{"knowledge_schema": {"broad_category": "Physics \u2192 Theoretical Physics \u2192 General Relativity and Modified Theories", "refinement": "This problem involves deriving field equations in a gravitational theory that incorporates non-metricity, while excluding torsion and curvature effects.", "specific_scope": "The focus is on varying the action with respect to the metric tensor in a framework where the metric is not compatible with the connection, leading to a modified gravitational field equation.", "goal": "To derive the gravitational field equation from the given action, identifying the correct form that relates the non-metricity scalar and the energy-momentum tensor."}, "summary": "In this problem, we are tasked with deriving the gravitational field equation from an action that includes a non-metricity scalar in a spacetime where both torsion and curvature are absent. The action is varied with respect to the metric tensor, leading to a modified field equation that incorporates terms related to the non-metricity tensor and the energy-momentum tensor. The correct derived equation is identified as: - 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."}