Schwarzschild-like metric and a quantum vacuum (original) (raw)

A quantum vacuum, represented by a viscous fluid, is added to the Einstein's vacuum, surrounding a spherical distribution of mass. This gives as a solution, in spherical coordinates, a Schwarzschild-like metric. The plot of g 00 and g 11 components of the metric, as a function of the radial coordinate, display the same qualitative behavior as that of the Schwarzschild metric. However, the temperature of the event horizon is equal to the Hawking temperature multiplied by a factor of two, while the entropy is equal to half of the Bekenstein one. 1-INTRODUCTION Einstein's theory of the general relativity has as one of its simplest applications, the case of determining the space-time structure in the neighborhood of an isolated, static, uncharged and non-rotating spherical mass M. In these conditions, the solutions of Einstein's equations were developed by Karl Schwarzschild [1] in 1916. A detailed calculation of vacuum spherical symmetric Einstein's equations leading to the Schwarzschild space-time can be found in a book by McMahon [2]. We refer also to another treatment of this problem reported by Schmude [3]. In this work, we first intend to "derive" the Schwarzschild metric by using special relativity, the equivalence principle, and the Newtonian law of gravity. However, the present treatment is somewhat different from others simple derivations of this subject, as are those discussed by Sacks and Ball [4]. Besides this, we propose a modification of the Schwarzschild problem, where the contribution of a viscous' fluid (a kind of quantum vacuum) acting on the test particle used to probe the gravitational field of the source M, will also be considered. 2-MAXIMUM FLOATING PRINCIPLE AND THE SCHWARZSCHILD RADIUS Derek Paul in studying the dispersion relation of the de Broglie waves [5], has used the equivalence principle in order to analyze the motion of a heavy (guided) photon in the gravitational field. Inspired in Paul's work [5], we developed an alternative way of estimating the Schwarzschild radius of a mass M.