Metric dependence of j^μ√(-g) and J^μ√(-g) under continuity equations

Determine whether the products j^μ√(-g) and J^μ√(-g) depend on the metric tensor g_{μν} in the covariant Lagrangian formalism for continuously distributed matter, given that the continuity equations V_μ j^μ = 0 and V_μ J^μ = 0 ensure only coordinate independence of these products.

Background

The paper analyzes two approaches to varying the action with respect to the metric tensor. In the second approach, the independent variables are taken to be the fields’ four-potentials, the position vector x^α, and the metric tensor, and it is asserted that the products j^μ√(-g) and J^μ√(-g) do not vary with the metric, implying their partial derivatives with respect to g_{μν} vanish.

The authors argue that the continuity equations only guarantee that j^μ√(-g) and J^μ√(-g) do not depend on the relevant coordinates, leaving their dependence on the metric unresolved. Clarifying this dependence is crucial for determining whether terms involving four-current–four-potential products should appear in the metric variation and thus in the resulting metric field equations.