Extremal limits and black hole entropy

TL;DR

The paper shows that the extremal limit of a non-extremal Reissner-Nordström black hole is discontinuous: the region between the inner and outer horizons becomes a patch of $AdS_2\times S^2$ even as the horizons merge. This leads to two coexisting classical spacetimes in the extremal limit—the extremal black hole and the $AdS_2\times S^2$ compactification—whose boundary conditions and global structures differ. The authors argue that this distinction can illuminate the long-standing entropy puzzle, proposing that the nonzero entropy in extremal contexts arises from the $AdS_2\times S^2$ region (via dual microstate counting or entanglement considerations) rather than the extremal black hole itself, and that the two frameworks may be describing different spacetime phases in the extremal limit. This perspective suggests a non-holographic interpretation of extremal entropy tied to bulk degrees of freedom in the compactification geometry and prompts further exploration of how the limiting procedure and boundary conditions influence black hole thermodynamics.

Abstract

Taking the extremal limit of a non-extremal Reissner-Nordström black hole (by externally varying the mass or charge), the region between the inner and outer event horizons experiences an interesting fate -- while this region is absent in the extremal case, it does not disappear in the extremal limit but rather approaches a patch of $AdS_2\times S^2$. In other words, the approach to extremality is not continuous, as the non-extremal Reissner-Nordström solution splits into two spacetimes at extremality: an extremal black hole and a disconnected $AdS$ space. We suggest that the unusual nature of this limit may help in understanding the entropy of extremal black holes.

Figures (3)

• Figure 1: The causal structure of the $AdS_2$ space (left), non-extremal Reissner Nordström black hole (center), and the extremal Reissner Nordström black hole (right). The non-extremal black hole possesses two event horizons at $r = r_+$ and $r = r_-$, while the extremal black hole possesses only one at $r=\rho$. The solid lines in Region II of the non-extremal black hole are timelike trajectories of constant $\psi$ (see Eq. \ref{['eq:regionIIcoords']}) extending from $r_+$ to $r_-$. The horizon of the extremal black hole solution indicated by the hatched red line is locally equivalent to the hatched red line of the $AdS_2$ diagram.
• Figure 2: A pictorial representation of the limiting procedure. The $AdS_2 \times S^2$ space (left) and extremal black hole (right) can be obtained from different regions of the non-extremal black hole (center). For fixed constant $r^* \sim \rho$ (dashed lines in the non-extremal black hole diagram), regions with smaller $r$ (the light shaded portions of the diagram comprising Region II and portions of Regions I and III) are approximated close to extremality by the corresponding light shaded regions of the $AdS_2 \times S^2$ diagram. These interfaces approach the timelike boundaries of the $AdS_2 \times S^2$ space when extremality is approached ($\epsilon \rightarrow 0$), as indicated by the arrows. The dark shaded regions on the non-extremal black hole diagram are approximated close to extremality by the corresponding dark shaded regions on the extremal black hole diagram. As extremality is approached, the extremal black hole approximation applies closer and closer to the horizon (as indicated by the arrows).
• Figure 3: The effective potential Eq. \ref{['eq:veffec']}. There is a solution that sits precisely at the maximum, corresponding to $AdS_2 \times S^2$. All other solutions correspond to some part of the RN spacetime; energies higher than the maximum of $V_{eff}$ are super-extremal, equal to the maximum are extremal, and below the maximum are sub-extremal. The sub-extremal region between the horizons at $r_+$ and $r_-$ corresponds to motion in an inverted potential between two turning points.