A6. Charges of the Same Sign
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1. The two charges nominally repel each other, instead of attracting each other as in the
Positronium case.
2. The radiation created by the system changes form to quadrupole, rather than dipole,
as it was in the Positronium case.
3. The torque on the system everywhere reverses sign from what it was in the
Positronium case.
The Positronium analysis identified intervals π < α < 3π , 5π < α < 7π , 9π < α < 11π ,
etc., within which the attraction between electron and positron works as if it were repulsion.
The complementing intervals 3π < α < 5π , 7π < α < 9π , 11π < α < 13π , etc., must play a
similar role for the same-charge pair: within these intervals, the repulsion between same
charges must work as if it were attraction. Note that figure A6.1 basically illustrated the first
point of the first such interval: α = π .
The radiation from the same-charge system is quadrupole, instead of dipole as it was for
the opposite-charge Positronium system. The kind of radiation does not matter very much
qualitatively, inasmuch as solutions are associated with dips to zero radiation that are in turn
keyed to speeds at multiples of πc , regardless of what kind of radiation it is.
The sign of torquing is more significant. The Positronium solutions displayed the
existence of regimes of negative torque within which both radiation and torquing cause
energy loss from the system considered. No solution for system balance between radiation
and torquing can exist in these regimes. The excluded intervals for Positronium were
2π < α < 4π , 6π < α < 8π , 10π < α < 12π ,... The complementing intervals 4π < α < 6π ,
8π < α < 10π , 12π < α < 14π ,… are excluded for the same-charge system.
The details of the analysis follow. The angles again satisfy:
α−β ≡β ≡ α / 2
(A6.1)
The initial formula for energy gain due to torquing changes slightly; i.e., from
PT = ( e4 / me ) c(2re )3 to PT = (q 4 / mq ) c(2rq )3
(A6.2)
The ‘same-charge’ condition causes several changes in the initial formula for energy loss
due to radiation. For Positronium we had (slightly rearranged for easy in conversion)
PR = 25 ( e6 / me2 ) 3c3 (2re )4 = 24 (2e6 / me2 ) 3c3 (2re )4
(A6.3)
For one thing, we have q ’s instead of e ’s, as in PT above. In addition, we have
quadrupole radiation instead of dipole radiation. To quantify this, we start with Eq. (9.52)
from Jackson [4], P = ck 6Q02 / 240 , and insert, Q0 = 2qd 2 with d = 2rq , and k = Ω / c with
Ω2 = q 2 / mrq d 2 , plus a factor of 26 for the Thomas rotation [28] that this system
experiences due to unbalanced forces. We end up with
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Cynthia Kolb Whitney
(
PR = 26 64q8 / mq3
) ⎡⎣⎢240c (2r ) ⎤⎦⎥ = 2 (4q / m ) ⎡⎣⎢15c (2r ) ⎤⎦⎥
5
5
6
8
q
3
q
5
5
q
(A6.4)
The condition defining the angle α becomes
α = (Vq / 2c ) × (2rq / rq ) → Vq / c
(A6.5)
The condition defining the angle β remains
β≡α/2
(A6.6)
mqVq2 / rq = e2 / (2rq )2
(A6.7)
The condition defining Vq becomes
which implies Vq = q
mq (2rq ) × (1 / 2) . The scaling of PR , is
2
PR → ⎡⎣cos(α / 2) ⎤⎦ PR
(A6.8)
PT → ⎡⎣ sin(α / 2) (α / 2) ⎤⎦ PT
(A6.9)
The scaling of PT is
The analog of figure A5.2 from Appendix 5 is then figure A6.3. Again, the plot starts at
system radius 10−16 cm, a progresses to 10−11 cm in 301 data points. Observe that the
radiation curve is generally steeper than it was for Positronium. That is because the radiation
is quadrupole rather than dipole. Observe too that there is no ‘ground-state’ solution for like
charges, but there are superluminal sub-ground solution doublets, just as for Positronium,
except that they occur at orbit speeds just slightly below V = 3πc,Ź7πc,Ź11πc,Ź15πc... instead
of V = πc,Ź5πc,Ź9πc,Ź13πc,... . Within a doublet, the solutions to the right are stable, and the
solution to the left is unstable.
For present purposes, the stable solution near V = 3πc is probably of greatest interest.
The system radius in this vicinity is about 1.58 × 10−15 cm. This is much smaller than the
believed Hydrogen atom radius re + rp = 5.28 × 10−9 cm, confirming the premise of this
model, that atomic charge clusters are small compared to atoms. It is smaller even than the socalled ‘classical radius of the electron’, rc = e2 / me c 2 = 2.82 × 10−13 cm. Note that this
Recent Progress in ‘Algebraic Chemistry’
59
‘classical radius’ involves the rest energy me c 2 , which is a concept from standard SRT,
which in turn assumes the ‘speed limit’ c . Its meaning in the expanded SRT, with its
unlimited superluminal speeds, is not clear.
40
35
30
25
Series1
Series2
20
15
10
5
0
1
21
41
61
81 101 121 141 161 181 201 221 241 261 281 301
Figure A6.3. Same-charge solutions.
A7. Systems of Many Charges
The analysis for two charges is a stepping-stone to analyses involving three, and more,
electrons. For three electrons, the analog of figure A6.1 is figure A7.1.
The condition expressed by figure A7.1 is that the light vector expansion at 2c across a
chord of the circle synchronizes with the particle transit halfway around the circumference of
the circle. Therefore, the particle transit has to be at Vq = πc × (diameter / chord length) =
πc / sin(π / 3) . The sin(π / 3) for the ternary case, in place of sin(π / 2) = 1 for the binary
case, recurs throughout the full analysis of ternary charge cluster.
Figure A7.1. Attraction between three like charges orbiting at speed V = πc / sin(π / 3) .
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Cynthia Kolb Whitney
The ternary analog for figure A6.2 is figure A7.2. The angle β is smaller than in the
binary case, and that affects both the torque energy gain and the radiation energy loss.
Figure A7.2. Angles involved in analysis of ternary charge cluster.
Another type of change required concerns the radiation. It is not the dipole formula, and
it is not the quadrupole formula. It is perhaps a ‘hexapole’ formula. In any case, it can be
worked out from the dipole formula, just as the quadrupole formula can be worked out in that
way. The quadrupole formula is the difference of two dipoles. To put it another way, it is the
sum of two dipoles at phases of 0o and 180o. Correspondingly, the radiation from this ternary
system is the sum of three dipoles, at phases 0o, 120o, and 270o. This exercise about the
radiation formula is interesting and challenging, but as we already know, does not affect the
solutions very much. Solutions will be doublets surrounding orbit speeds
Vq =
1
( 3πc, 7πc,11πc,13πc...)
sin( π / 3)
(A7.1)
Of these, the solution at orbit speed just below Vq = 3πc / sin(π / 3) is probably of
greatest interest.
The system radius for the ternary case can be expressed by scaling from the system radius
for the binary case. Two factors affect it. One is the increase of velocity for which radius is
sought, by the factor of 1 / sin(π / 3) . This decreases the solution radius by a factor of
sin 2 (π / 3) = 3 / 4 . The other factor is the change in the force attracting a subject charge,
which now has two contributing charges instead of one, both at closer distance, but at offradial direction. This changes the attractive force by a factor of 2 sin(π / 3) / sin 2 (π / 3) =
2 / sin(π / 3) = 2.3094 , and hence changes the solution radius by the inverse of that factor.
Altogether then, the solution radius decreases by 0.75 / 2.3094 = 0.3248 .
In the general case of N charges in a ring, the solution velocities are
Vq =
1
( 3πc, 7πc,11πc,15πc...)
sin( π / N )
(A7.2)