Narrow Results

We derive the equations governing static, spherically symmetric vacuum solutions to the Einstein equations, as modified by the frame-dependent effective action (derived from trace dynamics) that gives an alternative explanation of the origin of "dark energy". We give analytic and numerical results for the solutions of these equations, first in polar coordinates, and then in isotropic coordinates. General features of the static case are that: (i) there is no horizon, since g₀₀ is nonvanishing for finite values of the polar radius, and only vanishes (in isotropic coordinates) at the internal singularity, (ii) the Ricci scalar R vanishes identically, and (iii) there is a physical singularity at cosmological distances. The large distance singularity may be an artifact of the static restriction, since we find that the behavior at large distances is altered in a time-dependent solution using the McVittie Ansatz.

An algebraic extension of General Relativity is presented, which introduces pseudo-complex coordinates. We give a short review on the properties of pseudo-complex variables and their advantage of their use in field theory. Afterwards we extend General Relativity to pseudo-complex variables. The projection to real results is described and in particular the pseudo-complex Schwarzschild solution is discussed.

The Schwarzschild approach is applied to solve the field equations describing a Nexus graviton field. The resulting solutions are free from singularities which have been a problem in general relativity since its inception. Findings from this work also demonstrate that at the Hubble radius, the metric signature of space-time changes generating short-lived but intense bursts of energy during the transition process. The solutions in this paper also provide an explanation to the enigma of late time cosmic acceleration, the galaxy rotation curve problem and the coincidence problem.

We point out a sufficient condition for existence of a stable attractor in the two-body restricted problem. The result is strictly dependent on making reference to relativistic equations and could not be derived from classical analysis. The radius of the stable attractor equals the well known Schwarzschild radius of General Relativity (GR). So we establish a bridge between Special Relativity (SR) and GR via Stability Theory (ST). That opens one way to an innovative study of black-holes and of the cosmological problem. A distinguishing feature is that no singularities come into evidence. The application of the Direct Method of Lyapunov (with a special Lyapunov function that represents the local energy) provides us the theoretical background.

In this article, it will be discussed how we get the Einstein-Rosen wave spacetime and the Schwarzschild solution by solving the (2+2) Hamilton’s equations. And the local Hamiltonian and the field momentum densities corresponding to those solutions will be presented.