On the vierbein formalism of general relativity

TL;DR

This paper explores the vierbein formalism of general relativity, demonstrating the non-positivity of the kinetic energy term in the Einstein-Hilbert action and proposing coordinate conditions to achieve positive definiteness, facilitating Hamiltonian analysis and quantization.

Contribution

It introduces coordinate conditions that yield positive definite kinetic energy in the vierbein formalism, enabling Hamiltonian and canonical quantization approaches.

Findings

Kinetic energy term is non-positive in the standard formalism.

Proposed coordinate conditions produce positive definite kinetic energy.

Derived Hamiltonian constraints suitable for quantization.

Abstract

Both the Einstein-Hilbert action and the Einstein equations are discussed under the absolute vierbein formalism. Taking advantage of this form, we prove that the "kinetic energy" term, i.e., the quadratic term of time derivative term, in the Lagrangian of the Einstein-Hilbert action is non-positive definitive. And then, we present two groups of coordinate conditions that lead to positive definitive kinetic energy term in the Lagrangian, as well as the corresponding actions with positive definitive kinetic energy term, respectively. Based on the ADM decomposition, the Hamiltonian representation and canonical quantization of general relativity taking advantage of the actions with positive definitive kinetic energy term are discussed; especially, the Hamiltonian constraints with positive definitive kinetic energy term are given, respectively. Finally, we present a group of gauge conditions such that there is not any second time derivative term in the ten Einstein equations.