Solutions of Einstein Field Equations

I proposed a new expansion force to explain the flat rotation curves of plane galaxies, in a Newtonian approach in 2016. This force is validated here in General Relativity for an inhomogeneous radially symmetrical universe. It can be deduced from the solution of Einstein's equation with a cosmological constant. A pseudo-Schwarzschild "twin-potential" metric is considered, with two different potentials for the space and time dimensions. The expansion force results as a repulsive dual gravity. Negative masses are needed. The necessary mass repartitions are computed, in connection with the radial dynamics and the expansion rate's evolution. A linear model and a quadratic model are studied, connected with the observation representations of a universe in accelerated expansion. It is a first step towards a better representation of an inhomogeneous universe. The framework can be applied to deduce the universe mass repartition form the measured values of the expansion rate variation. Résumé J'ai proposé en 2016 une nouvelle force d'expansion pour expliquer les courbes de rotation plates des galaxies spirales dans une approche Newtonienne. Cette force est validée ici dans le cadre de la Relativité Générale pour un univers inhomogène à symétrie radiale. On peut la déduire comme solution de l'équation d'Einstein avec Constante Cosmologique. On considère une métrique à « potentiels jumelés », avec deux potentiels différents pour les dimensions spatiales et temporelles. Il en résulte une force d'expansion agissant comme une gravité duale répulsive. Il faut des masses négatives. On calcule les répartitions de masses nécessaires, en liaison avec la dynamique radiale et l'évolution du taux d'expansion. On étudie un modèle linéaire et un modèle quadratique, en lien avec les représentations d'un univers observé en expansion accélérée. Il s'agit là d'une première étape vers une meilleure représentation d'un univers inhomogène. Ce cadre peut être appliqué pour déduire la répartition de masses de l'univers à partir des valeurs mesurées du taux d'expansion.

The golden ring to which most physicists aspire is a unified field theory that incorporates all of modern and classical physics. Some scientists and others call this a TOE or ‘theory of everything’, but it is no more than false hubris to believe that humans could possibly know and explain everything about the universe at this time. Einstein chased this goal for the last three decades of his life, basing his theoretical research on his general theory of relativity. Meanwhile, the vast majority of scientists supporting the other major accomplishment of the Second Scientific Revolution were investing all of their time and efforts to advancing the quantum theory and their quest has been extremely successful. They originally had no interest in a unified field theory. After Einstein died in 1955, his efforts were all but abandoned because of his philosophical stance against the prevalent Copenhagen Interpretation of quantum theory even though he had been one of quantum theory’s founders. During the 1970s the tables started to turn and quantum theorists became interested in unifying physics, although not from the foundational principles of relativity theory. They claimed that quantum theory was more fundamental than relativity so they began the same quest from a totally different direction despite their claims to be continuing Einstein’s quest.  Throughout the development of the ensuing standard quantum model, superstring theory and many other theoretical schemes, quantum theorists have remained resolute in their conviction that the quantum and relativity are mutually incompatible so the quantum must completely do away with and replace relativity once and for all. However, the quantum theory and relativity are not actually incompatible and, in fact, say the some of the same things about the nature of physical reality. When the similarities are fully defined and studied and the basic assumptions behind each of the theories are altered to reflect the similarities instead of the incompatibilities, only then can the point of their compatibility be determined and act as a unifying principle resulting in a completed unified field theory of the type that Einstein once sought. The development of this physical model of reality is not without irony. Not only is the quantum theory incomplete as Einstein argued in EPR, but Einstein’s general relativity is also seriously incomplete and true unification cannot be rendered complete at any level of reality until all the theoretical models being unified are themselves complete.

“The general theory of relativity is regarded as the greatest intellectual achievement of any one person,” (Glendenning 2007), and 2015 will be the centenary of its publication. In part 1 of this report we explored the origins of the special theory of relativity, and the science leading up to its publication in 1905. The origin of the general theory of relativity is quite a different story and so this paper is about its development and the main steps that Einstein took to bring it to fruition. The special theory of relativity, was concerned mainly with the behaviour of light, or objects moving at speeds comparable to light, in a non-inertial reference frame; namely, a frame of reference that has no acceleration and does not include gravity. In order to include gravity and acceleration in relativity, Einstein would have to rewrite the law of gravity and use complex mathematical geometry to do so.

Albert Einstein is the greatest physicist of the 20th century, where thanks to his work, Physics has raised  to a very high level! Einstein has developed hundreds of thousands formulas, but the most important are the following expressions (presented in this paper). Let's discuss about these equations?!

This paper results from our investigation into novel means of electromagnetic propulsion. It requires the basis of our claims to be put on a sound theoretical footing regarding the purported momentum exchange with the electromagnetic field. One of these concerns is the huge discrepancy between the energy density of the Zeropoint and its purported manifestation as the Cosmological Constant. Here we state that it is manifestly wrong to introduce the zeropoint at zero order into the stress-energy tensor, because it is something which describes zero particle count. As a fluctuation, it belongs in a higher order Taylor expansion in frequency of the stress-energy tensor. Furthermore in the 3 rd order in the Einstein constant our procedure is some 9 orders of magnitude too small. We make up this difference by suggesting that vacuum energy is much higher still and that more degrees of freedom exist in physics beyond the Standard Model or that there is interaction energy between the modes.

The Einstein-Maxwell (E-M) equations in a curved spacetime that admits at least one Killing vector are derived, from a Lagrangian density adapted to symmetries. In this context, an auxiliary space of potentials is introduced, in which, the set of potentials associated to an original (seed) solution of the E-M equations are transformed to a new set, either by continuous transformations or by discrete transformations. In this paper, continuous transformations are considered. Accordingly, originating from the so-called γ A-metric, other exact solutions to the E-M equations are recovered and discussed.

The Super Principle Relativity Theory is a hydrodynamic field theory, which models “Gravitation” and the “Vacuum”, in the conceptual and mathematical framework of a variable density compressible ideal gaseous fluid. In contrast to General Relativity Theory which is a field theory, that models “Gravitation” and the “Vacuum”, in the conceptual and mathematical framework of a fixed density “frictionless incompressible fluid medium” or “incompressible liquid medium”. The new theory produces exact solutions to gravitation phenomena, and describes the Euler-Lagrangian Hydrodynamic Energy Density Equation, as a space-time frame of reference; and is mathematically the net sum of space-like and time-like energy density terms. The “Hydrodynamic Field Energy Density Frame of Reference” is used to model the interaction between seven (7) physical phenomena: gravitational, electromagnetic, isotropic space-time world line, gravitational sink, gravitational source, electromagnetic sink, and electromagnetic source energy density frames of reference. The “Hydrodynamic Field Theory” models physical phenomena: solar systems, galaxy rotation, quantum mechanical systems, etc.

In this paper we have described a sum of the reciprocal Fibonacci primes that converges to a new constant. Furthermore, in the Section 2, we have described also some new possible mathematical connections with the universal gravitational constant G , the Einstein field equations and some equations of String Theory and Supersymmetry Breaking linked to Φ and π

v3  -  09.03.2021

Some new classes of exact solutions (so-called functionally invariant solutions) of the elliptic and hyperbolic complex Monge–Ampère equations and of the second heavenly equation are found. Besides, non-invariance of the found classes of solutions of the hyperbolic complex Monge–Ampère equation and the second heavenly equation is investigated—these classes of solutions determine the new classes of metrics
without Killing vectors. A criterion of non-invariance of the solutions belonging to the found classes is also formulated.

The Einstein Field Equation (EFE) is widely accepted as the best description of gravity. We have shown in this series (Vol. 287, pp. 163-172, 2016, ISBN: 978-1-63484-193-1) that a slightly modified Schwarzschild relationship predicts the nature of gravitational attraction by subatomic and atomic particles. We discovered this during our search for solid evidence that photons exert gravitational attraction but found no support for this misconception. We formalized a simple equation describing the incorporation of spacetime into subatomic particles during particle creation as a new fundamental concept. Here we derive a more  general version of the Einstein Field Equation. This is useful for classical predictions and now describes some interesting properties of subatomic particles. We predict it possible for LIGO to observe the disappearance of gravity during a supernova type II explosion; these are thought to be symmetrical so very different than colliding black holes. We also speculate about the properties of matter when approaching lightspeed and of subatomic particles interacting with spacetime. We finally suggest the fundamental elements of our Universe are light and spacetime only, but not matter.

In this work, a spherically symmetric and static relativistic anisotropic fluid sphere solution of the Einstein field equations is provided. To build this particular model, we have imposed metric potential e 2λ(r) and an equation of state. Specifically, the so-called modified generalized Chap-lygin equation of state with ω = 1 and depending on two parameters, namely, A and B. These ingredients close the problem, at least mathematically. However, to check the feasibility of the model, a complete physical analysis has been performed. Thus, we analyze the obtained geometry and the main physical observables, such as the density ρ, the radial p r , and tangential p t pressures as well as the anisotropy factor . Besides, the stability of the system has been checked by means of the velocities of the pressure waves and the rel-ativistic adiabatic index. It is found that the configuration is stable in considering the adiabatic index criteria and is under hydrostatic balance. Finally, to mimic a realistic compact object, we have imposed the radius to be R = 9.5 [km]. With this information and taking different values of the parameter A the total mass of the object has been determined. The resulting numerical values for the principal variables of the model established that the structure could represent a quark (strange) star mixed with dark energy.

Dalam makalah ini ditelusuri perumusan persamaan dan solusi medan gravitasi-elektromagnet oleh sumber konformastatik bermuatan. Sistem yang ditinjau adalah persamaan medan Einstein-Maxwell dengan komponen tensor Einstein yang memerikan geometri ruangwaktu konformastatik dan tensor energi-momentum yang mendeskripsikan medan elektromagnet. Ditinjau pula persamaan Maxwell di ruangwaktu melengkung yang menjelaskan fenomena medan elektromagnet dengan kehadiran gravitasi. Adapun untuk memperoleh solusi dari persamaan medan diambil asumsi berdasar realita fisis bahwa kelengkungan ruangwaktu dipengaruhi oleh distribusi massa dan energi yang memberikan keterjalinan antara fungsi metrik dan potensial elektrostatik. Interpretasi fisis dari solusi yang diperoleh didiskusikan kemudian.

Kata Kunci: medan gravitasi, persamaan medan einstein, ruangwaktu konformastatik

“In Ferent Quantum Gravity theory the volume of the supermassive Black Hole from the center of our galaxy is 10^(-80) m^3”
Adrian Ferent

“Human can Not travel through the Black Hole with the volume V = 10^(-80) m^3 from the center of our galaxy”
Adrian Ferent

“Because the Black Hole from the center of our galaxy has the volume V = 10^(-80) m^3, the theories of Juan Maldacena, J. L. Blázquez-Salcedo, Carl P. Feinberg and Alexey Milekhin are wrong theories”
Adrian Ferent

“Human can Not travel through a wormhole”
Adrian Ferent

“Human can Not travel through a wormhole because the Black Holes have a very small volume, smaller than an atom”
Adrian Ferent

While researching quantum physics, I realized that I had just finished a book that was based on quantum theory. At the time, I didn’t quite realize that quantum theory and quantum physics were interrelated. Niels Bohr once said, anyone who is not shocked by quantum theory has not understood it. He believed this because quantum physics makes the common laws of classical physics false on small scales.First, quantum physics is the physics of the incredibly small. It tries to explain the behavior of even smaller particles such as protons, neutrons, electrons, and even the particles that make up those particles. Would you believe that the model of an atom taught to us in chemistry is about 70 years out of date? In fact, an atom isn’t just a nucleus with electrons looping around it. Instead of having a fixed place for the electrons to be, quantum physics gives us a statistical probability of the electron s location at any one moment.

In this paper, we found new exact solutions to the Einstein-Maxwell system of equations with charged anisotropic matter distribution considering quadratic equation of state. We specify the gravitational potential Z(x) that depends of a adjustable parameter n and that allow integrate analytical the field equations in order to calculate the energy density, the radial pressure, the anisotropy, charge density and the mass function for different values of n. The obtained solutions can be written in terms of elementary and polynomial functions.

In this paper we consider a multidimensional Kaluza-Klein (KK) model with a Ricci-flat internal space, e.g., a Calabi-Yau manifold. We perturb this background metrics by a system of gravitating masses, e.g., astrophysical objects such as our Sun. We suppose that these masses are pressureless in the external space but they have relativistic pressure in the internal space. We show that metric perturbations do not depend on coordinates of the internal space and gravitating masses should be uniformly smeared over the internal space. This means, first, that KK modes corresponding to the metric fluctuations are absent and, second, particles should be only in the ground quantum state with respect to the internal space. In our opinion, these results look very unnatural. According to statistical physics, any nonzero temperature should result in fluctuations, i.e. in KK modes. We also get formulae for the metric correction terms which enable to calculate the gravitational tests: the deflection of light, the time-delay of the radar echoes and the perihelion advance.

Since the 1970s, an ever growing number of theoretical physicists have become interested in unifying the quantum and relativity upon a quantum basis. Before the 1970s, only Einstein and a select few sought unification, but their theoretical work was based upon the continuity of relativity rather than quantum’s discrete nature of reality. To date neither paradigm has developed anything that would appear to unify physics, except for the work of one physicist and a few of his colleagues. H.T. Flint published more than thirty-five articles in well-known peer-reviewed journals over a period of four decades, extending relativity to include electromagnetism and the quantum. Yet his work and that of his close associates is almost completely unknown today. Flint published his complete unified field theory in the 1960s, well before most quantum theorists even began thinking along the lines of unification.  Strangely enough, Flint’s unification theory has been completely forgotten by a scientific community that has become enamored enough with the idea of unification that it would accept the most outrageous and non-intuitive ideas as long as they are based on the quantum, but then Kaluza’s five-dimensional unification of relativity and electromagnetism supposedly was not known until it was rediscovered by the superstring theorists, or so they claim. In reality, these ideas were lost in plain sight for decades simply because the physics community was unwilling to recognize accomplishments based on relativity theory.

Science from History to Future CONTENTS 1. INERTIA 2. Form of Intensity of the Moving Charge Electric Field is Asymmetrical. 3. Form of the Interference Field is Non-Linear 4. CORRECTED Maswell's equations 5. Corrected Newton´s Laws of Motion 6. Kinetic energy of a charge moving at the velocity of v has two different values: Kinetic energy against direction of motion as wave Tkin ad = mc2 [ln |1+v/c|- (v/c)/(1+v/c)] Kinetic energy in direction of motion as particle Tkin id = mc2 [ln|1-v/c|+ (v/c)/(1-v/c)] 7. Physics is Easy 8. Particles, Waves and Trends in Physics 9. Physics is Beautiful 10. Great Table of Elementary Particles 11. Movement Principles of the Fast-Spinning Bodies 12. Nuclear Fusion

Stellar models consisting of spherically symmetric distribution of charged matter locally anisotropic in strong gravitational fields have been widely considered in the frame of general relativity. These investigations require the generation of exact models through the resolution of the Einstein-Maxwell system of equations. The presence of charge produces values for redshifts, luminosity and mass for the stars different in relation to neutral matter. Some applications for dense charged matter we have them in the description of quark stars, spheres with linear or non-linear equation of state, hybrid stars and accreting process in compact objects where the matter acquires large amounts of electric charge. In this paper, we studied the behavior of relativistic compact objects with anisotropic matter distribution considering Van der Waals modified equation of state proposed in 2013 for Malaver and a gravitational potential Z(x) that depends on an adjustable parameter α in order to integrate analytically the field equations. They generalize the ideal gas law based on plausible reasons that real gases do not act ideally. New exact solutions of the Einstein-Maxwell system are generated and the physical variables as the energy density, radial pressure, mass function, anisotropy factor and the metric functions are written in terms of elementary and polynomial functions. We obtained expressions for radial pressure, density and mass of the stellar object physically acceptable with two different values of the adjustable parameter. The proposed models satisfy all physical features of a realistic star.

In Einstein Field Equations we find the Coupling Constant "kappa". This constant is very complicated and consists of many parameters. The explanations that I found are extremely difficult. If you follow the traces of the EFE in Wiki, you find connections with Jacobi field, with co-variant derivatives with respect to the Levi-Civita connection, Newton-Euler equations, and the Riemann curvature tensor.
However: My point of view is, that all these tensors together are summarized and compressed in a clear equation with just 3 parameters. Attached to the paper are equations and calculations. In invite you to read my paper "Around kappa". I like to know:
1.  Is my interpretation valid? If no: why not?
2. Is my interpretation too simple?
3. If my interpretation is right, does it just mean "So what?"

In this paper we consider the Universe at the late stage of its evolution and deep inside the cell of uniformity. At these scales, the Universe is filled with inhomogeneously distributed discrete structures (galaxies, groups and clusters of galaxies). Supposing that a small fraction of colored objects escaped hadronization and survived up to now in the form of quark-gluon nuggets (QNs), and also taking into account radiation, we investigate scalar perturbations of the FRW metrics due to inhomogeneities of dustlike matter as well as fluctuations of QNs and radiation. In particular, we demonstrate that the nonrelativistic gravitational potential is defined by the distribution of inhomogeneities/fluctuations of both dustlike matter and QNs. Consequently, QNs can be distributed around the baryonic inhomogeneities (e.g., galaxies) in such a way that it can solve the problem of the flatness of the rotation curves. We also show that the fluctuations of radiation are caused by both the inhomogeneities in the form of galaxies and the fluctuations of quark-gluon nuggets. Therefore, if QNs exist, the CMB anisotropy should contain also the contributions from QNs. Additionally, the spatial distribution of the radiation fluctuations is defined by the gravitational potential. All these results look physically reasonable.

The Super Principle Relativity Theory is a hydrodynamic field theory, which models “Gravitation” and the “Vacuum”, in the conceptual and mathematical framework of a variable density compressible ideal gaseous fluid. In contrast to General Relativity Theory which is a field theory, that models “Gravitation” and the “Vacuum”, in the conceptual and mathematical framework of a fixed density “frictionless incompressible fluid medium” or “incompressible liquid medium”. The new theory produces exact solutions to gravitation phenomena, and describes the Euler-Lagrangian Hydrodynamic Energy Density Equation, as a space-time frame of reference; and is mathematically the net sum of space-like and time-like energy density terms. The “Hydrodynamic Field Energy Density Frame of Reference” is used to model the interaction between seven (7) physical phenomena: gravitational, electromagnetic, isotropic space-time world line, gravitational sink, gravitational source, electromagnetic sink, and electromagnetic source energy density frames of reference. The “Hydrodynamic Field Theory” models physical phenomena: solar systems, galaxy rotation, quantum mechanical systems, etc.

We point out a weak side of the commonly used determination of scalar cosmological perturbations lying in the fact that their average values can be nonzero for some matter distributions. It is shown that introduction of the finite-range gravitational potential instead of the infinite-range one resolves this problem. The concrete illustrative density profile is investigated in detail in this connection.

We consider lattice Universes with spatial topologies $T\times T\times T$, $\; T\times T\times R\; $ and $\; T\times R\times R$. In the Newtonian limit of General Relativity, we solve the Poisson equation for the gravitational potential in the enumerated models. In the case of point-like massive sources in the $T\times T\times T$ model, we demonstrate that the gravitational potential has no definite values on the straight lines joining identical masses in neighboring cells, i.e. at points where masses are absent. Clearly, this is a nonphysical result since the dynamics of cosmic bodies is not determined in such a case. The only way to avoid this problem and get a regular solution at any point of the cell is the smearing of these masses over some region. Therefore, the smearing of gravitating bodies in $N$-body simulations is not only a technical method but also a physically substantiated procedure. In the cases of $\; T\times T\times R\; $ and $\; T\times R\times R$ topologies, there is no way to get any physically reasonable and nontrivial solution. The only solutions we can get here are the ones which reduce these topologies to the $T\times T\times T$ one.

In the framework of the concordance cosmological model, the first-order scalar and vector perturbations of the homogeneous background are derived in the weak gravitational field limit without any supplementary approximations. The sources of these perturbations (inhomogeneities) are presented in the discrete form of a system of separate point-like gravitating masses. The expressions found for the metric corrections are valid at all (sub-horizon and super-horizon) scales and converge at all points except at the locations of the sources. The average values of these metric corrections are zero (thus, first-order backreaction effects are absent). Both the Minkowski background limit and the Newtonian cosmological approximation are reached under certain well-defined conditions. An important feature of the velocity-independent part of the scalar perturbation is revealed: up to an additive constant, this part represents a sum of Yukawa potentials produced by inhomogeneities with the same finite time-dependent Yukawa interaction range. The suggested connection between this range and the homogeneity scale is briefly discussed along with other possible physical implications.

Using the well known “displace, cut and reflect” method used to generate disks from given solutions of Einstein field equations, we construct some relativistic models of time dependent thin disks of infinite extension made of a perfect fluid based on the Robertson-Walker metric. Two simple families of models of disks based on Robertson-Walker solutions admitting Matter and Ricci collineations are presented. We obtain disks that are in agreement with all the energy conditions.

MSC: 83C22; PACS: 04.40Nr

In this paper, we study the Einstein-Maxwell equations with anisotropic pressures considering a polytropic equation of state in presence of an electric field and a modified version of metric potential proposed by Korkina-Orlyanskii (1991). We obtain two new classes of exact models that satisfy all physical features expected in a realistic star. Graphical analysis indicates that physical properties as energy density, radial pressure, charge density and anisotropy profiles and the measures of anisotropy are consistent with seminal treatments which suggest great importance in the description of relativistic compact objects.

Information about “Mind boggling Gravitation. Applying the Lorentz Transformation of Mass-Energy”. Introducing the "Root Mean Square Velocity" (VRMS): an important parameter in our solar system and the basic velocity applied in the Lorentz Transformation of Mass-Energy.