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The linearized field equations of general relativity in harmonic coordinatesare given by an inhomogeneous wave equation. In the region exterior to thematter field, the retarded solution of this wave equation can be expanded interms of 10 Cartesian symmetric and tracefree (STF) multipoles inpost-Minkowskian approximation. For such a multipole decomposition only threeand rather weak assumptions are required:  1. No-incoming radiation condition.  2. The matter source is spatially compact.  3. A spherical expansion for the metric outside the matter source ispossible.  During the last decades, the STF multipole expansion has been established asa powerful tool in several fields of gravitational physics: celestialmechanics, theory of gravitational waves and in the theory of light propagationand astrometry. But despite its formidable importance, an explicit proof of thefundamental theorem of STF multipole expansion has not been presented thus far,while only some parts of it are distributed into several publications. In atechnical but more didactical form, an explicit and detailed mathematical proofof each individual step of this important theorem of STF multipole expansion isrepresented.

A detailed proof of the fundamental theorem of STF multipole expansion in linearized gravity