= Solution
The <dominant energy condition> says that, for every future <timelike vector> $u^a$, the energy current $J^a=-T^a{}_b u^b$ is future causal or zero. Choose <normal coordinates> whose future unit time vector at $p$ is $u^a=(1,0,0,0)$. With <metric signature> $(-,+,+,+)$ and symmetric <stress-energy tensor>,
$$
J^0=T_{00},\qquad J^i=-T_{i0}.
$$
A vector with these components is future causal or zero exactly when
$$
\boxed{T_{00}\ge\sqrt{\sum_iT_{0i}^2}.}
$$
This proves necessity. Conversely, every future unit <timelike vector> can be made the time axis of an orthonormal frame by a proper orthochronous <Lorentz transformation>, and that frame can be extended to <normal coordinates> at $p$. Requiring the displayed inequality in every such chart therefore gives the <dominant energy condition> for every future <timelike vector>; positive rescaling handles nonunit vectors. Future <null vectors> follow by continuity if they are included in the definition. \b[The condition must hold in every local Lorentz frame; one chart alone is insufficient.]
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