Schwarzschild solution

The Schwarzschild solution was the first known solution of the Gross-mann equation (beside the Minkowski ds2 of course). 

In an ehort to derive particle properties from general relativity Einstein and Rosen (1935) investigated spherically symmetrical solutions of the held equations, including the Schwarzschild solution. On the premise that every held theory should exclude singularities of the held, they found this to be possible provided the physical space is represented by two identical sheets connected by bridges at the position of a singular point of the unmodihed metric. The hnal solution was found only to pertain to massless particles, and quantum phenomena could not be demonstrated a priori. However, in... 

The separation of time-like and space-like events creates the impression of two types of response to increased curvature of space-time. If only space coordinates are curved it results in the inversion of the time coordinate and the conversion of matter into antimatter. This situation will be encountered in the Schwarzschild solution of the gravitational field equations, which serves as a model of a black hole, and assumed here to account for an inversion at Z/A = 1.04. It resembles the limitless time-like accumulation of matter, resulting in a space-time singularity. 

The Schwarzschild solution in nonempty space estimates the stress or energy-momentum tensor T p in terms of an incompressible perfect fluid medium with the same symmetry as before and serves as a simple model of a star. To get the complete picture the interior solution for a sphere of perfect fluid of radius ro is joined continuously with the free-space solution that applies at r > ro > 2m. As before m = nM/( , where M is the mass of the fluid sphere. 

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 first known solution was that of Schwarzschild that provides the origin of the notion of a black hole. Let us also mention the Kerr solution, plane gravitational waves, the general first order solution, the successive approximations of the two black hole problem hundreds of rigourous solutions are known today. 

The Kerr solution generalizes the Schwarzschild ds2 to a rotating black hole, with an angular momentum A and the corresponding length a = that always satisfies o < m. 

The only solution of the field equations without ruinous approximations was obtained by Schwarzschild. It serves as a model for isolated objects and is too localized for cosmology. A concise critical summary of the cosmological models was recently published by Mamone Capria (2005) and our more superficial treatise that follows will concentrate only on those aspects of immediate relevance. 

The gravitational field is described in general relativity by the set of equations (4.11). The right hand side depends on the description of matter in the system of interest and the corresponding solution consists of finding that form of the fundamental tensor that satisfies (4.11). The first successful solution of cosmological interest, obtained by Schwarzschild, is text-book material, described in detail by Adler et al. (1965). The time-independent spherically symmetric line element is of particular importance as a model of the basic one-body problem of classical astronomy. This element, of the form ... 

The solution to (6.1) is of direct importance for modelling planetary motion on a spherical orbit. It is noted that the coefficient of dr becomes infinite on the spherical shell r = 2m and, the solution therefore goes singular at both r = 0 and = 2m, known as the Schwarzschild radius. For any known macroscopic body the radius = 2/cM/c falls well inside the body where free-space equations are not valid and (6.1) does not yield an appropriate solution. 

Although the Godel solution is free of singularities the need to accommodate black holes in the cosmic model requires an interpretation of the Schwarzschild singularity which occurs with infinite curvature of space-time. A new interpretation is rather obvious. Such a high degree of curvature must clearly rupture the interface between adjacent sides of the postulated cosmic double cover. Rather than disappear into a singularity, the matter,... 

Another elegant approach was suggested by Meunier et. al. who presented a custom made objective based on a modified Schwarzschild objective. Its optic axis is perpendicular to the studied layer and consequently the complete area is in focus. The design is ideally suited for dynamic investigations and the alignment is easier than for the previously discussed solution. 

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