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I. INTRODUCTION

The spinor field can give explanation for the accelerating expansion of the universe, and may be a possible candidate of dark matter. It is studied by some researchers in recently years[1–6]. In these works, the space-time is usually Friedmann-Lemaitre-Robertson-Walker

type with diagonal metric. The energy-momentum tensor(EMT) Tµν of spinors is simple and can be directly derived from Lagrangian of the spinor field in this case. To consider the interaction of spinor and gravity in general case is as early as H. Weyl in 1929[7]. There are some approaches to the general expression of EMT of spinors in curved space-time[6, 8–10]. But the formalisms are usually quite complicated for practical calculation and different from each other. In [8, 9], according to the Pauli’s theorem

1 δγα = γ δgαβ + [γα, M], (1.1) 2 β e e e where M is a traceless matrix related to the frame transformation, the EMT for Dirac spinor φ was derived as follows,

1 T µν = φ† (γµi ν + γνi µ)φ , (1.2) 2ℜh ∇ ∇ i e e where φ† = φ+γ is the Dirac conjugation, µ is the usual covariant derivatives for spinor. ∇ A detailed calculation for variation of action was performed in [6], and the results were a little different from (1.1) and (1.2). The following calculation shows that, M is still related with δgµν, and provides nonzero contribution to T µν in general cases. Besides, the covariant derivatives operator i is not ∇µ parallel to the classical momentum pµ and the expression (1.2) is complicated for practical calculation and some important effects are concealed by the compact form. The exact EMT of spinor was actually not obtained before.

The derivation of Tµν is quite difficult due to non-uniqueness representation and com- plicated formalism of vierbein or tetrad frames. In this paper, we give a systematical and detailed calculation for EMT of Weyl and Dirac spinors. We clearly establish the relations between tetrad and metric at first, and then solve the Euler derivatives with respect to gµν to get explicit and rigorous Tµν . From the results we find some new and interesting conclusions. Besides the usual kinetic energy momentum term, we find three kinds of other additional terms in EMT of bispinor. 3

One is the self interactive potential, which acts like negative pressure. The other reflects the interaction of momentum pµ with tetrad, which vanishes in classical approximation. The α third is the spin-gravity coupling term Ωαs , which is a higher order infinitesimal in weak field, but becomes important in a neutron star. This term is the eye of a particle with location and navigation functions, and is closely related with magnetic field of a celestial body. All these results are based on the decomposition of usual spin connection Γµ into geometrical part Υµ and dynamical part Ωµ, which not only makes calculation simpler, but also highlights their different physical meanings. In addition, in the calculation of tetrad µν formalism we find a new tensor Sab which plays an important role in the interaction of spinor with gravity and appears in many places. The materials are organized as follows: In the next section, we specify all notations, conventions and relevant equations used in the discussion. In the third section, we provide the technical foundations for the following derivation of EMT. We derive the exact EMT of spinor and its classical approximation in section IV, and then we give some simple discussions and illustrations in the last section.

II. NOTATIONS AND EQUATIONS

Denote the metric and coordinates of the Minkowski space-time respectively by

η = diag( 1, 1, 1, 1), δXµ =(δX,δY,δZ,cδT ). (2.1) ab − − −

The Pauli matrices are expressed by

0 1 0 i 1 0 1 0 σµ =   ,  −  ,   ,   , (2.2)  1 0 i 0 0 1 0 1       −    σµ = ( ~σ, σ0), ~σ =(σ1, σ2, σ3).  (2.3) − e where a, µ 0, 1, 2, 3. In this paper, we use the Greek characters stand for indices of ∈ curvilinear coordinates and Latin characters for indices of local Minkowski coordinates. The element of the space-time can be expressed by

µ a dx = γµdx = γaδX , (2.4)

e 4

µ where γa and γ are tetrad expressed by Dirac matrices

e µ µ 0 σ µ µ a a γ =   , γ = h aγ , γµ = lµ γa, (2.5) µ σ e0   e e which satisfies the Cℓ(1, 3) Clifford algebra[11–15]

γµγν + γνγµ = ηµν , γµγν + γνγµ = gµν. (2.6)

In Minkowski space-time, we have Dirac equatione e e e

µ γ i∂µφ = mφ, (2.7)

or in chiral form[11]

µ σ i∂µψ = mψ, ψ  φ =   , (2.8) µ  σ i∂µψ = mψ,e ψ    e e where ψ, ψ are two Weyl spinors.e Denotee the Pauli matrices in curved space-time by

µ µ a a µ ̺ = h aσ , ̺µ = lµ σa, 0 ̺  γµ =   , (2.9) µ µ a a µ  ̺ = h aσ , ̺µ = lµ σa, ̺ e0 e   then we have Clifford e algebrae as followse e

̺µ̺ν + ̺ν ̺µ = ̺µ̺ν + ̺ν̺µ =2gµν. (2.10)

The Weyl spinor equation (2.8e) in curvede e space-timee becomes

µ ̺ i(∂µ +Γµ)ψ = mψ,  (2.11) µ  ̺ i(∂µ + Γµ)ψ = mψ,e  e e where Γµ and Γµ are the spinor affinee connections[1, 2, 4–6, 9, 14], e 1 1 Γ = ̺ ̺ν , Γ = ̺ ̺ν , (2.12) µ 4 ν ;µ µ 4 ν ;µ µ µ µ α e e e in which ̺;ν = ∂ν ̺ +Γαν̺ . In order to disclose the physical meanings of connection, the total covariant derivatives of spinor can be represented in the following form[16],

µ µ ̺ i(∂µ +Γµ)= ̺ [i(∂µ +Υµ) + Ωµ] , (2.13) ̺µi(∂ + Γ )= ̺µ [i(∂ +Υ ) Ω ] . (2.14) µ µ µ µ − µ e e e 5

In which, Υµ is related with the grade-1 Clifford algebra, which has only geometrical effect,

1 1 Υ (la∂ hν + ∂ ln √g)= hν(∂ la ∂ la). (2.15) µ ≡ 2 µ ν a µ 2 a µ ν − ν µ

But Ωµ is related with grade-3 Clifford algebra[17]. Similarly, for Dirac bispinor in curved space-time we have

1 φ =(∂ + Γˆ )φ, Γˆ = γ γν . (2.16) ∇µ µ µ µ 4 ν ;µ e e More generally, the Lagrangian corresponding to (2.11) with electromagnetic interaction and nonlinear self-potential F is given by

+ µ + µ + = φ α pˆ φ + Ω φ sˆ φ mφ γ0φ + F, (2.17) Lm ℜh µ i µ − + µ + µ + + + + = ψ e̺ pˆµψ + ψ ̺ pˆµψ + ψ Ωψ + ψ Ωψ m(ψ ψ + ψ ψ)+ F, (2.18) ℜh i − e e e e e e e e where means taking real part, w> 0 is the nonlinear coupling coefficient, ℜhi e1 F = wγˇ2, γˇ = φ+γ φ. (2.19) 2 0

µ µ e (ˆpµ, α , sˆ ) are respectively momentum, current and spin operators defined by

e pˆ = i(∂ +Υ ) eA , αµ = diag(̺µ, ̺µ), sˆµ = diag(̺µ, ̺µ). (2.20) µ µ µ − µ −

ae a e1 ~ e In local central coordinate system δX ,s ˆ equals to 2 in one direction but vanishes in other direction. Ω and Ω are two Hermitian matrix defined by

e i µ α α µ µ a Ω [̺ ̺ ∂µ̺α (∂µ̺α)̺ ̺ ] = Ωµ̺ = ωaσ ,  ≡ 8 − (2.21)  Ω i [̺µ̺α∂ ̺ (∂ ̺ )̺α̺µ]= Ω ̺µ = ω σa. ≡ 8 e µ α − µ α e − µ − a  e e e e e e e For any diagonal metric, it easy to check Ω = Ω = 0. By straightforward calculation we have[16] e

1 ~ α ~ β ~ ω0 = 4 (h h ) ∂αlβ,  − × · ~ω 1 ∂ l 0 ~hα ~hβ hα ~hβ hβ ~hα ∂ ~l ,  = 4 α β ( ) ( 0 0 ) α β (2.22)  −  × − − ×  Ω = 1 (~hα ~hβ) (l 0∂ ~l ~l ∂ l 0)+ ~l [(hα ~hβ hβ ~hα) ∂ ~l ] , µ − 4 × · µ α β − µ α β µ · 0 − 0 × α β     in which ~ω =(ω1,ω2,ω3). (2.22) defines the dynamical part of the spinor connection. 6

III. RELATIONS BETWEEN TETRAD AND METRIC

In this section, we give an explicit representation of tetrad formalism. The derivation of EMT is based on this representation. Different from the cases of vector and tensor, in general relativity the dynamical equations for spinor fields depend on the local tetrad frame, which make the representation of the spinor connection and the EMT quite complicated. Assume that xµ =(x,y,z,ct) is the coordinates and δXa is the element vector in the tangent space at fixed point xµ. The tetrad γµ cannot be uniquely determined by metric, which can be only determined to an arbitrarye local Lorentz transformation. Now we derive some important relations used below. Lemma 1. For Pauli matrices (2.2) and (2.3), we have relations

a b c c b a abcd σ σ σ σ σ σ =2iǫ σd,  − (3.1)  σaσbσc σcσbσa = 2iǫabcdσ , e − e − d  in which ǫabcd is the permutatione function.e e e e Lemma 1 can be easy checked. We have only 4 nonzero cases for each equation. µ 0 For metric gµν , not losing generality we assume that, in the neighborhood of x , dx is 1 2 3 time-like and (dx , dx , dx ) are space-like. This means g00 0 and g 0(k = 0), and the ≥ kk ≤ 6 following definitions of Jk are real numbers

g11 g12 g13 g g v v 11 12 u J1 = √ g11, J2 = u , J3 = u g21 g22 g23 , J0 = det(g). (3.2) − u g g u− − u 21 22 u p t u g31 g32 g33 u t Denote

g11 g12 g11 g12 g21 g22 u1 = , u2 = , u3 = , (3.3)

g31 g32 g01 g02 g31 g32

and

g12 g13 g10 g11 g13 g10 g11 g12 g10

v1 = g22 g23 g20 , v2 = g21 g23 g20 , v3 = g21 g22 g20 , (3.4)

g32 g33 g30 g31 g33 g30 g31 g32 g30

then we have the following conclusion. 7

Theorem 2. For LU decomposition of matrix (gµν),

+ µν + ∗ + −1 (gµν)= L(ηab)L , (g )= U(ηab)U , U = L =(L ) , (3.5)

with positive diagonal elements, we have the following unique solution

g11 0 0 0 − J1  g21 J2  a J1 J1 0 0 L =(Lµ )=  −  , (3.6)  g31 u1 J3 0   − J1 J1 J2 J2   01   g u2 v3 J0   − J1 J1 J2 − J2 J3 J3 

1 g21 u3 v1 J1 J1 J2 J2 J3 J3 J0  J1 u1 v2  µ 0 J2 J2 J3 J3 J0 U =(U a)=  − −  . (3.7)  0 0 J2 v3   J3 J3 J0     00 0 J3   J0  (3.6) and (3.7) can be checked directly. For any other solutions of (2.5), we have a µ Theorem 3. For any solution of tetrad (2.5) in matrix form (lµ ) and (h a), there exists ′a a b a local Lorentz transformation δX =Λ bδX independent of gµν , such that

a + µ −1 (lµ )= LΛ , (h a)= UΛ , (3.8)

a where Λ=(Λ b) stands for the matrix of Lorentz transformation. Theorem 3 can be checked as follows,

(g )= L(η )L+ =(l a)(η )(l a)+ = L−1(l a)(η )(L−1(l a))+ =(η ). (3.9) µν ab µ ab µ ⇒ µ ab µ ab a Then we have a Lorentz transformation matrix Λ = (Λ b), such that

L−1(l a)=Λ+ = (l a)= LΛ+, or l a = L bΛa . (3.10) µ ⇒ µ µ µ b µ −1 Similarly we have (h a)= UΛ . Remark. The decomposition (3.5) is nothing but a Gram-Schmidt orthogonalization for vectors dxµ = (dx,dy,dz,dt) in tangent space-time in the order dt dz dy dx, → → → namely in vector form δX = dxL or

2 µ ν a b ds = gµνdx dx = ηabδX δX = (L X dx + L X dy + L X dz + L X dt)2 − x y z t (L Y dy + L Y dz + L Y dt)2 (L Zdz + L Zdt)2 +(L T dt)2. (3.11) − y z t − z t t 8

(3.11) is a direct result of (3.6), but (3.11) manifestly shows the geometrical meanings of a the tetrad components Lµ . Obviously, (3.11) is much convenient for practical calculation of tetrad parameters. Define a spinor coefficient tensor by

1 Sµν (hµ hν + hν hµ )sgn(a b). (3.12) ab ≡ 2 a b a b −

µν Sab is symmetrical for Riemann indices (µ, ν) but anti-symmetrical for Minkowski indices (a, b). (3.12) is important for the following calculation. By representation of (3.6), (3.7) and relation (3.8), we can check the following results by straightforward calculation. Theorem 4. For any solution of tetrad (2.5), we have

n ∂lα 1 µ ν ν µ nm 1 µν a nb = (δαh m + δαh m)η + Sab lα η . (3.13) ∂gµν 4 2 α ∂h a 1 µ αν ν µα 1 µν α nb = (h ag + h ag ) Sab h nη . (3.14) ∂gµν −4 − 2

Or equivalently,

∂̺α 1 µ ν ν µ 1 µν a b = (δα̺ + δα̺ )+ Sab lα σ . (3.15) ∂gµν 4 2 α ∂̺ 1 µα ν να µ 1 µν α nb a = (g ̺ + g ̺ ) Sab h nη σ . (3.16) ∂gµν −4 − 2

In (3.13)-(3.16) we set ∂̺α = ∂̺α = 1 d̺α (µ = ν) to get the tensor form. d means the ∂gµν ∂gνµ 2 dgµν 6 dgµν α total derivative for gµν and gνµ. For given vector A , we have

α ∂̺α 1 µ ν ν µ 1 µν α a b A = (A ̺ + A ̺ )+ Sab A lα σ . (3.17) ∂gµν 4 2 α ∂̺ 1 µ ν ν µ 1 µν α nb a Aα = (A ̺ + A ̺ ) Sab Aαh nη σ . (3.18) ∂gµν −4 − 2

Similarly, for Dirac matrices we have Corollary 5. For Dirac matrices (2.5), we have

∂γα 1 µ ν ν µ 1 µν a b = (δαγ + δαγ )+ Sab lα γ . (3.19) ∂gµν 4 2 e e e Or equivalently, for any given vector Aµ, we have

α ∂γα 1 µ ν ν µ 1 µν α a b A = (A γ + A γ )+ Sab A lα γ . (3.20) ∂gµν 4 2 e e e 9

IV. ENERGY-MOMENTUM TENSOR OF SPINORS

Now we consider the coupling system of spinor and gravity. The Ricci tensor and scalar curvature is defined by

R = ∂ Γα ∂ Γα +Γα Γβ Γα Γβ , R gµνR . µν µ να − α µν µβ να − µν αβ ≡ µν The total Lagrangian of the system reads 1 1 = + = (R 2Λ) + , (4.1) L 2κLg Lm −2κ − Lm where κ = 8πG, Λ is the cosmological constant, the Lagrangian of spinors (2.18). Lm Variation of the Lagrangian (4.1) with respect to gµν we get Einstein’s equation 1 Rµν gµνR +Λgµν = κT µν, (4.2) − 2 where T µν is EMT of the spinors, which is defined by

µν δ( m√g) ∂ m γ ∂ m µν T = 2 L = 2 L (∂α +Γαγ) L g m, (4.3) − √gδgµν − ∂gµν − ∂(∂αgµν) − L

δ where g = det(gµν) and is the Euler derivatives. | | δgµν We take ψ as example to derive relations, because we have similar results for ψ. By(2.18) and (3.16) we have e

µν ∂ + α 1 + µ ν ν µ µν α nb a p ψ ̺ pˆαψ = ψ (̺ pˆ + ̺ pˆ +2Sab h nη σ pˆα)ψ . (4.4) E ≡ ∂gµν ℜh i −4ℜh i By (2.21), (3.1) and (3.17) we rewrite Ω as follows,

i µ α ∂̺α ∂̺α α µ Ω = ̺ ̺ ̺ ̺ ∂µgλκ, 8  ∂gλκ − ∂gλκ 

i ae b c c b a e µ λκ = σ σ σ σ σ σ h S ∂ g , by (3.17) 16 − a bc µ λκ 1
1 = ǫabcdeσ hµ Sλκe∂ g = ǫdabcσ hµ Sλκ∂ g . by (3.1) (4.5) −8 d a bc µ λκ 8 d a bc µ λκ α a By (4.5), we get (Ω ,ω ) expressed by ∂αgµν as follows, 1 1 ωd = ǫdabchα Sµν ∂ g , Ωα = ǫdabchα hβ Sµν ∂ g . (4.6) 8 a bc α µν 8 d a bc β µν Then we get + α λκ ∂(ψ Ωψ) 1 abcd ∂(h aSbc ) + = ǫ σˇd ∂αgλκ, σˇd ψ σdψ, (4.7) ∂gµν −8 ∂gµν ≡ + α µν β ∂(ψ Ωψ) 1 abcd α µν β ∂(h aSbc ) (∂α +Γαβ) = ǫ h aSbc (∂α +Γαβ)ˇσd +ˇσd ∂αgλκ , (4.8) ∂(∂αgµν ) −8  ∂gλκ  10 and then we have the Euler derivative for ψ+Ωψ as

+ + + µν δ(ψ Ωψ) ∂(ψ Ωψ) β ∂(ψ Ωψ) Ω = (∂α +Γαβ) , E ≡ δgµν ∂gµν − ∂(∂αgµν) α µν α λκ 1 abcd α µν β ∂(h aSbc ) ∂(h aSbc ) = ǫ h aSbc (∂α +Γαβ)ˇσd +ˇσd ∂αgλκ . (4.9) 8   ∂gλκ − ∂gµν  

In principle, we can substitute (3.14) into (4.9) and then simplify it to get the final expression of µν. However, the structure of µν has been clear, and we need not to do EΩ EΩ so complicated calculations now, because it is a second order tenser including only linear

first order derivatives ∂µσˇd and ∂µgαβ. The vectors and tensors constructed by metric and αβ its linear first order derivatives are only (Υµ, Ωµ) multiplied by gµν and g . Substituting β σˇd = h d̺ˇβ into (4.9), then the simplified expression should take the following covariant form

µν 1 abcd α β µν µν α α = ǫ h h S ̺ˇ ; + g (k1Ω + k2Υ )ˇ̺ , (4.10) EΩ 8 a b cd β α α where (k1,k2) are 2 constants to be determined. µµ µµ For diagonal metric we have Ω = S = = 0, and then by (4.10) we get k2 = 0. ab EΩ In Gaussian normal coordinate system g = ( g¯ , 1), we have Hamiltonian for linear µν − kl bispinor[16]

k µ H = α pˆ + eA0 + mγ0 Ω sˆ . (4.11) − k − µ e By (4.3) and (4.10), we have

00 00 00 µ T = 2( + e )+ = 2k1Ω sˇ + , (4.12) − EΩ EΩ ··· − µ ···

in which the omitted terms are irrelevant with the following comparison. Since T 00 is energy 1 of the spinors, in contrast (4.12) with (4.11) we get k1 = 2 . So we get

µν 1 abcd α β µν 1 µν α = ǫ h h S ̺ˇ ; + g Ω ̺ˇ . (4.13) EΩ 8 a b cd β α 2 α

Similarly, for ψ we have

e 1 1 µν = ǫabcdhα hβ Sµν ̺ˇ gµνΩα̺ˇ , ̺ˇ ψ+̺ ψ. (4.14) EΩe −8 a b cd β;α − 2 α α ≡ α e e e e e e For total spins ˆµ we have

µν µν µν 1 abcd α β µν 1 µν α + = + = ǫ h h S sˇ ; + g Ω sˇ , sˇ φ sφ.ˆ (4.15) Es EΩ EΩe 8 a b cd β α 2 α α ≡ 11

Substituting (4.4) and (4.15) into (4.3), we get the EMT for Weyl spinors

1 T µν = ψ+(̺µpˆν + ̺ν pˆµ)ψ + ψ+(̺µpˆν + ̺νpˆµ)ψ gµν 2ℜh i − Lm + µν α nb a + µν α nb a + ψ S h η σ pˆαψ +eψ eS h ηe σ pˆeαψ (4.16) ℜh ab n ab n i 1 abcd α β µν µν α ǫ h h S sˇβ;α g eΩ sˇα. e e −4 a b cd −

For nonlinear spinor with potential (2.19), by Dirac equation we have = F . Lm − Substituting the results into (4.16), we get the final EMT of bispinor φ as follows,

1 T µν = φ+(αµpˆν + ανpˆµ +2Sµνhα ηnbαapˆ )φ + gµνF 2ℜh ab n α i 1 abcd α β µν µν α ǫ he h S esˇ ; g Ω sˇ . (4.17) −4 a b cd β α − α

The physical meaning of

µν 1 abcd α β µν 1 abcd α β µν ǫ h h S sˇ ; = ǫ h h S (∂ sˇ ∂ sˇ ) (4.18) E ≡ 4 a b cd β α 8 a b cd α β − β α

µν is unclear. Its diagonal components vanish, so it should be very tiny term to keep T;ν = 0. Now we consider the classical approximation for (4.17),

φ+ανφ uν√1 v2δ3(~x X~ ), pˆµφ muµφ, F w√1 v2δ3(~x X~ ). (4.19) → − − → → − − e Substituting (4.19) into (4.17) and noticing Sµν = Sµν , we have ab − ba

φ+Sµν hα ηnbαapˆ φ mSµν uaub√1 v2δ3(~x X~ )=0. (4.20) ℜh ab n α i→ ab − −

Omitting the tiny spin-gravity coupling energy, we get the usual EMT for a classical particle with self-interactive potential

T µν (muµuν + wgµν Ω sαgµν)√1 v2δ3(~x X~ ). (4.21) → − α − −

w> 0 acts like negative pressure[3, 18]. By (4.21) and energy-momentum conservation law µν T ;ν = 0, we find only if w = 0 and m is a constant independent of v, the particle moves along geodesic and the principle of equivalence strictly holds[16, 19]. Some previous works usually use one spinor to represent matter field. This may be not the case, because spinor fields only has a very tiny structure. Only to represent one particle by one spinor field, the matter model can be comparable with general relativity, classical 12 mechanics and quantum mechanics[3, 16, 18, 20]. The many body system should be better described by the Lagrangian similar to the following one,

+ µ + µ + 1 2 = φ α pˆ φ + Ω φ sˆ φ mφ γ0φ + F , F = wγˇ . (4.22) Lm ℜh n µ ni µ n n − n n n n 2 n Xn e e The corresponding EMT is given by 1 T µν = φ+(αµpˆν + ανpˆµ +2Sµνhα ηbcαapˆ )φ + gµνF 2ℜh n ab c α ni n Xn 1 e e ǫabcdhα hβ Sµν (∂ sˇ ∂ sˇ ) gµνΩ sˇα . (4.23) −8 a b cd α nβ − β nα − α n The classical approximation becomes

T µν (m uµuν + w gµν sαΩ gµν) 1 v2δ3(~x X~ ), (4.24) → n n n n − n α − n − n Xn p which leads to the EMT for average field of spinor fluid as follows

T µν =(ρ + P )U µU ν +(W P )gµν. (4.25) − The self potential becomes the negative pressure W [21].

V. DISCUSSION AND CONCLUSION

In this paper, according to the explicit relations between tetrad and metric, we derived the

exact representation of Tµν of spinors. By splitting the spinor connection into geometrical part Υµ and dynamical part Ωµ, we find some new results. In this EMT, besides the usual kinetic energy momentum term, the nonlinear self-interactive potential W acts like negative pressure and may be important in astrophysics. The term φ+Sµνhα ηnbαapˆ φ reflects the ℜh ab n α i interaction of momentum pµ with tetrad, which vanishes in classical approximation. The α spin-gravity coupling term is described by Ωαs , which may be the origin for magnetic field of a celestial body. This term also appears in the dynamics of a spinor and has location and

navigation functions for the particle. To get a clear concept for vector Ωα, we make weak field approximation for Kerr metric in cylindrical coordinate system (t, ρ, ϕ, z). In this case, by (4.6) we have 4maρ2 ds2 dt2 dρ2 ρ2dϕ2 dz2 dtdϕ, r = ρ2 + z2. (5.1) → − − − − r3 1 ma p Ωα = (0, ∂ g , 0, ∂ g )= (0, 3ρz, 0, 2z2 ρ2). (5.2) 4ρ z tϕ − ρ tϕ 2r5 − 13

The vector Ωα is similar to the magnetic field B~ generated by a circular coil. The lines of flux of vector Ωα in spherical coordinate system are given by r = R sin2 θ, (R > 0). The spins of the particle will be arranged along these lines, and a macroscopical magnetic field B~ can be induced. Since dt is not orthogonal to dϕ, this may lead to procession of the magnetic field.

The decomposition of spin connection Γµ not only makes calculation simpler, but also highlights their different physical meanings. Only in this representation, we can discover and clarify the special effects as discussed above. In the calculation of tetrad formalism we get a µν new tensor Sab which plays an important role in the interaction of spinor with gravity and appears in many places. But it has not classical correspondences, and vanishes in classical approximation. The spinor has only a tiny but marvelous structure. It is an indivisible system and unsuitable to describe many body problem. Only by using one spinor to describe one particle, we get a harmonic picture for field theory and classical mechanics.

Acknowledgments

I’d like to acknowledge Prof. James M. Nester for his enlightening discussions and en- couragement. I have encountered the difficulty in derivation of exact energy-momentum tensor for many years. He encouraged “There may be a deeper wisdom in these things that we have not yet appreciated”.

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