Abstract
This study investigates the characteristics of the generalized gravitation equation in a complex spacetime manifold. The newly applied complex spacetime coordinates were designed to integrate peculiar velocity and the receding velocity of the particle into a single coordinate system. On this basis, the Schwarzschild metric solution was extended to a complexified version, and a generalized geodesic equation was derived in the complex spacetime manifold. It was found from the derived gravitation equation that the gravitation interaction depends on the space deceleration parameter \(q\) and can act as a repulsive force in the giant space scale that satisfies \(q>-2+\sqrt{2}\). These results require a rapid expansion of space in the early universe and suggest that the gravitational interactions of matter are fundamentally linked to the space expansion characteristics.
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Appendices
Appendix 1: Complex Bases, Complex Coordinates, and Derivatives Definition
Table 1 summarizes the definitions of basis components, coordinates, and partial derivatives in the complex manifold. The \(1/\sqrt{2}\) factor in the complex basis components is a normalization constant defined as \(\left|{e}_{\mu }\left(\boldsymbol{z}\right)\right|=1\).
Appendix 2: Complex Metric Tensors and Rotated Connection Coefficients
According to the complex basis definition presented in Appendix 1, the metric tensor components \({C}_{\mu \nu }\), \({C}_{\overline{\mu }\overline{\nu }}\), \({C}_{\overline{\mu }\nu }\), and \({C}_{\mu \overline{\nu }}\) and their inverse metric components \({C}^{\mu \nu }\), \({C}^{\overline{\mu }\overline{\nu }}\), \({C}^{\overline{\mu }\nu }\), and \({C}^{\mu \overline{\nu }}\) are listed in Tables 2 and 3. Additionally, the rotated connection coefficient in Eq. (12) is decomposed into eight sub-components; each full-expression is found in Table 4.
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Jun, HS. Theoretical Investigation of Deceleration Parameter-Dependent Gravitation in a Complex Spacetime Manifold. Found Phys 52, 71 (2022). https://doi.org/10.1007/s10701-022-00588-4
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DOI: https://doi.org/10.1007/s10701-022-00588-4