First-order tetrad gravity in Dirac's Hamiltonian formalism

Abstract

The Hamiltonian $\ensuremath{\int}{d}^{3}x(p\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{q}\ensuremath{-}\mathcal{L})$ is constructed without adding further terms. The velocities ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{e}}_{\mathrm{ia}}$ and ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\omega}}}_{\mathrm{iab}}$ are eliminated while ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{e}}_{0a}$ and ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\omega}}}_{0ab}$ remain arbitrary. The second-class constraints reduce the theory to second-order tetrad gravity. The first-class constraints differ from those in second-order formalism, but satisfy the same gauge algebra provided one uses Dirac brackets.