Curvature Singularities: the Good, the Bad, and the Naked

TL;DR

The paper investigates which curvature singularities in five-dimensional gravity coupled to scalars are physically admissible within the AdS/CFT framework, focusing on 3+1D Poincaré-invariant geometries and the role of the scalar potential V.A central conjecture is proposed: large curvatures are allowed only when the scalar potential remains bounded above along the solution, a criterion motivated by the existence of near-extremal black-hole generalizations and supported by Coulomb-branch analyses.Through finite-temperature extensions, explicit examples (one, two, and three massive adjoint chirals), and fluctuations analyses, the work links infrared bulk behavior to dual field theory properties, such as Coulomb-branch dynamics and phase structure, and discusses implications for cosmological constant scenarios and brane-world models.

Abstract

Necessary conditions are proposed for the admissibility of singular classical solutions with 3+1-dimensional Poincare invariance to five-dimensional gravity coupled to scalars. Finite temperature considerations and examples from AdS/CFT support the conjecture that the scalar potential must remain bounded above for a solution to be physical. Having imposed some restrictions on naked singularities allows us to comment on a recent proposal for solving the cosmological constant problem.

Figures (7)

• Figure 1: (a) Contour plot of $V$ and (b) contour plot of $W$, for $W({\vec{\varphi}}) = -{1 \over 4} e^{\sqrt{3} \varphi_1 + \varphi_2} - {1 \over 4} e^{-\sqrt{3} \varphi_1 + \varphi_2} - {1 \over 4} e^{-2 \varphi_2}$ and $V({\vec{\varphi}}) = {1 \over 8} \left( {\partial W \over \partial{\vec{\varphi}}} \right)^2 - {1 \over 3} W({\vec{\varphi}})^2$. $W$ was chosen arbitrarily, but $V$ and $W$ exhibit the same features in two dimensions that obtain in five for the $V({\vec{\varphi}})$ and $W({\vec{\varphi}})$ that describe Coulomb branch states for ${\cal N}=4$ super-Yang-Mills. Regions of higher elevation are lighter in shade. If the region of the plots were much larger, one would see that $V({\vec{\varphi}}) \to +\infty$ as $|{\vec{\varphi}}| \to \infty$, unless one proceeds in one of three special directions separated by $120^\circ$: north, east-southeast or west-southwest. The center of both plots is the local maximum representing the maximally supersymmetric $AdS_5$ vacuum. The three saddle points of $V({\vec{\varphi}})$ are analogs of the unstable $SO(5)$ critical points. The thick lines in (b) represent the set ${\cal P}$, to be defined in (\ref{['SolutionSet']}). This figure, as well as figures \ref{['figB']}, \ref{['figC']}, and \ref{['figA']}, may be easiest to read in color.
• Figure 2: (a) The Penrose diagram for $\zeta > \sqrt{2/3}$ is a strip. Curvatures diverge as one approaches the right edge. This is a naked timelike singularity. (b) The Penrose diagram for $\zeta \leq \sqrt{2/3}$ is a wedge. Curvatures diverge as one approaches either of the null edges. This is a null singularity.
• Figure 3: Far left: Coulomb branch states exhibit a wide range of $\zeta$. Middle: various expectations for the response of singularities to finite temperature. Right: a summary of the examples in section \ref{['Examples']}. Details can be found there regarding each entry.
• Figure 4: (a) Contours of $V(\varphi_2,\varphi_3)$, with $\varphi_2$ on the horizontal axis and $\varphi_3$ on the vertical axis. (b) Contours of $W(\varphi_2,\varphi_3)$, with the solution set ${\cal P}$ and some typical gradient flow trajectories superimposed. In both contour plots, lighter regions correspond to higher elevations. The undashed trajectories proceeding asymptotically due east are Coulomb branch states, and correspond to entry (B) in figure \ref{['figG']}. The trajectory which ends at the saddle point represents the infrared conformal fixed point, and corresponds to entry (C) in figure \ref{['figG']}. The dashed trajectories violate (\ref{['Conjecture']}).
• Figure 5: (a) Contours of $V(\varphi_2,\tilde{\varphi}_3)$, with $\varphi_2$ on the horizontal axis and $\tilde{\varphi}_3$ on the vertical axis. (b) Contours of $W(\varphi_2,\varphi_3)$, with the solution set ${\cal P}$ and some typical gradient flow trajectories superimposed. In both contour plots, lighter regions correspond to higher elevations. The solid trajectories which proceed asymptotically due west are states on the Coulomb branch of the ${\cal N}=2$ gauge theory, and correspond to entry (D) in figure \ref{['figG']}. The unique solid trajectory which proceeds northwest nearly along a curve in ${\cal P}$ is the best candidate for a vacuum which can support finite temperature. The dashed trajectories violate (\ref{['Conjecture']}).