Electromagnetic Field Equation and Lorentz Gauge in Rindler Space-time
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- Ἡρόδοτος Ξενοκράτης Καλογιάννης
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1 The fin Review of Phsis 6 : 55 leomgnei Fiel quion n Loen Guge in Rinle Spe-ime Sngwh-Yi * Depmen of Mhemis Teon Univesi Souh Koe In his ppe we eive eleomgnei fiel nsfomions n eleomgnei fiel equions of Mwell in Rinle spe-ime in he one of genel heo of elivi. We hen e he Loen guge nsfomion n he Loen guge fiing oniion in Rinle spe-ime n obine he nsfomion of iffeenil opeion he eleomgnei -veo poenil n he fiel. In iion hge ensi n he elei uen ensi in Rinle speime e eive. To view he invine of he guge nsfomion guge heo is pplie o Mwell equions in Rinle spe-ime. In ppeni we show h he eleomgnei wve funion nno eis in Rinle spe-ime. n impon poin we sse in his ile is he uniqueness of he elee fme. I is beuse in he elee fme one n e eleomgnei fiel equions.. Inouion In 7 el Csillio n Snhe isovee Mwell equions in vuum in unifoml elee fme n Mluf n Fi eive eleomgnei fiel nsfomions in Rinle spe-ime in. The use Mwell equions fo gvi fiel bu we isgee wih hei ppoh beuse Mwell equions fo unifoml elee fme hve o be ee in fl Minkowski spe-ime n no in he uve spe-ime whih implies he pesene of gviionl fiel. In his wok ou im is o fin eleomgnei fiel equions in Rinle spe-ime lso in vuum bu no in vuum of he genel elivi heo. In Se. fe woking ou eleomgnei fiel equions in Rinle spe-ime we eive he Loen guge nsfomion n he Loen fiing oniion in iion o nsfomions fo eleomgnei -veo poenil in Rinle spe-ime. In Se. we efine he eleomgnei fiel in Rinle spe-ime n we fin he nsfomion of he eleo-mgnei fiel. In Se. we obin he eleo-mgnei fiel equion in Rinle spe-ime n ppl he guge heo o Mwell equions woke ou in elie seions in Rinle spe-ime fo viewing he invins of he guge nsfomion. We hink i is impon o know he eleomgnei wve funion iion in Rinle spe-ime bu i is known h i oes no * sngwh@ne.om sisf eleo-mgnei wve equion mhemill see ppeni. Hene oing o ou gumens mn esuls publishe uing he peio 7 - see Refs. n espeill he ompuion of eleo-mgnei wve funion wee inoe. Howeve we o unesn h eleomgnei wve funion n eis in ineil fme s shown b Mwell n insein.. Tnsfomion of he leo-mgnei -veo Poenil Loen Guge Tnsfomion n Loen Guge Fiing Coniion The Rinle ooine nsfomion is osh The e e is see Refs. 5 τ η b b υ υ b υ υ ηbe e υ g υ
2 The fin Review of Phsis 6 : 56 e e osh The oienion of -is n -is is given s poenil is -veo τ 9 e e Whee he uni veo e is given s e osh 5 Theefoe osh osh 6 Now he veo nsfomion is V ' ' V U ' U 7 ' Theefoe he nsfomion of he eleomgnei -veo poenil is given b he following equions: ' ' e 8 e The nsfomion of iffeenil ooines is ' ' e 8b e n he equion of eleo-mgnei -veo Loen guge nsfomion in Rinle spe-ime is given b Whee is sl funion. g g g g g Hee is sl funion. Theefoe he Loen guge in Rinle speime n be wien s Γ ; Γ Γ g g υ υ g g g g Hene Loen guge nsfomion n Loen guge in Rinle spe-ime e s follows: ; Γ υ g υ Γ
3 The fin Review of Phsis 6 : 57 υ υ Γ g -i Hene one n obin Loen guge fiing oniion in Rinle spe-ime s Fom he esul obine bove we foun he nsfomion of he eleo-mgnei -veo poenil in n ineil fme n he eleo-mgnei -veo poenil in unifoml elee fme whih n be wien s osh osh If we ke he mi of he nsfomion in he iffeenil ooine hen we hve osh osh The invese mi of qn. n 5 n be obine s e osh osh 5 Hene we n obin he mi of nsfomion of iffeenil opeion s
4 The fin Review of Phsis 6 : 58 T T osh osh 6 Theefoe he nsfomion of iffeenil opeion is osh osh 7 The bove iffeenil opeion sisfies he following equions: 8. leo-mgnei Fiel in he Rinle Spe-ime Given he eleomgnei fiel in he ineil fme s 9 We nee o pefom ompuion in oe o efine he eleo-mgnei fiel in Rinle spe-ime whih equies h we lule eleomgnei fiel nsfomions in Rinle spe-ime. The ompuion is sighfow b using he eleomgnei -veo poenil nsfomion qn. n he nsfomion of iffeenil opeion qn. 7. One heefoe obins osh osh osh osh
5 The fin Review of Phsis 6 : 59 The -omponen of he elei fiel in he ineil fme is given s osh osh osh osh osh The -omponen of he elei fiel in he ineil fme n be wien s osh osh osh osh osh Now fo he mgnei fiel he n omponens e given b qns. o 5 s follows: osh osh
6 The fin Review of Phsis 6 : 6 osh osh osh osh 5 Hene we n efine he eleomgnei fiel in Rinle spe-ime. This is given s Now 6 We hen obin he nsfomion of he eleomgnei fiel s osh osh osh osh 7 Hene we n fin he mi of he nsfomion of he eleo-mgnei fiel. H osh osh osh osh H 8 Simill we n obin he mi of he invesensfomion of he eleomgnei fiel.
7 The fin Review of Phsis 6 : 6 H osh osh osh osh H 9 see lso Ref.. Hene he invesensfomion of he eleomgnei fiel is osh osh osh osh If we ppl Loen guge nsfomion qn. o eleomgnei fiel qn. 6 in Rinle spe-ime hen we ge Whee is sl funion. If we ppl Loen guge nsfomion qn. o he nsfomion of he eleomgnei - veo poenil n qn. hen we n obin he following esuls. osh osh. leomgnei Fiel quion in he Rinle Spe-ime Mwell equion is b
8 The fin Review of Phsis 6 : 6 We shll now pefom ompuion o eive Mwell equions in Rinle spe-ime. Fo his pupose we shll ompue i b using he eleomgnei fiel nsfomion qn. 7 n he nsfomion of iffeenil opeion s given in qn. 7. Le us fis el wih he nsfomion of - veo he hge ensi n he eleil uen ensi τ n Whee osh osh 5 Now -veo τ is efine in Rinle spe-ime. The fis of he Mwell equions is given in he ineil fme s follows:. osh osh osh osh osh osh 6. osh osh The -omponen is: osh osh osh osh Hene osh 7 The -omponen is: osh
9 The fin Review of Phsis 6 : 6 osh osh osh 8 n he -omponen is given s: osh osh osh osh Theefoe 9 The hi lw esibe b Mwell equions in ineil fme is:. osh osh osh osh Now he fouh equion of Mwell in he ineil fme is given s:. osh osh
10 The fin Review of Phsis 6 : 6 The -omponen is given s: osh osh osh osh Hene osh The -omponen is: osh osh osh osh The -omponen is given s: osh osh osh osh Theefoe we obin he eleomgnei fiel equion fom qns. 5- in Rinle speime see lso Ref. whih is given s
11 The fin Review of Phsis 6 : 65 b We know h el Csillio n Snhe le isovee Mwell equions in unifoml elee fme in vuum. Hene he nsfomion of -veo τ is osh osh 5 Now -veo τ. Fo insne we know h he spheil hge ensi of sion elee fme in hge huge sphee is R R Q R Q V Q Genell he oninui equion i.e. onsevion lws fo he hge ensi n he eleil uen in Rinle spe-ime is given b Γ ; g g Γ Γ g g g g 5 We now e he Loen guge nsfomion b using qn. fo he eleomgnei fiel equions n qns. o in Rinle spe-ime. The ompuion is sighfow b using he Loen guge nsfomion qn. n he Loen guge fiing oniion qn.. qn. is
12 The fin Review of Phsis 6 : 66 The Loen guge in Rinle spe-ime is given s Hene 6 If we ppl Loen guge nsfomion o qn. 6 hen we hve Whee is sl funion. 7 Now he Loen guge fiing oniion qn. in Rinle spe-ime 8
13 The fin Review of Phsis 6 : 67 Hene qn. is 9 qn. is invin une Loen guge nsfomion in Rinle spe-ime n qn. b is 5 Theefoe 5 5
14 The fin Review of Phsis 6 : 68 If we ppl he Loen guge nsfomion o qn. 5 we ge Whee is sl funion. 5 Now he Loen guge fiing oniion qn. in Rinle spe-ime is
15 The fin Review of Phsis 6 : 69 5 Theefoe qn. 5 is 55 Hene qn. b is invin une Loen guge nsfomion in Rinle spe-ime. Now qn. is 56 n qn. is 57 Hene qn. n qn. e invin une Loen guge nsfomion in Rinle spe-ime. Hene he eleomgnei fiel equions Mwell quions in Rinle speime qns. - e lso invin une Loen guge nsfomion.
16 The fin Review of Phsis 6 : 7 5. Conlusion Sine el Csillio n Snhe le lule Mwell equions in unifoml elee fme in vuum n Mluf n Fi obine eleomgnei fiel nsfomion in Rinle spe-ime 5 see Xiv pepin we ompue he eleomgnei fiel nsfomion n he eleo-mgnei equion in unifoml elee fme in single heo. Genell he ooine nsfomion of elee fme see Ref. 6 is I osh II ep 58 ep osh 59 If one uses qn. 59 o fin Mwell equions in Rinle spe-ime one fils o o so. In insein s ile see 7 he obine Loen nsfomions fo Mwell equions in ineil fme n i no use Glilei nsfomions in ineil fme. In n elee fme we hink ou hoie of Rinle ooine I is bee one n e eleomgnei fiel equions in mnne simil o insein s hoie. ppeni In -imensionl Rinle spe-ime if we use inoe lulion we hink h he eleomgnei wve funion will look like he epession given below: Now X sinω sinω ep sin Φ osh ep osh mus sisf he following equion. sin Φ sin Φ sin Φ sinω ep ω osφ ω ep sin Φ ep u his ompuionl siuion is iffeen. sin Φ sinω ep
17 The fin Review of Phsis 6 : 7 osφ ω ep Now D β βd Dβ osφ ω ep sin Φ ω Hene if we ompe qn. n qn. 5 osφ ω ep ep osφ ω ep 5 ω sin Φ sin Φ ep sin Φ ω ep osφ ω ep sin Φ 6 We onlue h i nno eis s he eleomgnei wve funion in Rinle speime. Refeenes G. F. Toes el Csillio n C. I. Pee Snhe Revis Mein De Fisi 5 7. J. W. Mluf n F. F. Fi The eleomgnei fiel in elee fmes iv:g-q/.567v. W. Rinle m. J. Phs S. Yi The fin Review of Phsis F. Shoi n. Shoi The equivlene piniple n he elivi veloi of lol ineil fme iv:g-q/55.669v 5. 6 Mssimo Pui n Mihele Vllisne Mke-Wheele ooines fo elee obseves in speil elivi iv:g-q/ insein Zu lekonmik bewege Köpe nnlen e Phsik Reeive: 6 Oobe 6 epe: 5 Novembe 6
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