We introduce an appropriate form and present an analysis of the elastic-gravitational system of equations describing the free oscillations of a rotating earth. We highlight the complications associated with the presence of an outer core. We discuss basic properties, including energy estimates, well-posedness, extraction of the elastic wave equation and a characterization and computation of the spectrum. We show how physics constrains the essential spectrum. We conclude with the analysis of an inverse problem for a spherically symmetric model (PREM) and give, via a trace formula, a spectral rigidity result. We will also show some recent observations. Joint research with S. Holman, J. Ilmavirta, S. Jimbo, G. Nakamura, H. Pham and J. Shi.