Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for ths filds hav th sam form as th strss-nrgy tnsor of lctromagntism Kywords: Einstin quations, ttrad filds, mtric tnsor, strss-nrgy tnsor, lctromagntism It is agrd that gravitation can b bst dscribd by gnral rlativity and that it cannot b xplaind by using filds as in lctromagntism or as in th cas of any othr intraction Furthrmor, it has bn assumd that th mtric tnsor is th bst mathmatical argumnt to us to study on gravitation Such opinions lad physicists to concntrat mor on only th mtric tnsor and, hnc, to chang it according to circumstancs As a rsult, this mthod provids som important rsults about gravitation Howvr, it is also obvious that ths rsults ar not nough to undrstand gravitation as wll as, prhaps, othr intractions In th prsnt papr, instad of concntrating on th mtric tnsor, w shall focus on ttrad filds Our first objctiv will b to find som
Apiron, Vol 13, No, Octobr 006 63 rasonabl mathmatical rsults with ths filds Th complt intrprtation of th rsults will b out of th scop of this papr Gravitation curvs th spac-tim and this ffct is rlatd to th lin lmnt or invariant intrval as ds = g dx dx whr g is th mtric tnsor and its lmnts ar som functions of th spac-tim Th mtric tnsor with ttrad filds is givn by [1], [] g = (1) whr ar basis vctors or ttrad filds, and ths ar som functions of th spac-tim also (, = 0,1,,3) Similar to (1), th invrs mtric tnsor can b writtn as g = whr ar basis vctors of th dual spac or cottrad filds Howvr, w will rfr to ths filds as invrs filds throughout this work Thr ar som usful faturs of and quations for th ttrad filds and invrs filds First g g = δ g = δ = δ () Othr quations and all dtaild calculations ar givn in th appndix sction
Apiron, Vol 13, No, Octobr 006 6 If th mtric tnsor is dtrmind, it is wll-known that it is dmanding work to find th Einstin quations Th Christoffl symbols for th mtric tnsor (1) ar 1 1 = f = f whr f = Th Rimann tnsor for th abov Christoffl symbols is R = f + f f + f f th Ricci tnsor is 1 R = j + f f, and th Ricci scalar is 1 R = j + f f 8 1 whr j = f = Finally th Einstin Tnsor can b xprssd as 1 1 G = g f f f f + j (3) Th xprssion in squar brackts is th sam as th strss-nrgy tnsor of lctromagntism xcpt for th innr products Dspit this diffrnc, th quations of motion of th ttrad filds hav th sam form as th Maxwll quations; that is = j Svral rsults can b obtaind from (3) Howvr, th most significant of ths is that th Einstin quations for th ttrad filds
Apiron, Vol 13, No, Octobr 006 65 crtainly giv th lctromagntic strss-nrgy tnsor Mor prcisly, th gnral rlativity rvals that thr ar som inhrnt constraints for ttrad filds This mans thr ar also dfinit limits for th mtric tnsor Sinc vry mtric tnsor can b writtn in trms of ttrad filds, mtric tnsors cannot b chosn or adjustd arbitrarily Instad, mtric tnsors must b found as innr products of ttrad filds aftr ths filds ar dtrmind to b consistnt with = j Appndix In this sction dtaild calculations and som usful quations ar givn for convninc, although som of ths can b found from svral sourcs and in diffrnt forms Sinc = δ, th partial drivativs of can b writtn as =ω ρ ρ So, whr ω ρ ar som cofficints Thn = ω = ω ρ ρ ρ = ω = ω, ρ ρ ρ ( ) = (A1) ρ ρ Similarly it can b shown that = (A) ( ) ρ ρ Anothr important quation can b drivd by starting from = Using (A1) ρ ρ ( ) = ρ ρ
Apiron, Vol 13, No, Octobr 006 66 ( ) ( ) ρ = ρ = ρ ρ Th dot product of th last quation with is = Sinc = δ ( ) ρ ρ = (A3) ρ ρ Similarly it can b found that = (A) ρ ρ Anothr quation can b drivd if th drivativ of () is rwrittn as = (A5) Now w can start to calculat th Einstin quations Th Christoffl symbols ar 1 = g + + 1 + g Using (A3), w gt 1 ( ) g = + (A6) Symmtris and charactristics of th ttrad filds nabl to driv som hlpful idntitis First using (A1)
Apiron, Vol 13, No, Octobr 006 67 (( ) ( ) ) 1 g ρ ρ = ρ + ρ Similarly, (A5) nabls us to writ ( ( ) ( ) ) 1 g ρ ρ = ρ ρ, ( ( ) ( ) ) 1 ρ ρ = ρ ρ Using (A3) and (A), w can obtain ( ( ) ( ) ) 1 ρ ρ = ρ ρ, (( ) ( ) ) 1 ρ ρ = ρ + ρ Also, by using (A3) and (A) 1 = ( + ) (A7) and using (A5) again 1 1 = ( ) = ( ) can b found Although it can b provd asily that =, for convninc lt f = Thn th Christoffl symbols ar 1 = f (A8)
Apiron, Vol 13, No, Octobr 006 68 With similar calculations, can b found in th following form 1 = f Or using (A), (A7) can b rwrittn as 1 = ( + ) Whn th Einstin Equations ar calculatd, on of ths can b usd Th Rimann tnsor dfind by R = + using th abov Christoffl symbols 1 R = f + f f f 1 1 + f ( + ) f ( + ), 1 1 R ( ) ( = f f ) 1 1 + ( f ) (( + ) ) ( f ) (( + ) ), = + + 1 1 + f ( + ), ( f f ) f ( ) R
Apiron, Vol 13, No, Octobr 006 69 = ( f + f ) + f f + f f R As a rsult of f + f + f = 0, th last Rimann tnsor can b simplifid For this, writ f + f = f = f Thus, R = f + f f + f f Th Ricci tnsor is = f + f f + f f R 1 Lt j = f = Thus th Ricci tnsor bcoms 1 R = j + f f and th Ricci scalar is 1 R = R g = j + f f 8 Hr f f is multiplid by 1 bcaus of twofold summation 1 Th Einstin tnsor is dfind as G = R g R Thn, G g = j + f f + j f f 8 Th last xprssion can b simplifid if w start with j = j
Apiron, Vol 13, No, Octobr 006 70 As a rsult of (), = So = 1 and w can writ ( ) ( ) j = j, (( ) ) 1 j = j, 1 j = ( j ) Th dot product of th last quation with yilds 1 1 j = ( j ) = ( j ) g Finally, th Einstin tnsor bcoms 1 1 G = g f f f f + j Rfrncs [1] CW Misnr, KS Thorn, JA Whlr, Gravitation, Ch 13, Frman (1973) 310 pp [] A Waldyr, Jr Rodrigus, Quintino A G d Souza, An Ambiguous Statmnt Calld Ttrad Postulat and th Corrct Fild Equations Satisfid by th Ttrad Filds, arxiv:math-ph/011085 v11 16 Aug 006