On the fate of black string instabilities: An Observation

TL;DR

The paper addresses the end-state of the Gregory-Laflamme instability for a Schwarzschild black string in 4+1 dimensions. It combines Horowitz–Maeda bounds on the neck radius decay, $R = A e^{-(\ln \lambda)^{\alpha}}$ with $0<\alpha<1/2$, and a modeled relation between affine parameter $\lambda$ and advanced time $v$ via an effective surface gravity $\kappa_{eff}$ to argue that pinch-off can occur at finite $v$ even though $R$ vanishes only as $\lambda \to \infty$. Under the plausible scaling $\kappa_{eff} \ge k/R$ (or $k/R^{\gamma}$), the integral for $v$ converges, resolving tensions between HM and other analyses that suggest a final state of separated holes for $d \le 13$. The results emphasize that numerical determination of $\kappa_{eff}$ and KK-censorship considerations are key to a full understanding of the end-state in higher-dimensional gravity.

Abstract

Gregory and Laflamme (hep-th/9301052) have argued that an instability causes the Schwarzschild black string to break up into disjoint black holes. On the other hand, Horowitz and Maeda (arXiv:hep-th/0105111) derived bounds on the rate at which the smallest sphere can pinch off, showing that, if it happens at all, such a pinch-off can occur only at infinite affine parameter along the horizon. An interesting point is that, if a singularity forms, such an infinite affine parameter may correspond to a finite advanced time -- which is in fact a more appropriate notion of time at infinity. We argue below that pinch-off at a finite advanced time is in fact a natural expectation under the bounds derived by Horowitz and Maeda.

Figures (1)

• Figure 1: A conformal diagram is shown for the case where the black string forms from gravitational collapse (so that there is a regular center of the rotational symmetry). Past and future null infinity $({\cal I}^\pm)$ are shown, along with the horizon $H$. The affine parameter $\lambda$ along $H$ and the advanced time $v$ along ${\cal I}^-$ are indicated, and can be compared on ingoing null surfaces (dotted lines). A similar retarded time $u$ can of course also be introduced along ${\cal I}^+$.