= Solution
The qualification concerns gravitational energy. Local matter energy exists: an observer $u$ measures $T_{\mu\nu}u^\mu u^\nu$. What is absent is a generally covariant local gravitational stress-energy density with the required universal conservation properties. The <Equivalence principle> allows the connection to vanish at a point in freely falling coordinates; standard gravitational energy expressions depend on coordinates. Curvature does remain, but supplies no unique gravitational energy tensor of the required kind. A nonstationary geometry also lacks a preferred time-translation <Killing vector field>. For a trial timelike $t$, <stress-energy conservation> gives $\nabla_\mu(T^{\mu\nu}t_\nu)=T^{\mu\nu}\nabla_{(\mu}t_{\nu)}$, not generally zero.
An internal charge instead has a current $\nabla_\mu j^\mu=0$ independently of spacetime time translations. Its hypersurface flux is conserved with suitable boundary conditions. For <electric charge>, the <Gauss law> expresses it as a surface flux. Observer-dependent charge density does not prevent a well-defined total charge.
For an <asymptotically flat spacetime>, take asymptotically Cartesian slice coordinates with $\gamma_{ij}=\delta_{ij}+h_{ij}$ and appropriate falloff. In $c=1$ units, \b[the <ADM energy> is]
$$
\boxed{E_{\rm ADM}=\frac1{16\pi G}\lim_{r\to\infty}\int_{S_r}(\partial_jh_{ij}-\partial_ih_{jj})n^i\,dS.}
$$
It is an asymptotic gravitational energy. The <dominant energy condition> requires $-T^\mu{}_\nu u^\nu$ to be future-directed nonspacelike or zero for every future timelike $u$, in particular $T_{\mu\nu}u^\mu u^\nu\geq0$. Together with the constraint equations, completeness and appropriate asymptotic/boundary hypotheses, it yields $E_{\rm ADM}\geq|\mathbf P_{\rm ADM}|\geq0$. The energy condition alone is not the complete positive-energy theorem.
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