Reconstructing the Goldberg-Palatini Action from Noether Identities
Abstract
We investigate the reconstruction of the first-order gravitational action from the Noether identities associated with diffeomorphism invariance. The independent fields are a symmetric perturbation ϕμν of the contravariant metric density and a trace-adjusted connection Gλμν, whose transformation laws are prescribed. For a local action polynomial in these fields, with the gravitational coupling as its only dimensionful constant and each additional field carrying one power of it, a dilatation, which is a particular diffeomorphism, shows that in any dimension d>2 the action terminates at most at cubic order, its only possible cubic term being a derivative-free ϕGG vertex. Expanding the kernels of the resulting Lorentz-covariant ansatz in tensor bases gives 31 coefficients for d≥ 4. The Noether identities with general diffeomorphisms, expanded order by order in the fields, yield a linear system that eliminates the ϕϕ kernel and determines the remaining coefficients up to an overall normalization, recovering the Goldberg-Palatini action. Since the connection can be eliminated exactly, this also fixes the complete second-order action, with its infinite set of graviton vertices. The construction requires no choice of energy-momentum tensor, the source of ambiguity in the self-coupling approach. The kernel bases, the identities and the solution are given in appendices, and the symbolic computations are publicly available.
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BibTeX
@misc{brandt2026reconstructing,
title = {{Reconstructing the Goldberg-Palatini Action from Noether Identities}},
author = {F. T. Brandt and Augusto C. B. Carvalho},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28065},
eprint = {2610.28065},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28065},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Reconstructing the Goldberg-Palatini Action from Noether Identities
AU - F. T. Brandt
AU - Augusto C. B. Carvalho
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28065
UR - https://arxiv.org/abs/2610.28065
AB - We investigate the reconstruction of the first-order gravitational action from the Noether identities associated with diffeomorphism invariance. The independent fields are a symmetric perturbation ϕ^μν of the contravariant metric density and a trace-adjusted connection G^λ_μν, whose transformation laws are prescribed. For a local action polynomial in these fields, with the gravitational coupling as its only dimensionful constant and each additional field carrying one power of it, a dilatation, which is a particular diffeomorphism, shows that in any dimension d>2 the action terminates at most at cubic order, its only possible cubic term being a derivative-free ϕGG vertex. Expanding the kernels of the resulting Lorentz-covariant ansatz in tensor bases gives 31 coefficients for d≥ 4. The Noether identities with general diffeomorphisms, expanded order by order in the fields, yield a linear system that eliminates the ϕϕ kernel and determines the remaining coefficients up to an overall normalization, recovering the Goldberg-Palatini action. Since the connection can be eliminated exactly, this also fixes the complete second-order action, with its infinite set of graviton vertices. The construction requires no choice of energy-momentum tensor, the source of ambiguity in the self-coupling approach. The kernel bases, the identities and the solution are given in appendices, and the symbolic computations are publicly available.
ER -