New Phase Diagram for Black Holes and Strings on Cylinders

TL;DR

The paper introduces a two-parameter phase diagram for neutral, static black holes and strings on cylinders, using the mass $M$ and a relative binding energy $n$. It derives a new Smarr formula, $T S = \frac{d-2-n}{d-1} M$, and shows how the first law leads to a straightforward determination of branch thermodynamics from the $n(M)$ curve; it also establishes an intersection rule to discern the dominant branch. By incorporating Wiseman's numerical non-uniform string data, the authors demonstrate how all known neutral solutions map into the $(M,n)$ diagram and discuss the implications for phase structure, bounds on $n$, and possible uniqueness questions. The framework is poised to generalize to multi-compact directions and to charged or rotating cases, offering a compact, observable-at-infinity representation of the black hole/string phase structure on cylinders.

Abstract

We introduce a novel type of phase diagram for black holes and black strings on cylinders. The phase diagram involves a new asymptotic quantity called the relative binding energy. We plot the uniform string and the non-uniform string solutions in this new phase diagram using data of Wiseman. Intersection rules for branches of solutions in the phase diagram are deduced from a new Smarr formula that we derive.

Figures (6)

• Figure 1: $(M,n)$ phase diagram for $d=5$ containing the uniform string branch and the non-uniform string branch of Wiseman. The black hole branch is sketched.
• Figure 2: Illustration of the Intersection Rule.
• Figure 3: Plot of $a_2$ and $b_2$ versus $\lambda$ using the data of Wiseman:2002zc.
• Figure 4: Plot of $M$ and $n$ versus $\lambda$. $M$ and $n$ are computed from the $a_2$ and $b_2$ data. The crosses mark the actual data-points.
• Figure 5: Plot of $M$ versus $n$ as computed from the $a_2$ and $b_2$ data. The crosses mark the actual data-points.