Description of Scientific Discovery: The Fundamental Wave Equation of Nature

In the chapter, the prerequisites and basic concepts of unified theory of space–time transformations of fundamental laws of nature in the form of three “great theorems” by Galileo, Lorentz and Poincare are considered in detail in the context of three “great geometries” by Euclid, Lobachevsky and Riemann using simple and obvious cases of description of free oscillations of string and the Earth. The author provides formulations and solutions of wave equation for free oscillations of string and radial, spheroidal, torsional free oscillations of the Earth as well. Considering these formulations and solutions from the point of view of new visible event horizons associated with the velocity of light propagation, gravitational radii and Planck’s quantities, the author comes up with the formulation of unified theory of space–time transformations of fundamental laws of nature. From the point of view of exact sciences—mathematics, geometry and physics, the unified theory of space–time transformations of fundamental laws of nature in the form of three theorems by Galileo, Lorentz and Poincare is supposed to be related to further development of infinitely large and infinitesimal in integral and differential calculus in the transition to new visible event horizons, determined by velocity of waves and light propagation, gravitational radii and Planck’s quantities. The chapter also includes the consideration of applied aspects of unified theory of space–time transforms of fundamental laws of nature, in particular, the relation of the Earth’s free oscillations to standing internal gravitational waves that determine the fractal hierarchy of dissipative structures of the Earth, which is associated with many aspects of its geological structure and evolution.