On the linearization of Regge calculus
This work establishes a rigorous convergence theory for linearized Regge calculus, benefiting numerical relativists and discrete geometers by validating the approximation of the Einstein-Hilbert action's quadratic part.
The paper provides an explicit formula for the quadratic form of linearized 3D Regge calculus around Euclidean metric, linking it to the curlTcurl operator, and proves convergence of eigenpairs approximated by Regge metrics using non-conforming finite element methods.
We study the linearization of three dimensional Regge calculus around Euclidean metric. We provide an explicit formula for the corresponding quadratic form and relate it to the curlTcurl operator which appears in the quadratic part of the Einstein-Hilbert action and also in the linear elasticity complex. We insert Regge metrics in a discrete version of this complex, equipped with densely defined and commuting interpolators. We show that the eigenpairs of the curlTcurl operator, approximated using the quadratic part of the Regge action on Regge metrics, converge to their continuous counterparts, interpreting the computation as a non-conforming finite element method.