Schwarzschild black hole as moving puncture in isotropic coordinates

The success of the moving puncture method for the numerical simulation of black hole systems can be partially explained by the properties of stationary solutions of the 1 + log coordinate condition. We compute stationary 1 + log slices of the Schwarzschild spacetime in isotropic coordinates in order to investigate the coordinate singularity that the numerical methods have to handle at the puncture. We present an alternative integration method to obtain isotropic coordinates that simplifies numerical integration and that gives direct access to a local expansion in the isotropic radius near the puncture. Numerical results have shown that certain quantities are well approximated by a function linear in the isotropic radius near the puncture, while here we show that in some cases the isotropic radius appears with an exponent that is close to but unequal to one.