How Do Axisymmetric Black Holes Grow Monopole and Dipole Hair?

TL;DR

This paper investigates how axisymmetric black holes develop scalar monopole and dipole hair during collapse, revealing the relationship between dipole modes and horizon formation through analytical and numerical methods.

Contribution

It provides new analytical insights and numerical simulations on the formation and dynamics of scalar monopole and dipole hair in axisymmetric black hole collapse within scalar Gauss-Bonnet and dynamical Chern-Simons theories.

Findings

Dipole hair mode is determined by mass and monopole hair.

Dipole mode dynamics are linked to event horizon formation.

Numerical simulations confirm analytical predictions.

Abstract

We study the dynamical formation of scalar monopole and dipole hair in scalar Gauss-Bonnet theory and dynamical Chern-Simons theory. We prove that the spherically-symmetric mode of the dipole hair is completely determined by the product of the mass of the spacetime and the value of the monopole hair. We then show that the dynamics of the $\ell=1$ mode of the dipole hair is intimately tied to the appearance of the event horizon during axisymmetric collapse, which results in the radiation of certain modes that could have been divergent in the future of the collapse. We confirm these analytical predictions by simulating the gravitational collapse of a rapidly rotating neutron star in the decoupling limit, both in scalar Gauss-Bonnet and dynamical Chern-Simons theory. Our results, combined with those of Ref.~\cite{R:2022cwe}, provide a clear physical picture of the dynamics of scalar monopole and dipole radiation in axisymmetric and spherical gravitational collapse in these theories.