String Instabilities in Black Hole Spacetimes

TL;DR

This paper investigates string instabilities in various black hole and de Sitter spacetimes, revealing that radial and spatial components often become unstable near horizons and singularities, especially under strong repulsive gravitational effects.

Contribution

It provides a detailed analysis of string fluctuations in D-dimensional black hole and de Sitter backgrounds, identifying conditions under which instabilities develop.

Findings

Radial components exhibit imaginary frequencies near horizons and singularities.

In Reissner-Nordström black holes, instabilities occur inside the horizon due to charge effects.

All spatial components are unstable in de Sitter space.

Abstract

We study the emergence of string instabilities in $D$ - dimensional black hole spacetimes (Schwarzschild and Reissner - Nordstr\o m), and De Sitter space (in static coordinates to allow a better comparison with the black hole case). We solve the first order string fluctuations around the center of mass motion at spatial infinity, near the horizon and at the spacetime singularity. We find that the time components are always well behaved in the three regions and in the three backgrounds. The radial components are {\it unstable}: imaginary frequencies develop in the oscillatory modes near the horizon, and the evolution is like $(\tau-\tau_0)^{-P}$, $(P>0)$, near the spacetime singularity, $r\to0$, where the world - sheet time $(\tau-\tau_0)\to0$, and the proper string length grows infinitely. In the Schwarzschild black hole, the angular components are always well - behaved, while in the Reissner - Nordstr\o m case they develop instabilities inside the horizon, near $r\to0$ where the repulsive effects of the charge dominate over those of the mass. In general, whenever large enough repulsive effects in the gravitational background are present, string instabilities develop. In De Sitter space, all the spatial components exhibit instability. The infalling of the string to the black hole singularity is like the motion of a particle in a potential $\gamma(\tau-\tau_0)^{-2}$ where $\gamma$ depends on the $D$ spacetime dimensions and string angular momentum, with $\gamma>0$ for Schwarzschild and $\gamma<0$ for Reissner - Nordstr\o m black holes. For $(\tau-\tau_0)\to0$ the string ends trapped by the black hole singularity.