Generalized Head Light Effect in 2D Dimensional Rotated frames using Euler rotation matrix. Asad Ali.

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1 Generalized Head Ligh Effe in D Dimensional Roaed frames g Eler roaion mari Absra Asad Ali aliasadhami1994@gmail.om Deparmen of phsis Karakoram inernaional niersi, Gilgi Pakisan in presen work,eqaions eplaining phenomena of Head Ligh effe is onsred for wo Dimensional roaed frames moing wih Relaiisi speed and i is shown ha he newl deeloped eqaion are more general han he one deried b Alber Einsein in 1905[1]. Wheneer, he inlininaion angle beween wo frames is se zero boh relaion beomes eqialen. Inrodion In 1905 Einsein eplained he phenomenon of headligh effe in his paper on he elerodnamis of moing bodies [1] ha a moing sore of Radiaion radiaing niforml in all direion in is res frame, appears o radiae predominanl along is direion of moion. This obsered Bnhing of Radiaion along forward direion is de o Aberraion and is alled Headligh effe. One of he eamples of Headligh Effe are ha mosl in Snhroron Radiaion appears o be srongl onenraed in he forward direion wih er lile Radiaion oming off in bakward direion. Wheneer Charge Pariles are aeleraed o er high Energ in Parile Aeleraors.Consider a Sore in frame S Radiaing niforml in all direions. Frame S is moing along posiie X-X ais wih eloi. In frame S he ra O P (in X Y ) has eloi omponen os (1) () Aording o Relaiisi Law of Addiion of eloi [3], in S-frame he X omponen of he eloi is

2 os 1 os (3) Whereas he eloi along - ais is zero.sine ais as obsered in frame S is gien b s os,he angle made b ra wih he X- os (4) os os (5) 1 os Where = From (5), if we se 0 or hen, we ge b os when os. Then as approahes 1, he angle approahes zero, whih means mos of he radiaion appears o be srongl onenraed in he forward direion wih er lile radiaion oming off in he bakward direion. The resl of (5) hold onl for he frames wih are ollinear o eah oher and moing wih relaiisi speed.in ne seion we hae deried relaions whih eplains head ligh effe among hose frames whih are no ollinear and where S frame is inlined a angle wih respe o X-ais of S frame. Deriaion of Generalized Head Ligh Effe Eqaion in dimensional roaed oordinae ssem Consider a sore in frame S radiaing niforml in all direions. Frame S is moing a inlined angle wih respe o posiie X- ais wih eloi. In frame S he ra O P (in X Y ) has eloi omponen os (6) (7) Sine S frame is roaed wih respe o X-ais of S frame, he Transformaion eqaion are

3 Figre.1 os 0 os os (8) Where gamma is, 1 1 Now se of Lorenz ransformaion eqaion for wo dimensional roaed frames beomes; ) os ( (9) ) os ( (10) ).. os ( (11)

4 Differeniaing (9) and (10) wih (11) ields; os os (1) Similarl, os os Now g (6) and (7) in (1) and (13) whih ields; (13) os os os os osos os os (14) os os os os (15) Eqaion (14) and (15) are desired generalized eqaions eplaining Head Ligh effe for wo Dimensional Roaed frames moing wih relaie eloi relaiisiall. To hek wheher hese are onsisen or no so we se = 0 in (14) and (15)

5 osos0 0 os0 0 os os 1 os (14) 0 This resl is eal wha we epe from (5). This learl erifies ha or approah is onsisen and reprodible. Conlsion We deried Generalized eqaions for omponens of eloi of ligh beam in wo frames eplaining Head Ligh Effe in wo Dimensional Roaed frames moing a relaiisi speed and onfirmed ha or resl regeneraes he resls eplained in [1] whih assres ha newl deeloped eqaion are Consisen and more general han one eplained in [1].This approah an be eended o hree dimensional roaed frames whih will appear in or sbseqen paper. Referenes [1] Alber Einsein, "On he Elerodnamis of Moing Bodies," (1905), reised and ranslaed in The Priniple of Relaii (Doer, NY, 193), pp