Analyzing time evolution of constraint equations of Einstein's equation

We propose a method of analyzing the time evolution of constraint equations of the Einstein's equation on the nonflat background by adding constraint terms to the evolution equations. In past studies, the eigenvalues of the coefficient matrix of the constraint equations was mostly obtained on the flat background. Since simulations do not always perform near the flat background, we need to calculate the eigenvalues on the appropriate background. We analyze eigenvalues numerically and predict the stability in the appropriate background. We also perform some numerical simulations and show consistency between the results of the eigenvalue analysis and the numerical stability.