2. SCHWARZSCHILD GEOMETRY REVISITED The metric for Schwarzschild Geometry is given by, ) (1) For constant values of time we have, c r

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1 urved Spae-Tie ad the Speed of Light aitra Palit uthor/teaher, P-54 Motijheel veue, Motijheel Housig ooperative soiety, Flat- 4, Kolkata , Idia, Eail: Keywords: Shwarzshild Geoetry, Geeral Relativity, Mikowski spae, Physial tie eleet. PS: 04 STRT That the speed of light i vauu is a uiversal ostat is a well established fat. ut is it really so? If two observers stad i urved spae-tie at two differet poits ad they easure the speed of light at soe other oo poit they are supposed to get differet results for the value of i y opiio. This beoes apparet i the view of the fat that loks ru at differet speeds at poits havig differet values of the gravitatioal potetial! Suh issues have bee ivestigated i relatio to the Shwarzshild Geoetry i this artile..introdution The ostay of the speed of light [] i vauu is oe of the ost vital oepts of oder physis. ut is it really so? The very questio is outrageous eough to be take ogizae of by the itelliget reader. ut I a ready to stad guaratee to the fat that the questio is iesely eaigful i relatio to urved spae-tie. I have explored the atter i relatio to the Shwarzshild Geoetry []. It is a well kow fat i Geeral Relativity[ [3] that loks ru slow [4] at plaes where the gravitatioal potetial has a saller value, that is, the tie itervals are shorter. I regios of higher gravitatioal potetial loks ru faster that is the tie itervals are log.(just thik of the fat that the lok had sweeps out a greater agle betwee the sae pair of evets if the lok is ruig fast. Let us osider two observers stadig at poits ad where the values of gravitatioal potetial are differet. d they observe a light ray flashig aross a a iterval at soe poit. The spatial lie eleet is the sae for both the observers while the teporal separatios are differet. So they easure differet values for the veloity of light! Now the basi questio that arises is that whih value orrespods to the speed of light i flat spae-tie(mikowski spae [5]. Ideed if a perso stadig i urved spaetie easures the speed of light at the very poit where he is stadig (with a lok i his had he gets the value of i flat spae-tie. This is ideed the sae value of oe fids i the Shwarzshild etri.. SHWRZSHILD GEOMETRY REVISITED The etri for Shwarzshild Geoetry is give by, GM GM ds ( ( ( dr r ( d si d ( For ostat values of tie we have, GM ds ( dr r ( d si d This is the spatial lie eleet whih we ay deote by d ad write,

2 GM d ( dr r ( d si d ( Usig ( we ay rewrite ( as, GM ds ( ( d (3 Now for the path of a light ray(ull geodesi we have a ds 0 b d => d dt dt Usig the above results i (3 we ay write, ds dt d Where dt is the physial tie eleet[ dt g00 ] For a ull geodesi, dt d For two separate spae-tie poits we ad we have, dt d d, dt d The observer easures the sae value of the veloity of light at the two poits. 3.THE PROLEM PROPER ut we ould have a iterestig exaple where oe is stadig with a lok at oe poit ad observig a light ray flashig aross a spatial iterval at soe other poit. Now we have, / dt d // dt d The observed veloity of light hages ad this does ot i ay way violate the Speial Theory of Relativity!. This is a effet of the urved ature of spae-tie(ad of ourse ot a optial effet. Now let us alulate the veloity of light at poit as observed by a perso stadig with a lok at poit at ad also by a perso stadig at poit (with his ow lok ad observig the light ray at. The spatial separatio(for the passage of the light ray for both the observers is the sae that is, GM d ( dr ( si r d d Teporal separatio for observer at is give by, GM dt ( ( d the teporal separatio for the observer at is give by,

3 dt GM ( ( Speed of light as reorded by is give by, GM ( dr r ( d si d d GM ( ( is idetial with the speed of light i flat spae-tie, that is (4 Speed of light as reorded by observer at is give by, GM ( dr r ( d si d d (5 GM ( ( Therefore we have, GM GM Now let us take GM r d, GM r We have, ( (6 ( If both the values of ad are large(ad oparable the above ratio is lose to uity. ut if oe is large while the other is sall the ratio is sigifiatly differet for oe. s a exaple let us take =000 ad =. Now we have,.998 That is, (7 If =0000 ad =.00 we have, 3

4 3.64 The above alulatios learly brig out the fat that a perso stadig loser to a blak hole will observe a light ray ovig i the outward diretio with a ireasig speed. Thus the speed barrier is broke. Explaatio: Let us osider two poits ad,with loser to the blak hole ad further away fro it, i.e, >(oforig to our previous alulatios. We osider a spatial separatio at aross whih a light ray passes, otig the fat that i our previous alulatios the spatial iterval was take at. It is the sae for both the observers.tie iterval is saller for observer at ad larger for observer at. Observer at easures the speed of light to be greater tha what easures. Ideed easures the value '' observed o flat spae-tie(sie he is stadig at the very poit where the light ray flashes past ad easures a greater value We have a sigifiat differee i the value of ad so we eed to be areful about our osideratio of spherial bodies like blak holes. perso stadig far away fro the Shwarzshild radius will observe a light ray deeleratig as it approahes a blak hole. We reaso out as follows: Explaatio: Let us osider two poits ad,with loser to the blak hole ad further away fro it, i.e, <(oforig to our previous alulatios. We osider a spatial separatio at aross whih a light ray passes, otig the fat that i our alulatios the spatial iterval was take at.it is the sae for both the observers.tie iterval is larger for observer at ad saller for observer at. Observer at easures the speed of light to be saller tha what easures. Ideed easures the value '' observed o flat spae-tie(sie he is stadig at the very poit where the light ray flashes past ad easures a greater value. osequetly photos would auulate o approahig a blak hole. Derease i their kieti eergy ould lead to further reatio of photos leadig to ireased luiosity. perso reotely plaed fro the blak hole observes a light ray i retardatio as it oves towards the blak hole. The veloity of light teds to zero as it approahes the blak hole( equatio (6 gives (---->0 as --->. This gives us the ipressio the othig should ever fall ito a blak hole.to get out of the proble we ay reaso out as follows: Matter auulatifg aroud a blak-hole would hage its size ireasig the Shwarzshild radius.thus atter gets egulfed ito the blak -hole i a peuliar way.this would irease the surfae area ad redue the teperature sie the ewly added ass osists of low eergy partiles.this is quite osistet with the existig otios Now keepig fixed let us alulate the liit of relatio (6 as teds to ifiity. ( li ( 4

5 li Therefore, ( ( (8 For sall values of (=+ε, where ε is a sall fratio the differee ould be quite rearkable! 4.HOW THE PROLEM PPLIES TO THE ERTH GM t poits lose to the earth s surfae we have,. Therefore the Shwarzshild etri ould be odified to: GM GM ds ( ( ( dr r ( d si d We osider observers at ad observig a light ray flashig aross a iterval at. We have, GM ( dr r ( d si d d (9 GM ( ( d, GM ( dr r ( d si d d (0 GM ( ( Therefore, GM GM GM GM Takig r ad r we have, ( ( ( 5

6 For poits outside the earth s surfae both ad are very large ad hee the above ratio is approxiately uity. 5. ONLUSIONS It is lear fro the above osideratios that the easured veloity of light depeds o two fators: Where the observer is stadig with his lok, Where the spatial iterval of observatio is beig osidered. Depedig o the above two fators differet values of the speed of light ould be observed i urve spae-tie. The flat spae-tie value of is observed oly if the observer observes the light ray at the poit where he is stadig. Sie the speed of light, the eletroageti ostats are also affeted by the spae-tie urvature i a relative way! There is a itiate oetio betwee gravitatio ad eletroagetis. 5. KNOWLEDGEMENTS I express y idebtedess towards all the authors whose texts have ispired e to aitai a ative iterest i the area of physis. 0 0 Referees []Robert Resik, Itrodutio to Speial Relativity, Wiley Idia Pvt. Ltd., Wiley Studet Editio, page [] Jaes. Hartle, Gravity, Itrodutio to Eistei s Geeral Relativity, Pearso Eduatio, page-0- [3] Steve Weiberg, Gravitatio ad osology, Joh Wiley ad Sos(sia Pvt. Ltd, Wiley Studet Editio -0 [4] Referee [3], page 80 [5] Referee [], page

ANOTHER PROOF FOR FERMAT S LAST THEOREM 1. INTRODUCTION

ANOTHER PROOF FOR FERMAT S LAST THEOREM Mugur B. RĂUŢ Correspodig author: Mugur B. RĂUŢ, E-mail: m_b_raut@yahoo.om Abstrat I this paper we propose aother proof for Fermat s Last Theorem (FLT). We foud