The Kerr-metric, mass- and light-horizons, and black holes' radii.

Transcription

1 006 Thiey De Mees The Ke-meti, mass- and light-hoizons, and blak holes' adii. (using the Analogue Maxwell theoy) T. De Mees - pandoa.be Abstat Blak holes an geneally be 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 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 outome fo otating and non-otating blak holes' hoizon, using the Maxwell Analogy fo Gavitation (MAG) (5,6,7,8) (o histoially moe oet: the Heaviside () Analogy fo Gavitation), find that the Ke meti must be inoet in elation to the definition of hoizons of otating blak holes. f the Maxwell Analogy fo Gavitation is supposed to be a good appoah of GRT, we may assume that the Maxwell Analogy is a valid analysis tool fo the sta's hoizon metis. athe find that the Ke meti does puely not defines typial hoizons at all, but only a kind of obit vanishing ondition of spinning stas ( all the Ke-type of hoizon futhe the mass-hoizon ). Moeove, the meti is not omplying with MAG but by a fato of two. find a seond meti fo the definition of hoizons, based on the bending of light ( all this futhe the light-hoizon ). 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 onditions fo explosion-fee blak holes. The dedutions ae based on the findings of my papes Did Einstein heat? (7), On the geomety of otay stas and blak holes (8) and On the obital veloities neaby otay stas and blak holes (9). Keywods. Maxwell Analogy gavitation otay sta blak hole Ke Meti toe gyotation hoizon Methods. analytial Gaphs. WZ-Gaphe ndex 1. The obital veloities neaby Rotay Stas and Blak Holes. ntodution: mass-hoizons and the Ke Meti. At what onditions will matte obit at the speed of light? The toe shape of fast spinning stas.. The bending of light into a iula obit. ntodution: what is a light-hoizon? What speifies the light-hoizon of blak holes?. Deiving the adius of Pue Blak Holes. The adius of a ing-shaped Pue Blak Hole. Ae Pue Blak Holes explosion-fee? 4. Disussion: Thee appoahes, thee impotant esults. Obiting masses at the speed of light, and the Ke meti. The bending of light and the Ke meti. Compaing both types of hoizons. Speial ases. 5. Conlusion. 6. Refeenes. 1

2 006 Thiey De Mees pandoa.be 1. The obital veloities neaby Rotay Stas and Blak Holes. ntodution: mass-hoizons and the Ke Meti. Shwazshild found one 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. 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's adius, we an speak of a 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. n my fome pape onening obital veloities (9), have deived the obital veloity neaby otay stas and blak holes, using the Maxwell Analogy fo gavitation (usually named 'gavitomagnetism' o 'gavitation and o-gavitation', o what pefe to all gyogavitation). The obital veloity is always highe than the lassial Kepleian obital veloity. The esult is: v = v obit = v k 1 + v k sω + v k s Ω (1.1) b g wheein have named the Keple veloity v k : v k = (1.) and wheein have defined the angula spead s Ω (dimension of invese veloity [s/m]) as : s Ω = θ (1.) Hee, θ is defined as the speifi angula density of the spheial sta (dimension of time [s]): θ sphee = R ω 10 (1.4) At what onditions will matte obit at the speed of light? By putting v obit =, we an find 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 (1.), (1.), (1.4) and v obit = in (1.1), we get: R ω + = 0 5 (1.5) This equation is quadati in if we multiply it by ². And of the two solutions, keep only the positive one fo now: Gω MH + = + H G K J + (1.6) May 006 elease 8/05/006

3 006 Thiey De Mees pandoa.be wheein we have eplaed the inetial moment of the sphee by (see 1.7). = m R 5 (1.7) Thus, the faste the sta spins, the fathe away fom it, the obital veloity of light an yet be eahed. And fo nonotating blak holes, the obit adius beomes: = MH, ω = 0 (1.8) whih is half the Shwazshild adius. t is pobable that (1.6) 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. n the following lines, simplify (1.6) fo fast spinning stas with masses of at least that of the sun. Equation (1.6) beomes afte some manipulation: Gω MH + = + H G K J + (1.9) The seond tem unde the oot sign is muh smalle than 1 fo all the known stas and blak holes. Thus, knowing that: 1 x << 1 1+ x 1+ x (1.10) it follows that: ω MH + + (1.11) Sine the definition of the Shwazshild adius is : R = s (1.1) the equation (1.11) an be e-witten as: MH Rs ω + + m (1.1) The equation (1.1) shows that the evolution of the mass-hoizon adius is 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. Remak that the negative solution of the quadati equation (1.5) 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 toi blak hole, but this option meits a lose study. Gω MH = H G K J + (1.14) n my fome pape On the shape of blak holes (9) demonstated, using MAG, the high pobability of toi blak holes when they spin fast. The two mass-hoizons that found hee ould signify the onfimation of my ealie finding. Both (1.6) and (1.14) esemble the Ke meti hoizon solutions. But they diffe with a fato of two fom the MAG meti. Hee, the equations desibe the limit onditions of an obital veloity of matte at the speed of light. n the disussion hapte, these mattes will be futhe explained. May 006 elease 8/05/006

4 006 Thiey De Mees pandoa.be n the following lines, simplify (1.14) fo fast spinning stas and blak holes. Equation (1.14) beomes afte some manipulation: = MH ω (1.15) The seond tem unde the oot sign is expeted to be fa smalle than 1. Hene, knowing that: 1 x << 1 1+ x 1+ x it follows that fo fast spinning stas, the seond mass-hoizon beomes: MH ω m (1.16) (1.17) Remak that (1.17) is independent of the mass, but dependent of its geomety. The toe shape of fast spinning stas n the pape On the shape of otay stas and blak holes dedut that fast spinning stas ae toe-shaped. Can this also be deduted fom the MAG mass-hoizon? ndeed, in the same pape, ome to the onlusion that when patiles aive in the toe's hole, the only steady motion is a iula equatoial obit whih is etogade to the toe's spin. When looking at (1.17), thee is a supising minus sign. And this is pefetly omplying with the pedited etogade obit. When (1.1) and (1.17) 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. R S / The spinning sta's mass-hoizons ig.1.1 Thus, aoding (1.1) and (1.17), the shape of the mass-hoizon at the fast spinning stas' equato is disk-like with an empty ental zone, and it an be expeted that suh spinning stas ae toe-like with a thikness below R S /. This hapte gave the solution fo the zone neaby the blak hole whee matte tends to obit at the speed of light and the onfimation of the toe-like shape of spinning stas. Befoe disussing the findings of this hapte moe in depth, fist study the geneal poblem of the bending of light neaby blak holes. May elease 8/05/006

5 006 Thiey De Mees pandoa.be. The bending of light into a iula obit. ntodution: what is a light-hoizon? Anothe appoah ould be the study of the bending of light by the spinning sta. 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 lighthoizon. o this pupose, we take the solution whih we have found in Did Einstein heat? (4), equation (6.14), witten in its geneal fom. = G mm + G mm + (.1) G mm R ω ϕ ϕα, os α os ϕ 1 5 v This equation desibes the bending of light, taking in aount thee foes and thus thee tems, based on : 1 the pseudo-gavitational effet, whih is two times the value of the Newton gavitation, the dagging veloity (in the pesent ase: of the Milky Way) and the sta's otation (in the pesent ase: the sun). And found this equation to be fa moe auate than the GRT deivation. The vey impotant 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 mass steam of the light wave itself, tavelling in the gavitation field of the sta. 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. The last tem in (6.14) (in Did Einstein heat? ) omes fom the basi equation (4.) fo sphees, in A oheent double veto field theoy fo Gavitation. Ω ext R 5 ω ω Combined with (1.1) of the same pape, Ω = m (g + v Ω), wheein g = 0, this beomes fo the equato plane: b g (.) R ω 5 Ω = (.) Besides staying at the equato level of the sta only, onside aeleations instead of foes. So, the aeleation beomes: a = G m +G mr ω (.4) 5 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 m G mr ω = + 5 (.5) The solutions fo the light hoizon's adius ae given by: LH G = Rs + 4 ω (.6) May elease 8/05/006

6 006 Thiey De Mees pandoa.be wheein R s is the Shwazshild adius, and the inetial moment of the sta (in (.) this was a sphee) : R = s m R (.7) = sphee (.8) 5 Equation (.4) is thus desibing the bending of light beams in iula obit about blak holes. Hoizons annot bette be defined than with this equation. n the disussion hapte, it will beome lea why this is so. R S / R S +G ω /( 4 ) The spinning sta's mass-hoizons and its light-hoizon ig..1 n figue.1, the light-hoizon's diamete is lage than the extenal mass-hoizon diamete. This is not always the ase, as will explained in the disussion hapte. Remak that (.6) an be expessed in tems of the sta's equato veloity v = ω R. Assuming that the inetial moment an be expessed as a fato λ, multiplied with m R², the expession beomes independent fom R. Setting = λ m R², equation (.6) beomes: λv LH = Rs 1 + (.9) 4 o ing-shaped blak holes, λ = 1.. Deiving the adius of Pue Blak Holes. The adius of a ing-shaped Pue Blak Hole. f, as found, (.6) 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's 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. n (.), the sphee's moment of inetia an be eplaed by (thee, it was fo a sphee). Equation (.5) an then be eplaed by: G m G ω = + (.1) As explained in fome haptes, blak holes ae pefeably toe-shaped, even pobably thin ing-shaped fo fast spinning stas. assume the latte in ode to simplify the alulations, and will hek the validity of the simplifiation aftewads. o thin ings, = m R², whee R is the adius at the equatoial level of the sta. May elease 8/05/006

7 006 Thiey De Mees pandoa.be n (.1) the value fo ω equals v/r. The solution to the light-hoizon fo ing-shaped blak holes is then: LH v = Rs whih is indeed idential to (.9) with λ = 1. Symbol R s is again the Shwazshild adius: R = s (1.1) f the adius of the blak hole oinides with the adius of the light-hoizon, one finds the pue blak hole's adius. This happens fo any veloity v in (.) smalle than, and (.) gives than the oesponding adius = R. The veloity at whih the sta's equatoial adius oinides with the light-hoizon is than expessed in tems of R : (.) v eq R eq = 1 λ R s (.) With (.), obtain a iula bending of light upon the equato of the sta itself. Light annot esape, and the hoizon is the sta's equato. Hene, an desibe patial blak holes, wheeof a pat is invisible, even when obseved fom the poles. As said befoe, all them pue blak holes. t follows immediately that fo non-spinning stas (v = 0), the adius R beomes the Shwazshild adius R s. f one puts v = 0 in (.), indeed the Shwazshild adius is found : The maximum veloity is. The esults ae then: v = 0 = R = R eq LH eq s λ v eq = LH = Req = Rs (.4.a) (.4.b) The esult of (.) needs moe laifiation. The gaphial pesentation of fig..1 will help us a lot. 5/4.R s R s 0 Ring-shaped blak hole (λ = 1) : Pue Blak Hole's light-hoizon adii. ig..1 The gaphi evolutes (fo λ = 1) fom R s to 4/5.R s with ineasing veloity of the ing-shaped blak hole's equato, until the equato's veloity eahes the (theoetial) maximum veloity of light. Equation (.) is beautifully desibing the equied adius at the equato level of pue blak holes (with spin o not). May elease 8/05/006

8 006 Thiey De Mees pandoa.be RR / S / s R < R < (1+λ²/4) R S.R s eq s The Pue Blak Hole's light-hoizon and the mass-hoizons ig..1 t is then lea that if depit this gaphially, get fig..1., wheein show the light-hoizon (lage dak bounday) and the mass-hoizons (ed boundaies) as well. mmediately, it beomes lea that the Ke meti will not omply with the shape of a toe-like blak hole, beause the Ke meti has a distane of R s between both mass-hoizons as well, instead what found with MAG: R s /, whih then omply without any poblem. Ae Pue Blak Holes explosion-fee? n a fome pape (8), have deduted the adius of ontinuous ompession of spheial spinning stas at the equato level (with negligible only-gavitation influene). This dedution was based on the gyotation field equations fo a sphee, and we use (.8) in ode to obtain a moe geneal equation. The minus sign is added fo the onvention of attation. 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 gyotation foe is given by the analogue Loenz foe a = ω R Ω (.6) and the last tem of (.5) is zeo fo Ω y. Ω ext G Hene, the aeleation due to gyotation at the equato plane is: ω ω b x g y (.5) a R G x = ω (.7) 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: G atot = ω R 1 whih beomes zeo at an equilibium when the aeleation a tot is zeo. The Compession Radius = R C an then be found fo blak holes with an inetial moment in the fom = λ m R² : (.8) = R = C λ Rs R 4 (.9) May elease 8/05/006

9 006 Thiey De Mees pandoa.be whih beomes explosion-fee at the equato if one puts R = = R C. R C R s = λ (.10) ω C R R/ S / s R< R S λ / / 4.R s The MAG explosion-fee Blak Hole with spin veloity ω C ig.. and any spin veloity ω C, lage enough, is valid. The non-explosion ondition (.9), 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. Sine the ondition (.9) always gives muh smalle adii than the ondition (.), explosion-fee blak holes ae always at the same time pue. ndeed, the minimum equiements fo the 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... Al these metis an oexist mathematially. 4. Disussion: Thee appoahes, thee impotant esults. Obiting masses at the speed of light, and the Ke meti. The fist deivation (1.6) fo finding hoizons esulted in the seah of the obit of matte tavelling 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. At 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. Gω MH + = + H G K J + (1.6) 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 (1.6) is vey pobably the oigin of these obsevations. Longe busts ae not likely, beause patly disintegated masses beome lighte, and will lookup slowe obits, laying at highe distanes fom the blak hole. Resuming, when one is puely speaking of the onept hoizon, whih is the iula bending of light, (1.6) is not exatly the expeted solution. May elease 8/05/006

10 006 Thiey De Mees pandoa.be So is the Ke meti in ontadition with (1.6) onening its hoizon onept, beause of the doubtful ompliane of hoizons with obiting masses at the speed of light. om (1.6) follows moeove that fo non-otating stas the limit adius of the mass-hoizon beomes: ω = 0 = MH 0 R s (4.1) Supisingly, the Ke meti is quasi idential to (1.6), 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 annot be deived by using the Maxwell Analogy. t ould howeve have been aeptable if the two onuent metis wee at the end almost idential, but a diffeene of a fato of two is not aeptable at all. The bending of light and the Ke meti Moe likely, the bending of light should be the oet appoah fo defining the onept of hoizon. This happens in (.6): Gω LH = Rs + 4 (.6) 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 hoizon. The hoizon onept of the Ke meti is in total disageement with the solution (.6). The mathematial expession (.6) has a vey simple set-up onsisting of a non-otating tem, and a tem, quadati in ω, when otation ous. Of ouse, the hoizon exists only at the ondition that its adius is lage than the sta's adius. Compaing both types of hoizons Compaing gaphially both equations (1.6) and (.6) gives a quite amazing pitue (fig. 4.1). The adius in the uppe gaphi (iula obit at the speed of light) aises 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. Gω MH + = + H G K J + R s a LH ω G = Rs + 4 ig. 4.1 R s May elease 8/05/006

11 006 Thiey De Mees pandoa.be The ossing point a is given by the equity of (1.6) and (.6) in that point, and by onsideing that fo low spin veloities ω, the seond tem of (.6) is negligible. Hene, ossing point a is defined by : ω << ω a 8 a Rs (4.) Remak that beyond the ossing point a of both gaphis, the obiting masses will disintegate (mass-hoizon) even befoe they ome in the light-hoizon. Befoe the ossing point a, the obit beomes invisible befoe the disintegation of the mass. But when stongly zooming out the figue 4.1 fo quite high spin veloities, say, milliseond blak holes, the lighthoizon gaphi hanges its shape (fig. 4.). LH G = Rs + 4 ω b Gω MH + = + H G K J + ig. 4. The value of ω and at the ossing point b also follow fom the equity of (1.6) and (.6) in that point. A simplifiation of the ossing-point-equation depends howeve fom the values of m, and ω. At lowe spins than the ossing point b, obiting masses will disintegate befoe they ould be invisible. This senaio is valid fo a vey lage ange of spinning veloities. uthe away fom the blak hole, beyond the ossing point b, the light-hoizon will be attained ealie than the mass-hoizon, although it is not known yet if suh fast spinning blak holes do eally exist. Speial ases. The speial ases of hapte onen pue blak holes, defined by its patial o full obsuation, due to the total bending of the light in an obit about the blak hole. The maximum adius fo pue blak holes is 4/5.R s, at equatoial veloities of the speed of light. At zeo veloity, still a adius of R s is needed. Heavie, moe ompat stas should be pue blak holes as well. So ae the explosion-fee spinning stas, whih have adii of λ R s /, sine it is aeptable that λ 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. May elease 8/05/006

12 006 Thiey De Mees pandoa.be The mass-hoizon type has two mathematial solutions, wheeof the one with negative sign vey pobably epesent the inne hole of a toi blak hole. This would totally omply with ou fome pape (8), whee found that fast spinning stas an patially explode, and that they nomally end up in toe-shaped blak holes. This fist type of hoizon (masshoizon) allows me to find a vey plausible oigin of the gamma busts whih last fo seveal seonds o minutes: the disintegation of mass at the speed of light into gamma ays, whih suddenly beome then visible to detetos, beause the light annot be bent as muh in ode to emain aptued by the spinning sta. The Ke meti is almost idential to the MAG mass-hoizon, exept fom the fato two, whih looks like being an atifie, in ode to get the Shwazshild adius as a limit fo non-otating blak holes. The MAG light-hoizon defines the hoizon of blak holes in its oet 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, as explained in an ealie pape (8). Beyond these dedutions, the adius of spinning and non-spinning pue blak holes ae found, as a speial ase of the light-hoizon. inally, the ondition fo pue blak holes with ontinuous ompession has been found. 5. Refeenes. 1. eynman, Leighton, Sands, 196, eynman Letues on Physis Vol.. Heaviside, O., A gavitational and eletomagneti Analogy, Pat, The Eletiian, 1, 81-8 (189). 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., 00, 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 May elease 8/05/006