Euler-Lagrange equations of the Einstein-Hilbert action. Faisal Amin Yassein Abdelmohssin 1

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1 Euler-Lrne eqution of the Eintein-Hilbert ction Fil Amin Yein Abdelmohin Sudn Intitute for Nturl Science, P.O.BOX 305, Khrtoum, Sudn July 6, 07 Abtrct: I pplied the method of the econd vrition of the Clculu of Vrition to the Eintein-Hilbert ction. Since the Eintein-Hilbert Lrnin for rvittionl field i proportionl to the icci curvture clr, contruction of the Euler-Lrne eqution require delin with the tenor quntitie from which the icci curvture clr i cooed rther thn the icci curvture clr itelf. A the reult of pplyin thi method two Euler- Lrne eqution emered; both eqution yielded the me Eintein field eqution in ence of enery-momentum field. PACS number: 0.0.-q, 0.0.Fy Keyword: Generl eltivity, Vritionl Principle, Euler-Lrne Eqution. Introduction: It i of interet to derive eqution of motion of vriou field (rvittionl, electromnetic, etc) from ction becue it h everl dvnte. Firt of ll, it llow for ey unifiction of different field theorie e.. rvittionl field with other clicl field theorie uch electromnetic field theory, which re lo formulted in term of n ction. In the proce the derivtion of eqution of motion of the rvittionl field from n ction identifie nturl cndidte for the ource term couplin the metric tenor to enery field. Moreover, the ction llow for the ey identifiction of conerved quntitie throuh Noether' theorem by tudyin ymmetrie of the ction. In the enerl theory of reltivity, the ction i uully umed to be functionl of the metric tenor (nd enery-momentum tenor field). In thi work we conider only the Euler-Lrne eqution in ence of enerymomentum field. f..y.delmohin@mil.com ()

2 The Eintein-Hilbert ction The Eintein-Hilbert ction for the rvittionl field my be written IEH d x ( ) ( ) () (.) Where,,, nd d x re Eintein contnt ( 8 G c ), icci curvture clr, determinnt of the metric tenor nd the invrint hyper-volume element of -dimenionl pcetime, repectively. The icci curvture clr i function of the metric tenor, it firt nd econd derivtive with repect to the pcetime i coordinte x ( x, x, x, x, x ) ( ct, x, x, x, x ) ; i.e. ( (,, c,, cd ) ). The ction principle then tell u tht the vrition of the ction in eqution (.) with repect to the rument of the internd i zero, yieldin 0 IEH [ ( ) ( d x)] ( ) ( ) d x we hve mde ue of the followin reltion where (.) i the contrvrint metric tenor nd i the icci curvture tenor. We my write the interl in eqution (.) um of two interl, nmely, 0 [ ( ) ( ) ( ) d x] [ ( ) ( ) ( ) d x] (.3) Second Vrition of the Clculu of Vrition of clr function It i well known tht the Euler-Lrne eqution reultin from pplyin the econd vrition of the Clculu of Vrition of Lrnin functionl L( x, y( x), y ( x), y( x )) of inle independent vrile y( x ), it firt nd econd derivtive of the followin ction I[ y( x)] L( x, y( x), y ( x), y( x)) dx (.) When vried with repect to the rument of internd nd the vrition re et to zero, i.e. i iven by 0 I[ y( x)] L( x, y( x), y ( x), y( x)) dx (.5) L d L d L 0 y dx y dx y (.6) provided tht the vrition y nd y vnih t the end point of the intertion.

3 Second Vrition of the Clculu of Vrition of tenor function We ume tht eqution (.6) to be vlid for tenor function whoe rument re the metric tenor, it firt derivtive nd it econd derivtive with repect to the pcetime coordinte, provided tht the vrition nd, c vnih t the end point of the intertion. A n exle, for tenor function K K(,, c,, cd ), the Euler-Lrne eqution my written K K K r x x x, 0 (.7) provided tht the vrition nd vnih t the end point of the intertion. Euler-Lrne eqution of the Eintein field eqution from the firt Eintein-Hilbert interl The Euler-Lrne eqution correpondin to the firt interl in eqution (.3) my be written ( ) ( ) ( ) ( ) [ ] 0 r x x x, Since nd (.8) don t depend on the firt nd the econd derivtive of the metric tenor with repect the pcetime coordinte, their prtil derivtive with repect to them ive zero. So eqution (.8) become ( ) [ ] 0 (.9) Thi i the Euler-Lrne eqution tht produce Eintein field eqution. It hould be noted tht in eqution (.9) the icci tenor houldn t be dropped out by divin both ide by it, for it would led to n urd reult. Derivtion of Eintein' field eqution from the Euler-Lrne eqution of the firt Eintein-Hilbert interl To proof tht eqution (.9) produce Eintein field eqution, we perform the prtil differentition nd ue the followin ueful eqution ( ) nd, (3)

4 m bp Then, eqution (.9) yield, ( ) 0 [ ] ( ) ( ) [ ] m bp [ ( ) ( ) ] m bp [ ( ( ) )] m bp [ ( ) ] [ ( ) ] Now, dividin both ide of the eqution (.0) by or either y (.0) i not zero, we rrive t the Eintein field eqution in ence of enery-momentum field. Euler-Lrne eqution of the Eintein field eqution from the econd Eintein-Hilbert interl The Euler-Lrne eqution correpondin to the econd interl in eqution (.3) my be written r x x x, ( ) ( ) [ ] 0 () (.) Differentition of the icci curvture tenor with repect to it rument To clculte the differentition of the icci curvture tenor with repect to the metric tenor, it firt derivtive nd econd derivtive we write firt iemnn curvture tenor nd icci tenor explicit function of the metric tenor, it firt derivtive nd econd derivtive. So, the iemnn curvture tenor i defined, d d d d e d e c c, b, c be c ce uin the Chritoffel ymbol of the firt nd the econd kind defined repectively, ijk ( ik, j jk, i ij, k )

5 nd l kl kl ij ijk ( ik, j jk, i ij, k ) The iemnn curvture tenor my be put in the followin form, d df ( ) de fh ( ) (.) c cf, b f, c cf beh f ceh We my now obtin the icci curvture tenor ily by contrctin two indice of the iemnn curvture tenor (puttin d c ) in eqution (.) nd ummin, we et c cf ( ) ce fh ( ) (.3) Differentitin c cf, b f, c cf beh f ceh repectively, we et,, in eqution (.3) with repect to,, nd,, cm p ce m p m p ( ) c c be ce { p c p cm pc p m p b ce bc m bc bc m ce m b ce ( ) ( ) } ce m p b p b pr b b ( ) { } r m r m m p (.) Performin the prtil differentition on the econd nd the third prt of eqution (.), we et cm p ce m p m p ( ) c c be ce x, {( p c p cm pc p m p b ce bc m bc bc m ce m ( ) ( ) ) b ce p c p cm ce m p bc m bc pc p m p b ce b ce bc m ce m ce m p ( ) ( ) )} r x x, ( ) { } b p b pr b b, r, m r, m, m p (.5) (5)

6 Performin the clcultion of the econd nd the third term in eqution (.5) nd multiplyin ech term in the me eqution by ( ) ( ), then eqution (.), yield e r q ( ) ( { 3 e r ( ) q ce m p m p m p q p q ( ) ( ) c be ce q bq ( ) ( ) ( ) ( ) mb p mb p q q, b bq q ( ) ( p mb mb p p ) ( ) ( ) m ( ) ( ) ( ) ( ) pq m t m t mr pq rq tq tq q q, r ( ) ( ) ( ) ( ) ( ) mr pq rq t t mr pq rq r, q tq r t rq m pq p mq p mr q mr pq ( ) q, r ( ) q, r ( ) q, r ( ) q, ( ) ( ) ( ) ( ) 0 p m rq r m pq rq r q q, r q, r q, r q, r (.6) In locl inertil frme of reference (LIF) t point, the followin expreion i true: nd, 0 Where ij i the metric tenor of the flt pcetime. ij ij The ove condition of LIF ilie tht ll Chritoffel ymbol re zero, nd, (we re uin pcetime with inture (,,, ) ), then eqution (.6) become ( ) ( { 3 e r ( ) q e r q } 0 (.7) Dividin both ide by ( ) ( ), nd collectin imilr term toether, we et e { ( ) } 0 (.8) e Which re in the Eintein field eqution in ence of enery-momentum field. Concluion Euler-Lrne eqution for the rvittionl field could only be contructed from tenor quntitie nd not by direct differentition of the clr quntitie on which the Lrnin i bed in ce of clr field theorie. ij k r (6)

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