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.
Highlights
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
We show that the curvature spinors for general gravitational fields in locally flat coordinates can be regarded as a sedon
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
Summary
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. 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. We can obtain the explicit representations of curvature spinors, the components of which can be identified by using new techniques, i.e., manipulating spinor indices and rotating sigma basis in locally flat coordinates [7]. We show that the components of Weyl conformal spinor can be represented as a simple combination of Wely tensors in flat coordinate. We show that the curvature spinors for general gravitational fields in locally flat coordinates can be regarded as a sedon. 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
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