[Paper Review] Mass and Riemannian Polyhedra

Pengzi Miao, Annachiara Piubello|arXiv (Cornell University)|Jan 7, 2021

Cosmology and Gravitation Theories15 references4 citations

TL;DR

This paper establishes a geometric interpretation of the ADM mass in asymptotically flat manifolds by showing it equals the limit of a combination of total mean curvature and total dihedral angle defect over large Riemannian polyhedra. The key result expresses the mass as an integral over faces and edges of the polyhedron boundary, linking general relativity to Gromov's scalar curvature comparison theory via polyhedral geometry.

ABSTRACT

We show that the concept of the ADM mass in general relativity can be understood as the limit of the total mean curvature plus the total defect of dihedral angle of the boundary of large Riemannian polyhedra. We also express the $n$-dimensional mass as a suitable integral of geometric quantities that determine the $(n-1)$-dimensional mass.

Motivation & Objective

• To connect the ADM mass in general relativity to geometric invariants of Riemannian polyhedra.
• To extend Gromov's scalar curvature comparison theory to asymptotically flat manifolds using polyhedral exhaustion.
• To provide a new integral formula for the ADM mass based on face mean curvature and edge dihedral angle defect.
• To establish a geometric interpretation of mass that generalizes known formulas for cubes and spheres in 3D.

Proposed method

• Uses a sequence of Euclidean polyhedra {P_k} in a coordinate chart of an asymptotically flat manifold (M^n, g) with increasing size.
• Imposes conditions on volume growth of faces and edges, and a uniform lower bound on |sin ᾱ| for dihedral angles to ensure metric control.
• Applies asymptotic expansion of metric coefficients to relate geometric quantities on P_k to those on coordinate spheres.
• Derives an expression for the ADM mass as a limit involving integrals of mean curvature H over faces and (α − ᾱ) over edges.
• Relies on the positive mass theorem and Bartnik/Chruściel's invariance of mass under coordinate exhaustion.
• Uses perturbation analysis of normal vectors and angles to control error terms in the asymptotic expansion.

Experimental results

Research questions

• RQ1 Can the ADM mass be expressed as a geometric limit involving polyhedral boundaries in asymptotically flat manifolds?
• RQ2 How do mean curvature and dihedral angle defect on polyhedral boundaries relate to the ADM mass?
• RQ3 What conditions ensure the convergence of the polyhedral mass formula to the ADM mass?
• RQ4 Does the formula hold for non-convex or mixed-type polyhedra in the exhaustion sequence?
• RQ5 Can the mass be computed via integration over faces and edges rather than spheres?

Key findings

• The ADM mass is given by the limit of a combination of total mean curvature and total dihedral angle defect over large Riemannian polyhedra: 𝔪(g) = 1/((n−1)ω_{n−1}) × [−∫_ℱ H dσ + ∫_ℰ (α − ᾱ) dμ] + o(1).
• The formula holds for any sequence of polyhedra {P_k} satisfying volume growth and dihedral angle conditions, including non-convex and mixed types.
• For coordinate cubes or scaled polyhedra P_{(r)} as r→∞, the formula reduces to known 3D expressions.
• The condition |sin ᾱ| ≥ c ensures control over metric-to-angle estimates in the asymptotic expansion.
• The mass formula is consistent with the positive mass theorem and geometric invariance under coordinate exhaustion.
• The result generalizes the 3D formula from [13] to arbitrary dimensions n ≥ 3 and provides a bridge between ADM mass and Gromov’s polyhedral scalar curvature comparison theory.