General Relativity in Three Lines
Abstract
We show that complex general relativity can be formulated directly in terms of three affine lines assigned to each spacetime point. The lines are represented by Plücker bivectors, and Urbantke's construction applied to the associated two-forms reconstructs the conformal spacetime metric. Starting from the chiral Plebański action, the simplicity constraints turn the three two-forms into a null internal basis of simple bivectors and thereby into three projective lines. A 3+1 decomposition then yields a first-order canonical theory whose configuration variable is the weighted triple of lines and whose conjugate variable is a line momentum. The resulting constraint system consists of three internal Gauss constraints, three vector constraints, and one Hamiltonian constraint. A notable feature of the line formulation is that the equation obtained by varying the line momentum is pointwise algebraic after the spatial integrations by parts. Consequently the momentum can be eliminated explicitly wherever the corresponding Legendre map is invertible, giving a second-order action whose only dynamical configuration variable is the weighted triple of lines. The line curvature separates naturally into a cosmological trace and the Weyl matrix of the selected chiral sector. The Legendre map becomes singular on a distinguished null-evolution locus where the spacetime metric remains nondegenerate. There its rank drops from nine to six, giving three algebraic compatibility conditions. We construct a finite invariant partial Legendre transform and a smooth constrained second-order continuation through this locus. This analysis provides a regular framework for regimes in which the evolution direction becomes null and may be useful in the study of near-horizon geometries and related gravitational phenomena.
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BibTeX
@misc{nurowski2026general,
title = {{General Relativity in Three Lines}},
author = {Pawel Nurowski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22396},
eprint = {2610.22396},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22396},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - General Relativity in Three Lines
AU - Pawel Nurowski
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.22396
UR - https://arxiv.org/abs/2610.22396
AB - We show that complex general relativity can be formulated directly in terms of three affine lines assigned to each spacetime point. The lines are represented by Plücker bivectors, and Urbantke's construction applied to the associated two-forms reconstructs the conformal spacetime metric. Starting from the chiral Plebański action, the simplicity constraints turn the three two-forms into a null internal basis of simple bivectors and thereby into three projective lines. A 3+1 decomposition then yields a first-order canonical theory whose configuration variable is the weighted triple of lines and whose conjugate variable is a line momentum. The resulting constraint system consists of three internal Gauss constraints, three vector constraints, and one Hamiltonian constraint. A notable feature of the line formulation is that the equation obtained by varying the line momentum is pointwise algebraic after the spatial integrations by parts. Consequently the momentum can be eliminated explicitly wherever the corresponding Legendre map is invertible, giving a second-order action whose only dynamical configuration variable is the weighted triple of lines. The line curvature separates naturally into a cosmological trace and the Weyl matrix of the selected chiral sector. The Legendre map becomes singular on a distinguished null-evolution locus where the spacetime metric remains nondegenerate. There its rank drops from nine to six, giving three algebraic compatibility conditions. We construct a finite invariant partial Legendre transform and a smooth constrained second-order continuation through this locus. This analysis provides a regular framework for regimes in which the evolution direction becomes null and may be useful in the study of near-horizon geometries and related gravitational phenomena.
ER -