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Dynamical Black Holes: Approach to the Final State

Abhay Ashtekar, Miguel Campiglia, Samir Shah

TL;DR

The work introduces a diffeomorphism-invariant framework of horizon multipole moments to characterize the dynamical, highly distorted phase of black holes and their approach to the Kerr equilibrium. By extending axisymmetric multipoles to general, non-axisymmetric dynamical horizons through a covariantly defined evolution vector field X^a and time-independent weighting fields, the authors derive generalized multipoles I_{l,m}, L_{l,m} and corresponding balance laws that link horizon geometry to fluxes. The formalism provides concrete, gauge-independent quantities that can be extracted from numerical simulations to probe universal features of black hole relaxation and to serve as internal checks on strong-field dynamics. In the late-time limit, the framework naturally recovers the Kerr IH multipoles, enabling direct comparison across simulations and with the final equilibrium state.

Abstract

Since black holes can be formed through widely varying processes, the horizon structure is highly complicated in the dynamical phase. Nonetheless, as numerical simulations show, the final state appears to be universal, well described by the Kerr geometry. How are all these large and widely varying deviations from the Kerr horizon washed out? To investigate this issue, we introduce a well-suited notion of horizon multipole moments and equations governing their dynamics, thereby providing a coordinate and slicing independent framework to investigate the approach to equilibrium. In particular, our flux formulas for multipoles can be used as analytical checks on numerical simulations and, in turn, the simulations could be used to fathom possible universalities in the way black holes approach their final equilibrium.

Dynamical Black Holes: Approach to the Final State

TL;DR

The work introduces a diffeomorphism-invariant framework of horizon multipole moments to characterize the dynamical, highly distorted phase of black holes and their approach to the Kerr equilibrium. By extending axisymmetric multipoles to general, non-axisymmetric dynamical horizons through a covariantly defined evolution vector field X^a and time-independent weighting fields, the authors derive generalized multipoles I_{l,m}, L_{l,m} and corresponding balance laws that link horizon geometry to fluxes. The formalism provides concrete, gauge-independent quantities that can be extracted from numerical simulations to probe universal features of black hole relaxation and to serve as internal checks on strong-field dynamics. In the late-time limit, the framework naturally recovers the Kerr IH multipoles, enabling direct comparison across simulations and with the final equilibrium state.

Abstract

Since black holes can be formed through widely varying processes, the horizon structure is highly complicated in the dynamical phase. Nonetheless, as numerical simulations show, the final state appears to be universal, well described by the Kerr geometry. How are all these large and widely varying deviations from the Kerr horizon washed out? To investigate this issue, we introduce a well-suited notion of horizon multipole moments and equations governing their dynamics, thereby providing a coordinate and slicing independent framework to investigate the approach to equilibrium. In particular, our flux formulas for multipoles can be used as analytical checks on numerical simulations and, in turn, the simulations could be used to fathom possible universalities in the way black holes approach their final equilibrium.

Paper Structure

This paper contains 11 sections, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quasi-local horizons
  3. Dynamical and isolated horizons
  4. Mutipole moments: The axi-symmetric case
  5. Multipole moments of general quasi-local horizons
  6. Main ideas
  7. Determining the dynamical vector field $X^a$
  8. Generalized multipoles
  9. Balance laws
  10. Approach to an axi-symmetric isolated horizon
  11. The setting

Figures (1)

  • Figure 1: A quasi-local horizon $M$. The past portion of $M$ consists of a dynamical horizon $H$: this portion is space-like and foliated by marginally trapped surfaces $S$. ${\widehat{\tau}}^a$ is the unit time-like normal to $H$ and ${\widehat{r}}^{\,a}$ the unit space-like normal within $H$ to the foliation. Although $H$ is space-like, motions along ${\widehat{r}}^{\,a}$ can be regarded as 'time evolution' with respect to observers at infinity. $H$ joins on to an isolated horizon $\Delta$ in the future, representing the equilibrium state of the black hole. $\Delta$ is null, endowed with a preferred null normal $\bar{\ell}^a$. The transition from $H$ to $\Delta$ occurs at $S_o$.