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Cauchy Data for Formation of Multiple Black Holes with Prescribed ADM Parameters

Dawei Shen, Jingbo Wan

TL;DR

The paper presents a construction of smooth, asymptotically flat vacuum initial data for the Einstein vacuum equations that model an $N$-body collapsing system with independently prescribed ADM energies, momenta, and angular momenta for each component, while satisfying the timelike condition $E>|P|$ and starting from data with no trapped surfaces. It combines Kerr initial data in Kerr-Schild coordinates, conic and annular gluing theorems, and a short-pulse interior to produce disjoint collapsing regions whose futures dynamically form trapped surfaces, potentially evolving into multiple black holes. A Poincaré-covariant framework is developed to understand the transformation of Kerr charges under asymptotic diffeomorphisms, with localized ADM charges and linearized charges ensuring compatibility between SR transformation laws and nonlinear GR constraints. The main result, Theorem, provides a relativistic $N$-body family of initial data, enabling analysis of multi-black-hole formation and offering a structured path toward a multi-black-hole end-state scenario within vacuum GR. The work also outlines an explicit two-body escape/merger threshold in a simplified setting, highlighting the interplay between conserved quantities and initial separations.

Abstract

We give a simple construction of smooth, asymptotically flat vacuum initial data modeling a relativistic collapsing $N$--body system, with independently prescribed ADM energy, linear momentum, and angular momentum for each component, subject to the timelike condition $\E>|¶|$. The initial data contain no trapped surfaces, and the future development contains multiple causally independent trapped regions that dynamically form from localized subsets of the initial slice. In particular, the maximal development of data with well-separated collapsing components and relative motion is expected to yield spacetimes containing multiple black holes.

Cauchy Data for Formation of Multiple Black Holes with Prescribed ADM Parameters

TL;DR

The paper presents a construction of smooth, asymptotically flat vacuum initial data for the Einstein vacuum equations that model an -body collapsing system with independently prescribed ADM energies, momenta, and angular momenta for each component, while satisfying the timelike condition and starting from data with no trapped surfaces. It combines Kerr initial data in Kerr-Schild coordinates, conic and annular gluing theorems, and a short-pulse interior to produce disjoint collapsing regions whose futures dynamically form trapped surfaces, potentially evolving into multiple black holes. A Poincaré-covariant framework is developed to understand the transformation of Kerr charges under asymptotic diffeomorphisms, with localized ADM charges and linearized charges ensuring compatibility between SR transformation laws and nonlinear GR constraints. The main result, Theorem, provides a relativistic -body family of initial data, enabling analysis of multi-black-hole formation and offering a structured path toward a multi-black-hole end-state scenario within vacuum GR. The work also outlines an explicit two-body escape/merger threshold in a simplified setting, highlighting the interplay between conserved quantities and initial separations.

Abstract

We give a simple construction of smooth, asymptotically flat vacuum initial data modeling a relativistic collapsing --body system, with independently prescribed ADM energy, linear momentum, and angular momentum for each component, subject to the timelike condition . The initial data contain no trapped surfaces, and the future development contains multiple causally independent trapped regions that dynamically form from localized subsets of the initial slice. In particular, the maximal development of data with well-separated collapsing components and relative motion is expected to yield spacetimes containing multiple black holes.
Paper Structure (20 sections, 16 theorems, 192 equations, 1 figure)

This paper contains 20 sections, 16 theorems, 192 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Initial data gluing
  4. Definition of charges relative to a domain
  5. Gluing theorems
  6. Localized ADM charges for Kerr initial data
  7. Localized ADM energy and center of mass
  8. Localized ADM momentum and angular momentum
  9. Kerr spacetime under Poincaré transformation
  10. Charges for linearized Einstein vacuum equations
  11. Kerr spacetime under Poincaré transformation
  12. Conclusion.
  13. Proof of Theorem \ref{['maintheorem']}
  14. Step 1. Construction of Cauchy data.
  15. Step 2. Free of trapped surfaces.
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $N\in\mathbb{N}$ and $s\ge3$. For each $I=1,\dots,N$, prescribe parameters and choose $N$ pairwise disjoint cones $C_{\omega_I,\theta_I}$ with $\omega_I\in{\mathbb{S}}^2$ and $0<\theta_I<\frac{\pi}{2}$. Then there exist parameters and an initial data set $(\mathbb{R}^3,g,k)$ that solves the Einstein constraint equations Econstraint, such that the following hold:

Figures (1)

  • Figure 1: An illustration for Theorem \ref{['maintheorem']}. Each collapsing component is supported in a disjoint conic sector $C_{\omega_I,\theta_I}(y_I)$. The data are exactly Euclidean inside $B_{(1-2\delta_I)R_I}(\mathbf{c}_I)$, coincide with a boosted Kerr initial data set with prescribed ADM parameters $(\mathbf{E}_I,\mathbf{P}_I,\mathbf{J}_I)$ in each conic region $B_{32R_I}^c(\mathbf{c}_I)\cap(C_{\omega_I,\frac{1}{2}\theta_I}(y_I)\cup B_{\frac{1}{2}}(y_I))$. The innermost short-pulse core replaces the Kerr interior and gives rise to a trapped surface in the future domain $D^+(B_{R_I}(\mathbf{c}_I))$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Conjecture 1.2: Two-body escape/merger threshold
  • proof
  • Definition 2.1
  • Proposition 2.2: Proposition 9 in MaoTao
  • Definition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • ...and 24 more