From E.G. Haug Escape Velocity To the Golden Ratio at the Black Hole. Branko Zivlak, Novi Sad, May 2018

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1 Fom E.G. Haug Esape eloity To the Golden Ratio at the Blak Hole Banko Zivlak, Novi Sad, May 018 Abstat Esape veloity fom the E.G. Haug has been heked. It is ompaed with obital veloity fomula fo an ideal iula path. The fomulas ae simified so that we have only one vaiable that ontains the Plank values and the mass of the ental body. In the ase of an abitay sta, the values of these veloities ae detemined duing its ompession to the blak hole. Unlike the standad and elativisti fomulas that ae appoimations fo a weak gavitational field, Haug's fomula is eat fo a weak and stong gavitational field. The elationships between fomulas showed the impotane of the golden atio below the Shwazshild adius. Key wods: Plank, Haug, esape veloity, golden atio, blak hole 1. Intodution We will use the known fomula fo the obital veloity fo the ideal iula path of the adius, body evolve aound the ental mass M, [1]: Fom the known elationship fo G - The univesal gavitational onstant, - light veloity, l - Plank length and m - Plank mass [] we have: G * l m Fom [3, fomula 4] we take the fomula hee in the fom (3). (1) () 1 / (1 / ) * (3) 1

2 Whee eh is esape veloity by E. G. Haug. Note that in [3], the tansition fom Haug s fomula 3 to 4, may be disayed in moe steps. The auay of the pass, I heked thanks to Wolfam Alpha, and onfimed the oetness of fomula. Substituting () into (3) we obtain: eh * 1 m (1 m / l / l M M ) (4) To note that my attempts with othe esape veloities, fom liteatue, esulted in ejetion of poposed fomulas. Only Haug s fomula (3) passed the test and also has a solid deivation and a ational eanation in [3].. Pepaation fo analysis Fo the sake of simiity we intodue (5): m l M It is obviously, dimensionless. If we inlude (1) we get: (5) (6) That is, is the atio of the squae of the speed of light to the veloity. The atio β = v / is often used in liteatue, so that fo v= is: = / v = 1/. Note that: "v is the elative veloity between inetial efeene fames", [4]. In ou ase, the veloity is defined elusively by equation (1), and theefoe we use instead, so that no onfusion aises. In the ase of the anet Eath, = 1.4 * 10 9, while fo the Sun and stas about fou odes of magnitude ae smalle. Now, (1) and (4) an be witten in the fom (7) and (8): (7) 1 1 * (8) Fo elestial bodies, fomula (7) multiied by, gives a standad value fo esape veloity, whih is not vey diffeent fom (8), until the adii of nea Shwazshild s eah, when in the standad appoah thee ae speeds lage of the speed of light. At even smalle adii thee is a ollapse of equations and blak hole singulaities, whih is not only a mathematially undefined

3 state, but also a physial nonsense, see [3, Table 1]. Fomulas (7) and (8) will be futhe analysed in the net setion. 3. An eame of a sta Let's take a definition fom [5]: Blak hole, osmi body of etemely intense gavity fom whih nothing, not even light, an esape. Thee is no blak hole in the above definition, beause thee is no situation "that light annot esape." We will, fo histoial easons, etain the tem "blak hole" in the meaning of "osmi body of etemely intense gavity", whih is in fat a body that tends towad a blak hole. We will analyse the esults obtained by the peeding fomulas in the ase of a gavitational ompession of a sta having a mass of about 5 times the mass of the Sun. We will fom Table 1, similaly to [3, Table 1], but fo the sake of simiity, instead of the Shwazshild adius, we use half of that value, b: b Let's all this value the basi adius, b, so we will epess othe adii as the poduts of this adius. Fo eame, Shwazshild's adius s = * b. Let's emphasize that the alulations hee ae based on the idealized situation in whih the mass of the ental body does not hange and thee ae no othe bodies in its viinity that make the alulations moe ome. At eal osmologial situations should take into aount as many neighbouing influenes as possible. The futue of a sta is a gavitational ompession, o smalle adii. The sta also had its own histoy, that is, lage adii, but hee is the theme of wok, the behaviou of the sta in its final stage of development, nea the state of the blak hole. Theefoe, let us eamine the equalization of the fomulas (7) and (8), ie = eh, that is: O by shotening with : 1 1 (9) * (10) (1 1 1 We got the equation with one dimensionless vaiable, whih thanks to the softwae "Wolfam Alpha" we easily find the solution: (11) 3

4 5 1 (1) This is a well-known value of the "Golden atio", whih we mak with φ. To summaize: the obital veloity fom (1) and Haug's esape veloity fom (3) ae equal at the adius whih is φ times geate than the basi adius, b, o: O, if we inlude (9):, fo * b * (13), fo (14) The adius that satisfies (14), let's all the "Golden adius" and mak it with φ. We an say that eah mass has its own Golden adius, if = / is geate than the lowe limit fo the length, fo that mass. If in the peeding equation we take the Plank mass than: = Gm / = l. That is, Plank's mass is the smallest mass that an tends to the blak hole and its golden adius is also the minimum adius of the blak hole: Gm ( ankmass) * * l.6151* m (15) The adius φ(plank mass) is φ times lage than Plank's length, whih is the lowe limit fo the length. These blak holes ae alled "mini blak holes" [6]. At this adius is also the maimum density ( ma = b / φ 3 ) and the maimum gavity (ama = ab / φ ) fo blak holes. It is easy to show that espeially fo the Plank mass, b and ab ae Plank's values fo density and aeleation ( b= and ab=a ). In Table 1. we hoose the eponents, k = 1 to 6 and k = to ove the Shwazshild adius, whee =. Note, the basi adius is fo k=0, =1, b=/ =7.4564*10 3 m. Table 1. eloities nea the blak hole fo a sta of kg =φ k =* b eh (eh/) (/eh) 17,944 1,335E+05 7,077E+07 9,611E+07 1, , ,090 8,35E+04 9,00E+07 1,194E+08 1, , ,854 5,0896E+04 1,145E+08 1,464E+08 1, , ,36 3,1456E+04 1,457E+08 1,76E+08 1, ,89447,618 1,9441E+04 1,853E+08,069E+08 1, ,099106,000 1,4851E+04,10E+08,35E+08 1, , ,618 1,015E+04,357E+08,357E+08 1, , ,000 7,4564E+03,998E+08,596E+08 0, ,

5 The veloities nea to the blak hole ae shown gaphially too. eloities [m/s] 4,000E+08 3,000E+08,000E+08 1,000E+08 1,000 1,618,000,618 4,36 6,854 11,090 17,944 0,000E+00 eh Figue 1. eloities nea to the blak hole (At absise ae the poduts of the adius b = / ) Fom (7) and (8) we get: (1 ) (16) Fom whee we analyse the atio of veloities to show the esults fo some haateisti adii: Fo lage, (eh / ) = (+ ) / (1+) / tend to, that is: eh tends to * In othe wods, fo a weak gavitational field, eat Haug's solution gives same esult as the standad and GR appoahes see [3, Chapte ]. Fo = ie. Shwazshild's adius, (eh / ) = ( + 8) / (1 + ) = 10/9, espetively: eh = (10/9) * o ( / eh) = 9/5 (see Table 1) o at [3] eh = * 5 / 3. Fo = φ, φ = φ * b, (eh / ) = 1/1, espetively: * (17) eh = =.357 *10 8 m/s, (/eh) = 1, = φ, o eh is always less than the speed of light, and theefoe thee is no blak hole. φ is the smallest adius that an eah some mass, simila to (15) fo the mini-blak hole. At this adius is also the maimum density and the maimum gavity fo the used mass. This is due to the fat that in (17) the mass and its gold adius ae popotional, and if in mass M thee ae n Plank masses, then thee is also n adii fom (15): Gm * ( M ) n* ( m ) * n* * n* l (18) 5

6 It follows that φ(m)=n*φ*l is the minimum adius fo the blak hole geneated by the mass M, beause fo a smalle adius it is neessay that some ingedient in a blak hole is smalle than the minimum, φ(plankmass). Fom the pevious we onlude that the golden adius φ=φ*/ is moe signifiant than the Swazhild adius s=* /. Haug laim [3, page 3]: at a adius onsideably below the Shwazshild adius, the esape veloity is appoahing. This is in shap ontast to the standad appoimate esape veloity of moden physis that pedits that esape veloity at the Shwazshild adius is and that the esape veloity inside the Shwazshild adius is. Although the Haug s statement is muh moe auate than the dominant position, it is even moe peise to say that the adius φ = φ * b, eh = =.357 * 10 8 m/s, o we an say: at a adius below the Shwazshild adius, the esape veloity appoahes / φ. It an be seen in Figue 1. As well as a mini-blak hole suh as in the eame of the sta in Table 1 the maimum density and a maimum of gavitational attation ae elated to the adius of φ. Fo 1, the adius is < φ * b and eh <. Than the adius is smalle than the smallest adius φ, suh a blak hole annot eist. It is inteesting to note that these elationships ae the same fo all bodies. These esults would be inteesting to ompae with eisting knowledge about the stas. 4. Conlusion Haug's fomula (3) was deived and eained in [3], while futhe possibilities of this fomula ae shown hee. In ontast to an iational moden undestanding involving speeds geate than the speed of light, blak hole singulaities and equations that beaks, we have a ational appoah in [3]. Hee it is suppoted by the mehanism shown by fomula (14) whih shows the estition of gowth of esape veloity. Two new adii at blak holes ae defined: the basi adius, b=/ and the golden adius, φ =φ*/ and the signifiane of these adii is shown. Golden atio obtained in fomulas elated to the blak hole is pobably the way to shift poesses in natue. Hee dominant gavity gives way to adiation and the poess that theatens to pass into infinite gavity and infinite density moves in the opposite dietion. In ode to ome up with pevious esults, I have been helped by a philosophial appoah epessed in seveal of my papes, fo eame in [7] with the following views: And: Pats ae dependent on the whole (Univese) and ae also an integal pat of the whole, theefoe, the whole is also dependent on the pats! Matte dominant Univese and adiation dominant Univese oeist in evey point in time! That an all be ombined unde elational appoah [8]. 6

7 Refeenes: [1] [] [3] E.G. Haug, The Collapse of the Shwazshild Radius: The End of Blak Holes, [4] [5] [6] [7] Banko Zivlak, Mathematis%0and%0Apied%0Mathematis/Download/707 [8] Relational theoy, 7