Curvature Spinors in Locally Inertial Frame and the Relations with Sedenion
universe
Article Curvature Spinors in Locally Inertial Frame and the Relations with Sedenion
In Ki Hong 1, Choong Sun Kim 1,2,* and Gyung Hyun Min1
1 Department of Physics and IPAP, Yonsei University, Seoul 03722, Korea; [email protected] (I.K.H.); [email protected] (G.H.M.) 2 Institute of High Energy Physics, Dongshin University, Naju 58245, Korea * Correspondence: [email protected]
Received: 30 January 2020; Accepted: 2 March 2020; Published: 6 March 2020
Abstract: In the 2-spinor formalism, the gravity can be dealt with curvature spinors with four spinor indices. Here we show a new effective method to express the components of curvature spinors in the rank-2 4 × 4 tensor representation for the gravity in a locally inertial frame. In the process we have developed a few manipulating techniques, through which the roles of each component of Riemann curvature tensor are revealed. We define a new algebra ‘sedon’, the structure of which is almost the same as sedenion except for the basis multiplication rule. Finally we also show that curvature spinors can be represented in the sedon form and observe the chiral structure in curvature spinors. A few applications of the sedon representation, which includes the quaternion form of differential Binanchi identity and hand-in-hand couplings of curvature spinors, are also presented.
Keywords: spinor formalism; general relativity; sedenion; quaternion; representation theory
1. Introduction In the 2-spinor formalism [1–3] all tensors with spacetime indices can be transformed into spinors with twice the number of spinor indices, i.e., a rank-2 tensor is changed into a spinor with four spinor indices. In addition, if the tensor is antisymmetric and real, it can be represented by a sum of two spinors with two spinor indices, and they are complex conjugate of each other, which indicates that a rank-2 antisymmetric tensor is equivalent to a spinor with two spinor indices. The Riemann curvature tensor is a rank-4 real tensor which describes gravitational fields and it has two antisymmetric characters. This means that the gravity can be described by two spinors with four spinor indices. Those two spinors are called curvature spinors: One of them is Ricci spinor and the other is Weyl conformal spinor [1,3–5]. At any points on a pseudo-Riemannian manifold, we can find a locally flat coordinate [6], whose metric is Minkowski. While the metric is locally Minkowski, the second derivative of the metric is not necessarily zero and the Riemann curvature tensor as well as curvature spinors do not have to be zero. Here we can obtain the explicit representations of curvature spinors, the components of which can be easily identified by using new techniques, i.e., manipulating spinor indices and rotating sigma basis in locally flat coordinates [7]. Then all the components of curvature spinors are represented with simple combinations of Riemann curvature tensors. Here the representations are not described by the four-dimensional basis but by the three-dimensional basis ⊗ three-dimensional basis, and thus it suggests a different interpretation of time. The process has been applied on both Ricci spinor and Weyl spinor, which are curvature spinors, and each spinor is described as the sum of two newly defined parts; one of which is a real part and the other is a pure imaginary part. The obtained representation can be used not only in a special flat coordinate but also for vielbein indices or in any other normal coordinates, like Riemann normal coordinate and Fermi coordinate [8–16]. By comparing
Universe 2020, 6, 40; doi:10.3390/universe6030040 www.mdpi.com/journal/universe Universe 2020, 6, 40 2 of 18 the final forms of Ricci spinor with the spinor form of Einstein equation, we figure out the roles of each component of Riemann curvature tensor, whose components serve as momentum, energy or stress of gravitational fields. Furthermore, we show that the components of Weyl conformal spinor can be represented as a simple combination of Wely tensors in flat coordinate. There are already quite a few papers that show the relation between gravitational fields and Cayley–Dickson algebras including sedenion; however, all papers are restricted to a weak gravitational field in a flat frame [17–22]. Here we express the basis of sedenion as a set of direct products of a quaternion basis, through which we can define a new algebra ‘sedon’, whose structure is similar to sedenion except for the basis multiplication rule. We show that the curvature spinors for general gravitational fields in locally flat coordinates can be regarded as a sedon. The spinors are described on the direct procduct of totally seperated left-handed basis and right-handed quaternion basis. From this, we can get a view of the gravitational effects as the combination of right-handed and left-handed rotational effects. We also introduce a few applications of the sedon form with multiplication techniques. One of the application is the quaternion form of differential Bianchi identity and, in the process, we introduce a new index notation with the spatially opposite-handed quantities.
2. Tensor Representation of a Field with Two Spinor Indices In this section we introduce the basics about the 2-spinor formalism, which have been already explained in detail in our earlier paper [7]. We use the front part of Latin small letters a, b, ..., h and Greek letters µ, ν, ρ... as four dimensional space-time indices, which can be 0, 1, 2 or 3. The later part of Latin small letters i, j, ..., which can be 1, 2 or 3, are used as three dimensional indices. Any tensor Tabc.. with spacetime indices a, b, c, .., can be inverted into a spinor with spinor indices 0 0 a A, A , B, B , .. like TAA0 BB0.. by multiplying Infeld-van der Waerden symbols gAA0 ,
a a TAA0 BB0.. = Tab..gAA0 gBB0 .. . (1)
a 1 a a 0 1 2 3 0 In Minkowski spacetime, g 0 is √ σ 0 , where σ 0 are four-sigma matrices (σ , σ , σ , σ ); σ AA 2 AA AA is 2 × 2 identity matrix and σ1, σ2, σ3 are Pauli matrices. Equation (1) can be written conventionally as
TAA0 BB0.. = Tab... (2)
Any arbitrary anti-symmetric tensor Fab = FAA0 BB0 can be expressed as the sum of two symmetric spinors as
FAA0 BB0 = ϕABε A0 B0 + ε ABψA0 B0 , (3)
0 1 C 0 0 1 C where ϕAB = 2 FABC0 and ψA B = 2 FCA0 B0 are symmetric spinors (unprimed and primed spinor AB A0 B0 indices can be switched back and forth each other), and ε , ε , ε AB, ε A0 B0 are the ε-spinors whose 12 21 A AB A components are ε = ε12 = +1, ε = ε21 = −1 [1]; k = ε kB, kB = k ε AB. If Fab is real, then ψA0 B0 = ϕ¯ A0 B0 (where ϕ¯ is the complex conjugate of ϕ) and
Fab = FAA0 BB0 = ϕABε A0 B0 + ε AB ϕ¯ A0 B0 . (4)
We have shown the components of ϕAB and ϕ¯ A0 B0 explicitly in flat spacetime in [7]. The sign conventions for the Minkowski metric is gµν = diag(1, −1, −1, −1). For any real anti-symmetric tensor FAA0 BB0 , we can write as
1 µ ν 1 µ ν C0C F 0 0 = F σ σ 0 = F σ σ¯ ε 0 0 ε , (5) AA BB 2 µν AA0 BB 2 µν AA0 C B CB Universe 2020, 6, 40 3 of 18 where σ¯ µ = (σ0, −σ1, −σ2, −σ3), then
1 A0 1 A0 B0 ϕ = F 0 = F 0 0 ε AB 2 AA B 2 AA BB 1 µ ν C0C A0 B0 1 µ ν A0C = F σ σ¯ ε 0 0 ε ε = F σ σ¯ ε , (6) 4 µν AA0 C B CB 4 µν AA0 CB 1 A 1 AB ϕ¯ 0 0 = F 0 0 = F 0 0 ε A B 2 AA B 2 AA BB 1 µC0C ν AB 1 µC0 B ν = F σ¯ σ 0 ε 0 0 ε ε = F ε 0 0 σ¯ σ 0 , (7) 4 µν BB A C AC 4 µν A C BB
ν 1 µ 1 ν C0C where we used the relation g 0 = √ σ 0 = √ σ¯ ε 0 0 ε . This can be established when the BB 2 BB 2 C B CB space-time metric is locally flat [1,23]. If we apply the relations to the general coordinates with relevant modifications, the results for arbitrary coordinates can be obtained. Since σ0σ0 −σ0σ1 −σ0σ2 −σ0σ3 σ0 −σ1 −σ2 −σ3 0 σ1σ0 −σ1σ1 −σ1σ2 −σ1σ3 σ1 −σ0 −iσ3 iσ2 µ ν A C = = σAA0 σ¯ 2 0 2 1 2 2 2 3 2 3 0 1 , (8) σ σ −σ σ −σ σ −σ σ σ iσ −σ −iσ 3 0 − 3 1 − 3 2 − 3 3 C 3 −i 2 i 1 − 0 C σ σ σ σ σ σ σ σ A σ σ σ σ A
D DB ϕA = ε ϕAB becomes
1 µ 0 ϕ D = F σ σ¯ ν A D A 4 µν AA0 0 1 2 3 T 0 −F10 −F20 −F30 σ −σ −σ −σ 1 F 0 F F σ1 −σ0 −iσ3 iσ2 = 10 12 13 2 3 0 1 4 F20 −F12 0 F23 σ iσ −σ −iσ F −F −F 3 −i 2 i 1 − 0 D 30 13 23 0 σ σ σ σ A 1 1 ij = (F σi − i e F σk) D, (9) 2 i0 2 k ij A
ij i j where i , j , k are the three-dimensional vector indices which have the value 1, 2 or 3, and e k is epqkδpδq for the Levi–Civita symbol eijk. Einstein summation convention is used for three-dimensional vector indices i, j and k. Similar to Equations (8) and (9), σ0σ0 σ0σ1 σ0σ2 σ0σ3 σ0 σ1 σ2 σ3 1 0 1 1 1 2 1 3 1 0 3 2 µC0 B ν −σ σ −σ σ −σ σ −σ σ −σ −σ −iσ iσ σ¯ σ 0 = = , (10) BB −σ2σ0 −σ2σ1 −σ2σ2 −σ2σ3 −σ2 iσ3 −σ0 −iσ1 0 0 − 3 0 − 3 1 − 3 2 − 3 3 C − 3 −i 2 i 1 − 0 C σ σ σ σ σ σ σ σ B0 σ σ σ σ B0
D0 D0 A0 1 µD0 B ν 1 i 1 ij k D0 ϕ¯ 0 = ε ϕ¯ 0 0 = − F σ¯ σ 0 = (F σ + i e F σ ) 0 . (11) B A B 4 µν BB 2 i0 2 k ij B
If we denote matrix representation of ε AB by ε, then
σµε = (σ0, σ1, σ2, σ3)ε = (iσ2, −σ3, iσ0, σ1), (12) εσµ = ε(σ0, σ1, σ2, σ3) = (iσ2, σ3, iσ0, −σ1). (13)
Let us define sµ and s¯µ as
s0 = iσ2, s1 = −σ3, s2 = iσ0, s3 = σ1, (14) Universe 2020, 6, 40 4 of 18
s¯0 = iσ2, s¯1 = −σ3, s¯2 = −iσ0, s¯3 = σ1, (15) where s¯µ is complex conjugate of sµ. Then
σµε = (s0, s1, s2, s3), (16) εσµ = (s0, −s1, s2, −s3) = (s¯0, −s¯1, −s¯2, −s¯3), (17) and
1 1 ij ϕ = ϕ Dε = (F si − i e F sk), (18) AB A DB 2 i0 2 k ij D0 D0 1 i 1 ij k ϕ 0 0 = ϕ 0 ε 0 0 = −ε 0 0 φ 0 = (F s¯ + i e F s¯ ), (19) A B B D A A D B 2 i0 2 k ij
i i i i i i where s have unprimed indices s = sAB and s¯ have primed indices s¯ = s¯A0 B0 .
3. Einstein Field Equations and Curvature Spinors In this section we introduce the basics about general relativity in the 2-spinor formalism, and flat coordinates on the pseudo-Riemannian manifold. For a (torsion-free) Riemann curvature tensor
µ µ µ λ µ λ µ R νρσ = ∂ρΓ νσ − ∂σΓ νρ + Γ νσΓ λρ − Γ νρΓ λσ, (20)
ρ where Γ µν is a Chistoffel symbol
ρ 1 Γ = gρλ(∂ g + ∂ g − ∂ g ). (21) µν 2 µ νλ ν µλ λ µν
Here Rµνρσ has follwing properties [6]
Rµνρσ = −Rνµρσ , (22)
Rµνρσ = −Rµνσρ , (23)
Rµνρσ = Rρσµν . (24)
In short, we can denote as
Rµνρσ = R([µν][ρσ]), (25) where parentheses ( ) and square brackets [ ] indicates symmetrization and anti-symmetrization of the indices [6]. The Riemann curvature tensor has two kinds of Bianchi identities
Rµ[νρσ] = 0, (26)
∇[λRµν]ρσ = 0, (27)
µ µ µ ν where ∇λ A = ∂λ A + Γνλ A . From the antisymmetric properties of Riemann curvature tensor, it can be decomposed into sum of curvature spinors, XABCD and ΦABC0 D0 , as
1 X0 1 X R = R 0 ε 0 0 + R 0 0 ε abcd 2 AX B cd A B 2 XA B cd AB = ΦABC0 D0 eA0 B0 eCD + Φ¯ A0 B0CDeABeC0 D0 + XABCDeA0 B0 eC0 D0 + X¯ A0 B0C0 D0 eABeCD, (28) Universe 2020, 6, 40 5 of 18 where
X0 Y0 X0 Y XABCD = RAX0 BCY0 D , ΦABC0 D0 = RAX0 BYC0 D0 . (29)
The totally symmetric part of XABCD
ΨABCD = XA(BCD) = X(ABCD) (30) is called gravitational spinor or Weyl conformal spinor, and ΦABC0 D0 is referred as Ricci spinor [1,4,5]. It is well known that
¯ a ΦAA0 BB0 = Φab = Φba = Φab, Φa = 0, (31) and Einstein tensor is 1 G = R − Rg = −Λg − 2Φ , (32) ab ab 2 ab ab ab
AB where Λ = XAB , which is equal to R/4 [1]. Therefore, the Einstein field equation
Gab + λgab = 8πGTab, (33) where λ is a cosmology constant, can be written in the form
1 q q Φ = 4πG(−T + T g ), Λ = −2πGT + λ. (34) ab ab 4 q ab q
Since any symmetric tensor Uab can be expressed as
Uab = UAA0 BB0 = SABA0 B0 + ε ABε A0 B0 τ, (35)
1 c where τ = 4 Tc and SABA0 B0 is traceless and symmetric [1], the traceless part of the energy-momentum 1 c (symmetric) tensor Tab can be written by Sab = Tab − 4 Tc gab. Therefore, the spinor form of Einstein Equations (34) becomes
AB ΦABA0 B0 = −4πGSABA0 B0 , XAB = −8πGτ + λ. (36)
Weyl tensor Cµνρσ which is another measure of the curvature of spacetime, like Riemann curvature tensor, is defined as [6,24]
1 1 C = R + (R g − R g + R g − R g ) + R(g g − g g ). (37) µνρσ µνρσ 2 µσ νρ µρ νσ νρ µσ νσ µσ 6 µρ σν µσ νρ It has the same propterties as Equations (22), (23) and (26). It is known [1] that Weyl tensor has the following relationship with Weyl conformal spinor ΨABCD:
Cabcd = ΨABCDε A0 B0 εC0 D0 + Ψ¯ A0 B0C0 D0 ε ABεCD. (38)
At any point P on the pseudo-Riemannian manifold, we can find a flat coordinate system, such that,
∂g ( ) = µν = gµν P ηµν, λ 0, (39) ∂x P Universe 2020, 6, 40 6 of 18
where gµν(P) is the metric at the point P and ηµν is the Minkowski metric. In this coordinate system, while the Christoffel symbol is zero, the Riemann curvature tensor is [25,26]
1 R = (∂ ∂ g − ∂ ∂ g + ∂ ∂ g − ∂ ∂ g ). (40) µνρσ 2 ν ρ µσ ν σ µρ µ σ νρ µ ρ νσ For future use we introduce Fermi coordinate, which is one of the locally flat coordinate whose time axis is a tangent of a geodesic. The coordinate follows the Fermi conditions
ρ gµν|G = ηµν, Γµν|G = 0, (41) along the geodesic G.
4. The Tensor Representation of Curvature Spinors In this section we show the process of representing curvature spinors in 4 × 4 matrices or 3 × 3 matrices. We discuss physical implications of those representations. From now on, we will always use locally flat coordinate for all spacetime indices. From Equations (4) and (28), we can lead to
Rabcd = φAB,cdε A0 B0 + ε ABφ¯A0 B0,cd
= ΦABC0 D0 eA0 B0 eCD + Φ¯ A0 B0CDeABeC0 D0 + XABCDeA0 B0 eC0 D0 + X¯ A0 B0C0 D0 eABeCD, (42) where 1 1 φ = (R si − i e R sk), (43) AB,cd 2 i0 cd 2 ijk ij cd 1 i 1 k φ¯ 0 0 = (R s¯ + i e R s¯ ), (44) A B ,cd 2 i0 cd 2 ijk ij cd
ij from Equations (18) and (19). We write here the form of e k as eijk for convenience; it is not so difficult to recover the upper- and lower-indices. By decomposing φAB,cd one more times, we get
1 i j 1 i r 1 k l 1 k r Φ 0 0 = (R s s¯ + ie R s s¯ − ie R s s¯ + e e R s s¯ ), ABC D 4 i0 j0 2 pqr i0 pq 2 ijk ij l0 4 ijk pqr ij pq 1 1 1 1 = (R + ie R − ie R + e e R )sks¯l, (45) 4 k0 l0 2 pql k0 pq 2 ijk ij l0 4 ijk pql ij pq
1 1 1 1 X = (R sisj − ie R sisr − ie R sksl − e e R sksr) ABCD 4 0i 0j 2 pqr 0i pq 2 ijk ij 0l 4 ijk pqr ij pq 1 1 1 1 = (R − ie R − ie R − e e R )sksl. (46) 4 0k 0l 2 pql 0k pq 2 ijk ij 0l 4 ijk pql ij pq We note that Φ and X are expressed with two three-dimensional basis like the form in 3 × 3 basis. Even though there is no 0-th base, which may be related to the curvature of time, Φ and X can fully describe the spacetime structure. If Φ and X are represented in Fermi coordinate, the disappearance of 0-th compomonents of the si basis may come from the fact that time follows proper time. However, since here Equations (45) and (46) are expressed not only in Fermi coordinate but also in general locally flat coordinates, whose 0-th coordinate may not be time direction, the representations like Equations (45) and (46) may demand a new interpretation of the curvature of time, which is not just as a component of fourth (or 0-th) axis in 4-dimensional space-time. Technically the interpretation of the space-time structure, which was considered to be a bundle of four directions, might be reconsidered. Universe 2020, 6, 40 7 of 18
We can divide Equation (45) into two terms by defining
1 1 P ≡ e R − e R , (47) ij 2 pqj i0 pq 2 pqi j0 pq 1 S ≡ R + e e R , (48) ij 0i 0j 4 pqi rsj pq rs 1 1 1 Θ ≡ R + ie R − ie R + e e R = S + i P , (49) ij i0 j0 2 pqj i0 pq 2 pqi pq j0 4 pqi rsj pq rs ij ij where Pij is anti-symmetric and Sij is symmetric for i, j. Then Equation (45) is represented as
1 i j Φ 0 0 = Θ s s¯ . (50) ABC D 4 ij AB C0 D0
The components of Pij and Sij can be simply expressed as
1 1 − e P = − (e e R − e e R ) = R , (51) 2 ijk ij 4 ijk pqj i0 pq ijk pqi j0 pq 0i ki Si j = R0i 0j + εi pqεj rsRpq rs , (52)
Si i = R0i 0i + |εi pq|Rpq pq , (53) where the underlined symbols in subscripts are the value-fixed indices which does not sum up for dummy indices; one of example is S11 = R01 01 + R23 23. AC0 We can express ΦABCD as a tensor by multiplying gµ , which is
AB0 AC B0 D0 ν 1 0 1 2 3 AB0 1 t AB0 1 ∗ AB0 gµ = ε ε gµνg 0 = √ (σ , σ , −σ , σ ) = √ σ µ = √ σ µ , (54) CD 2 2 2 where sigma matrices with superscript σt and σ∗ mean the transpose and the complex conjugate of σ. 0 0 j 0 0 0 0 AC BD i AC BD To calculate ΦABC D gµ gν = (1/4 ΘijsABs¯C0 D0 )gµ gν , let us define
k l AC0 BD0 1 k l ∗ AC0 ∗ BD0 f (k, l) = σ σ 0 0 g g = (σ σ 0 0 )σ σ . (55) µν AB C D µ ν 2 AB C D µ ν
Values of f (k, l)µν are shown in Table1.
k l AC0 BD0 Table 1. The lists of f (k, l)µν = σABσC0 D0 gµ gν .
0 1 0 0 0 1 0 0 0 0 i 0 1 0 0 0 1 0 0 0 0 0 0 1 f (0, 1)µν = , f (1, 0)µν = , f (3, 1)µν = , 0 0 0 i 0 0 0 −i i 0 0 0 0 0 i 0 0 0 −i 0 0 1 0 0 0 0 −i 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 −i 0 0 0 i 0 f (1, 3)µν = , f (0, 3)µν = , f (3, 0)µν = , −i 0 0 0 0 −i 0 0 0 i 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 −1 0 0 f (0, 0)µν = , f (1, 1)µν = , f (3, 3)µν = . 0 0 −1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 −1 0 0 0 1
Using this table, we get 4 × 4 representation of ΦABC0 D0 as
AC0 BD0 Φµν = ΦABC0 D0 gµ gν Universe 2020, 6, 40 8 of 18
1 2 2 1 (Θ12s s¯ + Θ21s s¯ ) 1 +(Θ s2s¯3 + Θ s3s¯2) 0 0 = 23 32 AC BD 3 1 1 3 gµ gν 4 +(Θ31s s¯ + Θ13s s¯ ) +( s1s1 + s2s2 + s3s3) Θ11 ¯ Θ22 ¯ Θ33 ¯ ABC0 D0 (iΘ12 f (3, 0) − iΘ21 f (0, 3)) +( ( ) − ( )) 1 iΘ23 f 0, 1 iΘ32 f 1, 0 = 4 +(−Θ31 f (1, 3) − Θ13 f (3, 1)) +( f ( ) + f ( ) + f ( )) Θ11 3, 3 Θ22 0, 0 Θ33 1, 1 µν 1 2 (S11+S22+S33) −P23 −P31 −P12 1 −P 1 (−S +S +S ) −S −S = 23 2 11 22 33 12 31 1 , (56) 2 −P31 −S12 2 (S11−S22+S33) −S32 1 −P12 −S31 −S32 2 (S11+S22−S33)
µν which is a real tensor and Φµνη = 0, as expected. From Equations (32), (33) and (56), we can find that Pij and Sij are also non-diagonal components of Gµν and Tµν. By comparing Equation (34) with Equation (56), we can interpret Pij/(8πG) as a momentum and Sij/(8πG) as a stress of a spacetime fluctuation. We can also observe from Equations (47) and (48) that the component of the Riemann curvature tensor of the form Rj0 pq is linked to a momentum, and the form Ri0 j0, Rpq rs linked to a stress-energy. Now we investigate XABCD and ΨABCD more in detail. Before representing X and Ψ in matrix AB form, we can check Equation (46) to find out whether Λ = XAB = R/4 or not. From the properties of Riemann curvature tensor, Ricci scalar is
µν 0i ij ρ0 σi ρi σj R = Rµν = 2R0i + Rij = 2R0iρσg g + Rijρσg g . (57)
For Minckowski metric gµν = ηµν, R becomes
R = −2Ri0i0 + Rijij. (58)
Because
k l CA DB k P l Q CA DB CA k D l Q sABsCDε ε = σ A εPBσ C εQDε ε = (ε σ A )(σ C εQD) ! −2 (k = l) = −Tr(s¯ksl) = , (59) 0 (k 6= l) we can finally see that
AB CA DB XAB = XABCDε ε 1 1 1 R = (−2R + e e R ) = (−2R + R ) = (60) 4 0l 0l 2 ijl pql ij pq 4 0l 0l ij ij 4 from Equation (46). We have used epql R0l pq = 0 by Bianchi identity. To represent the spinors X and Ψ in simple matrix forms, we first define
1 1 Q ≡ e R + e R , (61) ij 2 pqj i0 pq 2 pqi j0 pq 1 E ≡ R − e e R , (62) ij 0i 0j 4 pqi rsj pq rs 1 1 1 Ξ ≡ R − e e R − ie R − ie R = E − i Q , (63) ij i0 j0 4 pqi rsj pq rs 2 pqj i0 pq 2 pqi pq j0 ij ij Universe 2020, 6, 40 9 of 18
where Qij and Eij both are symmetric for i, j. Then, we have
1 j X = Ξ si s , (64) ABCD 4 ij AB CD from Equation (46). This can be expressed in a 4 × 4 matrix form by multiplying the factors in a ∗ AC ∗ BD AC BD similar way to the Equation (56), but here it is useful to multiply by (σε) µ (σε) ν = s¯µ s¯ν for AC BD AC µ simplicity, instead of σµ σν , where the components of s¯µ is equal to s¯ defined in Equation (15):
1 j X s¯ ACs¯ BD = ( Ξ si s )s¯ ACs¯ BD ABCD µ ν 4 ij AB CD µ ν −Ξ11 − Ξ22 − Ξ33 −iΞ23 + iΞ32 iΞ13 − iΞ31 −iΞ12 + iΞ21 − + − − + + 1 iΞ23 iΞ32 Ξ11 Ξ22 Ξ33 Ξ12 Ξ21 Ξ13 Ξ31 = . (65) 2 iΞ13 − iΞ31 Ξ12 + Ξ21 −Ξ11 + Ξ22 − Ξ33 Ξ23 + Ξ32 −iΞ12 + iΞ21 Ξ13 + Ξ31 Ξ23 + Ξ32 −Ξ11 − Ξ22 + Ξ33
Since Ξij is symmetric for i, j, it becomes −Ξ11 − Ξ22 − Ξ33 0 0 0 − − 0 Ξ11 Ξ22 Ξ33 2Ξ12 2Ξ13 = . (66) 0 2Ξ12 −Ξ11 + Ξ22 − Ξ33 2Ξ23 0 2Ξ13 2Ξ23 −Ξ11 − Ξ22 + Ξ33
1 For Wely conformal spinor ΨABCD = 3 (XABCD + XACDB + XADBC),
1 j j j Ψ s¯ ACs¯ BD = Ξ (si s + si s + si s )s¯ ACs¯ BD ABCD µ ν 12 ij AB CD AC DB AD BC µ ν 0 −iΞ23 + iΞ32 iΞ13 − iΞ31 −iΞ12 + iΞ21 − − + + 1 0 2Ξ11 Ξ22 Ξ33 2Ξ12 Ξ21 2Ξ13 Ξ31 = . (67) 3 0 Ξ12 + 2Ξ21 −Ξ11 + 2Ξ22 − Ξ33 2Ξ23 + Ξ32 0 Ξ13 + 2Ξ31 Ξ23 + 2Ξ32 −Ξ11 − Ξ22 + 2Ξ33
Considering the symmetricity of Ξ, it becomes 0 0 0 0 − − 1 0 2Ξ11 Ξ22 Ξ33 3Ξ12 3Ξ13 = . (68) 3 0 3Ξ12 −Ξ11 + 2Ξ22 − Ξ33 3Ξ23 0 3Ξ13 3Ξ23 −Ξ11 − Ξ22 + 2Ξ33
The components of XABCD, ΨABCD are expressed as symmetric tensors. As we can see on Equation (67) and Matrix (68), Ξij includes all information of ΨABCD. Because of Wely tensor Cabcd = ΨABCDε A0 B0 εC0 D0 + Ψ¯ A0 B0C0 D0 ε ABεCD, we may conclude that all informations of Weyl tensor are comprehended in Ξij. The form of Matrix (68) is similar to the tidal tensor Tij with a potential U = −U0/r = p 2 2 2 −U0/ x + y + z : 2x2 − y2 − z2 3xy 3xz U0 2 2 2 Tij = 3xy −x + 2y − z 3yz , (69) r5 3xz 3yz −x2 − y2 + 2z2
a 2 i j where Tij = Jij − Ja δij and Jij = δ U/δx δx [27,28]. The similarity may come from the link between tidal forces and Weyl tensor. The tidal force in general relativity is described by the Riemann curvature tensor. The Riemman curvature tensor Rabcd can be decomposed to Rabcd = Sabcd + Cabcd, where Cabcd Universe 2020, 6, 40 10 of 18
is a traceless part which is a Weyl tensor and Sabcd is a remaining part which consists of Ricci tensor c a Rab = R acb and R = Ra [6]. In the Schwartzchild metric, since R = Rab = Sabcd = 0 but Cabcd 6= 0, the tidal forces are described by Weyl tensor. This shows that Cabcd, ΨABCD and Ξij are all related to the tidal effects. The components of Ψ and Ξ can be represented with Weyl tensors. In a flat coordinate, by using
νσ Rµρ = Rµνρσg = Rµ0ρ0 − Rµiρi (70) and Equation (58), the components of Weyl tensor referred to as Equation (37) can be expressed as
1 1 C = C = R + R (for p 6= q), (71) 0p0q j p j q 2 0p0q 2 pjqj 1 1 1 1 C = −C = R − R + R − R , (72) 0p0p i j i j 2 0p0p 2 0k0k 2 pkpk 2 klkl 1 C = R − R , (73) p0pq p0pq 2 0iqi Ci0pq = Ri0pq . (74)
Comparing Equations (71)—(74) with Equations (61) and (62), we find that
1 C = C = E (for p 6= q), (75) 0p0q j p j q 2 pq 1 C = −C = (3E − E − E − E ) , (76) 0p0p i j i j 6 pp 11 22 33 Qi p C = e , (77) p0pq i p q 2 Qii C = e (for i 6= p and i 6= q). (78) i0pq i p q 2 Therefore, Matrix (68) can be rewritten to
i Ψ = Ψ s¯ ACs¯ BD = 2C − ieipqC + elpqC . (79) ij ABCD i j 0i0j j0pq 3 l0pq
lpq Since e Cl0pq = Q11 + Q22 + Q33 is zero by Bianchi identity, it becomes
AC BD ipq Ψij = ΨABCDs¯i s¯j = 2C0i0j − ie Cj0pq . (80)
Equation (63) can be reformulated to
R Ξ = 2C − δ − ieipqC , (81) ij 0i0j 6 ij j0pq where R = −2Eii = −2Ξii = −2(E11 + E22 + E33). Therefore, we can finally find the relation
R Ξ = Ψ − δ . (82) ij ij 6 ij
Here we can see the equivalence and the direct correspondences among ΨABCD, Ξij and Weyl tensor.
5. Definition of Sedon and Relations among Spinors, Sedenion and Sedon In this section, we investigate the basis of sedenion and we define a new algebra which is a similar structure to sedenion. Sedenion is 16 dimensional noncommutative and nonassociative algebra, which can be obtained from Cayley–Dickson construction [29,30]. The multiplication table of sedenion basis µ 0µ0 µµ0 is shown in Table2. The elements of sedenion basis can be represented in the form ei = q ⊗ q = q Universe 2020, 6, 40 11 of 18
0 with i = µ + 4ν, where qµ = (1, i, j, k), q0µ = (1, i0, j0, k0). The multiplication rule can be written by µµ0 νν0 µµ0 νν0 0 0 ei ∗ ej = q ∗ q = sµµ0νν0 q q , where sµµ0νν0 is +1 or −1, which is determined by µ, µ , ν, ν [7].
Table 2. The multiplication table of sedenion. For convenience, ‘eN’s are represented as ‘eN’ ; e.g., e3 → e3.
e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15 e0 e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15 e1 e1 −e0 e3 −e2 e5 −e4 −e7 e6 e9 −e8 −e11 e10 −e13 e12 e15 −e14 e2 e2 −e3 −e0 e1 e6 e7 −e4 −e5 e10 e11 −e8 −e9 −e14 −e15 e12 e13 e3 e3 e2 −e1 −e0 e7 −e6 e5 −e4 e11 −e10 e9 −e8 −e15 e14 −e13 e12 e4 e4 −e5 −e6 −e7 −e0 e1 e2 e3 e12 e13 e14 e15 −e8 −e9 −e10 −e11 e5 e5 e4 −e7 e6 −e1 −e0 −e3 e2 e13 −e12 e15 −e14 e9 −e8 e11 −e10 e6 e6 e7 e4 −e5 −e2 e3 −e0 −e1 e14 −e15 −e12 e13 e10 −e11 −e8 e9 e7 e7 −e6 e5 e4 −e3 −e2 e1 −e0 e15 e14 −e13 −e12 e11 e10 −e9 −e8 e8 e8 −e9 −e10 −e11 −e12 −e13 −e14 −e15 −e0 e1 e2 e3 e4 e5 e6 e7 e9 e9 e8 −e11 e10 −e13 e12 e15 −e14 −e1 −e0 −e3 e2 −e5 e4 e7 −e6 e10 e10 e11 e8 −e9 −e14 −e15 e12 e13 −e2 e3 −e0 −e1 −e6 −e7 e4 e5 e11 e11 −e10 e9 e8 −e15 e14 −e13 e12 −e3 −e2 e1 −e0 −e7 e6 −e5 e4 e12 e12 e13 e14 e15 e8 −e9 −e10 −e11 −e4 e5 e6 e7 −e0 −e1 −e2 −e3 e13 e13 −e12 e15 −e14 e9 e8 e11 −e10 −e5 −e4 e7 −e6 e1 −e0 e3 −e2 e14 e14 −e15 −e12 e13 e10 −e11 e8 e9 −e6 −e7 −e4 e5 e2 −e3 −e0 e1 e15 e15 e14 −e13 −e12 e11 e10 −e9 e8 −e7 e6 −e5 −e4 e3 e2 −e1 −e0
Table3 shows the multiplication table of an algebra which is similar to sedenion. It consists of µ 0µ0 µ 0µ0 µ 0µ0 16 bases ei = q ⊗ q with i = µ + 4ν and the multiplication rule ei ∗ ej = (q ⊗ q ) ∗ (q ⊗ q ) = 0 0 (qµqµ ⊗ q0µ q0µ ). The table is almost the same as the multiplication table of sedenion basis, but just differs in signs. The signs of red colored elements in Table3 differ from Table2. We will call this algebra as ‘sedon’. Sedon can be written in the form
~ ~ ←→ i i ij S = A0 + |B} + {C| + { D } = A0 + BiqR + CiqL + Diju , (83) ←→ i i i i ij i j ~ i ~ i ij where q = 1 ⊗ q , q = q ⊗ 1, u = q ⊗ q , |B} = Biq , {C| = Ciq , and { D } = Diju . We can R L ←→ R L name |~B} as ‘right svector’, {C~ | as ‘left svector’, and { D } as ‘stensor’. The coefficient of sedon can be represented as in Table4. For example, D13 is a coefficient of i ⊗ k term. Now we will see the relation between Ricci spinors and the sedon. Since
i C i C0 i σA εCB = sAB, ε A0C0 σ B0 = −s¯A0 B0 , (84) therefore
i B i CB i C0 C0 A0 i σA = −sACε , σ B0 = ε s¯A0 B0 . (85)
Equation (50) can be reformulated as
0 QP0 P0C0 BQ 1 P0C0 i j BQ 1 i Q j P Φ = ε Φ 0 0 ε = Θ ε s s¯ ε = − Θ σ σ . (86) AD0 ABC D 4 ij AB C0 D0 4 ij A D0
Since qi = (i, j, k) is isomorphic to −iσi = (−σ1i, −σ2i, −σ3i), we can set qi = −iσi. Equation (86) can be written as
0 1 j P0 Φ QP = Θ qi Qq , (87) AD0 4 ij A D0 which can be regarded as a sedon. In a similar way, XABCD can be written as
1 X BD = Ξ qk Bql D. (88) AC 4 kl A C Universe 2020, 6, 40 12 of 18
From Equation (87), a Ricci spinor can be interpreted as a combination of a right-handed and a left-handed rotational operations, since the basis has the form ‘left-handed quaternion ⊗ right- handed quaternion’. Following the rotational interpretation of Cayley–Dickson algebra [7], it can be interpreted as the twofold rotation ⊗ twofold rotation. i j For two quaternions A. = Aiq = a1i + a2j + a3k and B. = Bjq = b1i + b2j + b3k, which i D j D can be represented in the 2 × 2 matrix representation with spinor indices (Aiq C and Bjq C ), the multiplication of them can be written as
i D j E E k E A. B. = Aiq C Bjq D = −AiBiδC + eijk AiBjq C . (89)
We can use this to express multiplications of spinors. One of the example is
0 BC0 ED0 i B j C r E s D0 ΦAD0 ΦBF0 = θijq A q D0 θrsq B q F0 0 E u E j C s D0 = (−θljθlsδA + epquθpjθqsq A )q D0 q F0 E C0 v C0 E = θlkθlk δA δ F0 − emnvθlmθln q F0 δA u E C0 u E v C0 −epquθplθql q A δ F0 + emnvepquθpmθqn q A q F0 , (90)
1 where θij = 4 Θij. The result is also a sedon form. Above example shows not only multiplications of BC0 ΦAD0 but also the general multiplication of stensor. Here is an another example: An antisymmetric differential operator ∇[a∇b] can be divided into two parts
∆ab = 2∇[a∇b] = eA0 B0 AB + eABA0 B0 , (91)
1 A0 1 A where AB = 2 ∆AA0 B and A0 B0 = 2 ∆AA0 B0 . As we can see in Equations (9) and (11), each term can be considered as a quaternion.
1 µ 0 1 1 ij B = ∆ σ σ¯ ν A B = (i ∆ + e ∆ ) qk B (92) A 4 µν AA0 2 k0 2 k ij A A0 1 µA0C ν 1 1 ij k A0 ¯ 0 = − ∆ σ¯ σ 0 = (i ∆ − e ∆ ) q 0 . (93) B 4 µν CB 2 k0 2 k ij B
B CD0 Then, A ΦBE0 can be considered as a multiplication of a quaternion and a sedon.
0 B CD0 k B i C j D A ΦBE0 = [k q A θijq B q E0 0 C j D p C j P = −[p θpjδA q E0 + ekip[k θijq A q T j D0 1 j D0 = −i ∆ θ δ Cq − e ∆ θ δ Cq p0 pj A E0 2 rsp rs pj A E0 p C j D0 1 p C j D0 +i e ∆ θ q q + e ∆ e θ q q , (94) kip k0 ij A E0 2 lqk lq kip ij A E0
1 where [k ≡ i∆k0 + 2 eijk∆ij. elqk∆lqekipθij in the last term can be changed as ∆qpθqj − ∆pqθqj = 2∆qpθqj. The result in Equation (94) is in a sedon form. Using those expressions, we can represent the quantities with spinor indices as sedon forms whose elements are components of tensors. Universe 2020, 6, 40 13 of 18
Table 3. The multiplication table of sedon.
e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15 e0 e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15 e1 e1 −e0 e3 −e2 e5 −e4 e7 −e6 e9 −e8 e11 −e10 e13 −e12 e15 −e14 e2 e2 −e3 −e0 e1 e6 −e7 −e4 e5 e10 −e11 −e8 e9 e14 −e15 −e12 e13 e3 e3 e2 −e1 −e0 e7 e6 −e5 −e4 e11 e10 −e9 −e8 e15 e14 −e13 −e12 e4 e4 e5 e6 e7 −e0 −e1 −e2 −e3 e12 e13 e14 e15 −e8 −e9 −e10 −e11 e5 e5 −e4 e7 −e6 −e1 e0 −e3 e2 e13 −e12 e15 −e14 −e9 e8 −e11 e10 e6 e6 −e7 −e4 e5 −e2 e3 e0 −e1 e14 −e15 −e12 e13 −e10 e11 e8 −e9 e7 e7 e6 −e5 −e4 −e3 −e2 e1 e0 e15 e14 −e13 −e12 −e11 −e10 e9 e8 e8 e8 e9 e10 e11 −e12 −e13 −e14 −e15 −e0 −e1 −e2 −e3 e4 e5 e6 e7 e9 e9 −e8 e11 −e10 −e13 e12 −e15 e14 −e1 e0 −e3 e2 e5 −e4 e7 −e6 e10 e10 −e11 −e8 e9 −e14 e15 e12 −e13 −e2 e3 e0 −e1 e6 −e7 −e4 e5 e11 e11 e10 −e9 −e8 −e15 −e14 e13 e12 −e3 −e2 e1 e0 e7 e6 −e5 −e4 e12 e12 e13 e14 e15 e8 e9 e10 e11 −e4 −e5 −e6 −e7 −e0 −e1 −e2 −e3 e13 e13 −e12 e15 −e14 e9 −e8 e11 −e10 −e5 e4 −e7 e6 −e1 e0 −e3 e2 e14 e14 −e15 −e12 e13 e10 −e11 −e8 e9 −e6 e7 e4 −e5 −e2 e3 e0 −e1 e15 e15 e14 −e13 −e12 e11 e10 −e9 −e8 −e7 −e6 e5 e4 −e3 −e2 e1 e0
Table 4. The representation of coefficients of sedon.
∼ ⊗ 1 ∼ ⊗ i ∼ ⊗ j ∼ ⊗ k 1 ⊗ ∼ A0 B1 B1 B1 i ⊗ ∼ C1 D11 D12 D13 j ⊗ ∼ C2 D21 D22 D23 k ⊗ ∼ C3 D31 D32 D33
6. A Few Examples of Curvature Spinors in a Locally Flat Coordinate
6.1. Weyl Conformal Spinor for the Schwarzschild Metric: An Example of Section IV It is known that the Schwarzschild metric can be represented in Fermi normal coordinate [31]. 0 In Schwarzschild coordinate xµ = (T, R, Θ, Φ), the metric is displayed in the form
2 µ0 ν0 2 −1 2 2 2 2 2 2 ds = gµ0ν0 dy dy = − f dT + f dR + R dΘ + R sin Θ dΦ , (95) where f = 1 − 2GM/R. The basis of a constructed Fermi coordinate xµ = (t, x, y, z) is
0 0 e0 = ∂/∂t|G = T ∂/∂T + R ∂/∂R, −1 0 0 e1 = ∂/∂x|G = f R ∂/∂T + f T ∂/∂R,
e2 = ∂/∂y|G = 1/R ∂/∂Θ,
e3 = ∂/∂z|G = 1/R sinΘ ∂/∂Φ, (96) where the primes indicate derivatives with respect to proper time t. The non-zero components of the Riemann curvature tensor Rµ0ν0ρ0σ0 in Schwarzschild coordinate are
2 R10001000 = 2GM/R , 2 R30003000 = −( f GM/R)sin Θ,
R10201020 = GM/( f R),
R20002000 = − f GM/R, 2 R20302030 = −2GMRsin Θ, 2 R10301030 = (GM/R f )sin Θ. (97)
Then the Riemman curvature tenor Rµνρσ in the Fermi coordinate is
3 R10 10 = 2GM/R , 3 R20 20 = R30 30 = −GM/R , Universe 2020, 6, 40 14 of 18
3 R12 12 = R13 13 = GM/R , 3 R23 23 = −2GM/R . (98)
From Equations (47), (48), (61) and (62), we can observe that Pij = Sij = Qij = 0, but E11 = 3 3 4GM/R and E22 = E33 = −2GM/R . Classically, the tidal acceleration of black hole along the radial line is −2GM/r3δX, and the acceleration perpendicular to the radial line is GM/r3δX, where δX is the separation distance of two test particles. In this example, the link between Ξij and tidal accelerations has been shown.
6.2. The Spinor Form of the Einstein–Maxwell Equation: An Example of Section V Einstein–Maxwell Equations, which is Einstein Equations in presence of electromagnetic fields, is known [32] as
1 1 R − g R = 8πG(F Fσ − g F Fρσ), (99) µν 2 µν µσ ν µν 4 ρσ
σ 1 ρσ where Fµσ Fν − gµν 4 Fρσ F is the electromagnetic stress-energy tensor. The spinor form of the Einstein–Maxwell equation [1] is
ΦABA0 B0 = 8πGϕAB ϕ¯ A0 B0 , (100) where ϕAB, ϕ¯ A0 B0 are decomposed spinors of electromagnetic tensor Fµν, as following Equations (18) and (19). This can be deformed to
BA0 B A0 ΦAB0 = 8πGϕA ϕ¯ B0 . (101)
From Equation (86),
1 k B l A0 − Θ σ σ 0 4 kl A B 1 1 ij k B 1 1 pq l A0 = 8πG[ (F − i e F )σ ] × [ (F + i e F )σ 0 ] 2 k0 2 k ij A 2 l0 2 l pq B 1 ij pq i pq ij k B l A0 = 2πG[(F F + e e F F ) + (F e F − F e F )]σ σ 0 . (102) k0 l0 4 k l ij pq 2 k0 l pq l0 k ij A B Comparing the first line with the third line in Equation (102), we get
1 ij pq S = −8πG(F F + e e F F ), (103) kl k0 l0 4 k l ij pq pq ij Pkl = −4πG(Fk0e l Fpq − Fl0e k Fij), (104) and, from Equation (104) we get
mkl mk e Pkl = −16πGFk0F . (105)
For Fµν such that 0 −E1 −E2 −E3 − µν E1 0 B3 B2 F = , (106) E2 B3 0 −B1 E3 −B2 B1 0 Universe 2020, 6, 40 15 of 18
mk ~ ~ m 1 ij pq we have Fk0F = (E × B) and (Fk0Fl0 + 4 e ke l FijFpq) = EkEl + BkBl. From Equations (51) and (104) we get
1 − emkl P = R mi = 8πG(~E × ~B)k . (107) 2 kl 0i
This is a momentum of electromagnetic tensor and it shows that Pij/(8πG) is related to momentum. From Equations (49) and (103),
1 S = R + e e R = 8πG(−E E − B B ), (108) kl 0i 0j 4 pqi rsj pq rs k l k l ~ 2 ~ 2 Sl l = 8πG(|E| + |B| ). (109)
Those are the shear stress and the energy of electromagnetic field. It shows that Sij/(8πG) is related to stress-energy.
6.3. The Quaternion form of Differential Bianchi Identity: Another Example of Section V The spinor form of Bianchi identity referred to as Equation (27) is known [1] as
A A0 ∇B0 XABCD = ∇B ΦCDA0 B0 , (110) which can be deformed to
B0 A BD BA0 DB0 ∇ XAC = ∇ ΦCA0 . (111)
0 In flat coordinate, ∇A A equals to
A0 A µ A0 A 1 µA0 A 1 µ˜ A0 A ∂ = g ∂µ = √ σ ∂µ = √ σ ∂µ˜ 2 2 1 µ A0 A ˜ 1 µ A0 A 0 = √ q¯ ∂µ = √ q ∂µ , (112) 2 2
µ˜ 0 1 2 3 where µ˜ is tilde-spacetime indices which is defined as O = (O , i O , i O , i O ), Oµ˜ = µ 0 1 2 3 µ µ (O0, −i O1, −i O2, −i O3) for any O = (O , O , O , O ), Oµ = (O0, O1, O2, O3) [7]; q¯ is q¯ = µ˜ A0 A 0 1 2 3 σ = (σ , iσ , iσ , iσ ) which is isomorphic to (1, −i, −j, −k), ∂˜ µ = ∂µ˜ = (∂0, −i∂1, −i∂2, −i∂3), 0 µ µ˜ A0 A and ∂µ = (∂0, i∂1, i∂2, i∂3). We used the property AµB = Aµ˜ B [7]. ∂ can be expanded to
A0 A A0 A 0 k A0 A ∂ = ∂0δ + ∂kq (113)
0 and, considering matrix representation, ∂AA can be represented as
0 0 0 0 0 AA = AA + 0 k¯ AA = AA + 0 k AA ∂ ∂0δ ∂kq ∂0δ ∂k¯ q , (114)
¯ where the bar index Ak means the opposite-handed quantity of Ak, which is A1¯ = A1, A2¯ = −A2, A3¯ = A3 ; when k = 2, k¯ index change sings of Ak. It has following properties,
k¯ k k¯ k q¯ r¯ q r A Bk = A Bk¯ , A Bk¯ = A Bk, ε pqr A B = −ε pqr¯ A B , ε p¯q¯r¯ = −ε pqr. (115)
Then Equation (111) can be written as
0 0 0 0 0 ( B A + 0 k B A) i B r D = ( BA + 0 k BA ) r D s B ∂0δ ∂kq Ξirq A q C ∂0δ ∂k¯ q Θrs¯q C q A0 , (116) Universe 2020, 6, 40 16 of 18
0 0 0 0 s¯ B B D 0 0 s C since q A0 = ε εC A q D0 . Using Equation (89),
i B0 B 0 B0 B 0 p B0 B ∂0Ξirq + ∂kΞkrδ + ε pki∂kΞirq 0 0 0 = s¯ BB + 0 BB + 0 p B B ∂0Θrsq ∂kΘrkδ ε pks∂k¯ Θrs¯q , (117) which can be rearranged as
0 B0 B 0 s B0 B ∂k(Ξkr − Θrk)δ + [∂0(Ξsr − Θrs) + εski∂k(Ξir + Θri)]q = 0 . (118)
This is the quaternion form of Bianchi identities. Using Θij = Θ¯ ji and denoting Ξij as Ξj, Equation (118) can be written as
i ∇ · (Ξ − Θ¯ )r − i σ · (Ξ − Θ¯ )r + σ · (∇ × Ξ + ∇ × Θ¯ )r = 0 , (119) where σ are Pauli matrices.
6.4. Spatial-Handedness of Graviton and Spin of Matters in Gravitational Phenomena
k B Considering q A ,
k A AC k D A q B = −ε iσ C εDB = −i(σ1, −σ2, σ3) B, (120)
k¯ A which can be understood as a spatially left-handed quaternion basis q B, interchanging of up-down positions for spinor indices of quaternion basis indicates the interchanging of spatial-handedness of spatial index. As examples, Equations (87) and (88) can also be written as
0 AC0 1 i A j C Φ 0 = Θ¯ q q 0 , (121) BD 4 ij B D
AC 1 k A j C X = Ξ¯ ¯ q q . (122) BD 4 kl B D
In modified gravity or quantum gravity [33,34], the action could include higher order terms like 2 ab abcd 2 2 R , RabR , RabcdR . Therefore, we can also include terms like X , Φ , ΦXΦ... into the action of gravity. In such a case, following the representation like Equations (87) and (88) to couple those spinors together, the changes of spartial-handedness could occur, which means that spartial-right-handed particles couple to the spartial-left-handed and vice versa. For an example,
1 j D 1 ¯ ¯ X2 = X BDXAC = ( Ξ qi Bq )( Ξ qk A ql C ) (123) AC BD 4 ij A C 4 kl B D = ΞijΞi¯j¯. (124)
This can affect the drawing of Feynman diagrams of graviton. When matters couple with spinors, it can be thought that spins of matter fields contribute to make a gravitational effects on matter, since i B A A q A ψ gives the spin of ψ .
7. Conclusions We established a new method to express curvature spinors, which allows us to grasp components of the spinors easily in a locally inertial frame. During such a process, we technically utilized modified sigma matrices as a basis, which are sigma matrices multiplied by ε, and calculated the product of sigma matrices with mixed spinor indices. Using those modified sigma matrices as a basis can be regarded as Universe 2020, 6, 40 17 of 18 the rotation of the basis of four sigma matrices (σ0, σ1, σ2, σ3) to (s0, s1, s2, s3) defined in Equation (14), similar to a rotation of quaternion basis as shown in our previous work [7]. By comparing the Ricci spinor with the spinor form of Einstein equation, we could appreciate the roles of each component of the Riemann curvature tensor. The newly defined (3,3) tensors related to curvature tensors were introduced, and furthermore, from the representation of Weyl conformal spinor, we find that the components of Weyl tensor can be replaced by complex quantities Ξij, which are defined in Equation (63). We represented the elements of sedenion basis as the direct product of elements of the quaternion bases themselves. Then we defined a new algebra ‘sedon’, which has the same basis representation except for a slightly modified multiplication rule from the multiplication rule of sedenion. The relations between sedon and the curvature spinors are derived for a general gravitational field, not just for a weak gravitational field. We calculated multiplications of spinors with a quaternion form, and observed that the results of the multiplications are also represented in a sedon form. The relations among quaternion, sedon and curvature spinors may imply that gravity could be the consequence of combination of right-handed and left-handed abstract rotational operations. A few applications of the sedon representations were also introduced. One of the applications represented the Bianchi identity in the quaternion form, which might give the fluidic interpretation of the identity. We also suggested further possible application in modified gravity and quantum gravity. This may suggest that the spin-handedness, spatial-handedness, and spins of matter field would affect gravitational phenomena. The handedness structure of gravitational force has not yet been considered seriously. The sedon representation can be primarily a tool in which the handedness in gravity could be considered in detail. Finally we note that the research of this paper can be extended to general coordinates by considering vielbein formalism [35] and/or by establishing a connection to either self-dual or anti-self-dual variables [36,37]. Author Contributions: Conceptualization, I.K.H. and C.S.K.; investigation, I.K.H.; validation, I.K.H. and G.H.M.; writing—original draft preparation, I.K.H.; writing—review and editing, C.S.K. and G.H.M.; supervision, C.S.K.; funding acquisition, C.S.K. Funding: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP) (NRF-2018R1A4A1025334). Acknowledgments: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP) (NRF-2018R1A4A1025334). Conflicts of Interest: The authors declare no conflict of interest.
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