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A Lorentz frame at a spacetime event is an orthonormal tangent basis with metric $\mathrm{diag}(-1,1,1,1)$. Future-oriented frames are related by proper orthochronous Lorentz transformations. Any such basis can be realized at the event by normal coordinates.

Adopt the printed Riemann curvature tensor convention and contract $R_{ab}=R^c{}_{acb}$. At an arbitrary point choose normal coordinates for the original metric tensor. There the original connection coefficients vanish. Varying the coordinate curvature formula leavesAt that point these partial derivatives equal the covariant derivatives of the tensor $C$. Both sides are tensors, so the result is valid in every coordinate system:Contracting its first and third indices proves the Palatini identityThe normal-coordinate argument is applied independently at every point; it does not assume a flat background or set derivatives of the original connection to zero.

At any chosen point take normal coordinates for $g$, so $\Gamma =0$ there. Substitute $\tilde{\Gamma }=\Gamma +T$ into the stated Riemann curvature tensor formula and subtract the original curvature. At that point the mixed products with $\Gamma$ vanish and ordinary derivatives of $T$ equal its covariant derivatives. The result isIn unit-weight antisymmetrization notation this is $\boxed{2T^a{}_{b[d;c]}-2T^a{}_{e[d}{T}^e{}_{c]b}}$. Every term is a tensor, so the identity established in normal coordinates holds in every coordinate system.

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,A vector with these components is future causal or zero exactly whenThis 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. The condition must hold in every local Lorentz frame; one chart alone is insufficient.