Curvature Spinors in Locally Inertial Frame and the Relations with Sedenion

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 $4 \times 4$ 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', whose structure is almost the same as sedenion except 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 indentity, are also presented.


I. INTRODUCTION
In the 2-spinor formalism [1][2][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. It means that the gravity can be described by two spinors with four spinor indices. Those two spinors are called as curvature spinors: one of them is Ricci spinor and the other is Weyl conformal spinor [1,[3][4][5].
At any points on a pseudo-Riemannian manifold, we can find a locally flat coordinate [6], whose metric is Minkowski. Though 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, whose components 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. The obtained representation can be used not only in a speciallt flat coordinate but also for vielbein indices or in any other normal coordinates, like Riemann normal coordinate and Fermi coordinate [8][9][10][11][12][13][14][15][16]. By comparing the final forms of Ricci spinors with the spinor forms of Einstein equation, we are able to figure out the roles of each component of Riemann curvature tensor, whose components serve as momentum, energy or stress of gravitational fields. Furthermore, we find that the Weyl sipnor can be analyzed by dividing into real part and pure imaginary part, henceforth, the components of Weyl conformal spinor can be represented as a simple combination of Wely tensors in flat coordinate.
There are already a quite few papers that show the relation between sedenion and gravitational field, however, all are restricted to a weak gravitational field in a flat frame [17][18][19][20].
Here we express the basis of sedenion as a set of direct product of quaternion basis, through which we can define a new algebra 'sedon', whose structure is similar to sedenion except the basis multiplication rule. We will show that the curvature spinors for general gravitational fields in a locally flat coordinates can be regarded as a sedon. From the sedon form of curvature spinors, we can get a view of the curvature spinors as the combination of righthanded and left-handed rotational effects. And 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.

II. TENSOR REPRESENTATION OF A FIELD WITH TWO SPINOR INDICES
In this section we introduce the basics about the 2-spinor formalism, which are explained in detail in our earlier paper [7].
For any real anti-symmetric tensor F AA ′ BB ′ , we can write as Since where i , j , k are the 3-dimensional vector indices which have the value 1, 2 or 3, and ǫ ij k is ǫ pqk δ i p δ j q for the Levi-Civita symbol ǫ ijk . Einstein summation convention is used for 3-dimensional vector indices i, j and k. Similar to (8) and (9), If we denote matrix representation of ε AB by ε, then Let us define s µ ands µ as wheres µ is complex conjugate of s µ . Then and where s i have unprimed indices s i = s i AB ands i have primed indicess i =s i A ′ B ′ .

III. 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 where Γ ρ µν is a Chistoffel symbol Here R µνρσ has follwing properties [6] In short, we can denote as where parentheses ( ) and square brackets [ ] indicates symmetrization and anti-symmetrization of the indices [6]. The Riemann curvature tensor has two kinds of Bianchi identities where ∇ λ A µ = ∂ λ A µ + Γ µ νλ A ν . From the antisymmetric properties of Riemann curvature tensor, it can be decomposed into sum of curvature spinors, X ABCD and Φ ABC ′ D ′ , as The totally symmetric part of X ABCD is called gravitational spinor or Weyl conformal spinor, and Φ ABC ′ D ′ is referred as Ricci spinor [1,4,5]. It is well known that and Einstein tensor is where Λ = X AB AB , which is equal to R/4 [1]. Therefore, the Einstein field equation where λ is a cosmology constant, can be written in the form Since any symmetric tensor U ab can be expressed as where τ = 1 4 T c c and S ABA ′ B ′ is traceless and symmetric [1], the traceless part of the energymomentum (symmetric) tensor T ab can be written by S ab = T ab − 1 4 T c c g ab . Therefore, the spinor form of Einstein equations (34) becomes Weyl tensor C µνρσ which is another measure of the curvature of spacetime, like Riemann curvature tensor, is defined as [6,21] It has the same propterties with (22), (23) and (26). It is known [1] that Weyl tensor has the following relationship with Weyl conformal spinor Ψ ABCD : At any point P on the pseudo-Riemannian manifold, we can find a flat coordinate system, such that, where g µν (P ) is the metric at the point P and η µν is the Minkowski metric. In this coordinate system, while the Christoffel symbol is zero, the Riemann curvature tensor is [22,23] 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 along the geodeic G.

IV. 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. And we discuss physical implications of those representations. From now on, we will always use locally flat coordinate for spacetime indices, and use small letters i, j...z as a three dimensional indices, which can be 1, 2 or 3; while small letters a, b...h as a four dimensional indices, which can be 0, 1, 2 or 3. From (4) and (28), we can lead to from (18) and (19). We write here the form of ǫ ij k as ǫ ijk for convenience; it is not so difficult to recover the upper-and lower-indices. By decomposing φ AB,cd one more times, we get We note that Φ and X are expressed with two 3-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, We can divide (45) into two terms by defining where P ij is anti-symmetric and S ij is symmetric for i, j. Then (45) is represented as The components of P ij and S ij can be simply expressed as where the underlined symbols in subscripts are the value-fixed indices which does not sum up for dummy indices; one of example is S 11 = R 01 01 + R 23 23 .
We can express Φ ABCD as a tensor by multiplying g AC ′ µ , which is where sigma matrices with superscript σ t and σ * mean the transpose and the complex Values of f (k, l) µν are shown in Table I. Using this table, we get 4 × 4 representation of which is a real tensor and Φ µν η µν = 0, as expected.
From Eqs. (32, 33, 56), we can find that P ij and S ij are also non-diagonal components of G µν and T µν . By comparing Eq. (34) with Eq. (56), we can interpret P ij /(8πG) as a momentum and S ij /(8πG) as a stress of a spacetime fluctuation. We can also observe from (47) and (48) For Minckowski metric g µν = η µν , R becomes Because we can finally see that from Eq. (46). We have used ǫ pql R 0l pq = 0 by Bianchi identity.
To represent the spinors X and Ψ in simple matrix forms, we first define where Q ij and E ij both are symmetric for i, j. Then, we have  (15): Since Ξ ij is symmetric for i, j, it becomes For Wely conformal spinor Ψ ABCD = 1 3 (X ABCD + X ACDB + X ADBC ), Considering the symmetricity of Ξ, it becomes The components of X ABCD , Ψ ABCD are expressed as symmetric tensors. As we can see on (67) and (68), Ξ ij includes all information of Ψ ABCD . Because of Wely tensor C abcd = The form of (68) is similar to the tidal tensor T ij with a potential U = −U 0 /r = −U 0 / x 2 + y 2 + z 2 : where T ij = J ij − J a a δ ij and J ij = δ 2 U/δx i δx j [24,25]. 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 R abcd can be decomposed to R abcd = S abcd + C abcd , where C abcd is a traceless part which is a Weyl tensor and S abcd is a remaining part which consists of Ricci tensor R ab = R c acb and R = R a a [6]. In the Schwartzchild metric, since R = R ab = S abcd = 0 but C abcd = 0, the tidal forces are described by Weyl tensor. This shows that C abcd , Ψ ABCD and Ξ ij 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 and Eq. (58), the components of Weyl tensor (37) can be expressed as Comparing Eq. (71) -Eq. (74) with (61) and (62), we find that Therefore, (68) can be rewritten to Since ǫ lpq C l0pq = Q 11 + Q 22 + Q 33 is zero by Bianchi identity, it becomes And Eq. (63) can be reformulated to where R = −2E ii = −2Ξ ii = −2(E 11 + E 22 + E 33 ). Therefore, we can finally find the relation Here we can see the equivalence and the direct correspondences among Ψ ABCD , Ξ ij and Weyl tensor.
For two quaternions A . = A i q i = a 1 i + a 2 j + a 3 k and B . = B j q j = b 1 i + b 2 j + b 3 k, which can be represented in the 2 × 2 matrix representation with spinor indices (A i q i D C and B j q j D C ), the multiplication of them can be written as We can use this to express multiplications of spinors. One of the example is where θ ij = 1 4 Θ ij . The result is also a sedon form. Above example shows not only multiplications of Φ BC ′ A D ′ but also the general multiplication of stensor. Here is an another example: An antisymmetric differential operator ∇ [a ∇ b] can be divided into two parts As we can see in (9) and (11), each term can be considered as a quaternion. Then, E ′ can be considered as a multiplication of a quaternion and a sedon.
where ♭ k ≡ i∆ k0 + 1 2 ǫ ijk ∆ ij . ǫ lqk ∆ lq ǫ kip θ ij in the last term can be changed as ∆ qp θ qj −∆ pq θ qj = 2∆ qp θ qj . The result (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.

VI. A FEW EXAMPLES OF CURVATURE SPINORS IN A LOCALLY FLAT CO-ORDINATE
A. 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 [28]. In Schwarzschild coordinate x µ ′ = (T, R, Θ, Φ), the metric is displayed in the form where f = 1 − 2GM/R. The basis of a constructed Fermi coordinate x µ = (t, x, y, z) is where the primes indicate derivatives with respect to proper time t. The non-zero components of the Riemann curvature tensor R µ ′ ν ′ ρ ′ σ ′ in Schwarzschild coordinate are Then the Riemman curvature tenor R µνρσ in the Fermi coordinate is Einstein-Maxwell Equations, which is Einstein Equations in presence of electromagnetic fields, is known [29] as where F µσ F σ ν − g µν 1 4 F ρσ F ρσ is the electromagnetic stress-energy tensor. The spinor form of the Einstein-Maxwell equation [1] is where ϕ AB ,φ A ′ B ′ are decomposed spinors of electromagnetic tensor F µν , as following (18) and (19). This can be deformed to From Eq. (56), Comparing the first line with the third line in (102), we get and, from Eq. (104) we get For F µν such that This is a momentum of electromagnetic tensor and it shows that P ij /(8πG) is related to momentum. From (49) and (103), Those are the shear stress and the energy of electromagnetic field. It shows that S ij /(8πG) is related to stress-energy.
C. The quaternion form of differential Bianchi identity: Another Example of Section V The spinor form of Bianchi identity (27) is known [1] as which can be deformed to In flat coordinate, ∇ A ′ A equals to whereμ is tilde-spacetime indices which is defined as , and ∂ ′ µ = (∂ 0 , i∂ 1 , i∂ 2 , i∂ 3 ). We used the property A µ B µ = AμBμ [7]. ∂ A ′ A can be expanded to and, considering matrix representation, ∂ AA ′ can be represented as where the bar index Ak means the opposite-handed quantity of A k , which is A1 = A 1 , A2 = −A 2 , A3 = A 3 ; when k = 2,k index change sings of A k . It has following properties, Then Eq. (111) can be written as which can be rearranged as This is the quaternion form of Bianchi identities.

VII. CONCLUSION
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 the rotation of the basis of four sigma matrices (σ 0 , σ 1 , σ 2 , σ 3 ) to (s 0 , s 1 , s 2 , s 3 ) defined in Eq. (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. Furthermore, from the representation of Weyl conformal spinor, we find that the components of Weyl tensor can be replaced by complex quantities Ξ ij , which are defined in Eq. (63).
We represented the elements of sedenion basis as the direct product of elements of quaternion basis themselves. And then we defined a new algebra 'sedon', which has the same basis representation except slightly modified multiplication rule from the multiplication rule of sedenion. The relations between sedon and the curvature spinors are derived for a general gravitaional 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.
We observed that many gravitational quantities can be represented with 3-dimensional basis. It shows that time and space may be interpreted differently from conventional interpretations in which time and space are treated as the same. And the relations among quaternion, sedon and the representations of curvature spinors imply that gravity may come from a combination of right-handed and left-handed abstract rotational operations.