Natural coordinate system in curved space-time. (English) Zbl 1400.83005

Gu, Ying-Qiu

J. Geom. Symmetry Phys. 47, 51-62 (2018).

Summary: In this paper we establish a generally and globally valid coordinate system in curved space-time with the simultaneous hypersurface orthogonal to the time coordinate. The time coordinate can be presented according to practical evolving process and keep synchronous with the evolution of the realistic world. In this coordinate system, it is convenient to express the physical laws and to calculate physical variables with clear geometrical meaning. We call it “natural coordinate system”. The constructing method for the natural coordinate system is concretely provided, and its physical and geometrical meanings are discussed in detail. In natural coordinate system, we make classical approximation of spinor equation to get Newtonian mechanics, and then make weak field approximation of Einstein’s equation and low speed approximation of particles moving in the space-time. From the analysis and examples we find it is helpful to understand the nature of space-time.

MSC:

83C05	Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C60	Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
53Z05	Applications of differential geometry to physics
83C10	Equations of motion in general relativity and gravitational theory

Keywords:

canonical metric; coordinate system; orthogonalization; simultaneity; spinor equation; particles moving in the space-time