Theoretical Derivation: Center of Mass ≠ Center of Gravity Separation in Electromagnetic Resonant Cavities

We derive the theoretical relationship between electromagnetic energy density gradients in resonant cavities and the resulting separation between center of mass and center of gravity. This represents the first application of General Relativistic effects to controllable electromagnetic systems for laboratory-scale gravitational experiments. Using the linearized Einstein field equations and the electromagnetic stress-energy tensor, we show that asymmetric resonant cavities produce position-dependent self-gravitational fields that shift the center of gravity away from the center of mass. The separation scales as Delta_z ~ G*Q*P*L/(c^3*g*V) * Delta_R, where Q is the quality factor, P is input power, L is cavity length, V is cavity volume, and Delta_R is the reflectivity asymmetry. While the separation magnitude (~10^-27 m for realistic parameters) is far below direct position measurement capability, we demonstrate that the associated time dilation effect (~10^-20 to 10^-18) falls within the sensitivity of modern atomic clocks, providing a viable experimental pathway.