Einstein gravitational field equations

This procedure leads to the Einstein gravitational field equations, one form of which, without cosmological term, is... 

The symmetry between curvature and matter is the most important result of Einstein s gravitational field equations. Both of these tensors vanish in empty euclidean space and the symmetry implies that whereas the presence of matter causes space to curve, curvature of space generates matter. This reciprocity has the important consequence that, because the stress tensor never vanishes in the real world, a non-vanishing curvature tensor must exist everywhere. The simplifying assumption of effective euclidean space-time therefore is a delusion and the simplification it effects is outweighed by the contradiction with reality. Flat space, by definition, is void. 

Expanding universe cosmologies, in contrast, are based on a special metric that assumes a imiversal time, independent of the curved spatial manifold, as a solution of Einstein s gravitational field equations. In reality this is a second-generation generalization of Einstein s static-universe cosmology, once assumed to be the only possible solution of the field equations on a cosmic scale. However, as emphasized by Fuller and Wheeler (1962) ... 

The biggest surprise about de Sitter s solution was that, although a legitimate solution of Einstein s gravitational field equations, it violates Mach s principle, which Einstein considered to be the pillar that supports the general theory of relativity. 

Kaluza and Klein managed to formulate a unified theory of gravitation and electromagnetism in terms of Einstein s field equations in five-dimensional space, but with the metric tensor defined to be independent of the fourth space dimension. Without this restriction, solution of the equations in apparent 5D vacuum ... 

A planet in orbit therefore looses energy, and like a classical orbiting electron (section 3.4.2), spirals in towards the nucleus of the system. In the case of an atom this is prevented by the quantum potential and there is no reason why a cosmological quantum effect could not be responsible for the stabilization of satellite orbits. In fact, there is the evidence of planets and moons in the solar system on orbits characterized by integers, as discussed in section 5.3.1. The cosmological term. A, which Einstein included in the gravitational field equations (6.4) describes exactly such an effect. 

Cosmology in its present form developed as a by-product of Einstein s gravitational field equations, based on the astronomical data of the previous millenium, which established the heliocentric model of the solar system. The struggle against the authority of Ptolemy, Aristotle and the Inquisition, the rivalry between Kepler and Galileo, and the intrigue between Newton and his contemporaries, Descartes, Leibniz, Hooke and others, overshadow the important theoretical advances that produced the mechanical clockwork model of Laplace. 

Einstein granted that the (Dirac) equation was "the most logically perfect presentation" of quantum mechanics yet found, but not that it got us any closer to the "secret of the Old One". It neither described the real world phenomena that he wanted to understand nor proposed new concepts that would make the real world accessible to understanding. Furthermore, Dirac s unification of quantum mechanics with special theory of relativity left out Einstein s later success with general relativity and the gravitational field. 

We now turn to Einstein s full gravitational equation. There being ten metric components, there are ten partial differential equations to determine them. One is a fanciful elaboration of Poisson s equations with the relativistic energy density—as opposed to rest mass density—as source. Pressure and energy fluxes become the sources of the others. If we are mostly interested in the external gravitational field of a spherically symmetric body, then the sources can be dropped and the unique exact solution is Schwarzschild s metric (not Martin Schwarzschild but his dad Karl Schwarzschild, also the father of photographic photometry) ... 

The Kerr solution (1963) of the Einstein s vacuum field equations describes the gravitational field of a material source at rest having mass and angular momentum. The angular momentum determines a physically significant di-... 

We have seen in this section that the factorization of Einstein s symmetric, second-rank tensor field equations (10 relations) to a quaternion form (16 relations) not only yields the gravitational and electromagnetic manifestations of matter in a unified field theory but also reveals a feature of quantum mechanics. In particular, it was found that in the flat-space approximation to the curved-space representation in general relativity, the time component of the electromagnetic four-current density corresponds in a one-to-one way with the probability density of quantum mechanics. Its integration over all of space in this limit is found to be unity. 

This calculation precludes development of the Einstein diffusion equation for forced diffusion in the presence of a gravitational field. The coefficient of (ge — gA) in equation (25-77) for the diffusional mass flux of species A can be evaluated via thermodynamics. The extensive Gibbs free energy of a one-phase binary mixture with 3 degrees of freedom requires four independent variables for complete description of this thermodynamic state function. Hence, ( (T, p, N/, Nb) is postulated where Ni represents the mole numbers of species i, and the total differential of is... 

As mentioned before it is conjectured that in projective relativity theory the coefficients gij of the conic equation are gravitational potentials and the coefficients of the hyperplane equation are electromagnetic potentials. We shall see, in fact, that the closest field equations for the 7, 3 are a combination of the classical Einstein gravitation equations and the Maxwell field equations. 

Equations (1.59,1.60,1.62) agree with the results of Einstein s theory of general relativity for the perihelion movement of Mercury and the law that a photon deviates in a gravitational field twice the amount as predicted by Newton s gravitational law. 

Einstein discovered the celestial mechanical consequences of GRT in 1914, just before his completion of GRT. General relativity produces a change in the gravitational field in the vicinity of a massive body. The distortion is equivalent to introducing an additional term in the harmonic oscillator equation... 

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