[Paper Review] Fuzzballs in general relativity: a missed opportunity

Lorenzo Pieri|arXiv (Cornell University)|Nov 16, 2016

Black Holes and Theoretical Physics29 references4 citations

TL;DR

This paper investigates whether regular, horizonless 'fuzzball' solutions in four-dimensional general relativity can be constructed by extending known asymptotically $AdS_2 \times S^2$ bubbling solutions to asymptotically flat spacetimes. Using an axially symmetric ansatz within the IWP family and a circular profile combined with a single-center charged Taub-NUT geometry, the study finds no regular asymptotically flat solutions exist under these constraints, indicating a fundamental obstruction to realizing 4D fuzzballs directly in general relativity.

ABSTRACT

Recently some 4d asymptotically $AdS_2 imes S^2$ regular bubbling solutions written in terms of an arbitrary profile function has appeared in literature. We discuss the possibility of extending these solutions to asymptotically flat spaces, therefore building a model for an horizonless and singularity-free fuzzball with the same asymptotic charges of the associated black hole, directly in general relativity. A negative conclusion is reached within an axial-symmetric ansatz inside the IWP family.

Motivation & Objective

• To determine whether regular, horizonless fuzzball solutions with the same asymptotic charges as black holes can be constructed directly in four-dimensional general relativity.
• To extend recently discovered $AdS_2 \times S^2$-asymptotic, regular bubbling solutions to asymptotically flat spacetimes.
• To test the viability of the IWP family of solutions—specifically with a circular profile and a charged Taub-NUT center—as a candidate for 4D fuzzballs within pure general relativity.
• To assess whether the topological complexity of $AdS_2$ boundaries, which suggest wormhole-like structures, can be leveraged to construct regular asymptotically flat geometries.

Proposed method

• The study employs the IWP (Israel-Wilson-Perjes) solution framework, which is parameterized by a complex harmonic function $ H = H_1 + iH_2 $, to model stationary, electrovacuum spacetimes.
• A specific ansatz is introduced combining a circular profile function $ \vec{F}(v) $ with a single-center charged Taub-NUT geometry to form a 'circle-NUT' solution.
• The analysis focuses on axial symmetry, assuming the profile and source are symmetric about the axis, to simplify the equations and preserve regularity conditions.
• Regularity conditions are enforced at $ \chi = \pm 1 $ (the poles of the coordinate system) to eliminate Dirac-Misner string singularities, requiring constraints on the harmonic function parameters.
• The asymptotic behavior of the metric and gauge fields is analyzed to verify flatness at infinity, with fall-off rates of $ r^{-1} $, $ r^{-2} $, and $ r^{-3} $ for mass, dipole, and higher multipole terms.
• The solution is derived in Weyl-Papapetrou coordinates, and the resulting expressions for the metric functions $ w $, $ \beta $, and $ V $ are evaluated for regularity and asymptotic flatness.

Experimental results

Research questions

• RQ1 Can the known class of regular, asymptotically $AdS_2 \times S^2$ bubbling solutions be extended to asymptotically flat spacetimes within general relativity?
• RQ2 Does the inclusion of a charged Taub-NUT center in a circular profile geometry yield a regular, horizonless solution with asymptotically flat behavior?
• RQ3 Are there obstructions in the IWP family of solutions that prevent the construction of 4D fuzzballs directly in four-dimensional general relativity?
• RQ4 Can the topological structure of $AdS_2$ boundaries, which suggest non-trivial wormhole-like geometries, be used to construct regular asymptotically flat solutions?
• RQ5 Do the stringent theorems against regular asymptotically flat solutions in 4D general relativity still apply when considering non-trivial spacetime topologies?

Key findings

• No regular, asymptotically flat solution was found within the axially symmetric IWP ansatz when combining a circular profile with a single-center charged Taub-NUT geometry.
• The requirement to eliminate Dirac-Misner strings forces the vanishing of the real and imaginary parts of the harmonic function's constant term, leading to a constraint that prevents asymptotic flatness.
• The asymptotic expansion of the metric shows that the mass and dipole terms fall off as $ r^{-1} $ and $ r^{-2} $, respectively, but the higher-order multipole contributions prevent the solution from approaching Minkowski spacetime smoothly.
• The analysis confirms that the standard theorems against regular asymptotically flat solutions in 4D general relativity remain effective even when considering non-trivial topologies and profile functions.
• The failure to construct such a solution suggests a fundamental obstruction to realizing 4D fuzzballs directly in general relativity, even with the inclusion of Taub-NUT structures.
• The result implies that the regularization of black hole microstates in 4D may require going beyond pure general relativity, possibly involving higher-dimensional compactifications or stringy effects.