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Modified General Relativity and dark matter

Gary Nash

TL;DR

Modified General Relativity (MGR) extends GR by introducing a line element field that yields a connection-independent gravitational energy-momentum tensor ${oldsymbol{ m oldPhi}}_{ ho au}$, thereby restoring a complete total energy-momentum tensor and preserving the equivalence principle. The theory derives an extended Schwarzschild solution, predicts a gravitational energy density that includes a dark-energy-like term, and provides a modified Newtonian force with two extra terms interpreted as dark matter and dark energy effects, enabling a Tully-Fisher-type relation and flat rotation curves without ad hoc DM profiles. MGR treats dark matter as both a geometrical object (line element covectors) and a particle-like entity obeying a Klein-Gordon-type equation, allowing a dual, consistent description that reproduces galactic rotation curves and lensing phenomena (e.g., Bullet Cluster) without invoking standard CDM. Cosmologically, Phi dynamically replaces the cosmological constant and coexists with a dark-energy/dark-matter skeleton in curved spacetime, offering a richer framework to address inflation, galactic anisotropies, and CMB features beyond ΛCDM. Overall, MGR provides a mathematically coherent and testable route to unify gravity with dark sectors while preserving the core structure of the Einstein equation.

Abstract

Modified General Relativity (MGR) is the natural extension of General Relativity (GR). MGR explicitly uses the smooth regular line element vector field $(\bm{X},-\bm{X}) $, which exists in all Lorentzian spacetimes, to construct a connection-independent symmetric tensor that represents the energy-momentum of the gravitational field. It solves the problem of the non-localization of gravitational energy-momentum in GR, preserves the ontology of the Einstein equation, and maintains the equivalence principle. The line element field provides MGR with the extra freedom required to describe dark energy and dark matter. An extended Schwarzschild solution for the matter-free Einstein equation of MGR is developed, from which the Tully-Fisher relation is derived, and the gravitational energy density is calculated. The mass of the invisible matter halo of galaxy NGC 3198 calculated with MGR is identical to the result obtained from GR using a dark matter profile. Although dark matter in MGR is described geometrically, it has an equivalent representation as a particle with the property of a vector boson or a pair of fermions; the geometry of spacetime and the quantum nature of matter are linked together by the unit line element covectors that belong to both the Lorentzian metric and the spin-1 Klein-Gordon wave equation. The three classic tests of GR provide a comparison of the theories in the solar system and several parts of the cosmos. MGR provides the flexibility to describe inflation after the Big Bang and galactic anisotropies.

Modified General Relativity and dark matter

TL;DR

Modified General Relativity (MGR) extends GR by introducing a line element field that yields a connection-independent gravitational energy-momentum tensor , thereby restoring a complete total energy-momentum tensor and preserving the equivalence principle. The theory derives an extended Schwarzschild solution, predicts a gravitational energy density that includes a dark-energy-like term, and provides a modified Newtonian force with two extra terms interpreted as dark matter and dark energy effects, enabling a Tully-Fisher-type relation and flat rotation curves without ad hoc DM profiles. MGR treats dark matter as both a geometrical object (line element covectors) and a particle-like entity obeying a Klein-Gordon-type equation, allowing a dual, consistent description that reproduces galactic rotation curves and lensing phenomena (e.g., Bullet Cluster) without invoking standard CDM. Cosmologically, Phi dynamically replaces the cosmological constant and coexists with a dark-energy/dark-matter skeleton in curved spacetime, offering a richer framework to address inflation, galactic anisotropies, and CMB features beyond ΛCDM. Overall, MGR provides a mathematically coherent and testable route to unify gravity with dark sectors while preserving the core structure of the Einstein equation.

Abstract

Modified General Relativity (MGR) is the natural extension of General Relativity (GR). MGR explicitly uses the smooth regular line element vector field , which exists in all Lorentzian spacetimes, to construct a connection-independent symmetric tensor that represents the energy-momentum of the gravitational field. It solves the problem of the non-localization of gravitational energy-momentum in GR, preserves the ontology of the Einstein equation, and maintains the equivalence principle. The line element field provides MGR with the extra freedom required to describe dark energy and dark matter. An extended Schwarzschild solution for the matter-free Einstein equation of MGR is developed, from which the Tully-Fisher relation is derived, and the gravitational energy density is calculated. The mass of the invisible matter halo of galaxy NGC 3198 calculated with MGR is identical to the result obtained from GR using a dark matter profile. Although dark matter in MGR is described geometrically, it has an equivalent representation as a particle with the property of a vector boson or a pair of fermions; the geometry of spacetime and the quantum nature of matter are linked together by the unit line element covectors that belong to both the Lorentzian metric and the spin-1 Klein-Gordon wave equation. The three classic tests of GR provide a comparison of the theories in the solar system and several parts of the cosmos. MGR provides the flexibility to describe inflation after the Big Bang and galactic anisotropies.
Paper Structure (33 sections, 1 theorem, 145 equations, 1 figure)

This paper contains 33 sections, 1 theorem, 145 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Lorentzian spacetimes and Modified General Relativity
  3. Lorentzian spacetimes
  4. Modified General Relativity
  5. The extended Schwarzschild solution and the energy density of the gravitational field
  6. The energy density of the gravitational field
  7. Dark matter
  8. The dark parameters $a_{0}$ and $b$ of the line element field
  9. The radial gravitational force and galactic rotation curves in MGR
  10. Modeling galactic rotation curves with MGR
  11. Baryonic galaxies
  12. Galaxy NGC 3198
  13. The Bullet Cluster
  14. The geometrical and particle descriptions of dark matter
  15. Tests of GR compared to MGR
  16. ...and 18 more sections

Key Result

Theorem A.1

A non-divergenceless (0,2) symmetric tensor $\gamma_{\alpha\beta}$ in the symmetric cotangent bundle $\Gamma^{\infty}_{c} (S^{2}T^{\ast}\mathcal{M})$ with smooth sections of compact support on an n-dimensional noncompact paracompact boundaryless time-oriented Lorentzian manifold $\mathcal{M}$ with a

Figures (1)

  • Figure 1: Angular arrangement for the gravitational lensing by the Milky Way

Theorems & Definitions (2)

  • Theorem A.1
  • proof