Generalized transformations and coordinates for static spherically symmetric general relativity

We examine a static, spherically symmetric solution of the empty space field equations of general relativity with a non-orthogonal line element which gives rise to an opportunity that does not occur in the standard derivations of the Schwarzschild solution. In these derivations, convenient coordinate transformations and dynamical assumptions inevitably lead to the Schwarzschild solution. By relaxing these conditions, a new solution possibility arises and the resulting formalism embraces the Schwarzschild solution as a special case. The new solution avoids the coordinate singularity associated with the Schwarzschild solution and is achieved by obtaining a more suitable coordinate chart. The solution embodies two arbitrary constants, one of which can be identified as the Newtonian gravitational potential using the weak field limit. The additional arbitrary constant gives rise to a situation that allows for generalizations of the Eddington–Finkelstein transformation and the Kruskal–Szekeres coordinates.

A Spectral Element Method for the Numerical Simulation of Axisymmetric Flow in Czochralski Crystal Growth

In this paper, a spectral element method (SEM) is applied to simulate the axisymmetric flow in Czochralski crystal growth. The coordinate singularity for 1/r at r=0 is avoided by using the Gauss-Radau type quadrature points for the spatial discretization. The SEM solver is validated by its application to the benchmark problems suggested by Wheeler [1]. The stream function is solved by an artificial Poisson equation with the present method. The results show that it agrees well with available data in the literatures.

Convective processes and hydromagnetic instabilities in core collapse supernova simulations

Hydrodynamic and potentially also hydromagnetic instabilities play an important role in core-collapse supernovae. We report on simulations performed to investigate the effects of magnetic fields in simplified models. Furthermore, we present efforts to get rid of problems arising from the coordinate singularity at the symmetry axis and the associated boundary conditions.

DISCRETE SYMMETRIES AND GENERAL RELATIVITY, THE DARK SIDE OF GRAVITY

The parity and time reversal invariant actions, equations and their conjugated metric solutions are obtained in the context of a general relativistic model modified in order to suitably take into account discrete symmetries. The equations are not covariant however the predictions of the model, in particular its Schwarzschild metric solution in vacuum, only start to differ from those of General Relativity at the Post-Post-Newtonian order. No coordinate singularity (black hole) arises in the privileged coordinate system where the energy of gravity is found to vanish. Vacuum energies have no gravitational effects. A flat universe accelerated expansion phase is obtained without resorting to inflation nor a cosmological constant. The context may be promising to help us elucidate several outstanding enigmas such as the Pioneer anomalous blue-shift, flat galactic rotation curves or the universe voids.

INEVITABILITY OF SPACE–TIME SINGULARITIES IN THE CANONICAL METRIC

We discuss the question of whether the existence of singularities is an intrinsic property of 4D space–time. Our hypothesis is that singularities in 4D are induced by the separation of space–time from the other dimensions. We examine this hypothesis in the context of the so-called canonical or warp metrics in 5D. These metrics are popular because they provide a clean separation between the extra dimension and space–time. We show that the space–time section, in these metrics, inevitably becomes singular for some finite (nonzero) value of the extra coordinate. This is true for all canonical metrics that are solutions of the field equations in space–time-matter theory. This is a coordinate singularity in 5D, but appears as a physical one in 4D. At this singular hypersurface, the determinant of the space–time metric becomes zero and the curvature of the space–time blows up to infinity. These results are consistent with our hypothesis.