Wherever and whenever the space-time Lagrangian is rationalized the result resembles a double action Lagrangian. For example, the classical Lagrangian (L) is simply the difference between kinetic energy (T) and potential energy (V): L=T-V. Therefore, rationalizing L simply means that the domains of both T and V are exclusively the set of rational numbers. This rational closure property implies that the product of T and V equals the difference of T and V: TV=T-V Without a reasonable renormalizing factor this simple equation fails all the tests of dimensional analysis. On the one hand, if a renormalizing multiplicative factor of unit energy is applied on the right hand side then the units on both sides become squares of energy. On the other hand, if a divisible factor of unit energy is applied on the left hand side then the units on both sides become simply energy. A successfully rationalized Lagrangian seems to require both multiplicative and divisible factors of unit energy. Which is best depends on the physical theory. Classical theories prefer the divisible factors. Quantum theories prefer the multiplicative factors. The first decreases the physical dimension by 1 while the second increases it by 1. However, quantum field theories use both in order to satisfy both Abelian and non-Abelian local or global gauge invariance. If the rationalized Lagrangian is multiplied by the Hamiltonian (H): H=T+V then T/V=(1+T)/(1-V), if T=0 then V=0, if T=2 then V=-2, if T=1 then V=1/2. For all T≠V there are an infinite number of rational solutions.
The logic of dimensional rational normalization and renormalization had been useful since Newton and were used in a subtle way by both Maxwell and Einstein. For Newton’s second law of motion defining inertial forces the divisible dimensional factor is the inertial mass while for Newton’s universal law of gravitation the dimensional factor was the gravitational mass. For Maxwell’s electromagnetic field equations the dimensional factor is the speed of light squared. For Einstein’s special theory of relativity the dimensional factor is also the square of light speed. However, for Einstein’s general theory of relativity the dimensional factor is just divisible light speed. Why did Einstein choose these distinct dimensional factors (divisible �� and multiplicative ��) remain a mystery?