Abstract
In this paper, we show that a minimally coupled 3-form endowed with a proper potential can support a regular black hole interior. By choosing an appropriate form for the metric function representing the radius of the 2-sphere, we solve for the 3-form field and its potential. Using the obtained solution, we construct an interior black hole spacetime which is everywhere regular. The singularity is replaced with a Nariai-type spacetime, whose topology is text {dS}_2 times text {S}^2, in which the radius of the 2-sphere is constant. So long as the interior continues to expand indefinitely, the geometry becomes essentially compactified. The 2-dimensional de Sitter geometry appears despite the negative potential of the 3-form field. Such a dynamical compactification could shed some light on the origin of de Sitter geometry of our Universe, exacerbated by the Swampland conjecture. In addition, we show that the spacetime is geodesically complete. The geometry is singularity-free due to the violation of the null energy condition.
Highlights
Singularities are commonplace in general relativity, as established by the singularity theorems [1,2,3,4]
The analysis regarding the black hole solutions conducted in Ref. [41] only focuses on the exterior region of the spacetime
Since we are interested in formulating regular black hole solutions in this theory, we will mainly focus on the spacetime inside the event horizon, which, as we have mentioned, can be described by the Kantowski–Sachs metric (2)
Summary
Singularities are commonplace in general relativity, as established by the singularity theorems [1,2,3,4]. Where d 2 is the standard metric on a 2-sphere, and m, q are related to the mass and charge, respectively This solution behaves like a Reissner–Nordström black hole asymptotically as r → ∞. Unlike the aforementioned charged Bardeen black hole, our solution has only one horizon, so that the interior is a cosmological spacetime with time extending eternally into the infinite future, instead of a static “core”. As we will discuss in more detail, its topology is dS2 × S2, with the size of the 2-sphere being constant as the de Sitter part expands, effectively giving rise to some sort of compactification This is different from the Frolov–Markov–Mukhanov model in which a Schwarzschild interior transits into a 4-dimensional de Sitter spacetime [30,31].
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