The theory of relativistic spontaneous emission from hydrogen atom in Schwarzschild Black hole

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1 Amica Jual f Astmy ad Astphysics 01; (6): Publishd li Dcmb 9, 01 ( di: /j.ajaa IN: (Pit); IN: (Oli) Th thy f lativistic sptaus missi fm hydg atm i chwazschild lack hl Jahagi A. Da Ivat at UIC Dpatmt, Uivsity f Kashmi, iaga, J&K, Idia addss: Jahagiahmad6@gmail.cm T cit this aticl: Jahagi A. Da. Th Thy f Rlativistic ptaus Emissi fm Hydg Atm i chwazschild lack Hl. Amica Jual f Astmy ad Astphysics. Vl., N. 6, 01, pp di: /j.ajaa Abstact: Th mchaism f sptaus missi adiatd by th lativistic hydg atm fallig adially twads chwazschild black hl, th basis f Nwtia mchaics ad h s atmic thy is pstd h. Th gy adiatd by this hydg atm is calculatd as, ζ= A R (m τ/ )-m c, wh, A R is a cstat. Th lati f Ltz factγ f lativity with mass f cllapsd sta ad m iitial mass f paticl is als divd. Futhm, Hawkig s gy lati f black hls has b divd als fm th sptaus gy latis usig sam buday cditis th Hawkig adiati pssss. Th fqucy f gy spctum has b fud fall i gamma gi f lctmagtic spctum with ag f 10 ad 10 Hz. Kywds: avitatial Radiati, Elctmagtic Radiati, chwazchild lack Hl, Hawkig Radiati, Nwtia chaics, h Atmic mdl ad pcial Rlativity 1. Itducti Th gavitati adiati ith i th cllisi pcss fall f lativistic paticls [1,, 1, 1, 17] has b f csidabl itst fm last fw ctuis. It is blivd als that lctmagtic adiati is pssibl if th chagd paticls lik lct is mvig i stg gavitatial fild [1,,5,6,7, 1, 1, 17] lik chwazschild black hl [] K black hl [], but th lctmagtic pcss is still i d t b studid m. T kw m abut th phm f lctmagtic adiati fm a chagd paticl mvig i stg gavitatial fild, th wks a d i Rff. N. 1-7 is advisabl t ad. T calculat adiatd gy fm lativistic paticl lik Ruffii [6] ad Cads, Ls ad Yshida[1] did, Ltz fact cmmly calld as gamma fact f lativity, γ, has b usd with its difft valus lik, γ << 1by Ruffii [6] ad γ 10 by Cads, Ls ad Yshida[1]. F ay paticl with γ, it is much difficult t umically btai th ttal adiatd gy missi [1]. It must b td that f th ttal gavitatial gy, γ fact is tatd sam as i lctmagtic adiati gy. Th lati f gavitatial gy adiatd is ξ g = 0.6m γ ad f lctmagtic gy adiatd is ξ = ( ) γ 0.08q γ lg (wh γ ), h, m is mass f paticl, is th mass f lack hl, q is th chag f th paticl ad γ = ( 1 β ) 1/, β = v / c wh v is th vlcity ad c = x 10-8 ms -1 is spd f light i vacuum. Th bauty f this pst pap is that th lati f γ with th facts lik mass f gavitatial suc ad iitial mass f paticl itslf is dtmid. m f physicists lik m bliv that v sptaus missi is pssibl fm th lativistic paticls, i stg gavitatial fild but th is t ay wll dfid pcss which xplais th missi. Th ly slight ida abut th sptaus missi i lack-hl-ystm is availabl i Hawkig s adiati pcss [10, 1], wh it is agud that black hl mit thmal adiati via a quatum sptaus missi [10] but it is t th ppty f paticl as it ds i lctmagtic gavitatial adiati missis. Th mai mtiv f this pap is t dfi th phm f sptaus missi by lativistic sigl lct atm lik hydg i a stg gavitatial fild lik chwazschild black hl. This idd is th fist attmpt that has b v tak i th scic t dfi th pcss f sptaus missi fm such atm i ay stg gavitatial fild by usig cmbiati f Nwtia mchaics, lativistic ffcts ad h s atmic mdl [, ]. a whil, with sptaus gy missi ξ s by th

2 Amica Jual f Astmy ad Astphysics 01; (6): atm, th lati f its lativistic Ltz fact γ, i ay gavitatial fild with th mass f its suc ad th iitial mass f lct m f sigl lct atm has b als dtmid i th pap, als s that t dtmi th Hawkig s gy lati, ħc ξh =. 8π. Fmulati Csid ay atm adially fallig twads ay tatig cllapsd sta lik chwazhild black hl f mass. Duig its adial fall its distac x fm th ct f cllapsd sta dcass ad at ay istat f tim du t Nwtia mchaics [11, 15], th fc f attacti, F, is giv as; m F = (1) x wh, is th gavitatial cstat, m is th mass f atm ad x is th distac btw th gavitatial cts f th tw bdis. Takig th gavitatial ffct th cstituts f atm, i.. lcts ad uclus idpdtly, thf, w hav gavitatial attactis f lct f mass m ad uclus f mass m as; m F = () x m F = () x Th ttal attacti F btw th atm ad th cllapsd sta is F = F + F. () ic th adial f fall f atm its vlcity xcds with dcas i x, thf du t spcial lativity th mass f atm ad thus f its cstituts xcds. This xcd i mass [15, 16] is giv as; ad f its cstituts as, ( ) 1/ m = m 1 β = m γ (5) * ( 1 β ) 1/ m γ m = m = * ( 1 β ) 1/ m γ m = m = Nw, csid th hydg atm, 1 H, csistig f a lct, (6) (7), ad pt, p, fallig adially twads a chwazchild black hl. th ad p will pssss gavitatial attacti with th chwazschild black hl f mass. Ttal gavitatial fc will b giv by th q. (1). Rlativistic mass f its cstituts is giv by q. (6) ad (7). I h mdl [, ] f 1 H atm, th adius ccupid by a lct vlvig aud th uclus is giv as; Wh lativistic ffcts a tak a a πε ħ = (8) m πεħ = (9) m γ Rlativistic cditis qui a < a, sic m γ > m. Thus th is ctacti i 1 H atm duig its f fall twads ay stg gavitatial sta. uch ctacti ffct has b studid f may atms lik Hg, Ag, tc, s f.. [18,19]. Hwv, th th ffct is du t th bital mti f aud th uclus [18,19]. If th dcas i bital adius a is quit high, as pssibl if th atm fall twads stg gavitatial fild as i u 1 H, it will ld t th tasacti f lct fm gy lvl t ath. This tasacti f lct fm high gy stat t lw gy stat ld t th las f sptaus missi fm th 1 H atm. Th amut f gy lasd i th fm f sptaus missi is qual t th diffc btw gis pssssd by lct at its iitial gy lvl, wh x, ad fial gy lvl at x, i.. R ξ = ξ f ξ i (10) Usig h s mdl f atm [, ], w hav, gy ξ ccupid by lct at ay gy lvl as; ξ = m π ε ħ (11) i q. (11), ε is pmittivity f f spac, is th gy lvl als calld as picipal quatum. i quatum mchaics, ħ is ducd Plack s cstat ad π is.1 (appx.). Thf, q. (10) tasfms as; wh, λm lvl f at ξ = m ( γ 1) π ε ħ ( is th mass pssssd by 1 ) (1) at its w gy R x ad m mass at iitial gy stat i at x y Eisti s gy-mass quivalc lati, ξ = mc [0, 1]. Usig this quivalc lati ad istig th q. (11) i q. (10), th tasfmd lati cms ξ = ( γm m)c = m( γ 1) c (1)

3 68 Jahagi A. Da: Th Thy f Rlativistic ptaus Emissi fm Hydg Atm i chwazschild lack Hl wh, m is th ttal chag i mass f by th lativistic ffcts, qual t th diffc btw th lativistic mass γm ad iitial mass m f. Th lativistic ffctiv mass f pt has b glctd sic th adiati ca b assumd ly by th fall f lct fm gy lvl t th, ad ttal chag i distac btw th cts f tw paticls is sam t th chag i + distac f fm th ct f p.. Divig Valu f Ltz Fact As statd ali i all calculatis latd t gy missi by lativistic paticl i stg gavitatial fild, th valu f Ltz fact [] is assumd. H w will div it lati with th stgth f cllapsd sta. Du t th ifluc f f gavitatial fc adially fallig 1 H, its mmtum chags with tim τ f f fall, ad thus mmtum f chags. Takig γ m v, as fial mmtum, say smwh a vt hiz ad m v as iitial mmtum smwh at ifiity ad τ as tim tak f ttal mmtum chag by, th Nwtia mchaics [11] says, F γmv mv γmv = (1) τ τ m = Th valu f iitial mmtumm has b glctd v by assumig th atm was iitially at st it was mvig twads ath dicti th tha th dicti f its adial fall. I th ma tim lct as it satisfis q. (1), it must als satisfy th gavitatial lati i q. (), i.. F = F. m Aft sm tasfmatis th cditi divs th lati f Ltz fact as; τ γ = (15) x v Ay ifmati abut ay paticl fallig twads black hl ca b btaid as lg as it sids byd vt hiz f adius R, what happs t paticl aft it csss it physics is kw t dfi it, thf, th valu f Ltz fact ca g maximum at x = R. Wh R is chwazschild s adius, giv as R = c. Itchagig x by R i q. (15), w aiv th sult c τ γ = (16) v What th smat cclusis w hav i q. (15-16)! F ay adially fallig paticl twads th cllapsd sta, Ltz fact dpds als τ ad, t ly vlcity vas w kw s fa.. Tasfmati ptaus Egy Emissi Rlatis Eq. (15) ad (16), has ld us t w valus f Ltz factγ, ths w valus f γ ca b implatd it u pviusly divd latis f sptaus missis, as q. (1 & 1), f 1 H duig its adial f fall du t th chwazschild black hl. Th tasfmd latis a; ξ = τ m 1 x v π ε ħ ( ξ = 1 c τ m 1 v π ε ħ ( ) τ x v ξ = m( 1) c at 1 c τ ), at R x (17) ξ = m ( 1) v c at x, at R x (18) R x (19) R (0) Eq. (17 & 18) giv th lati f sptaus gy missi by 1 H duig its adial fall twads chwazschild black hl, i tms f h s atmic mdl at cditis wh it is a t vt hiz fa fm it, whil at th sam tw cditis, Eq. (19 & 0) giv th lati f sptaus missi i tms f Eisti s mass-gy quivalc lati [0, 1]. 5. Divati f Hawkig s lack Hl Egy Emissi Rlati tph Hawkig suggstd that black hls pssss tmpatu [1, 5] wll kw as Hawkig tmpatu, TH = ħc /8π K, wh k is ltzma s cstat, thf, mit gy [1, 5], ħc ξh = kth = (1) 8π Th ly pssibility f th fmati f Hawkig adiati is at vt hiz, that is, f Hawkig adiati it must satisfy th cditi, ξ H = ξh ( R ). Ou latis 18 & 0, f sptaus gy missi du t lativistic atm als satisfis th sam cditi, ξ = ξ ( x R ). Nw, csid i q. (0), v c ad τ as sm cstat ad iitial gy m c = 0, thf, th said lati gts tasfmd as; c ξ = m ( ħ c c m j ) τ = () 8π

4 Amica Jual f Astmy ad Astphysics 01; (6): Wh, j = πc τ / ħ is a cstat ad τ is tak as ay kw quatity. It is cla fm q. (), that th Hawkig gy missi is a simpl cas f u sptaus missi wh th mass m is glctd. Elswh, Hawkig s gy missi is th ppty f th mass f black hl ly, it ds t dfi th fact that by quatum mchaics as all typs f pai paticls lik lct-uti, tau-atitau, tc, ca b catd by vacuum du t stg gavitatial fild as p Hawkig pcss th what is th ffct adiatd gy du t th fall f ths difft paticls pssssig difft pptis is t dfid ad this idd is f th gatst dawbacks f Hawkig s adiati pcss. Nw lk at u q. (), it dfis claly th ffct adiati missi du t th fall f difft paticls with difft pptis lik mass twads black hl. Hwv, i Hawkig pcss th black hl lsss sm f its mass [1, 5], but i u sptaus missi it is t dfid ad has b mad mystius h. I my upcmig wk am gig t xplai ad slv th mysty. T div Hawkig adiati w hav glctd sm valus f q. [0] ad ducd it t Hawkig s lati, hwv th actual lati ca b tasfmd ad ducd t simpl lati as; m τ ξ = AR mc at x R v = c =, () wh, AR is a cstat f valu = 1.50 x L Kg s -5 N -1. Nw lk at u q. (), it dfis claly th ffct adiati missi du t th fall f difft paticls havig difft st masss twads black hl, mass f black hl, tim τ ad th iitial gy f paticl. Claly, th adiati du t th fall f paticl mvig with th spd f light at vt hiz dpds fuctis m, ad τ, that is, ξ ξ ( m, τ ) =., 6. Numical Rsults Fig. 1. Th lctmagtic spctum f th hydg atm fallig it chwazschild black hl at x = R ad by csidigτ =1. I d t btai umical sults as wll bad spctum adiatd by sigl lct hydg atm duig its adial fall i chwazschild black hl, i tms f fqucy, q. () is csidd, that is, wh adiatd gy fllws th cditi x = Rs. F calculatig th sults, th valuτ =1 ad th mass m= 9.1 x 10-1 Kg as th stadad valu f lct has b tak. Fig. 1 shws th fqucy υ ag f this lct H atm wh it fall it chwazschild black hl f mass tak i sla masss.i divig th gy spctum Plack s cstat h has b als itducd f lati υ = ξ h. Th gy spctum fm th calculatis is calculatd t fall i gamma gi f lctmagtic spctum f ag 10 Hz ad 10 Hz. I Fig., a typical sult f th valu f Ltz fact γ is O shw, by csidig th sam as cditis as usd f gy spctum. Th quati. 16 has b csidd f sults. H m assumpti has b tak, that is, v c. It is bvius fm th tw figus that th slp f sults is simila, this mas th gy spctum dpd th γ fact ly ad is idpdt th iitial gy m c f th fallig hydg atm it th chwazschild black hl. Thf th tm m c ca b glctd fm th quati ad thus th quati ca b simply witt as; m τ ξ = A at x = R, v = c () R

5 70 Jahagi A. Da: Th Thy f Rlativistic ptaus Emissi fm Hydg Atm i chwazschild lack Hl If that is, m c culd hav b gat tha th tm m c > A R A m τ, R m τ, th th slp i FI. 1 shuld divg twads th staight li, wh it is t. Thf th iitial gy tm ca b glctd i th Eq. (). Fig.. Vaiati f Ltz fact λ f th hydg atm fallig it chwazschild black hl at x = ad by csidig τ =1. R 7. Cclusi Th sptaus gy missi, has b divd th basis f Nwtia ad Rlativistic mchaics with th hlp f h s Atmic dl. Egy latis as quatis [17, 18, 19, 0, ] has b divd usig ths classical idas, at difft cditis. It is fud i sm gads, th gy missi is simila as hav b pdictd by Hawkig s agumts. Th ly diffc btw Hawkig s ad this classical lativistic adiati missi fm black hl systm is i th fm gy missi is idpdt th ppty f fallig paticl ad i th lat th adiati missi is dpdt th iitial mass as wll iitial gy f th fallig paticl, as bvius fm q. [1, ]. Ackwldgmts I pfmd this wk withut th suppt f ay gat fm ay pivat gvt. istituti/agcy. I am thakful t my cllg Pf. D. hashikath upta at Amity Uivsity Hayaa f pvidig m th quid ifmati gadig plttig f gaphs, ad am dply thakful t my family mmbs f pvidig m all quid fiacial suppt ad spcially duig my stay at Amity Uivsity Hayaa, Idia. Rfcs [1] V. Cads, J. P.. Lms,. Yshida, Phys. Rv. D 68, (00). [] V. Cads, J. P.. Lms, Phys. Ltt. 58, 1 (00). [] V. Cads, J. P.. Lms,. Rl. avitati 5, 7- (00); Phys. Rv. D 67, (00). []. Chadaskha,. Dtwil, Pc. R. c. Ld A, 1 (1975);. Chadaskha, Th Thy f lack athmatical Hls, (Oxfd Uivsity Pss, Nw Yk, 198). [5]. Jhst, R. Ruffii, F. Zilli, Phys. Ltt. 9, 185 (197). [6] R. Rufii, Phys. Ltt. 1, (197). [7] F. Zilli, Phys. Rv. D, 11 (1970); Phys. Rv. D9, 860 (197). [8] Hawkig,. W., Cmmu. ath. Phys., 199-0(1975)

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