Gaussian curvature of spherical shells: A geometric measure of complexity

TL;DR

This paper introduces a new geometric measure of complexity for spherically symmetric spacetimes based on the Gaussian curvature of spherical shells, linking geometry to physical complexity and classifying various metrics.

Contribution

It develops a hyperbolic equation for Gaussian curvature and uses it to define and demonstrate a geometric complexity measure for spherically symmetric spacetimes.

Findings

The measure depends critically on Gaussian curvature.

It classifies well-known spherically symmetric metrics.

An order structure based on complexity is proposed.

Abstract

In this paper we consider a semitetrad covariant decomposition of spherically symmetric spacetimes and find a governing hyperbolic equation of the Gaussian curvature of two dimensional spherical shells, that emerges due to the decomposition. The restoration factor of this hyperbolic travelling wave equation allows us to construct a geometric measure of complexity. This measure depends critically on the Gaussian curvature, and we demonstrate this geometric connection to complexity for the first time. We illustrate the utility of this measure by classifying well known spherically symmetric metrics with different matter distributions. We also define an order structure on the set of all spherically symmetric spacetimes, according to their complexity and physical properties.