On the Plebanski Formulation with Energy Momentum
In Plebanski's formulation the coupling of matter is less direct than in the metric formulation since the energy-momentum tensor $T_{μν}$ is symmetric, while the Plebanski variables are naturally valued in the self-dual/anti-self-dual Hodge decomposition of 2-forms. An explicit construction of the Plebanski matter source $T^i$ is obtained by lifting the trace-free energy-momentum tensor $\hat T_{μν}$ into the $(1,1)$ component of the algebraic curvature space using the Kulkarni-Nomizu product, and then extracting its chiral components. This construction reproduces the definition for $T^i$ in terms of the self-dual basis $Σ^i$ and $\hat T_{μν}$ introduced by Krasnov. We also verify that the matter-coupled chiral field equations imply the usual conservation law $\nabla_μT^{μν}=0$ through the chiral Bianchi identity $d^A F^i=0$. As an application, the construction is applied to a spherically symmetric electromagnetic stress-energy tensor, where the anti-self-dual part of the matter-coupled Plebanski field equations yields the Reissner-Nordström-de Sitter solution. The result gives a systematic prescription for translating metric matter sources into the anti-self-dual source terms required by the Plebanski field equations.