We develop a comprehensive theoretical framework for analyzing electromagnetic modes in spherical optical cavities with time-dependent radii, providing systematic treatment of parametric instabilities in three-dimensional oscillating resonators. We derive explicit closed-form coupling coefficients for spherical Bessel function modes, incorporating angular momentum structure and extending previous treatments of mode coupling in oscillating cavities. Our approach yields exact analytical solutions for cavities undergoing sinusoidal oscillations, revealing how boundary motion induces mode coupling and parametric amplification. We derive complete stability criteria showing that driving frequencies $\mathrm{\Omega }=2{\omega }_{nl}/k$ ($k=1,2,3...$) trigger exponential mode growth with rates $\gamma =\varepsilon {\omega }_{nl}/2$ for $k=1$ and $\gamma \approx \varepsilon {\omega }_{nl}/\left(2k\right)$ for odd $k\ge 3$, with even $k$ resonances suppressed by phase cancellation. The framework predicts frequency shifts $\mathrm{\Delta }\omega /\omega \approx -\varepsilon cos\left(\mathrm{\Omega }t\right)$ and establishes quantitative scaling laws for parametric effects. The systematic mathematical treatment provides analytical tools for cavity optomechanical systems and establishes theoretical foundation for analyzing parametric phenomena in diverse optical systems. While experimental validation presents significant challenges, the framework offers quantitative predictions that may guide future studies as technology enables access to the required parameter regimes. This work assumes lossless cavities and focuses on sinusoidal boundary motion, with extensions to dissipative and nonlinear regimes outlined as future directions.

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