The discussion explores whether E=mc^2 applies to gravitational potential energy, particularly in the context of an object dropped from a tower. In classical mechanics, potential energy converts to kinetic energy without changing total energy, while in relativity, energy gain can imply mass increase, though this is a nuanced concept. The conversation emphasizes that relativistic mass is largely abandoned in favor of invariant mass, which does not change regardless of an object's energy state. The total energy of a system, including gravitational interactions, remains constant, but energy is frame-dependent, complicating measurements of mass during motion. Ultimately, the discussion highlights the need for clarity in definitions and understanding of energy in relativity.