Mass- and light-horizons, black holes' radii, the Schwartzschild metric and the Kerr metric

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1 Thiey De Mees Mass- and light-hoizons, blak holes' adii, the Shwatzshild meti and the Ke meti mpoved alulus. (using gavitomagnetism) T. De Mees - thieydm@pandoa.be Abstat Blak holes geneally ae defined as stella objets whih do not elease any light. The Shwazshild adius, deived fom GRT, defines the hoizon adius fo non-otating blak holes. The Ke meti is supposed to define the event hoizon of otating blak holes, and this meti is deived fom geneally aeptable piniples. The limit fo the Ke meti's hoizon fo non-otating blak holes is the Shwazshild adius. By analysing the hoizon outome fo otating and non-otating blak holes, using the Maxwell Analogy fo Gavitation (MAG) [5,6,7,8] (o histoially moe oetly: the Heaviside [] Analogy fo Gavitation, often alled gavitomagnetism), find that the Ke meti must be inomplete in elation to the definition of event hoizons of otating blak holes. f the Maxwell Analogy fo Gavitation (gavitomagnetism) is supposed to be a good appoah of GRT, we may assume that it is a valid analysis tool fo the sta hoizon metis. The Ke meti only defines the hoizons fo light, but not the mass-hoizons. find both the light-hoizons and the the mass-hoizons based on MAG. Moeove, dedut the equatoial adii of otating blak holes. The pobable oigin of the minutes-lasting gamma busts nea blak holes is unveiled as well. inally, dedut the spin veloity of blak holes with a 'Citial Compession Radius'. The dedutions ae based on the findings of my papes Did Einstein heat?, On the geomety of otay stas and blak holes and On the obital veloities neaby otay stas and blak holes. Keywods. Maxwell Analogy gavitation otay sta blak hole Ke Meti tous gyotation hoizon methods : analytial Gaphs. WZ-Gaphe ndex 1. The obital veloities neaby Rotay Stas and Blak Holes. ntodution: the meaning of mass-hoizons. At what onditions will matte obit at the speed of light? The tous shape of fast spinning stas.. The bending of light into a iula obit. ntodution: the meaning of light-hoizons and the Ke Meti. What speifies the light-hoizon of blak holes? 3. Deiving the adius of Pue Blak Holes. Evolution of the Pue Blak Hole's adii. Spin veloity of Blak Holes at the Citial Compession Radius. The Pue Blak Hole's adius and the Pefet Blak Hole 4. Disussion: Thee appoahes, thee impotant esults. Obiting masses at the speed of light. The bending of light and the Ke meti. Compaing both types of hoizons. The blak hole's adius. 5. Conlusion. 6. Refeenes. 1

2 1. The obital veloities neaby Rotay Stas and Blak Holes. ntodution: the meaning of mass-hoizons. The hoizon an -unhappily- be defined as the ultimate possible obit of masses about the spinning sta. n ode to find the hoizon's adius in this hapte, look afte the obit whih has an obital veloity of the speed of light. This hoizon all the mass-obit hoizon o simply the mass-hoizon. f the hoizon's adius is geate than the sta adius, we an speak of a blak hole of the mass-hoizon-type, o at least of a equato blak hole (o patial blak-hole ) of the mass-hoizon-type. ndeed, the egion of the poles of spinning stas do not espond to the same equiements than the equato, and thus is not emission-fee. Let us look at the bending of objets about stas into obits. istly, we have the Newtonian gavitation foe. Seondly, we have the attating foe due to the spin of the sta. Theefoe, we fist need to find the seond gavitation field ( magneti pat of gavitomagnetism, what all gyotation ). om my pape A oheent double veto field theoy fo Gavitation, we have the basi equation of the gyotation pat Ω ( magneti pat) of gavitomagnetism fo sphees : o, in geneal: R Ω ext 5 G Ω ext 3 ω ω wheein we have eplaed the inetial moment of the sphee = m R by a geneal inetia momentum. 5 3 b g 3 ω ω 3 b g (1.1) (1.) This equation follows fom the integation of equation (1.5) below, fo onstant gavity, ove the whole sphee. The set of Maxwell equations fo Gavitomagnetism is given by the equations (1.3) to (1.10) below. m' ( g + v Ω ) (1.3). g ρ / ζ (1.4) ² Ω j / ζ + g / t (1.5) whee j is the mass flow though a fititious sufae. The tem g/ t is added fo same the easons suh as Maxwell did: the ompliane of fomula (.3) with the equation : div j - ρ / t (1.6) t is also expeted that: div Ω. Ω = 0 (1.7) and g - Ω / t (1.8) t is possible to speak of gyogavitation waves with tansmission speed. = 1 / ( ζ τ ) (1.9) wheein τ = 4π G/ (1.10). Equations (1.3) till (1.10) below fom a oheent ange of equations, simila to the Maxwell equations. The eleti hage is then substituted by mass, the magneti field by gyotation, and the espetive onstants ae also substituted (the gavitation aeleation is witten as g, the so-alled gyotation field as Ω, and the univesal gavitation onstant out of G -1 = 4π ζ, whee G is the "univesal" gavitation onstant. We use sign instead of = beause the ighthand side of the equations auses the left-hand side. This sign will be used when we want insist on the indution popety in the equation. is the esulting foe, v the speed of mass m' with density ρ. Combined with (1.3) = m' (g + v Ω ), this beomes fo the equato plane (α = 0) : new edition 13 Ap 010 fist elease 10 Ap 006

3 m' ' ωv m' ' ω = + = (1.11) wheein v is in this ase the veloity of the light, v =. The sign fo has been omitted beause we onside quantities hee, no vetos. G ω nstead of foes, pefe to use aeleations by putting a = /m'. Hene : a = + 3 (1.1) This aeleation foms a iula obit if a = v²/, wheein v is the obital veloity of the objet : v obit = v. v G ω = + 3 (1.13) By putting v obit =, we an find the obit adius whee the obit veloity should eah the speed of light. This dedution is puely theoetial, beause vey pobably this ase will lead to a disintegation of the obiting matte into gamma ays. o any obit lose to the blak hole, no matte obits will still subsist. By filling v obit = in (1.1), we get: G ω = + (1.14) The positive solution of (1.14) This equation is quadati in if we multiply it by ². And of the two solutions, we only keep the positive one: G ω 3 MH = ± H G K J + (1.15) Thus, the faste the sta spins, the lage the matte-hoizon-adius = MH beomes. t is pobable that (1.15) gives the ondition of disintegation of matte nea a spinning sta, due to the high enegies involved fo masses eahing the speed of light, and it seems easonable to take in aount this possibility. And fo non-otating blak holes, the obit adius (matte hoizon) beomes: MH = = Rs if ω = 0 (1.16) whih is half the Shwazshild adius : = s (1.17) Equation (1.16) means that if an objet is obiting at (almost) the speed of light about a sta without a spin, that sta must not be lage than half the diamete of a Shwazshild blak hole. n the following lines, simplify (1.15) fo fast spinning stas with masses of at least that of the sun. Equation (1.15) beomes afte some manipulation: = MH ω The seond tem unde the oot sign is smalle than 1. Thus, knowing that: R (1.18) = (1.15) new edition 13 Ap 010 fist elease 10 Ap 006

4 x << x x (1.19) it follows that: ω MH 1 + ω <<1 fo (1.0.a) (1.15) (1.0.b) The expession (1.0.b) is valid fo all the known elestial objets. Sine the definition of the Shwazshild adius is : R = s (1.17) the equation (1.0.a) an be e-witten as: MH Rs ω + m (1.1) (1.15) The equation (1.1) shows that the evolution of the mass-hoizon adius is nealy linea in ω. The faste the sta spins, the wide away fom its ente the mass-hoizon obit beomes. This equation means that no mass an 'suvive' fo that adius, no smalle adii. Moeove, when mass obits as lose as the matte-hoizon-adius = MH, the obit speed must eah and matte must disintegate. The negative solution of (1.14) Remak that the negative solution of the quadati equation (1.14) does not have yet a lea physial meaning hee. t would be quite speulative to assoiate this equation with the empty inne spae of a tous blak hole, but this option meits a lose study. G ω 3 = H G K J + (1.) n my fome pape On the shape of blak holes demonstated, using MAG, the high pobability of tous blak holes when they spin fast. These two mass-hoizons ould signify the onfimation of my ealie finding. Hee, the equations desibe the (quite unusual) onditions of an obital veloity of matte at the speed of light. n the disussion hapte, these issues will be futhe explained. n the following lines, simplify (1.) fo fast spinning stas with masses of at least that of the sun. Equation (1.) beomes afte some manipulation: = 1 1+ ω The seond tem unde the oot sign is expeted to be fa smalle than 1. Hene, knowing that: (1.3) = (1.) x << x x (1.19) it follows that fo fast spinning stas, the seond mass-hoizon beomes: new edition 13 Ap 010 fist elease 10 Ap 006

5 ω MH m (1.4) t might be vey possible that equation (1.4) has no physial meaning. Remak that it is mass-independent. The tous shape of fast spinning stas n the pape On the shape of otay stas and blak holes dedut that fast spinning stas ae tous-shaped. Can this also be deduted fom the MAass-hoizon? ndeed, in the same pape, ome to the onlusion that when patiles aive in the tous' hole, the only stable motion is a iula equatoial obit whih is etogade to the tous' spin. When looking at (1.4), thee is a supising minus sign. And this is pefetly omplying with a etogade obit. When (1.1) and (1.4) ae gaphially epesented (fig.1.1), it beomes lea that the two mass-hoizons (ed boundaies) diffe only with the width of half the Shwazshild adius. - ω / ( m ) R S / / + ω / ( m ) ig.1.1. The spinning sta mass-hoizons (ed lines) Thus, aoding to an ealie pape [8], the shape of the mass-hoizon of fast spinning stas is tous-like, and it an be expeted that suh spinning stas ae tous-like as well with a thikness muh below Rs /. This hapte gives the solution fo the zone neaby the blak hole whee matte tends to obit at the speed of light. Befoe disussing the findings of this hapte moe in depth, fist study the geneal poblem of the bending of light neaby blak holes.. The bending of light into a iula obit. ntodution: the meaning of a light-hoizon and the Ke Meti. Anothe appoah ould be the study of the bending of light by the spinning sta. Shwazshild found one event hoizon fo non-otation blak holes by applying GRT. With the Ke meti, whih gives the onditions neaby blak holes, two hoizons ae found. Hee, look fo hoizons via the Maxwell Analogy. Although this hapte seems to be quite idential to the fome one, thee is an impotant diffeene. Hee, speak of the bending of light in the gyogavitation field, and not about matte in an obit. And the esult of iula light-bending is alled the light-hoizon new edition 13 Ap 010 fist elease 10 Ap 006

6 o this pupose, we take the solution whih we have found in Did Einstein heat? [7], equation (6.14), witten in its geneal fom. = m + m + v m R ω ϕ ϕ, α os α os ϕ 1 5 (.1) This equation desibes the bending of light, taking in aount thee foes and thus thee tems, based on : 1 the pseudo-gavitational effet fo light, whih is two times the value of the Newton gavitation; the gyotation foe due to the obit veloity of the sta in its galaxy (in the pesent ase: of the Milky Way, whee α is the angle between the obiting objet at veloity v1 and the axis between the ente of the Milky way and the sun) and 3 the sta otation (in the pesent ase: the sun) while the light passes at a etain latitude ϕ. And found this equation to be fa moe auate than the GRT deivation. The finding in this deivation was that light is not bent by gavitational effets (beause the est mass of light is zeo), but only by the gyotation field of the mass beam of the light wave itself, taveling in the gavitation field of the sta. The equation (.1) has been witten fo light that is gazing the sun (o any massive objet). This must be hanged into an equation that is valid fo any distane of the light to the ente of the elestial objet and fo any type of inetial moment, not only fo spheial objets. Below, this will be adapted by stating fom the following onepts : the fist tem of (.1) emains valid, the seond tem will not be onsideed futhe and the thid tem will be adapted as said befoe. What speifies the light-hoizon of blak holes? n this ase, of ouse, do not onside the Milky Way's dagging veloity v 1, whih assume to be insignifiant neaby the blak holes we want to study. Besides staying at the equato level of the sta only, onside aeleations instead of foes. So, the pependiula aeleation upon the light beomes, in analogy with equation (1.1), wheein only the Newtonian tem gets a double value : G ω a = + 3 Sine this aeleation is a bending, thus, adial aeleation, and sine we look at the light pefoming a iula obit, the aeleation a is supposed to also omply with the entipetal aeleation v²/, whih is a puely geometial fomula. o light, we eplae the speed v by. (.) Hene: G ω = + 3 (.3) By making this equation quadati in the adius of the light-hoizon = MH, we get the following solutions: LH = ± H G K J + G ω 3 = ± + ω LH 1 1 o (.4.a) = (.4.b) The seond tem unde the oot sign is expeted to be fa smalle than 1. Hene, knowing that: x << x x (1.19) we an wite this as a positive and a negative solution : ω LH + = ω ω (.5) (.4) + and if (.6) (.4) - LH = 4m << new edition 13 Ap 010 fist elease 10 Ap 006 (.7)

7 Remak that LHis independent fom the mass. Hene, it is vey possible that (.6) has no physial meaning, but it might have the meaning of a etogade obit inside the hole of the tous. Equation (.5) also an be witten as : LH R + ω + s 4m (.8) (.4) + wheein is the Shwazshild adius. Equation (.8) is thus desibing the bending of light beams in a iula obit about blak holes. Hoizons annot be defined bette than with this equation. n the disussion hapte, it will beome lea why this is so. ig..1. The spinning sta mass-hoizons (ed lines) and its light-hoizon (dak line). R S / - ω / (4 m ) + ω / (4 m ) As shown in fig..1, the extenal light-hoizon's diamete is always smalle than the extenal mass-hoizon diamete. 3. Deiving the adius of Pue Blak Holes. Evolution of the Pue Blak Hole's adii. f, as found, (.8) desibes the hoizon of blak holes, thee is a speial ase whih even goes beyond that esult: when the light-hoizon oinides with the sta equato, a pat of the sta is invisible, even when looking fom the poles to the sta, wheeas this obsuation was not the ase in the fome hoizons. speak of Pue Blak Holes at the limit whee the equato of the sta is obsued. Light annot esape, and the light hoizon is the sta equato. Hene, an desibe patial blak holes, wheeof a pat is invisible, even obseved fom the poles. To manage this, we need to adapt the paametes of equation (.8) as follows : o thin ings and thin touses in geneal, = λ m R², whee R is the adius at the equatoial level of the sta, and the fato λ 1. By putting LH = R, obtain a iula bending of light upon the equato of the sta itself. Sine we look fo the ase whee ω º, equation (.10) an then be eplaed by: wheein is again the Shwazshild adius. R pue = b 1 λ 4 g (3.1) new edition 13 Ap 010 fist elease 10 Ap 006

8 We see immediately that, fo a ing blak hole, when the light hoizon eahes the ing's adius itself, this ing's adius must have eahed about 4/3 th of the Shwazshild adius (the Shwazshild adius stands fo the theoetial spheial non-otating blak hole). Note that the value fo the spin ate of that Pue Blak Hole equals to ω º /, as defined ealie. Remak that the onept of Pue Blak Hole is only theoetial. f the spin veloity beomes lose to the speed of light, disintegation of the matte patiles is extemely pobable. The gaphi evolutes as expeted: the highe the spin, the smalle the adius of the light ile beomes. Equation (3.1) is beautifully desibing the equied adius at the equato level of otating Pue Blak Holes. R S / < R S ig.3.1 The Pue Blak Hole's light-hoizon and mass-hoizons t is then lea that if depit this gaphially, get fig.3.1., wheein show the light-hoizon (lage dak bounday) and the mass-hoizons (ed boundaies) as well. Spin veloity of Blak Holes at the Citial Compession Radius. n a fome pape [8], have deduted the adius of ontinuous mass ompession at the equato level of spheial stas (with negligible Newtonian-gavitation influene). This dedution was based on the gyotation field equations fo a sphee, and we use (1.) in ode to obtain a moe geneal equation. The minus sign signifies attation. G Ω ext Heein is the distane to the ente of the sphee, R is the adius of the sphee and ω is the spin veloity. The equatoial ompessive gyotation foe is given by the analogue Loenz foe ax = ωr Ω y (3.) and the last tem of (1.) is zeo in the dietion of the spin axis, so Ω y = 0. Hene, the aeleation due to gyotation at the equato plane is: b g 3 ω ω 3 (1.) a R G x = ω 3 (3.3) At the othe hand, we have the following foes: the entifugal foe and the gavitation foe. o fast spinning stas, the gavitation foe an be negleted, and we find that, in geneal: a = G R + tot ω 1 3 (3.4) new edition 13 Ap 010 fist elease 10 Ap 006

9 whih beomes zeo at an equilibium at the Compession Radius = R = R C. The angula veloity at whih this ous is given by : ω C = R G λm R 4R s (3.5) wheein have put = λ m R² as a simplifiation. The dimensionless paamete λ geneally has a value between 0 and 1. Remak that R must omply with 4R < Rs. When that angula veloity has been eahed, and the blak hole beame explosion-fee, we all the blak hole Pefet. Sine in this ase, the value of the angula veloity is high, the Newtonian gavitation is muh smalle than the gyotational one. By negleting the Newtonian gavitation, we find that a tot is zeo, fo thin ing-shaped pue blak holes, if: R C = λr 4 s (3.6) ω C R S / 4 R S / ig.3.. The Pefet MAG Blak Hole with spin veloity ω C, when the Citial Compession has been eahed. The non-explosion ondition (3.5), valid fo all ing-shaped stas, defines the exteio adius of the ing-shaped spinning sta fo a total ontinuous ompession at the equatoial level. By ompaing (3.6) with (3.1), thee is no way by finding a spinning blak hole that is simultaneously Pue and Pefet. Thus, blak holes annot be at the same time pue, and explosion-fee. ndeed, the minimum equiements fo the pefet spinning blak hole, whih annot explode and whih an disintegate obiting matte, would then be given by the ombination of the metis, given by fig.3.. Al these metis an oexist mathematially. 4. Disussion: Thee appoahes, thee impotant esults. Obiting masses at the speed of light. The fist deivation (1.15) fo finding hoizons esulted in the seah of the obit of matte taveling at the speed of light about the spinning sta. The meaning of this obit is howeve not vey lea. Could this be the hoizon of the sta? Not eally, beause this equation goes about matte instead of light. On the othe hand, is seems to be oet that no moe light an ovepass this bounday, as fa as matte effetively disintegate at that plae new edition 13 Ap 010 fist elease 10 Ap 006

10 MH Rs ω + m (1.1) But when the matte disintegates, and when it tansfom to gamma ays, these ays obey to othe ules. The gamma ays will be emitted and will in most of the ases not be ahed by the sta. The disintegation of an obiting objet nea suh a sta will indeed emit enomous gamma busts duing seonds o minutes. Suh gamma busts ae obseved and (1.1) is vey pobably the oigin of these obsevations. Longe busts ae not likely, beause patly disintegated masses beome lighte, and will look up slowe obits, laying at highe distanes fom the blak hole. Resuming, when one is puely speaking of the onept event hoizon, whih is the iula bending of light, (1.15), o (1.1), is not exatly the expeted solution. n the fist plae, the Ke meti is in ontadition with (1.15) onening its hoizon onept, beause of the doubtful ompliane of hoizons with obiting masses at the speed of light. om (1.15) follows moeove that fo non-otating stas the limit adius of the mass-hoizon beomes: ω = 0 = (4.1) = (1.16) Supisingly, the Ke meti is quasi idential to (1.15), apat fom a onstant fato, whih allows the Ke meti to obtain the Shwazshild adius as a limit fo ω = 0. But this seems moe to be an atifie. The onlusion is that the Ke meti simply has not to be onsideed as a matte hoizon. The bending of light and the Ke meti. Moe likely, the bending of light should be the oet appoah fo defining the onept of event hoizon. This happens in (.8): LH ω + = Rs + 4m (.8) Heein, the Shwazshild adius is obtained fo the limit whee ω = 0. As explained befoe, it seems muh moe logial to onside the iula bending of light as the oet definition of the event hoizon. The onept of the Ke meti is in disageement with the solution (.4), o (.8), but in ageement with (1.15). The mathematial expession (.4) has a vey simple set-up onsisting of a non-otating tem, and a tem, linea in ω, when otation ous. Of ouse, the hoizon exists only at the ondition that its adius is lage than the sta adius. Compaing both types of hoizons Compaing gaphially both equations (1.1) and (.8) gives the pitue (fig. 4.1). The adius in the uppe gaphi (iula obit at the speed of light) aises vey quikly with ineasing spin veloity. The lowe gaphi (iula bending of the light), whih is baely ineasing, stats at the Shwazshild adius. So, fo blak holes with a elatively slow otation veloity, the light-hoizon is nealy onstant at that same adius. The mass-hoizon gaphi howeve moves immediately towads highe adii new edition 13 Ap 010 fist elease 10 Ap 006

11 G ω 3 MH = ± H G K J + LH = ± H G K J + G ω 3 ig Compaing the adii of matte hoizon (MH) and light hoizon (LH). A peise alulus shows that fo an inoming objet nea a spinning blak hole, the matte hoizon always follows afte the light hoizon at a fixed distane of Rs/, whateve the spin ate is. This means that we neve an see a disintegation of matte (exept by tidal foes), beause fistly the limit of the light hoizon has to be passed. Howeve, sine spinning blak holes ae tous-like, matte disintegation at the matte hoizon an be made visible at the side of the poles of the sta. 4. Conlusion. Thee exist two types of hoizons: the fist one is based on the obital veloities of matte, obiting at the speed of light, (alled: mass-hoizon) and the seond is based on the bending of light towads a iula obit (alled: light-hoizon). Both ae puely deduted fom the Maxwell Analogy theoy fo Gavitation (gyogavitation). The mass-hoizon type has two mathematial solutions, wheeof the negative signed one isn't totally lea, but whih might epesent the inne hole of a tous blak hole. This would totally omply with ou fome pape. n an ealie pape [8], found indeed that fast spinning stas an patially explode, and that they nomally end up in tous-shaped blak holes. This fist type of hoizon (mass-hoizon) allows me to find a vey plausible oigin of gamma busts whih last fo seveal seonds o minutes: the disintegation of mass at the speed of light (whih beame invisible to the eye) into gamma ays, whih suddenly beome then visible, beause the light annot be bent as muh in ode to emain aptued. The Ke meti is almost idential to the MAG light-hoizon, in ode to get the Shwazshild adius as a limit fo non-otating blak holes. The MAG light-hoizon defines the event hoizon of blak holes in its pue fom, as the ultimate iula bounday of visible light about the blak hole. Both hoizon types an oexist, but at some vey low and vey high spin veloities, the light-hoizon obsues the masshoizon, so that even gamma busts might totally be aptued by the spinning blak hole, whih might hold these busts invisible, unless they an esape via the poles of the ing (tous) blak hole, as explained in an ealie pape [8]. Beyond these dedutions, the adii of spinning and non-spinning blak holes ae found, as a speial ase of the lighthoizon. inally, the spin veloity of blak holes with ontinuous ompession has been found new edition 13 Ap 010 fist elease 10 Ap 006

12 5. Refeenes. 1. eynman, Leighton, Sands, 1963, eynman Letues on Physis Vol.. Heaviside, O., A gavitational and eletomagneti Analogy, Pat, The Eletiian, 31, 81-8 (1893) 3. Jefimenko, O., 1991, Causality, Eletomagneti ndution, and Gavitation, (Eletet Sientifi C y, 000). 4. Jefimenko, O., 1997, Eletomagneti Retadation and Theoy of Relativity, (Eletet Sientifi C y, 004). 5. De Mees, T., 003, A oheent double veto field theoy fo Gavitation. 6. De Mees, T., 004, Cassini-Huygens Mission. 7. De Mees, T., 004, Did Einstein heat? 8. De Mees, T., 005, On the shape of otay stas and blak holes. 9. De Mees, T., 006, On the obital veloities neaby otay stas and blak holes Ke Meti new edition 13 Ap 010 fist elease 10 Ap 006