Non-Spherically Symmetric Collapse in Asymptotically AdS 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.
Introduction.—The effort to understand the dynamics of gravity in asymptotically anti-de Sitter (AdS) spacetimes is driven by a series of open questions that has generated intense interest in recent years. As the maximally symmetric solution of the Einstein equations with a negative cosmological constant , AdS space is as fundamental as Minkowski space , the non-linear stability of which was established by the work of Ref. Christodoulou and Klainerman (1993). Similar stability results for de Sitter space were proven in Friedrich (1986, 1995). A question related to stability is that of black hole formation: how do black holes form in these spacetimes? The answer to this question has implications for several open problems in general relativity, including the validity of the weak cosmic censorship conjecture Penrose (1969). Black hole formation in asymptotically flat spacetimes was studied mathematically by Ref. Christodoulou (1987, 1991) in spherical symmetry, and by Ref. Choptuik (1993) using numerical techniques that led to the discovery of the celebrated critical phenomena in gravitational collapse; see Ref. Jałmużna and Gundlach (2017) for a recent study. In the asymptotically flat setting, considerable progress has been made with no symmetry assumptions by Ref. Christodoulou (2008); see also Ref. Klainerman and Rodnianski (2009); Klainerman et al. (2013).
The dynamics of gravity in asymptotically AdS spacetimes is not as well-understood as the asymptotically flat case. One reason for this is that when , solving the Einstein equations constitutes an initial boundary value problem. In contrast to the asymptotically flat case, the boundary of AdS is timelike and in causal contact with its interior. For the reflecting boundary conditions that are commonly used in the literature, Ref. Dafermos and Holzegel (2006); Dafermos (2006) conjectured that AdS is non-linearly unstable to the formation of black holes; see also Ref. Anderson (2006). In spherical symmetry, Ref. Moschidis (2017a, b) provided a proof for this instability in the specific setting of the Einstein-null dust system with an inner mirror. Black hole formation in AdS acquires an added significance in light of the AdS/CFT correspondence Maldacena (1998); Gubser et al. (1998); Witten (1998), according to which black hole formation in AdS corresponds to thermalization in a dual conformal field theory (CFT). The question of whether or not a black hole generically forms in AdS is then related to the question of whether or not a state in a strongly interacting CFT generically thermalizes.
Gravitational collapse in AdS was first studied in three dimensions in Ref. Pretorius and Choptuik (2000). Asymptotically AdS spacetimes in spacetime dimensions (AdS) are special for : there is a mass gap, so configurations with a total mass below a certain threshold cannot undergo gravitational collapse 111This was revisited by Ref. Jałmużna and Gundlach (2017) that studied critical scalar collapse with a rotating complex scalar field in AdS spacetimes, where a rotational Killing vector reduces the evolution to a dimensional problem., even though such solutions may not remain close to AdS in any reasonable norm Bizon and Jalmuzna (2013). For , configurations that collapse into a black hole after several bounces from the AdS boundary were found in Ref. Bizon and Rostworowski (2011); see also Ref. Liebling (2013); Buchel et al. (2012). In those studies, spherical symmetry was imposed, and the dynamics was driven by the presence of a massless scalar field. Initial data with an amplitude were numerically evolved in time and were found to develop features at progressively smaller spatial scales, in a turbulent process that terminates in gravitational collapse on a timescale of . On the other hand, within the same model in spherical symmetry, it was also found that there exist open sets of initial data that lead to solutions that are non-linearly stable against gravitational collapse Dias et al. (2012); Maliborski and Rostworowski (2013); Balasubramanian et al. (2014).
Efforts have begun to extend these studies beyond spherical symmetry Horowitz and Santos (2015); Dias and Santos (2016); Rostworowski (2017); Martinon et al. (2017); Dias and Santos (2017). The majority of results to date have been perturbative in nature 222The notable exceptions are the numerical construction of time-periodic AdS geon solutions in Ref. Horowitz and Santos (2015); Martinon et al. (2017); these solutions never collapse, by construction., and as such, cannot directly address the question of whether or not black holes form. In this Letter, we present the first study of gravitational collapse in AdS with inhomogeneous deformations away from spherical symmetry. For the numerical simulations presented here, we specialize to the case of global AdS, and we address the following question: “does gravitational collapse in AdS occur earlier or later away from spherical symmetry?” We answer this question by constructing fully non-linear time-dependent solutions of the Einstein equations starting with initial data sourced by a massless real scalar field.
Numerical Scheme.—The results presented in this Letter are based on a new numerical code to solve the Einstein equations for asymptotically AdS spacetimes. This code uses Cartesian coordinates in global AdS and is based on generalized harmonic evolution Pretorius (2005); see also Ref. Bantilan et al. (2012); Bantilan and Romatschke (2015). We define Cartesian coordinates in the following way: consider the metric of global AdS,
where is the AdS radius, which is related to the cosmological constant by , and is the metric on the unit round 3-sphere. We compactify the radial coordinate by defining so that the AdS boundary, , is included in the computational domain. Here, is an arbitrary compactification scale, independent of the AdS radius .
In polar coordinates, the Courant-Friedrichs-Lewy (CFL) condition at the origin imposes a severe restriction on the size of the time step. We bypass this issue by introducing Cartesian coordinates and . Setting and , the metric (1) in these Cartesian coordinates becomes
where , and is the metric on the unit round 2-sphere.
In moving away from pure AdS, we preserve an symmetry that acts to rotate the 2-spheres parametrized by and . This implies that there are seven independent metric components , each of which depends on . With this symmetry in five dimensions, the general form of the full metric away from pure AdS is
In our Cartesian coordinates, the axis of symmetry where the 2-sphere shrinks to zero size is at . To ensure that the spacetime remains smooth there, we impose suitable regularity conditions on our evolved variables at those points. See Supplemental Material for details of our evolution scheme together with the gauge choice, boundary conditions, and axis regularity conditions sm_ ().
We couple gravity to a massless real scalar field , and construct time-symmetric data on the initial time slice by solving the Hamiltonian constraint subject to a freely chosen initial scalar field profile. We use a family of profiles that smoothly interpolate between spherically symmetric and non-spherically symmetric configurations
where and are transition functions that spatially vary in the range , for some arbitrary , ; see Fig. 1. The spatial gradients of these data are compactly supported in the radial direction in the shape of an annulus centered around . The constants and measure the strength of the spherically symmetric and the non-spherically symmetric terms of the initial data respectively. Our results are not sensitive to the specific choice of and . See Supplemental Material for the details of the typical transition functions that we use in our simulations. Given our choice of initial data, the total angular momentum of the spacetime is zero. Therefore, if weak cosmic censorship holds in our setup, the final state of each of our simulations should be a global Schwarzschild-AdS black hole.
We monitor the evolution of black holes by keeping track of trapped surfaces. We excise a portion of the interior of any apparent horizon (AH) that forms, to remove any singularities from the computational domain. No boundary conditions are imposed on the excision surface; instead, the Einstein equations are solved there using one-sided stencils. We use Kreiss-Oliger dissipation Kreiss et al. (1973) to damp unphysical high-frequency modes that can arise at grid boundaries, with a typical dissipation parameter of .
We numerically solve the Einstein equations and the Hamiltonian constraint using the PAMR/AMRD libraries PAM (), and discretize the equations using second order finite differences. The evolution equations for the metric and scalar field are integrated in time using an iterative Newton-Gauss-Seidel relaxation procedure. The numerical grid is in with , , . A typical unigrid resolution has , grid points with equal grid spacings in the Cartesian directions. We use a typical Courant factor of . The results presented here were obtained with unigrid or with fixed refinement although the code has adaptive mesh refinement capabilities. See Supplemental Material for convergence tests.
Results.—For a fixed total mass and radial compactness, the time evolution of initial configurations (4) further away from spherical symmetry consistently exhibit gravitational collapse within fewer bounces. Throughout, we choose initial data where the spatial gradients of the initial scalar field profile are non-vanishing in an annular region bounded by and . These configurations form black holes after a variable number of bounces that depend on the values of the coefficients and in (4). The size and location of the first AH to form also depend on the values of and . For the special case of , i.e., spherically symmetric data, gravitational collapse leads directly to the formation of a black hole centered at the origin . Non-zero cases correspond to a one-parameter family of non-spherically symmetric configurations sourced by a scalar field whose initial profile has a dependence.
Fig. 1 (top) shows several snapshots of the normalized difference between the Kretschmann scalar and its pure AdS value for one such configuration with , and a total mass in units of . Fig. 1 (bottom) shows the corresponding profiles of the scalar field. These data collapse after two bounces into two distinct black holes centered at antipodal points on the axis . Aside from the last snapshot, which corresponds to a time slice shortly before collapse at in units of , the snapshots are taken after each bounce to emphasize the quasi-periodic nature of the evolution. Deviations away from strictly periodic behavior are evidenced by a gradual sharpening of spatial gradients after each bounce, which eventually leads to the formation of two black holes on the axis. See Supplemental Material for the energy density in the dual conformal field theory on which provides another view of this evolution.
Fig. 2 depicts the effect of moving away from spherical symmetry in two complementary ways. In Fig. 2 (left), we consider a one-parameter family of initial data obtained by varying and in (4) while keeping the total mass of the spacetime fixed for two representative cases with total masses (blue squares) and (red circles). The left-most point for each mass case corresponds to a spherically symmetric initial configuration with . For these spherically symmetric points, collapse occurs in the manner that had been found in Ref. Bizon and Rostworowski (2011). Data obtained for , , and show similar qualitative behavior. Increasing in (4) has the effect of deforming the initial data away from spherical symmetry. As is increased, is decreased in order to keep the total mass fixed. For larger , collapse occurs earlier: at certain critical values of , the collapse time decreases by roughly , i.e., the two AdS light-crossing times that it takes for a bounce. The center of collapse also shifts further away from the origin as is increased. The coefficient eventually reaches a maximum value corresponding to a maximally non-spherically symmetric initial configuration . These appear in Fig. 2 (left) as the right-most points. For these points, the initial data collapse into two distinct black holes on the axis, in fewer bounces than it takes their spherically symmetric counterparts to collapse into a single black hole at the origin.
Fig. 2 (right) depicts a different way of visualizing this faster collapse as one moves away from spherical symmetry. Here, the values for and are obtained in such a way that the data are at the cusp of collapse after one bounce (yellow squares) and at the cusp of collapse after two bounces (black circles), i.e., increasing either or would result in collapse one bounce earlier. In practice, this entails increasing while keeping fixed, thereby increasing the total mass of the configuration, until one finds the value for where collapse time decreases by . Fig. 2 (right) shows that for configurations with larger , less mass is required to stay at the cusp. Hence, for the family (4) of initial profiles, non-spherically symmetric configurations require less mass to collapse within a given collapse time than their spherically symmetric counterparts.
Gravitational collapse is preceded by the appearance of large curvatures near the axis. This provides a way to anticipate when collapse will occur, even before the first trapped surface is detected. We quantify the curvature deformation of any given time slice away from pure AdS in the following way. For some arbitrary threshold value , consider the spatial region where on a given time slice with intrinsic metric . Construct the quantity
and compare it to the spatial volume of the region . Fig. 3 shows how the dimensionless number behaves over time for various representative cases with different collapse times. The oscillations in are inherited from the quasi-periodic evolution that is depicted in Fig 1. In all cases, trapped surfaces are formed within a bounce of exceeding roughly unity. We have begun to extend this study to initial data where the deformation away from spherical symmetry is parametrized not just in terms of the lowest spherical harmonic on , but also in terms of the higher harmonics. In these cases, continues to be a useful quantity to signal collapse, i.e., collapse is preceded by a time where . It is also important to note that the time at which peaks is robust under changes in the threshold , even though the actual value of that precedes collapse does depend on . See Supplemental Material for the data corresponding to these statements.
Configurations of the form (4) with and generically collapse into two black holes on the axis which subsequently merge to form a single Schwarzschild-AdS black hole. The time it takes to reach this final state corresponds to the thermalization time of the dual CFT. The discrepancy between thermalization time and collapse time is therefore the time it takes for the two black holes to merge and ring down to a Schwarzschild-AdS black hole. This discrepancy is bounded: the merger time is bounded from above by , i.e., the length of a timelike geodesic joining a given point to the origin, while the ring-down is exponential and thus takes a negligibly short period of time. There is a potential caveat however: Ref. Holzegel and Smulevici (2013) conjectured that Schwarzschild-AdS, or more generally Kerr-AdS, is dynamically unstable for generic perturbations. Away from spherical symmetry, it is not known whether the system equilibrates to a stable black hole. Therefore, it is possible that the dual CFT may not thermalize at all 333Ref. Dias et al. (2012) argued that Schwarzschild-AdS should remain stable for long times for sufficiently regular perturbations..
Discussion.—We have presented the first study of gravitational collapse in AdS with inhomogeneous deformations away from spherical symmetry. Our results show that the evolution towards smaller spatial scales that leads to collapse persists in this setting, and in fact, we find that moving away from spherical symmetry facilitates collapse for a particular family of deformations. For a fixed total mass and radial compactness, these non-spherically symmetric configurations collapse earlier than their spherically symmetric counterparts, and for fixed radial compactness and collapse time, they require less mass to collapse within the same number of bounces. This faster collapse may have already been guessed from the smaller spatial scales that are present in the initial profiles to begin with. For the annular configurations that we construct, breaking spherical symmetry amounts to introducing spatial gradients along the annulus. Nevertheless, it was far from clear whether faster collapse as was observed would result from full non-linear evolution, especially in cases where collapse occurs after multiple bounces from the AdS boundary. This may have consequences for specific sets of initial data that are non-linearly stable against collapse (“islands of stability”). Namely, it is conceivable that the faster collapse we observe is an indication that these islands of stability are shrinking when spherical symmetry is broken. Systematic studies are required to investigate this point.
Our results further suggest that there exists a condition on the strength of the curvature deformation away from AdS in a given spatial volume, which separates data that collapse within one bounce of the condition being satisfied from data that will undergo further bounces. Such a condition was obtained in the asymptotically flat case in spherical symmetry Christodoulou (1991) and can be easily generalized to AdS Figueras (2010). It would be interesting to extend this analysis to the non-spherically symmetric setting. We have also repeated parts of the present study in four spacetime dimensions and have obtained qualitatively similar results.
We have broken spherical symmetry while preserving an symmetry in five dimensions. In this setting, we see that collapse is shifted away from the origin and leads to two distinct trapped regions on the axis. This can be understood as a focusing of energy density on the poles of the boundary . In the general case with no symmetry and arbitrary initial data, we expect this behavior to migrate away from the axis. A straightforward extension is to solve the momentum constraint equations along with the Hamiltonian constraint to generate configurations with non-zero total angular momentum. Based on results from perturbation theory, Ref. Dias and Santos (2017) suggests that angular momentum may further enhance the non-linear instability of AdS (though see Ref. Choptuik et al. (2017) for results of delayed collapse with the inclusion of angular momentum). As we noted in the main text, it is not clear whether AdS black holes themselves are non-linearly stable, so completely general perturbations of AdS may not settle down. We leave the question of stability of the black holes in this more general setting for a future work.
Acknowledgements.—We thank Andrzej Rostworowski, Luis Lehner, Alex Buchel, Frans Pretorius, and Maciej Maliborski for valuable discussions and comments. We gratefully acknowledge the computer resources and the technical support provided by Princeton University and the Barcelona Supercomputing Center (FI-2016-3-0006 “New frontiers in numerical general relativity”). H. B. and P. F. are supported by the European Research Council Grant No. ERC-2014-StG 639022-NewNGR. P. F. is also supported by a Royal Society University Research Fellowship (Grant No. UF140319). M. K. is supported by an STFC studentship. P. R. was supported in part by the Department of Energy, DOE award No. DE-SC0008132. Simulations were run on the Perseus cluster at Princeton University and the MareNostrum cluster at the Barcelona Supercomputing Center.
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Equations of motion.— Here we collect details of the scheme we use to evolve asymptotically AdS spacetimes. We obtain initial data that are time symmetric and conformal to the pure AdS metric by solving the Hamiltonian constraint equation for an initial spatial metric sourced by a massless real scalar field. The spatial transition functions used in the construction of the scalar profiles in the main text are
We obtain subsequent times by solving the Einstein equations in generalized harmonic form with constraint damping
where is the spacetime metric with Christoffel symbols , are the source functions with constraints , is the unit (timelike) vector normal to the slices, and and are constants. In the simulations presented here, we choose and . These equations are coupled to the equation of motion for a massless real scalar field,
with stress tensor
The evolved variables are constructed out of the full spacetime metric and the pure AdS metric by
where and are the Cartesian coordinates defined in the main text. For the invariant 2-sphere piece of the metric, we define a single variable so that
The evolved variables are similarly constructed out of the full spacetime source functions and the values that they take on in pure AdS according to
Finally, the evolved variable for the scalar field is constructed out of a real scalar field by
The gauge choice is made by specifying the following form of the source functions:
where are the initial values of the source functions, are components of the target gauge obtained by the procedure outlined in Ref. Bantilan et al. (2012)
and are the lapse damping terms used in Ref. Bantilan and Romatschke (2015)
Here, is the lapse function, are the shift vector components, and
where , and , , , , , , , , , are constants. On a typical run, we set , , , , , , , , , .
We set Dirichlet boundary conditions at spatial infinity for the metric, source functions, and scalar field:
In practice, because of our Cartesian grid, does not necessarily lie on a grid point aside from the exceptional points and . We thus implement (Non-Spherically Symmetric Collapse in Asymptotically AdS Spacetimes) on the grid points that are closest to , i.e., points that are at most one grid point away from . These near-boundary points are set using (Non-Spherically Symmetric Collapse in Asymptotically AdS Spacetimes) at and data at the point that are one grid point interior to the point in question by linear interpolation in the Cartesian directions: along for near-boundary points with , and along for near-boundary points with .
On the symmetry axis we impose regularity conditions on all evolved variables. These are:
In addition, we demand that there be no conical singularities at , which amounts to requiring that . We impose this condition on at instead of the corresponding regularity condition for in (Non-Spherically Symmetric Collapse in Asymptotically AdS Spacetimes).
CFT Stress Tensor.—
Here we briefly spell out how we extract the expectation value of the CFT stress energy tensor from the asymptotic behavior of the metric by
Here, is the extrinsic curvature of the surface, is a space-like outward pointing unit vector normal and is the induced 4-metric on the surface, is the covariant derivative operator and is the Einstein tensor associated with . The last two terms in (S16) are counterterms designed to cancel the divergent boundary behavior of the first two terms of (S16) evaluated in pure AdS. The stress tensor (S16) is non-vanishing even in pure global AdS: the CFT is defined on the boundary which has topology , and so can have a non-vanishing Casimir vacuum energy. Since this is a constant non-dynamical contribution, we consider it as part of our background vacuum and simply subtract it from (S16).
The conserved mass of the spacetime is computed from the quasi-local stress tensor (S16) as follows: we take a spatial slice of the surface, with induced 3-metric , lapse and shift such that and we compute
where is the time-like unit vector normal to In particular, because of background subtraction described above, we recover a vanishing mass for pure AdS.
The energy density in terms of the CFT stress tensor components is
The evolution of the energy density as a function of the global time for the simulation with , and a total mass is shown in Fig. S1. The energy density exhibits quasi-periodic behavior until it localizes at the poles. The latter corresponds to the formation of two black holes on the axis in the bulk.
Higher Harmonics and Different Thresholds.— Fig. S2 (left) shows three simulations of spacetimes with the same total mass, but deformed from pure AdS by spatial profiles with different dependence. The initial data deformed with the higher harmonics collapse earlier. Fig. S2 (right) shows the effect of changing the threshold for the simulation with , and a total mass . For smaller values of , i.e., for a less strict condition and thus a larger integration region, the maximum value of decreases, indicating that these points are associated with configurations where the Kretschmann scalar is very sharply peaked in smaller regions. Changing does not affect the time attains this maximum.
Numerical tests.— We check our solutions using a pair of standard convergence tests. To check for stability and consistency, we compute the rate of convergence at each point on the grid for a given evolved variable
Here, denotes one of from a simulation with mesh spacing . We use second-order accurate finite difference stencils, with refinement in between successive resolutions. Since we keep the CFL factor constant at , the time step is decreased by the same ratio. We thus expect to asymptote to in the limit .
To check that our numerical solutions are converging to a solution of the Einstein equations, we compute an independent residual by taking the numerical solution and substituting it into a discretized version of . Since the numerical solution was found solving the generalized harmonic form of the Einstein equations, we expect the independent residual to only be due to numerical truncation error and thus converge to zero. We can compute a convergence factor for the independent residual by
Here, denotes a component of . Again, given our second-order accurate finite difference stencils and with refinement in between successive resolutions, we expect to approach as .