In their paper [1] Evans & Eckardt start with the ODE
∂²Φ/∂r² +
(2/r+ωr) ∂Φ/∂r +
(2rωr + r² ∂ωr/∂r)
Φ/r² = − 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/r² = − 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:
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/r² Φ = − ρ/εo
(1''')
A simple calculation yields the eigensolutions
Φ1(r) = 1/rα
and
Φ2(r) = 1/r ,
if α ≠ 1
(1'''a)
and
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",
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
References
http://www.aias.us/documents/uft/a92ndpaper.pdf
http://www2.mathematik.tu-darmstadt.de/~bruhn/RemEvansPaper90.html
http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.4433v1.pdf
http://www2.mathematik.tu-darmstadt.de/~bruhn/Evans-Resonance.html
HOME