Instabilities of Black Strings and Branes

Abstract

We review recent progress on the instabilities of black strings and branes both for pure Einstein gravity as well as supergravity theories which are relevant for string theory. We focus mainly on Gregory-Laflamme instabilities. In the first part of the review we provide a detailed discussion of the classical gravitational instability of the neutral uniform black string in higher dimensional gravity. The uniform black string is part of a larger phase diagram of Kaluza-Klein black holes which will be discussed thoroughly. This phase diagram exhibits many interesting features including new phases, non-uniqueness and horizon-topology changing transitions. In the second part, we turn to charged black branes in supergravity and show how the Gregory-Laflamme instability of the neutral black string implies via a boost/U-duality map similar instabilities for non- and near-extremal smeared branes in string theory. We also comment on instabilities of D-brane bound states. The connection between classical and thermodynamic stability, known as the correlated stability conjecture, is also reviewed and illustrated with examples. Finally, we examine the holographic implications of the Gregory-Laflamme instability for a number of non-gravitational theories including Yang-Mills theories and Little String Theory.

Figures (14)

• Figure 1: The Gregory-Laflamme $\Omega(k)$ curve in $d$ dimensions reprinted from Ref. Gregory:1993vy, where $D \vert_{\rm figure} =d$ used in this review.
• Figure 2: Illustration of the Intersection Rule.
• Figure 3: Black hole and string phases for $d=4$ (left) and $d=5$ (right), drawn in the $(\mu,n)$ phase diagram. The horizontal (red) line at $n=1/2$ and 1/3 respectively is the uniform string branch. The (blue) branch emanating from this at the Gregory-Laflamme mass is the non-uniform string branch. For $d=4$ and $d=5$ this branch was obtained numerically in Kleihaus:2006ee and Wiseman:2002zc. The (purple) branch starting at the point $(\mu,n)=(0,0)$ is the black hole branch which was obtained numerically by Kudoh and Wiseman Kudoh:2004hs. For both dimensions the results strongly suggest that the black hole and non-uniform black string branches meet.
• Figure 4: Entropy $\mathfrak{s}/\mathfrak{s}_{\rm u}$ versus the mass $\mu/\mu_{GL}$ diagram for the uniform string (red), non-uniform string (blue) and localized black hole (purple) branches.
• Figure 5: $(\mu,n)$ phase diagrams for five (left figure) and six (right figure) dimensions. We have drawn the $(p,q)=(1,1)$, $(1,2){}_{\mathfrak{t}}$ and $(2,1)$ solutions. These curves lie in the region $1/2 < n \leq 2$ for the five-dimensional case and $1/3 < n \leq 3$ for the six-dimensional case. The lowest (red) curve corresponds to the $(1,1)$ solution. The (blue) curve that has highest $n$ for high values of $\mu$ is the equal temperature $(1,2){}_{\mathfrak{t}}$ solution. The (green) curve that has highest $n$ for small values of $\mu$ is the $(2,1)$ solution. The entire phase space of the $(1,2)$ configuration is the wedge bounded by the equal temperature $(1,2){}_{\mathfrak{t}}$ curve and the $(1,1)$ curve. For completeness we have also included the uniform (orange) and non-uniform (cyan) black string branch, and the small black hole branch (magenta) displayed in Figure \ref{['fig1']}.