Pointwise Maxwell and matter energy-condition vocabulary

Use the smooth oriented, time-oriented Lorentzian four-manifold, signature $(-,+,+,+)$, metric $g$, and future cone of Oriented Lorentzian fields, observers and coordinate conventions. Here timelike means $g(v,v)<0$, null means $v\mathrm{\ne }0$ and $g(v,v)=0$, causal means timelike or null, and a nonzero causal vector is future-directed when $g(v,N)<0$, as in that supplier. A future unit observer is a vector $n$ with $g(n,n)=-1$ and $g(n,N)<0$. At a point $p$, let $T_p:T_{p}M\times T_{p}M\to \mathbb{R}$ be a continuous real symmetric bilinear form, represented by the covariant tensor $T_{ab}$. Define its metric trace by ${\mathrm{tr}}_{g}T:=g^{ab}{T}_{ab}$ and its raised mixed tensor by $T^a{}_b:=g^{ac}{T}_{cb}$.

• The weak energy condition (WEC) holds at $p$ when $T_{ab}{v}^a{v}^b\ge 0$ for every future timelike $v\in T_{p}M$.
• The null energy condition (NEC) holds at $p$ when $T_{ab}{\mathrm{\ell }}^a{\mathrm{\ell }}^b\ge 0$ for every nonzero future-directed null $\mathrm{\ell }\in T_{p}M$.
• The strong energy condition (SEC) holds at $p$ when the WEC holds for the trace-reversed tensor $T_{ab}-\frac{1}{2}({\mathrm{tr}}_{g}T)g_{ab}$.
• The dominant energy condition (DEC) holds at $p$ when, for every future-directed timelike $v\in T_{p}M$, the vector $J^a:=-T^a{}_b{v}^b$ is future causal or zero.

These are pointwise algebraic conditions. They are independent of an Einstein equation or an assertion that a physical material realizes $T$. The continuous regularity suffices for the definitions; any differentiability needed for a conservation identity is a separate hypothesis.

For a future unit observer $n$, observer energy density means $T_{ab}{n}^a{n}^b$. The observer energy-current vector is $-T^a{}_b{n}^b$, and its spatial energy flux is the projection of this vector into the positive rest space $n^{\perp }$.

Tong's lecture notes express SEC through a curvature inequality after invoking the Einstein equation. The definition here instead names the pointwise trace-reversed-tensor inequality itself, so no physical field equation is a prerequisite.

In the length-coordinate SI convention, if $T$ has stress-energy units $\mathrm{J}{\mathrm{m}}^{-3}$, the spatial projection of $-T^a{}_b{n}^b$ is an energy-current component with the same units. The corresponding physical energy-flux vector is $c$ times that component, in $\mathrm{W}{\mathrm{m}}^{-2}$.