The Cheshire Cap

TL;DR

The paper investigates the interior structure of slightly nonextremal three-charge black holes through the microstate geometries program, proposing that long-string degrees of freedom on the Higgs branch reside near the inner horizon while the Coulomb branch captures the inter-horizon exterior. By drawing a detailed analogy with fivebrane dynamics and employing quiver quantum mechanics, it argues that the long string sector resolves null singularities at the inner horizon and communicates information nonlocally to the exterior, potentially addressing the information paradox. A central thrust is that the covariant entropy bound locates most entropy in the region between horizons, and that a two-phase (long-string vs short-string) description naturally accounts for entropy, horizon thermodynamics, and Hawking radiation without locality violations at macroscopic scales. The work further connects four- and six-dimensional descriptions via dualities, analyzes singularities in microstate geometries, and suggests a broader relevance to cosmology and the fate of information in gravitational settings.

Abstract

A key role in black hole dynamics is played by the inner horizon; most of the entropy of a slightly nonextremal charged or rotating black hole is carried there, and the covariant entropy bound suggests that the rest lies in the region between the inner and outer horizon. An attempt to match this onto results of the microstate geometries program suggests that a `Higgs branch' of underlying long string states of the configuration space realizes the degrees of freedom on the inner horizon, while the `Coulomb branch' describes the inter-horizon region and beyond. Support for this proposal comes from an analysis of the way singularities develop in microstate geometries, and their close analogy to corresponding structures in fivebrane dynamics. These singularities signal the opening up of the long string degrees of freedom of the theory, which are partly visible from the geometry side. A conjectural picture of the black hole interior is proposed, wherein the long string degrees of freedom resolve the geometrical singularity on the inner horizon, yet are sufficiently nonlocal to communicate information to the outer horizon and beyond.

Figures (8)

• Figure 1: Penrose diagram for charged/rotating black holes, taking into account the instability of the inner horizon, after Marolf:2010ndMarolf:2011dj. Instabilities of the analytically continued stationary solution preclude the existence of a regular geometry beyond the inner horizon.
• Figure 2: (a) Near-horizon causal structure of Reissner-Nordstrom and BTZ geometries in Eddington-Finkelstein coordinates. Outward going null trajectories are depicted in blue, ingoing in red. The radius of the horizon of the extremal geometry carrying the same conserved charges ($r_+r_-=r_{\rm ext}^2$ in the BTZ case) is the green dashed line. (b) The covariant entropy bound applied to an outgoing light sheet that begins just inside the outer horizon and ends just outside the inner horizon. The interior of the light sheet is shaded, and the flow of entropy across it indicated.
• Figure 3: (a) Geometry of a shockwave excitation (the brown null trajectory) of the extremal geometry. (b) Qualitative picture of the evaporation process when quantum effects are included.
• Figure 4: Two perturbations of a ${\mathbb Z}_{n}$ symmetric arrangement of type IIB fivebranes on a circle, dual to type IIA string theory on ${\mathbb C}^2/{\mathbb Z}_n$: (a) moving the fivebranes on ${\mathbb S}^1$ is related to changing NS $B$-field fluxes through vanishing cycles on the IIA side; (b) moving them in ${\mathbb R}^3$ is dual to turning on the triplets of geometrical blow up modes of the vanishing cycles on the IIA side.
• Figure 5: Sources in equation (\ref{['FPSource']}) for the two-charge solution. Putting a macroscopic number of quanta in the lowest mode (the laconic source shown in black, making a single turn in the $X_1$-$X_2$ plane as one moves along the $v$ circle) constitutes a macroscopic ring source whose geometry turns out to be the spectral flow of the global $AdS_3\times{\mathbb S}^3$ geometry. Putting a single quantum in the highest mode (the tightly coiled spiral shown in red) makes an orbifold geometry $(AdS_3\times {\mathbb S}^3)/{\mathbb Z}_{n_1n_5}$. All the two-charge BPS geometries are specified by such a coiling long string source, which when separated in space describes a state on the Coulomb branch of D1-D5 system.