Gravitational Effects of Sources Inspired by ideal Electromagnetic Fields in Spherical Painlev\'e-Gullstrand Coordinates

TL;DR

This paper constructs and analyzes static spherically symmetric spacetimes in general relativity sourced by classical electromagnetic-like fields, exploring their properties and energy conditions in a tractable analytical framework.

Contribution

It introduces a class of solutions with electromagnetic-inspired sources in Painlevé-Gullstrand coordinates, linking matter distributions to spacetime geometry without exotic matter.

Findings

Energy conditions are systematically evaluated for each model.

Singular behaviors are identified using Israel junction conditions.

The framework provides insights into gravitational effects of simple electromagnetic sources.

Abstract

We construct and analyze a class of static spherically symmetric spacetimes in general relativity sourced exclusively by classical electrostatic configurations. Using a spherically symmetric Painlev\'e-Gullstrand-like metric with unit lapse and a radial shift function, we develop piecewise-defined solutions where the interior geometry is flat and the exterior is supported by several sources inspired by electromagnetic distributions. These include point-charge-like fields, Yukawa-screened electric fields, dielectric layers, and Hulth\'en-type field. The Einstein equations naturally impose a relation between the energy density and radial pressure, while the tangential pressure is derived from the metric. We systematically evaluate the classical energy conditions in each model and study the appearance of singular behavior using Israel junction conditions. This framework offers an analytically tractable setting to explore the gravitational effects of physically simple, well-understood sources without resorting to exotic matter.