[Paper Review] Pure Gauss-Bonnet NUT black hole with and without non-central singularity

Sajal Mukherjee, Naresh Dadhich|arXiv (Cornell University)|Dec 31, 2020

Black Holes and Theoretical Physics27 references10 citations

TL;DR

This paper presents a new exact solution to the pure Gauss-Bonnet Λ-vacuum equation with NUT charge, revealing three physically viable spacetime configurations: (1) a black hole with both event and cosmological horizons hiding a non-centric singularity; (2) a regular, horizon- and singularity-free spacetime for 24 < Λl⁴ < 36; and (3) a black hole with only an event horizon and no cosmological horizon or singularity for Λl⁴ > 36. The solution features product S²×S² horizon topology and non-central singularities due to the Gauss-Bonnet term and NUT parameter.

ABSTRACT

It is known that NUT solution has many interesting features and pathologies like being non-singular and having closed timelike curves. It turns out that in higher dimensions horizon topology cannot be spherical but it has instead to be product of $2$-spheres so as to retain radial symmetry of spacetime. In this letter we wish to present a new solution of pure Gauss-Bonnet $\Lambda$-vacuum equation describing a black hole with NUT charge. It has three interesting cases: (a) black hole with both event and cosmological horizons with singularity being hidden behind the former, (b) a regular spacetime free of both horizon and singularity, and (c) black hole with event horizon without singularity and cosmological horizon. Singularity here is always non-centric at $r eq 0$.

Motivation & Objective

• To derive and analyze a new exact solution of the pure Gauss-Bonnet Λ-vacuum equation with NUT charge in higher dimensions.
• To investigate the interplay between the NUT parameter, cosmological constant, and Gauss-Bonnet term in determining horizon and singularity structure.
• To identify physically viable parameter windows where singularities are hidden or absent, avoiding naked singularities.
• To explore the role of product S²×S² horizon topology in enabling non-centric singularities in pure Gauss-Bonnet gravity.
• To clarify the physical viability of NUT spacetimes in higher-dimensional Gauss-Bonnet gravity with positive cosmological constant.

Proposed method

• Derives an exact metric solution with S²×S² horizon topology for pure Gauss-Bonnet gravity coupled to a positive cosmological constant and NUT charge.
• Introduces a dimensionless parameter λ = Λl⁴ to unify the effects of the cosmological constant and NUT parameter.
• Analyzes the discriminant h(r) in the metric function f(r), which must be non-negative for spacetime reality and diverges at h(r) = 0, indicating a non-centric singularity.
• Determines horizons as positive roots of the equation X(r) = 0, derived from ∆ = 0, with X(r) = r⁶Λ + 5Λl²r⁴ + 15r²(λ − 12) + 60Mr − 5l²(λ − 36).
• Solves the horizon and singularity conditions numerically and analytically across different λ ranges to classify viable spacetime types.
• Uses scaled plots of h(r) and X(r) to visualize singularity and horizon locations, with radial distance normalized by mass M.

Experimental results

Research questions

• RQ1 What are the physical conditions under which a pure Gauss-Bonnet NUT black hole with S²×S² horizon topology remains free of naked singularities?
• RQ2 How does the dimensionless parameter λ = Λl⁴ govern the existence and structure of horizons and singularities in this solution?
• RQ3 Can a regular, horizon-free, and singularity-free spacetime emerge in pure Gauss-Bonnet gravity with NUT charge and positive Λ?
• RQ4 Why does the cosmological horizon disappear for λ > 12, and how does this affect the black hole structure?
• RQ5 What constraints are imposed on the NUT parameter l for each viable configuration, and how narrow are these parameter windows?

Key findings

• For λ < 12, the solution describes a black hole with both event and cosmological horizons, and the non-centric singularity is hidden behind the event horizon, satisfying cosmic censorship.
• For 24 < λ < 36, the spacetime is regular everywhere with no horizons and no singularities, representing a compact, horizon-free NUT object with positive Λ.
• For λ > 36, the solution describes a black hole with only an event horizon, no cosmological horizon, and no singularity, provided h(r) = 0 has no positive roots.
• The NUT parameter l is tightly constrained: for λ < 12, l ∈ (1.79M, 1.82M) when ΛM⁴ = 0.05; for 24 < λ < 36, l ∈ (4.68M, 5.18M); and for λ > 36, l ∈ (5.18M, 5.41M), indicating narrow viable windows.
• The range 12 ≤ λ ≤ 24 is strictly forbidden due to unavoidable naked singularities, as no horizon forms to cover the singularity.
• The absence of a cosmological horizon for λ > 12 arises because the large-r expansion of the horizon equation has no positive roots, while the small-r limit shows no event horizon for 12 < λ < 36, explaining the regular case (b).