Modified general relativity

Abstract

In a Lorentzian spacetime there exists a smooth regular line element field $(\mathbf{X},-\mathbf{X})$ and a unit vector $\mathbf{u}$ collinear with one of the pair of vectors in the line element field. An orthogonal decomposition of symmetric tensors can be constructed in terms of the Lie derivative along $\mathbf{X}$ of the metric and a product of the unit vectors; and a linear sum of divergenceless symmetric tensors. A modified Einstein equation of general relativity is then obtained by using the principle of least action, the decomposition and a fundamental postulate of general relativity. The decomposition introduces a new symmetric tensor ${\Phi }_{\alpha \beta }$ which describes the energy-momentum of the gravitational field. It completes Einstein's equation and addresses the energy localization problem. Variation of the action with respect to $X^{\mu }$ restricts $u_{\mu }$ to a particular value, which defines the possible Lorentzian metrics. $\Phi$, the trace of ${\Phi }_{\alpha \beta }$, describes dark energy. The cosmological constant is dynamically replaced by $\Phi$. A cyclic universe that developed after the Big Bang is described. The dark energy density provides a natural explanation of why the vacuum energy density is so small, and why it dominates the present epoch. Assuming dark matter does not exist, a solution to the modified Einstein equation introduces two additional terms into the Newtonian radial force equation, from which the baryonic Tully-Fisher relation is obtained.

This update corrects ${\Phi }_{\alpha \beta }$ in the FLRW metric, adds the variations of the action functional, and simplifies the manuscript

References in corpus (10)

• Type Ia Supernova Discoveries at z>1 From the Hubble Space Telescope: Evidence for Past Deceleration and Constraints on Dark Energy Evolution
• Quintom Cosmology: Theoretical implications and observations
• Cosmic Evolution in a Cyclic Universe
• Emergent Gravity and the Dark Universe
• A galaxy lacking dark matter
• Alternatives to Dark Matter and Dark Energy
• One Law To Rule Them All: The Radial Acceleration Relation of Galaxies
• Bimetric MOND gravity
• A Tale of Two Paradigms: the Mutual Incommensurability of LCDM and MOND
• Testing Verlinde's Emergent Gravity with the Radial Acceleration Relation

Cited by in corpus (6)

• Modified General Relativity and dark matter
• New Nonrelativistic Quantum Theory of Cold Dark Matter
• Modified General Relativity and quantum theory in curved spacetime
• The unification of gravity and the spin-1 field
• Study of the large scale structure through modified gravity theory using statistical mechanics
• Pathology of the Unified Dark Sectors in Modified General Relativity