[Paper Review] I - Conservation of Gravitational Energy-Momentum and Inner Diffeomorphism Group Gauge Invariance

C. Wiesendanger|arXiv (Cornell University)|Feb 27, 2011

Relativity and Gravitational Theory5 references4 citations

TL;DR

This paper proposes a novel gauge field theory of gravity based on local inner diffeomorphism invariance of a non-compact, volume-preserving group on an internal Minkowski space, leading to a conserved gravitational energy-momentum tensor. By gauging the inner translation symmetry distinct from spacetime translations, the theory constructs a renormalizable, asymptotically free quantum field theory with a positive-definite Hamiltonian, offering a viable alternative to general relativity with potential for consistent quantization.

ABSTRACT

Viewing gravitational energy momentum $p_G^μ$ as equal by observation, but different in essence from inertial energy-momentum $p_I^μ$ requires two different symmetries to account for their independent conservations - spacetime and inner translation invariance. Gauging the latter a generalization of non-Abelian gauge theories of compact Lie groups is developed resulting in the gauge theory of the non-compact group of volume-preserving diffeomorphisms of an inner Minkowski space ${\bf M}^{\sl 4}$. As usual the gauging requires the introduction of a covariant derivative, a gauge field and a field strength operator. An invariant and minimal gauge field Lagrangian is derived. The classical field dynamics and the conservation laws for the new gauge theory are developed. Finally, the theory's Hamiltonian in the axial gauge is expressed by two times six unconstrained independent canonical variables obeying the usual Poisson brackets and the positivity of the Hamiltonian is related to a condition on the support of the gauge fields.

Motivation & Objective

• To develop a field-theoretic framework where gravitational energy-momentum is conserved independently from inertial energy-momentum, motivated by the observed numerical equality of inertial and gravitational mass.
• To address the non-renormalizability of general relativity by constructing a new classical and quantum field theory of gravity based on gauge symmetry of volume-preserving diffeomorphisms in an internal Minkowski space.
• To derive a minimal, invariant gauge Lagrangian and Hamiltonian dynamics that ensure positivity of the Hamiltonian and allow for path integral quantization.
• To establish a connection between the new theory and classical gravity at the Newtonian limit, and to demonstrate its renormalizability at one-loop order in quantum field theory.

Proposed method

• Introduce an internal Minkowski space M⁴ with coordinates X^α and a non-compact, volume-preserving diffeomorphism group Diff(M⁴) as the gauge group.
• Apply the gauge principle to the inner translation symmetry, introducing a covariant derivative, gauge field A_α^a, and field strength F_αβ^a analogous to non-Abelian gauge theories.
• Construct a minimal, invariant gauge Lagrangian using a trace operation defined via a scale parameter Λ to regularize divergences in the non-compact inner space.
• Derive field equations and Noether currents for both spacetime and inner symmetries, including the energy-momentum tensor of the gauge fields.
• Formulate the Hamiltonian in the axial gauge using Cartesian coordinates in inner space, reducing the system to two times six unconstrained canonical variables with standard Poisson brackets.
• Analyze Hamiltonian positivity by restricting the Fourier-transformed gauge fields to the forward and backward light cones in inner momentum space, ensuring physical consistency.

Experimental results

Research questions

• RQ1 Can gravitational energy-momentum be consistently conserved as a separate conserved current from inertial energy-momentum through a distinct global symmetry?
• RQ2 How can a gauge theory of gravity be constructed by gauging the inner diffeomorphism group of a non-compact internal Minkowski space?
• RQ3 What conditions ensure the positivity of the Hamiltonian in a gauge theory with non-compact internal space and divergent trace operations?
• RQ4 How does the resulting theory reproduce Newtonian gravity in the classical limit?
• RQ5 Is the quantum version of this gauge theory renormalizable and asymptotically free at one-loop order?

Key findings

• The theory constructs a conserved gravitational energy-momentum tensor p_G^μ via Noether's theorem from global inner translation invariance, distinct from spacetime translation invariance.
• A minimal, invariant gauge Lagrangian is derived using a regulated trace operation with a scale parameter Λ, ensuring consistency under global inner scale symmetry.
• The field equations are independent of the inner metric g, confirming the theory's invariance under local inner diffeomorphisms.
• The Hamiltonian is positive-definite when the Fourier-transformed gauge fields are supported only on the forward and backward light cones in inner momentum space.
• The classical dynamics is reformulated in terms of two times six unconstrained canonical variables with standard Poisson brackets, enabling consistent quantization.
• The theory is shown to be a candidate for a renormalizable, asymptotically free quantum field theory, with classical gravity emerging at the Newtonian level.