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We reproduce and extend the previous studies of Lehner and Pretorius of the endpoint of the Gregory-Laflamme instability of black strings in five space-time dimensions. We consider unstable black strings of fixed thickness and different lengths, and in all cases we confirm that at the intermediate stages of the evolution the horizon can be interpreted as a quasistationary self-similar sequence of black strings connecting spherical black holes on different scales. However, we do not find any evidence for a global timescale relating subsequent generations. The endpoint of the instability is the pinch off of the horizon in finite asymptotic time, thus confirming the violation of the weak cosmic censorship conjecture around black string spacetimes., Comment: 12 pages, 10 figures plus one appendix. v2: Minor corrections, matches published version

The COVID-19 pandemic presents challenges to psychological wellbeing, but how can we predict when people suffer or cope during sustained stress? Here, we test the prediction that specific types of momentary emotional experiences are differently linked to psychological wellbeing during the pandemic. Study 1 used survey data collected from 24,221 participants in 51 countries during the COVID-19 outbreak. We show that, across countries, wellbeing is linked to individuals’ recent emotional experiences, including calm, hope, anxiety, loneliness, and sadness. Consistent results are found in two age, sex, and ethnicity-representative samples in the United Kingdom (N = 971) and the United States (N=961) with pre-registered analyses (Study 2). A prospective 30-day daily diary study conducted in the United Kingdom (N = 110) confirms the key role of these five emotions, and demonstrates that emotional experiences precede changes in wellbeing (Study 3). Our findings highlight differential relationships between specific types of momentary emotional experiences and wellbeing, and point to the cultivation of calm and hope as candidate routes for wellbeing interventions during periods of sustained stress.

We show that the most general scalar-tensor theory of gravity up to four derivatives in $3+1$ dimensions is well-posed in a modified version of the CCZ4 formulation of the Einstein equations in singularity-avoiding coordinates. We demonstrate the robustness of our new formulation in practise by studying equal mass black hole binary mergers for different values of the coupling constants. Although our analysis of well-posedness is restricted to cases in which the couplings are small, we find that in simulations we are able to push the couplings to larger values, so that a certain weak coupling condition is order one, without instabilities developing. Our work provides the means for such simulations to be undertaken by the many numerical relativity codes that rely on the moving puncture gauge to evolve black hole singularities., 5 pages, 2 figures, 2 appendices. v2: Clarified the different physical polarizations. References added. v3: Convergence test added. Matches published version

We study black hole binary mergers in certain cubic Horndeski theories of gravity, treating them fully non-linearly. In the regime of validity of effective field theory, the mismatch of the gravitational wave strain between Horndeski and general relativity (coupled to a scalar field) can be as large as $10-13\%$ in the Advanced LIGO mass range. The initial data and coupling constants are chosen such the theory remains in the weakly coupled regime throughout the evolution. In all cases that we have explored, we observe that the waveform in the Horndeski theory is shifted by an amount much larger than the smallness parameter that controls the initial data. This effect is generic and may be present in other theories of gravity involving higher derivatives., 17 pages, 12 figures, 1 appendix. v2: updated figures 3 and 7, updated table I

Around the globe, people are living under the threat of the COVID-19 pandemic, and many are experiencing curtailed opportunities for sustainable livelihoods and real-world social interactions. These conditions present a significant challenge to wellbeing1,2 and it is therefore essential to examine the factors associated with psychological wellbeing during this pandemic and other collective crises3. Here, we used survey data collected from 26,684 participants in 51 countries from 17 April to 15 May 2020 to examine the individual and societal factors that predict the degree of individuals’ suffering or thriving during the COVID-19 outbreak. We show that wellbeing is linked to an individual’s recent experiences of specific momentary positive and negative emotions, including calm, loneliness, love, and determination, as well as socioeconomic status. Around the world, calm, connection, and control were found to be central to wellbeing during the COVID-19 outbreak. The results are consistent with findings from two representative samples using pre-registered analyses, and a prospective 30-day daily diary study. Our findings highlight differential relationships between socio-affective processes and wellbeing during collective crises, and point to the importance of momentary experiences of calm, love and determination, particularly for those in less privileged positions in society. Funding Statement: This research was supported by ERC Starting grant 714977 awarded to D.A.S. Declaration of Interests: The authors have no competing interests to declare. Ethics Approval Statement: The study received approval from the University of Amsterdam Department of Psychology Ethics Committee and all participants provided digital informed consent.

We introduce new methods to numerically construct for the first time stationary axisymmetric black hole solutions in Einstein-aether theory and study their properties. The key technical challenge is to impose regularity at the spin-2, 1, and 0 wave mode horizons. Interestingly we find the metric horizon, and various wave mode horizons, are not Killing horizons, having null generators to which no linear combination of Killing vectors is tangent, and which spiral from pole to equator or vice versa. Existing phenomenological constraints result in two regions of coupling parameters where the theory is viable and some couplings are large; region I with a large twist coupling and region II with also a (somewhat) large expansion coupling. Currently these constraints do not include tests from strong field dynamics, such as observations of black holes and their mergers. Given the large aether coupling(s) one might expect such dynamics to deviate significantly from general relativity, and hence to further constrain the theory. Here we argue this is not the case, since for these parameter regions solutions exist where the aether is "painted" onto a metric background that is very close to that of general relativity. This painting for region I is approximately independent of the large twist coupling, and for region II is also approximately independent of the large expansion coupling and normal to a maximal foliation of the spacetime. We support this picture analytically for weak fields, and numerically for rotating black hole solutions, which closely approximate the Kerr metric., Comment: 61 pages, 26 figures; v2. minor amendments and corrections, references added; v3. minor additions, journal version

Plasma balls are droplets of deconfined plasma surrounded by a confining vacuum. We present the first holographic simulation of their real-time evolution via the dynamics of localized, finite-energy black holes in the five-dimensional anti–de Sitter (AdS) soliton background. The dual gauge theory is four-dimensional N=4 super Yang-Mills theory compactified on a circle with supersymmetry-breaking boundary conditions. We consider horizonless initial data sourced by a massless scalar field. Prompt scalar field collapse produces an excited black hole at the bottom of the geometry together with gravitational and scalar radiation. The radiation disperses to infinity in the noncompact directions and corresponds to particle production in the dual gauge theory. The black hole evolves toward the dual of an equilibrium plasma ball on a time scale longer than naively expected. This feature is a direct consequence of confinement and is caused by long-lived, periodic disturbances bouncing between the bottom of the AdS soliton and the AdS boundary.

We study spherically symmetric gravitational collapse in cubic Horndeski theories of gravity. By varying the coupling constants and the initial amplitude of the scalar field, we determine the region in the space of couplings and amplitudes for which it is possible to construct global solutions to the Horndeski theories. Furthermore, we identify the regime of validity of effective field theory as the sub-region for which a certain weak field condition remains small at all times. We evolve the initial data using the CCZ4 formulation of the Einstein equations and horizon penetrating coordinates without assuming spherical symmetry., 27 pages + 5 appendices (11 pages) + 16 figures. v2: additional figure added. Updated to match published version

We present the first proof-of-principle Cauchy evolutions of asymptotically global AdS spacetimes with no imposed symmetries, employing a numerical scheme based on the generalized harmonic form of the Einstein equations. In this scheme, the main difficulty in removing all symmetry assumptions can be phrased in terms of finding a set of generalized harmonic source functions that are consistent with AdS boundary conditions. In four spacetime dimensions, we detail an explicit set of source functions that achieves evolution in full generality. A similar prescription should also lead to stable evolution in higher spacetime dimensions, various couplings with matter fields, and on the Poincare patch. We apply this scheme to obtain the first long-time stable 3+1 simulations of four dimensional spacetimes with a negative cosmological constant, using initial data sourced by a massless scalar field. We present preliminary results of gravitational collapse with no symmetry assumptions, and the subsequent quasi-normal mode ringdown to a static black hole in the bulk, which corresponds to evolution towards a homogeneous state on the boundary., Comment: Changes to match published version

We report on the end state of nonaxisymmetric instabilities of singly spinning asymptotically flat Myers-Perry black holes. Starting from a singly spinning black hole in D=5, 6, 7 dimensions, we introduce perturbations with angular dependence described by m=2, m=3, or m=4 azimuthal mode numbers about the axis of rotation. In D=5, we find that all singly spinning Myers-Perry black holes are stable, in agreement with the results from perturbation theory. In D=6 and 7, we find that these black holes are nonlinearly stable only for sufficiently low spins. For intermediate spins, although the m=2 bar mode becomes unstable and leads to large deformations, the black hole settles back down to another member of the Myers-Perry family via gravitational wave emission; surprisingly, we find that all such unstable black holes settle to what appear to be the same member of the Myers-Perry family. The amount of energy radiated into gravitational waves can be very large, in some cases more than 30% of the initial total mass of the system. For high enough spins, the m=4 mode becomes the dominant unstable mode, leading to deformed black holes that develop local Gregory-Laflamme instabilities, thus forming a naked singularity in finite time, which is further evidence for the violation of the weak cosmic censorship conjecture in asymptotically flat higher-dimensional spacetimes.

We numerically simulate gravitational collapse in asymptotically anti-de Sitter spacetimes away from spherical symmetry. Starting from initial data sourced by a massless real scalar field, we solve the Einstein equations with a negative cosmological constant in five spacetime dimensions and obtain a family of non-spherically symmetric solutions, including those that form two distinct black holes on the axis. We find that these configurations collapse faster than spherically symmetric ones of the same mass and radial compactness. Similarly, they require less mass to collapse within a fixed time.

We determine the end point of the axisymmetric ultraspinning instability of asymptotically flat Myers-Perry black holes in D = 6 spacetime dimensions. In the non-linear regime, this instability gives rise to a sequence of concentric rings connected by segments of black membrane on the rotation plane. The latter become thinner over time, resulting in the formation of a naked singularity in finite asymptotic time and hence a violation of the weak cosmic censorship conjecture in asymptotically flat higher-dimensional spaces., Comment: Matches the published version. 4 pages plus references and supplemental material, 8 figures

We present the first example of a linearized gravitational instability of an asymptotically flat vacuum black hole. We study perturbations of a Myers-Perry black hole with equal angular momenta in an odd number of dimensions. We find no evidence of any instability in five or seven dimensions, but in nine dimensions, for sufficiently rapid rotation, we find perturbations that grow exponentially in time. The onset of instability is associated with the appearance of time-independent perturbations which generically break all but one of the rotational symmetries. This is interpreted as evidence for the existence of a new 70-parameter family of black hole solutions with only a single rotational symmetry. We also present results for the Gregory-Laflamme instability of rotating black strings, demonstrating that rotation makes black strings more unstable., Comment: 38 pages, 13 figures

We produce the first concrete evidence that violation of the weak cosmic censorship conjecture can occur in asymptotically flat spaces of five dimensions by numerically evolving perturbed black rings. For certain thin rings, we identify a new, elastic-type instability dominating the evolution, causing the system to settle to a spherical black hole. However, for sufficiently thin rings the Gregory-Laflamme mode is dominant, and the instability unfolds similarly to that of black strings, where the horizon develops a structure of bulges connected by necks which become ever thinner over time., Final published version: main paper of 5 pages and 2 figures + supplemental material of 3 pages and 3 figures

We present in detail the Einstein equations in the Baumgarte-Shapiro-Shibata-Nakamura formulation for the case of $D$ dimensional spacetimes with $SO(D-d)$ isometry based on a method originally introduced in Ref.1. Regularized expressions are given for a numerical implementation of this method on a vertex centered grid including the origin of the quasi-radial coordinate that covers the extra dimensions with rotational symmetry. Axisymmetry, corresponding to the value $d=D-2$, represents a special case with fewer constraints on the vanishing of tensor components and is conveniently implemented in a variation of the general method. The robustness of the scheme is demonstrated for the case of a black-hole head-on collision in $D=7$ spacetime dimensions with $SO(4)$ symmetry., Comment: 25 pages, 2 figures, Special Issue on Selected Papers of the III Amazonian Symposium on Physics

Previously the DeTurck 'trick' has been used to render the stationary Einstein's equation a well posed elliptic system that may be solved numerically by geometric flow or directly. Whilst in the static case for pure gravity with zero or negative cosmological constant there is a simple proof that solving the modified "harmonic" Einstein's equation leads to a solution of the original Einstein system - i.e. not a Ricci soliton - in the stationary case this argument no longer works. Here we provide a new argument that extends the static result to the case of stationary spacetimes that possess a "$t$-$\phi$" reflection symmetry. Defining a "soliton charge" from the asymptotic behaviour of the solution, we show that this quantity is always non-positive. Provided asymptotic conditions are chosen such that this charge vanishes, then stationary solitons cannot exist., Comment: 15 pages, no figures

We construct five dimensional black rings in global anti-de Sitter space using numerical methods. These rings satisfy the BPS bound $| J | < M \ell$, but the angular velocity always violates the Hawking-Reall bound $| \Omega_H \ell | \leq 1$, indicating that they should be unstable under superradiance. At high temperatures, the limit $| \Omega_H \ell | \searrow 1$ is attained by thin rings with an arbitrarily large radius. However, at sufficiently low temperatures, this limit is saturated by a new kind of rings, whose outer circle can still be arbitrarily long while the hole in the middle does not grow proportionally. This gives rise to a membrane-like horizon geometry, which does not have an asymptotically flat counterpart. We find no evidence for thin AdS black rings whose transverse $S^2$ is much larger than the radius of AdS, $\ell$, and thus these solutions never fall into the hydrodynamic regime of the dual CFT. Thermodynamically, we find that AdS black rings never dominate the grand canonical ensemble. The behaviour of our solutions in the microcanonical ensemble approaches known perturbative results in the thin-ring limit., Comment: 26 pages, 10 figures

We study six-dimensional rotating black holes with bumpy horizons: these are topologically spherical, but the sizes of symmetric cycles on the horizon vary non-monotonically with the polar angle. We construct them numerically for the first three bumpy families, and follow them in solution space until they approach critical solutions with localized singularities on the horizon. We find strong evidence of the conical structures that have been conjectured to mediate the transitions to black rings, to black Saturns, and to a novel class of bumpy black rings. For a different, recently identified class of bumpy black holes, we find evidence that this family ends in solutions with a localized singularity that exhibits apparently universal properties, and which does not seem to allow for transitions to any known class of black holes., 24 pages, 33 figures. v2: published version

In this paper we numerically construct localised black hole solutions at the IR bottom of the confining geometry of the AdS soliton. These black holes should be thought as the finite size analogues of the domain wall solutions that have appeared previously in the literature. From the dual CFT point of view, these black holes correspond to finite size balls of deconfined plasma surrounded by the confining vacuum. The plasma ball solutions are parametrised by the temperature. For temperatures well above the deconfinement transition, the dual black holes are small and round and they are well-described by the asymptotically flat Schwarzschild solution. On the other hand, as the temperature approaches the deconfinement temperature, the black holes look like pancakes which are extended along the IR bottom of the space-time. On top of these backgrounds, we compute various probes of confinement/deconfinement such as temporal Wilson loops and entanglement entropy., 24 pages, 8 figures, 2 appendices

In this article we review the present status of the numerical construction of black holes in the Randall–Sundrum II braneworld model. After reviewing the new numerical methods to solve the elliptic Einstein equations, we numerically construct a black hole solution in five-dimensional anti-de Sitter (AdS5) space whose boundary geometry is conformal to the four-dimensional Schwarzschild solution. We argue that such a solution can be viewed as the infinite radius limit of a braneworld black hole, and we provide convincing evidence for its existence. By deforming this solution in AdS we can then construct braneworld black holes of various sizes. We find that standard 4d gravity on the brane is recovered when the radius of the black hole on the brane is much larger than the radius of the bulk AdS space.

We explore use of the harmonic Einstein equations to numerically find stationary black holes where the problem is posed on an ingoing slice that extends into the interior of the black hole. Requiring no boundary conditions at the horizon beyond smoothness of the metric, this method may be applied for horizons that are not Killing. As a non-trivial illustration we find black holes which, via AdS-CFT, describe a time-independent CFT plasma flowing through a static spacetime which asymptotes to Minkowski in the flow's past and future, with a varying spatial geometry in-between. These are the first explicit examples of stationary black holes which do not have Killing horizons. When the CFT spacetime slowly varies, the CFT stress tensor derived from gravity is well described by viscous hydrodynamics. For fast variation it is not, and the solutions are stationary analogs of dynamical quenches, with the plasma being suddenly driven out of equilibrium. We find evidence these flows become unstable for sufficiently strong quenches, and speculate the instability may be turbulent., Comment: 12 pages, 11 figures

We use AdS/CFT to construct the gravitational dual of a 5D CFT in the background of a non-extremal rotating black hole. Our boundary conditions are such that the vacuum state of the dual CFT corresponds to the Unruh state. We extract the expectation value of the stress tensor of the dual CFT using holographic renormalisation and show that it is stationary and regular on both the future and the past event horizons. The energy density of the CFT is found to be negative everywhere in our domain and we argue that this can be understood as a vacuum polarisation effect. We construct the solutions by numerically solving the elliptic Einstein--DeTurck equation for stationary Lorentzian spacetimes with Killing horizons., Comment: 20 + 13 pages, 3 appendices. (Updated to match the content of published version. One extra appendix added.)

The higher-dimensional vacuum Einstein equation admits translationally non-uniform black string solutions. It has been argued that infinitesimally non-uniform black strings should be unstable in 13 or fewer dimensions and otherwise stable. We construct numerically non-uniform black string solutions in 11, 12, 13, 14 and 15 dimensions. Their stability is investigated using local Penrose inequalities. Weakly non-uniform solutions behave as expected. However, in 12 and 13 dimensions, strongly non-uniform solutions appear to be stable and can have greater horizon area than a uniform string of the same mass. In 14 and 15 dimensions all non-uniform black strings appear to be stable., Comment: 26 pages, 11 figures. V2: reference added, matches published version

We use a mix of analytic and numerical methods to exhaustively study a class of asymptotically global AdS solitons and hairy black hole solutions in negative cosmological constant Einstein Maxwell gravity coupled to a charged massless scalar field. Our results depend sensitively on the charge 'e' of the scalar field. The solitonic branch of solutions we study hit the Chandrashekhar limit at finite mass at small 'e', but extends to arbitrarily large mass at larger 'e'. At low values of 'e' no hairy black holes exist. At intermediate values of 'e' hairy black holes exist above a critical charge. At large 'e' hairy black holes exist at all values of the charge. The lowest mass hairy black holes is a smooth zero entropy soliton at small charge, but a (probably) singular nonzero entropy hairy black hole at larger charge. In a phase diagram of solutions, the hairy black holes merge with the familiar Reissner-Nordstrom-AdS black holes along a curve that is determined by the onset of the superradiant instability in the latter family., 80 pages. 25 Figures. RevTex4 format. v2: added discussions on second soliton branch and on planar limit; matches published version

Various higher-dimensional black holes have been shown to be unstable by studying linearized gravitational perturbations. A simpler method for demonstrating instability is to find initial data that describes a small perturbation of the black hole and violates a Penrose inequality. An easy way to construct initial data is by conformal rescaling of the unperturbed black hole initial data. For a compactified black string, we construct initial data which violates the inequality almost exactly where the Gregory-Laflamme instability appears. We then use the method to confirm the existence of the "ultraspinning" instability of Myers-Perry black holes. Finally we study black rings. We show that "fat" black rings are unstable. We find no evidence of any rotationally symmetric instability of "thin" black rings., 35 pages, 12 figures. v2: typos corrected, matches published version

The elliptic Einstein-DeTurck equation may be used to numerically find Einstein metrics on Riemannian manifolds. Static Lorentzian Einstein metrics are considered by analytically continuing to Euclidean time. Ricci-DeTurck flow is a constructive algorithm to solve this equation, and is simple to implement when the solution is a stable fixed point, the only complication being that Ricci solitons may exist which are not Einstein. Here we extend previous work to consider the Einstein-DeTurck equation for Riemannian manifolds with boundaries, and those that continue to static Lorentzian spacetimes which are asymptotically flat, Kaluza-Klein, locally AdS or have extremal horizons. Using a maximum principle we prove that Ricci solitons do not exist in these cases and so any solution is Einstein. We also argue that Ricci-DeTurck flow preserves these classes of manifolds. As an example we simulate Ricci-DeTurck flow for a manifold with asymptotics relevant for AdS_5/CFT_4. Our maximum principle dictates there are no soliton solutions, and we give strong numerical evidence that there exists a stable fixed point of the flow which continues to a smooth static Lorentzian Einstein metric. Our asymptotics are such that this describes the classical gravity dual relevant for the CFT on a Schwarzschild background in either the Unruh or Boulware vacua. It determines the leading O(N^2) part of the CFT stress tensor, which interestingly is regular on both the future and past Schwarzschild horizons., Comment: 48 pages, 7 figures; Version 2 - section 2.2.1 on manifolds with boundaries substantially modified, corrected and extended. Discussion in section 3.1 amended. References added and minor changes

We show how to construct low energy solutions to the Randall Sundrum II (RSII) model using an associated AdS_5-CFT_4 problem. The RSII solution is given in terms of a perturbation of the AdS-CFT solution, with the perturbation parameter being the radius of curvature of the brane metric compared to the AdS length \ell. The brane metric is then a specific perturbation of the AdS-CFT boundary metric. For low curvatures the RSII solution reproduces 4d GR on the brane. The leading correction is from local higher derivative curvature terms. The subleading correction is derived from similar terms and also the dual CFT stress tensor. Recently AdS-CFT solutions with 4d Schwarzschild boundary metric were numerically constructed. We modify the boundary conditions to introduce the RSII brane, and use elliptic numerical methods to solve the resulting boundary value problem. We construct large RSII static black holes with radius up to ~ 20 \ell. For large radius the RSII solutions are indeed close to the associated AdS-CFT solution. In this case the local curvature corrections vanish, and we confirm the leading correction is given by the AdS-CFT solution stress tensor. We also follow the black holes to small radius << \ell, where as expected they transition to a 5d behaviour. Our numerical solutions indicate the RSII black holes are dynamically stable for axisymmetric perturbations for all radii., 5 pages, 4 figures; v2: references added, minor improvements