The equation you mentioned, γ = 1 / √(1 - v²/c²), commonly known as the Lorentz factor or gamma factor, is an important component of Einstein's theory of special relativity. It arises from the mathematics of the theory and is derived through rigorous mathematical reasoning, rather than being a mere declaration by Einstein.
Einstein didn't simply declare γ to be a physical thing; rather, he derived it from his theory of special relativity, which revolutionized our understanding of space, time, and motion. Special relativity is based on two postulates: the principle of relativity and the constancy of the speed of light in a vacuum (c).
One of the key insights of special relativity is that the laws of physics should be the same in all inertial reference frames, which are frames of reference moving at constant velocities relative to each other. However, to ensure this consistency, it was necessary to introduce the concept of time dilation and length contraction. These phenomena, predicted by the Lorentz transformations, are what lead to the appearance of γ in the equation you mentioned.
The Lorentz factor, γ, describes the relationship between an object's velocity (v) and its observed properties such as time dilation, length contraction, and relativistic mass increase. It quantifies the effects of special relativity on physical phenomena and has been experimentally verified through various observations and measurements.
So, rather than being declared as a physical thing arbitrarily, γ is an integral part of the mathematical framework of special relativity, which has been extensively tested and confirmed by experimental evidence. Its inclusion in the theory allows for the consistent description of the physical world at high velocities and forms the basis of many phenomena observed in relativistic systems.