A New Wave Equation Emerges from Rational Trigonometry and Universal Hyperbolic Geometry -Defining a First Order Derivative of a E = m^2 * C^4 + P^2* C^2 and Taking the Matrix Free Square Root (the Basis of the Dirac Equation)

This paper presents an approach to deriving the wave equation from the energy equation E = m^2 * c^4 + p^2 * c^2, using the framework of Rational Trigonometry (RT) and Universal Hyperbolic Geometry (UHG). By expressing Maxwell's equations in terms of the RT and UHG concepts of quadrance and spread, we demonstrate how the fundamental electromagnetic relationships can be preserved in a more algebraic and geometric form, without the need for the additional mathematical structures required by the standard Dirac equation approach[4][5].The key advantage of the RT and UHG framework is its ability to eliminate irrational numbers and infinite sums, providing a more computationally efficient and intuitive way of working with physical equations and theories[1][2][3]. Importantly, this approach aligns with Dirac's original intention to find a direct square root solution for the energy equation, avoiding the complexity of the 4x4 matrices and Clifford algebra used in the Dirac equation[7].The paper outlines the step-by-step derivation of the direct square root solution, which leverages the RT and UHG formulation of Maxwell's equations[5]. This method allows for a simpler and more straightforward way to obtain the wave equation, while maintaining consistency with the underlying physics.The validity of the direct square root approach is demonstrated through the reproduction of well-known physical observables, such as the ground state energy of the hydrogen atom and the electron g-factor. These results reinforce the potential of the RT and UHG framework to contribute to the ongoing development and understanding of quantum mechanics and electromagnetic theory. The paper also discusses the limitations and caveats of the direct square root method, including its dependence on accurate eigenvalue solutions, the need for further validation and experimental comparison, and the challenges of integrating the RT and UHG approach with existing mathematical and physical frameworks.