Comments on Evans' Resonance Paper 92

Gerhard W. Bruhn, Darmstadt University of Technology

1. The general case

In their paper [1] Evans & Eckardt start with the ODE

                ∂²Φ/∂r² + (2/rr) ∂Φ/∂r + (2rωr + r² ∂ωr/∂r) Φ/ = − ρ/εo                                 (1)

which is a linear ODE with r-dependant coefficients. They assert:

Eq.(1) can be reduced straightforwardly to the basic structure of the damped oscillator equation, which was discovered in the eighteenth century {11}

                d²x/dr² + 2β dx/dr + κo² x = A cos (κr)                                                                 (2)

In Eq.(2) β takes the role of the friction coefficient and κo is a Hooke law type wave number. . . .

We remark that apparently Evans has given up his method of attaining resonance by a dubious complex "Euler transform" (see [2]) and instead of this failed method he now tries to get resonance by pure "definition". He continues:

Eq.(1) reduces to Eq.(2) when

                ωr = 2 (β − 1/r)                                                                                                 (3)

                κo² = 4/r (β − 1/r)                                                                                                 (4)

This is incorrect: The damped oscillator equation has constant coefficients while the coefficients of Evans' ODE (2) are given by Eqs.(3-4) and are evidently non-constant if we assume β≠1/r . However, already the case β=1/r shows the essential difference between the ODEs (1) and (2): Due to ωr=0 and κo=0 we obtain from Eq.(1)

                ∂²Φ/∂r² + 2/r ∂Φ/∂r + 0 Φ/ = − ρ/εo                                                                 (1')

with the non-oscillatory eigensolutions

                Φ1(r) = 1         and         Φ2(r) = 1/r ,                                                                 (1")

while the damped oscillator equation (2) (with constant coefficient β) takes the form

                d²x/dr² + 2β dx/dr = A cos (κr)                                                                                 (2')

with the eigensolutions

                x1(r) = 1         and         x2(r) = e−2βr                                                                 (2")

The comparison of the eigenfunctions (1") and (2") shows that the ODEs (1) and (2) disagree essentially.

In addition, if one takes the cofficients of ODE (2) at a fixed point r=ro then the eigenfunctions of Eq.(2) depend on the value ro. More, for β<0 or for β>0 and r<1/β the ODE (2) has no oscillatory eigenfunctions and therefore cannot yield resonance. Therefore it turns out that:

The resonance behavior of Eq.(1) cannot be discussed by means of the ODE (2), constant coefficients β and κo assumed.

2. The Euler case

There is a special choice of the function ωr that reduces the ODE (1) to Euler type:

                ωr = α/r                 where α is some constant.                                                                 (5')

Then Eq.(1) reduces to the Euler type ODE

                ∂²Φ/∂r² + (2+α) 1/r ∂Φ/∂r + α 1/ Φ = − ρ/εo                                                                 (1''')

A simple calculation yields the eigensolutions

                Φ1(r) = 1/rα         and         Φ2(r) = 1/r ,                 if α ≠ 1                                         (1'''a)


                Φ1(r) = 1/r log r         and         Φ2(r) = 1/r ,           if α = 1                                           (1'''b)

Since these eigenfunctions are non-oscillatory there cannot occur resonance with the oscillatory driving term of Eq.(2) whatever the value of the real constant α may be.


[1] M.W. Evans and H. Eckardt, Development of Spin Connection Resonance in the Coulomb Law,

[2] G.W. Bruhn, Rejection of Evans' "Refutation of Comment by Jadczyk et Alii",

[3] G.W. Bruhn, F.W. Hehl and A. Jadczyk, Comments on "Spin Connection Resonance in Gravitational General Relativity",