Date

Abstract

Calculate the values of the Hamiltonian and Momentum Constraints on the computational grid for output.

### 1 Physics

This analysis thorn calculates the values of the Hamiltonian and momentum constraint equations on the computational grid during a spacetime evolution. The Hamiltonian Constraint (grid function: ham) is given by

 $H=R-{K}_{\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}j}^{i}{K}_{\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}i}^{j}+tr{K}^{2}-16\pi \rho$ (1)

The Momentum Constraints (grid functions momx, momy, momz) are given by

 ${M}_{i}={\nabla }_{j}{K}_{i}^{\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}j}-{\nabla }_{i}trK-8\pi {S}_{i}$ (2)

where

 ${S}_{i}=-{g}_{ia}{n}_{b}{T}^{ab}=-{g}_{i}^{a}{n}^{b}{T}_{ab}=-\frac{1}{\alpha }\left({T}_{i0}-{\beta }^{j}{T}_{ij}\right)$ (3)