= Chern-Simons source stress convention
{c}
{title2=$T_{\mu\nu}=2cJ_{(\mu}A_{\nu)}-cg_{\mu\nu}J_\rho A^\rho$}
The pure <Abelian Chern--Simons theory> has zero metric stress. For a source written as $-c\int\star J\wedge A$, holding the covariant components of the <one-forms> $J,A$ fixed makes the source equal to $-c\int\sqrt{-g}\,g^{\mu\nu}J_\mu A_\nu\,d^3x$. Variation using $\delta\sqrt{-g}=-\sqrt{-g}g_{\mu\nu}\delta g^{\mu\nu}/2$ gives the displayed <stress-energy tensor>. If the independent source is instead the fixed <two-form> $\mathcal J=\star J$, the source action is metric-independent and its stress vanishes. The variational data must therefore be stated.
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