A divergence-free symmetric tensor with negative measured energy density

Given: The time-oriented flat Lorentzian four-manifold $({\mathbb{R}}^4,g,N)$ with identity chart, $g=\mathrm{diag}(-1,1,1,1)$, time orientation $N:={\mathrm{\partial }}_0$ and the standard coordinate orientation, with metric-selected connection $\mathrm{\nabla }$; the open set $U:={\mathbb{R}}^4$; constants $E_0>0$ with units $\mathrm{J}{\mathrm{m}}^{-3}$ and $L_0>0$ with units $\mathrm{m}$; the smooth function $\phi (x):=E_0[1+(x^1\mathrm{/}{L}_0{)}^2]>0$, with energy-density units; and the smooth symmetric covariant $2$-tensor field $T$ with components $T_{ab}=-\phi {\delta }_a^0{\delta }_b^0$.

On the identity chart the Christoffel symbols vanish and $\mathrm{\nabla }$ is the coordinate connection, so ${\mathrm{\nabla }}^a{T}_{ab}={\eta }^{ac}{\mathrm{\nabla }}_c{T}_{ab}={\eta }^{ac}{\mathrm{\partial }}_c{T}_{ab}$ and $T^{ab}={\eta }^{a\alpha }{\eta }^{b\beta }{T}_{\alpha \beta }$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, The inverse metric, musical isomorphisms and index raising).

$T$ is a smooth symmetric covariant $2$-tensor field on $U$: its components are smooth functions of $x$ and symmetric in $a,b$; equivalently, ${\mathbb{R}}^4$ is a smooth manifold and the component prescription defines a smooth tensor field (Smooth covariant 2-tensor fields on a smooth manifold, An open subset of a smooth manifold has a canonical restricted smooth structure, $C^r$ and smooth maps between smooth manifolds).

The measured energy density is $\mu =T_{ab}{n}^a{n}^b$ (The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress clause (b), Physical reading of the observer split: measured energy density, energy flux and momentum density, and measured stress clause (a)).

The smooth field $N={\mathrm{\partial }}_0$ is timelike since $g(N,N)=-1<0$ and therefore supplies a time orientation. For $n:=N$, one has $g(n,n)=-1$ and $g(n,N)=-1<0$, so $n$ is a smooth future-directed observer congruence on ${\mathbb{R}}^4$, with $n^a={\delta }_0^a$ and $n_a=-{\delta }_a^0$ (Observer congruences, their observer curves and adapted charts, Timelike, null, spacelike and causal vectors of a Lorentz vector space, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

The weak energy condition of The pointwise energy conditions WEC, NEC, DEC and SEC is an additional pointwise assumption on $T$, not a consequence of conservation (The pointwise energy conditions WEC, NEC, DEC and SEC, Stress-energy tensor field of a model).