Derivation Of The Schwarzschild Radius Without General Relativity

Transcription

1 Derivation Of Te Scwarzscild Radius Witout General Relativity In tis paper I present an alternative metod of deriving te Scwarzscild radius of a black ole. Te metod uses tree of te Planck units formulas: te Planck mass, te Planck momentum and te Planck lengt. Terefore it is sown tat General Relativity (GR) is not necessary to derive te formula for te radius of a black ole. Tus, compared to GR, te simplification acieved by tis derivation is enormous. by R. A. Frino Electronics Engineer Degree from te National University of Mar del Plata rodolfo_frino@yaoo.com.ar Keywords: quantum gravity, universal uncertainty principle, Planck units, Planck mass, Planck momentum, Planck lengt, de Broglie law, Scwarzscild radius. 1. Introduction and I stumbled on te key: te Planck units, a primitive but marvelous quantum gravity teory In 1783 Jon Mitcell derived te correct formula for te Scwarzscild radius using a wrong formulation [1] based on classical or Newtonian pysics. He applied te principle of conservation of energy to a poton emitted from a star of mass M 1 2 mv 2 = G M m r (1.1) Were m denotes te equivalent mass of a poton (Mitcell supposed, at least implicitly, tat te equivalent inertial mass of te poton was identical to its equivalent gravitational mass. Tus, from an implicit point of view, Einstein was not te first to introduce te equivalence principle). In equation (1.1) Mitcell replaced te escape velocity v of a poton wit te speed of ligt, c. Tus, e was able to write 1 2 c2 = G M r (1.2) Ten e solved tis equation for r to get te following correct result for te Scwarzscild radius of a black ole Derivation Of Te Scwarzscild Radius Witout General Relativity - v1. Copyrigt Rodolfo A. Frino. All rigts reserved. 1

2 r = 2G M c 2 (1.3) However, we must empasize tat, despite te correctness of te above result, Mitcell's formulation was wrong and it sould not be considered te proof of te formula of te Scwarzscild radius for black oles. If we want to obtain a correct derivation of tis formula we sould use eiter GR, quantum gravity or te Planck units (an extremely primitive form of quantum gravity). In tis paper I ave cosen te metod based on te Planck units. In 2014 I publised an article entitled Te Special Quantum Gravitational Teory of Black Holes (Wormoles and Cosmic Time Macines) [2] were I derived (a) a general formula for te temperature of a black ole and (b) te formula for te black ole entropy. In tat formulation I used te universal uncertainty principle (UUP) [3] as te cornerstone of te teory along wit te Scwarzscild radius formula discovered by Carl Scwarzscild in Te issue was tat I wanted to derive te termodynamic properties of black oles witout using any results from Einstein's General Teory of Relativity; and altoug, at tat time, I could ave included te derivation presented ere, I did not do so because I wanted to write a relatively sort paper. Now I feel it is time to introduce te metod I left out at tat time so tat my previous formulation on black oles does not rest on Einstein's GR. Te matematics used in tis paper are simple enoug so tat te derivation may be easily understood by te non-expert. Appendix I includes te nomenclature used trougout tis paper. 2. Derivation of te Scwarzscild Radius Te idea is to find, if possible, a relation based on (but not identical) te de Broglie law from te Planck units so tat te formula will ave te following form R = K (2.1) Were = Planck's constant R = radius of a black ole K = dimensionless constant to be determined = Planck momentum From ere we sall derive a special formula for te radius of a black ole (a black ole of minimum size). Ten, we sall generalize te formula so tat it coincides wit te famous black ole radius formula found by Carl Scwarzscild. I sall start te derivation wit te formula for te Planck mass, M P Planck mass M P c 2π G (2.2) and te formula for te Planck momentum, Derivation Of Te Scwarzscild Radius Witout General Relativity - v1. Copyrigt Rodolfo A. Frino. All rigts reserved. 2

3 Planck momentum c3 2 π G (2.3) Ten we multiply tese two equations as follows M P = c c 3 2π G 2π G (2.4) Wic gives M P = c2 2π G (2.5) Now we invert eac side. Tis yields Wic may also be expressed as follows 1 = 2 π G (2.6) M P c 2 2π = G M P c 2 (2.7) Now we multiply and divide te second side of tis equation by te mass, M, of a star = 2G M M P π c 2 M (2.8) (We assume tat tis star will is sufficiently massive to collapse into a black ole wen it runs out of fuel). But, wat lengt does te quantity /π represents? In order to answer tis question we replace te Planck momentum on te first side of equation (2.8) by te second side of equation (2.3). Tis gives π = c3 2π G (2.9) Wic, after some work, leads to = 2 G (2.10) π 2π c 3 Were te square root of te second side is te definition of te Planck lengt, Derivation Of Te Scwarzscild Radius Witout General Relativity - v1. Copyrigt Rodolfo A. Frino. All rigts reserved. 3

4 Planck lengt G 2π c 3 (2.11) From equations (2.10) and (2.11) we write π = 2 (2.12) From equations (2.7) and (2.12) we may write te following equation Or 2 = 2G M P c 2 (2.13) = G M P c 2 (2.14) But from previous studies carried out in Appendix 1 of reference [4] we know te mass of a black ole of minimum size (wic, by te way, coincides wit te initial mass of te universe). In oter words, we know tat (a) te minimum radius of a black ole is equal to te Planck lengt and (b) tat te minimum mass of a black ole is equal to te Planck mass divided by 2, tis is M b min = M P 2 (2.15) Ten we multiply and divide te second side of equation (2.14) by 2. Tis yields = 2G c 2 But according to equation (2.15) we may write M P 2 (2.16) Minimum radius of a black ole = 2G M b min c 2 (2.17) Terefore te quantity on te second side of tis equation must be a radius (because is a radius) and not a diameter. Terefore, it follows tat te more general expression 2G M /c 2 must also be a radius and not a diameter. Wit tese considerations in mind we may generalize te last equation by replacing M b min by M (te mass of a star) and te radius by R (te radius of a collapsed star or black ole). Wit tese two canges equation (2.17) becomes te general equation for te radius of a black ole (because equation 2.17 is a special equation for te radius of a black ole). Tus we ave Radius of a black ole R 2 G M c 2 (2.18) Derivation Of Te Scwarzscild Radius Witout General Relativity - v1. Copyrigt Rodolfo A. Frino. All rigts reserved. 4

5 Terefore R must be te Scwarzscild radius, R S, of te black ole and M must be its mass (M was, originally, te mass of a star tat collapsed into a black ole). Tus, we finally write te formula for te Scwarzscild radius of a black ole as it is generally written in te literature R S 2G M c 2 (2.19) 3. Determination of te Constant K Equation (2.8) may be rewritten as Tus te dimensionless constant K may be defined as M = 2 G M (3.1) π M P c 2 K Terefore te formula we want it is (see equation 2.1) M π M P (3.2) K = R S = 2 G M c 2 (3.3) 4. Conclusions Tis paper sows te power of te metodology based on te Planck units: we do not need General Relativity to obtain te formula for te Scwarzscild radius of a black ole. Tus, quantum gravity is someow embedded in tese units. Due to its simplicity, te metod presented ere is suitable to be taugt in senior courses of secondary scools. Before I finis tis article I would like to say tat tere is at least anoter metod of deriving te Scwarzscild radius witout GR. But tis is a subject for anoter article. Appendix 1 Nomenclature = Planck's constant G = Newton's gravitational constant c = speed of ligt in vacuum K = a constant to be determined = Planck momentum m = equivalent mass of a poton Derivation Of Te Scwarzscild Radius Witout General Relativity - v1. Copyrigt Rodolfo A. Frino. All rigts reserved. 5

6 M P = Planck mass = Planck lengt M b min = minimum mass of a black ole M = mass of a black ole r = radius of a black ole (black ole radius or Scwarzscild radius) R = radius of a black ole (black ole radius or Scwarzscild radius) R S = Scwarzscild radius REFERENCES [1] A. Hamilton, More about te Scwarzscild Geometry, Colorado University, retrieved from: ttp: //casa.colorado.edu/~ajs/scwp.tml, (2006). [2] R. A. Frino, Te Special Quantum Gravitational Teory of Black Holes (Wormoles and Cosmic Time Macines), vixra.org: vixra , (2014). [3] R. A. Frino, Te Universal Uncertainty Principle, vixra.org: vixra , (2014). [4] R. A. Frino, Te Quantum Gravitational Cosmological Model witout Singularity, Appendix 1, pp , vixra.org: vixra , (2015). Derivation Of Te Scwarzscild Radius Witout General Relativity - v1. Copyrigt Rodolfo A. Frino. All rigts reserved. 6