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Gauge Symmetry and Geometric Discretization: A Path to the Numerical Discretization of General Relativity

Joe Salamon


Gauge symmetries play a central role in every physical theory, from fluid dynamics to general relativity (GR). Unfortunately, their role both within theoretical and numerical applications is often overlooked, as one must pick a gauge in order to proceed with a well-defined problem in hand. As a result, most discretization schemes choose a gauge and hence fail to preserve the full symmetry structure of the Lagrangian. This problem is perhaps the most egregious within the framework of GR, as the gauge symmetry of the theory is the group of diffeomorphisms of spacetime; in other words, spacetime coordinate transformations lead to physically indistinguishable dynamics. In this talk, I will showcase the first baby steps that have been made towards a geometric, symmetry-preserving discretization scheme for GR. I will present an extension of finite element exterior calculus (FEEC) to flat, pseudo-Riemannian manifolds of arbitrary dimension. In particular, I will apply this framework to electromagnetism (EM) in Minkowski spacetime, and highlight the automatic imposition of a self-consistent gauge condition at first order. Furthermore, using the homothetic subdivision and refinement scheme given by Rapetti and Bossavit, I will also demonstrate that this scheme can be extended to arbitrary order. Within the context of EM, this leads to a much richer approximation to the full gauge group.

Tuesday, October 22, 2013
11:00AM AP&M 2402