Black Holes: a Different Perspective

International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.6.4.14 [Vol-6, Issue-4, Apr- 2019] ISSN: 2349-6495(P) | 2456-1908(O) Black Holes: a Different Perspective Carmine Cataldo Independent Researcher, PhD in Mechanical Engineering, Battipaglia (SA), Italy Email: [email protected] Abstract—In this paper we propose a full revised version of a simple model, which allows a formal derivation of an infinite set of Schwarzschild-Like solutions (non-rotating and non-charged “black holes”), without resorting to General Relativity. A new meaning is assigned to the usual Schwarzschild-Like solutions (Hilbert, Droste, Brillouin, Schwarzschild), as well as to the very concepts of “black hole” and “event horizon”. We hypothesize a closed Universe, homogeneous and isotropic, characterized by a further spatial dimension. Although the Universe is postulated as belonging to the so-called oscillatory class (in detail, we consider a simple-harmonically oscillating Universe), the metric variation of distances is not thought to be a real phenomenon (otherwise, we would not be able to derive any static solution): on this subject, the cosmological redshift is regarded as being caused by a variation over time of the Planck “constant”. Time is considered as being absolute. The influence of matter/energy on space is analysed by the superposition of three three-dimensional scenarios. A short section is dedicated to the so-called gravitational redshift which, once having imposed the conservation of energy, may be ascribable to a local variability of the Planck “constant”. Keywords—Black Holes, Schwarzschild, Hilbert, Droste, Brillouin, Extra Dimension, Weak Field, Redshift. I. INTRODUCTION We hypothesize a closed Universe, homogeneous and isotropic, belonging to the so-called Oscillatory Class [1]. The existence of a further spatial dimension is postulated. Although space, as we are allowed to perceive it, is curved, since it can be approximately identified with a HyperSphere (the radius of which depends on the state of motion) [2], the Universe in its entirety, assimilated to a FourDimensional Ball, is to be considered as being flat. All the points are replaced by straight line segments [3] [4]: in other terms, what we perceive as being a point is actually a straight-line segment crossing the centre of the 4-Ball. Consequently, matter is not to be regarded as evenly spread on the (Hyper)Surface of the 4-Ball, but rather as homogeneously filling the 4-Ball in its entirety. We have elsewhere [4] deduced the following identity: 2𝐺𝑀𝑡𝑜𝑡,𝑚 𝑅𝑚 = 𝑐2 www.ijaers.com (1) G represents the Gravitational Constant, c the Speed of Light, Rm the mean value of the radius of the 4-ball, and Mtot,m the corresponding mass. According to our model, Rm and Mtot,m can be conventionally considered as being real values, since the metric variation of the cosmological distances is not thought to be a real phenomenon (in other terms, we hypothesize that the real amount of space between whatever couple of points remains constant with the passing of time) [4] [5]. In this regard, we specify how, in order to legitimize the so-called Cosmological Redshift, the Plank Constant may vary over time [6] [7]. Replacing, for convenience, Mtot,m with Mtot, and Rm with Rs (the Schwarzschild Radius), from (1) we have: 𝑅𝑠 = 2𝐺𝑀𝑡𝑜𝑡 𝑐2 (2) The Universe we have hypothesized may be approximately described, with obvious meaning of the notation, by the following inequality: 𝑥12 + 𝑥22 + 𝑥32 + 𝑥42 ≤ 𝑅𝑠2 (3) 𝑥12 + 𝑥22 + 𝑥32 + 𝑥42 = 𝑅𝑠2 (4) 𝑅𝑝 (𝜒) = 𝑅𝑠 𝜒 (5) 𝑅𝑐 (𝜒) = 𝑋(𝜒) = 𝑅𝑠 sin 𝜒 (6) The Universe we are allowed to perceive (static configuration) can be assimilated to the Hyper-Surface defined by the underlying identity: Let us denote with C the centre of the 4-ball, with O and P two points on the surface, the first of which taken as origin, and with O’ the centre of the so-called Measured Circumference, to which P belongs. Both O and O’ are considered as belonging to x4. The Angular Distance between O and P, as perceived by an ideal observer placed in C, is denoted by . The arc bordered by O and P, denoted by Rp, represents the so-called Proper Radius (the measured distance between the above-mentioned points). We have: The straight-line segment bordered by O’ and P, denoted by Rc, represents the so-called Predicted (or Forecast) Radius (the ratio between the perimeter of the Measured Circumference and 2). We have: From the previous we immediately deduce: Page | 119 International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.6.4.14 𝑋 𝜒 = arcsin ( ) 𝑅𝑠 (7) 𝑑𝑋 √1 − ( 𝑋 ) 𝑅𝑠 2 (8) The scenario is qualitative depicted in Figure 1. 𝑋 2 1−( ) 𝑅𝑠 − 𝑋 2 (𝑑𝜃 2 + sin2 𝜃 𝑑𝜑 2 ) 2𝜋 ∫ 𝜋 = 0 4𝜋𝑅𝑠2 sin2 𝜒 The above-mentioned surface is simultaneously border of a 3-Ball, denoted by V3, and of a Hyper-Spherical Cap, denoted by S3. V3 represents the Predicted (or Forecast) Volume, S3 the Proper Volume. We have: 𝑅𝑐 𝑋 4 𝑉3 (𝜒) = ∫ 𝑆2 (𝜒)𝑑𝑅𝑐 = 4𝜋 ∫ 𝑋 2 𝑑𝑋 = 𝜋𝑋 3 3 0 0 𝑅𝑝 4 = 𝜋𝑅𝑠2 sin3 𝜒 3 𝜒 𝑆3 (𝜒) = ∫ 𝑆2 (𝜒) 𝑑𝑅𝑝 = 4𝜋𝑅𝑠3 ∫ sin2 𝜒 𝑑𝜒 0 0 = 2𝜋𝑅𝑠3 (𝜒 − sin 𝜒 cos 𝜒) (11) We can generalize the foregoing as follows: 𝑆3 (𝑅, 𝜒) = 2𝜋𝑟 3 (𝜒 − sin 𝜒 cos 𝜒) www.ijaers.com 𝑅 ∈ [0, 𝑅𝑠 ] 0 (14) (13) 4𝜋 sin 𝜃𝑑𝜃𝑑𝜑 = 𝑋 2 ∫ 𝑑𝛺 = 4𝜋𝑋 2 (10) 𝜑=0 𝜃=0 1 = 𝜋𝑅𝑠4 (𝜒 − sin 𝜒 cos 𝜒) 2 2. Three-Dimensional Scenarios From (3), by setting equal to zero, one at a time, x1, x2 and x3, we obtain the following three-dimensional scenarios: (9) Let us denote with S2 the 2-Sphere characterized by a radius of curvature equal to Rc. In order to simplify the notation, from now onwards we shall denote with the same symbol both the geometrical object and the corresponding surface area or volume. Consequently, we have: 𝑆2 (𝜒) = 𝑋 2 ∫ 0 (12) At this point, for the Hyper-Surface defined in (4), the Friedmann–Robertson–Walker metric [8] can be written: 𝑑𝑋 2 𝑅𝑠 𝑉4 (𝜒) = ∫ 𝑆3 (𝑅, 𝜒)𝑑𝑅 = 2𝜋(𝜒 − sin 𝜒 cos 𝜒) ∫ 𝑅3 𝑑𝑅 II. GRAVITY: HOW MASS “BENDS” SPACE 1. Gravitational “Singularities” As previously stated, the (curved) space we are allowed to perceive can be approximately identified with a HyperSphere, the radius of which depends on our state of motion: at rest, this radius equates Rs. In our simple model the total amount of mass is constant: in other terms, mass can only be redistributed. Let us consider a generic point Q, belonging to the surface of the 4-Ball, and let us denote with max the angular distance between this point and the origin O. In order to create a “gravitational singularity” in correspondence of the origin, we have to ideally concentrate in O, from the point of view of an observer at rest (who is exclusively allowed to perceive a threedimensional curved universe), all the mass enclosed in the 2-Sphere defined by (10) (with =max). This surface represents the border of the Hyper-Spherical Cap defined in (12) (with =max) which, in turn, is associated to the hyper-spherical sector defined by (14) (with =max). According to our theory, in enacting the ideal procedure previously expounded, we actually hypothesize that all the mass of the Hyper-Spherical Sector earlier defined may be concentrated (and evenly spread) along the material segment bordered by C and O. The procedure entails a linear mass (energy) density increment, no longer compatible with the previous radial extension: consequently, both the segment and the corresponding space undergo a radial contraction (the segment shortens together with space) and the surrounding spatial lattice, the integrity of which must be in any case preserved, results deformed. We want to determine the new radial extension of the segment (that represents the singularity) and the shape of the deformed spatial lattice. It is worth specifying how, abiding to the global symmetry elsewhere introduced [2] [4] and herein taken for granted, the procedure previously exploited is symmetric with respect to the centre of the 4-Ball: consequently, we should have actually considered two opposite Hyper-Spherical Sectors, characterized by the same amplitude, and a single material segment, crossing the centre C, bordered by O and its antipodal point. Figure 1. 4-Ball 𝑑𝑠 2 = 𝑐 2 𝑑𝑡 2 − The Hyper-Surface S3 defined in (12) is associated to a Hyper-Spherical Sector, denoted by V4. We have: 𝑅𝑠 Consequently, we have: 𝑑𝑅𝑝 = 𝑅𝑠 𝑑𝜒 = [Vol-6, Issue-4, Apr- 2019] ISSN: 2349-6495(P) | 2456-1908(O) Page | 120 International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.6.4.14 2 𝑥4,1 + 𝑥22 + 𝑥32 ≤ 𝑅𝑠2 (15) 2 𝑥12 + 𝑥22 + 𝑥4,3 ≤ 𝑅𝑠2 (17) (16) 2 𝑥12 + 𝑥4,2 + 𝑥32 ≤ 𝑅𝑠2 Evidently, if we take into consideration one among the static scenarios we have just obtained, the procedure previously discussed (the creation of the singularity) is equivalent to concentrating along a segment the mass of a spherical sector. Let us denote with S2-1, the Circumference defined by the following relation: (18) 𝑆2−1 (𝜒) = 2𝜋𝑋 In the three dimensional scenario we have been considering, S2-1 “plays the role” of S2, defined in (10). The circumference defined in (18) is simultaneously border of a Disc, denoted by V3-1, and of a Sphere, denoted by S3-1. In the three-dimensional scenario we have been considering, the first “plays the role” of the Predicted (or Forecast) Volume V3, defined in (11), while the second “plays the role” of the Proper Volume S3, defined in (12). We have: 𝑅𝑐 𝑉3−1 (𝜒) = ∫ 𝑆2−1 (𝜒)𝑑𝑅𝑐 0 (19) 𝑋 = 2𝜋 ∫ 𝑋𝑑𝑋 = 𝜋𝑋 2 = 𝜋𝑅𝑠2 sin2 𝜒 0 𝑅𝑝 𝜒 𝑆3−1 (𝜒) = ∫ 𝑆2−1 (𝜒) 𝑑𝑅𝑝 = 2𝜋 ∫ sin 𝜒 𝑑𝜒 0 = 2𝜋𝑅𝑠2 (1 0 − cos 𝜒) (20) 𝑅 ∈ [0, 𝑅𝑠 ] (21) Consequently, S3-1 is associated to a Spherical Sector, denoted by V4-1, characterized by a volume provided by the following relation: 𝑅𝑠 𝑅𝑠 𝑉4−1 (𝜒) = ∫ 𝑆3−1 (𝑅, 𝜒)𝑑𝑅 = 2𝜋(1 − cos 𝜒) ∫ 𝑅2 𝑑𝑅 0 2 = 𝜋𝑅𝑠3 (1 − cos 𝜒) 3 0 𝜒 𝑑𝑟 2 𝑅𝑝,𝑔 = ∫ √( ) + 𝑟 2 𝑑𝜒 = 𝑅𝑠 𝜒 = 𝑅𝑝 𝑑𝜒 0 (23) 𝑑𝑟 2 𝑅𝑠2 = ( ) + 𝑟 2 𝑑𝜒 (24) 𝑑2 𝑟 +𝑟 = 0 𝑑𝜒 2 (25) 𝑑𝑟 )=0 (𝜒 𝑑𝜒 𝑚𝑎𝑥 (26) From the previous we easily obtain the following banal differential equation: The boundary conditions can be easily determined by resorting to the well-known shell theorem: in other terms, we have to impose that, for all the points belonging to the circumference defined in (18) once having set =max (actually, for all the points belonging to the 2-Sphere defined in (10), once having set =max), there must be no difference between the initial condition and the final one (matter concentrated in a single point). Therefore, we have: 𝑟(𝜒𝑚𝑎𝑥 ) = 𝑅𝑠 (27) 𝑟(𝜒) = 𝑅𝑠 cos(𝜒𝑚𝑎𝑥 − 𝜒) (28) 𝑟𝑚𝑖𝑛 = 𝑟(0) = 𝑅𝑠 cos 𝜒𝑚𝑎𝑥 (29) From (25), taking into account (26) and (27), we obtain: The scenario is qualitative depicted in Figure 2 (22) In the three dimensional scenario we have been considering, V4-1 “plays the role” of V4, defined in (14). As previously highlighted, the new radial extension of the segment (that represents the singularity) is still unknown, as well as the shape of the deformed spatial lattice. Let us carry out some hypotheses. Let us denote with r the Radial Coordinate of a generic point of the warped surface. Now, let’s suppose that, notwithstanding the deformation of the spatial lattice induced by the mass, if the angular distance between whatever couple of points does not vary, the corresponding www.ijaers.com measured distance remains constant. Actually, there is no point in hypothesizing a different behaviour. From now onwards, we shall resort to the subscript “g” every time we refer to a quantity measured after the creation of the singularity. We must impose the following: From the previous, we can immediately deduce: We can generalize the foregoing as follows: 𝑆3−1 (𝑅, 𝜒) = 2𝜋𝑟 2 (1 − cos 𝜒) [Vol-6, Issue-4, Apr- 2019] ISSN: 2349-6495(P) | 2456-1908(O) Figure 2. Gravitational Singularity Figure 2 qualitatively shows how space results in being deformed due to the Gravitational Singularity, perceived as being placed in Og. At the beginning, the origin coincides Page | 121 International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.6.4.14 with O. If we concentrate in O (actually along the segment bordered by C and O) the mass of the Spherical Sector (actually a Hyper-Spherical Sector) with an amplitude equal to 2max, space undergoes a contraction. The new origin coincides with Og, and the surrounding space is symmetrically warped. The initial radial coordinate of a generic point P (actually its initial radial extension) is represented by the segment bordered by C and P. The corresponding angular distance is denoted by . The final coordinate (actually the final radial extension), represented by the segment bordered by C and Pg, is shorter than the initial one, and its value is provided by (28). The proper radius does not undergo any modification: the arc bordered by O and P, in fact, is evidently equal to the one bordered by Og and Pg. If we denote with x the Reduced “Flat” Coordinate (the Reduced Forecast Radius), we have: 𝑅𝑐,𝑔 = 𝑥 = 𝑟 sin 𝜒 = 𝑅𝑠 sin 𝜒 cos(𝜒𝑚𝑎𝑥 − 𝜒) (30) 𝛿(𝜒) = 𝑅𝑠 − 𝑟(𝜒) = 𝑅𝑠 [1 − cos(𝜒𝑚𝑎𝑥 − 𝜒)] (31) [Vol-6, Issue-4, Apr- 2019] ISSN: 2349-6495(P) | 2456-1908(O) Figure 3 shows once again how the singularity, perceived as being placed in Og, does not influence the measured distance (the proper radius). The arc bordered by O and P, as previously underlined, is evidently equal to the one bordered by Og and Pg. On the contrary, the “Flat” Coordinate (the Forecast Radius) undergoes a reduction. The segment bordered by B and P represents the Forecast Radius (X) when matter is evenly spread; the segment bordered by Bg and Pg represents the Reduced Forecast Radius (x). III. QUANTIZATION If mass homogeneously fills the 4-Ball with which we identify the Universe (static configuration), by virtue of the symmetry [3] [4], the Energy of a Material Segment, provided with a mass M, can be written as follows: The Linear Mass Density [3] [4] is defined as follows: Moreover, with obvious meaning of the notation, we can immediately write: 𝛿𝑚𝑎𝑥 = 𝛿(0) = 𝑅𝑠 (1 − cos 𝜒𝑚𝑎𝑥 ) (32) If we denote with Mtot the mass of the Ball (that “plays the role” of the 4-Ball with which we identify our Universe), and with M,max the mass contained in the spherical sector with an amplitude equal to 2max (which, as previously remarked, “plays the role” of a Hyper-Spherical Sector), we can write, taking into account (32), the following: 𝑀𝑚𝑎𝑥 𝑀𝑡𝑜𝑡 𝛿𝑚𝑎𝑥 = 𝑅𝑠 = 1 − cos 𝜒𝑚𝑎𝑥 = 𝑀𝑚𝑎𝑥 𝑀𝑡𝑜𝑡 = 2𝐺𝑀𝑚𝑎𝑥 𝑐2 𝛿𝑚𝑎𝑥 𝑅𝑠 = 𝑅𝑠,𝑚𝑎𝑥 (35) 𝐸 = 𝑀𝑐 2 (33) (34) In other terms, the procedure entails a reduction of the radial coordinate of O (actually, the material segment bordered by C and O undergoes a contraction) the size of which is equal to the Schwarzschild radius of Mmax. The scenario is qualitatively portrayed in the following figure, where the singularity (as we can perceive it) coincides with the point Og. 𝑀 𝑅𝑠 (36) 𝐸 𝑀𝑐 2 ̅ 𝑐2 = =𝑀 𝑅𝑠 𝑅𝑠 (37) ̅= 𝑀 By virtue of the foregoing, the (Linear) Energy Density can be defined as follows: 𝐸̅ = If we denote with ∆Rm the (Radial) Quantum of Space [4], the Punctual Mass, denoted by m, is defined as follows: 𝑀 ∆𝑅 𝑅𝑠 𝑚𝑖𝑛 (38) 𝑀𝑐 2 ∆𝑅𝑚𝑖𝑛 = 𝑚𝑐 2 𝑅𝑠 (39) ̅ ∆𝑅𝑚𝑖𝑛 = 𝑚=𝑀 As for the corresponding Energy, by virtue of (37) and (38), we can immediately write: 𝐸𝑚 = 𝐸̅ ∆𝑅𝑚𝑖𝑛 = Let us denote with Mmin the Minimum Linear Mass. The corresponding Energy can be obviously written as follows: (40) 𝐸𝑚𝑖𝑛 = 𝑀𝑚𝑖𝑛 𝑐 2 As for the Minimum Linear Mass Density we have: 𝑀𝑚𝑖𝑛 𝑅𝑠 (41) 𝐸𝑚𝑖𝑛 𝑀𝑚𝑖𝑛 𝑐 2 ̅𝑚𝑖𝑛 𝑐 2 = =𝑀 𝑅𝑠 𝑅𝑠 (42) 𝑀𝑚𝑖𝑛 ∆𝑅𝑚𝑖𝑛 𝑅𝑠 (43) 𝑀𝑚𝑖𝑛 𝑐 2 ∆𝑅𝑚𝑖𝑛 = 𝑚𝑚𝑖𝑛 𝑐 2 𝑅𝑠 (44) ̅𝑚𝑖𝑛 = 𝑀 The Minimum (Linear) Energy Density is clearly provided by the following: 𝐸̅𝑚𝑖𝑛 = The Minimum Punctual Mass, denoted by mmin, is defined as follows: ̅𝑚𝑖𝑛 ∆𝑅𝑚𝑖𝑛 = 𝑚𝑚𝑖𝑛 = 𝑀 Consequently, as for the Energy related to the abovementioned mass, we have: Figure 3. Gravitational Singularity (Particular) www.ijaers.com 𝐸𝑚,𝑚𝑖𝑛 = 𝐸̅𝑚𝑖𝑛 ∆𝑅𝑚𝑖𝑛 = Page | 122 International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.6.4.14 By virtue of (34), we can write the expression for the Minimum Schwarzschild Radius: 𝑅𝑠,𝑚𝑖𝑛 = 2𝐺𝑀𝑚𝑖𝑛 𝑐2 (46) Denoting with h, as usual, the Planck Constant, we can determine the Minimum (Perceived) Energy: 𝐸𝑝ℎ𝑜𝑡𝑜𝑛,𝑚𝑖𝑛 = ℎ𝑐 (47) 𝜆𝑚𝑎𝑥 From (44) and (47), we can easily obtain the expression for the Minimum Punctual Mass: 𝐸𝑝ℎ𝑜𝑡𝑜𝑛,𝑚𝑖𝑛 = ℎ𝑐 𝜆𝑚𝑎𝑥 𝑚𝑚𝑖𝑛 = = ℎ𝑐 = 𝑚𝑚𝑖𝑛 𝑐 2 𝜋𝑅𝑠 ℎ 𝜋𝑐𝑅𝑠 (48) (49) For a (linear) mass to induce a spatial deformation (a radial contraction), the value of the corresponding Schwarzschild Radius must be greater than or equal to the value of the (Radial) Quantum of Space. Consequently, we have: 𝑅𝑠,𝑚𝑎𝑥 ≥ ∆𝑅𝑚𝑖𝑛 (50) If we banally impose that Mmin represents the value of linear mass, still unknown, below which no deformation of spatial lattice (no radial contraction) occurs, we can carry out the following (upper-limit) position: 𝑅𝑠,𝑚𝑖𝑛 = ∆𝑅𝑚𝑖𝑛 (51) 𝑅𝑠 = 𝒩∆𝑅𝑚𝑖𝑛 (52) When mass homogeneously fills the 4-Ball, denoting with 𝒩 an integer (the Number of Radial Quanta), we have: Now, from (43), (45) and (51) we have: 𝑚𝑚𝑖𝑛 = 2 𝑀𝑚𝑖𝑛 𝑀𝑚𝑖𝑛 2𝐺𝑀𝑚𝑖𝑛 ∆𝑅𝑚𝑖𝑛 = 𝑅𝑠,𝑚𝑖𝑛 = 𝑅𝑠 𝑅𝑠 𝑅𝑠 𝑐 2 (53) = ℎ 𝜋𝑐𝑅𝑠 𝑚𝑚𝑖𝑛 = ∆𝑅𝑚𝑖𝑛 1 1 ℏ𝑐 𝑀𝑚𝑖𝑛 = 𝑀𝑚𝑖𝑛 = √ 𝑅𝑠 𝒩 𝒩 𝐺 (57) 𝑅𝑠,𝑚𝑖𝑛 = ℏ𝐺 2𝐺𝑀𝑚𝑖𝑛 = 2√ 3 = 2ℓ𝑃 = ∆𝑅𝑚𝑖𝑛 2 𝑐 𝑐 (58) Finally, from (45) and (56), we obtain the value of the (Radial) Quantum of Space: At this point, we can also carry out a Time Quantization. Taking into account the previous, denoting with tp the socalled Planck Time, we define the Quantum of Time as follows: ∆𝑡𝑚𝑖𝑛 = ℏ𝐺 ∆𝑅𝑚𝑖𝑛 ∆𝑟𝑚𝑖𝑛,𝜋/2 (59) = = ∆𝑡𝑚𝑖𝑛,𝜋/2 = 2√ 5 = 2𝑡𝑃 𝑐 𝑐 𝑐 We can now start concretely building our simple model of (non-rotating and non-charged) “Black Hole”. IV. “BLACK HOLES” 1. Short Introduction Let us suppose that the total available mass may be concentrated in O. Abiding by our model, from (27) and (28), by setting max=/2, we can write the following: 𝜋 𝑟(𝜒) = 𝑅𝑠 cos ( − 𝜒) = 𝑅𝑠 sin 𝜒 2 𝜋 𝑟𝑚𝑎𝑥 = 𝑟 ( ) = 𝑅𝑠 2 (60) 𝑅𝑐 = 𝑋 = 𝑟 (62) 𝑅𝑐,𝑔 = 𝑅𝑐 sin 𝜒 (63) (61) Evidently, the value of the Radial Coordinate (the Reduced Radial Extension) coincides, for any χ, with the one of the Predicted Radius provided by (6): For the Reduced Predicted Radius, we have: 𝑅𝑐,𝑔 = 𝑥 = 𝑋 sin 𝜒 = 𝑅𝑠 sin2 𝜒 (64) The scenario is qualitatively portrayed in Figure 4. From the previous, by virtue of (49), we obtain: 2 2𝐺𝑀𝑚𝑖𝑛 𝑅𝑠 𝑐 2 From (56), taking into account (43) and (52), for the Minimum Punctual Mass we have: (45) Now, taking into account the symmetry, the Maximum Wavelength for a photon can be written as follows: 𝜆𝑚𝑎𝑥 = 𝜋𝑅𝑠 [Vol-6, Issue-4, Apr- 2019] ISSN: 2349-6495(P) | 2456-1908(O) (54) From the previous, taking into account the definition of Reduced Planck Constant, we finally obtain: ℎ 𝑐 ℏ𝑐 2 𝑀𝑚𝑖𝑛 =( ) = 2𝜋 𝐺 𝐺 𝑀𝑚𝑖𝑛 = √ ℏ𝑐 = 𝑀𝑃 𝐺 (55) (56) The previous represents the Minimum Value for Linear Mass. It is worth underlining how this value formally coincides with the one of the so-called Planck Mass, herein denoted with Mp. www.ijaers.com Figure 4. “Black Hole” Page | 123 International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.6.4.14 As for S2, V3 and S3, the Singularity induces the following modifications: 𝑆2,𝑔 (𝜒) = 4𝜋𝑥 2 = 4𝜋𝑅𝑠2 sin4 𝜒 𝑉3,𝑔 (𝜒) = ∫ 𝑅𝑐,𝑔 0 𝑆2,𝑔 (𝜒)𝑑𝑅𝑐,𝑔 𝑥 4 = 4𝜋 ∫ 𝑥 2 𝑑𝑥 = 𝜋𝑥 3 3 0 4 = 𝜋𝑅𝑠3 sin6 𝜒 3 𝑅𝑝 = (66) 0 𝜋 3 𝑅 (𝜒 − sin 𝜒 cos 𝜒 + 2 𝑠 (67) − 2 sin3 𝜒 cos 𝜒) 2. Variable Space-Quantum We want to carry out a quantization of the coordinate r. As shown in (60), this coordinate depends on the angular distance χ: the more we approach the “Singularity”, the more the value of r decreases. However, once again, r does not shorten within space: it shortens together with space, since space itself undergoes a progressive (radial) contraction in approaching the “singularity”. Consequently, we consider a Variable (Radial) SpaceQuantum, denoted with ∆rχ,min, the value of which depends on the angular distance χ. If 𝒩 represents the same integer introduced in (52), we impose the following: 𝑟 = 𝒩∆𝑟𝑚𝑖𝑛,𝜒 (68) 𝑟𝑚𝑎𝑥 = 𝒩∆𝑟𝑚𝑖𝑛,𝜋/2 = 𝑅𝑠 = 𝒩∆𝑅𝑚𝑖𝑛 (69) ∆𝑟𝑚𝑖𝑛,𝜋/2 = ∆𝑅𝑚𝑖𝑛 = 2ℓ𝑃 (70) Consequently, by virtue of (58), we can write: From (68) and (69) we immediately obtain: 𝒩= ∆𝑟𝑚𝑖𝑛,𝜒 𝑅𝑠 𝑅𝑠 = = ∆𝑅𝑚𝑖𝑛 ∆𝑟𝑚𝑖𝑛,𝜋/2 (71) From the foregoing, taking into account (60), we have: ∆𝑟𝑚𝑖𝑛,𝜒 ∆𝑟𝑚𝑖𝑛,𝜒 𝑟 = = = sin 𝜒 ∆𝑟𝑚𝑖𝑛,𝜋/2 𝑅𝑠 ∆𝑅𝑚𝑖𝑛 (72) In the light of the previous relation, we can now introduce the following Non-Dimensional Parameter, which represents nothing but a simple Scale Factor: 𝜂∆𝑟𝑚𝑖𝑛 ∆𝑟𝑚𝑖𝑛,𝜋/2 ∆𝑅𝑚𝑖𝑛 𝑅𝑠 1 = = = = ∆𝑟𝑚𝑖𝑛,𝜒 𝑟 sin 𝜒 ∆𝑟𝑚𝑖𝑛,𝜒 (73) Now, from (58) and (72), we immediately obtain: ∆𝑟𝑚𝑖𝑛,𝜒 = sin 𝜒∆𝑅𝑚𝑖𝑛 = 2 sin 𝜒 √ www.ijaers.com ℎ𝜒 ℎ𝜒 𝑟 2 = ( ) = sin2 𝜒 = ℎ 𝑅𝑠 ℎ𝜋/2 (75) The Variable Quantum of Time is defined as follows: ∆𝑟𝑚𝑖𝑛,𝜒 ℏ𝜒 𝐺 = 2√ 5 = 2𝑡𝑃,𝜒 𝑐 𝑐 (76) By virtue of (59) and (73), from the previous we obtain: ∆𝑡𝑚𝑖𝑛,𝜒 ∆𝑡𝑚𝑖𝑛,𝜒 1 = = sin 𝜒 = ∆𝑡𝑚𝑖𝑛,𝜋/2 𝜂∆𝑟𝑚𝑖𝑛 ∆𝑡𝑚𝑖𝑛 (77) 3. “Gravitational” Mass In case of singularity, a material segment does not undergo any radial reduction (in other terms, it does not shorten within space): as previously remarked, both the segment and the corresponding space undergo a radial contraction (the segment shortens together with space). Consequently, if we denote with M the Mass of a “Test” Material Segment, the (Variable) Linear Mass Density, in case of gravitational singularity, con be defined as follows: ̅= 𝑀 𝑀 𝑟 (78) As for the Mass of a Test Particle (the mass we perceive), by virtue of (71) and (78), we can write, with obvious meaning of the notation, the following: ̅ ∆𝑟𝑚𝑖𝑛,𝜒 = 𝑀 𝑚𝜒 = 𝑀 ∆𝑟𝑚𝑖𝑛,𝜒 ∆𝑟𝑚𝑖𝑛,𝜋/2 =𝑀 = 𝑚𝜋/2 𝑟 𝑅𝑠 (79) From the previous, by virtue of (38) and (52), we have: According to the previous, taking into account (52) and (61), we must have: 𝑟 In other terms, we have been hypothesizing a local variability of the Planck “Constant”. From the previous, taking into account (60), we easily deduce the following: ∆𝑡𝑚𝑖𝑛,𝑔 (𝜒) = ∆𝑡𝑚𝑖𝑛,𝜒 = 𝜒 𝑆3,𝑔 (𝜒) = ∫ 𝑆2,𝑔 (𝜒) 𝑑𝑅𝑝 = 4𝜋𝑅𝑠3 ∫ sin4 𝜒 𝑑𝜒 0 (65) [Vol-6, Issue-4, Apr- 2019] ISSN: 2349-6495(P) | 2456-1908(O) ℏ𝜒 𝐺 (74) ℏ𝐺 = 2√ 3 = 2ℓ𝑃,𝜒 𝑐3 𝑐 𝑚𝜒 = 𝑚𝜋/2 = 𝑀 𝑀 ∆𝑅 = =𝑚 𝑅𝑠 𝑚𝑖𝑛 𝒩 (80) In other terms, thanks to the position in (68) (the meaning of which should now be clearer), the “Gravitational” Mass and the inertial one coincide (as requested by the Equivalence Principle) [9]. 4. Conservation of Energy As elsewhere deduced, the Conservation of Energy for a Free Material Segment can be written as follows [3] [4]: 𝑟 2 𝐸 = 𝑀𝑐 2 = 𝑀𝑟 𝑣 2 + ( ) 𝑀𝑟 𝑐 2 + (𝑀 − 𝑀𝑟 )𝑐 2 𝑅𝑠 (81) 𝑀 = 𝑀𝑟 (82) 𝑟 2 𝐸 = 𝑀𝑐 2 = 𝑀𝑣 2 + ( ) 𝑀𝑐 2 𝑅𝑠 (83) In our case, by virtue of what has been specified in the previous paragraph, bearing in mind the meaning of r, we have to banally impose: As a consequence, for a Test Material Segment, the motion of which is induced by a (Gravitational) Potential, from (81) and (82) we immediately obtain: Page | 124 International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.6.4.14 From the previous relation, taking into account (80), we immediately obtain the Conservation of Energy for a (Free-Falling) Test Particle: 𝐸 𝑀 𝑀 𝑟 2𝑀 2 𝐸𝑚 = = 𝑐2 = 𝑣2 + ( ) 𝑐 𝒩 𝒩 𝒩 𝑅𝑠 𝒩 𝑟 2 𝐸𝑚 = 𝑚𝑐 2 = 𝑚𝑣 2 + ( ) 𝑚𝑐 2 𝑅𝑠 (84) (85) 5. The (Gravitational) Potential and the Coordinate R* From (60) and (85) we can easily deduce: 𝑟 2 𝑣 = 𝑐√1 − ( ) = 𝑐 cos 𝜒 𝑅𝑠 𝑟 𝑣 2 = √1 − ( ) = sin 𝜒 𝑅𝑠 𝑐 (86) (87) 2 1 𝑟 1 1 1 (88) 𝑚𝑣 2 − 𝑚𝑐 2 [1 − ( ) ] = 𝑚𝑣 2 − 𝑚𝑐 2 cos2 𝜒 = 0 2 𝑅𝑠 2 2 2 From (2) we immediately obtain: 𝑐2 = Consequently, we have: 2𝐺𝑀𝑡𝑜𝑡 𝑅𝑠 1 𝐺𝑀𝑡𝑜𝑡 − 𝑐 2 cos2 𝜒 = − cos2 𝜒 2 𝑅𝑠 (89) [Vol-6, Issue-4, Apr- 2019] ISSN: 2349-6495(P) | 2456-1908(O) 6. Speed of a Free-Falling Particle From (2), (97) and (99), we have: 2𝐺𝑀𝑡𝑜𝑡 𝑅𝑠 2𝐺𝑀𝑡𝑜𝑡 (100) 𝑣 = √2𝜙 = √ = 𝑐 √ 2 ∗ = 𝑐 √ ∗ = 𝑣𝑒𝑠𝑐𝑎𝑝𝑒 ∗ 𝑅 𝑅 𝑐 𝑅 (101) 𝑣 = 𝑐 cos 𝜒 The Speed consists of two Components, denoted by vI and vII. We can evidently write: (102) 𝑣 = √𝑣𝐼 2 + 𝑣𝐼𝐼 2 From (100) we can easily deduce: 𝑅𝑠 2𝜙 = 𝑅∗ 𝑐 2 (103) Consequently, vI and vII assume the following forms: 𝑣𝐼 = 𝑐 sin 𝜒 cos 𝜒 = 𝑐 √(1 − 𝑅𝑠 𝑅𝑠 2𝜙 2𝜙 2 (104) √ −( ) ) = 𝑐 𝑅∗ 𝑅∗ 𝑐2 𝑐2 𝑣𝐼𝐼 = 𝑐 cos2  = 𝑐 𝑅𝑠 2𝜙 = 𝑅∗ 𝑐 (105) The components of speed are depicted in Figure 5 (90) Let us introduce a New Coordinate [10], denoted by R*, defined as follows: 𝑅∗ (𝜒) = 𝑅𝑠 cos2 𝜒 (91) Obviously, from the previous we have: 𝑅∗ (0) = 𝑅𝑠 lim 𝑅∗ = +∞ →𝜋/2 𝑅𝑠 cos 𝜒 = √ ∗ 𝑅 sin 𝜒 = √1 − 𝑅𝑠 𝑅∗ (92) (93) Figure 5. Speed of a Free-Falling Particle (94) (95) From (), taking into account (), we obtain: 1 𝐺𝑀𝑡𝑜𝑡 𝑚𝑣 2 − 𝑚=0 2 𝑅∗ (96) Let us define the Pseudo-Newtonian Potential, denoted by φ, as follows: − 𝐺𝑀𝑡𝑜𝑡 1 = − 𝑐 2 cos2 𝜒 = 𝜙(𝜒) 𝑅∗ 2 (97) Evidently, with obvious meaning of the notation, from the previous we have: 1 𝑅∗ = 𝑅𝑠 ⟹ 𝜙(𝜒) = 𝜙(0) = 𝜙𝑚𝑖𝑛 = − 𝑐 2 2 (98) From (), taking into account (), we immediately obtain: 1 𝑚𝑣 2 + 𝜙𝑚 = 0 2 www.ijaers.com (99) Figure 5 shows how, when a test particle approaches the singularity, the value of vI decreases while, on the contrary, the value of vII increases. It is commonly said that, in approaching the singularity, the Space-Like Geodesics become Time-Like, and vice-versa. In our case, the abovementioned interpretation is not correct, since the radial coordinate is nothing but the extension of a material segment, that we perceive as being a material point (the Test Particle). The straight-line segment bordered by C (that evidently coincides with Og) and Pg represents the radial extension of the particle, the one bordered by Og’ and Pg represents the Reduced “Flat Coordinate” (the Radius of the Reduced Circumference). 7. Parameterization We want to find two new coordinates, related to each other, that could “play the role” of Rs and r. Firstly, in the light of (23), we must impose: Page | 125 International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.6.4.14 𝜒 𝑑𝑟𝐾∗ 2 ∗ = 𝑅∗ ∫ √( ) + 𝑟𝐾∗2 𝑑 = 𝑅𝑝,𝑔 𝑝 𝑑 0 (106) [Vol-6, Issue-4, Apr- 2019] ISSN: 2349-6495(P) | 2456-1908(O) In Figure 6 a useful comparison between old and new (Parameterized) Coordinates, once having set K=Rs, is qualitatively displayed. 𝜒 𝑑𝑅𝐾∗ 2 = ∫ √( ) + 𝑅𝐾∗ 𝑑 𝑑 0 Secondly, in the light of (60), we must additionally impose: 𝑟𝐾∗ = 𝑅𝐾∗ sin  (107) 𝑑𝑅𝐾∗ = 2 tan  𝑑 𝑅𝐾∗ (108) From (106) and (107) we easily obtain the following: The general solution of the foregoing, denoting with K an arbitrary constant, is: 𝑅𝐾∗ = 𝐾 cos2 𝜒 (109) Figure 6. Parameterization (K=Rs) From the previous we immediately deduce the underlying noteworthy identity: 𝐾 𝑅𝐾∗ (110) 𝑑𝑅𝐾∗ sin 𝜒 = 2𝐾 𝑑𝜒 cos3 𝜒 (111) sin 𝜒 = √1 − From (109) we have: = 𝑅𝐾∗ sin 𝜒 sin 𝜒 = 𝐾 cos2 𝜒 1 + sin2 𝜒 𝑑𝑟𝐾∗ =𝐾 cos3 𝜒 𝑑𝜒 (112) (113) 𝑅𝑐∗ = 𝑋𝐾∗ = 𝑅𝐾∗ sin 𝜒 = 𝑟𝐾∗ (114) ∗ = 𝑅 ∗ sin 𝜒 𝑅𝑐,𝑔 𝑐 (115) ∗ 𝑅𝑐,𝑔 = 𝑥𝐾∗ = 𝑅𝑐∗ sin 𝜒 = 𝑋𝐾∗ sin 𝜒 = 𝑟𝐾∗ sin 𝜒 = 𝑅𝐾∗ sin2 𝜒 (116) Therefore, as for the Reduced Predicted Radius we have: ∗ ∆𝑅𝑚𝑖𝑛 = 1 ∗ 𝑑𝑅 ∗ = 𝑑𝑅𝑝,𝑔 = √1 + 4 tan2 𝜒 𝐾 www.ijaers.com (122) (123) ∗ 𝑟𝐾∗ = 𝒩∆𝑟𝑚𝑖𝑛,𝜒 From the previous, taking into account (112), we obtain: ∗ ∆𝑟𝑚𝑖𝑛,𝜒 = 𝑟𝐾∗ 𝐾 sin 𝜒 = 𝒩 𝒩 cos2 𝜒 (124) If we set K=Rs, taking into account (68), the foregoing can be written as follows: ∗ ∆𝑟𝑚𝑖𝑛 = ∆𝑟𝑚𝑖𝑛,𝜒 𝑅𝑠 sin 𝜒 1 = 2 𝒩 cos 𝜒 cos2 𝜒 (125) (117) Evidently, by virtue of (121) and (124), we can write: (118) ∗ ∗ ∆𝑟𝑚𝑖𝑛,𝜋/2 = ∆𝑅𝑚𝑖𝑛 According to (106), the Proper Radius is not influenced by the singularity. Therefore, from (109) and (111) we obtain: 1 𝑑𝑅𝐾∗ 2 𝑑𝑅𝐾∗ √1 + 𝑑𝑅𝑝∗ = √( ) + 𝑅𝐾∗ 2 𝑑𝜒 = 𝑑𝜒 4 tan2 𝜒 𝑑𝜒 𝑑𝜒 𝑅𝑠 1 ∆𝑅𝑚𝑖𝑛 = 𝒩 cos2 𝜒 cos2 𝜒 If we set K=Rs, taking into account (52), the foregoing can be written as follows: From the previous, taking into account (109), we obtain: 𝑑𝑥𝐾∗ 𝑑𝑅𝐾∗ = 𝑑𝜒 𝑑𝜒 (121) Obviously, by virtue of (68), we must also impose: In the light of (63), the relation between the Predicted Radiuses with (additional subscript “g”) and without (no additional subscript) the Singularity must be the following: 𝐾 − 𝐾 = 𝑅𝐾∗ − 𝐾 cos2 𝜒 𝑅𝐾∗ 𝐾 1 = 𝒩 𝒩 cos2 𝜒 ∗ ∆𝑅𝑚𝑖𝑛 = As for the Predicted Radius, coherently with (62), we have: 𝑥𝐾∗ = 𝐾 tan2 𝜒 = (120) ∗ ∗ 𝑅𝐾∗ = 𝒩∆𝑅𝑚𝑖𝑛 = 𝒩∆𝑟𝑚𝑖𝑛,𝜋/2 From the previous, taking into account (109), we obtain: From (107) and (109), we have: 𝑟𝐾∗ 8. Parameterized Quantization The parameterization also affects the quantization. Obviously, it is not a real phenomenon. Coherently with the parameterization we have been resorting to, by virtue of (52) we must now impose: (119) (126) From (72), (122) and (124), taking into account the foregoing, we have: ∗ ∆𝑟𝑚𝑖𝑛,𝜒 ∗ ∆𝑟𝑚𝑖𝑛,𝜋/2 = ∗ ∆𝑟𝑚𝑖𝑛,𝜒 ∗ ∆𝑅𝑚𝑖𝑛 = ∆𝑟𝑚𝑖𝑛,𝜒 ∆𝑟𝑚𝑖𝑛,𝜒 = = sin 𝜒 ∆𝑅𝑚𝑖𝑛 ∆𝑟𝑚𝑖𝑛,𝜋/2 (127) In the light of the previous, resorting to (110), we can now introduce the new following Parameterized Scale Factor: ∗ 𝜂∆𝑟 = 𝑚𝑖𝑛 ∗ ∆𝑅𝑚𝑖𝑛 ∆𝑅𝑚𝑖𝑛 1 = = = ∗ ∆𝑟𝑚𝑖𝑛,𝜒 ∆𝑟𝑚𝑖𝑛,𝜒 sin 𝜒 1 √1 − 𝐾 𝑅𝐾∗ (128) Page | 126 International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.6.4.14 [Vol-6, Issue-4, Apr- 2019] ISSN: 2349-6495(P) | 2456-1908(O) The Parameterized Quantum of Time is defined as follows: √𝐶(𝑅𝐾∗ ) = 𝑅𝐾∗ ∗ ∗ (𝜒) = ∆𝑡𝑚𝑖𝑛,𝜒 = ∆𝑡𝑚𝑖𝑛,𝑔 ∗ ∆𝑟𝑚𝑖𝑛,𝜒 (129) 𝑐 Thanks to the previous, (139) can be written as follows: 𝑑𝑠 2 = 𝐴∗ (𝑅𝐾∗ )𝑐 2 𝑑𝑡 2 − 𝐵∗ (𝑅𝐾∗ )𝑑𝑟 2 Taking into account (77), (127) and (128), from the previous we obtain: ∗ ∆𝑡𝑚𝑖𝑛,𝜒 ∗ ∆𝑡𝑚𝑖𝑛,𝜋/2 = ∗ ∆𝑟𝑚𝑖𝑛,𝜒 ∗ ∆𝑅𝑚𝑖𝑛 ∆𝑡𝑚𝑖𝑛,𝜒 ∆𝑡𝑚𝑖𝑛,𝜒 = = = sin 𝜒 ∆𝑡𝑚𝑖𝑛 ∆𝑡𝑚𝑖𝑛,𝜋/2 = 1 ∗ 𝜂∆𝑟 𝑚𝑖𝑛 (130) 𝐾 = √1 − ∗ 𝑅𝐾 It is worth highlighting how, from (73), (87) and (128), denoting with γ the so-called Relativistic Factor, we have: ∗ 𝜂∆𝑟 𝑚𝑖𝑛 = 𝜂∆𝑟𝑚𝑖𝑛 1 = = sin 𝜒 1 √ 1 − (𝑣 ) 𝑐 2 =𝛾 𝑑𝑠 ∗2 = 𝑐 2 𝑑𝑡 ∗2 − 𝑑𝑅𝑝∗2 − 𝑅𝑐∗2 (𝑑𝜃 2 + sin2 𝜃 𝑑𝜑 2 ) (132) 𝑋𝐾∗ 𝑟𝐾∗ lim ∗ = lim ∗ = lim sin 𝜒 = 1 →𝜋/2 𝑅𝐾 →𝜋/2 𝑅𝐾 →𝜋/2 (133) 𝑅𝑐∗ ≅ 𝑅𝐾∗ (134) Bearing in mind the definition of Predicted Radius provided by (114), we have: Consequently, far from the origin, Predicted Radius and Radial Coordinate are interchangeable. We can write: Now, we evidently have: lim √1 + 1 =1 4 tan2 𝜒 𝑑𝑅𝑝∗ ≅ 𝑑𝑅𝐾∗ (136) 𝑑𝑠 ∗2 = 𝑐 2 𝑑𝑡 ∗2 − 𝑑𝑅𝐾∗2 − 𝑅𝐾∗2 (𝑑𝜃 2 + sin2 𝜃 𝑑𝜑 2 ) (137) Finally, far from the origin, (132) becomes: It is fundamental to underline how the approximation in (134) prevents the Predicted Radius from assuming a null value. In detail, by virtue of (109), we have: = 𝑅𝑐∗ (0) = 𝑅𝐾∗ (0) =𝐾 (138) 2. Schwarzschild-Like Metric: Conventional Derivation As is well known, the General Static, Spherically (and Time) Symmetric Solution can now be written as follows: 𝑑𝑠 2 = 𝐴(𝑅𝐾∗ )𝑐 2 𝑑𝑡 2 − 𝐵(𝑅𝐾∗ )𝑑𝑅𝐾∗2 − 𝐶(𝑅𝐾∗ )(𝑑𝜃 2 + sin2 𝜃 𝑑𝜑 2 ) 𝐴(𝑅𝐾∗ ), 𝐵(𝑅𝐾∗ ), 𝐶(𝑅𝐾∗ ) > 0 Let us carry out the following position [11]: www.ijaers.com 𝐴∗ (𝑅𝐾∗ ), 𝐵∗ (𝑅𝐾∗ ) (139) >0 (141) As for the Metric Tensor, from (141) we obtain: 𝐴∗ (𝑅𝐾∗ ) 0 0 0 −𝐵∗ (𝑅𝐾∗ ) 0 𝑔𝑖𝑗 = 0 0 −𝑅𝐾∗2 0 0 [ 0 𝑔𝑖𝑗 = 1 𝐴∗ (𝑅𝐾∗ ) 0 0 [ − 0 1 𝐵∗ (𝑅𝐾∗ ) 0 0 0 0 − 0 0 0 − 𝑅𝐾∗2 sin2 𝜃] 0 0 1 𝑅𝐾∗2 0 (142) 0 − (143) 0 1 𝑅𝐾∗2 sin2 𝜃] Let’s deduce the Christoffel Symbols. Generally, we have: 1 𝜕𝑔ℎ𝑖 𝜕𝑔ℎ𝑗 𝜕𝑔𝑖𝑗 𝛤𝑖𝑗𝑘 = 𝑔𝑘ℎ ( 𝑗 + − ℎ) 2 𝜕𝑥 𝜕𝑥 𝑖 𝜕𝑥 (144) The indexes run from 0 to 3. Clearly, 0 stands for t, 1 for r, 2 for θ, and 3 for φ. Setting k=0, from (142), (143) and (144), we obtain: 0 0 𝛤01 = 𝛤10 = 1 𝑑𝐴∗ 2𝐴∗ 𝑑𝑅𝐾∗ (145) All the other symbols (if k=0) vanish. Setting k=1, from (142), (143) and (144), we obtain: 1 𝛤00 = (135) Far from the origin, therefore, by virtue of (119), Proper Radius and Radial Coordinate are interchangeable: ∗ (𝜒) 𝑅𝑐,𝑚𝑖𝑛 − 𝑅𝐾∗2 (𝑑𝜃 2 + sin2 𝜃 𝑑𝜑 2 ) (131) V. METRICS 1. Initial “Flat” Metric (no singularity) We can immediately write the following general metric: →𝜋/2 (140) 1 𝛤11 = 1 𝑑𝐴∗ 2𝐵∗ 𝑑𝑅𝐾∗ 1 𝑑𝐵 ∗ 𝑅𝐾∗ 𝑅𝐾∗ 1 1 2 ∗ , 𝛤12 = − ∗ , 𝛤13 = − ∗ sin 𝜃 ∗ 2𝐵 𝑑𝑅𝐾 𝐵 𝐵 (146) All the other symbols (if k=1) vanish. Setting k=2, from (142), (143) and (144), we obtain: 2 2 𝛤12 = 𝛤21 = 1 , 𝛤 2 = − sin 𝜃 cos 𝜃 𝑅𝐾∗ 33 (147) All the other symbols (if k=2) vanish. Setting k=3, from (142), (143) and (144), we obtain: 3 3 𝛤13 = 𝛤31 = 1 1 3 , 𝛤 3 = 𝛤23 = 𝑅𝐾∗ 23 tan 𝜃 (148) All the other symbols (if k=3) vanish. Let’s now deduce the components of the Ricci Tensor. Generally, with obvious meaning of the notation, we have: 𝑅𝑖𝑗 = 𝜕𝛤𝑖𝑘𝑘 𝜕𝛤𝑖𝑗𝑘 − + 𝛤𝑖𝑘𝑙 𝛤𝑗𝑙𝑘 − 𝛤𝑖𝑗𝑙 𝛤𝑘𝑙𝑘 𝜕𝑥 𝑗 𝜕𝑥 𝑘 (149) By means of some simple mathematical passages, omitted for brevity, we obtain all the non-vanishing components: 𝑅00 = − 1 𝑑 2 𝐴∗ 1 𝑑𝐴∗ 1 𝑑𝐴∗ 1 𝑑𝐵∗ + ∗ ∗ ( ∗ ∗ + ∗ ∗) ∗2 ∗ 2𝐵 𝑑𝑅𝐾 4𝐵 𝑑𝑅𝐾 𝐴 𝑑𝑅𝐾 𝐵 𝑑𝑅𝐾 1 𝑑𝐴∗ − ∗ ∗ ∗ 𝑅𝐾 𝐵 𝑑𝑅𝐾 (150) Page | 127 International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.6.4.14 𝑅11 = 1 𝑑2 𝐴∗ 1 𝑑𝐴∗ 1 𝑑𝐴∗ 1 𝑑𝐵∗ − ∗ ∗ ( ∗ ∗ + ∗ ∗) ∗2 ∗ 2𝐴 𝑑𝑅𝐾 4𝐴 𝑑𝑅𝐾 𝐴 𝑑𝑅𝐾 𝐵 𝑑𝑅𝐾 𝑅22 1 𝑑𝐵∗ ∗ ∗ 𝑅𝐾 𝐵 𝑑𝑅𝐾∗ − 1 𝑅𝐾∗ 1 𝑑𝐴∗ 1 𝑑𝐵 ∗ = ∗ + ∗( ∗ ∗ − ∗ ∗)−1 𝐵 2𝐵 𝐴 𝑑𝑅𝐾 𝐵 𝑑𝑅𝐾 𝑅33 = sin2 𝜃 [ (153) sin2 𝜃 𝑅22 If we denote with R the Ricci Scalar and with Tij the generic component of the Stress-Energy Tensor, the Einstein Field Equations [9] [12] can be written as follows: 1 8𝜋𝐺 𝑅𝑖𝑗 − 𝑅𝑔𝑖𝑗 = 4 𝑇𝑖𝑗 2 𝑐 (154) If we impose that, outside the mass that produces the field, there is the “absolute nothing” (neither matter nor energy), the first member of (154), that represents the so-called Einstein Tensor, must vanish. Consequently, we have: 1 𝑅𝑖𝑗 − 𝑅𝑔𝑖𝑗 = 0 2 From (150), (151) and (156), we immediately obtain: − ∗ ∗ ∗ 1 𝑑 𝐴 1 𝑑𝐴 1 𝑑𝐴 1 𝑑𝐵 + ( + ) 2𝐴∗ 𝐵 ∗ 𝑑𝑅𝐾∗2 4𝐴∗ 𝐵∗ 𝑑𝑅𝐾∗ 𝐴∗ 𝑑𝑅𝐾∗ 𝐵 ∗ 𝑑𝑅𝐾∗ 𝑑𝐴∗ 1 − ∗ ∗ ∗ ∗ =0 𝑅𝐾 𝐴 𝐵 𝑑𝑅𝐾 1 𝑑𝐴∗ 1 𝑑𝐴∗ 1 𝑑𝐵∗ 1 𝑑2 𝐴∗ − ( + ) 2𝐴∗ 𝐵∗ 𝑑𝑅𝐾∗2 4𝐴∗ 𝐵∗ 𝑑𝑅𝐾∗ 𝐴∗ 𝑑𝑅𝐾∗ 𝐵∗ 𝑑𝑅𝐾∗ − 𝑑𝐵 ∗ =0 ∗ ∗ 2 𝑑𝑅 ∗ 𝑅𝐾 𝐵 𝐾 1 𝐵∗ =− 𝐵∗ = 𝑑𝐴∗ 𝐴∗ 𝐾1 𝐴∗ (158) (159) (160) 𝑟→∞ From (160), taking into account (161), we obtain: 𝐵∗ = 1 𝐴∗ 𝑔00 𝑔11 = −1 (162) (163) From (152) and (156) we have: 𝐴∗ + 𝑅𝐾∗ 𝐴∗ 2 [ 𝑑𝐴∗ 1 𝑑 1 − 𝐴∗ ∗ ( ∗ )] − 1 = 0 𝐴∗ 𝑑𝑅𝐾∗ 𝑑𝑅𝐾 𝐴 𝑑𝐴∗ 𝑑 ∗ ∗ 𝐴∗ + 𝑅𝐾∗ ∗ − 1 = 𝑅𝐾 (𝑟𝐴 ) − 1 = 0 𝑑 𝑑𝑅𝐾 www.ijaers.com 𝑔00 = (1 + 𝜙 2 ) 𝑐2 (167) The value of K2 can be directly deduced by resorting to the so-called Weak Field Approximation: (1 + 𝜙 2 𝜙 ) ≅1+2 2 𝑐2 𝑐 (168) Far from the source from (97), (110) and (168) we have: 𝐴∗ = 𝑔00 ≅ 1 + 2 𝜙 𝐾 = 1 − cos2 𝜒 = 1 − ∗ 𝑐2 𝑅𝐾 (169) If we set K=Rs, the foregoing can be written as follows: 𝐴∗ = 1 − 2𝐺𝑀𝑡𝑜𝑡 𝑅𝑠 =1− ∗ 𝑐 2 𝑅∗ 𝑅 (170) From (162) and (169), we have: 𝐵∗ = 1 1− 𝐾 𝑅𝐾∗ (171) 𝑅𝑠 𝑅𝐾∗ (172) If we set K=Rs, the previous can be written as follows: 𝐵∗ = 1 1− At this point, the Schwarzschild-Like Metric can be immediately written by substituting into (141) the values of A* and B* deduced, respectively, in (169) and (171). 3. Schwarzschild-Like Metric: Alternative Derivation According to our model, taking into account (106) and (115), from (137) we can deduce, in case of Singularity, the following solution: 𝑑𝑠𝑔∗2 = 𝑐 2 𝑑𝑡 ∗ 2 − 𝑑𝑅𝐾∗2 − 𝑅𝐾∗ 2 sin2 𝜒 (𝑑𝜃 2 + sin2 𝜃𝑑𝜑 2 ) The value of the constant K1 can be deduced by imposing that, at infinity, the Flat Metric in (137) must be recovered. In other terms, we must impose the following condition: lim 𝐴∗ (𝑅𝐾∗ ) = lim 𝐵 ∗ (𝑅𝐾∗ ) = 1 (161) ∗ 𝑅𝐾 →∞ (166) (157) From (157) and (158), we have: 𝑑𝐵 ∗ 𝐾2 𝑅𝐾∗ Now, if ϕ represents the Gravitational Potential, for an arbitrary metric we have: (155) From (155), exploiting the fact that the Einstein Tensor and the Ricci Tensor are trace-reverse of each other, we have: 𝑅𝑖𝑗 = 0 (156) 2 ∗ 𝐴∗ = 1 + (152) 1 𝑅𝐾∗ 1 𝑑𝐴∗ 1 𝑑𝐵∗ + ( − ) − 1] 𝐵∗ 2𝐵∗ 𝐴∗ 𝑑𝑅𝐾∗ 𝐵∗ 𝑑𝑅𝐾∗ = (151) [Vol-6, Issue-4, Apr- 2019] ISSN: 2349-6495(P) | 2456-1908(O) (164) (165) (173) The previous represents an analytic solution, built without taking into account the modified value of the SpaceQuantum. The above-mentioned condition is expressed by means of g00, the value of which is manifestly unitary: Space and Time Quanta, in fact, are related to each other by means of (129). Obviously, t* represents the proper time (the time measured by an observer ideally placed at infinity, where the singularity has no longer effect). We can rewrite (173) in the underlying form: 𝑑𝑠𝑔∗2 | 𝑔00 =1 ∗2 = 𝑐 2 𝑑𝑡 ∗ 2 − 𝑑𝑅𝑝,𝑔 | 𝑔00 =1 ∗2 (𝑑𝜃 2 − 𝑅𝑐,𝑔 | 𝑔00 =1 (174) + sin2 𝜃 𝑑𝜑 2 ) In other terms, we have carried out the following positions: ∗ | 𝑅𝑐,𝑔 = 𝑅𝑐∗ sin 𝜒 = 𝑅𝐾∗ sin 𝜒 (175) 𝑔00 =1 ∗ | 𝑑𝑅𝑝,𝑔 𝑔00 =1 = 𝑑𝑅𝑝∗ = 𝑑𝑅𝐾∗ (176) Now, from (130) we immediately obtain: Page | 128 International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.6.4.14 ∗ ∆𝑡𝑚𝑖𝑛,𝜋/2 ∗ ∗ (𝜒) = ∆𝑡𝑚𝑖𝑛,𝜒 = ∆𝑡𝑚𝑖𝑛,𝑔 ∗ 𝜂∆𝑟 𝑚𝑖𝑛 = 𝐾 ∗ = ∆𝑡𝑚𝑖𝑛 √1 − ∗ 𝑅𝐾 ∗ ∆𝑡𝑚𝑖𝑛 ∗ 𝜂∆𝑟𝑚𝑖𝑛 (177) In the light of the previous, we can write: 𝑑𝑡𝑔∗ = ∗ 𝑑𝑡𝜋/2 ∗ 𝜂∆𝑟 𝑚𝑖𝑛 = 𝑑𝑡 ∗ ∗ 𝜂∆𝑟 𝑚𝑖𝑛 = 𝑑𝑡 ∗ √1 − 𝐾 𝑅𝐾∗ (178) From (175), taking into account (120) and (127), we have: ∗ 𝑅𝑐,𝑔 | 𝑔00 =1 (179) = 𝒩 sin 𝜒 ∆𝑟𝑚𝑖𝑛,𝜋/2 = 𝒩∆𝑟𝑚𝑖𝑛,𝜒 Exploiting (134) and (175), we can temporarily introduce to following Non-Dimensional (Normalized) Coordinates: 𝑅̅𝑐∗ = 𝑅𝑐∗ 𝑅𝑐∗ = ∗ =𝒩 ∗ ∆𝑅𝑚𝑖𝑛 ∆𝑟𝑚𝑖𝑛,𝜋/2 ∗ | 𝑅̅𝑐,𝑔 𝑔00 =1 = (180) ∗ 𝑅𝑐,𝑔 =𝒩 ∗ ∆𝑟𝑚𝑖𝑛,𝜒 (181) Evidently, the value of the Predicted Radius, as long as it is expressed in terms of Space-Quanta, can be regarded as being constant. Consequently, from (180) and (181) we can banally write: (182) ∗ | 𝑅̅𝑐∗ = 𝑅̅𝑐,𝑔 𝑔00 =1 Now, if we replace dt* with dtg*, taking into account (178), we obtain a new value for g00: 𝑔00 = 1 − 𝐾 1 = ∗2 𝑅𝐾∗ 𝜂∆𝑟 𝑚𝑖𝑛 𝑑𝑠𝑔∗2 | 𝑔00 =1/𝜂 2 = (1 − − 𝐾 ∗2 ) 𝑐 2 𝑑𝑡 ∗ 2 − 𝑑𝑅𝑝,𝑔 | 𝑔00 =1/𝜂 2 𝑅𝐾∗ ∗2 (𝑑𝜃 2 𝑅𝑐,𝑔 | 𝑔00 =1/𝜂 2 2 + sin 𝜃 𝑑𝜑 (184) 2) From (175), (176) and (183), we can write, with obvious meaning of the notation, the following: ∗ ∗ | ∗ 𝑅𝑐,𝑔 = 𝜂∆𝑟 𝑅∗ | = 𝑅𝐾∗ = 𝑅𝑐,𝑔 (185) 𝑚𝑖𝑛 𝑐,𝑔 𝑔00 =1 𝑔00 =1/𝜂 2 𝑑𝑅𝑝∗ |𝑔 00 =1/𝜂 2 ∗ ∗ | = 𝜂∆𝑟 𝑑𝑅𝑝,𝑔 𝑚𝑖𝑛 = 𝑑𝑅𝐾∗ 𝑔00 =1 𝐾 √1 − ∗ 𝑅𝐾 = 𝜂∆𝑟𝑚𝑖𝑛 𝑑𝑅𝐾∗ www.ijaers.com ∗ 𝑅𝑝,𝑔 =∫ = 𝑑𝑅𝑝∗ 𝑑𝑅𝐾∗ √1 − 𝐾 𝑅𝐾∗ =∫ √(𝑅𝐾∗ − 𝐾) + 𝐾 2 √𝑅𝐾∗ − 𝐾 𝑑(𝑅𝐾∗ − 𝐾) (188) 2 = 2 ∫ √(√𝑅𝐾∗ − 𝐾) + (√𝐾) 𝑑(√𝑅𝐾∗ − 𝐾) We have just found an integral of the following kind: 2 ∫ √𝑦 2 + (√𝐾) 𝑑𝑦 = 𝐾 ln (𝑦 + √𝑦 2 + 𝐾) 2 (189) 𝑦 + √𝑦 ′ 2 + 𝐾 + 𝐶𝐾 2 Consequently, from (188) and (189) we have: ∗ =∫ 𝑅𝑝,𝑔 𝑑𝑅𝐾∗ 𝐾 √1 − ∗ 𝑅𝐾 = ln(√𝑅𝐾∗ + √𝑅𝐾∗ − 𝐾) (190) + √𝑅𝐾∗ (𝑅𝐾∗ − 𝐾) + 𝐶𝐾 As for the constant Ck we have: 𝑅𝑝∗ (𝐾) = 0 ⟹ 𝐶𝐾 = −𝐾 ln √𝐾 (191) √𝑅𝐾∗ − 𝐾 + √𝑅𝐾∗ 𝑅𝑝,𝑔 = 𝐾 ln ( ) + √𝑅𝐾∗ (𝑅𝐾∗ − 𝐾) √𝐾 (192) Finally, from (190) and (191) we have: The previous, by virtue of (117), can be written as follows: √𝑥𝐾∗ + √𝑥𝐾∗ + 𝐾 𝑅𝑝,𝑔 = 𝐾 ln ( ) + √𝑥𝐾∗ (𝑥𝐾∗ + 𝐾) √𝐾 (193) 𝑥𝐾∗ 𝑅𝐾∗ − 𝐾 = lim =1 ∗ ∗ →𝜋/2 𝑅𝐾 →𝜋/2 𝑅𝐾 (194) 4. Generalization Taking into account (117), we have: lim By virtue of the previous, we can write: 𝑥𝐾∗𝑎 𝑥𝐾∗𝑎 + 𝐾 𝑎 = lim ∗𝑎 →𝜋/2 𝑅𝐾 →𝜋/2 𝑅𝐾∗𝑎 lim (195) (𝑅𝐾∗ − 𝐾)𝑎 + 𝐾 𝑎 =1 →𝜋/2 𝑅𝐾∗𝑎 = lim (186) We can finally write the so-called Droste Solution [13]: 𝐾 𝑑𝑅𝐾∗2 2 ∗2 𝑑𝑠𝑔∗2 | − 1 = (1 − ∗ ) 𝑐 𝑑𝑡 𝑔00 = 2 𝐾 𝑅𝐾 𝜂 1− ∗ (187) 𝑅𝐾 ∗2 2 2) ∗ 2 (𝑑𝜃 2 − 𝑅𝐾 + sin 𝜃𝑑𝜑 = 𝑑𝑠𝑔 𝑅𝐾∗ > 𝐾 The Singularity is not a point, but a 2-Sphere characterized by a radius equal to K. However, this strange phenomenon is anything but real, since it is clearly and exclusively caused by the approximation in (134). According to the new scenario, the value of the Escape Speed is now provided by (104): it is easy to verify how this value formally coincides with the one that can be derived by resorting to the Geodesic Equation. As for the New Proper Radius, we have: (183) The value of g00 reveals how we measure time (which is still considered as being absolute) and space and nothing else. In other words, we have simply changed the Units of Measurement (we have modified the Scale Factor). By virtue of (183), we can rewrite (173) by changing the Scale Parameter: [Vol-6, Issue-4, Apr- 2019] ISSN: 2349-6495(P) | 2456-1908(O) 𝑎 ∈ ℛ+ Therefore, far from the source, we obtain: 𝑎 𝑅𝐾∗𝑎 ≅ (𝑅𝐾∗ − 𝐾)𝑎 + 𝐾 𝑎 𝑎 (196) ∗ 𝑅𝐾∗ ≅ √(𝑅𝐾∗ − 𝐾)𝑎 + 𝐾 𝑎 = √𝑥𝐾∗𝑎 + 𝐾 𝑎 = 𝑅𝐾,𝑎 (197) Evidently, moreover, we have: ∗ 𝜒 = 0 ⟹ 𝑅𝐾,𝑎 =𝐾 (198) Page | 129 International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.6.4.14 ∗ ∗ 1−𝑎 𝑑𝑅𝐾,𝑎 𝑎 𝑎 ∗𝑎−1 𝑑𝑥𝐾 = (𝑥∗𝑎 𝑥𝐾 >0 𝐾 +𝐾 ) 𝑑𝜒 𝑑𝜒 0<𝜒< (199) 𝜋 2 From (195), (198) and (199) we deduce how the New Parametric Coordinate defined in (197) and the one defined in (109) are fully interchangeable (since they behave exactly the same way). In other terms, we have: (200) ∗ 𝑅𝐾,𝑎 ≅ 𝑅𝐾∗ Taking into account the foregoing, by setting a=1 in (197), from (187) we obtain: 𝑑𝑠𝑔∗2 = (1 − 𝑑𝑠𝑔∗2 = 𝐾 𝑑𝑥𝐾∗2 ) 𝑐 2 𝑑𝑡 ∗ 2 − ∗ 𝐾 𝑥𝐾 + 𝐾 1− ∗ 𝑥𝐾 + 𝐾 − (𝑥𝐾∗ + 𝐾)2 (𝑑𝜃 2 + sin2 𝜃𝑑𝜑 2 ) (201) 𝑥𝐾∗ > 0 𝑐 2 𝑑𝑡 ∗ 2 𝐾 − (1 + ∗ ) 𝑑𝑥𝐾∗2 𝐾 𝑥 𝐾 1+ ∗ 𝑥𝐾 − (𝑥𝐾∗ + 𝐾)2 (𝑑𝜃 2 + sin2 𝜃𝑑𝜑 2 ) (202) 𝑥𝐾∗ > 0 The previous represents the original form of the so-called Brillouin Solution [14]. From (187), by setting a=3 in (197), we have: 3 ∗ 𝑅𝐾,3 = √𝑥𝐾∗3 + 𝐾 3 (203) By substituting the previous into (187), we can finally obtain the real Schwarzschild Form [15]. VI. GRAVITATIONAL REDSHIFT If we impose the Conservation of Energy, we can write, with obvious meaning of the notation, the following: 𝐸𝑝ℎ𝑜𝑡𝑜𝑛,𝜒 = ℎ𝜒 𝜈𝜒 = ℎ𝜋/2 𝜈𝜋/2 = 𝐸𝑝ℎ𝑜𝑡𝑜𝑛,𝜋/2 (204) ℎ𝜒 𝜈𝜋/2 = = sin2 𝜒 ℎ𝜋/2 𝜈𝜒 (205) 𝑐 = 𝜆𝜒 𝜈𝜒 = 𝜆𝜋/2 𝜈𝜋/2 (206) 𝜆𝜋/2 ℎ𝜋/2 1 = = = 𝜂𝜆 sin2 𝜒 ℎ𝜒 𝜆𝜒 (207) 𝜆𝜋/2 − 𝜆𝜒 𝜆𝜋/2 = − 1 = 𝜂𝜆 − 1 𝜆𝜒 𝜆𝜒 (208) [Vol-6, Issue-4, Apr- 2019] ISSN: 2349-6495(P) | 2456-1908(O) ∗ 𝜂∆𝑟 𝜂∆𝑟𝑚𝑖𝑛 𝑚𝑖𝑛 = lim = lim sin 𝜒 →𝜋/2 𝜂𝜆 →𝜋/2 𝜂𝜆 →𝜋/2 lim (209) 𝑣 = lim √1 − ( ) = 1 𝑣→0 𝑐 2 Consequently, far from the source, we can write: 𝑧 ≅ 𝜂∆𝑟𝑚𝑖𝑛 − 1 = 1 −1 sin 𝜒 (210) From the foregoing, taking into account (110), we have: 𝑧= 𝜆∞ − 𝜆𝑅 ∗ = 𝜆𝑅 ∗ 1 √1 − 𝐾 𝑅𝐾∗ −1 (211) If we set K=Rs, according to (2) and (109), the previous can be written in the following well-known form: 1 𝑧= −1 (212) 𝑡𝑜𝑡 √1 − 2𝐺𝑀 𝑐2𝑅∗ VII. BRIEF CONCLUSIONS The coordinate deduced in (109), which appears both in the metrics and at the denominator of the pseudo-Newtonian relation we have obtained for the gravitational potential, does not represent a real distance nor a real radius of curvature. In fact, it is clear how the expression of the above-mentioned coordinate arises from a banal parameterization, by means of which we are able to write the initial “Flat” Metric in (137). From the latter, it is possible to derive an infinite set of Schwarzschild-like Metrics, suitable for non-rotating and non-charged “Black Holes”, without resorting to Relativity. According to the simple model herein proposed, the minimum value for the coordinate in (109) equates the Schwarzschild Radius. When this coordinate equates the Schwarzschild radius, both the Proper Radius and the Forecast Radius are equal to zero: in other terms, we are exactly placed in correspondence of the “Singularity”. From the previous, by virtue of (75), we obtain: If we impose the Speed of Light Constancy, we have: The two foregoing relations allows to immediately define a New Scale Parameter: According to the definition of Gravitational Redshift [9], usually denoted by z, from the previous we have: 𝑧= From (131) and (207) we have: www.ijaers.com ACKNOWLEDGEMENTS I would like to dedicate this paper to my very little friend Carmine Vasco Costa, sincerely hoping he may preserve his great interest, already astonishingly deep despite his age, towards mathematics and physics. REFERENCES [1] Harrison, E.R. (1967). Classification of Uniform Cosmological Models. Monthly Notices of the Royal Astronomical Society, 137, 69-79. https://doi.org/10.1093/mnras/137.1.69 [2] Cataldo, C. (2016). Faster than Light: again on the Lorentz Transformations. Applied Physics Research, 8(6), 17-24. http://dx.doi.org/10.5539/apr.v8n6p17 [3] Cataldo, C. (2016). From the Oscillating Universe to Relativistic Energy: a Review. 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Sitzungsber. der Deutschen Akad. der Wiss. zu Berlin, 189-196 (On the Gravitational Field of a Mass Point according to Einstein’s Theory, translat. by Antoci and Loinger, 1999). Retrieved from: http://zelmanov.pteponline.com/papers/zj-2008-03.pdf Page | 131