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Merger Transitions in Brane--Black-Hole Systems: Criticality, Scaling, and Self-Similarity

Valeri P. Frolov

TL;DR

This work introduces a toy brane–black-hole system in which a bulk $N$-dimensional black hole interacts with a test $D$-dimensional brane via the Dirac–Nambu–Goto action to model merger transitions. It identifies two static phases (brane crossing the horizon vs remaining outside) and a critical solution that acts as an attractor, enabling a detailed study of near-critical, self-similar scaling near the horizon. The analysis reveals a critical dimension $D_*=6$ separating discrete (oscillatory) from continuous scaling regimes, with explicit expressions for scaling exponents and log-periodic modulations in the subcritical regime. The results connect the BBH merger phenomenology to Choptuik critical collapse and to higher-dimensional black-hole–black-string transitions, offering analytic scaling laws, phase-portrait structure, and implications for time-dependent dynamics and finite-thickness brane models.

Abstract

We propose a toy model for study merger transitions in a curved spaceime with an arbitrary number of dimensions. This model includes a bulk N-dimensional static spherically symmetric black hole and a test D-dimensional brane interacting with the black hole. The brane is asymptotically flat and allows O(D-1) group of symmetry. Such a brane--black-hole (BBH) system has two different phases. The first one is formed by solutions describing a brane crossing the horizon of the bulk black hole. In this case the internal induced geometry of the brane describes D-dimensional black hole. The other phase consists of solutions for branes which do not intersect the horizon and the induced geometry does not have a horizon. We study a critical solution at the threshold of the brane-black-hole formation, and the solutions which are close to it. In particular, we demonstrate, that there exists a striking similarity of the merger transition, during which the phase of the BBH-system is changed, both with the Choptuik critical collapse and with the merger transitions in the higher dimensional caged black-hole--black-string system.

Merger Transitions in Brane--Black-Hole Systems: Criticality, Scaling, and Self-Similarity

TL;DR

This work introduces a toy brane–black-hole system in which a bulk -dimensional black hole interacts with a test -dimensional brane via the Dirac–Nambu–Goto action to model merger transitions. It identifies two static phases (brane crossing the horizon vs remaining outside) and a critical solution that acts as an attractor, enabling a detailed study of near-critical, self-similar scaling near the horizon. The analysis reveals a critical dimension separating discrete (oscillatory) from continuous scaling regimes, with explicit expressions for scaling exponents and log-periodic modulations in the subcritical regime. The results connect the BBH merger phenomenology to Choptuik critical collapse and to higher-dimensional black-hole–black-string transitions, offering analytic scaling laws, phase-portrait structure, and implications for time-dependent dynamics and finite-thickness brane models.

Abstract

We propose a toy model for study merger transitions in a curved spaceime with an arbitrary number of dimensions. This model includes a bulk N-dimensional static spherically symmetric black hole and a test D-dimensional brane interacting with the black hole. The brane is asymptotically flat and allows O(D-1) group of symmetry. Such a brane--black-hole (BBH) system has two different phases. The first one is formed by solutions describing a brane crossing the horizon of the bulk black hole. In this case the internal induced geometry of the brane describes D-dimensional black hole. The other phase consists of solutions for branes which do not intersect the horizon and the induced geometry does not have a horizon. We study a critical solution at the threshold of the brane-black-hole formation, and the solutions which are close to it. In particular, we demonstrate, that there exists a striking similarity of the merger transition, during which the phase of the BBH-system is changed, both with the Choptuik critical collapse and with the merger transitions in the higher dimensional caged black-hole--black-string system.

Paper Structure

This paper contains 11 sections, 80 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Brane equations
  3. Far distance solutions
  4. Brane configurations near the horizon
  5. Critical solutions as attractors
  6. Near-critical solutions in the vicinity of the horizon
  7. Scaling and self similarity
  8. $D\le 6$ case
  9. $D> 6$ case
  10. Merger transitions: Inside brane story
  11. Discussion

Figures (2)

  • Figure 1: A brane interacting with a black hole: The round circle is the black hole horizon. We schematically show the profiles of the brane for 3 different cases. The brane which touches the horizon at one point and has a cone-like profile near it is critical. Two other solid lines show branes close to the critical one. One of them (sub-critical) does not intersect the horizon, while the other (super-critical) one crosses the horizon at the spherical surface with a small radius $R_0$. At far distance the brane surfaces are parallel to the equatorial plane of the black hole.
  • Figure 2: Phase portraits of the system (\ref{['sys1']})-(\ref{['sys2']}) for $n=2$ (left plot) and for $n=4$ (right plot).