Technical Notes 2007

Gravitational Bohr Radius [Sept 2007]


According to pre-SSCP physics, one can calculate what the radius of the hydrogen atom would be if the atom is governed by the conventional gravitational interaction between the proton and the electron.  This radius is referred to as the Gravitational Bohr Radius (R), and it can be determined by:

R = ħ2/Gm2M                                      (1)

where ħ is Planck's constant divided by 2π, G is the gravitational coupling factor, m is the mass of the electron and M is the mass of the proton.  The conventional calculation of R, using 6.67 x 10-8 cm3/g sec2 as the appropriate value for G, yields

R = 1.20 x 1031 cm.

This radius is larger than the radius of the observable universe.  It is clearly a ridiculously large value and is usually cited as iron-clad proof that Atomic Scale systems are primarily bound by electrostatic rather than gravitational interactions.

Unfortunately, this conclusion is almost certainly incorrect.

The SSCP argues that when one understands the discrete scale invariance of nature and the appropriate cosmological scaling rules, then one realizes that 6.67 x 10-8 cm3/g sec2 is not the correct value of G to use in Eq. (1).  Rather, we know that G-1 = Λ2.174 G0 = 2.18 x 1031 cm3/g sec2 is the G-value that applies within Atomic Scale systems.  The scaling of GΨ is explained near the beginning of Paper #12 (“Discrete Scale Relativity”) of the Selected Papers section of this website.

When G-1 is used in Eq. (1), we find that

R = 3.67 x 10-8 cm,

which is roughly 2π times the Bohr Radius (a0) for the hydrogen atom.  Whereas a0 equals the radius at which the probability of finding the electron is maximized, the value 2πa0 is a much better estimate of the physical radius of the hydrogen atom, i.e., the limiting radial extent of the electron's wavefunction.

As expected, the SSCP leads once again to the conclusion that the dynamics within atoms are governed by gravitational interactions which are ~ 1038 times stronger than those assumed by conventional physics.  As we have learned from the exploration of the fine structure constant (see New Developments – 2007), within Atomic Scale systems, or their analogues on any cosmological Scale, gravitational interactions tend to be stronger than electrostatic interactions by a factor of 1/α = 137.036.