Second Order Perturbations of a Macroscopic String; Covariant Approach

TL;DR

This paper develops a covariant formalism for second order perturbations of macroscopic strings, providing explicit results for contracting strings, quantization, and implications for string behavior near black holes.

Contribution

It generalizes first order perturbation results to second order, derives the mass spectrum including low-frequency modes, and discusses applications to cosmic strings and black hole horizons.

Findings

Second order perturbations are essential for consistent energy calculations.

The string mass spectrum is recovered at all frequencies with an appropriate vacuum choice.

Hamiltonian becomes non-diagonal at low frequencies, affecting quantization.

Abstract

Using a world-sheet covariant formalism, we derive the equations of motion for second order perturbations of a generic macroscopic string, thus generalizing previous results for first order perturbations. We give the explicit results for the first and second order perturbations of a contracting near-circular string; these results are relevant for the understanding of the possible outcome when a cosmic string contracts under its own tension, as discussed in a series of papers by Vilenkin and Garriga. In particular, second order perturbations are necessaary for a consistent computation of the energy. We also quantize the perturbations and derive the mass-formula up to second order in perturbations for an observer using world-sheet time $\tau $. The high frequency modes give the standard Minkowski result while, interestingly enough, the Hamiltonian turns out to be non-diagonal in oscillators for low-frequency modes. Using an alternative definition of the vacuum, it is possible to diagonalize the Hamiltonian, and the standard string mass-spectrum appears for all frequencies. We finally discuss how our results are also relevant for the problems concerning string-spreading near a black hole horizon, as originally discussed by Susskind.