Holographic Signatures of Critical Collapse

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 AdS, 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.

Introduction -Nearly 30 years ago, Choptuik [1] discovered that gravitational systems on the threshold of collapse exhibit universal features (see [2] for a review). First, as initial conditions are tuned towards criticality, the mass of the produced black hole scales like where parameterizes the initial data, with a black hole forming for supercritical data > * . The exponent γ is universal in that it is independent of initial conditions. Second, the critical solution itself exhibits self-similarity that can either be continuous or discrete. For discrete self-similarity, for suitably chosen time and radial coordinates t and r, the critical solution is periodic in ln r and ln(t−t c ) with a universal period ∆. This "scale echoing," means that as t → t c , the critical solution oscillates on increasing finer scales with a naked singularity at t = t c . Holographic duality [3][4][5][6] relates quantum theories of gravity to quantum field theories (QFT) without gravity in lower dimensions. The mapping is most useful for strongly coupled QFTs with a large number of colors N , where the dual gravitational theory is classical. The most widely studied example of holographic duality is AdS/CFT, which relates gravitational dynamics in antide Sitter (AdS) space to the dynamics of a conformal field theory (CFT) that lives on the AdS boundary. Gravitational collapse has a natural QFT interpretation as thermalization: black holes have O(N 2 ) entropy [7], at least O(∼ N 2 ) lifetime, and yield dual correlation functions that satisfy the fluctuation dissipation theorem [20][21][22].
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 1 manifest themselves in QFT observables? While critical phenomena has been studied in AdS [8][9][10][11][12], and attempts have been made to compute γ from the QFT side [13,14], the dynamics of QFT observables themselves remain largely unexplored.
We seek to fill this gap by computing the expectation value of the QFT stress tensor T µν from a dual gravity state near criticality. We focus on dynamics in global AdS 5 , dual to N = 4 supersymmetric Yang-Mills theory on R × S 3 . We find that T µν exhibits scale echoing. For both subcritical and supercritical solutions, we find that scale echoing in T µν terminates when the echo frequency (in linear boundary time t) is f ∼ | − * | −γ/2 . Moreover, we find evidence that as → * , scale echoing terminates with T µν diverging, meaning the naked singularity in the gravity description manifests itself as a singular stress tensor in the dual QFT.
Gravitational description -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 massles 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 SO(4) symmetry of global AdS 5 by a homogeneous squashing of the sphere that preserevs a SO(3)×U (1) symmetry. This squashing becomes a dynamical degree of freedom which evades Birkhoff's theorem. Previous studies in the asymptotically flat version of this setup [15] show that the Minkowski and Schwarzschild solutions serve as endpoints in the evolution of generic initial data, and that critical phenomena can be found by fine tuning initial data. Like the massless scalar in spherical symmetry, this model also has a universal critical exponent and discrete self similarity.
To obtain the boundary observables of both supercritical and subcritical collapse, we must propagate the signals associated to 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 K = 0.
We now describe our metric ansatz:
For initial data, we take with fixed σ = 0.1 and let parameterize our initial data. The remaining metric components are determined by constraint equations. From studies of AdS instability [12], this family of initial data is expected to contain an infinite number of critical * , corresponding to the number of times the configuration bounces back and forth between the origin and the boundary before eventually approaching the critical solution. For simplicity, we take to be near the highest critical * , so that the first signal we see at infinity corresponds to critical behavior.
The holographic stress tensor T µ ν is determined by the near-boundary asymptotics of the metric [16]. The stress tensor is covariantly conserved with respect to the boundary metric (3), which just reflects energy and momentum conservation in the dual field theory. Since the metric functions α, β, a, b do not depend on angles, T µ ν only depends on time. It follows that T µ ν has a single dynamical degree of freedom which, together with the conserved energy, determines all other components of the stress tensor. We will take the this degree of freedom to be the normalized pressure anisotropy where G 5 = π 2N 2 is the five dimensional Newton constant. Since T µ ν is order N 2 , B is independent of N in the large N limit. In terms of the bulk geometry, Now we broadly describe our numerical evolution scheme. We are studying a sub-sector of the model in [17] with the same setup, nomenclature, and numerical methods. We therefore refer the reader to [17] for further details and for numerical validation of the code. There are also first-order spatial constraint equations for β, a and α, as well as temporal constraint equations for a and β. There is also one spatial constraint equation and one temporal constraint equation for b, obtained by reducing its wave equation to first order. Our main evolution system consists of the wave equation for b (in first-order form), and the temporal constraint for b. The spatial constraint for b is used as added constraint damping terms to improve numerical stability. Evolution proceeds by a fourth order Runge-Kutta method (RK4). Spatial constraints, which consists of a sequence of linear problems, are solved via Gaussian elimination at each RK4 step for β, a and α. The remaining equations are not solved directly and verified afterwards. For all data presented here, relative energy violation and constraint violation are within or below 10 −6 . 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 inter-element coupling. We utilise 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.
Results and discussion -In Fig. 1, we show the metric function b at the time of apparent horizon formation for finely-tuned supercritical initial data. We see that b exhibits nearly equally spaced oscillations in ln ρ. The self-similar structure in b subsequently propagates outwards towards the boundary, where via (6), it influences the evolution of the pressure anisotropy B .
In Fig. 2, we show B as a function of t for one instance of subcritical and supercritical data. As the selfsimilar waveform in b approaches the boundary, B be- gins to oscillate, with increasing frequency and amplitude as time progresses. For subcritical data, the oscillations abruptly terminate at some maximum frequency f max . In contrast, for supercritical data, the oscillations reach a maximum frequency f max and then decay in amplitude while continuing to oscillate at frequency ∼ f max . This is due to the newly formed horizon making causal contact with the boundary, with the subsequent attenuation of B reflecting the ringdown of the black hole.
Both subcritical and supercritical simulations show a time t * about which B has self similar structure. In Fig. 3, we plot B as a function of − ln(1 − t/t * ) for various amplitudes . We choose t * such that the oscillations in Fig. 3 have constant period in − ln(t * − t). Note that at early times all curves in both plots overlap, and that fine-tuning closer to criticality adds more echos with successively increasing amplitudes. The echoing period is ∆ ≈ 0.46, which is comparable to the flat space result [15] of ∆ ≈ 0.47. As extracting the echoing period involves a loss of fidelity due to fitting, our result for ∆ is consistent with that of [15]. Evidently, the self-similar structure in the gravity description gets imprinted on the boundary where t * is chosen to preserve self-similarity, for subcritical (top) and supercritical (bottom) initial data. Colored and dotted/dashed lines are less fine-tuned data. Note that all curves agree at early times, and that successive echoes appear with more fine tuning.
stress tensor T µ ν . t * has the natural interpretation as the time at which the naked singularity of the critical solution makes causal contact with the AdS boundary.
The maximum frequency f max reached by B at the termination of scale echoing can be related to the critical exponent γ. For supercritical initial data, scale echoing terminates when a horizon is formed, where f max has the same scale as the horizon radius: f max ∼ r −1 H . Subsequently, the black hole should ring down with a frequency also set by this scale. But in five dimensions, r H ∝ √ M H , so the scaling relation (1) implies that To test (7) on the boundary, we perform a short-time Fourier transform of B with Gaussian windowing. We then locate the dominant peak in the windowed Fourier transform f (t), which gives the local echo frequency of B . f (t) takes its maximum value f max when scale echoing terminates. In Fig. 4 we plot f max as a function of for both subcritical (top) and supercritical (bottom) solutions [25]. The red lines in each plot are fits to (7). For subcritical and supercrtical data the fits yield γ = 0.32 and γ = 0.33, respectively [26]. This should be compared to the flat space result of γ = 0.33 reported in [15]. Evidently, near the threshold of thermalization, the stress tensor oscillates at f ∼ f max . Remarkably, the scaling (7) applies for both subcritical and supercritical data. As far as we are aware, this agreement on both sides of criticality has not previously been demonstrated in numerical time evolution. Nevertheless, this result should be expected, since the critical exponent γ is related to the only growing mode of perturbations of the critical solution, which is insensitive to whether the perturbation is supercritical or subcritical [18].
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 [19]. Presumably, scale echoing in the news function also terminates at maximum frequency f max . In four dimensions, the horizon radius r H ∼ M H , which means that scale echoing should terminate at f max ∼ | − * | −γ .
We now turn to the envelope of B . In Fig. 5, we plot ln | B | as a function of − ln(1 − t/t * ). We see that the amplitude of each echo in ln | B | appears linear in − ln(1 − t/t * ) in the self-similar region. (There are also transition regions to early times and times very close to t * which do not show self-similar behavior.) Our numerics are consistent with Since B can be obtained from the derivative of the metric near the AdS boundary, the power-law divergence (8) is naturally interpreted as consequence of self-similarity of the bulk geometry. Evidently, the stress tensor of the critical solution diverges when the naked singularity in the bulk makes causal contact with the boundary. To summarize, the evidence we have presented shows that the primary features of critical collapse, including the self-similar scale echoing with its period ∆, the critical scaling with its exponent γ, and the appearance of a naked singularity, can all be captured from simple onepoint functions in the dual field theory: in the case, the stress tensor. The ubiquity and universality of critical collapse suggests that similar behaviour should be found in a wide variety of holographic field theories, for wideranging dynamical scenarios near thermalization.
While we have focused on T µν , signatures of critical phenomena should also be present in other QFT observables. One interesting observable to study is the photoemission spectrum, which is dual to Hawking radiation [20][21][22]. After the horizon forms, the black hole should emit Hawking radiation with wavelength on the order of the horizon radius. Correspondingly, during thermalization it is natural to expect the photoemmision spectrum to be peaked at frequency f ∼ ( − * ) −γ/2 . It would also be interesting to study critical phenomena from the QFT side of the duality, as attempted in [13,14]. Do weakly coupled large N quantum field theories also exhibit self-similarity and scalings near the thermalization threshold? Finally, it would be interesting to explore critical phenomena within the context of holographic models of heavy-ion collisions, such as those studied in [23,24], where localized gravitational waves were collided with some finite impact parameter. One could probe the critical state by merely adjusting the impact parameter, thereby studying peripheral collisions. We leave the study of these interesting topics for future analysis.