The Simplification of Gauge conditions in General Relativity Theory

We proposed an equation of motion as a constraint condition on the gauge degree of freedom in general relativity theory based on a co-moving framework. This constraint equation transformed the gauge degree freedom of general relativity theory into an natural equation of motion, and provide an explicit theoretical understanding on the Mach's principle and rotating bucket experiment. The galaxy rotation curves is derived directly from general relativity theory under this constraints equation. This work may inspires different understanding on dark matter, dark energy, and the quantization of gravity.