Graphs

We present here some plots of our central equation as a function of the variable r_x , which gives a multiple of our r_t = (2\pi/\sqrt{\alpha})\,l_P .

f(r_x) =  \frac{\alpha\hbar c}{8\pi\,l_P(\frac{2\pi}{\sqrt{\alpha}})r_x} - \frac{3\pi\hbar c}{2\,l_P(\frac{2\pi}{\sqrt{\alpha}}r_x)^3}

In the above plot, r_x ranges from 1 to 10.  We can see that we have a maximum at around 5.56 x 10^13 GeV right before 2\,r_x.

In the above plot we have expanded the horizontal axis out to 100 times r_t .  This makes it easier to see that the right hand solution asymptotically approaches zero.  This is the positive solution responsible for the Compton wavelength while the left hand slope gives the observed fermion masses.  If higher energy fermions do exist, then we can see that they are limited to energies near the GUT scale.