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.
With appropriate modifications, the multi-spin Klein–Gordon (KG) equation of quantum field theory can be adapted to curved space–time for spins 0, 1, 1/2. The associated particles in the microworld then move as a wave at all space–time coordinates. From the existence in a Lorentzian space–time of a line element field [Formula: see text], the spin-1 KG equation [Formula: see text] is derived from an action functional involving [Formula: see text] and its covariant derivative. The spin-0 KG equation and the KG equation of the outer product of a spin-1/2 Dirac spinor and its Hermitian conjugate are then constructed. Thus, [Formula: see text] acts as a fundamental quantum vector field. The symmetric part of the spin-1 KG equation, [Formula: see text], is the Lie derivative of the metric. That links the multi-spin KG equation to Modified General Relativity (MGR) through its energy–momentum tensor of the gravitational field. From the invariance of the action functionals under the diffeomorphism group Diff(M), which is not restricted to the Lorentz group, [Formula: see text] can instantaneously transmit information along [Formula: see text]. That establishes the concept of entanglement within a Lorentzian formalism. The respective local/nonlocal characteristics of MGR and quantum theory no longer present an insurmountable problem to unify the theories.
Modified General Relativity (MGR) is the natural extension of General Relativity (GR). MGR explicitly uses the smooth regular line element vector field [Formula: see text], 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 nonlocalization 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.
A classical field model is proposed, involving a scalar field interacting with the electromagnetic field, that has discrete particle-like solutions corresponding to any desired mass spectrum, all such solutions having exactly the same electric charge.
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