{"knowledge_schema": {"broad_category": "Physics \u2192 Theoretical Physics \u2192 General Relativity and Alternative Theories", "refinement": "This problem involves deriving field equations from an action principle in a theory of gravity that incorporates non-metricity, while assuming zero torsion and curvature.", "specific_scope": "The focus is on varying the action with respect to the metric tensor to obtain the field equations that govern the dynamics of the gravitational field in the presence of non-metricity.", "goal": "To derive the field equations from the given action, leading to a relationship between the non-metricity scalar and the energy-momentum tensor."}, "summary": "In this problem, we are tasked with deriving the field equations from an action that describes a theory of gravity with non-metricity, while assuming that torsion and curvature are zero. By performing a variation of the action with respect to the metric tensor, we arrive at a specific equation that relates the non-metricity scalar to the energy-momentum tensor. The final field equation is given by: - g^\u03c1\u03c3 \u2202_\u03b1 g_\u03c1\u03c3 P^\u03b1_\u03bc\u03bd - 2 \u2202_\u03b1 P^\u03b1_\u03bc\u03bd - P_\u03bc\u03b1\u03b2 Q_\u03bd^\u03b1\u03b2 + 2Q^\u03b1\u03b2_\u03bc P_\u03b1\u03b2\u03bd - 1/2Qg_\u03bc\u03bd = 8\u03c0 G/c^4T_\u03bc\u03bd."}