Dimensionless numbers in relativistic theory

Question

Dimensionless numbers allow physicists and engineers to extend the physical modeling landscape by reducing otherwise complex mathematics to a simple proportional relationship. For example by assuming specific categorical constraints one can derive a simpler pressure flow relationship and dimensionless numbers, such as the Reynolds number can provide a map of the flow regime .

It occurred to me that dimensionless numbers also exist in special relativity. At least the factor calculated as the square root of 1 minus the velocity squared divided by the speed of light squared. In some books they name this factor beta.

Is that the official name of this dimensionless number?

Can one derive this number using the method of Buckingham? (Pi Theorem)

Can the techniques of dimensionl analysis offer additional insights into the theory of relativity?

Answer 1

Velocity is intrinsically a dimensionless quantity in special relativity. This is because the four dimensional symmetry of Minkowsky space suggests that all four coordinates should have the same units. The constant c=299792458 m/s is then just a conversion constant, no different than 5,280 ft/mi in American topographical maps. The non-relativistic limit is then v<<1. The constant \gamma is another intrinsically dimensionless quantity. The extreme relativistic limit is \gamma>>1. Charge (either electric, weak, or strong) is also an intrinsically dimensionless quantity, whether in special relativity, or even nonrelativisticly. The charge of the electron is given by the dimensionless number e^2=1/137.036. This is why SI units, with dimensional constants epsilonzero, and muzero, and the notion that charge is a new dimension, are so awkward for modern physics.