Scalar test-wave stress and multiplier current

Assume the axiom of countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$ for the smooth tangent-bundle conventions. Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold of signature $(-,+,+,+)$ with metric-selected connection $\mathrm{\nabla }$ and inverse metric $g^{ab}$. Let $U\subseteq M$ be open, let $\psi \in C^{\mathrm{\infty }}(U,\mathbb{R})$ be a real scalar field, and let $X$ be a smooth vector field on $U$. Define the wave operator ${\mathrm{\square }}_g\psi :=g^{ab}{\mathrm{\nabla }}_a{\mathrm{\nabla }}_b\psi$ and the symmetric covariant tensor

$Q_{ab}[\psi ]:=({\mathrm{\nabla }}_a\psi )({\mathrm{\nabla }}_b\psi )-\frac{1}{2}{g}_{ab}{g}^{cd}({\mathrm{\nabla }}_c\psi )({\mathrm{\nabla }}_d\psi )\mathrm{.}$

Define the multiplier covector by $J_a^X[\psi ]:=Q_{ab}[\psi ]X^b$. The associated energy-current vector is $j_X^a[\psi ]:=-g^{ab}{J}_b^X[\psi ]$; this sign convention makes its flux through a future spacelike slice equal to the density $Q_{ab}{n}^a{X}^b$ when $n$ is the future unit normal. The test-wave equation is ${\mathrm{\square }}_g\psi =0$ on $U$. These objects are defined on the fixed background $g$; $Q$ is not automatically an SI matter stress-energy tensor, is not inserted into an Einstein equation here, and does not evolve $g$.

If $\psi$ is dimensionless and the coordinates have length units, the components of $Q$ have units of inverse length squared. This dimensional fact does not identify $Q$ with a physical stress tensor without an independently specified field normalization and coupling.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors

This $Q_{ab}$ is the stress tensor of a mathematical test scalar on a fixed background. It is not identified with the physical Einstein-matter tensor. The page supplies no gauge-invariant Regge–Wheeler, Zerilli, or Teukolsky master variables, no reconstruction theorem, and no mode exclusions or gravitational boundary conditions. Those definitions and reductions remain prerequisites for any theorem about linear gravitational perturbations. A scalar-wave estimate here is not a gravitational or nonlinear Kerr stability result.