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Closed-Form Canonical-Scalar Black Hole with de Sitter Core and Schwarzschild-(A)dS Exterior

Thomas Schürmann

TL;DR

This work constructs a fully analytic, static, spherically symmetric solution to the Einstein--Klein--Gordon system with a minimally coupled canonical scalar that yields a regular de Sitter core smoothly matched to a Schwarzschild--(A)dS exterior. By adopting a two-function metric gauge and a monotone $n$-kink scalar profile, the scalar potential $V(\phi)$ is reconstructed in closed form, and for $M=0$ it reduces to a quintic polynomial in the compact interpolation variable $x$; the exterior mass term $M$ appears as a universal, sign-controlled correction $\Delta V$. The solution fixes the redshift algebraically, requires no thin shells or modified gravity, and provides explicit horizon locations through a simple polynomial condition $N_n(r)=0$, complemented by an extremality relation for the coalescence of horizons. The model offers a compact analytic benchmark for horizon thermodynamics, stability analyses, and AdS applications within a canonical-matter sector, with tunable core and exterior cosmological scales and transparent energy-condition behavior.

Abstract

Regular (non-singular) black holes with de Sitter cores are often constructed by prescribing effective mass functions or anisotropic vacuum fluids, or by invoking phantom scalars that violate energy conditions. Constructions with canonical self-interacting scalars in asymptotically flat spacetimes and inverse reconstruction schemes do exist, but they rarely produce a fully closed-form potential that smoothly matches a de Sitter core to a Schwarzschild-(A)dS exterior. Here we present a fully analytic, static, spherically symmetric solution of the Einstein-Klein-Gordon system with a minimally coupled canonical scalar. Working in a two-function metric gauge and adopting a simple monotone kink profile that interpolates between two vacuum energies, we reconstruct the scalar potential directly from the field equations; for zero mass parameter it reduces to a quintic polynomial in a natural interpolation variable. The resulting configuration is everywhere regular, fixes the redshift algebraically, requires neither thin shells nor modified gravity, and realizes a de Sitter core matched to a Schwarzschild-(A)dS exterior with explicitly characterizable horizons. This compact closed-form model furnishes a convenient analytic benchmark for studies of horizon structure, thermodynamics, stability, and AdS applications within a canonical matter sector.

Closed-Form Canonical-Scalar Black Hole with de Sitter Core and Schwarzschild-(A)dS Exterior

TL;DR

This work constructs a fully analytic, static, spherically symmetric solution to the Einstein--Klein--Gordon system with a minimally coupled canonical scalar that yields a regular de Sitter core smoothly matched to a Schwarzschild--(A)dS exterior. By adopting a two-function metric gauge and a monotone -kink scalar profile, the scalar potential is reconstructed in closed form, and for it reduces to a quintic polynomial in the compact interpolation variable ; the exterior mass term appears as a universal, sign-controlled correction . The solution fixes the redshift algebraically, requires no thin shells or modified gravity, and provides explicit horizon locations through a simple polynomial condition , complemented by an extremality relation for the coalescence of horizons. The model offers a compact analytic benchmark for horizon thermodynamics, stability analyses, and AdS applications within a canonical-matter sector, with tunable core and exterior cosmological scales and transparent energy-condition behavior.

Abstract

Regular (non-singular) black holes with de Sitter cores are often constructed by prescribing effective mass functions or anisotropic vacuum fluids, or by invoking phantom scalars that violate energy conditions. Constructions with canonical self-interacting scalars in asymptotically flat spacetimes and inverse reconstruction schemes do exist, but they rarely produce a fully closed-form potential that smoothly matches a de Sitter core to a Schwarzschild-(A)dS exterior. Here we present a fully analytic, static, spherically symmetric solution of the Einstein-Klein-Gordon system with a minimally coupled canonical scalar. Working in a two-function metric gauge and adopting a simple monotone kink profile that interpolates between two vacuum energies, we reconstruct the scalar potential directly from the field equations; for zero mass parameter it reduces to a quintic polynomial in a natural interpolation variable. The resulting configuration is everywhere regular, fixes the redshift algebraically, requires neither thin shells nor modified gravity, and realizes a de Sitter core matched to a Schwarzschild-(A)dS exterior with explicitly characterizable horizons. This compact closed-form model furnishes a convenient analytic benchmark for studies of horizon structure, thermodynamics, stability, and AdS applications within a canonical matter sector.

Paper Structure

This paper contains 31 sections, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Misner--Sharp mass and algebraic reconstruction
  4. Closed-form scalar field model
  5. Metric function and asymptotics
  6. Redshift in closed form
  7. Potential in closed form
  8. Closed form for the $M=0$ branch ($V_0$) for arbitrary $n\ge 2$.
  9. Universal mass correction $\Delta V$ (all $n\ge 3$).
  10. Endpoint values and near-centre behaviour.
  11. Reading $V$ as a function of $\phi$.
  12. Verification of the field equations
  13. Klein--Gordon equation: detailed check
  14. Regularity at the centre and Ricci scalar (for $\mathbf{n\geq 3}$)
  15. Horizons and extremality
  16. ...and 16 more sections

Figures (2)

  • Figure 1: Decomposition of the reconstructed scalar potential into the $M=0$ background $V_{0}$ (blue), the universal mass correction $\Delta V$ (orange), and the full $V=V_{0}+\Delta V$ (green), plotted as functions of the areal radius $r$ for interpolation order $n=10$ and parameters $\Lambda_{1}=2$, $\Lambda_{2}=-1$, $\Delta\phi=0.5$, $R=2$, $M=1.5$. The transition is localized near $r\simeq R$, yielding a narrow peak--dip structure with a characteristic width of about $\simeq 2.2R/n$ (see text).
  • Figure 2: Radial potential $V(r)$ for $\Lambda_1=2$, $\Lambda_2=-1$, $M=1.5$, $\Delta\phi=0.5$, and $R=2$. Solid curves show $V(r)$ for $n=3,4,10,15$. As ${r\to 0}$, the derivative follows $V'(r)\propto (n-3) r^{\,n-4}$, that is zero for $n=3$, finite for $n=4$, and vanishing for $n>4$ (see text).