From ADM Geometrodynamics to Ashtekar Connection Dynamics: A Compact Pedagogical Abstract-Index Derivation

Abstract

These notes present a systematic derivation of the Hamiltonian formulation of general relativity and Ashtekar’s new variables. We begin with the Dirac–Bergmann theory of constrained Hamiltonian systems, emphasizing how singular Lagrangians give rise to primary and secondary constraints and why this structure is essential for general relativity. After establishing the required notation, including Penrose’s abstract index convention and vielbein-based internal indices, we develop the $3+1$ decomposition of spacetime in detail. The lapse function, shift vector, spatial metric, spatial projection, extrinsic curvature, spatial covariant derivative, and Lie evolution of spatial tensors are introduced as the geometric ingredients needed to derive the ADM Hamiltonian formulation from the Einstein–Hilbert action.

The ADM canonical variables $(h_{ab},\pi^{ab})$ are then derived explicitly, and the Hamiltonian and momentum constraints are shown to arise from the consistency of the primary constraints associated with lapse and shift. Finally, we reformulate the canonical theory using Ashtekar’s new variables, replacing metric dynamics by connection dynamics through the canonical pair $(A^i_a,E^a_i)$. In this language, the gravitational constraints become the Gauss, vector, and scalar constraints, with the scalar constraint taking a polynomial form after introducing the densitized lapse. The notes conclude by emphasizing the gauge-theoretic interpretation of the Ashtekar formulation and the role of the reality conditions required to recover real Lorentzian general relativity.

Citation

@misc{Lee2026ADMtoAshtekar,
  author    = {Lee, Qian-Rui},
  title     = {From {ADM} Geometrodynamics to {Ashtekar} Connection Dynamics: A Compact Pedagogical Abstract-Index Derivation},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.20606513}
}