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Gravitational Effects of Sources Inspired by ideal Electromagnetic Fields in Spherical Painlevé-Gullstrand Coordinates

G. Abellán, N. Bolívar, I. Vasilev

TL;DR

This work builds static, spherically symmetric spacetimes in general relativity sourced exclusively by classical electrostatic configurations using a unit-lapse Painlevé–Gullstrand metric where all gravitational dynamics are carried by the radial shift β(r). The authors derive the Einstein equations in a local flat frame, showing p_r = -ρ and linking p_⊥ to ρ through p_⊥ = - (1/2 r) d/dr (r^2 ρ), then analyze piecewise interiors and exteriors joined without thin shells via Israel junction conditions. They explore four explicit EM-based examples—charged shell, Yukawa screening, dielectric layer, and Hulthén field—computing β(r) and the full energy–momentum tensor in each region and assessing the classical energy conditions. Across the examples, regular spacetimes emerge with distinct patterns: the Coulomb case fully satisfies energy conditions, while screened or layered configurations introduce anisotropy that commonly violates the dominant energy condition, yet do not produce singularities, illustrating how electromagnetic sources shape spacetime without requiring exotic matter.

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 Painlevé-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 Hulthén-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.

Gravitational Effects of Sources Inspired by ideal Electromagnetic Fields in Spherical Painlevé-Gullstrand Coordinates

TL;DR

This work builds static, spherically symmetric spacetimes in general relativity sourced exclusively by classical electrostatic configurations using a unit-lapse Painlevé–Gullstrand metric where all gravitational dynamics are carried by the radial shift β(r). The authors derive the Einstein equations in a local flat frame, showing p_r = -ρ and linking p_⊥ to ρ through p_⊥ = - (1/2 r) d/dr (r^2 ρ), then analyze piecewise interiors and exteriors joined without thin shells via Israel junction conditions. They explore four explicit EM-based examples—charged shell, Yukawa screening, dielectric layer, and Hulthén field—computing β(r) and the full energy–momentum tensor in each region and assessing the classical energy conditions. Across the examples, regular spacetimes emerge with distinct patterns: the Coulomb case fully satisfies energy conditions, while screened or layered configurations introduce anisotropy that commonly violates the dominant energy condition, yet do not produce singularities, illustrating how electromagnetic sources shape spacetime without requiring exotic matter.

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 Painlevé-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 Hulthén-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.

Paper Structure

This paper contains 29 sections, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Painleve Gullstrand Metric
  3. Einstein equations in a local flat frame
  4. Matter Content
  5. Junction Conditions
  6. Extrinsic Curvature Calculation
  7. Israel Junction Conditions
  8. Energy Conditions
  9. Summary of general energy conditions
  10. Examples
  11. Charged shell
  12. Region I: $0\leq r < R_1$.
  13. Region II: $r\geq R_1$.
  14. Energy conditions
  15. Yukawa type field
  16. ...and 14 more sections

Figures (1)

  • Figure 1: (Top) Electromagnetic energy density profiles $\rho(r)$, (bottom) shift functions $\beta^2(r)$. The piecewise structure is evident: flat interior regions ($r < R_1$) and electromagnetically sourced mid and exterior regions. Parameters: $Q^2 = 8\pi$, $R_1 = 1$, $R_2 = 2$ (dielectric), $k = 1$ (screening), $B^2 = 8\pi$ (Hulthén).