A new approach to isometric embedding of 2-surfaces, with applications in numerical relativity

The isometric embedding of 2-surfaces such as apparent horizons in flat geometry is useful both for visualizing horizon structure and for computing quasilocal charges such as mass, energy, and angular momentum. Despite a very long history of approaches, the problem still remains difficult, particularly for the highly deformed horizons that occur when numerical relativity simulations are most interesting. Here we will introduce a radically new approach to the problem that factors the solving of the highly nonlinear embedding PDEs into a sequence of quasilinear and fully linear PDEs that are much more straightforward to handle. We will present an implementation of this method in the SpEC code and compare it with more conventional approaches.