[Paper Review] The N-body Problem in Tetrad Gravity: a First Step towards the Unified Description of the Four Interactions

Luca Lusanna|ArXiv.org|Oct 9, 1998

Cosmology and Gravitation Theories19 references3 citations

TL;DR

This paper introduces a novel formulation of tetrad gravity that enables a canonical reduction to the rest-frame Wigner-covariant instant form, even in the presence of N scalar particles. By modifying the ADM formalism to resolve the deparametrization problem in general relativity, it achieves a consistent Hamiltonian description of the N-body problem, laying a foundational step toward a unified theory of the four fundamental interactions.

ABSTRACT

After a review of the canonical reduction to the rest-frame Wigner-covariant instant form of standard theories in Minkowski spacetime, a new formulation of tetrad gravity is introduced. Its canonical reduction, also in presence of N scalar particles, is done. The modification of the ADM formulation to solve the deparametrization problem of general relativity (how to recover the rest-frame instant form for G=0), is presented.

Motivation & Objective

• To develop a canonical formulation of tetrad gravity that allows for a consistent rest-frame Wigner-covariant description of N-body systems.
• To address the deparametrization problem in general relativity, which prevents the recovery of the instant form when Newton's constant G is set to zero.
• To extend the standard ADM formalism to maintain physical degrees of freedom and time evolution in a manifestly covariant framework.
• To provide a Hamiltonian framework for N scalar particles coupled to tetrad gravity, preserving spacetime covariance and enabling quantization.

Proposed method

• Adopting the tetrad formalism of general relativity, the paper formulates the gravitational action in terms of orthonormal tetrads and spin connections.
• Performing a canonical analysis of the tetrad gravity action, identifying first-class constraints and gauge symmetries.
• Implementing a canonical reduction to the rest-frame Wigner-covariant instant form, fixing the 3+1 slicing and time gauge via constraints.
• Introducing a modified ADM-like formalism that ensures the correct number of physical degrees of freedom and allows for a consistent time evolution in the G→0 limit.
• Integrating N scalar particles into the canonical framework by coupling them to the tetrad fields through minimal coupling.
• Using the resulting Hamiltonian system to derive equations of motion that are manifestly Lorentz-covariant and consistent with the instant form of dynamics.

Experimental results

Research questions

• RQ1 How can tetrad gravity be canonically reduced to the rest-frame Wigner-covariant instant form in the presence of N scalar particles?
• RQ2 What modifications to the ADM formalism are necessary to resolve the deparametrization problem in general relativity?
• RQ3 Can a consistent Hamiltonian formulation of the N-body problem be achieved in tetrad gravity that preserves spacetime covariance and physical degrees of freedom?
• RQ4 How does the introduction of scalar particles affect the canonical structure of tetrad gravity in the instant form?
• RQ5 Is it possible to recover the standard instant form of dynamics in the limit G→0 using this modified tetrad-ADM approach?

Key findings

• The paper successfully formulates a canonical reduction of tetrad gravity to the rest-frame Wigner-covariant instant form, even with N scalar particles present.
• A modified ADM-like formalism is derived that resolves the deparametrization problem, ensuring consistency in the G→0 limit.
• The canonical structure preserves the correct number of physical degrees of freedom and maintains explicit Lorentz covariance.
• The N-body system is described by a Hamiltonian that is manifestly covariant and compatible with the instant form of dynamics.
• The framework provides a consistent starting point for the quantization of gravity coupled to matter, paving the way for a unified description of the four interactions.
• The results demonstrate that tetrad gravity in this formulation allows for a well-defined transition to the non-relativistic limit and supports a unified field-theoretic approach.