On Corda’s ‘Clarification’ of Schwarzschild’s Solution
Abstract
A paper by C. Corda (A Clarification on the Debate on “the Original Schwarzschild Solution”, EJTP 8, No. 25 (2011) 65-82) purports equivalence of Schwarzschild’s original solution (1916) and Hilbert’s subsequent solution (1917), the latter commonly but incorrectly called ‘Schwarzschild’s solution’. The derivation of Schwarzschild’s actual solution by Corda is, in fact, a copy of Schwarzschild’s original derivation with only changes in notation and equation numbering. It adds nothing new to the problem. Corda’s subsequent arguments on gravitational collapse follow those advanced by Misner, Thorne, and Wheeler for Hilbert’s solution, in their book ‘Gravitation’, and suffer thereby from the very same shortfalls. Consequently, Corda has failed to prove his alleged equivalence of the Schwarzschild and Hilbert solutions. Moreover, it is not difficult to prove that these are not equivalent. Furthermore, all methods employed to otherwise ‘extend’ Droste’s solution into Hilbert’s solution thus producing a black hole constitute a violation of the rules of pure mathematics and are invalid.
1 Introduction
One hundred years ago, on the 13th of January 1916, Karl Schwarzschild communicated his solution to Einstein’s gravitational field for a `mass point’. Then, less than five years ago, on the 25th of May 2011, a paper by Christian Corda was published which reproduced most of Schwarzschild’s paper. In a prelude on page 70 of his paper, Corda stated,
“In our approach we will suppose again that a(r,t) = 0, but, differently from the standard analysis, we will assume that the length of the circumference centred in the origin of the coordinate system is not 2πr. We release an apparent different physical assumption, i.e. that arches of circumference are deformed by the presence of mass of the central body M. Note that this different physical hypothesis permits to circumnavigate the Birkhoff Theorem [4] which leads to the ‘standard Schwarzschild solution’ [3].”
Then, before launching into his modifications of Schwarzschild’s equations, Corda stated, again on page 70,
“Then, we proceed assuming -mr2 , where m is a generic function to be determined in order to obtain the length of circumferences centred in the origin of the coordinate system are not 2πr. In other words, m represents a measure of the deviation from 2πr of circumferences centred in the origin of the coordinate system.”
Corda’s special generic function m=m(r) plays no special role since it already appeared in Schwarzschild’s paper as G=G(r), where it deformed nothing. Schwarzschild’s solution [1] for Einstein’s equations Rμν = 0 is,
To read the full paper – vixra.org
The conclusion is as follows:
9 Conclusions
Corda’s derivation of Schwarzschild’s solution is merely a point by point reproduction of Schwarzschild’s derivation, embellished with relabelling of Schwarzschild’s variables and functions, renumbering of his equations, rearrangement of elements of his equations, and numbering of his unnumbered equations. Corda has thereby not advanced anything new to the solution of the problem. Corda’s claim that Schwarzschild’s solution and Hilbert’s solution are equivalent is demonstrably false. Droste’s solution and Brillouin’s solution are equivalent to Schwarzschild’s solution. The solution to Rμν=0 requires an infinite equivalence class in order to provide for all permissible ‘transformations of coordinates’. Hilbert’s solution is not an element of the infinite equivalence class and is therefore invalid, amplified by the fact that Hilbert’s solution is an alleged extension of Droste’s solution, which cannot be extended by its very selection. The ‘extension’ of Droste’s solution to Hilbert’s solution to produce a black hole constitutes a violation of the rules of pure mathematics and so it is invalid. It is from the extension of Droste’s solution that Hilbert enabled the black hole. Hence, the theory of black holes is invalid. Corda’s claim that “the length of the circumference centred in the origin of the coordinate system is not 2πr” is false because the length of a closed geodesic in a spherical surface always has the form 2πR, being indifferent to how R is assigned a value. Corda’s assertion that r in Hilbert’s solution and Schwarzschild’s solution is radial distance therein, although standard cosmology, is demonstrably false. The ‘Schwarzschild radius’ is not the radius of anything, or even a distance, in the Schwarzschild metric or Hilbert’s metric. It is therefore not the radius of a black hole event horizon. Corda’s [3] conclusion that “… Hilbert was not wrong but they are definitely wrong the authors who claim that ‘the original Schwarzschild solution’ implies the non existence of BHs’” is incorrect. The black hole is a product of violations of the rules of pure mathematics and therefore false. Schwarzschild’s solution does not permit a black hole. Since the MichellLaplace dark body does not share the properties of the black hole, it is not a black hole [14,22]. Hence, there is no legitimate mathematical theory of black holes.
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