1#q Copyright 1995 by Michael E. Heffron. All rights reserved. Page  Subatomic Black Holes 1995 by Michael E. Heffron (102636.3314@compuserve.com) There is a black hole at the nucleus of every atom. The speed of light is not the speed limit of the universe. All known subatomic particles routinely travel faster than the speed of light. I. Introduction An object attempting to permanently escape from the gravitational pull of a body X of mass mX must meet or exceed the escape velocity ______ / 2GXmX v = _ / ------, (1) |/ r where GX is the gravitational constant of the body; and r is the radius from which the attempted escape begins, measured from the center-of-mass of the body to the center-of-mass of the escapee.[1] Squaring both sides of Eq.(1), we find that 2GXmX v2 = -----. (2) r Rearranging Eq.(2) to solve for r yields the equation 2GXmX r = -----. (3) v2 The Schwarzchild radius,[2] also known as the "event horizon" of a black hole, 2GXmX rS = -----, (4) c2 is simply Eq.(3) written to solve for the radius that corresponds to the velocity of light c. The theory is that nothing can escape from within the Schwarzchild radius, because nothing can travel faster than the speed of light--hence the name "black hole." II. Subatomic black holes Given the gravitational constant of the proton,[3] Gp = 1.5141720 x 1029 m3/kg*s2; the mass of the proton, mp = 1.6726231 x 10-27 kg; and the velocity of light, c = 2.99792458 x 108 m/s, we find, from Eq.(4), that the Schwarzchild radius of the hydrogen atom is 5.6358818 x 10-15 m. The radius of the nucleus of any given atom is R (1.3 x 10-15 m)A1/3, (5) where A is the number of nucleons in the nucleus.[4] Thus, the radius of the nucleus of the hydrogen atom (A = 1) is about 1.3 x 10-15 m. Since the Schwarzchild radius of the hydrogen atom is much larger than the radius of the nucleus, subatomic particles cannot escape from the nucleus unless they travel faster than the speed of light. From Eq.(1), we find that the escape velocity from the surface of the hydrogen nucleus is about 6.2 x 108 m/s (roughly twice the speed of light). III. More subatomic black holes Chemical ionization energies[5] reveal that the potential energy of a single electron in orbit about a nucleus containing some number of protons Z is EZ = mev2 = Z2(4.3597482 x 10-18 kg*m2/s2). (6) In his dissertation of 1924, French physicist Louis de Broglie suggested that any given particle of mass m in motion with velocity v has a wavelength l of h l = --, (7) mv where h is Plancks constant. In 1927, de Broglies hypothesis was confirmed by the following two independent experiments: 1. Clinton S. Davisson and Lester H. Germer diffracted slow electrons using a single nickel crystal; and 2. George P. Thomson and A. Reid diffracted fast electrons using a thin celluloid film.[6] We can derive Plancks constant from the hydrogen atom, as follows: h = 2pr0mev0, (8) where r0 = 5.29177249 x 10-11 m is the "Bohr radius"; me = 9.1093897 x 10-31 kg is the mass of the electron; and v0 = 2.1876914 x 106 m/s is the velocity of the electron in orbit at the Bohr radius of the hydrogen atom. Substituting the equivalent of h from Eq.(8) into Eq.(7), then solving for me and v0, we find that 2pr0mev0 l0 = -------- = 2pr0, (9) mev0 where l0 is the wavelength of an electron in orbit at the Bohr radius of the hydrogen atom. Substituting the equivalent of l0 from Eq.(9) back into Eq.(7), then again solving for me and v0, we find that h 2pr0 = ----. (10) mev0 Plancks constant determines the wavelength of the particle regardless of any atoms it may interact with. Since me is constant, there is only one variable on each side of the equation. Thus, we can generalize Eq.(10) to solve for the radius of orbit r of any given electron of velocity v. Generalizing, and dividing both sides of Eq.(10) by 2p, we find that h r = -----. (11) 2pmev Dividing the sides of Eq.(6) by the mass of the single electron me, we find that the variable velocity squared is v2 = Z2(4.7859937 x 1012 m2/s2). (12) Multiplying Eq.(11) by Eq.(12), we find that hZ2(4.7859937 x 1012 m2/s2) v2r = ---------------------------. (13) 2pmev Since we know, from the square root of Eq.(12), that the variable velocity of the electron is v = Z(2.1876914 x 106 m/s), we find that hZ2(4.7859937 x 1012 m2/s2) v2r = ---------------------------. (14) 2pmeZ(2.1876914 x 106 m/s) h(4.7859937 x 1012 m2/s2) Combining the constants, -------------------------, and dividing 2pme(2.1876914 x 106 m/s) Z2 by Z reduces Eq.(14) to v2r = Z(2.5326390 x 102 m3/s2). (15) The author revealed that the gravitational constant GX of any given body X is found by the equation, v2XSrXS GX = -------, (16) mx where vXS is the sidereal velocity of any satellite S in orbit about the body; rXS is the mean radius between the center-of-mass of the body and the center-of-mass of the satellite; and mX is the mass of the body.[3] Multiplying both sides of Eq.(16) by mX, we find that GXmX = v2XSrXS. (17) Thus, replacing GXmX of Eq.(4) by its equivalent from Eq.(15), we find that the Schwarzchild radius for the nucleus of any given atom is 2Z(2.5326390 x 102 m3/s2) rS = -------------------------. (18) c2 2(2.5326390 x 102 m3/s2) By combining the constants, -----------------------, Eq.(8) reduces to c2 rs = Z(5.6358818 x 10-15 m). (19) Replacing the variables of Eq.(1) by their equivalents from Eq.(5) and Eq.(15), we find that the escape velocity from the surface of the nucleus of any given atom is _________________________ / 2Z(2.5326390 x 102 m3/s2) v _ / -------------------------. (20) |/ (1.3 x 10-15 m)A1/3 Again combining the constants, we can simplify Eq.(20) into ___________________ / Z(3.9 x 1017 m2/s2) v _ / -------------------, (21) |/ A1/3 where A is the number of nucleons; and Z is the number of protons in the nucleus. For the helium atom, A = 4 and Z = 2. From Eq.(19), the Schwarzchild radius of the helium atom is rS = 1.1271764 x 10-14 m. From Eq.(21), the escape velocity is about 7.0 x 108 m/s (well over twice the speed of light). Eq.(19) reveals that the Schwarzchild radius increases as the atomic number of the element increases. Likewise, Eq.(21) reveals that the escape velocity from the surface of the nucleus increases as the atomic number of the element increases. IV. Much faster than the speed of light The heaviest naturally occurring element is plutonium, 242Pu (A = 242 and Z = 94).[7] Its Schwarzchild radius is 5.2977289 x 10-13 m, and the escape velocity from the surface of its nucleus is about 2.4 x 109 m/s (over eight times the speed of light). 242Pu is radioactive, with a half-life of 3.763 x 105 years.[8] Thus, one gram of 242Pu emits about 1.048 x 108 particles per second from its nucleus, at escape velocities of at least eight times the speed of light. It is well established that all known subatomic particles escape from the nucleus during the innumerable myriad of atom-smashing experiments. Furthermore, many elements radioactively decay, spontaneously emitting electrons, neutrons, and protons from the nucleus. Thus, from section III, we know that all such particles must routinely travel faster than the speed of light to accomplish their escape. V. Conclusion For all elements, comparing the Schwarzchild radius of the nucleus, Eq.(19), to the physical radius of the nucleus, Eq.(5), reveals that the nucleus of every atom is a black hole. Considering that particles routinely escape from the nuclei of atoms, perhaps "dark hole" is a better term than black hole. The escape velocity of the nucleus, Eq.(21), reveals that all subatomic particles must exceed the speed of light to escape from the nucleus. Thus, the speed of light is not the speed limit of the universe. References [1] Paul A. Tipler, Physics, 2nd ed. (Worth Publishers, New York, 1976, 1982), pp.207-213. [2] Ian D. Lawrie, A Unified Grand Tour of Theoretical Physics (Adam Hilger, New York, 1990), pp.78-81. [3] Michael E. Heffron, "Gravity Unified!" (self-published), p.6. [4] Rita G. Lerner and George L. Trigg, Encyclopedia of Physics, 2nd ed. (VCH Publishers, New York, 1990), p.839. [5] John A. Dean, Langes Handbook of Chemistry, 14th ed. (McGraw-Hill, New York, 1992), pp.4.6-4.7. [6] Reference 4, pp.303-305. [7] Isaac Asimov, Asimovs Guide to Science (Basic Books, New York, 1972), pp.240-242. [8] Reference 5, pp.4.42-4.52. body; rXS is the mean radius between the center-of-mass of the body and the cenwsojfb`][WURMP$@ MNxWvXsYljgeb[XQOMP xvsqngeb`]VSMP trohfc a^,\-Y.R/OMP  /0t2r3oHmIjJca^WTMKM  xqole#c$`^[ Y!V"OM  "3y5r7p8i=g>`Z^[[\TmRpKIM     xvomf d`A^B[CYDVNTM    NQtTrUoXhgfhca^\U S! LM    ! 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