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This is the first of the two papers devoted to the study of global regularityof the 3+1 dimensional Einstein-Klein-Gordon system with a $U(1)\times\mathbb{R}$ isometry group. In this first part, we reduce the Cauchy problem ofthe Einstein-Klein-Gordon system to a 2+1 dimensional system. Then, we willgive energy estimates and construct the null coordinate system, under which wefinally show that the first possible singularity can only occur at the axis.

Global regularity for Einstein-Klein-Gordon system with $U(1) \times \mathbb{R}$ isometry group, I