Pursuing both disciplinary authenticity and personal relevance in the teaching and learning of science in school generates tensions that should be acknowledged and resolved. This paper problematizes and explores the conceptualizations of these tensions by considering personal relevance, disciplinary authenticity, and common school science as three perspectives that entail different educational goals. Based on an analysis of the literature, we identify five facets of the tensions: "content fidelity," "content coverage," "language and discursive norms," "epistemic structure and standards," and "significance." We then explore the manifestations of these facets in two different examples of the instruction and learning of physics at the advanced high school level in Israel and Italy. Our analysis suggests that (1) the manifestations of these tensions and their resolution are highly contextual. (2) While maintaining personal relevance and disciplinary authenticity requires some negotiation, the main tension that needs to be resolved is between personal relevance and common school science. (3) Disciplinary authenticity, when considered in terms of its full depth and scope, can be equipped to resolve this tension within the discipline. (4) To achieve resolution, teachers' expertise should include not only pedagogical expertise but also a deep and broad disciplinary understanding.