General theory of relativity (GTR) in the formalism of a hypercomplex system of numbers

Abstract

The intervals$d\hat S = \hat \gamma _\mu (x){\mathbf{ }}dx^\mu ,{\mathbf{ }}d\hat S^2 = \hat g_{\mu \nu } {\mathbf{ }}(x){\mathbf{ }}dx^\mu {\mathbf{ }}d^\nu$ are hypercomplex numbers. In curved space, one can decompose them according to the 16-element Dirac algebra {Γ_α}, α=0, 1, 2, ..., 15, into the form$\hat \gamma _\mu = Z_\mu ^\alpha (x){\mathbf{ }}\Gamma _\alpha ,{\mathbf{ }}\hat g_{\mu \nu } = {\mathbf{ }}h_{\mu \nu }^\alpha (x){\mathbf{ }}\Gamma _\alpha$ Then, the gravitational field is described by the components h ^μν_a . In the decomposition one can limit oneself to a quaternion, biquaternion, or consider the complete algebra. Using the projection operators, one can switch to a spin-tensor description of gravitation$\hat g_{\mu \nu } = h_{\mu \nu }^\alpha {\mathbf{ }}\Gamma _\alpha {\mathbf{ }}D_4^\varepsilon {\mathbf{ }}D_{13}^s = {\mathbf{ }}(\varphi _{\mu \nu }^0 {\rm I} + \varphi _{\mu \nu }^1 {\mathbf{ }}\mathop \gamma \limits^0 _1 + \varphi _{\mu \nu }^2 {\mathbf{ }}\mathop \gamma \limits^0 _2 + \varphi _{\mu \nu }^3 {\mathbf{ }}\mathop \gamma \limits^0 _1 \mathop \gamma \limits^0 _2 ){\mathbf{ }}D_4^\varepsilon D_{13}^s$. In the hypercomplex discussion, the Einstein equations are transformed to the system$R_{\mu \nu }^\alpha - \frac{1}{2}{\mathbf{ }}R^\beta h_{\mu \nu }^\gamma {\mathbf{ }}\varepsilon _{\beta \gamma }^\alpha = \chi {\rm T}_{\mu \nu }^\alpha$. As an example, the external Schwarzchild problem is solved in quaternions:$d\hat S^2 = \Phi 0I + \Phi ^1 \mathop {\gamma _1 }\limits^0 + \Phi ^2 \mathop {\gamma _2 + }\limits^0 \Phi ^3 \mathop {\gamma _1 }\limits^0 \mathop {\gamma _2 }\limits^0$. From the definition of the 4-velocity$\hat U_\mu = dx^\mu /d\hat S$ follows$\hat U_\mu = \hat \gamma _\mu$, and the equation of a geodesic is converted into a condition of the covariant constancy of$\hat \gamma _\mu$.