In this work, based on the $3+1$ decomposition in [24, 33], we present a fully exterior calculus breakdown of spacetime and Einstein's equations. Links to the orthonormal frame approach [38] are drawn to help understand the variables in this context. Two formulations are derived, discretised and tested using the exterior calculus discrete de Rham complex [13], and some discrete quantities are shown to be conserved in one of the cases.

翻译：本文基于文献[24, 33]中的$3+1$分解，提出了时空与爱因斯坦方程的完全外微积分表述框架。通过建立与正交标架法[38]的联系，阐明了该语境下各变量的物理意义。我们推导了两种数值表述形式，并采用外微积分离散德拉姆复形[13]进行离散化与数值验证，其中一种情形下证明了若干离散量的守恒性。