SEARCH

SEARCH BY CITATION

Abstract

An elementary criterion of the stability of a matter sphere against gravitational collapse is given by the circular velocity condition of POINCARÉ: In a space with a spherically symmetric gravitation potential ϕ (r) and with a spherically symmetric metric gik (e.g., a SCHWARZSCHILD space time) the circular velocity V* of a particle on the surface r = R of the matter-sphere must be

  • equation image

(This condition is a consequence of the virial theorem and of the POINCARÉ theorem.) - However, EINSTEIN's axiom of causality implies that this velocity V* must be smaller than the local velocity of light v: V*2 < v2. And this local velocity v is a function of the gravitation potential ϕ, too: v = v [ϕ].

In the case of NEWTON's or EINSTEIN's theory the spherically symmetric gravitation potential is given by the NEWTONian function ϕ = fM/r. In the special theory of relativity, we would have v = c (c = EINSTEIN's fundamental velocity) and grr = 1. Therefore, the specialrelativistic stability condition is R > fMc−2. - But in the NEWTONian theory v is depending of the gravitation potential and depends of the boundary condition for the light propagation, also. According to the ansatz of LAPLACE (1799) we have:

  • equation image

(emanation-theory of light). But, according to SOLDNER (1801), we have

  • equation image

Therefore, we are finding in the case of LAPLACE the same condition R > fMc−2 as in the SRT. But, in the case of SOLDER's ansatz non condition for stability is resulting. - In the general relativistic theories the local velocity of light is given by EINSTEIN 's expression

  • equation image

According to EINSTEIN's theory of “static gravitation” (1911/12) we have grr = 1 and therefore the formula

  • equation image

and according to the GRT (with - gω = grr−1) we have the formula

  • equation image

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 LAPLACE “black-holes” with bounding states of light result for R ≤ 2fM−2. But, no “black-holes” are existing according to SOLDNER's ansatz. However, in GRT each black-hole must be a “collapsar”. But according to the classical theory of LAPLACE we have uncollapsed “black- holes” for the domain

  • equation image

.