Considerations on Gravitational Effects Stimulated by Gravitational Fields via Classical Field Theory Helmut Kling * (Dated: October 7, 2013) Abstract A mass distribution is analyzed in terms of classical gravitational field theory. Newton’s law of gravitation is consistently applied on the assumption that the equivalence of energy and mass according to Einstein’s theory of relativity is valid for gravitational fields as well. Differ- ent from standard approaches the gravitational field, via its associated field energy, is han- dled as a source of gravitation by itself. Starting from these principles a gravitational self- shielding phenomenon is derived as a common characteristic of all mass/energy distribu- tions. Moreover, it is demonstrated that even in the absence of any stimulating mass the ex- istence of independent static gravitational fields is fully consistent with Newton’s law of grav- itation as long as the equivalence of energy and mass is respected. From a distance such gravitational fields appear as negative masses. Keywords: Gravitation, Gravitational Field, Gravitational Self-shielding 1. Introduction When analyzing the gravitation of a mass distribution by means of classical field theory New- ton’s law of gravitation is applied to the stimulating mass distribution. According to Einstein’s theory of special relativity energy and inertial mass are equivalent. Einstein’s theory of gen- eral relativity extended this principle to the gravitational mass. Consequently, the term stimu- lating mass has not only to include classical masses but any energy distribution. In classical field theory, however, the associated gravitational field is usually not seen as a source of gravitation although it represents an energy distribution. In this study a mass distribution is analyzed in terms of classical gravitational field theory in which the gravitational field energy is handled as a mass, which is acting as a source of gravitation by itself. In that sense equivalence of energy and mass is applied more conse- quently as is usually done. Based on these principles a field equation for static gravitational 1 fields is developed and applied to a spherical mass distribution and also to the special case where no stimulating mass is present at all, i.e. to vacuum. Beyond the described equivalence of energy and gravitational mass further aspects of Ein- stein’s theory of general relativity, its formalism including Einstein’s field equations are not in the scope of this work. 2. Gravitational Field Strength and Energy Density Analysis shall be started from Newton’s law of gravitation, which gives the following force to a particle of mass 푚 at the position 풙, which is exposed to the gravitation of a second mass 푚2 at the position 풙ퟐ 풙 − 풙2 푭품 = − 퐺 푚 푚2 3 , (1) |풙 − 풙2| where G is the gravitational constant. We get 풙 − 풙′ 푭 (풙) = − 퐺 푚 ∫ 휌(풙′) 푑3푥′ , (2) 품 |풙 − 풙′|3 if the origin of the gravitational force is not a single point mass but a mass distribution repre- sented by 휌(푥). This mass distribution is associated with a gravitational field described by its field strength 풙 − 풙′ 푬 (풙) = − 퐺 ∫ 휌(풙′) 푑3푥′ (3) 품 |풙 − 풙′|3 3 so that 푭푔 = 푚 푬푔. Using 훻(1/|풙 − 풙′ |) = −(풙 − 풙′)/|풙 − 풙′| leads to 휌(풙′) 푬 (풙) = 퐺 훻 ∫ 푑3푥′ . (4) 푔 |풙 − 풙′| The gravitational field carries energy at a density of 1 푢 = − 푬2 . (5) 푔 8휋퐺 푔 This expression can easily be obtained in full analogy to the corresponding electrostatic case. Detailed evaluations of the electrical case are given in any textbook on electrody- namics such as [1] or [2]. However, even if well known [3] it has to be noted that different from electrostatics gravitational field energy is always negative, which is a consequence of 2 Newton’s law of gravitation when respecting the law of the conservation of energy due to the opposite sign compared to Coulomb’s law. 3. Evaluation of the Gravitational Field Strength Every mass/energy distribution is accompanied by a gravitational field carrying an energy density, which is, according to Einstein’s correlation E = mc2 (E energy, m mass, c vacuum speed of light), equivalent to a mass distribution. As Visser pointed out in his paper “A Clas- sical Model for the Electron” [4]1, it is logic and consistent to assume that the gravitational field acts again as a source of gravitation and the overall mass density to be taken into ac- 2 count is given by 휌푡표푡 = 휌 + 휌푓푖푒푙푑 = 휌 + 푢푔/푐 , where 휌 represents the density of the stimulating mass as introduced in section 1 and 휌푓푖푒푙푑 the equivalent mass density resulting from the gravitational field energy given by equation (5). It is obvious that this field-induced contribution, due to the property of the gravitational field energy to be negative, is equivalent to a negative mass distribution and acts opposite to the stimulating mass distribution: It weakens its gravitational effect. The gravitational field strength can therefore be expressed as 3 ′ 푢푔 푑 푥 푬 (풙) = 퐺 훻 ∫ [휌(풙′) + ] (6) 푔 푐2 |풙 − 풙′| resulting in the following field equation on 푬품 1 푑3푥′ 푬 (풙) = 퐺 훻 ∫ [휌(풙′) − 푬2(풙′)] . (7) 푔 8휋퐺푐2 푔 |풙 − 풙′| Let us now concentrate our analysis on a spherical mass distribution having a radius a and a mass M, whose center is located at the origin of our coordinate system. Spherical symmetry implies that 푬푔(풙) is not dependent on polar angles and always pointing into the direction of 풆푟 , which is the radial unit vector. When orientating our coordinate system in such a way that 풙 points along the z axis, i.e. 휗 =0, equation (7) can be rewritten as ∞ 2휋 휋 푀 1 2 2 푠푖푛 휗′ 푑 휗′ 퐸푔(푟) = − 퐺 − 훻푟 ∫ 퐸푔 (푟′)푟′ 푑푟′ ∫ 푑휑′ ∫ (8) 푟2 8휋푐2 √푟2 − 2 푟 푟′ 푐표푠 휗′ + 푟′2 푎 0 0 for r ≥ a, where r =|x|, 휑 and 휗 spherical coordinates and 푬푔(풙) = 퐸푔(푟) 풆푟. 1 In [4] M. Visser considers the gravitational field as well as the electrical field as a source of gravita- tion and applies this principle to his classical model for the electron, which assumes the electron to be a point particle. Other arrangements are not investigated in this paper. 3 Performing integration on 휑 and using 휋 ′ 2 ′ √ 2 ′ ′ ′ ′ ′ ′ (9) ∫0 (푠푖푛 휗 / 푟 − 2 푟 푟 푐표푠 휗 + 푟 ) 푑 휗 = (푟 + 푟 − |푟 − 푟 |)/푟 푟 we end up at 푟 ∞ 푀 1 1 퐸 (푟) = − 퐺 − 훻 [ ∫ 퐸2(푟′)푟′ 2푑푟′ + ∫ 퐸2(푟′)푟′푑푟′] 품 푟2 2푐2 푟 푟 푔 푔 푎 푟 푟 푀 1 1 = − 퐺 + ∫ 퐸2(푟′)푟′ 2푑푟′ . (10) 푟2 2푐2 푟2 푔 푎 Equation (10) reflects the well known fact that the gravitational field at a distance r from the center of a spherical mass distribution is equal to the field of a point mass located at the cen- ter whose magnitude is identical to the total mass enclosed by the sphere of radius r. In or- 2 der to simplify the evaluation of this integral equation we substitute 휅(푟) = − 푟 퐸푔(푟)/퐺푀 assuming M>0, which leads to 푟 2 ′ 2 휅(푟) = 1 − 푅푔 ∫ 휅 (푟 )/푟′ 푑푟′ , (11) 푎 2 where 푅푔 = 퐺푀/2푐 is one fourth of the Schwarzschild radius [5]. Differentiation yields 푑휅(푟) 휅2(푟) = −푅 . (12) 푑푟 푔 푟 2 Separation of the variables 휅 and r results in 푑휅(푟) 푑푟 = −푅 . (13) 휅2 푔 푟 2 Integration then leads to 1 휅(푟) = , (14) 푑 − 푅푔/r where d is an integration constant. It can easily be calculated by turning back to equation (11). Finally we get 1 휅(푟) = (15) 푅푔 푅푔 1 + 푎 − 푟 4 and 1 푀 퐸푔(푟) = − 퐺 2 . (16) 푅푔 푅푔 푟 1 + 푎 − 푟 Evidently 휅(푟) describes the influence of the gravitational field itself. Figure 1 shows 휿 ver- sus r/a for different Rg/a. Obviously at the surface of the sphere no self-contribution of the 1 R /a 0,9 g 0,8 0.01 0,7 0.1 0,6 0.2 0,5 0.5 0,4 1 0,3 2 5 0,2 10 0,1 0 1 10 100 Figure 1: 휿 versus r/a for different Rg/a gravitational field is at work, which is confirmed by equation (15), i.e. 휅(푟 = 푎) = 1. For Rg > a another interesting reference point is r = Rg. In order to achieve a given gravitational field strength at this point according to equation (16) a mass is needed which is a factor Rg/a big- ger than it would be expected when ignoring the contribution of the gravitational field itself. Let us now investigate the resulting behavior at long distances from the sphere, in particular for r ≫ Rg. Expansion of 퐸푔(푟) in powers of 푅푔⁄푟 yields 2 1 1 푅푔 1 푅푔 푀 퐸푔(푟) = − [1 + + ( ) + . ] 퐺 2 . (15) 푅푔 푅푔 푟 푅푔 푟 푟 1 + 푎 1 + 푎 1 + 푎 Restriction to the first term leads to 1 푀 퐸푔(푟) = − 퐺 2 . (16) 푅푔 푟 1 + 푎 5 When observing the gravitational interaction at a long distance the sphere will appear as an object having a mass Mobs according to classical theory of gravitation, i.e. when ignoring the contribution of the gravitational field itself, which is 푀 푀표푏푠 = = (1 − 휎) 푀 , (17) 푅푔 1 + 푎 2 where 푅푔 = 퐺푀/2푐 and 휎 = 1 – 1/(1 + 푅푔⁄푎). 휎 is a self-shielding coefficient measur- ing, which amount of the gravitational field strength is shielded by the gravitational field itself.