What do you think about the following derivation of energy distribution of a localized energy whirl:

In any radial direction away from center, the energy decreases. At each point [tex]r[/tex], energy has a certain value [tex]E(r)[/tex]. Along the radius, the energy does change: [tex]\frac{dE}{dr}[/tex]. This is force, by definition.

At some given radius [tex]r[/tex], that force has a certain value [tex]\frac{dE(r)}{dr}[/tex]. Both force and energy do not change along the circle of that given radius [tex]r[/tex]. And that is the case in each given value [tex]r[/tex] of a radius, along which energy is distributed. Hence:

[tex]\frac{dE(r)}{dr} = - a(r) \cdot m(r) = - \frac{v^2}{r} \cdot \frac{E(r)}{c^2} \Rightarrow \frac{dE(r)}{E(r)} = - \frac{v^2}{c^2} \cdot \frac{dr}{r}[/tex]

[tex]v = \omega \cdot r, dv = \omega \cdot dr \Rightarrow \frac{dr}{r} = \frac{dv}{v}[/tex]

[tex]\Rightarrow \frac{dE(r)}{E(r)} = - \frac{v \cdot dv}{c^2} \Rightarrow \frac{E(v)}{E_0} = e^{ - \frac{1}{2} \cdot \frac{v^2}{c^2}}[/tex]