On Black Hole Scalar Hair in Asymptotically Anti de Sitter Spacetimes

TL;DR

The work addresses whether scalar hair can exist for black holes in asymptotically Anti-de Sitter spacetimes and how this differs from the asymptotically flat case. It develops a static, spherically symmetric Einstein–scalar framework for a minimally coupled field, proving a no-hair result for exact AdS asymptotics while showing hair appears when the scalar potential shifts the effective cosmological constant to $\Lambda_{\text{eff}}=\Lambda+8\pi V(\phi_\infty)$ with $\phi_\infty$ at a local maximum of $V$ (satisfying $\partial V/\partial\phi|_{\phi_\infty}=0$ and $0>\partial^2V/\partial\phi^2|_{\phi_\infty}>-9\lambda/4$). The analysis also provides a mass integral $M_2 = m(r_H) + 4\pi \int_{r_H}^{\infty} r^2 [ V(\phi)-V(\phi_\infty) + (1/2)\mu (\phi')^2 ] dr$, whose sign determines the energetic preference for hair and connects to stability considerations. The results have implications for AdS/CFT by suggesting bulk instability corresponds to instability of the dual thermal state, motivating further numerical and holographic investigations into hairy AdS black holes.

Abstract

The unexpected discovery of hairy black hole solutions in theories with scalar fields simply by considering asymptotically Anti de-Sitter, rather than asymptotically flat, boundary conditions is analyzed in a way that exhibits in a clear manner the differences between the two situations. It is shown that the trivial Schwarzschild Anti de Sitter becomes unstable in some of these situations, and the possible relevance of this fact for the ADS/CFT conjecture is pointed out.