Radiation Reaction and the Vacuum Gauge Electron. June 10, 2016

Transcription

1 Radiation Reaction and the Vacuum Gauge Electron June 10, 016

2 ...This tremendous edifice which is such a beautiful success in explaining so many phenomena, ultimately falls on its face. R.P. Feynman, The Feynman Lectures, chapter 8

3 Abstract The velocity field of the vacuum gauge electron is a vacuum radiation field. A discussion of the emission of electromagnetic radiation by a charged particle must therefore be modified accordingly. In addition, the motion of the velocity field provides a new perspective by which the well known problem of the radiation reaction can be addressed. The violation of energy conservation forces out the Lorentz-Dirac equation and generates solutions for straight line and circular motions without the difficulties associated with the conventional theory. 3

4 Contents 1 Covariant Radiation Theory Applications of the Divergence Theorem Accelerated Motions of the Electron Theory of the Radiation Reaction 13.1 Applications to Straight Line Motion Application to Circular Motion: List of Figures 1 Gauss law applied to the vacuum tensor Gauss law applied to the total stress tensor Transverse power distribution for straight line motion Vacuum gauge electron in a harmonic oscillator potential Transverse power distribution for circular motion Perpetual radiation cyclotron List of Tables 1 Inertia estimates for an accelerating electron

5 1 Covariant Radiation Theory The premise of a causal velocity field generates a theory of the vacuum as a deformable medium. In close proximity to a charged particle, stresses and strains are characterized by the vacuum tensor µν = η µν g µν η (1.1) where η µν is determined from velocity potentials obeying the vacuum gauge condition. An appropriate lagrangian is constructed as L vac 1 8π µν η µν 1 c jν ea ν (1.) and leading to the Maxwell-Lorentz equations of motion. Propagation of vacuum radiation thru the velocity fields can be accomodated by implementing a symmetry operation on the vacuum tensor of the form µν µν 4πσ e g µν (1.3) which leads to the modified lagrangian L vac 1 8π µν η µν + 1 σ e η 1 c jν ea ν (1.4) The symmetric stress tensor for the velocity fields, inclusive of the propagation stress tensor Λ µν, follows from equation (1.4) by calculating the Noether current under the infinitesimal translation group. The calculation requires the removal of a single superfluous divergence free term yielding the total tensor T µν = 1 [ 1 4π gµν η η µλ η λ] ν + Λ µν (1.5) Stresses associated with accelerations of the particle can then be addressed with the inclusion of the acceleration strain tensor ɛ µν = A µ a ν providing small changes to the velocity strain of the form η µν η µν ɛ µν (1.6) Inserting (1.6) into the symmetric stress tensor leads to the generalized total tensor T µν = 1 [ 1 4π gµν η η µλ η ν λ + η µλ ɛ ν λ ɛ µλ ɛ ν λ + ɛ µλ η λ] ν + Λ µν (1.7) but a more concise formulation can be constructed from the definition R µν 1 [ η µλ η ν λ η µλ ɛ ν λ + ɛ µλ ɛ ν λ ɛ µλ η ν 4π λ] (1.8) The equation representing the most general motions of the particle is then T µν = 1 4 gµν R R µν + Λ µν (1.9) This tensor may be referred to as the total electromagnetic stress tensor. It can be shown to be completely equivalent to the electromagnetic theory except for the added capability for propagating longitudinal electromagnetic modes thru the velocity field. 5

6 1.1 Applications of the Divergence Theorem Components of T µν which fall off as ρ 4 are independent of accelerations; They characterize stress energy of the deformed vacuum in the neighborhood of a vacuum portal of radius ρ = r e. An application of the Gauss integral theorem must respect this boundary. In contrast, components of Λ µν are defined both inside and outside the radius. Both tensors are also constrained by a limited causal reach, but the ρ 4 velocity fields may be extended to infinity for reasonably large times. Vacuum Tensor: In the simplest case of constant velocity motion Gauss law can be applied to the vacuum tensor over the oblique hyper-cylinder shown in the central part of figure 1. Three surface integrals replace the four dimensional volume integral producing µ Λ µν d 4 x = V cyl Λ µν β µ d 3 σ + S 1 Λ µν β µ d 3 σ S Λ µν U µ d 3 ς S 3 (1.10) Figure 1: Dashed red line is the world line of the particle. The cylinder has a radius ρ = r e. But the divergence on the left side of the equation is zero along with the surface integral over S 3. This implies that integrals over the two hyper-ellipses are the same 6

7 to within a sign. If the radius is ρ = r e these integrals also determine the energymomentum four-vector of the particle: S 1, Λ µν β µ d 3 σ = ±mc β µ (1.11) A slightly more complicated problem is to re-write equation (1.10) with contractions on the second index. In this case the divergence on the left side of the equation is a delta function so the volume integral is non-zero. The integrals over the ends of the cylinder still cancel and this requires the integral over S 3 to be non-zero. In fact, to within a sign, each of the remaining integrals is equal to the energy-momentum four-vector of the electron multiplied by an appropriate time interval: mc β µ τ τ e = ν Λ µν d 4 x = V cyl Λ µν U ν d 3 ς S 3 (1.1) In words, this equation relates a point source to a flux integral over S 3 which determines the inertia of the particle. Covariant integrals of the vacuum tensor can also be applied over the causal light cone shown in figure 1, and defined by the two surfaces S 4 and S 5. The analysis is similar to the hyper-cylinder and includes contractions on either index. Choosing the second index renders the equation ν Λ µν d 4 x = Λ µν β ν d 3 σ + Λ µν R ν d ω (1.13) V lc S 4 S 5 A little work then shows that for arbitrary proper time τ the integrals on the right are related by 1 Λ µν R ν d ω = Λ µν β ν d 3 σ = mc β µ τ (1.14) S 5 S 4 τ e This result can now be compared with equation (1.1). If the initial and final proper times are such that τ = τ τ o then the general result can be written τ Λ µν U ν d 3 ς = Λ µν β ν d 3 σ (1.15) τ In words, equation (1.15) says that the total flux of energy thru the particle radius over a proper time τ is equal to the energy density in the vacuum at time τ integrated over the particles now plane. Total Electromagnetic Stress Tensor: Minkowski space integrals of the vacuum tensor can be generalized to include the total tensor T µν inclusive of accelerated motions. Accelerated motions appear to be a complicated issue so it is worthwhile to begin by assuming constant velocities only. 7

8 The divergence formula for T µν is µ T µν = 1 c ηλν je λ (1.16) and the application of Gauss law over the causal light cone is I = µ T µν d 4 x = T µν β µ d 3 σ + T µν R µ d ω (1.17) S 1 S However, contacting on indices on the right side of the equation leads to Figure : Surfaces of the causal light cone inclusive of the dashed line surface surrounding the world line of the particle and defined by ρ = r e. I = S 1 [ 1 8π η 1 ] [ 1 σ eη β ν d 3 σ + S 8π η 1 ] σ eη R ν d ω (1.18) Both integrands vanish at the radius of the electron (ρ = r e ) so at no cost, a tube-like three-surface can be included about the world line of the particle. Moreover, the introduction of this surface necessarily avoids the point charge current density je λ so the value of the integral I is zero. Introducing this extra surface as shown in figure and noting that the integrands are exactly the vacuum lagrangian leads to L vac β ν d 3 σ + L vac R ν d ω + L vac U ν d 3 ς = 0 (1.19) S 1 S ρ=r e If each of the three-surfaces are viewed collectively as S ν then a re-application of Gauss law, derives the fundamental integral statement of vacuum gauge electrodynamics for a particle having a radius r e : ν L vac d 4 x = L vac ds ν = 0 (1.0) V 8 S

9 The constant velocity theory can be extended to accelerated motions by including acceleration terms in the symmetric stress tensor. Writing then various contractions are Θ µν vac = Θ µν 1 + Θ µν + Θ µν 3 (1.1) Θ µν R ν = 0 Θ µν 3 R ν = 0 (1.) Θ µν β ν = e 4πρ 3 aν Θ µν 3 β ν = e 4πρ 3 (ξ + a λ a λ )R µ (1.3) Θ µν U ν = e 4πρ 3 aν Θ µν 3 U ν = e 4πρ 3 (ξ + a λ a λ )R µ (1.4) Showing that equation (1.0) holds for accelerated motions will then hinge on equality of the two spacelike and timelike integrals Θ µν,3 β ν d 3 σ = S 1 Θ µν,3 U ν d 3 ς S (1.5) 9

10 1. Accelerated Motions of the Electron Theory of Inertia: One of the most striking features of vacuum gauge electrodynamics is the emergence of a theory of inertia arising naturally from an evaluation of the total power radiated by the electron. For general accelerated motions, a differential quantity of radiated energy can be labeled de rad which follows immediately from the covariant integral de rad = β µ T µν U ν ρ dω dτ = β µ [ 1 4 gµν R R µν + Λ µν] U ν ρ dω dτ (1.6) Evaluating the double contraction shows that the only surviving terms are those proportional to ρ. These are the radiation fields of the electron and the total energy radiated by the particle per unit proper time then follows as de rad [ = β µ ɛ µλ ɛ ν λ + Λ µν] U ν ρ dω (1.7) dτ Evaluating individual terms removes the dependence on ρ giving the solid angle integral [ ] de rad 1 = dτ σ ee e 4π (ξ + a λ a λ ) dω (1.8) A final integration then gives the fundamental result de rad dτ = P in + P acc (1.9) where P in is the inertial power radiated by the velocity field and P acc is the well known Liénard formula for the acceleration fields. The inertia of the particle drops out when the total power is integrated over a proper time interval τ e. Except for exceedingly large accelerations, this time is short enough that it is sufficient to simply multiply it by the radiated power. The instantaneous mass-energy of the accelerating particle is then adequately determined from Now eliminate the classical radius to obtain P τ e = πσ e e r e e 3 c a λa λ r 4 e = m c (1.30) e 4 m = m e 1 3 m e c a λa λ (1.31) 8 10

11 According to this formula the mass of the electron is a relativistic invariant even during accelerations. Although the mass appears to decrease, the additional term is actually a positive definite quantity. The departure m from the inertial value can be readily evaluated for specific accelerated motions. The easiest way to do this is to write the inertia as [ m c = m e c 1 + P ] acc (1.3) P in and then calculate radiative outputs. Since P in is constant the largest deviations will occur when the transverse output is maximized. Bremstrahlung is a possible candidate except that the deviation only exists over collision times δt which are small. A better indicator might be the rough estimates given in the last column of Table 1 for three well known facilities. The SLAC result was estimated near the end of the Facility Length Parameter Energy P acc P in m SLAC L = mi. 50 GeV 976 W 1.74E10 W 5.11E-38 kg LEP Circ. = 7 km 91 GeV 1.55E-7 W 1.74E10 W 8.1E-48 kg CESR Circ. = 768 m 10 GeV 4.54E-7 W 1.74E10 W.38E-47 kg Table 1: Estimates of m in particle accelerators. linac. It is not known whether m could be detected by a local experiment. Four-Vector Radiation Rate: An extension of the scalar power formula is a calculation of the four momentum rate of radiation emission of an electron undergoing arbitrary accelerated motions. Following Rohrlich; Chapter 5, a differential radiation rate four-vector dϱ µ is given by the limiting integral dϱ µ 1 = lim T µν d 3 ς ν ρ c ds (1.33a) 1 [ = lim 1 ρ 4 c gµν R R µν + Λ µν] d 3 ς ν (1.33b) ds 1 [ = lim ɛ µλ ɛ ν λ + Λ µν] U ν ρ dω c dτ (1.33c) ρ c ds Equation (1.33b) includes all terms of the symmetric stress tensor but at large values of ρ only those terms falling off like ρ will contribute to the integral. The final result in(1.33c) is similar to equation (1.7) depending only on the vacuum tensor 11

12 and the accleration strain. Evaluation of the integral leads to dϱ µ tot dτ = dϱµ vel dτ + dϱµ acc dτ (1.34a) = 1 c [ πσ e e c ] e 3 c a λa λ β µ (1.34b) 3 = 1 c [P in + P acc ] β µ (1.34c) and re-produces the scalar power formula in (1.9) from the contraction P = c dϱµ tot dτ β µ (1.35) Energy Flux in the Rest Frame: In the rest frame the rate at which energy and momentum is leaving the particle per unit solid angle is given by [ ] [ ] dp µ 1 e dτdω = σ e ee, σ e e ˆnˆṋn + 4πc 4 a, 4πc 4 a ˆnˆṋn (1.36) Both vectors on the right side of this equation are four-vectors. The second term is a null vector representing radiated power from electromagnetic undulations which are transverse to the direction of power flow. Of particular interest is the observation that the energy flux and the momentum flux in a given direction are related by the simple expression E = p c. A quantized theory of this radiation will therefore follow by application of the de Broglie relation leading to the formulas E = ω p = k (1.37) Clearly the theory of transverse radiation is only applicable for accelerated motions of the particle relative to an arbitrary Lorentz frame of reference. In contrast, the first term on the right, represents radiated power in the form of longitudinal electromagnetic modes. It is propagated by the vacuum gauge field A ν l and must be considered as an unobservable phenomena. Unlike the acceleration term, the energy flux is only one-half the scalar momentum flux (times c) in a given direction and this will require a quantized theory to relate energy and momentum thru the formulas E = 1 ω p = k (1.38) Clearly the quantized theory can only be derived from the velocity fields of the particle. 1

13 Theory of the Radiation Reaction The theory of the radiation reaction addresses the reaction of a particle to its own fields. like all other theories in physics, its design is rooted in conservation of energy. On the other hand, causality applied to the velocity field of the electron generates vacuum gauge theory which precludes conservation of energy from the outset. Consequently, the vacuum gauge electron will require an extensive re-assessment of the self-force problem. This will not be an injustice to the theory however which is plagued with a plethora a unsuccessful attempts to design a model for a charged particle using traditional classical concepts. Replacement for the Lorentz-Dirac Equation: The equation describing the motion of a charged particle based on energy conservation is the Lorentz-Dirac equation. It is specifically designed to include the reactive effects of radiation and can be written ma µ = F µ ext + ef µν em β ν + Γ µ (.1) where the Abraham four-vector is given by Γ µ = e [ȧ µ + 1c ] 3 c 3 aλ a λ β µ (.) Unfortunately, there are difficulties in the application of this equation. For example, the term containing ȧ µ is problematic since the resulting equation is third order in time derivatives; this is the source of acausal motions of the particle. Another serious problem is the vanishing of the Abraham four-vector for a particle in hyperbolic motion. The implication here is that a particle under the action of a constant force neither radiates nor experiences a radiation reaction. Based on this short review the immediate goal will be to re-fashion equation (.1) for the radiation based theory to eliminate the inherent difficulties and generate reasonable solutions for all possible motions of the electron. One modification already apparent from section 1. will be the need to replace m with m. We also propose the elimination of the Abraham four-vector altogether in favor of a radiation reaction four-force: F µ rad = [γβ F rad, γf rad ] (.3) where the three-force F rad is determined by examination of the distribution of radiation about the charge. In the rest frame, the flow of the vacuum field about the center of radiation is symmetric and cannot exert a force. Since this is true in the rest frame it must also be true relative to any other moving frame. To make the argument quantitative, consider the momentum flux π vel associated with the velocity fields. The net force exerted on the charge by the emission of radiation thru the particle radius (or any other radius) will be F vel = π vel R dω = 0 (.4) R=r e 13

14 A similar integral for the acceleration fields, which radiate preferentially in the direction of motion of the particle, will not produce a null result. In this case the momentum flux about the source can be derived from the Poynting vector and is proportional to the square of the acceleration field vector π acc = 1 4π E acc ˆnˆṋn (.5) The net force exerted on the charge by the emission of transverse radiation can be calculated from F rad = π acc (1 ˆnˆṋn β)r dω (.6) R=r e This equation is closely related to Liénards generalization of the Larmor power formula which derives by integrating the angular power distribution dp acc dω = c π acc ˆnˆṋnR (1 ˆnˆṋn β) (.7) over solid angle. A practical replacement for the reaction force F rad is then F rad = 1 dp acc c dω ˆnˆṋn dω = 1 c P accβ (.8) Ω Like the power formula itself, the proof of the right side of this equation for general motions of the particle is lengthy. Once proved however a covariant formulation of the reaction force from (.3) is and a revised version of the Lorentz-Dirac formula is F µ rad = 1 c P acc ( γβ, γβ ) (.9) m a µ = F µ ext + ef µν em β ν + F µ rad (.10) This equation includes two separate reactions on the electron brought about by the emission of transverse radiation, and therefore only occuring when the particle accelerates: A scalar inertial reaction calculated from the total field energy within the particle radius causes a slight increase in the mass of the particle; and a vector self-force reaction resulting from the asymmetry of transverse energy flux crossing the particle radius provides a damping force opposite the direction of motion. These two quantities may be referred to collectively as the radiation reaction. For the vacuum gauge electron, they both represent violations of energy conservation which can be detailed by considering specific examples. 14

15 .1 Applications to Straight Line Motion For straight line motion the first two terms on the right side of (.10) have the same form and can be grouped together to represent a general external force. The entire equation can also be reduced to having a single time and space component each of which generates the single equation γ 3 m a = F ext 1 c P accβ (.11) The force F ext is generally a function of both x and t. The angular distribution of transverse power radiated by the particle is indicated in figure 3 and is a special case of equation (.7) when the velocity and acceleration are co-linear. The simple formula Figure 3: Transverse power distribution for straight line motion. characterizing this distribution is dp acc dω = e a sin θ (.1) 4πc 3 (1 β cos θ) 5 and can be integrated immediately over solid angle to derive 1 P acc = γ 6 m e τa where τ = (.13) 3 m e c 3 1 The time τ is defined in Jackson, chapter 17 as the characteristic time having a value s for electrons. e 15

16 Functional forms of P acc and m can now be inserted into (.11) resulting in the cubic force equation [ ] [ ] 3 γ 9 τ m e γ a 3 6 m e τβ + a + γ 3 m 4 c e a = F ext (.14) c At this stage it is appropriate to define the unitless variables y = γ3 aτ c and arrive at the simplified cubic and y o = F extτ m e c (.15) 3 4 y3 + βy + y = y o (.16) This expression may seem alarming at first since the general cubic has three roots. However, the theory of cubic equations, summarized in the Appendix, will show that it exhibits exactly one real root. This is a very desirable feature of a radiation reaction theory since it eliminates the well known problem of runaway solutions in the conventional theory. The root is easily extracted from (.16) by setting the force to zero leaving y = 0 and a quadratic equation with only complex roots. The general procedure for solving this equation will be to first solve the cubic for y in terms of the external force F ext. A general expansion of the solution valid to any order in F ext will then generate an appropriate second order (probably non-linear) differential equation. Linear Motion Under a Constant Force: An exact solution for the linear acceleration of a charged particle is available by solving equation (.16) for y using the theory of cubic equations. While the motion x(t) may have no analytic solution, it is still possible to determine the final velocity of the particle. This is simply accomplished from the observation that the solution y = g(y o, β) is finite for any value of β. The value of y o is obviously constant and the acceleration of the particle can then be related to its velocity by a = c τ g(y o, β) (1 β ) 3/ (.17) As the particle approaches light speed the value of the acceleration tends to zero. This means that the radiation reaction evaporates for large velocities returning the motion to the hyperbolic curve. Another important point is to determine when inertial and self-force reactions are comparable at high energies. From (.16) it is evident that this will occur when y 1 or when c γ 3 a τ (.18) A 50 GeV electron at SLAC crashing into a hard target with a m/s comes pretty close but only for a short time. 16

17 For constant forces F ext which are not too large it seems reasonable that the motion of the electron might be accurately described by neglecting the inertial reaction in equation (.16) altogether. This leaves a much more manageable quadratic form βy + y y o = 0 (.19) Solving for y and expanding the square root leads to the infinite series [ a = F ] ext C γ 3 n (βy o ) n m e n=0 (.0) where the C n are numbers. In fact, with C 0 = 1 = C 1 a first order approximation to the acceleration is a F (1 βy γ 3 o ) (.1) m e This equation indicates that a significant change to the acceleration will occur when βy o 1. In this limit the reactive power delivered to the electron by the radiation reaction force will be the same order as the inertial power delivered to the vacuum by the electron. Simple Harmonic Oscillator: Another important application of the radiation reaction is the problem of a charged particle responding to a linear restoring force. Without a driving force an oscillating charge of initial energy E o will undergo radiation damping until the motion stops completely. The problem is best formulated at nonrelativistic speeds so that the radiated transverse power may be accurately calculated from the Larmor power formula. The equation of motion derived from (.16) is then 3τ 4c ẍ3 + βτ c ẍ + ẍ + ωox = 0 (.) Probably the simplest algorithm to solve this equation for ẍ is to simply insert the unperturbed equation ẍ = ω ox (.3) and this produces the non-linear solution ẍ + ωox + τ c ω4 ox ẋ 4 τ 3 c ω6 ox 3 = 0 (.4) According to the literature, this equation is an example of Liénard system having the general form ẍ + f(x)ẋ + g(x) = 0 (.5) It has all the properties consistent with the radiation based theory since both nonlinear terms vanish at x = 0 when the acceleration is zero, and since the self-force 17

18 term is zero when ẋ = 0 indicating that no radiation is emitted along the direction of motion. A crude approximation for the oscillator problem is to linearize (.4) by removing the inertial reaction term all together and averaging the motion in the self-force term as x ẋ 1 x oẋ (.6) This replacement will increase the rate of radiation damping somewhat but should give a reasonable estimate of the decay rate. The resulting linearized theory is ẍ + ɛ τ ẋ + ω ox = 0 where ɛ = E o mc (ω oτ) (.7) Inserting a trial solution x(t) = x o e αt and solving the quadratic equation gives α = ɛ τ ± iω o [1 ɛ 4ω oτ ] 1/ (.8) The decay constant and level shift of the oscillator can then be written Γ = E o mc ω oτ ω = [ Eo mc ] ω 3 oτ (.9) Although the solution here is typical of damped oscillator equation, a problem presents itself because the power delivered to the oscillator by the self-force to slow it down is not the same as the power dispensed to the vacuum by the acceleration fields. In short, energy conservation must be violated to bring the particle to rest. The total work done by the self-force is W = 1 kx o (.30) and this compares to the total energy radiated by the particle E rad P avg t = P avg Γ = m ec (.31) It could be argued at this point that a calculation such as this for an electron will not be applicable since the microscopic system will only obey the laws of quantum mechanics. One might also suggest that energy violation in the classical theory might be accomodated by a quantum mechanical theory whereby the excess of radiated energy is counterbalanced by energy absorbed from the vacuum. The path shown in figure 4 for a quantum oscillator is an example of such an effect which might be called spontaneous absorption. There is a non-zero matrix element for it from the electric dipole approximation [ n n ex n = δn,n m e ω +1 + ] n δ n,n 1 (.3) o One would assume that if the effect is legitimate that it is probably small. Equations for decay constant and level shift can be compared with values calculated in Jackson; Second Edition, section

19 Figure 4: Possible quantum mechanical decay curve for the vacuum gauge electron in a harmonic oscillator potential.. Application to Circular Motion: Caclulation of power radiated for motion in a circular orbit follows by orienting the instantaneous velocity vector along the z-axis. The power radiated can then be determined from equation (.7) with the requirement that β a = 0. Figure 5 shows the power distribution for circular motion described by the formula dp acc dω = e a 4πc 3 [ 1 (1 β cos θ) (1 β ) sin θ cos φ 3 (1 β cos θ) 5 Integrations over solid angle are straight forward and lead to the power formula ] (.33) P acc = γ 4 m e τa (.34) Motion in a Magnetic Field: For simplicity, it will be assumed that circular motion is determined by an electron moving through a constant magnetic field directed perpendicular to the trajectory. Based on the work in section 1. assume also that m m e. The governing equation for the particle orbit is then m e a µ = F µ ext + ef µν em β ν + F µ rad (.35) In the absence of an electric field this equation de-couples. The acceleration term on the left side and the Lorentz force on the right are ef µν em β ν = (0, ecγβ B) m e a µ = (0, γ m e a) (.36) Using r as the orbit radius the equality of these four-vectors leads to the relativistic cyclotron formula evb = γmv (.37) r 19

20 Figure 5: Transverse power distribution for circular motion. A pre-requisite for a circular orbit will then be determined by the remaining two terms F µ ext + F µ rad = 0 (.38) and this identifies the required external force as F ext = γ4 m e τa β (.39) c Once again, energy conservation is violated; the power supplied by the external force to maintain a circular orbit is less than the power radiated by the particle. This phenomenon should also be apparent for motion in a synchrotron. However, for high energy facilities like LEP and CESR, it can be asserted that the discrepancy will not be observable. A rough calculation for CESR where β 1 to about three parts in a billion gives unaccounted power per electron of P = P (1 β ) Watts (.40) This is about 10 9 times the power radiated by the electron. Experiment to Measure Energy Violation: One possibility for measuring energy violation is to construct a specialized low energy cyclotron. A constant magnetic field of B = 10 G and a beam radius r = 1.5 m sets the relativistic quantities β = and γ =.099 (.41) From equation (.34) the power inititally radiated by an electron in a pre-determined beam current is P = Watts (.4) 0

21 If energy is conserved, the beam will slowly spiral inward to be detected by a counter placed at radius r in some time T CE later. If the reaction force is given by equation (.39), then the spiral inward will be much slower and detection will occur at a time T V G where T V G > T CE (.43) A rough calculation of the time difference is easy to calculate. The initial kinetic energy of an electron is K = 0.56 MeV so T CE K P 1, 156 s T V G K β P 1, 496 s (.44) If technical difficulties can be removed this descrepancy should be easy to detect. Perpetual Energy Production: Vaccum gauge theory allows for the possibility of building a perpetual energy machine out of cyclotron or storage ring. In an ideal situation one supposes that permanent magnets produce the necessary field required for circular motion. Furthermore, it must also be assumed that the total power radiated from the beam can be converted to useful work. Some of this work must be re-routed to maintain the beam in a circular orbit the rest can be used as a perpetual energy source. Figure 6: Diagram illustrating the operation of a perpetual radiation cyclotron. 1

22 A Cubic Equations The most general cubic equation is y 3 + py + qy + r = 0 (A.1) where p, q, and r are real and constant coefficients. It can be shown however, that any cubic equation such as this can alway s be reduced to the form x 3 + ax + b = 0 by making the substitution y = x p/3 This substitution then implies the following relations among the constants (A.) (A.3) a = 1 3 (3q p ) and b = 1 7 (p3 9pq + 7r) (A.4) The exact solutions to the cubic equation are given by x 1 = R + S (A.5a) x = 1 3i (R + S) + (R S) (A.5b) Where x 3 = 1 3i (R + S) (R S) (A.5c) R = [ (b 4 + a3 7 ) ] 1/ b 1/3 [ (b S = 4 + a3 7 ) ] 1/ + b 1/3 (A.6) Now note the following properties of solutions to the cubic equation for given values of a and b: If b 4 + a3 7 If b 4 + a3 7 If b 4 + a3 7 > 0, there will be one real root and two complex roots. = 0, there will be three real roots of which at least two are equal. < 0, there will be three unequal, real roots.

23 Power Series Solution For b a: A solution may be sought for a cubic equation where b a. If this is the case, then a power series solution may be appropriate. For the case of a single real root the solution can be approximated by x b a + b3 a 3b5 4 a + 7 This solution can be verified by inserting it into the cubic equation and setting all higher orders to zero. suppose next that b a but there are three unequal real roots. In this case both R and S can be power expanded: R = ( a 3 )1/ 1 a b + 1 7( a b )5/ a 4 b S = ( a 3 )1/ + 1 a b + 1 7( a b 1 3 )5/ a 4 b ( a b4 3 3 )11/ a 7 b5 + 1 ( a b )11/ a 7 b5 + 3