Volume elements of spacetime and a quartet of scalar fields

Starting with a “bare” 4-dimensional differential manifold as a model of spacetime, we discuss the options one has for defining a volume element which can be used for physical theories. We show that one has to prescribe a scalar density σ. Whereas conventionally $\sqrt{|\det g_{\mathrm{ij}}|}$ is used for that purpose, with $g_{\mathrm{ij}}$ as the components of the metric, we point out other possibilities, namely σ as a “dilaton” field or as a derived quantity from either a linear connection or a quartet of scalar fields, as suggested by Guendelman and Kaganovich.