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A5. An Example with Equal Masses: Positronium
Cynthia Kolb Whitney
3) The condition defining Ve becomes
meVe2 / re = e2 / (2re )2
The solution is then Ve ≈ e
me (2re ) × (1 / 2) .
4) The factors for removing the small-angle approximations from PT and PR simplify.
The expression for energy loss rate due to radiation changes from
PR → ⎨ ⎡⎣cos(α − β) + cos(β) ⎤⎦ ⎬ PR to PR → ⎡⎣cos(α / 2) ⎤⎦ PR
5) The expression for the energy gain rate due to torquing changes from
PT → ⎨ ⎡⎣sin(α − β) (α − β) + sin(β) β′ ⎤⎦ ⎬ PT to PT → ⎡⎣ sin(α / 2) (α / 2) ⎤⎦ PT (A5.8)
116 139 162 185 208 231 254 277 300
Figure A5.2. Positronium solutions.
Being formally analogous to Hydrogen, Positronium also has a ‘ground-state’, low-speed
solution, plus a family of high-speed sub-ground solutions. Figure A5.2 illustrates the
solutions. The range of system radii starts at 10−16 cm, and goes to 10−11 cm via 301 points.
The ground-state solution occurs between points 269 and 270, at system radius of about
3 × 10−12 cm (quite a bit smaller than the 5.28 × 10−9 cm for the Hydrogen atom), with orbit
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speed of about 0.22 c . The other solutions are superluminal, occurring in the vicinity of
V = πc,5πc,9πc,13πc,...
The ‘ground-state’ solution for Positronium occurs at about point 270, where system
radius is about 3.04 × 10−12 cm and V = 0.22c . This state is qualitatively like the groundstate solution for Hydrogen: it is unstable. Any perturbation to the right, to higher energy, is
amplified by energy gain due to torquing, and any perturbation to the left, to lower energy, is
amplified by energy loss due to radiation. As we know from experience, Positronium is
indeed quite unstable; it tends to annihilate itself quite quickly.
The sub-ground solutions for Positronium are startlingly different from the corresponding
sub-ground solutions for Hydrogen: they occur where the energy gain rate due to torquing is
near maximum, not near where it is passing through zero. This is a consequence of the perfect
mass equality of the two particles in Positronium, opposite to the extreme mass inequality of
the particles in Hydrogen. While both Positronium and Hydrogen have dips in the energy loss
rate due to radiation at angle α = π,3π,5π, 7π,.... , Hydrogen has its energy gain rate due to
torquing pass through zero there, whereas Positronium has it passing through maximum there.
The reason for this difference is visible within the mathematics describing energy gain rate
due to torquing:
1. Hydrogen always has angle β ≈ 0 , so for α = π,3π,5π, 7π,.... , both the sin β and the
sin(α − β) involved in the expression for energy gain rate due to torquing pass
2. Positronium always has angle β = α / 2 , so sin β = sin(α − β) = sin(α / 2) . The
sin(α / 2) passes through unity at α = π,3π,5π, 7π,....
As a result of this difference between Hydrogen and Positronium, all of the Positronium
superluminal solutions are actually doublets: a solution to the right with higher system radius,
lower orbit speed, and a solution to the left with lower system radius and higher orbit speed.
Within each doublet, the solution to the right is extremely stable: the plot shows that any
perturbation further right, to higher energy, is driven back by energy loss due to radiation, and
any small perturbation left, to lower energy, is driven back by energy gain due to torquing.
The solution to the left is then just the reverse, extremely unstable: any small perturbation
right, to higher energy, is encouraged by energy gain due to torquing, and any perturbation
further left, to lower energy, is encouraged by energy loss due to radiation. Something even
worse is not obvious in the plot, although it is obvious in the mathematics. For this solution to
the left, the factor cos(α / 2) that enters the radial force components is actually slightly
negative. So no such circular orbit can exist at all, unless a retaining radial force arises from
some external circumstance.
The conclusion to remember for future reference is that the Positronium solutions occur
at orbit speeds just slightly below V = (πc,5πc,9πc,13πc) .
Cynthia Kolb Whitney
A6. Charges of the Same Sign
Technical journals occasionally feature reports and commentary about the apparently
incomprehensible phenomenon of electrons clustering together. Some of these articles have
been in the Galilean Electrodynamics journal I now edit [29-31]. The phenomenon is widely
known; related literature cited in the third of those references is quite extensive, and some of
it appears in the most widely circulated physics journals.
The non-central forces illustrated in figure A5.1 for Positronium can also provide a
mechanism that can explain charge clusters. As is emphasized by Appendix 2 (A2), there is
no limitation on Galilean speed V ; it can exceed c . Figure A6.1 illustrates a case where two
like charges are orbiting each other at speed V = πc . Because of the ‘mid-point’ feature of
Eqs. (A1.2b), the Coulomb ‘repulsion’ between the two charges is half-retarded, and because
of the super luminal orbit speed, the ‘repulsion’ actually works as attraction.
Figure A6.1. Attraction between two like charges orbiting at superluminal speed V = πc .
Figure A6.1 just constitutes a ‘proof of existence’: there do exist circumstances in which
like charges attract. More detailed analysis of the binary same-charge system begins with
figure A6.2, which is similar to the figure A5.2 for Positronium. The angle α is the sum of
the half retardation angles, both of which are β = α / 2 .
Figure A6.2. Binary charge cluster.
Let us identify exactly what changes are needed to adapt the existing Positronium
analysis to the binary charge cluster. The change from an opposite-charge to a same-charge
system requires that:
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1. The two charges nominally repel each other, instead of attracting each other as in the
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
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
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
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
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 , 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  that this system
experiences due to unbalanced forces. We end up with