Note on the natural system of units

Abstract

We propose to substitute Newton's constant G_N for another constant G₂, as if the gravitational force would fall off with the 1/r law, instead of the 1/r²; so we describe a system of natural units with G₂, c and ħ. We adjust the value of G₂ so that the fundamental length L=L_Pl is still the Planck's length and so G_N=L×G₂. We argue for this system as (1) it would express longitude, time and mass without square roots; (2) G₂ is in principle disentangled from gravitation, as in (2+1) dimensions there is no field outside the sources. So G₂ would be truly universal; (3) modern physics is not necessarily tied up to (3+1)-dim. scenarios and (4) extended objects with p=2 (membranes) play an important role both in M-theory and in F-theory, which distinguishes three (2, 1) dimensions. As an alternative we consider also the clash between gravitation and quantum theory; the suggestion is that non-commutative geometry [x_i, x_j]=Λ²θ_ij would cure some infinities and improve black hole evaporation. Then the new length Λ shall determine, among other things, the gravitational constant G_N.