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Critical phenomena in gravitational collapse exhibit the universal features of self-similarity, critical scaling, and the appearance of a naked singularity. We study critical collapse in anti–de Sitter space, focusing on holographic field theory observables. We demonstrate that the echoing period, critical exponent, and signatures of the naked singularity can all be extracted from the holographic stress tensor.

Nearly 30 years ago, Choptuik

Holographic duality

The ubiquity of critical collapse together with holographic duality suggests that the dynamics of quantum systems on the threshold of thermalization show universal features. A natural question arises: within the framework of holographic duality, how do critical collapse and its associated scale echoing and scaling relation

We seek to fill this gap by computing the expectation value of the QFT stress tensor

Most studies of critical collapse in AdS utilize a massless scalar field in spherical symmetry. While we have obtained results within this model, the echoing period is rather large, which limits the number of echoing periods that can be captured by numerical time evolution. We therefore present results from a different model with a shorter echoing period. Our results for the massless scalar are qualitatively similar.

The model we have chosen actually lies in a theory that is more universal than the massless scalar: pure gravity. We break the spherical

To obtain the boundary observables of both supercritical and subcritical collapse, we must propagate the signals associated with the formation of a horizon out to the AdS boundary. It is therefore necessary to use a gauge that allows for evolution past horizon formation. For this reason, rather than the more common radial Schwarzschild-like gauge, we use a maximal slicing gauge, where the trace of the extrinsic curvature

We now describe our metric ansatz:

For initial data, we take

The holographic stress tensor

Now we broadly describe our numerical evolution scheme. We are studying a subsector of the model in Ref.

The Einstein equation and maximal slicing gauge condition

Our main evolution system consists of the wave equation for

Spatial discretization is supplied by a spectral element mesh with Legendre-Gauss-Lobatto nodes, using a discontinuous Galerkin method with Lax-Friedrichs flux to handle interelement coupling. We utilize adaptive mesh refinement to maintain numerical control over the large gradients in the fields. When an apparent horizon forms, a portion of the grid inside the horizon is excised before continuing with the numerical evolution.

In Fig.

Supercritical metric function

In Fig.

Pressure anisotropy

Both subcritical and supercritical simulations show a time

Pressure anisotropy

The maximum frequency

To test Eq.

Maximum echo frequency

Note that the scaling

We also note that studies of critical collapse in asymptotically flat spacetime indicate the Bondi news function inherits the self-similar structure of the critical solution

We now turn to the envelope of

To understand how the envelope

An expansion at the boundary gives

The divergence at

It is striking that while we have started in a state that appears isotropic and close to equilibrium, the pressure anisotropy subsequently grows like

It is a pleasure to thank Gary Horowitz, Laurence Yaffe, and our referees for helpful comments. P. M. C. is supported by the Black Hole Initiative at Harvard University, which is funded by a grant from the John Templeton Foundation. B. W. is supported by NSERC. We thank the Yukawa Institute for Theoretical Physics at Kyoto University, where this work was initiated during the YITP-T-18-05 workshop on “Dynamics in Strong Gravity Universe.”

We have removed data points corresponding to runs with overly large constraint violation.

Because of the black hole ring down, there are more oscillations in