Firstly we derive peculiar spherical Weyl solutions, using a general spherically symmetric metric due to a massive charged object with definite mass and radius. Afterwards, we present new analytical solutions for relevant cosmological terms, which appear in the metrics. Connecting the metrics to a new geometric definition of a charged Black Hole, we numerically investigate the effective potentials of the total dynamical system, considering massive and massless test particles, moving on such Black Holes.

In this paper we report a few examples of algebraically solvable dynamical systems characterized by 2 coupled Ordinary Differential Equations which read as follows:with P (n) (x1, x2) specific polynomials of relatively low degree in the 2 dependent variables x1 ≡ x1 (t) and x2 ≡ x2 (t) . These findings are obtained via a new twist of a recent technique to identify dynamical systems solvable by algebraic operations, themselves explicitly identified as corresponding to the time evolutions of the zeros of polynomials the coefficients of which evolve according to algebraically solvable (systems of) evolution equations

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It has been shown that the negative-norm states necessarily appear in a covariant quantization of the free minimally coupled scalar field in de Sitter spacetime. In this processes ultraviolet and infrared divergences have been automatically eliminated. A natural renormalization of the one-loop interacting quantum field in Minkowski spacetime (λϕ 4 ) has been achieved through the consideration of the negative-norm states defined in Krein space. It has been shown that the combination of quantum field theory in Krein space together with consideration of quantum metric fluctuation, results in quantum field theory without any divergences. Pursuing this approach, we express Wick's theorem and calculate Møller scattering in the one-loop approximation in generalized Krein space. The mathematical consequence of this method is the disappearance of the ultraviolet divergence in the one-loop approximation.quantum field theory, Krein space, negative energy states In view of the appearance of infrared divergence in the two-point function for the minimally coupled scalar field in de Sitter spacetime, a new method of field quantization called "Krein quantum field theory (QFT)" has been proposed, which uses negative-norm states [1][2][3][4] .Consideration of the negative-norm states was proposed by Dirac in 1942 [5] . In 1950 Gupta applied the idea in quantum electrodynamics (QED) [6] . The presence of higher derivatives in the Lagrangian also leads to ghosts; states with negative norm [7] . The auxiliary negative-norm states were primarily introduced in de Sitter spacetime to achieve covariant quantization. However, their presence has other consequences, too. For example, in QED the negative-energy photon disappears [6] , and in de Sitter spacetime the infrared divergence of minimally coupled scalar field is eliminated [8] . In similarity with linear gravity, this divergence does not manifest itself in the quadratic part of the effective action in the one-loop approximation. This means that the pathological behavior of the graviton propagator is gauge dependent and so should not appear in an effective way as a physical quantity [9,10] . Ignoring the positivity condition (for norm and energy), similar to Gupta-Bleuler quantization of the electrodynamics in Minkowski spacetime, the quantization of free boson and spinor fields has been performed in Krein space [11]1) . Following this scheme, the normal ordering procedure is rendered useless since the vacuum energy remains convergent [11]1) .It is worthwhile to note that the unphysical (negative-energy) states for boson fields have both negative and positive norms and are, therefore, defined in Krein space. In Gupta-Bleuler quantization, the unphysical states with positive and negative norms are introduced in order to preserve the Lorentz invariance. The negative-norm states appear due to the sign of the Minkowskian metric. However, in Krein space quantization the appearance of the additional negative-norm states owes itself to the negative-frequency solutions. The two sets of solu...