This thesis establishes a geometric framework for deriving energy conservation laws in curved spacetime. By combining the stress-energy-momentum tensor of the KleinGordon field with the divergence theorem on Lorentzian manifolds, we prove that spacetime symmetries, encoded by Killing vector fields, yield rigorous conservation laws. The results are shown to extend to a broad class of hyperbolic partial differential equations and remain valid under weakened regularity assumptions, demonstrating the universality of this geometric approach to energy conservation

We aim to prove existence and uniqueness of solutions to the initial value problem of the Klein-Gordon (K-G) equation $(\square+m^2)\psi=0$. For this we properly introduce the mathematical framework and motivation for deriving the K-G equation, including the basics of quantum mechanics and special relativity. We derive the energy to show uniqueness and use Fourier transformations for existence.

We aim to prove existence and uniqueness of solutions to the initial value problem for the wave equation and its application to Maxwell’s equations. For this, we introduce the mathematical framework for the homogeneous wave equation in various dimensions—incorporating d’Alembert’s, Poisson’s, and Kirchhoff’s formulas—and analyze the implications of the Huygens principles. We derive the inhomogeneous solution via Duhamel’s principle and Green’s functions, utilize energy estimates to show uniqueness, and finally reformulate Maxwell’s equations into a wave-form system to establish the initial value theorem for electromagnetic fields.