[Paper Review] Einstein's Curvature for Nonlocal Gravitation of Gesamt Energy Carriers

I. É. Bulyzhenkov|ArXiv.org|Mar 13, 2006

Relativity and Gravitational Theory7 references9 citations

TL;DR

This paper proposes a nonlocal, flat-space formulation of general relativity where gravitational effects arise from continuous $ r^{-4} $ energy distributions rather than point masses, preserving strict 3D spatial flatness. By reinterpreting Einstein’s curvature via nonlocal energy-charge densities and enforcing universal metric symmetries, the model reproduces key relativistic effects—like perihelion precession and light bending—without curved spatial geometry, offering a singularity-free, Machian alternative to standard GR.

ABSTRACT

The intrinsic metric symmetries for energy-momentum in warped space-time universally reinforce strict spatial flatness in the GR metric formalism. The passive/active energy-charge for the 1686, 1913, and 1915 gravitational laws maintains the universal free fall in non-empty material space of nonlocal elementary (radial) energies. The known planetary perihelion precession, radar echo delay, and gravitational light bending can be explained by the singularity-free metric solution without departure from Euclidean spatial geometry. Non-Newtonian flatspace precessions are expected for non-point orbiting gyroscopes exclusively due to the GR inhomogeneous time in the Earth's radial energy-charge. The self-contained SR-GR relativity of {\it gesamt} particle-field carriers of inertial energy relies on the Principle of Equivalence for geometrization of the r^{-4} continuous particle without references to Newton's mass-to-mass attraction. The post-Newtonian logarithmic potential for distributed particle densities is also the exact solution to Maxwell's equations with the analytical r^{-4} electric charge density instead of the delta-operator density.

Motivation & Objective

• To reformulate general relativity using continuous, nonlocal energy carriers instead of point masses, avoiding singularities and preserving spatial flatness.
• To demonstrate that planetary precession, radar delay, and light bending can be explained via a singularity-free metric without departing from Euclidean spatial geometry.
• To uphold the Principle of Equivalence and geometrization of energy-momentum for nonlocal 'gesamt' particles, replacing Newtonian mass-to-mass attraction with energy-to-energy gravitation.
• To unify gravitational and electromagnetic potentials under a vector-tensor framework for chiral, confined gravitational waves in non-empty space.
• To test the viability of flat-space relativity in light of Gravity Probe B and LISA, particularly for nonlocal energy-charge interactions and chiral graviton dynamics.

Proposed method

• Adopts the 1913 Entwurf metric formalism as a foundation, extending it to nonlocal, continuous energy sources with $ r^{-4} $ density distributions.
• Imposes universal 3D metric symmetry $ ar{oldsymbol{ ho}}_{ij} = oldsymbol{ ho}_{ij} = oldsymbol{ ho}_{ij} $, enforcing spatial flatness $ ar{oldsymbol{ ho}}_{ij} = oldsymbol{ ho}_{ij} $ in the GR framework.
• Reinterprets the time element $ d au $ as a non-linear function of spatial displacement $ dl $, embedding field corrections into $ d au $ rather than $ dl $, preserving flat spatial intervals.
• Applies the Einstein tensor curvature $ oldsymbol{ ho}_{ u}^{ u} $ to continuous energy densities, requiring $ oldsymbol{ ho}_{ u}^{ u} = 0 $ and $ oldsymbol{ ho}_{ u; u}^{ u} = 0 $ for consistency with nonlocal sources.
• Derives a logarithmic post-Newtonian potential for distributed mass-energy, showing it satisfies Maxwell’s equations with $ r^{-4} $ charge density instead of delta functions.
• Introduces a vector-potential approach to gravitational waves, where paired vector waves form a tensor graviton, enabling chiral confinement and energy conservation without outward flux.

Experimental results

Research questions

• RQ1 Can key relativistic effects like perihelion precession and light bending be reproduced without curved spatial geometry?
• RQ2 Does a nonlocal, continuous $ r^{-4} $ energy-charge distribution yield a singularity-free solution consistent with GR and the Equivalence Principle?
• RQ3 Can the 3D spatial flatness be preserved universally in a relativistic framework when gravitational fields are non-local and non-point-like?
• RQ4 How do nonlocal energy-to-energy gravitation and chiral wave confinement explain energy conservation in binary pulsars without gravitational wave energy loss?
• RQ5 Can the observed precessions of Gravity Probe B gyroscopes be explained by nonlocal time-dilation effects in flat space, rather than 3D metric curvature?

Key findings

• The model achieves a singularity-free solution to the gravitational field equations by replacing point-like mass sources with continuous $ r^{-4} $ energy distributions.
• Planetary perihelion precession, radar echo delay, and gravitational light bending are reproduced without assuming curved 3D space, relying instead on non-linear time dilation and flat spatial intervals.
• The spatial interval $ dl $ remains universally flat and independent of gravitational fields, while field effects are fully contained within the non-linear time element $ d au(dl, v) $.
• The post-Newtonian logarithmic potential for distributed energy densities exactly satisfies Maxwell’s equations when $ r^{-4} $ charge density replaces the Dirac delta function.
• Chiral confinement of paired vector gravitational waves in a tensor graviton structure explains the observed energy loss in binary pulsars like B1913+16 without outward gravitational radiation.
• The Gauss energy flux is strictly conserved in flat space for two-body systems, supporting the absence of 3D metric ripples and reinforcing the non-local, non-singular nature of the theory.