Authors apply multisymplectic and polysymplectic formalisms to the known Palatini-Cartan model, recovering torsion-free and Einstein equations, constructing momentum maps and Noether currents, and performing a space-time decomposition into instantaneous Hamiltonian form.
De Donder-Weyl Hamiltonian formalism of MacDowell-Mansouri gravity
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abstract
We analyse the behaviour of the MacDowell-Mansouri action with internal symmetry group $\mathrm{SO}(4,1)$ under the covariant Hamiltonian formulation. The field equations, known in this formalism as the De Donder-Weyl equations, are obtained by means of the graded Poisson-Gerstenhaber bracket structure present within the covariant formulation. The decomposition of the internal algebra $\mathfrak{so}(4,1)\simeq\mathfrak{so}(3,1)\oplus\mathbb{R}^{3,1}$ allows the symmetry breaking $\mathrm{SO}(4,1)\to\mathrm{SO}(3,1)$, which reduces the original action to the Palatini action without the topological term. We demonstrate that, in contrast to the Lagrangian approach, this symmetry breaking can be performed indistinctly in the covariant Hamiltonian formalism either before or after the variation of the De Donder-Weyl Hamiltonian has been done, recovering Einstein's equations via the Poisson-Gerstenhaber bracket.
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gr-qc 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
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Geometric formulation for Palatini-Cartan gravity
Authors apply multisymplectic and polysymplectic formalisms to the known Palatini-Cartan model, recovering torsion-free and Einstein equations, constructing momentum maps and Noether currents, and performing a space-time decomposition into instantaneous Hamiltonian form.