Spinor moves along geodesic. In some sense, only vector prospective is strictly compatible with Newtonian mechanics and Einstein’s principle of equivalence. Clearly, the additional acceleration in (81) 3 is different from that in (1), which is in two . The approximation to derive (1) h 0 might be inadequate, simply because h can be a universal constant acting as unit of physical variables. If w = 0, (81) naturally holds in all coordinate program on account of the covariant kind, although we derive (81) in NCS; however, if w 0 is massive sufficient for dark spinor, its trajectories will manifestly deviate from geodesics,Symmetry 2021, 13,13 ofso the dark halo AZD4625 Purity & Documentation within a galaxy is automatically separated from ordinary matter. Besides, the nonlinear prospective is scale dependent [12]. For many physique issue, dynamics in the technique need to be juxtaposed (58) resulting from the superposition of Lagrangian, it (t t )n = Hn n , ^ Hn = -k pk et At (mn – Nn )0 S. (82)The coordinate, speed and momentum of n-th spinor are defined by Xn ( t ) =Rxqt gd3 x, nvn =d Xn , dpn =R ^ n pngd3 x.(83)The classical approximation condition for point-particle model reads, qn un1 – v2 3 ( x – Xn ), nundXn = (1, vn )/ dsn1 – v2 . n(84)Repeating the derivation from (72) to (76), we acquire classical dynamics for every spinor, d t d pn p un = gen F un wn ( – ln n ) (S ) . n dsn dt 5. Energy-Momentum Tensor of Spinors Similarly to the case of metric g, the definition of Ricci tensor also can differ by a negative sign. We take the definition as follows R – – , (85)R = gR.(86)To get a spinor in gravity, the Lagrangian from the coupling technique is given byL=1 ( R – 2) Lm ,Lm =^ p – S – m 0 N,(87)in which = 8G, will be the cosmological constant, and N = 1 w2 the nonlinear possible. 2 Variation with the Lagrangian (87) with respect to g, we acquire Einstein’s field equation G g T = 0, whereg( R g) 1 G R- gR = – . 2 gg(88)will be the Euler derivatives, and T is EMT on the spinor defined by T=(Lm g) Lm Lm -2 = -2 2( ) – gLm . ggg( g)(89)By detailed calculation we have Theorem 8. For the spinor with nonlinear prospective N , the total EMT is offered by T K K = = =1 2 1 2 1^ ^ ^ (p p 2Sab a pb ) g( N – N ) K K ,abcd ( f a Sbc ) ( f a Sbc ) 1 f Sg Sd – g , a bc two g g (90) (91) (92)abcd Scd ( a Sb- b S a ),S S.Symmetry 2021, 13,14 of^ Proof. The Keller connection i is anti-Hermitian and truly vanishes in p . By (89) and (53), we receive the component of EMT related for the kinematic energy as Tp-2 =1g^ p = -(i – eA ) g(93)^ ^ ^ (p p 2Sab a pb ) ,where we take Aas independent variable. By (54) we obtain the variation associated with spin-gravity coupling possible as ( d Sd ) 1 = gabcdSd( f Sbc ) a g , g(94)( )1 ( d Sd ) = ( g) Sbc a Sd Sdabcd ( )( f Sbc Sd ) a =1abcd( f Sbc ) 1 a g . f a Sbc g g(95)Then we have the EMT for term Sas Ts = -d ( d Sd ) ( Sd ) 2( ) = K K . g( g)(96)Substituting Dirac Equation (18) into (87), we get Lm = N – N . For nonlinear 1 two possible N = two w , we’ve Lm = – N. Substituting each of the benefits into (89), we prove the theorem. For EMT of compound systems, we’ve got the following useful theorem [12]. Theorem 9. Assume matter consists of two subsystems I and II, namely Lm = L I L I I , then we have T = TI TI I . When the subsystems I and II BI-0115 medchemexpress haven’t interaction with every single other, namely, L I = L I I = 0, (98)(97)then the two subsystems have independent energy-momentum conservation laws, respectively, TI; = 0,.