The self-attractive ultralight axion and its time scale of collapsing in general relativistic framework

The formation of supermassive black holes (SMBHs) at high red shift $z$ by ultralight axion dark matter (DM) is discussed in the general relativistic framework. The critical condition of collapsing of self-attracting DM in non-relativistic treatment corresponds to dust particle, which allows to estimate the time scale of the SMBH formation and the its length scale in the general relativistic framework. It gives the support of ultralight axion model based on the string model as the origins of the early universe SMBH and the Little Red Dots (LRDs) etc. observed by JWST.