Breaking an Abelian gauge symmetry near a black hole horizon (0801.2977v1)

Abstract: I argue that coupling the Abelian Higgs model to gravity plus a negative cosmological constant leads to black holes which spontaneously break the gauge invariance via a charged scalar condensate slightly outside their horizon. This suggests that black holes can superconduct.

• The paper shows that black hole horizons can trigger spontaneous breaking of Abelian gauge symmetry, leading to superconducting states.
• It employs an Abelian Higgs model in AdS4 to analyze marginally stable modes that indicate second-order phase transitions under specific conditions.
• Numerical results reveal that certain charge and temperature parameters induce a negative effective mass, challenging the classical no-hair theorem.

Breaking an Abelian Gauge Symmetry Near a Black Hole Horizon: An Analysis

In the paper "Breaking an Abelian Gauge Symmetry near a Black Hole Horizon" by Steven S. Gubser, the author explores the intriguing concept of gauge symmetry breaking near black hole horizons through the introduction of the Abelian Higgs model coupled with gravity and a negative cosmological constant. This framework allows the investigation into whether black holes can exhibit superconductivity due to spontaneous breaking of gauge invariance via a charged scalar condensate forming near the event horizon.

Summary

Gubser begins by building on previous suggestions that black hole horizons could induce spontaneous symmetry breaking under the right circumstances, bringing about a superconducting state near the horizon. The model utilized in this paper includes gravity, a negative cosmological constant, and a lagrangian resembling the Abelian Higgs model, which notably excludes the typical quartic interaction but still allows for the novel outcome of symmetry breaking under specific conditions.

The main focus is on charged black holes in a four-dimensional anti-de Sitter space ($AdS_4$), where the formation of a scalar condensate potentially occurs just outside the black hole horizon. The derived effective mass of the scalar field, $m_{\text{eff}}^2$, is crucial in this context, as it becomes negative near the horizon, suggesting the instability of the $\psi=0$ solution. This condition facilitates symmetry breaking when certain parameters—charge of the black hole, electric charge $q$, and mass $m^2$ of the scalar field—are met. The work challenges and expands the classical no-hair theorems by suggesting that black holes are not merely inert objects characterized by their mass, charge, and spin, but can also possess this scalar field 'hair.'

Strong Numerical Results and Claims

One of the paper's strong numerical illustrations is the identification of marginally stable modes, which are explored through perturbations considered around charged black hole solutions in $AdS_4$. These modes indicate the presence of second-order phase transitions under specific conditions, leading to an ordered state where the scalar field $\psi$ acquires a non-zero expectation value. It was observed that for certain parameter values, marginally stable modes exist on co-dimension one surfaces of parameter space, suggesting the possibility of an actual instability.

Moreover, it is speculated that charged black holes with high enough charge and low enough temperature can highlight the limitations of the no-hair theorem in the presence of this unconventional matter field configuration.

Implications and Future Directions

The implications of this research potentially extend into both the theoretical understanding of black holes and the fundamental nature of superconductivity. If black holes can support a superconducting state, it would significantly impact theories of quantum gravity and the paper of strongly correlated systems via the AdS/CFT correspondence.

Looking forward, further research might delve into the practical construal of these states in laboratory settings, probing how the analogies set out between black holes and superconductors might contribute to our understanding of high-temperature superconductivity. Moreover, the existence of these condensates could hint at new types of phase transitions in the context of quantum field theories dual to gravitational backgrounds.

Ultimately, the paper calls for a deeper exploration into the stability of these systems, beyond the confines of asymptotically anti-de Sitter space, possibly extending to more realistic cosmological models. One notable point of discussion remains: whether these effects are merely theoretical curiosities or bear realistic connections to experimentally accessible phenomena.

This research opens various intriguing questions about the interplay between fundamental fields and gravitational systems and suggests new horizons for studying complex quantum phenomena through the lens of classical general relativity.