Null plane-wave scalar field and its conserved stress-energy tensor

The field solves the massless equation. By [F1], ${\mathrm{\partial }}_a\phi =f^{\mathrm{\prime }}(\theta )k_a$ and ${\mathrm{\partial }}_a{\mathrm{\partial }}_b\phi =f^{\mathrm{\prime }\mathrm{\prime }}(\theta )k_a{k}_b$; hence $\mathrm{\square }\phi ={\eta }^{ab}{\mathrm{\partial }}_a{\mathrm{\partial }}_b\phi =f^{\mathrm{\prime }\mathrm{\prime }}(\theta ){\eta }^{ab}{k}_a{k}_b=0$, because $k$ is null. With $V\equiv 0$ this is exactly the scalar-field equation $\mathrm{\square }\phi =V^{\mathrm{\prime }}(\phi )$ of [F2].

The stress-energy tensor. Substituting [F1] into the Hilbert formula of [F2] with $V\equiv 0$ gives ${\mathrm{\partial }}^a\phi {\mathrm{\partial }}^b\phi =f^{\mathrm{\prime }}(\theta {)}^2{k}^a{k}^b$ and ${\mathrm{\partial }}_d\phi {\mathrm{\partial }}^d\phi =f^{\mathrm{\prime }}(\theta {)}^2{k}_d{k}^d=0$, hence $T^{ab}=K{f}^{\mathrm{\prime }}(\theta {)}^2{k}^a{k}^b$, which is symmetric. Wherever $f^{\mathrm{\prime }}(\theta )\mathrm{\ne }0$ the coefficient $K{f}^{\mathrm{\prime }}(\theta {)}^2$ is positive and $T$ is a positive multiple of the rank-one tensor $k^{\mathrm{♯}}\otimes k^{\mathrm{♯}}$ with nonzero future null generating vector $k^{\mathrm{♯}}$; wherever $f^{\mathrm{\prime }}(\theta )=0$ all components vanish. For every future unit $n$, $T^{ab}{n}_a{n}_b=K{f}^{\mathrm{\prime }}(\theta {)}^2(k_a{n}^a{)}^2\ge 0$; in the inertial frame, $T^{00}=K{f}^{\mathrm{\prime }}(\theta {)}^2(k^0{)}^2\ge 0$ with $k^0>0$.

Conservation. Since $\phi$ solves the field equation by step 1.1, [F3] gives ${\mathrm{\partial }}_a{T}^{ab}=0$: the plane-wave tensor is conserved on the whole chart domain.

The example is an explicit conditional computation for the massless scalar model with a null plane-wave profile; it asserts no wave observation, no energy condition beyond the displayed nonnegativity and no measurement.