Stefan-Boltzmann s Law under Relativistic Conditions

Transcription

1 Psial Rviw & Rsar tratioal 4(): 7-, 4 SCENCEDOMAN itratioal Stfa-Boltzma s Law ur Rlativisti Coitios E. V. Vitsma * Vitsma s Si Projt, 8 Apartmt, 5 Klimaski Str., Mosow, 557, Russia. Autor s otributio ol autor prform t wol rsar work. Autor EVV wrot t first raft of t papr. Autor EVV ra a approv t fial mausript. Rsar Artil Riv 5 t Ma Apt t Sptmbr Publis 6 t Otobr ABSRAC A xprssio was obtai for t rg sit of t movig blak-bo raiatio, i.., t Stfa-Boltzma law vali i t itrval of objt vloitis from zro to t vloit of ligt i vauo. objt tmpratur is sow to ompris two parts. first o is a salar ivariat ur t Lortz trasformatios. so o is a vtor pig o t vloit of sstm motio. salar ompot of t tmpratur is a otratio of two tsor ompots of rak. Ur ormal oitios tis matmatial objt is a salar. akig aout of a tsor aratr of t tmpratur a w formulatio is giv for t so trmoamis law. rsults obtai ar of t grat pratial importa, i partiular, wil sigig vis to masur t raiatio tmpratur of movig osmi objts,.g., quasars. Kwors: Blak bo; raiatio; Stfa-Boltzma law; tmpratur; spial rlativit.. NRODUCON problm of t movig blak-bo raiatio aros i 97 almost immiatl aftr t ratio of Spial rlativit (SR). t is i tis ar tat Kur vo Mosgil s big artil was publis i r Aal r Psik []. is work suprvis b Max Plak urlis is rlativisti trmoamis []. grat sitist osir t tor of t blakbo raiatio to b wll-stui a t most suitabl for formulatig fouatios of * Corrspoig autor: v_vitsma@mail.ru;

2 Psial Rviw & Rsar tratioal, 4(): 7-, 4 trmoamis orrt ovr t tir wol itrval of objt vloitis v, i.., ragig from zro to t vloit of ligt i vauo. artil [] a sstm is stui omprisig a raiator of ltromagti wavs, rivr a rfltor (mirror). raiators ar rivrs at t sam tim. tr lmts ar movig uiforml a rtiliarl i spa wit a rlativisti vloit formig a aut agl wit o aotr. As a rsult, t tmpratur trasformatio law was obtai ur rlativisti oitios:, () wr is t tmpratur if v<< (r a blow ix mas tat t giv quatit ors ormal oitios); v /. For mor ta 5 ars formula () a ot b all i qustio util X.Ott s artil was publis [], i wi t rlativisti tmpratur was sow to trasform followig aotr law: /. () xprssio () was obtai b X.Ott for a varit of psial prosss iluig ltromagti raiatio. Howvr ulik Mosgil, X.Ott lt aotr approa for stuig t pross of ltromagti wav raiatio ur rlativisti oitios. H xami wav missio of iiviual atoms, wras Mosgil stui blak-bo raiatio, as w av oti abov. partiular, i [] Stfa-Boltzma s law was obtai: E ε, () 4 a V bas o t famous Plak formula riv first smimpiriall: (, ) 8 ρ, (4) wr E is t raiatio rg of t blak-bo; V is t volum; а is Boltzma s ostat (J/с gra 4 ); (, ) Stpa- ρ is t raiativ rg sit (J/с); k is Boltzma s ostat; is t frqu of osillator raiatio. 8

3 Psial Rviw & Rsar tratioal, 4(): 7-, 4 As kow, Stfa-Boltzma s ostat quals: 4 k a. (5) 5 X.Ott s artil as iu a log-trm polmi o t tmpratur trasformatio ur rlativisti oitios. Som rsarrs ar to Plak-Eisti s viwpoit; t otrs ar to X.Ott s. Som sitists osir t tmpratur to b a rlativisti ivariat [4]. r appar absolutl xoti opiios. For xampl, t autors of Rf. [5] arriv at a olusio of t tmpratur ur rlativisti oitios big ag bot aorig to Plak, a to Ott, a to Call a Horwitz as t abl situatio rquirs. Morovr, P. Lasbrg a G. Matsas av i to put to t log-tim isput [6,7]. partiular, t writ ( it): t propr tmpratur alo is lft as t ol tmpratur of uivrsal sigifia. is sms to omplt a stor start 9 ars ago [8] (mor ta ars toa E.V.) of ow usual tmpratur trasforms, a to olu a otrovrs [] of ars staig. (5 ars toa). Wat is autors opiio [6,7] bas o? ir basis is as follows. First of all, t autors us a Uru-D Witt ttor, i.., a two-lvl moopol, wit a uit itrval of t raiatio rg. t autors [6,7] suppos tat blak-bo raiatio wit t propr tmpratur is at rst i som irtial rfr fram S. xitatio rat of t ttor movig wit a ostat vloit v is fou from quatum fil tor. t is proportioal to t partil umbr sit (,, v). As a rsult, t followig formula was obtai: ( / )/ / v / v v, v l ( / ) v / v / 4 v (, ), (6) wi, as t autors of [6,7] ot, oul ot b ru at v to t wll-kow formula (, v), / / ( ). (7) opiio of P. Lasbrg a G. Matsas, formula (6) is absolutl orrt, tus it is ussar to spak about a uifi law of tmpratur trasformatio ur rlativisti oitios. Howvr it is ot ompltl t as. Bot t rsults obtai b Mosgil (a soo us b Plak), a t matmatial mostr (6) ar iorrt. t is ssar to amit tat t mai raso of su a ramati situatio wit a rlativisti tmpratur is a giat sitifi autorit of Max Plak first a Albrt Eisti. Naturall, aftr publisig X.Ott s artil tis work was arfull k. Errors a ot b fou. But obo ar k t works [,,8]. s artils wr arri out just aftr t ratio of Spial Rlativit (SR) w obo a kow o t Bos-Eisti istributio. As w av oti abov, Plak s wll-kow formula, orig blak-bo raiatio, was obtai b a smi mpirial wa witout ivolvig tis istributio. Aftr t isovr of tis istributio, i t twtis of last tur, Plak s formula was alra obtai wit its 9

4 Psial Rviw & Rsar tratioal, 4(): 7-, 4 lp. Howvr if t raiator of ltromagti wavs is movig wit a rlativisti vloit, t form of Bos-Eisti istributio ags it boms at last a futio of two variabls, wi immiatl follows from SR ltroamis. Fig.. X,X,X a X,X,X ar t laborator rfr fram a tat movig uiforml a rtiliarl wit t vloit v. is t obsrvr at rst; is t raiatig blak bo, xami t simplst as rprst i t Figur. As s, tr ar two rfr frams. O of tm (wit prims) is movig uiforml a rtiliarl wit t vloit v. A poto raiator is at rst i t movig rfr fram. A obsrvr is at rst i t laborator rfr fram. obsrvr is ttig potos. f t agl btw v a t obsrvr is zro, t t raiatio frqu of t osillator will b qual to. (8) f t agl wr /, so t formula for t frqu trasformatio woul av aotr form, aml:. (9) for t obsrvr i t laborator rfr fram.

5 Psial Rviw & Rsar tratioal, 4(): 7-, 4 us witout takig ito osiratio (8) a (9), w aot vitl us t wll-kow Bos-Eisti istributio for obtaiig t Stfa-Boltzma law w t objt ur stu is movig wit rlativisti vloitis. Hr w must a t followig. Attmpts av b ma to obtai t law otig t raiatio itsit wit t tmpratur w rlativisti ffts ar ivolv [,]. For xampl, i [] a ultrarlativisti plasma is xami otaiig ltros a positros. ir aiilatio grats ltromagti raiatio. ts itsit is fi, i partiular, wit t lp of a o-imsioal Bos-Eisti istributio. t is proportioal to t plasma tmpratur to t fourt powr, wit t vloit of t objt as a wol big qual to zro. t is plasma partils tat ar i motio. aforsai allows us to formulat a mai goal of our work obtaiig a raiatio law for t blak-bo movig wit a rlativisti vloit w t agl btw t movig vloit v a t obsrvr is zro (s Figur). A solutio of t problm will b prform b t mtos giv i [9].. ELECRODYNAMCS AND HERMODYNAMCS OF HE OBJEC UNDER SUDY. Dfiitio of t Numbr of Fil Osillators wit a Giv Frqu w t Agl is Zro (Fig. ) Assum tat w av a opaqu objt wit a ir lirial avit. ts surfa is a blak bo at up to som tmpratur. r is a trmoamial quilibrium i t avit btw its ir surfa a ltromagti raiatio. r is a vr small ol i t objt ovr, troug wi ltromagti wavs raiat out of t avit (Fig. ). objt is movig uiforml a rtiliar wit t vloit v togtr wit t rfr fram. raiatio from t avit is tt wit a vi big at rst i a laborator rfr fram. First of all, w will sow tat t Stfa-Boltzma law () is iorrt ovr t wol rag of objt motio vloitis, i.., from zro up to v., aorig to X. Ott [], t raiatio rg i t avit is qual to: E,,,.l () t t ltromagti rg sit ε V ( ) ( ) E V wr is a osillator srial umbr,,,, l, () is t frqu of its osillatios. No mattr ow t tmpratur of t sstm trasforms, i.., aorig to Plak or to Ott or to Call a Horwitz, w sall alwas arriv at t poit of absurit., lt t

6 Psial Rviw & Rsar tratioal, 4(): 7-, 4 tmpratur trasform,.g., aorig to Plak, i.., to (). tis as t rigt si of () a., as s from (), t rigt si of () appars to t to zro as v, wil t lft si of tis formula to iras ifiitl. is iiats a los otio btw t raiatio law of a movig blak bo a t tmpratur trasformatio ur rlativisti oitios. 4 will av t followig form Now fi t umbr of osillators g(, ) wit frquis i itrvals, a, a a giv polarizatio i t avit usig t wllkow prour [9]. followig fat soul b poit out at o. umbr of ts osillators is a futio of two variabls. raso for tat was xplai abov but r t followig soul b ot. f a sprial ooriat sstm is us for t as v<<, t i our as it is ovit to us a lirial o takig aout of formula (8) a (9). lassial approa to fiig t quatit g is bas o usig t umbr spa follow b trasitio to a sprial spa of t wav vtor k k, wr L is t L ormaliz ub g, a fiall to t sprial spa of frquis. t as stui w us a lirial spa rprstabl as two spas flat, irular a liar prpiular to o aotr. to fi t ssar quatit w sall us two ooriat sstms: polar a o-imsioal Eulia, i.., a straigt li. amout of umbrs witi t sprial lar of t sprial spa is 4 [9] (t sprial ooriat sstm). amout of umbrs i t irular lar is qual to (t polar ooriat sstm). As to i a liar itrval of o-imsio spa, it will b qual to. As a rsult, w av for t wol sstm: ) g(,. () urig from a umbr spa to a wav vtor spa a fiall to a frqu o, w sall av: k k k g (, ) L L L L V ( ). () as of ltromagti wavs, soul b tak ito aout two polarizatios, t w sall av: g (, ) V. (4)

7 Psial Rviw & Rsar tratioal, 4(): 7-, 4 Hr it is importat to mpasiz tat formula (4) is orrt for t obsrvr at rst i a ral spa moitorig, from t rfrrig fram, t objt movig t uiforml a rtiliarl wit t rlativisti vloit v. Si t raiatio is trmal t avrag volum of t osillators wit a giv polarizatio will almost b ipt of tim. tis as, it is ussar to fi osillator umbrs i Mikowski spa.. Rlativisti mpratur as itr a Vtor or a sor Now w soul mak a w attmpt to solv som problms ot wit t rlativisti tmpratur. First of all, w soul larif if tis trmoami paramtr is a salar or appars to b a vtor or a tsor. tis otio w soul first rall t formula for vloit aitio i SR. As kow, t ompots of t total vloit i t irtios X or X will t to zro for t obsrvr i t laborator rfr fram as v (Fig. ). tur, t ompot paralll to axs t X will ot o. is suggsts immiatl tat t tmpratur boms a matmatial objt iffrt from a salar. Wat is t objt? Util vr rtl t tmpratur i t abov as is osir to b itr a salar or a quatit formig a vtor wit otr quatitis. For xampl, i [] V. Hamit rprsts tis trmoamial paramtr as wr i.., α v µ v Θ µ,,,,, ˆ µ (5) µ v is a uit 4-vtor i Mikowski spa, morovr µ v [ v,v α v is a vloit vtor i Eulia spa; ], α,,, (6) v. (7) µ vµ Furtr, vlopig t ia of tmpratur vtor rprstatio, t autor of [] fiall oms to t followig xprssio: v /, (8) wit µ (,,, ), t µ µ /, µ δ µ ν δ µ. (9)

8 Psial Rviw & Rsar tratioal, 4(): 7-, 4 Otr autors,.g., [], also tri to rprst t rlativisti tmpratur xlusivl as a vtor. Howvr, i our opiio, tis approa to t problm is iorrt, si t poto gas i t avit is a otiuous mium. a xpa tsor approa is ssar to srib rg prosss i it. tis as t so trmoamis law a b rprst i Mikowski spa as ijk δq g jk δσ ; i, j, k,,,4; α,,,,4, iα g α () wr t at Q a t tmpratur ar tsors of rak, but fuamtal tsors. Formula () s a spial xplaatio. σ σ g jk, gα ar ovariat As kow, M.Plak assum tat (v), i.., t trop of t sstm varis xlusivl owig to trmoamial prosss i t objt ur stu a is ipt of its vloit rlativ to t obsrvr i t laborator rfr fram []. As will b sow blow, t law () agrs wit t Plak statmt. Furtr, t otratio of t at a tmpratur tsors wit t fuamtal tsors trasforms tm to t vtors multipli ito salar quatitis. lattr ar ivariat parts of t abov tsors tat o ot var w passig from o rfr fram to aotr. As to t vtors, tir ompots ar qual to uit w t movig sstm 4-vloit quals to zro, i..,, () i 4 4 4,,,, () wr i is imagiar uit; v /, t otratio i () of two vtor quatitis i iis i givs a salar quatit, wi is ivariat ur t Lortz trasformatios. As to at a t tmpratur, tir ivariat parts var xlusivl owig to purl trmoami rasos. tur, t vtor ompots var xlusivl, w passig from o rfr fram to aotr. bot ass itr t at or t tmpratur ar ivrsl proportioal to t quatit. t trop will ot ag i t abs of at iput ito t sstm. lattr is i a full aor wit t rsults obtai i works [,4, a 5] wr t tmpratur was sow to trasform ur rlativisti oitios i ivrs proportio to t quatit. w a rprst t tmpratur i Mikowski spa as 4

9 Psial Rviw & Rsar tratioal, 4(): 7-, 4 i iα g α Τ i Τ, () wr Τ is t ivariat part of t tsor magitu of rak, i..,. t ral spa formula () a () rmai uag wit t ol iffr tat, first, w ow us affi tsors, so, t ps () a () tak t form:, (4) iα,, (5) At v t spatial ompots of ompots i Mikowski spa. i oii i Eulia spa wit t sam spa-tim t ompots of squar sum of t vtor quatit Τ ra Τ x Τ Τ z Τ τ Τx Τ Τz Τ τ, (6) ivariat i all irtial rfr frams. O t otr a t ivariat of tis sort givs i Eulia spa Τ Τ Τ Τ Τ Τ Τx Τ Τz ivar, (7) takig ito osiratio tat (affi tsors), i.., t spatial part of t ivariat ot wit t tmpratur 4-tsor is ompltl itial to t ivariat ot wit t tmpratur -tsor. t is vr importat si it allows o to solv our problm irtl i Eulia spa. As to t ultrarlativisti plasma osir i [], t aforsai will b vali i tis as as wll, wi will b isuss blow.. HE RADAON OF HE MOVNG BLACK BODY. Gral Dp for t Raiatio tsit of t Movig Blak- Bo Bas o t aforsai as wll as o t lassial mtos of solvig t problm (i.., for v<<, s,.g., [9,6]) w a ow start its solvig. For tis purpos w sall writ ow a xprssio for t avrag total rg ε of t liar osillators as follows (lirial spa, zro osillatios ar glt): 5

10 Psial Rviw & Rsar tratioal, 4(): 7-, 4 6 [ ] ) (, ε ε ε, (8) wr a ar t frquis of osillators i t irtio prpiular a paralll to t vloit of t movig objt; a ar positiv (quatum) itgrs for t osillators i t first a so irtios. tis as, si potos ar bosos, t a b i o quatum stat;, ar t valus of t tmpratur tsor ompots. t avrag volum of t total rg ε of t ltromagti fil pr uit volum i t movig avit provs to qual ε V E. (9) t is t gral p for t raiatio itsit of t movig blak-bo. Obtaiig t p for Blak-Bo Raiatio ur Rlativisti Coitios As a rsult, w av obtai, i fat, four impropr itgrals, tr of tm ovrg. last two itgrals i (9) iffr ol b variabls. ar asil alulat usig variabl trasformatios as follows: () () () () (), ()

11 Psial Rviw & Rsar tratioal, 4(): 7-, a k, () wr a is t Stfa-Boltzma ostat, i.., 4 5 k a. (), () Γ ζ z z,5498., (4) wr z Γ is t gamma futio, z [7], k z K z ζ,5498;. ; Γ Γ Γ Γ z z [ ] l. (5) As s, t itgral (5) ivrgs a w tak aout of a ig frqu raiatio of t objt. Havig lt - from t omiator, w obtai; ) (. (6) As a rsult, w av for : 9.. a, (7)

12 Psial Rviw & Rsar tratioal, 4(): 7-, 4 a ε V E.8a for t rg sit of raiatio ur rlativisti oitios. Formula (8) will tak t followig form ur ormal oitios: 4 (8) ε.447a. (9) issimilarit of formula (9) from t Stfa-Boltzma law is quit atural, if o taks ito aout of t abov assumptio. o obtai a mor xat xprssio for t blak-bo raiatio ur rlativisti oitios, it is ssar to rormaliz t p (9). valu of t umrial offiit i (7) soul b su tat t w offiit plus t offiit from formula () woul giv uit. w av fiall: ε V E.8a.79a.8a.79a ( ) is is t raiatio law of t blak-bo big i a uiform a rtiliar motio w t agl btw t vloit vtor of t objt v a t obsrvr is qual to zro. 4. RESUL AND DSCUSSON law (4) is t mai rsult of t rsar. t soul b ot at o tat t p (4) os ot la to t poit of absurit a otraitios. t is, i fat, ol t first stp o t wa of obtaiig a mor gral a ivolv p btw t itsit of t blak-bo raiatio a its tmpratur. is p also taks aout of a ozro agl. Kowlg of tis ratio soul pla a importat rol i sigig t vis masurig t tmpratur of t raiatig sours movig wit rlativisti vloitis,.g., quasars. p (4) provis ratr a probabl aswr to t qustio orig t tmpratur trasformatio ur rlativisti oitios. t is vit tat t p () is iorrt a woul b rjt ma ars ago a witout all matmatial ivolvmts if it wr ot a giat autorit of Plak a Eisti. Rall, ow is t p () b follow if su osmi objts as quasars o xist, wos vloit v of motio a b qual,9 wit t lumiosit raig ormous valus? t most gral as t tmpratur is a omplx matmatial objt. t ompriss a ivariat part ipt of t motio vloit a a part pt o t vloit a orit i spa. Ur ormal oitios t tmpratur boms a salar, t sam os for t at. trop problm as ot b stui ompltl. t is ot improbabl. (4) 8

13 Psial Rviw & Rsar tratioal, 4(): 7-, 4 tat t trop a b a tsor objt i wi iis ar otrat wi rsults i a salar ipt of t sstm motio vloit. Evitl, it is for xprimt to solv tis problm. Howvr o xprimt as b prform si t birt of rlativisti trmoamis i 97. Of utmost itrst is to osir if t p () rmais vali for t as of a ultrarlativisti ig-tmpratur sprial plasma (firball) []. Aorig to t autor of (J/) u to t aiilatio of ε γ * [], t sptrum of its quilibrium raiatio ltros a positros is srib b t p ε i t firball, wr γ * * ( ) * * 4 * / f *, (4) is t imsiolss frqu; Т>>m is t rg, i.., appartl, (k is t Boltzma ostat, is ow t absolut tmpratur; ot to ofus wit t agl similarl sigat (s abov); p, rl / ; p, rl is t rlativisti frqu of t plasma osillatios; f is a imsiolss ostat. Formula () is vali for t as (4) wit t vtor part of t tmpratur pt o t total vloit of ltros a positros i t firball but ot o t vloit of its tr of mass. f tir vloitis ar vr ig, t w av t wll-kow as srib,.g., i [9]. is is t as of a sstm of partils big wil apart a movig wit vr ig vloitis. t soul b ot tat ts two ass ar ot full itial, si t miropartils i [9] ar ot itial bfor a aftr t ollisio. artil [], a ltropositro ollisio rsults i tir aiilatio. Howvr ts ass ar vr similar, tus t sstm rg ε ma b giv as wr m i ε ~, (4) i vi m i is t miroprtil mass, v i is its vloit. t vtor part of t tmpratur i t ultrarlativisti as will trasform i ivrs proportio of t roots v i /. Hr w immiatl arriv at t olusio tat t p (4) is vr oubtful, si t rigt si os ot trasform itiall to its lft si ur t rlativisti oitios. t soul b also ot tat t objt stui i [] is, i fat, a stabl firball. Evitl, w t sit of ltros a positros xs a rtai limit, t stabilit will b brok, a a xplosio will our. 5. CONCLUSON A law was obtai for t blak-bo raiatio i t tir itrval of its (blak-bo) movmt sp, i.., from zro up to t sp of ligt i vauum. 9

14 Psial Rviw & Rsar tratioal, 4(): 7-, 4 COMPENG NERESS Autor as lar tat o omptig itrsts xist. REFERENCES. Mosgil vo K. ori r statioar Stralug i im gliformig bwgt Holraum. Aal r Psik. 97;(5): Duts.. Plak M. Zur Diamik bwgtr Sstm. Aal r Psik. 98;6(6):-4. Duts.. Ott X. Lorz-rasformatio r Warm a r mpratur. Zitsrift f. Ps. 96;75():7-4. Duts. 4. Call H, Horwitz G. Rlativistik rmoamis. Am. J. Ps. 97;9(8): Cavallri G, Salgarlli G. Rvisio of t rlativisti amis wit variabl rst mass a appliatio to rlativisti trmoamis. t uovo imto. Orgao lla soita italiao i fisia. 969;sr.,A6(-4): Lasbrg P, Matsas GEA. Laig t gost of t rlativisti trasformatio. Psis Lttrs A. 996;: Lasbrg P, Matsas GEA. impossibl of a uivrsal rlativisti tmpratur trasformatig, Psia A. 4;4( ): Eisti A. Ubr as Rlativitatsprizip u i aus mslb gzog Folgrug, Jarbu uts Raioaktivitat u Elktroik. 97;4:4-46. Duts. 9. Lvi VG. Cours of ortial Psis, V.. Mosow: Fizmatgiz; 96. Russia.. Hamit VH. Rlativisti rmoamis. Ps. Rv. 969;87(5): Mvv MV. rmoamis of poto i rlativisti - γ plasma. Ps. Rviw, E. 999:59(5): Hakim R, Mag A. Rmarks o Rlativisti rmoamis. Lttr al Nuovo Cimto. Rvista itrazioal Soit lla italiaa Di Fisia, sria. 969;(9); Vitsma EV. Som Problms i Rlativisti rmoamis. J. Exprimtal a ortial Ps. 7;5(5): Vitsma EV. O Rlativistik Surfa sio. J. Colloi trfa Si. ;65(): Vitsma EV. Spifi trmoamial pottials o surfas ur rlativisti oitios. J Colloi trfa Si. 9;7(): rltsk YP. Statistial Psis. Mosow: Vissaa Skola; 966. Russia. 7. Pruikov AP, Brkov UA, Mariv O. tgrals a Sris, V. (Pag 77). Mosow: Fizmatgiz;. Russia. 4 Vitsma; is is a Op Ass artil istribut ur t trms of t Crativ Commos Attributio Lis (ttp://rativommos.org/liss/b/.), wi prmits urstrit us, istributio, a rproutio i a mium, provi t origial work is proprl it. Pr-rviw istor: pr rviw istor for tis papr a b ass r: ttp://