Second Approximation of the Generalized Planetary Equation Based upon Golden Metric Tensors

In this paper, we consider the Post Einstein Planetary equation of motion. We succeeded in offering a solution using second approximation method, in which we obtained eight exact mathematical solutions that rebel amazing theoretical results. To the order of C-2, two of these exact solutions are reduced to the approximate solutions from the method of successive approximations.

Limit Treatment of Space-Time Foam Calculations If Non Pathological Big Bang Singularity Is Accessible to Modified Einstein Equations

When initial radius if Stoica actually derived Einstein equations in a formalism which removes the big bang singularity pathology, then the reason for Planck length no longer holds. We follow what Ng derived as limit calculations as to a space time length factor Without the drop off of the vacuum energy as given by is at least the value of . We review the work by Ng as to quantum foam as to how that affects a general expression as to energy when , with determined at least approximately by arguments he presented in 2008 in the Dark side of the universe conference. Well before certain effects make themselves apparent, in ways which are illustrated in the manuscript. Having at a point singularity would remove expansion by the scale factor, so that the extreme version of Stoica’s treatment in an isolated 4-dimensional universe would be no expansion at all.

Friedmann-Like Cosmological Equations for the Accelerated Expansion of the Universe

In recent papers [1] [2] [3], we framed suitable axioms for Space called Super Space by Wheeler [4]. Using our axioms in Newtonian formalism and considering the density of the universe to be constant in time, we showed in the above references that at t = 0 the radius of the universe need not be zero. And thus, we avoided the problem of singularity. We further showed that the Hubble factor is no longer constant in time and goes on decreasing as confirmed by experiments. We pointed out in the above references that Space is the source of dark energy which is responsible for the accelerated expansion of the universe. With a view to improving the above-mentioned results quantitatively, in this paper, we are discussing the consequences of our axioms using Einstein’s field equations of general theory of relativity. Friedmann-like Cosmological equations with Dark Energy built-in are derived. This derivation is obtained using Robertson-Walker line element and by introducing a suitable expression for Energy-Momentum tensor in terms of matter and Dark energy contents of the universe. The solutions of our cosmological equations obtained here, show that the radius of the universe cannot reach zero but has a minimum value and there is also maximum value for the radius of the universe. The inflationary expansion of the very early universe emerges from our theory.

Einstein转盘引力场精确解

本文解爱因斯坦场方程得出了转盘的度规。

微扰法解Einstein场方程

首先从已知具有对角型度规的Einstein场方程的精确解出发,近似推导了含有微扰条件下的场方程形式;其次,利用这一微扰形式具体计算了静态球对称引力场的外部微扰解,并进而讨论了球状星系外部的引力特征.结果表明,该微扰解不仅可以与内部解衔接,而且在消除微扰的情况下还可以自动恢复到Schwarzchild解的形式.

Two-Dimensional Numerical Simulation of the DC Glow Discharge in the Normal Mode and with Einstein’s Relation of Electron Diffusivity

This paper presents an investigation of a DC glow discharge at low pressure in the normal mode and with Einstein＇s relation of electron diffusivity. Two-dimensional distributions in Cartesian geometry are presented in the stationary state, including electric potential, electron and ion densities, longitudinal and transverse electrics fields as well as electron temperature. Our results are compared with those obtained in existing literature. The model used in this work is based on the first three moments of Boltzmann＇s equation. They serve as the continuity equation, the momentum transfer and the energy equations. The set of equations for charged particles presented in monatomic argon gas are coupled in a self-consistent way with Poisson＇s equation. A parametric study varying the cathode voltage, gas pressure, and secondary electron emission coefficient predicts many of the well-known features of DC discharges.

Einstein’s Pseudo-Tensor in <i>n</i>Spatial Dimensions for Static Systems with Spherical Symmetry

It was noted earlier that the general relativity field equations for static systems with spherical symmetry can be put into a linear form when the source energy density equals radial stress. These linear equations lead to a delta function energymomentum tensor for a point mass source for the Schwarzschild field that has vanishing self-stress, and whose integral therefore transforms properly under a Lorentz transformation, as though the particle is in the flat space-time of special relativity (SR). These findings were later extended to n spatial dimensions. Consistent with this SR-like result for the source tensor, Nordstrom and independently, Schrodinger, found for three spatial dimensions that the Einstein gravitational energy-momentum pseudo-tensor vanished in proper quasi-rectangular coordinates. The present work shows that this vanishing holds for the pseudo-tensor when extended to n spatial dimensions. Two additional consequences of this work are: 1) the dependency of the Einstein gravitational coupling constant κ on spatial dimensionality employed earlier is further justified;2) the Tolman expression for the mass of a static, isolated system is generalized to take into account the dimensionality of space for n ≥ 3.

New Planck’s Law

In quantum mechanics, there are two very famous formulas. One is the energy formula of the bose particle, called Planck’s law. The other is the wavelength formula, which is called the de Broy wavelength. According to Einstein’s mass-energy equation, we have studied Planck’s law and De Bloy’s wavelength, and generalized it to the De Bloy’s wavelength formula from low speed to light speed. Then, on this basis, the smallest particle is defined as mass quantum. The new wavelength formula is obtained from the mass quantum and converted into the frequency formula. The generalized Planck’s law is obtained.

The Quantization of Space

In the present work, it will be shown that the dimensionless number 137 of the fine-structure constant α demands a quantization of space. For this purpose, we refer to a volume constant of electromagnetic processes, which takes effect as a volume quantum. This involves not only a re-evaluation of the Dirac equation but also, and above all, a determination of Einstein’s velocity vector as the fundamental property of these processes. A prerequisite is the linking of the hydrogen spectrum with the hydrogen nucleus.

Time-periodic solutions of the Einstein's field equations Ⅰ:general framework

In this paper,we develop a new algorithm to find the exact solutions of the Einstein's field equations.Time-periodic solutions are constructed by using the new algorithm.The singularities of the time-periodic solutions are investigated and some new physical phenomena,such as degenerate event horizon and time-periodic event horizon,are found.The applications of these solutions in modern cosmology and general relativity are expected.

Time-periodic solutions of the Einstein's field equations II:geometric singularities

In this paper,we construct several kinds of new time-periodic solutions of the vacuum Einstein's field equations whose Riemann curvature tensors vanish,keep finite or take the infinity at some points in these space-times,respectively.The singularities of these new time-periodic solutions are investigated and some new physical phenomena are discovered.

The Singularities of Gravitational Fields of Static Thin Loop and Double Spheres Reveal the Impossibility of Singularity Black Holes

In the classical Newtonian mechanics, the gravity fields of static thin loop and double spheres are two simple but foundational problems. However, in the Einstein’s theory of gravity, they are not simple. In fact, we do not know their solutions up to now. Based on the coordinate transformations of the Kerr and the Kerr-Newman solutions of the Einstein’s equation of gravity field with axial symmetry, the gravity fields of static thin loop and double spheres are obtained. The results indicate that, no matter how much the mass and density are, there are singularities at the central point of thin loop and the contact point of double spheres. What is more, the singularities are completely exposed in vacuum. Space near the surfaces of thin loop and spheres are highly curved, although the gravity fields are very weak. These results are inconsistent with practical experience and completely impossible. By reasonable analogy, black holes with singularity in cosmology and astrophysics are something illusive. Caused by the mathematical description of curved space-time, they do not exist in real world actually. If there are black holes in the universe, they can only be the types of the Newtonian black holes without singularities, rather than the Einstein’s singularity black holes. In order to escape the puzzle of singularity thoroughly, the description of gravity should return to the traditional form of dynamics in flat space. The renormalization of gravity and the unified description of four basic interactions may be possible only based on the frame of flat space-time. Otherwise, theses problems can not be solved forever. Physicists should have a clear understanding about this problem.

A Spherical Relativistic Anisotropic Compact Star Model

We provide solutions to Einsteins field equations for a model of a spherically symmetric anisotropic fluid distribution, relevant to the description of compact stars. The central matter-energy density, radial and tangential pressures, red shift and speed of sound are positive definite and are decreasing monotonically with increasing radial distance from the center of matter distribution of astrophysical object. The causality condition is satisfied for complete fluid distribution. The central value of anisotropy is zero and is increasing monotonically with increasing radial distance from the center of the distribution. The adiabatic index is increasing with increasing radius of spherical fluid distribution. The stability conditions in relativistic compact star are also discussed in our investigation. The solution is representing the realistic objects such as SAXJ1808.4-3658, HerX-1, 4U1538-52, LMC X-4, CenX-3, VelaX-1, PSRJ1614-2230 and PSRJ0348+0432 with suitable conditions.

Dark Matter and the Energy-Momentum Relationship in a Hydrogen Atom

Einstein derived the energy-momentum relationship which holds in an isolated system in free space. However, this relationship is not applicable in the space inside a hydrogen atom where there is potential energy. Therefore, in 2011, the author derived an energy-momentum relationship applicable to the electron constituting a hydrogen atom. This paper derives that relationship in a simpler way using another method. From this relationship, it is possible to derive the formula for the energy levels of a hydrogen atom. The energy values obtained from this formula almost match the theoretical values of Bohr. However, the relationship derived by the author includes a state that cannot be predicted with Bohr’s theory. In the hydrogen atom, there is an energy level with n = 0. Also, there are energy levels where the relativistic energy of the electron becomes negative. An electron with this negative energy (mass) exists near the atomic nucleus (proton). The name “dark hydrogen atom” is given to matter formed from one electron with this negative mass and one proton with positive mass. Dark hydrogen atoms, dark hydrogen molecules, other types of dark atoms, and aggregates made up of dark molecules are plausible candidates for dark matter, the mysterious type of matter whose true nature is currently unknown.

Class of Charged Fluid Balls in General Relativity

In the present study, we have obtained a new analytical solution of combined Einstein-Maxwell field equations describing the interior field of a ball having static spherically symmetric isotropic charged fluid within it. The charge and electric field intensity are zero at the center and monotonically increasing towards the boundary of the fluid ball. Besides these, adiabatic index is also increasing towards the boundary and becomes infinite on it. All other physical quantities such as pressure, density, adiabatic speed of sound, charge density, adiabatic index are monotonically decreasing towards the surface. Causality condition is obeyed at the center of ball. In the limiting case of vanishingly small charge, the solution degenerates into Schwarzchild uniform density solution for electrically neutral fluid. The solution joins smoothly to the Reissner-Nordstrom solution over the boundary. We have constructed a neutron star model by assuming the surface density . The mass of the neutron star comes with radius 14.574 km.

The Incompleteness of Quantum Mechanics Demonstrated by Considerations of Relativistic Kinetic Energy

The energy levels of a hydrogen atom, derived by Bohr, are known to be approximations. This is because the classical quantum theory of Bohr does not take the theory of relativity into account. In this paper, the kinetic energy and momentum of an electron in a hydrogen atom are treated relativistically. A clearer argument is developed while also referring to papers published in the past. The energy levels of a hydrogen atom predicted by this paper almost match the theoretical values of Bohr. It is difficult to experimentally distinguish the two. However, this paper predicts the existence of an n = 0 energy level that cannot be predicted even with Dirac’s relativistic quantum mechanics. The only quantum number treated in this paper is n. This point falls far short of a finished quantum mechanics. However, even in discussion at the level of this paper, it can be concluded that quantum mechanics is an incomplete theory.

A NEW CLASS OF SOLUTIONS OF VACUUM EINSTEIN'S FIELD EQUATIONS WITH COSMOLOGICAL CONSTANT

In this paper, a new class of solutions of the vacuum Einstein＇s field equa- tions with cosmological constant is obtained. This class of solutions possesses the naked physical singularity. The norm of the Riemann curvature tensor of this class of solutions takes infinity at some points and the solutions do not have any event horizon around the singularity.

Vaidya Solution in Non-Stationary de Sitter Background: Hawking’s Temperature

In this paper we propose a class of non-stationary solutions of Einstein’s field equations describing an embedded Vaidya-de Sitter solution with a cosmological variable function Λ(u). Vaidya-de Sitter solution is interpreted as the radiating Vaidya black hole which is embedded into the non-stationary de Sitter space with variable Λ(u). The energymomentum tensor of the Vaidya-de Sitter black hole may be expressed as the sum of the energy-momentum tensor of the Vaidya null fluid and that of the non-stationary de Sitter field, and satisfies the energy conservation law. We also find that the equation of state parameter w= p/ρ = -1 of the non-stationary de Sitter solution holds true in the embedded Vaidya-de Sitter solution. It is also found that the space-time geometry of non-stationary Vaidya-de Sitter solution with variable Λ(u) is type D in the Petrov classification of space-times. The surface gravity, temperature and entropy of the space-time on the cosmological black hole horizon are discussed.

Mass as the Fifth Dimension of the Universe

Newton considered three-dimensional universe endowed with flat space Euclidean geometry, and treated the time as an outside parameter and established his dynamics of the universe. Einstein along with space, considered time, and generated a four-dimensional universe endowed with non-Euclidean curved space-time geometry with time as its fourth dimension, and set up his field equations. Schwarzschild solved Einstein’s field equations around a star in space, which is, otherwise, flat, and obtained a solution. We, along with space and time, considered mass which also included energy according to Einstein’s mass-energy equivalence relation: E = mc2, and generated a five-dimensional universe with the mass as its fifth dimension, and solved the Einstein’s field equations, in some simple cases, and obtained solutions around a star in space, which is otherwise, flat.

Spherical Symmetric Solitons of Interacting Spinor, Scalar and Gravitational Fields in General Relativity

The concept of soliton as regular localized stable solutions of nonlinear differential equations is being widely utilized in pure science for various aims. In present analysis, the soliton concept is used as a model in order to describe the configurations of elementary particles in general relativity. To this end, our study deals with the spherical symmetric solitons of interacting Spinor, Scalar and Gravitational Fields in General Relativity. Thus, exact spherical symmetric general solutions to the interaction of spinor, scalar and gravitational field equations have been obtained. The Einstein equations have been transformed into a Liouville equation type and solved. Let us emphasize that these solutions are regular with localized energy density and finite total energy. In addition, the total charge and spin are limited. Moreover, the obtained solutions are soliton-like solutions. These solutions can be used in order to describe the configurations of elementary particles.