fequencies κ + δ an κ δ. The gaph of this function is shown in Fig fo δ κ =.5. Calculating the solution in the same way as in Eqs.(22.21, 22.22) above leads to the esult afte some simplifications: Φ p2 = sin(2κ)a 16ɛ κ 3 + cos(2κ)a 8ɛ κ 2 sin(2δ)a 16δ 3 ɛ + cos(2δ)a 8δ 2 ɛ (22.26) This shows the following behaviou as gaphed in Fig Obviously the cuve follows the heteodyne fom of ρ(), cf. Fig We ae inteested in the behaviou δ, i.e. whethe thee is a esonance behaviou fo this case. The thid and fouth tem of Eq. (22.26) ae of the fom sin(ax) x 3 + cos(ax) x 2 with a constant a. Both tems divege fo δ but cancel each othe in pat so that the limit emains finite: Φ p2 (3 sin(2κ) 6κ cos(2κ)+8κ3 3 )A 48ɛ κ 3 (22.27)

8 33 22 Development of Spin Connection Resonance in the Coulomb Law Φ p2 () Fig Paticula solution Φ p2 (). 1 1 Φ p2 (δ ) δ Fig Paticula solution Φ p2 (δ) fo a constant adius = 6. The gaph fo Φ p2 (δ) fo a constant adius = 6 (Fig. 22.5) shows that Φ p2 neveveless gows significantly fo δ :

9 22.3 Diect Solution of the SCR Equation fo the Coulomb Law ρ() Fig Heteodyne function of Eq. (22.28). As a thid example we modify the diving foce by using the sine instead of the cosine function in the second facto: ρ() :=A cos((κ + δ)) sin((κ δ)) (22.28) The -dependence of this ρ() is shown in Fig Compaed with Fig. 22.2, this is only a phase shift. The paticula solution of this diving foce is Φ p3 = sin(2κ)a 8ɛ κ 2 + cos(2κ)a 16ɛ κ 3 sin(2δ)a 8δ 2 ɛ cos(2δ)a 16δ 3 ɛ (22.29) which at a fist glance looks simila to Φ p2 (Eq ) of the second example. Fom the gaph Φ p3 (Fig. 22.7) we can see, howeve, that the function diveges fo. In addition we obtain esonances fo any if delta appoaches zeo. This can be seen fom Fig. 22.8, which was calculated fo = 45 and has to be compaed to Fig The eason fo the unbound esonance is that the two diveging tems in Eq. (22.29) have the same sign. Compaed to Eq. (22.26), no compensation is pesent. We have to state that thee is an unsteady tansition in the limit δ. Then Eq. (22.28) tansfoms to ρ() :=A cos(κ)sin(κ) = sin(2κ)a 2 (22.3)

10 Development of Spin Connection Resonance in the Coulomb Law 2 Φ p3 () Fig Paticula solution Φ p3 () Φ p3 (δ ) δ Fig Paticula solution Φ p3 (δ) fo a fixed value of = 45.

11 22.3 Diect Solution of the SCR Equation fo the Coulomb Law δ =.5 δ =.3 δ =.2 δ = 1 Φ p3 () Fig Paticula solution Φ p3 () fosevealδ values. i.e. the diving foce is a pue sinoidal function with a high fequency, compaed to the ocsillating δ tems of Eq. (22.29). Fo deceasing δ the amplitude gows by the same facto ove the whole ange of, see Fig At the same time the wavelength inceases coespondingly. This behaviou collapses fo δ =. Intepeting this physically, diving heteodyne beats evoke an electical SCR potential which exhibits vey stong fluctuations fom and to the spacetime backgound. The amplitude of the oscillations does not go to zeo fo lage adii in contast to the Coulomb potential. This is a completely diffeent behaviou. Thee ae now two methods to constuct cicuits fom Eq. (22.16). Eithe one uses Eq. (22.16) diectly as shown in this pape o the equation is tansfomed by the Eule method so that the cicuit can be constucted simply as a LC esonance cicuit. The Eule tansfomation changes the diving foce so that the tansfomed foce has to be implemented in that case. This was the method descibed in [1 8]. The cuent method could have the advantage of being applicable moe diectly. Howeve, the convesion of Eq. (22.16) to a cicuit is not staightfowad. As a thid independent method fo evoking spacetime esonances, so-called vacuum fluctuations expeienced in the Lamb shift [1 8] have been identified. Acknowledgments The Bitish Govenment is thanked fo a Civil List Pension to MWE and the staff of AIAS and many colleagues fo inteesting discussions.

12 Refeences [1] M. W. Evans, Geneally Covaiant Unified Field Theoy (Abamis Academic, Suffolk, 25 to pesent, softback) vols. 1 5 (papes 1 92 on [2] L. Felke, The Evans Equations of Unified Field Theoy (Abamis Academic, Suffolk, 27). [3] M. W. Evans, Acta Phys. Pol., 38, 2211 (27). [4] ibid., M. W. Evans and H. Eckadt, in pess (27); M. W. Evans, in pess (27). [5] H. Eckadt, L. Felke, S. Cothes, D. Indanu, K. Pendegast and G. J. Evans, aticles on S. Cothes, Poc. Roy. Soc., submitted fo publication. [6] M. W. Evans and L. B. Cowell, Classical and Quantum Electodynamics and the B(3) Field (Wold Scientific, 21), Found. Phys. Lett. and Found. Phys., 1994 to pesent, see Omnia Opea section and hypelinks on [7] M. W. Evans (ed.), Moden Non-Linea Optics, a special topical issue in thee pats of I. Pigogine and S. A. Rice, Advances in Chemical Physics (Second Edition, Wiley Intescience, New Yok, 21), vols. 119(1) to 119(3); ibid. M. W. Evans and S. Kielich (eds.), Fist Edition, (Wiley Intescience, New Yok, 1992, epinted 1993 and 1997), vols. 85(1) to 85(3), see Omnia Opea section and hypelinks on [8] M. W. Evans and J.-P. Vigie, The Enigmatic Photon (Kluwe, Dodecht, 1994 to 22 hadback and softback), in five volumes, (see Omnia Opea Section and hypelinks on [9] J. D. Jackson, Classical Electodynamics (Wiley, 1999, Thid Edition). [1] All About Tesla, a film diected by Michael Kauss which pemieed in Belin in May 27, and has been shown in the Cannes film festival and Dylan Thomas Cente in Swansea, geneal elease. [11] J. B. Maion and S. T. Thonton, Classical Dynamics of Paticles and Systems (HBC, New Yok, Thid Edition, 1988), Chapte 3, Section 3.6. ff. [12] Maxima compute algeba system (