Lagrangian of the Unified Gravitational–Electromagnetic Field

Given the distribution of inertial mass density of some continuum medium \(\mu :=\mu (\textbf{x},t)\) , the field of velocities of this medium \(\textbf{u}:=\textbf{u}(\textbf{x},t)\) , the charge density \(\rho :=\rho (\textbf{x},t)\) and the current density \(\textbf{j}:=\textbf{j}(\textbf{x},t)\) , consider a Lagrangian density L defined by where \(\Phi \) is an ancillary proper scalar field and \(\textbf{p}\) is an ancillary proper vector field. Then L is invariant under the change of inertial or non-inertial Cartesian coordinate system.