Gravitationskollaps und Lichtgeschwindigkeit im Gravitationsfeld

An elementary criterion of the stability of a matter sphere against gravitational collapse is given by the circular velocity condition of P OINCARÉ : In a space with a spherically symmetric gravitation potential ϕ (r) and with a spherically symmetric metric gik (e.g., a S CHWARZSCHILD space time) the circular velocity V * of a particle on the surface r = R of the matter‐sphere must be\documentclass{article}\pagestyle{empty}\begin{document}$$ V^{*2} = \frac{1}{{g_{rr} (R)}}R|\frac{{\partial \Phi}}{{\partial {\rm r}}}|_{r = R} $$\end{document}(This condition is a consequence of the virial theorem and of the P OINCARÉ theorem.) ‐ However, E INSTEIN 's axiom of causality implies that this velocity V * must be smaller than the local velocity of light v : V *2 < v 2 . And this local velocity v is a function of the gravitation potential ϕ, too: v = v [ϕ]. In the case of N EWTON 's or E INSTEIN 's theory the spherically symmetric gravitation potential is given by the N EWTON ian function ϕ = fM/r . In the special theory of relativity, we would have v = c ( c = E INSTEIN 's fundamental velocity) and grr = 1. Therefore, the specialrelativistic stability condition is R > fMc −2 . ‐ But in the N EWTON ian theory v is depending of the gravitation potential and depends of the boundary condition for the light propagation, also. According to the ansatz of L APLACE (1799) we have:\documentclass{article}\pagestyle{empty}\begin{document}$$ v = c\sqrt {1 + \frac{{2fM}}{{C^2}}\left({\frac{1}{r} - \frac{1}{R}} \right) \le c\ for\ r \ge R} $$\end{document}(emanation‐theory of light). But, according to S OLDNER (1801), we have\documentclass{article}\pagestyle{empty}\begin{document}$$ v = c\sqrt {1 + \frac{{2fM}}{{C^2 r}}} \ge c $$\end{document}Therefore, we are finding in the case of L APLACE the same condition R > fMc −2 as in the SRT. But, in the case of S OLDER 's ansatz non condition for stability is resulting. ‐ In the general relativistic theories the local velocity of light is given by E INSTEIN 's expression\documentclass{article}\pagestyle{empty}\begin{document}$$ v = c\sqrt {\frac{{- g_{00}}}{{g_{rr}}}} \le c $$\end{document}According to E INSTEIN 's theory of “static gravitation” (1911/12) we have grr = 1 and therefore the formula\documentclass{article}\pagestyle{empty}\begin{document}$$ v = c\sqrt {- g_{00}} = c\sqrt {1 - \frac{{2fM}}{{c^2 r}}} $$\end{document}and according to the GRT (with ‐ g ω = grr −1 ) we have the formula\documentclass{article}\pagestyle{empty}\begin{document}$$ v = c\left({1 - \frac{{2fM}}{{c^2 r}}} \right) $$\end{document}Therefore, the Hilbert‐Laue condition r= R > 3fMc −2 results as stability condition. From the gravo‐optical point of view, in GRT and for the classical ansatz of L APLACE “black‐holes” with bounding states of light result for R ≤ 2fM −2 . But, no “black‐holes” are existing according to S OLDNER 's ansatz. However, in GRT each black‐hole must be a “collapsar”. But according to the classical theory of L APLACE we have uncollapsed “black‐ holes” for the domain\documentclass{article}\pagestyle{empty}\begin{document}$$ \frac{{fM}}{{c^2}} R \le \frac{{2fM}}{{c^2}} $$\end{document} .