Families of regular spacetimes and energy conditions

TL;DR

This paper introduces a systematic method to construct regular, static, spherically symmetric spacetimes in general relativity that satisfy the weak energy condition, unifying known solutions and deriving new families with explicit formulas.

Contribution

It provides a unified framework for regular black hole solutions, classifies density profiles, and derives new geometries involving special functions with explicit forms.

Findings

Recoveries of Bardeen, Hayward, and Dymnikova models within the framework.

Derivation of new regular geometries with explicit hypergeometric and Gamma function expressions.

Analysis of horizons, photon spheres, and matching conditions to Schwarzschild exterior.

Abstract

We present a systematic method for constructing static, spherically symmetric regular spacetimes in general relativity satisfying the weak energy condition. Our approach relies on physically reasonable assumptions on the matter energy density, together with the boundedness of the Kretschmann scalar. The latter property ensures the finiteness of all curvature invariants and, for the configurations considered, is equivalent to the completeness of causal geodesics. By classifying admissible density profiles according to their complexity, we recover well-known regular black hole solutions such as the Bardeen, Hayward, and Dymnikova models, which are thus naturally embedded in a unified and broader framework. Within this setting, we also derive closed-form expressions for several new families of regular geometries involving hypergeometric or incomplete Gamma functions, which in many cases reduce to elementary functions including algebraic, logarithmic, arctangent, and exponential forms. The emergence of horizons and photon spheres, as well as matching conditions to a Schwarzschild exterior, are also investigated.