Curvature Spinors in Locally Inertial Frame and the Relations with Sedenion

• Main Authors: In Ki Hong, Choong Sun Kim, Gyung Hyun Min
• Format: Article
• Language: English
• Published: MDPI AG 2020-03-01
• Series: Universe
• Subjects: spinor formalism; general relativity; sedenion; quaternion; representation theory
• Online Access: https://www.mdpi.com/2218-1997/6/3/40

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 <inline-formula> <math display="inline"> <semantics> <mrow> <mn>4</mn> <mo>&#215;</mo> <mn>4</mn> </mrow> </semantics> </math> </inline-formula> 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&#8217;, 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.