Article

# Gravitationskollaps und Lichtgeschwindigkeit im Gravitationsfeld

Article first published online: 16 MAR 2006

DOI: 10.1002/andp.19744860404

Copyright © 1974 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Additional Information

#### How to Cite

Treder, H.-J. (1974), Gravitationskollaps und Lichtgeschwindigkeit im Gravitationsfeld. Ann. Phys., 486: 325–334. doi: 10.1002/andp.19744860404

#### Publication History

- Issue published online: 16 MAR 2006
- Article first published online: 16 MAR 2006
- Manuscript Received: 4 JAN 1974

- Abstract
- References
- Cited By

### 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

(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} < *v*^{2}. 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:

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

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

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

and according to the GRT (with - g_{ω} = grr^{−1}) we have the formula

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

.