On the Plebanski Formulation with Energy Momentum

Abstract: 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.

• The paper demonstrates the explicit construction of the anti-self-dual 2-form T^i from the symmetric energy-momentum tensor using the Kulkarni-Nomizu product.
• It verifies that the matter-coupled Plebanski field equations naturally imply standard energy-momentum conservation through the chiral Bianchi identity.
• The approach is validated by deriving the Reissner-Nordström-de Sitter solution from a spherically symmetric electromagnetic source, bridging abstract theory and practical phenomenology.

On the Plebanski Formulation with Energy-Momentum: A Technical Summary

Overview

This work systematically addresses the long-standing problem of matter coupling in the Plebanski formulation of general relativity (GR), elucidating the translation of a metric energy-momentum tensor $T_{\mu\nu}$ into the anti-self-dual 2-form matter source $T^i$ required in the chiral, self-dual/anti-self-dual (SD/ASD) formalism. By invoking the Kulkarni-Nomizu (KN) algebraic product, the authors give an explicit, algorithmically tractable construction for $T^i$, reconciling the metric- and 2-form-based approaches to GR, and demonstrating that the matter-coupled Plebanski field equations naturally imply conventional energy-momentum conservation.

The Plebanski Formalism and the Matter Coupling Problem

Plebanski's approach recasts GR as a constrained chiral BF theory where the primary variables are a triple of self-dual 2-forms $\Sigma^i$ (encoding spacetime geometry) and an $\mathfrak{so}(3,\mathbb{C})$ connection $A^i$ [krasnov_formulations_2020, capovilla_self-dual_1991]. The formulation is predicated on the fact that in four dimensions, $\mathfrak{so}(1,3)_\mathbb{C}$ splits into SD and ASD components, a structure exploited to yield a minimal, purely differential-forms-based representation of gravity.

While vacuum GR admits an elegant Plebanski description, the incorporation of generic matter sources is less direct: $T_{\mu\nu}$, a symmetric 2-tensor, does not sit naturally within the SD/ASD decomposition of 2-forms. The complexity arises because the trace-free part of $T_{\mu\nu}$ sources the ASD component of the curvature, while the trace shifts the effective cosmological constant (i.e., enters the SD sector). Constructing a 2-form $T^i$ corresponding to a given $T^i$0, and verifying that energy-momentum conservation survives in the chiral formalism, are nontrivial nonholonomic translation problems [krasnov_plebanski_2011].

Construction of the Matter Source via the Kulkarni-Nomizu Product

The key technical result is the explicit prescription: $T^i$1 where $T^i$2 is the trace-free part of $T^i$3, $T^i$4 is the metric, $T^i$5 is the self-dual projection operator acting on indices in the curvature tensor algebra, and $T^i$6 denotes the KN product [besse_einstein_1987, hughes_encoding_2026]. This construction yields the precise anti-self-dual 2-form needed to source the ASD sector in the Plebanski field equations: $T^i$7 The approach is grounded in the representation theory of $T^i$8 and the algebraic decomposition of curvature tensors. The result is shown to be equivalent to, and computationally preferable over, the earlier definition due to Krasnov, $T^i$9, especially for implementation in computer algebra systems [krasnov_formulations_2020].

Energy-momentum Conservation via the Chiral Bianchi Identity

The matter-coupled chiral Plebanski field equations, with sources constructed as above, are shown to imply the standard conservation law $T^i$0 via the chiral Bianchi identity $T^i$1. The technical derivation demonstrates that the contracted chiral Bianchi identity, when expressed in the SD/ASD formalism and with the KN-based source term, contracts precisely onto the divergence of $T^i$2, confirming that diffeomorphism symmetry (and hence conservation) is maintained in the chiral approach. This is nontrivial—unimodular gravity and certain BF-like theories fail to fully recover all Einstein sector constraints in the absence of explicit conservation assumptions, as the authors discuss in light of recent work [montesinos_diffeomorphism-invariant_2023, gielen_unimodular_2024, hughes_encoding_2026].

Application: Reissner-Nordström-de Sitter from Plebanski with Electromagnetic Source

As a demonstration, the authors apply their formalism to the spherically symmetric electromagnetic (EM) energy-momentum tensor, reconstructing the full Reissner-Nordström-de Sitter solution using the matter-coupled Plebanski field equations. The construction leverages the explicit computation of $T^i$3 for the EM stress tensor under spherical symmetry, confirming that only $T^i$4 is nonzero (with $T^i$5 by symmetry), and that the ASD part of the field equations yields the correct metric functions when supplemented with the appropriate trace constraint. This operationalizes the theoretical framework and demonstrates compatibility with standard coordinate-based results, bridging abstraction and phenomenology.

Implications and Future Prospects

The clarified, algorithmic translation from symmetric metric energy-momentum tensors to chiral 2-form sources makes the Plebanski formalism and related chiral/BF formulations more tractable for practical computations in non-vacuum spacetimes. It unlocks the possibility of studying matter couplings—including phenomenological and astrophysical sources—in canonical variables suitable for quantization and for studying alternative and topological gravity theories.

The work prompts several theoretical investigations:

• Quantum Gravity and Spin Foam Models: Explicit matter coupling promotes Plebanski-type variables as a foundation for Lorentzian and Euclidean quantum gravity, particularly in spin foam quantization and in the investigation of chiral/gravitational instantons [baez_introduction_1999, krasnov_pure_2011].
• Unimodular and Trace-Free Gravity: By contrasting the trace-full and trace-free field content in Plebanski-inspired and unimodular approaches, this analysis further clarifies the status of the cosmological constant and the necessity of external conservation postulates [montesinos_trace-free_2025].
• Extension to Higher-Spin and Nontrivial Topologies: The protocol may be extensible, for instance, to higher-spin analogs or to the inclusion of nontrivial topological or bundle structures, especially given the modularity of the group-theoretic approach.
• Automatization and Computational Applications: The prescription is readily implemented in algebraic geometry software (e.g., SageManifolds), enabling systematic derivations for a broad class of matter sources [gourgoulhon_tensor_2015, joudyfjb_joudyfjbplebanski-formulation--energy-momentum_2026].

Notable Claims and Results

• The prescription for $T^i$6 via the KN product and self-dual projection constitutes a robust mapping from metric energy-momentum to chiral matter sources, reconciling the metric and 2-form languages.
• The energy-momentum conservation law follows directly from the chiral Bianchi identity in the matter-coupled Plebanski system using this prescription—no additional constraints are needed.
• The procedure gives an explicit derivation of the Reissner-Nordström-de Sitter solution from chiral variables and matter coupling alone.

Conclusion

This paper closes a significant technical gap in the implementation of matter coupling within the Plebanski formalism and allied chiral approaches to gravity. The explicit construction of $T^i$7 and the demonstration of conservation are of foundational value in both the classical and quantum gravitational contexts. The interface between symmetric tensor and chiral 2-form languages provided by the Kulkarni-Nomizu apparatus enhances the accessibility of self-dual gravity methods for applications involving phenomenologically relevant matter sectors, and it lays a precise foundation for further theoretical developments in chiral, BF, unimodular, and higher gauge theories of gravity.

References:

(2606.20042) (this manuscript), [krasnov_formulations_2020], [capovilla_self-dual_1991], [hughes_encoding_2026], [gielen_unimodular_2024], [montesinos_diffeomorphism-invariant_2023], [baez_introduction_1999], [krasnov_plebanski_2011], [gourgoulhon_tensor_2015], [joudyfjb_joudyfjbplebanski-formulation--energy-momentum_2026].