Towards a Novel no-hair Theorem for Black Holes

TL;DR

This work proposes a no-scalar-hair conjecture for gravity coupled to a single scalar field: spherical scalar hair cannot exist on static black holes if the Positive Energy Theorem (PET) holds. The authors derive PET as a condition that potentials must be expressible via a superpotential $P(\phi)$, with critical potentials identified where PET is at risk, and they show, through extensive numerics, that hair appears only when PET fails and disappears as the PET boundary is approached. They extend the analysis to asymptotically AdS spacetimes and explore designer gravity, demonstrating that AdS-invariant boundary conditions preserve the PET and rule out hair, while certain boundary deformations can admit hairy black holes and metastable vacua in the dual CFT. The findings unify no-hair expectations across asymptotically flat and AdS settings and highlight the PET as a decisive criterion, with implications for Calabi–Yau compactifications and holographic dualities.

Abstract

We provide strong numerical evidence for a new no-scalar-hair theorem for black holes in general relativity, which rules out spherical scalar hair of static four dimensional black holes if the scalar field theory, when coupled to gravity, satisfies the Positive Energy Theorem. This sheds light on the no-scalar-hair conjecture for Calabi-Yau compactifications of string theory, where the effective potential typically has negative regions but where supersymmetry ensures the total energy is always positive. In theories where the scalar tends to a negative local maximum of the potential at infinity, we find the no-scalar-hair theorem holds provided the asymptotic conditions are invariant under the full anti-de Sitter symmetry group.

Figures (11)

• Figure 1: A potential $V(\phi)$ that is on the verge of violating the Positive Energy Theorem for solutions that asymptotically approach the local AdS minimum at $\phi_0$.
• Figure 2: The function $B_c(A)$ represents a set of critical potentials of the form (3.2), with a local AdS minimum at $\phi_0 <0$ and a global minimum at $\phi=0$. The function $B_f(A)$ corresponds to potentials with a local minimum at $\phi_0$ where $V(\phi_0)=0$. The combination $(A,B)=(57.6,10.87)$ where $B_c=B_f$ yields a critical potential with a local minimum at zero.
• Figure 3: The function $h_{c} (\Lambda)$ that specifies a set of critical potentials of the form (3.3), which are unbounded from below.
• Figure 4: A potential $V(\phi)$ that is on the verge of violating the the Positive Energy Theorem for solutions that asymptotically approach the local minimum at $\phi =0$.
• Figure 5: The left panel shows the functions $\beta(\alpha)$ obtained from the solitons and from hairy black holes of two different sizes. The full line shows the soliton curve $\beta_{s}(\alpha)$, the dot-dashed line shows the $\beta_{R_e}(\alpha)$ curve for $R_e=.2$ black holes and the dashed line is the $R_e=1$ curve. One sees the boundary condition function $\beta_{bc}(\alpha)=-{1 \over 5} \alpha^2 +{1 \over 30} \alpha^3$, given by the dotted line, intersects the curves $\beta_{R_e}(\alpha)$ twice for small $R_e$. The right panel shows the mass of the hairy black holes that obey these boundary conditions. The full line gives the masses of the second (perturbatively stable) branch of solutions, which are associated with the second intersection point of the curves $\beta_{R_e}(\alpha)$ with $\beta_{bc}(\alpha)$, and hence have more hair.