[Paper Review] Non-Singular Spherically Symmetric Solution in Einstein-Scalar-Tensor Gravity

J. W. Moffat|ArXiv.org|Feb 13, 2007

Cosmology and Gravitation Theories3 references3 citations

TL;DR

This paper proposes a non-singular, static, spherically symmetric solution in Einstein-scalar-tensor gravity with a scalar field potential, avoiding both black hole event horizons and spacetime singularities. The solution relies on a repulsive scalar field potential that generates negative energy density near r=0, violating the weak energy condition but stabilizing the metric against collapse, with the solution remaining non-singular even when ordinary matter is added.

ABSTRACT

A static spherically symmetric metric in Einstein-scalar-tensor gravity theory with a scalar field potential $V[ϕ]$ is non-singular for all real values of the coordinates. It does not have a black hole event horizon and there is no essential singularity at the origin of coordinates. The weak energy condition $ρ_ϕ> 0$ fails to be satisfied for $r\lesssim 1.3r_S$ (where $r_S$ is the Schwarzschild radius) but the strong energy condition $ρ_ϕ+3p_ϕ> 0$ is satisfied. The classical Einstein-scalar-tensor solution is regular everywhere in spacetime without a black hole event horizon. However, the violation of the weak energy condition may signal the need for quantum physics anti-gravity as $r o 0$. The non-singular static spherically symmetric solution is stable against the addition of ordinary matter.

Motivation & Objective

• To construct a static, spherically symmetric solution in Einstein-scalar-tensor gravity that avoids spacetime singularities and event horizons.
• To investigate whether a scalar field potential can prevent gravitational collapse into a black hole, even for masses exceeding the Chandrasekhar limit.
• To analyze the energy conditions (weak and strong) in the solution and assess their physical viability.
• To examine the stability of the non-singular solution under the addition of ordinary matter.
• To explore the implications for the black hole information paradox and Hawking radiation in the absence of an event horizon.

Proposed method

• Formulates the action for Einstein-scalar-tensor gravity with a scalar field potential $V[\phi]$, including gravitational, scalar field, and matter terms.
• Derives the field equations from the action using the metric ansatz $ds^2 = B(r)dt^2 - A(r)dr^2 - r^2 d\Omega^2$ for a static, spherically symmetric spacetime.
• Solves the gravitational and scalar field equations under the condition $T_{M\mu\nu} = 0$ and $\Lambda = 0$, yielding exact solutions for $A(r)$, $B(r)$, and $\phi(r)$.
• Imposes boundary conditions at spatial infinity and at $r=0$ to ensure regularity and finiteness of curvature invariants.
• Analyzes the energy conditions by computing $\rho_\phi = \frac{1}{2}\phi'^2 + V[\phi]$ and $p_\phi = \frac{1}{2}\phi'^2 - V[\phi]$, showing violation of the weak energy condition for $r \lesssim 1.3r_S$.
• Tests stability by perturbing the solution with ordinary matter density $\rho_M$, showing that the scalar field barrier grows with mass, preserving non-singularity.

Experimental results

Research questions

• RQ1 Can a scalar-tensor gravity theory produce a non-singular, static, spherically symmetric solution without an event horizon or spacetime singularity?
• RQ2 What are the energy conditions (weak and strong) satisfied by the scalar field in the non-singular solution, and what do their violations imply?
• RQ3 How does the addition of ordinary matter affect the stability and structure of the non-singular solution?
• RQ4 Does the absence of an event horizon eliminate Hawking radiation and the associated information loss paradox?
• RQ5 Can this solution describe a stable, compact astrophysical object (e.g., a 'dark grey star') even for masses above the neutron star limit?

Key findings

• The solution is non-singular throughout spacetime, with all curvature invariants, $\rho_\phi$, $p_\phi$, and $V[\phi(r)]$ remaining bounded.
• The weak energy condition $\rho_\phi > 0$ is violated for $r \lesssim 1.3r_S$, indicating negative energy density near the origin.
• The strong energy condition $\rho_\phi + 3p_\phi > 0$ is satisfied, ensuring attractive gravity on large scales.
• The scalar field potential $V[\phi(r)]$ acts as a repulsive barrier that prevents singularity formation and maintains non-singularity under mass addition.
• The solution remains stable when ordinary matter is added, as the scalar field barrier grows with $r_S$, preserving the non-singular structure.
• The absence of an event horizon implies no Hawking radiation and thus no black hole information loss paradox for such compact objects.