Classical And Extended Electrodynamics

114Physics Essays 32, 1 (2019)“Gauge” is used to describe Eqs. (15a) and (15b), originally arising from the nonstandard width of railroad track inthe 1800 s (a synonym for “arbitrary”). Equations (15a) and(15b) leave B and E unchanged, which is termed “gaugeinvariance.” A @2-infinitude of choices exists for the gaugefunction19 ðKÞ. For example, the velocity gauge isr A þ bel@U @t ¼ 0. If b ¼ 1, a charge source propagatesat the speed of light (Lorenz gauge). For b ¼ 0, U propagatesat infinite speed (Coulomb gauge). For 0 b 1; U propagates at a speed, c b. Other conditions are equivalent to theLorenz gauge with different physical meanings.5,6CED invariance under Eqs. (15a) and (15b) can be castin four-vector form as A ! A þ @ K. Here, A ¼ ðU c; AÞand @ ¼ ð @ @ct; rÞ with a metric signature of (–,þ,þ,þ).K is a harmonic, scalar function of space and time that satisfies @ @ K ¼ 0. Thus, the four-gradient component of A isgauged away under CED.20,21The assumption of no four-gradient in A is contrary toexperiments that have measured an irrotational vector potential.22–25 Section VI shows that an irrotational vector potential implies an irrotational current density. Moreover, anirrotational current been observed in (for example): arc discharges,26 ion-concentration-gradient-driven current acrossliving-cell walls,27 and irrotational, human, electroencephalogram current.28 This section has demonstrated three inconsistencies in CED: (a) the interface matching conditionbetween two different media is inconsistent at the theoreticallevel for a surface charge (1) and surface current (2); and (b)the irrotational component of the vector potential is gaugedaway, contrary to experimental measurements (3). Thesethree inconsistencies may not seem compelling in the lightof the success of Maxwell theory in modern physics. However, falsifiability29 states that a hypothesis (theory) cannotbe proved by favorable evidence, but can only be disproved,even by a single failure. These disparities require resolutionvia EED, as discussed next.IV. EXTENDED ELECTRODYNAMICSThe Helmholtz theorem8 uniquely decomposes anythree-vector into irrotational and solenoidal parts. For example, electrical current density has the form, J ¼ rj þ r G, with j and G, as scalar and vector space-time functions, respectively. Smooth, Minkowski four-vector fieldsalso can be uniquely decomposed into four-irrotational andfour-solenoidal parts with tangential and normal componentson the bounding three-surface.30 Woodside31 subsequentlyused the Stueckelberg Lagrangian density32 2ec2cec2 Fl Fl þ Jl Al @l Al42 ec2 k2 Al Al : 2L¼ (16)Fl is the Maxwell field tensor; c is the speed of light (notnecessarily vacuum); Jl ¼ ðcq; JÞ is the 4-current; the fourpotential is Al ¼ ðU c; AÞ; the Compton wave number for aphoton with mass ðmÞ is k ¼ 2pmc h; and h is Planck’s constant. The fully relativistic Stueckelberg Lagrangian32includes both A and U, and resolves many issues with previous CED Lagrangians. For c ¼ 0 and m 0, Eq. (16) yieldsthe Maxwell-Proca theory, for which a 2012 test measuredm 10 54 kg (equivalent to 10 18 eV), consistent withmassless photons.33 Equation (16) for c ¼ 1 and m ¼ 0 is31" # 2ec2 1@A2L¼rU þ ðr AÞ2 c2@t 2ec21 @Ur A þ 2:(17) qU þ J A 2c @tEquation (17) allows only two classes of four-vectorfields.31 One class of fields has zero four-curl of Al :Fl ¼ @ l A @ Al ¼ 0, with a solution,31 Al ¼ @ l K,together with a nonzero, dynamical, scalar field, C ¼ @l Al¼ @l @ l K. K is a scalar function of space-time. The secondsolution31 has zero four-divergence of Al , C ¼ @l Al ¼ 0, asthe Lorenz gauge, consistent with CED.4 Woodside7 laterproved the uniqueness of Eqs. (18)–(22) [7 equations in 7unknowns ðB; C; EÞ] that form EED:E ¼ rU @A;@tB ¼ r A;C ¼ r A þr B r E þ(18)(19)1 @U;c2 @t1 @E rC ¼ lJ;c2 @t@C q¼ :@te(20)(21)(22)Equation (18) is equivalent to Faraday’s law; Eq. (19) isequivalent to the no magnetic-monopoles equation. Equation(21) uniquely decomposes J into solenoidal ðr BÞand irrotational ðrCÞ parts, in accord with the Helmholtztheorem.8 Equation (17) implies34 Eqs. (18)–(22). Moreover,Eqs. (18)–(22) imply7,30,31 Eq. (17). Equation (17) is thennecessary and sufficient for Eqs. (18)–(22). The Lagrangianfor curved space-time35 reduces to Eq. (17) in Minkowskifour-space.A long history of work exists on EED.7,14,30,31,34–44 Fockand Podolsky36 wrote the new Lagrangian in 1932 with adynamical, scalar field, C ¼ r A þ el@U @t, withoutderiving the resultant equations. Ohmura37 first wrote thedynamical equations in 1956. Aharonov and Bohm34 gavethe revised Lagrangian and Hamiltonian, and derived thedynamical equations therefrom in 1963. Munz et al.14showed the use of EED in computational simulations in1999. Van Vlaenderen and Waser38 used EED to derive awave equation for C, and revised forms for momentum andenergy conservation in 2001. Woodside7,30,31 rigorouslyderived EED (1999–2009), assuming only Minkowskifour-space. Jim enez and Maroto35 used Eq. (16) with c ¼ 1and m ¼ 0 to model quantum, curved space-time, electrodynamics for an expanding universe in 2011. Thesepapers7,14,30,31,34–38 cited no previous EED work, andserve as seven independent verifications of EED theory.

Physics Essays 32, 1 (2019)115Modanese44 studied a nonrelativistic, nonlocal, EED quantum source.Equations (18)–(22) use the least-action principle,7requiring a finite, lower bound on the Lagrangian, Eq. (17).The Planck scale45 provides such a bound. Another term ðr AÞ2 2l, in Eq. (17), has the same requirement for afinite, lower bound, and has been well validated.4,8V. BASIC EED PREDICTIONSThe B wave equation arises from the curl of Eq. (21).Note that10 r r B ¼ rðr BÞ r2 B, which forEq. (19), B ¼ r A, gives r r B ¼ r2 B. Faraday’slaw, r E ¼ @B @t, along with10 r rC ¼ 0 yieldsthe EED B wave equation, identical to CED8r2 B @2B w2 B ¼ lr J:@c2 t2(23)The E wave equation relies on the equivalence ofEq. (18) to Faraday’s law, to which the curl is applied with10r r E ¼ rðr EÞ r2 E; replacing r B fromEq. (21); substituting for r E via Eq. (22); and noting that@rC @t r@C @t ¼ 0. The EED E wave equation is theCED result46r2 E @2Erq@Jþl :¼22@c te@t(24)The A-wave equation is obtained by: replacing B; E, andC in Eq. (21) with Eqs. (18)–(20); using the vector calculusidentity,10 r r A ¼ rðr AÞ r2 A; and noting that@rC @t r@C @t ¼ 0. The result is the CED A waveequation,8 Eq. (9), without the use of a gauge condition.The U wave equation can be obtained by: substitutingE and C from Eqs. (18) and (20) into Eq. (22); and notingthat @r A @t r @A @t ¼ 0. The result is the CEDU wave equation,8 Eq. (8), without the use of a gaugecondition. The usual wave equations for A and U are thenrigorously derivable without a gauge condition. Thus, EEDis gauge-free and predicts the same wave equations forA; B; E, and U as CED.Section III described inconsistent interface matchingconditions, which are resolved by this gauge-free theory.Namely, Eqs. (21) and (22) are alternative forms of the waveequations for A and U, respectively, as shown above. Thus,the interface boundary conditions for the A and U waveequations are the appropriate forms: Eqs. (12) and (14).A wave equation for C can be obtained from the divergence of Eq. (21); application of elð@ @tÞ to Eq. (22); andsumming the two results with10 r r B ¼ 0. The resultis18,38 @2C@qþr J :r C 2 2 ¼ l@c t@t2(25)The rigorous derivation of Eq. (25) eliminates the ad hocassumptions that were described in Section II to avoid theoverdetermination of Maxwell’s equations.Equation (25) is an instantaneous equation. But, allexperiments are performed over a finite time, DT, i.e., a timeaverage. A long-time average gives @q @t þ r J ¼ 0 onthe right-hand side (RHS) of Eq. (25), in accord withlong-standing experiments that validate classical chargebalance.47 For example, the lower bound on electron lifetimefor charge balance has been carefully measured as 6:6 1028 years48 (decay into two c-rays, each atme c2 2).Long-time charge conservation is not inconsistent withcharge nonconservation over short-time scales, DT Dt, perthe Heisenberg uncertainty relation, DEDt h 2. Here, DEis the charged-quantum-fluctuation energy; and h is Planck’sconstant divided by 2p. Equation (25) can be interpreted ascharge nonconservation driving C, and vice versa, not unlikeenergy fluctuations driving mass fluctuations in quantumtheory and vice versa.18 Thus, Eq. (25) predicts charge conservation on long time-scales (consistent with CED), andexchange of energy between C and quantum fluctuationsfor DT Dt. Confirmation of these quantum charge fluctuations involves tests, consistent with the Heisenberg uncertainty relation. One possible test could use the electron[DE(electron) ¼ me c2 ¼ 0.51MeV] corresponding to a time,Dt 6 10 22 s. Subzeptosecond dynamics have been measured,49 so a direct measurement of this prediction is feasible. Moreover, quantum fluctuations can control chargequantization,50 in accord with Eq. (25).The homogeneous solution to Eq. (25) is wavelike, withthe lowest-order form in a spherically psymmetricgeometry,4ffiffiffiffiffiffiffiC ¼ Co exp ½jðkr xtÞ r. Here, j ¼ 1; k is the wavenumber ð2p kÞ for a wavelength, k; x ¼ 2pf for a frequency, f ; and r is the spherical radius. Boundary conditionsfor Eq. (25) include Cðr ! 1Þ ! 0, which is triviallysatisfied. Equation (44) predicts that the energy density ofthe C field is ðC2 2lÞ, yielding a constant energy,4pr2 ðC2 2lÞ, through a spherical boundary around a sourcein arbitrary media, as required.18 The interface matchingcondition for Eq. (25) uses a Gaussian pill box with the endfaces parallel to the interface in regions 1 and 2. Noting thatr2 C ¼ r rC, use of the divergence theorem in the limit ofzero pill-box height yields continuity in the normal component (‘n’) of rC l for long times rCl ¼1nrCl :(26)2nThe subscripts, 1 and 2, denote medium 1 and medium 2 forl not necessarily in vacuum.VI. EED PREDICTION OF SLWSection III showed that the four-gradient component ofA is gauged away under CED,20,21 which is inconsistentwith experiments.22–28 Gauge-free EED eliminates this disparity by explicitly including solenoidal (or transverse,denoted by superscript, “T”) and irrotational (or longitudinal,denoted by superscript, “L”) parts in Eq. (21). Then, a longitudinal vector potential, AL ¼ ra, yields10

116Physics Essays 32, 1 (2019)B ¼ r AL ¼ r ra ¼ 0; or AL ¼ ra ) B ¼ 0:(27)Here, a is a scalar function of space-time. The inverse isBT ¼ 0 ¼ r AL ) AL ¼ ra; or(28)BT ¼ 0 ) AL ¼ ra:Combining Eqs. (27) and (28) givesAL ¼ ra () BT ¼ 0:(29)A longitudinal vector potential from Eq. (27) alsoimplies10r r AL ¼ r r ra ¼ 0¼ rðr AL Þ r2 AL ) rðr AL Þ¼ r2 AL ¼ r2 ra ¼ rr2 a:(30)Insertion of Eqs. (29) and (30) into the A wave equationgivesw2 AL ¼ w2 ra ¼ rw2 a ¼ lJ ) JL ¼ rj; orAL ¼ ra ) JL ¼ rj:(31)A corollary to Eq. (31), rw2 a ¼ lrj, isw a ¼ lj. The inverse is also true; J ¼ rj þ r GTallows decomposition of Eq. (21) into transverse and longitudinal parts2r BT T 1 @ETTT¼lr G)r B lGc2 @t@ 2 ATþ 2 2 ¼ 0;(32a)@c t1 @EL@ 2 AL rC¼lrj)@c2 t2c2 @t 1 @Uþ C þ lj :¼r 2c @t(32b)Rearrangement of Eq. (32b) with substitution of C fromEq. (20) and cancellation of terms, gives the same corollaryto Eq. (31) as cited above. Since time and spatial derivativescommute, the right-hand portion of Eq. (32b) results inJL ¼ rj ) AL ¼ ra:(33)Combining Eqs. (31) and (33) yieldsAL ¼ ra () JL ¼ rj:(34)The combination of Eqs. (29) and (34) relates B ¼ 0 andJLJL ¼ rj () BT ¼ 0:Equation (37) assumes spherical waves in linear conductive media: EL ¼ Er r exp ½jðkr xtÞ r, together withC ¼ Co exp ½jðkr xtÞ r. The unit vector in the radialdirection is r ; eo and lo are the free-space permittivity andpermeability, respectively; e0 and l0 are the relative permittivity and permeability (not necessarily vacuum), respectively; tan ðde Þ ¼ e00 e0 . Here, the same definitions are usedfor k; r; t; and x as in Section V. From Eq. (20), C has thesame dimensions as B ¼ lH. Consequently, Eq. (37) usesjEj ðC lÞ to obtain the SLW impedance, like the CEDform, Z ¼ jEj jHj. The C and E field energies fromEq. (44), 4pr 2 ðeE2 2Þ and 4pr 2 ðC2 2lÞ, are constant througha spherical boundary and Cðr ! 1Þ ¼ jEL ðr ! 1Þj ! 0.pffiffiffiffiffiffiffiffiffiffiffiEquation (37) predicts Zo ¼ lo eo , in free-space(e0 ¼ l0 ¼ 1 and e00 ¼ l00 ¼ 0). The SLW radiation patternfrom a monopole antenna is isotopic, and attenuates as r 2 infree space; see Appendix B,rffiffiffiI2 rl:(38)POUT ¼2eð4pr ÞThe SLW has not been previously observed, becausetransverse electromagnetic (TEM) antennas detect onlywaves that produce a circulating current. A TEM transmitterproduces only circulating currents, yielding C ¼ 0 and EL ¼0 with no SLW power output or reception; see Appendix C.The wave equations for A; B; E; and U are unchangedunder time reversal. A sign change occurs on both sides ofEq. (25) for t ! t that also gives time invariance, and indicates the pseudo-scalar nature of C. Time-reversibility ofEED implies that reciprocity holds. Then, a SLW transmittercan act as a receiver, and vice versa.VII. EED PREDICTION OF SCALAR WAVE(35)The net result of Eqs. (29), (34), and (35) isAL ¼ ra () BT ¼ 0 () JL ¼ rj:Equation (36) is consistent with the above-citedtests22–28 and drives the SLW, which is also called anelectro-scalar wave.38,51,52 Clearly, Eq. (36) holds onlywhere J is nonzero. We prefer the more descriptive phrase,“scalar-longitudinal wave,” or SLW, which is used throughout this paper. The explanation for JL driving a SLW is asfollows. The resultant electric field is EL ¼ JL r ¼ rj rfor media with a linear conductivity, r. EL and JL for theSLW are curl-free. Faraday’s law becomes r EL ¼ B ¼r rj r ¼ 0: The overdot denotes a partial-time derivative. Thus, no eddy currents occur, so the SLW is unimpededby the skin effect for propagation through linearly conductive media. See also Graham et al.53 and Appendix A.C; EL ; and JL are all related by Eq. (32b) with54 r ¼ eo e00 xand e ¼ eo ðe0 je00 Þ for a SLW impedance, Z,rffiffiffiffiffirffiffiffiffi0jEr jlo l 1 1 ðjkrÞ¼:(37)Z¼eo e0 1 j tan ðde ÞjC lo l0 j(36)EED also predicts a scalar wave (that has only a scalarfield, and is distinct from the SLW) under the two conditions: E ¼ 0 and Eq. (36).18 E ¼ 0 corresponds to zero onthe left-hand side (LHS) of Eq. (18). Then, the condition(AL ¼ ra) from Eq. (36) can be combined with the RHS of

Physics Essays 32, 1 (2019)117Equation (20) for the scalar fieldEq. (18) giving U ¼ a.can subsequently be rewritten by the replacements, U ¼ aand AL ¼ ra :C ¼ r Aþ1 @U1 @2a¼ r2 a 2¼ w2 a:2c @tc @tSubstitution of Eq. (39) into Eq. (25) yields18 @q224þ r J :w w a ¼ w a ¼ l@t(39)(40)Here, w2 is the wave operator as defined in Eq. (23). Use ofE ¼ 0 in Eq. (22) yields@C q¼ ) q ¼ eC ¼ ew2 a:@te(41)The last form in Eq. (41) comes from the partial-timederivative of Eq. (39). Equation (42) arises from Eq. (40) byreplacing @q @t with the partial-time derivative of Eq. (41),use of JL ¼ rj from Eq. (36) to evaluate r J, and rearrangement of the terms, resulting in the left-hand form1w a þ 2 w2 a þ lr2 jc a a22¼ w r a 2 þ 2 þ lr2 jcc 2 2¼ r w a þ lj ¼ 0:4(42)The second line of Eq. (42) arises from expansion ofthe wave operator. The third line of Eq. (42) results fromcancellation of the positive and negative a -terms along withinterchange of the wave- and Laplacian (r2 ) operators,which commute. One solution to Eq. (42) involves settingthe term inside the parentheses to zero, which is anotherform for the longitudinal component of the A wave equation [Eq. (9)] with AL ¼ ra and JL ¼ rj: This solutionarises, because the gradient and Laplacian operators commute for AL , per Eq. (30). A second solution to Eq. (42)involves setting the entire left-hand form to zero, whichresults in the AL wave equation being set equal to a harmonic function55 [HðrÞejxt ]w2 aðr; tÞ ¼ HðrÞejwt ljðr; tÞ:Substitution of Eq. (43) into Eq. (39) gives the same 1 rdependence for C in free space, as for the SLW. Revisedenergy balance, Eq. (44), shows that the scalar-wave energyis C2 2l 1 r2 . Thus, the total scalar-wave energy,4pr2 ðC2 2lÞ, is constant through a spherical boundaryaround a source and Cðr ! 1Þ ! 0, as expected. Revisedmomentum balance, Eq. (45), shows that the scalar wave hasa pressure of rC2 2l, but no momentum density.VIII. REVISED BALANCE EQUATIONSEED predicts a revised energy balance,18,38 Eq. (44),from the sum of: ðC lÞ times Eq. (22); ðB lÞ applied toFaraday’s law; and application of ðE lÞ applied toEq. (21)! @ B2 C2E B CE2þþ eE þ r þ@t 2l 2lllþ J E ¼qC:el(44)Equation (44) has new energy density terms: scalar fieldenergy ðC2 2lÞ, SLW energy ðCE lÞ, and a power sourceðqC elÞ.EED predicts revised momentum balance,18,38 Eq. (45),as the sum of: ðB lÞr B ¼ 0; the cross product of ðeEÞwith Faraday’s law; Eq. (21) ð B lÞ; ðC lÞ Eq. (21);and ð eEÞ Eq. (22). Equation (45) has new density terms:SLW momentum flux ðCE lÞ, TEM-SLW mixed mode flux½ðr BCÞ l , a force ðJCÞ parallel to the current density,and scalar-field pressure ðrC2 2lÞ. The last term in Eq. (45)is the divergence of the CED Maxwell stress tensor4 @ E B CEr BC þ qE þ J B þel@tlll¼ JC þ rC2þ r T :2l(45)Equation (44) predicts a power gain ðþCE lÞ withmomentum loss ð CE lÞ in Eq. (45), and vice versa. Thissign difference means that a SLW emission (power loss) drivesa momentum gain in a massive object that emits the SLW.(420 ÞIX. PRELIMINARY SLW EXPERIMENTSHere, the space and time dependences are shown explicitly. r2 HðrÞ ¼ 0 is typically solved by separation ofvariables.56 For example, in Cartesian coordinates, HðrÞPinvolves the sum HðrÞ ¼ Xn ðxÞYw ðyÞZf ðzÞ, where n2 þw2 þ f2 ¼ 0 and [Xn ðxÞ; Yw ðyÞ; Zf ðzÞ] are an appropriate set ofscalar functions of (x,y,z), respectively. The time-dependentterm (ejxt ) that is associated with HðrÞ is unaffected by theLaplacian operator. Nonhomogeneous solutions to Eq. (420 )are beyond the scope of the present work, being dependent onthe specific boundary conditions, geometry, and form forjðr; tÞ. The lowest-order, homogeneous solution to Eq. (420 ) isa¼ao jðxt krÞe:r(43)The present work eliminates sources of error in previoustests,51,52 which are discussed in Appendix D. High frequency ( 8 GHz) experiments facilitate an indoor, controlled test environment. Figure 1 shows the test layout. Thetransmitter and receiver are identical (inverted triangles inthe lower left of Fig. 1), since time-reversal symmetryallows the transmitter to act as a receiver, as discussed inSection VI. The directional couplers act as 45-dB isolators.Grounding to a single point avoids current loops. Moderndigital instrumentation allows accurate measurement of signal amplitudes and distances for comparison of test results toEED predictions.Figure 2 shows the linear, monopolar, SLW antennawith the coaxial center conductor as the radiator. The outer

118Physics Essays 32, 1 (2019)FIG. 1. (Color online) Test layout, with numbered items in the diagram corresponding to the tabular description above. This figure does not show Items 3,and 6–8, which are discussed in the text.coaxial conductor is electrically connected to the top of theskirt balun.57,58 The skirt balun length ðk 4Þ causes a phaseshift in the current flow along the guided path from the bottom (inside surface) of the skirt balun conductor (0 ) to thetop (inside surface) of the skirt balun (90 ) and back downthe outer surface of the coax outer conductor to the end ofthe skirt balun (180 ). The 180 -phase shift attenuates thereturn current along the outside of the outer coaxial conductor to form a monopole antenna, thus eliminating the imagecharge and image current of previous tests.51,52 The resultantfar-field contours of constant jEj from an HFSS electrodynamic simulation are essentially spherical, as expected for amonopolar antenna (top of Fig. 2). The RG-405/U coaxialcabling uses a solid, outer conductor to minimize stray fields;the presence or absence of an outer insulating jacket makesno difference in the results of the electrodynamic simulation.(A 3k-diameter ground-plane disk at the feed-point givesessentially the same jEj contours, thus confirming the linearmonopolar, counter-poise design.)The return-current attenuation (23 dB) of the previousparagraph is quantified in Fig. 3, as a sharp null at 7.94 GHz(shown in red online). Return-current attenuation allows themonopole antenna to draw charge from the ground plane(top of the skirt balun) and also creates an impedance matchbetween the antenna-balun (49.76–j0.24 X) and the source(50 X). Thus, the skirt balun reduces the return current alongthe outside of the outer coax conductor, so that essentially allof the electrical current goes into charging and dischargingthe antenna (an irrotational current) to drive the SLW, as predicted in Eq. (36). The test result for a single skirt balun(shown in green online in Fig. 3) shows the same trend as theHFSS simulation with a minimum of 23 dB at 8.00 GHz;the test result for a double balun was 42 dB (not shown).The difference in null depth arises from inaccu

Classical and extended electrodynamics Lee M. Hively1,a) and Andrew S. Loebl2 14947 Ardley Drive, Colorado Springs, Colorado 80922, USA 29325 Briarwood Blvd., Knoxville, Tennessee 39723, USA (Received 21 November 2018; accepted 8 February 2019; published online 25 February 2019) Abstract: Classical electrodynamics is modeled by Maxwell's equations, as a system of eight