Black Hole Critical Collapse in Infinite Dimensions: Continuous Self-Similar Solutions

TL;DR

This work extends the study of black hole critical collapse to the limit of large spacetime dimensions, using continuous self-similarity for a massless scalar field to generate analytically tractable, region-decomposed solutions via matched asymptotics. The authors develop a background spacetime by a $1/D$ expansion, revealing region-specific dynamics (weak, transition, strong gravity) and a precise FLRW-like interpretation that connects shell evolution to a closed cosmology. They compute the mass scaling exponents for both CSS-preserving and CSS-breaking perturbations, uncovering dimension-dependent structures and identifying two critical dimensions that alter the perturbation spectrum; in the infinite-dimensional limit the CSS-breaking exponent remains finite, aligning with expectations from DSS universality. The results establish large-$D$ as a powerful analytic tool in gravitational collapse, with potential extensions to other self-similar systems and broader gravitational phenomena such as fluids and black-string mergers.

Abstract

We investigate the dynamics of black hole critical collapse in the limit of a large number of spacetime dimensions, $D$. In particular, we study the spherical gravitational collapse of a massless, scale-invariant scalar field with continuous self-similarity (CSS). The large number of dimensions provides a natural separation of scales, simplifying the equations of motion at each scale where different effects dominate. With this approximation scheme, we construct matched asymptotic solutions for this family, including the critical solution. We then compute the mass critical exponent of the black hole for linear perturbations that break CSS, finding that it asymptotes to a constant value in infinite dimensions. Additionally, we present a link between these solutions and closed Friedmann--Lemaître--Robertson--Walker (FLRW) cosmologies with a dimension-dependent equation of state and cosmological constant. The critical solution corresponds to an unstable Einstein-like universe, while subcritical and supercritical solutions correspond to bouncing and crunching cosmologies respectively. Our results provide a proof of concept for the large-$D$ expansion as a powerful analytic tool in gravitational collapse and suggest potential extensions to other self-similar systems.

Figures (26)

• Figure 1: The effective potential $V(\mathcal{R})$, as defined by the right-hand side of the Friedmann Equation \ref{['eqn:Friedmann1']}, at a value of $q$ corresponding to criticality in two different numbers of dimensions. Each distinct region has been labelled for the $D=30$ case.
• Figure 2: A sketch of a continuously self-similar spacetime. It "looks the same", but rescaled, when the coordinates are rescaled by any amount.
• Figure 3: A sketch of a discretely self-similar spacetime. It only "looks the same", but rescaled, when the coordinates are rescaled by a discrete set of values. There is a period associated to the magnification lines, indicated by the colour.
• Figure 4: A Conformal Diagram demonstrating the fluid lines and HVF. Region I corresponds to flat Minkowski space due to our choice of boundary conditions. Region II is the dynamical region in which we solve the equations, with lines of constant $x$ indicated in blue and lines of constant $\omega$ in purple. The existence of Region III is dependent on the outcome of the dynamics in Region II. When it exists, it is also Minkowski space. Light grey lines in Region I and Region III indicate lines of constant $x$ beyond the self-similar horizon where the HVF becomes null.
• Figure 5: The phase portrait for \ref{['eqn:y4']}. The physical flows are denoted with solid lines and some of these are colour coded as representatives of the flows that lead to the different end states detailed in the introduction to this section. In naming these flows, we have also introduced a notion of how far from criticality the system is for later convenience.