Anthr Explanatin f th Csmlgical Rdshift April 6, 010. Jsé Francisc García Juliá C/ Dr. Marc Mrncian, 65, 5. 4605 Valncia (Spain) E-mail: js.garcia@dival.s h lss f nrgy f th phtn with th tim by missin f hat t th intrgalactic spac might xplain th csmlgical rdshift. Ky wrds: csmlgical rdshift, missin f hat. 1. Intrductin Gnrally, it is cnsidrd that th univrs was riginatd in th ig ang, and sinc thn it is xpanding. In that thry, th rdshift f th light mittd frm distant galaxis, th s-calld csmlgical rdshift, is intrprtd as a Dpplr ffct and thn cnsidrd as an indicatin f th xpansin f th univrs, fllwing th law f Hubbl. h light rdshift paramtr is dfind as z (1.1) bing and th light frquncis mittd and bsrvd, rspctivly; but as c λ, bing c th light spd in vacuum, λ th wavlngth and th frquncy, thn als z λ λ (1.) λ bing λ and λ th light wavlngths mittd and bsrvd, rspctivly. Fr th rlativistic Dpplr ffct [1] (p. 166) 1 v csθ c z 1 1 v c (1.3) 1
bing v th spd f th light surc and θ th angl f th mtin. Fr a mtin in th lin f sight ( θ 0 ) and with v << c (which crrspnds t lw rdshift, z << 1) h Hubbl s law is statd as [1] (p. 486) v z (1.4) c v r Hd (1.5) whr v r is th vlcity f rcssin, namly th spd at which a light surc mvs away frm th bsrvr, du t th xpansin f th spac btwn bth; H is th Hubbl s cnstant, and d is th distanc btwn th bsrvr and th light surc. Fr lw rdshift ( z << 1) [1] (p. 486) vr Hd z (1.6) c c thrfr, th rdshift f th galaxis is prprtinal t thir distancs t th bsrvr. hat is, th gratr th distanc, th gratr th rdshift. Frm (1.4) and (1.6) w wuld hav that v r v. As th distanc frm a galaxy t us, fr a light signal, is bing t th tim, thn d ct (1.7) z (1.8) Hwvr, in this simpl nt, w ar ging t cnsidr, using nly vry lmntary argumnts, that th rdshift in th light cming frm th stars might b prducd by th lss f nrgy f th phtn with th tim by missin f hat.. h lss f nrgy f th phtn with th tim h first principl f thrmdynamics stats that U Q W (.1) bing U th intrnal nrgy, Q th hat and W th wrk, all f thm rfrrd t a givn systm. It xprsss that th incrmnt f th intrnal nrgy f th systm is qual t th hat absrbd minus th wrk dn. Applying this principl t a phtn travling in th intrgalactic spac (IGS), w wuld hav that
U Q < 0 (.) bcaus w cnsidr that th phtn ds nt mak any wrk ( W 0) and mits hat t th IGS. h phtn lss nrgy. Frm th principl f quipartitin f th nrgy, fr a phtn U ( 3 ) k h (.3) bing h th Planck s cnstant, k th ltzmann s cnstant and th abslut tmpratur (Klvin s tmpratur). Frm (.) and (.3) and th phtn undrgs a rdshift. Frm (1.1) and (1.8) ( 3 ) h < 0 (.4) k (.5) which xprsss a linar dcras f th frquncy f th phtn with th tim. Frm (.3) ( 3 ) k h (.6) ( 3 ) k h (.7) bing and th abslut tmpraturs f th phtn at th pints f missin and bsrvatin, rspctivly, and substituting (.6) and (.7) int (.5) (.8) and th dcras in frquncy is a dcras in tmpratur, r in thr wrds, th phtn lss nrgy by missin f hat. As (.5) is valid nly fr z << 1, th sam happns with (.8). Fr any rdshift, th hat quatin wuld b t a x y z (.9) whr a is a cnstant, and x, y and z th spatial crdinats. And fr nly t and x 3
t a x (.10) y sparatin f variabls ( t x) f g( x), (.11) Substituting (.11) int (.10) df af dt g( x) ( x) dx d (.1) g Sinc th lft hand sid dpnds nly n t and th right hand sid nly n x, bth sids ar qual t sm cnstant valu b df af dt d g g( x) ( x) dx b (.13) frm which w wuld hav bing A, and C intgratin cnstants. As f A (.14) ( x) ( b 1 x C) g cs (.15) x λ, ( t, x) ( t,0) f g( 0) AcsC. Fr t 0, max. ab ( 0) A csc 0 A csc ut as ( 0) and and ( ) 0 (.16) (.17) abt (.18) Fr abt << 1 ( abt) abt 1 (.19) 4
which is th sam as (.8) with hrfr, fr lw rdshift w may us (.5), (.8) and ab H (.0) z (.1) And fr high rdshift (.) (.3) z 1 (.4) whr (.3) is btaind substituting (.6) and (.7) int (.). Substituting (1.7) int (.4) z ( H c) d 1 (.5) and th rdshift incrass xpnntially with th distanc. Fr a rdshift valu f z 1, 088 fr th csmic micrwav backgrund radiatin (CMR), which wuld crrspnd t th rdshift f th light mittd by th light surcs lcatd n a circumfrnc cntrd at th Earth f radius d c H ln 1 z c H ln1,, th crrspnding rlatin f tmpraturs wuld ( ) ( ) ( ) 089 b ( z) 1, 089 34 1. Fr h 6.65 10 juls sc, 3 14 k 1.380 10 juls/ºk and 10 cycls/sc, frm (.6), 3, 00 ºK, and.9 ºK. In summary, th ky is t cnsidr H as a cnstant rlatd with th lss f nrgy f th phtn but du t th missin f hat nt t th xpansin f th univrs. 3. Cnclusin W cnclud that th lss f nrgy f th phtn with th tim by missin f hat t th IGS might xplain th csmlgical rdshift. 5
Rfrncs [1] L. D. Landau and E. M. Lifshitz, ría clásica d ls camps, in Spanish, scnd ditin, Editrial Rvrté, S. A., arclna (1973). Original ditin by Nauka, Mscw. 6