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A polytopal discrete de Rham scheme for the exterior calculus Einstein's equations

Todd A. Oliynyk, Jia Jia Qian

Abstract

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.

A polytopal discrete de Rham scheme for the exterior calculus Einstein's equations

Abstract

In this work, based on the 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.
Paper Structure (36 sections, 3 theorems, 158 equations, 49 figures)

This paper contains 36 sections, 3 theorems, 158 equations, 49 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. $3+1$ exterior calculus decomposition
  4. Vector fields
  5. Differential forms
  6. Exterior derivative
  7. Hodge star
  8. Einstein's equations
  9. Differential form formulation
  10. Closing equations
  11. $3+1$ Einstein's equations
  12. Non differential relations
  13. Two formulations
  14. The exterior calculus discrete de Rham complex
  15. Notations
  16. ...and 21 more sections

Key Result

Theorem 2

If $(D^i, \theta^i)\in C^1\Lambda^{2}_{\perp}(M)\times C^2\Lambda^{1}_{\perp}(M)$ solves eq:form.ev1 where the initial conditions $D^i(0)$, $\theta^i(0)$ satisfy the Einstein constraints eq:form.cst, then they generate a consistent solution to Einstein's equations.

Figures (49)

  • Figure 1: Mesh 1
  • Figure 2: Mesh 2
  • Figure 3: Mesh 3
  • Figure 4: Mesh 4
  • Figure 5: Mesh 5
  • ...and 44 more figures

Theorems & Definitions (11)

  • Remark 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Remark 4
  • Remark 5
  • Proposition 6: Preservation of discrete constraints for the three-field scheme \ref{['eq:scheme2']}
  • proof
  • Remark 7
  • ...and 1 more