A spin on Hagedorn temperatures and string stars

Abstract

We discuss the correspondence between highly excited strings and black holes in the presence of angular momentum. At fixed imaginary angular velocity $ν$, we show that free strings exhibit a Hagedorn instability, explained, as in Atick-Witten, by a thermal-winding mode turning tachyonic. This allows us to determine the exact Hagedorn temperature $β_H(ν)$ for bosonic, type II, and heterotic strings. Using the effective field theory for the thermal-winding mode around the $ν$-dependent background, we find a novel `rotating string star' saddle (perturbatively in the angular velocity) and study its properties. This configuration describes a self-gravitating bound state of highly excited rotating strings. As in the non-rotating case, the saddle is qualitatively shown to interpolate between the rotating strings phase and a rotating black hole. We also comment on the implications of these results for anti-de Sitter space.

Figures (4)

• Figure 1: A schematic summary of the suggested black hole/string star correspondence. The Schwarzschild black hole (left) shrinks as the temperature (vertical axis) increases, and turns into a string star close to the string theory Hagedorn temperature $T\lesssim T_H$. Turning on angular momentum (horizontal axis), we find a 'rotated' version of the correspondence (right): the rotated black hole turns into a rotating string star at higher temperatures.
• Figure 2: The zeroth order solution for $d=3,4,5$ as a function of $m_0 r$. The solutions have been obtained numerically in Mathematica, with a shooting method to get the correct asymptotic behaviour.
• Figure 3: The first order corrections to winding mode and the gravitational potential profiles for $d=3,4,5$, as a function of $m_0 r$. The solutions have been obtained numerically by solving equations \ref{['eq:mon_quad_eqs']} using a shooting method.
• Figure 4: A schematic drawing of the effect of rotation, both in the $\nu$ (left) and $\Omega$ (right) ensembles. Depicted is the cross-section of the string star, with the vertical direction corresponding to $z$ and the horizontal direction to $\rho$. Red areas are of increased density, while blue areas are less dense compared to the nonrotating solution. For real $\Omega$, the string star becomes oblate, as intuitively expected.