Volume and Boundary Face Area of a Regular Tetrahedron in a Constant Curvature Space

TL;DR

This paper derives exact formulas for the volume and face area of regular tetrahedra in spherical and hyperbolic spaces, relevant for quantum gravity models with a cosmological constant.

Contribution

It provides explicit formulas for curved tetrahedra's volume and face area as functions of curvature and edge length, advancing geometric understanding in curved spaces.

Findings

Exact formulas for volume and face area in spherical and hyperbolic geometries

Formulas depend explicitly on scalar curvature and edge length

Applicable to models in loop quantum gravity and Regge calculus

Abstract

An example of the volume and boundary face area of a curved polyhedron for the case of regular spherical and hyperbolic tetrahedron is discussed. An exact formula is explicitly derived as a function of the scalar curvature and the edge length. This work can be used in loop quantum gravity and Regge calculus in the context of a non-vanishing cosmological constant.