Complex scalar stress tensor with a supplied potential

Let $(M,g,N,o)$ have the smooth, boundaryless, oriented and time-oriented Lorentzian four-manifold structure, signature $(-,+,+,+)$, metric-selected connection ${\mathrm{\nabla }}^g$, and length-coordinate convention of Oriented Lorentzian fields, observers and coordinate conventions. Let $A\in {\mathrm{\Omega }}^1(M)$ be a supplied smooth real one-form, let $e\in \mathbb{R}$ be a fixed real coupling, and let $\psi \in C^{\mathrm{\infty }}(M,\mathbb{C})$ be a smooth complex-valued function. In the length-coordinate convention, assign dimensionless components to $A_a$ and unit ${\mathrm{m}}^{-1}$ to $e$, so $e{A}_a$ has derivative unit ${\mathrm{m}}^{-1}$; assign $\psi$ an arbitrary fixed unit $[\psi ]$. Define the complex covector

$D_a\psi :={\mathrm{\nabla }}_a^g\psi -ie{A}_a\psi \mathrm{.}$

Since $\psi$ is a scalar, ${\mathrm{\nabla }}_a^g\psi ={\mathrm{\partial }}_a\psi$. Define the real symmetric covariant tensor

${\mathsf{T}}_{ab}[\psi ;g,A,e]:=\overline{D_a\psi }{D}_b\psi +\overline{D_b\psi }{D}_a\psi -g_{ab}{g}^{cd}\overline{D_c\psi }{D}_d\psi ,$

with indices raised using $g^{-1}$. Its covariant components have unit $[\psi {]}^2{\mathrm{m}}^{-2}$. This is a mathematical tensor assignment. Its interpretation as a stress-energy tensor, and any field equation or conservation law, require a separate physical model.

Oriented Lorentzian fields, observers and coordinate conventions

The definition uses a globally supplied real one-form $A$ and a chosen complex trivialization. It asserts no global-bundle trivialization for a general charged field.