Modified general relativity

Abstract

A modified Einstein equation of general relativity is obtained by using the principle of least action, a decomposition of symmetric tensors on a time oriented Lorentzian manifold, and a fundamental postulate of general relativity. The decomposition introduces a new symmetric tensor Φ αβ which describes the energy-momentum of the gravitational field itself. It completes Einstein's equation and addresses the energy localization problem. The positive part of Φ, the trace of the new tensor with respect to the metric, describes dark energy. The cosmological constant must vanish and is dynamically replaced by Φ. A cyclic universe which 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 of the universe. The negative part of Φ describes the attractive self-gravitating energy of the gravitational field. Φ αβ introduces two additional terms into the Newtonian radial force equation: the force due to dark energy and the 1 r "dark matter" force. When the dark energy force balances the Newtonian force, the flat rotation curves and the baryonic Tully-Fisher relation are obtained. The Newtonian rotation curves for galaxies with no flat orbital curves, and those with rising rotation curves for large radii are described as examples of the flexibility of the orbital rotation curve equation. This is an update of an article published in General Relativity and Gravitation. The original authenticated version is available online at: https://doi.

Modified General Relativity and quantum theory in curved spacetime

“…This forbids the presence of closed causal curves [14]. The Lorentzian metric g αβ is constructed [14,10] from a Riemannian metric g + αβ , which always exists, and the unit vectors u:…”

“…as shown in [10] where f is the length of an arbitrary line element vector X α and Φ is the trace, with respect to the metric, of Φ αβ . Φ > 0 represents dark energy and Φ < 0 is the attractive energy of the gravitational field itself [10]. Φ cannot vanish because even weak gravitational fields gravitate.…”

“…The symmetric spin-2 field must therefore involve the metric as a field variable to enable gravitons to be described by the spin-2 KG equation. An expression for Ψαβ can be obtained from the Orthogonal Decomposition Theorem in [10] which involves a linear sum of (0,2) divergenceless tensors as one part of the orthogonal splitting of an arbitrary (0,2) symmetric tensor. This collection of tensors can be defined to consist of the union of the set of Lovelock tensors and a set of symmetric divergenceless tensors which are not Lovelock tensors.…”

“…They are independent of both the metric and the Einstein tensor, which are the Lovelock tensors in a four-dimensional spacetime. With Ψαβ defined to be a linear combination of symmetric tensors consisting of the matter energy-momentum tensor, the energy-momentum tensor of the gravitational field, the Lovelock tensors and the non-Lovelock tensors with Λ = 0 as discussed in [10]:…”

“…Einstein developed GR in a four-dimensional Riemannian spacetime. Recently, the natural extension of GR to a four-dimensional time-oriented Lorentzian spacetime was developed in the article: Modified general relativity (MGR) [10]. A symmetric tensor Φ αβ is introduced in MGR which describes the energy-momentum of the gravitational field itself and completes GR.…”