[Paper Review] Maxwell-like picture of General Relativity and its Planck limit

Piotr Ogonowski, Piotr Skindzier|arXiv (Cornell University)|Jan 13, 2013

Noncommutative and Quantum Gravity Theories18 references3 citations

TL;DR

This paper proposes a Lorentz-invariant, Maxwell-like formulation of General Relativity using Geroch decomposition in (3+1) spacetime, introducing a gravitational four-potential $ V^\mu $, field tensor $ F^{\mu\nu} $, and four-current $ J^\mu $ that reproduce the Einstein tensor and energy-stress tensor. The key result is a consistent framework for covariant quantization of gravity at the Planck scale, yielding quantum values for rest mass, photon energy, and the fine structure constant.

ABSTRACT

We show that Geroch decomposition leads us to Maxwell-like representation of gravity in $(3+1)$ metrics decomposition that may be perceived as Lorentz invariant version of GEM. For such decomposition we derive four-potential $V^μ$ and gravitational field tensor $F^{μν}$ that is associated with gravitational interaction. Next we show that gravitational four-current $J^μ$ derived for introduced four-potential produce energy-stress tensor and reproduce main General Relativity formula. Next we introduce valid Lagrangian and equations of motion that explains obtained results. At the end we introduce new approach to quantization of gravity that results in proper quantum values and is open to further generalization.

Motivation & Objective

• To develop a Lorentz-invariant, GEM-like reformulation of General Relativity that bridges Newton-Cartan and full GR.
• To derive a gravitational four-potential $ V^\mu $, field tensor $ F^{\mu\nu} $, and four-current $ J^\mu $ from Geroch decomposition of the Schwarzschild metric.
• To show that the energy-stress tensor derived from $ J^\mu $ reproduces the main equations of General Relativity.
• To introduce a new quantization scheme for gravity in the Planck limit, yielding physically meaningful quantum values.
• To explore the wave nature of matter and the role of the gravitational field in quantum gravity.

Proposed method

• Apply Geroch decomposition to the Schwarzschild metric, isolating a timelike Killing vector field and deriving the metric in (3+1) form with $ \lambda $ and $ \gamma_{ij} $.
• Define the gravitational scalar field $ \Phi = 1/\gamma_r $, identified as the inverse of the gravitational time dilation factor, and interpret curved spacetime as a flat manifold with a refractive index $ \gamma_r $.
• Construct a gravitational four-potential $ V^\mu $, field tensor $ F^{\mu\nu} $, and four-current $ J^\mu $ analogous to electromagnetism, satisfying $ \partial_\mu F^{\mu\nu} = J^\nu $.
• Derive the energy-stress tensor from $ J^\mu $, showing it reproduces the Einstein tensor and thus the full GR field equation in the new formalism.
• Introduce a Lagrangian for the gravitational field and derive equations of motion consistent with the field tensor and current.
• Perform covariant quantization of the scalar field $ \Phi $ in the Planck limit, yielding rest mass, photon energy, and fine structure constant values consistent with quantum theory.

Experimental results

Research questions

• RQ1 Can General Relativity be reformulated in a Maxwell-like, Lorentz-invariant form using Geroch decomposition in (3+1) spacetime?
• RQ2 Does the derived gravitational four-potential and field tensor reproduce the Einstein tensor and energy-stress tensor of GR?
• RQ3 Can the Planck-scale limit of the gravitational field yield physically meaningful quantum values such as rest mass and photon energy?
• RQ4 Is the wave nature of matter naturally encoded in this gravitational field formulation, especially in the local, massless limit?
• RQ5 Can this framework serve as a viable intermediate step toward a consistent quantum gravity theory, distinct from standard GR?

Key findings

• The Geroch decomposition of the Schwarzschild metric leads to a (3+1) spacetime structure where gravity is equivalent to a flat spacetime with a refractive index $ \gamma_r $, interpreted as the gravitational time dilation factor.
• The gravitational field is described by a four-potential $ V^\mu $ and field tensor $ F^{\mu\nu} $, analogous to electromagnetism, with a conserved four-current $ J^\mu $.
• The energy-stress tensor derived from $ J^\mu $ reproduces the Einstein tensor, confirming consistency with General Relativity's main field equation.
• In the Planck limit, the quantization of the gravitational scalar field $ \Phi $ yields rest mass and photon energy values consistent with quantum theory.
• The model reproduces a Coulomb-like potential for elementary charges in the infinity limit, suggesting a unification of gravity and electromagnetism at the classical level.
• The local quantum solution for the field is a massless plane wave, indicating that gravity in this framework acts as a wave-like field with no intrinsic mass in the local frame.