As first discovered by Choptuik, the black hole threshold in the space of initial data for general relativity shows both surprising structure and surprising simplicity. Universality, power-law scaling of the black hole mass, and scale echoing have given rise to the term "critical phenomena". They are explained by the existence of exact solutions which are attractors within the black hole threshold, that is, attractors of codimension one in phase space, and which are typically self-similar. Critical phenomena give a natural route from smooth initial data to arbitrarily large curvatures visible from infinity, and are therefore likely to be relevant for cosmic censorship, quantum gravity, astrophysics, and our general understanding of the dynamics of general relativity.

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Critical phenomena in gravitational collapse

We present a formalism to study the metric perturbations of the Schwarzschild spacetime. The formalism is gauge invariant, and it is also covariant under two-dimensional coordinate transformations that leave the angular coordinates unchanged. The formalism is applied to the typical problem of calculating the gravitational waves produced by material sources moving in the Schwarzschild spacetime. We examine the radiation escaping to future null infinity as well as the radiation crossing the event horizon. The waveforms, the energy radiated, and the angular-momentum radiated can all be expressed in terms of two gauge-invariant scalar functions that satisfy one-dimensional wave equations. The first is the Zerilli-Moncrief function, which satisfies the Zerilli equation, and which represents the even-parity sector of the perturbation. The second is the Cunningham-Price-Moncrief function, which satisfies the Regge-Wheeler equation, and which represents the odd-parity sector of the perturbation. The covariant forms of these wave equations are presented here, complete with covariant source terms that are derived from the stress-energy tensor of the matter responsible for the perturbation.

Gravitational perturbations of the Schwarzschild spacetime: A Practical covariant and gauge-invariant formalism

We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations (“linearised gravity”) around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior, and in fact decay to a linearised Kerr metric. We express the equations in a suitable double null gauge. To obtain decay, one must in fact add a residual pure gauge solution which we prove to be itself quantitatively controlled from initial data. Our result a fortiori includes decay statements for general solutions of the Teukolsky equation (satisfied by gauge-invariant null-decomposed curvature components). These latter statements are in fact deduced in the course of the proof by exploiting associated quantities shown to satisfy the Regge–Wheeler equation, for which appropriate decay can be obtained easily by adapting previous work on the linear scalar wave equation. The bounds on the rate of decay to linearised Kerr are inverse polynomial, suggesting that dispersion is sufficient to control the non-linearities of the Einstein equations in a potential future proof of non-linear stability. This paper is self-contained and includes a physical-space derivation of the equations of linearised gravity around Schwarzschild from the full non-linear Einstein vacuum equations expressed in a double null gauge.

https://www.intlpress.com/site/pub/files/_fulltext/journals/acta/2019/0222/0001/ACTA-2019-0222-0001-a001.pdf

The linear stability of the Schwarzschild solution to gravitational perturbations

We survey rigorous, formal and numerical results on the formation of point-like singularities (or blow-up) for a wide range of evolution equations. We use a similarity transformation of the original equation with respect to the blow-up point, such that self-similar behaviour is mapped to the fixed point of a dynamical system. We point out that analysing the dynamics close to the fixed point is a useful way of characterizing the singularity, in that the dynamics frequently reduces to very few dimensions. As far as we are aware, examples from the literature either correspond to stable fixed points, low-dimensional centre-manifold dynamics, limit cycles or travelling waves. For each 'class' of singularity, we give detailed examples.

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The role of self-similarity in singularities of partial differential equations*

In Paper I in this series we constructed evolution equations for the complete gauge-invariant linear perturbations of a time-dependent spherically symmetric perfect fluid spacetime. A key application of this formalism is the interior of a collapsing star. Here we derive boundary conditions at the surface of the star, matching the interior perturbations to the well-known perturbations of the vacuum Schwarzschild spacetime outside the star.

Gauge-invariant and coordinate-independent perturbations of stellar collapse. II: Matching to the exterior

At the threshold of black hole formation in the gravitational collapse of a scalar field a naked singularity is formed through a universal critical solution that is discretely self-similar. We study the global spacetime structure of this solution. It is spherically symmetric, discretely self-similar, regular at the center to the past of the singularity, and regular at the past light cone of the singularity. At the future light cone of the singularity, which is also a Cauchy horizon, the curvature is finite and continuous but not differentiable. To the future of the Cauchy horizon the solution is not unique, but depends on a free function (the null data coming out of the naked singularity). There is a unique continuation with a regular center (which is self-similar). All other self-similar continuations have a central timelike singularity with negative mass.

Global structure of Choptuik’s critical solution in scalar field collapse

We summarize the consequences of the twin assumptions of (discrete) self-similarity and spherical symmetry for the global structure of a spacetime. All such spacetimes can be constructed from two building blocks: the "fan" and "splash." Each building block contains one radial null geodesic that is invariant under the self-similarity (self-similarity horizon).

Kinematics of discretely self-similar spherically symmetric spacetimes

We investigate numerical methods for wave equations in $n+2$ spacetime dimensions, written in spherical coordinates, decomposed in spherical harmonics on $S^n$, and finite-differenced in the remaining coordinates $r$ and $t$. Such an approach is useful when the full physical problem has spherical symmetry, for perturbation theory about a spherical background, or in the presence of boundaries with spherical topology. The key numerical difficulty arises from lower-order $1/r$ terms at the origin $r=0$. As a toy model for this, we consider the flat space linear wave equation in the form $\dot\pi=\psi'+p\psi/r$, $\dot\psi=\pi'$, where $p=2l+n$, and $l$ is the leading spherical harmonic index. We propose a class of summation by parts (SBP) finite differencing methods that conserve a discrete energy up to boundary terms, thus guaranteeing stability and convergence in the energy norm. We explicitly construct SBP schemes that are second and fourth-order accurate at interior points and the symmetry boundary $r=0$, and first and second-order accurate at the outer boundary $r=R$.

Summation by parts methods for the spherical harmonic decomposition of the wave equation in arbitrary dimensions

After reviewingthe basics of Critical Phenomena in Gravitational Collapse of a spherically symmetric perfect fluid system, we address the relevance of adding angular momentum to the process. We study two different examples: the same perfect fluid but now with angular momentum, and Vlasov matter (collisionless particles, each with angular momentum). Using linear perturbation theory we show that in the former case there are still critical phenomena, explicitly predictingthe associated scalingla ws. We show that, on the contrary, critical phenomena are not generic for Vlasov matter.

José M. Martín-García

Papers

Gauge-invariant and coordinate-independent perturbations of stellar collapse. II: Matching to the exterior

Global structure of Choptuik’s critical solution in scalar field collapse

Kinematics of discretely self-similar spherically symmetric spacetimes

Summation by parts methods for the spherical harmonic decomposition of the wave equation in arbitrary dimensions

Critical Phenomena in Gravitational Collapse: The Role of Angular Momentum