= Einstein-equation coupling under reversed curvature
{c}
{title2=$R^a{}_{bcd}\mapsto-R^a{}_{bcd}\implies\kappa\mapsto-\kappa$}
With the metric and <stress-energy tensor> fixed and zero <cosmological constant>, changing the sign convention for the <Riemann curvature tensor> changes the signs of the <Ricci tensor>, <scalar curvature> and <Einstein tensor>. The same physical field equation is therefore written with the opposite coupling sign. For signature $(+---)$, the convention $R^a{}_{bcd}=\partial_d\Gamma^a_{bc}-\partial_c\Gamma^a_{bd}-\Gamma^a_{ec}\Gamma^e_{bd}+\Gamma^a_{ed}\Gamma^e_{bc}$ and $R_{ab}=R^c{}_{acb}$ gives $R_{00}\simeq-\Delta\Phi$ in a static <Newtonian limit>. The trace-reversed equation and $\Delta\Phi=4\pi G\rho$ then require $G_{ab}=-8\pi GT_{ab}$.
Back to article page
