Numerical Approaches to Spacetime Singularities

2.1.1 Overview

Perhaps, the first numerical approach to study the cosmic censorship conjecture consisted of attempts to create naked singularities. Many of these studies were motivated by Thorne’s “hoop conjecture” [239] that collapse will yield a black hole only if a mass M is compressed to a region with circumference C ≤ 4πM in all directions. (As is discussed by Wald [244], one runs into difficulties in any attempt to formulate the conjecture precisely. For example, how does one define C and M, especially if the initial data are not at least axially symmetric? Schoen and Yau defined the size of an arbitrarily shaped mass distribution in [228]. A non-rigorous prescription was used in a numerical study by Chiba [75]. If the hoop conjecture is true, naked singularities may form if collapse can yield C ≥ 4πM in some direction. The existence of a naked singularity is inferred from the absence of an apparent horizon (AH) which can be identified locally. Although a definitive identification of a naked singularity requires the event horizon (EH) to be proven to be absent, to identify an EH requires knowledge of the entire spacetime. While one finds an AH within an EH [166, 167], it is possible to construct a spacetime slicing which has no AH even though an EH is present [246]. Methods to find an EH in a numerically determined spacetime have only recently become available and have not been applied to this issue [179, 184]. A local prescription, applicable numerically, to identify an “isolated horizon” is under development by Ashtekar et al. (see for example [7]).

2.1.2 Naked spindle singularities?

In the best known attempt to produce naked singularities, Shapiro and Teukolsky (ST) [230] considered collapse of prolate spheroids of collisionless gas. (Nakamura and Sato [195] had previously studied the collapse of non-rotating deformed stars with an initial large reduction of internal energy and apparently found spindle or pancake singularities in extreme cases.) ST solved the general relativistic Vlasov equation for the particles along with Einstein’s equations for the gravitational field. They then searched each spatial slice for trapped surfaces. If no trapped surfaces were found, they concluded that there was no AH in that slice. The curvature invariant I = R _μνρσ R ^μνρσ was also computed. They found that an AH (and presumably a BH) formed if C ≤ 4πM > 1 everywhere, but no AH (and presumably a naked singularity) in the opposite case. In the latter case, the evolution (not surprisingly) could not proceed past the moment of formation of the singularity. In a subsequent study, ST [231] also showed that a small amount of rotation (counter rotating particles with no net angular momentum) does not prevent the formation of a naked spindle singularity. However, Wald and Iyer [246] have shown that the Schwarzschild solution has a time slicing whose evolution approaches arbitrarily close to the singularity with no AH in any slice (but, of course, with an EH in the spacetime). This may mean that there is a chance that the increasing prolateness found by ST in effect changes the slicing to one with no apparent horizon just at the point required by the hoop conjecture. While, on the face of it, this seems unlikely, Tod gives an example where a trapped surface does not form on a chosen constant time slice — but rather different portions form at different times. He argues that a numerical simulation might be forced by the singularity to end before the formation of the trapped surface is complete. Such a trapped surface would not be found by the simulations [241]. In response to such a possibility, Shapiro and Teukolsky considered equilibrium sequences of prolate relativistic star clusters [232]. The idea is to counter the possibility that an EH might form after the time when the simulation must stop. If an equilibrium configuration is non-singular, it cannot contain an EH since singularity theorems say that an EH implies a singularity. However, a sequence of non-singular equilibria with rising I ever closer to the spindle singularity would lend support to the existence of a naked spindle singularity since one can approach the singular state without formation of an EH. They constructed this sequence and found that the singular end points were very similar to their dynamical spindle singularity. Wald believes, however, that it is likely that the ST slicing is such that their singularities are not naked — a trapped surface is present but has not yet appeared in their time slices [244].

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Another numerical study of the hoop conjecture was made by Chiba et al. [76]. Rather than a dynamical collapse model, they searched for AH’s in analytic initial data for discs, annuli, and rings. Previous studies of this type were done by Nakamura et al. [196] with oblate and prolate spheroids and by Wojtk1ewicz [251] with axisymmetric singular lines and rings. The summary of their results is that an AH forms if C ≤ 4πM ≤ 1.26. (Analytic results due to Barrabès et al. [10, 9] and Tod [241] give similar quantitative results with different initial data classes and (possibly) an alternative definition of C.)

There is strong analytical evidence against the development of spindle singularities. It has been shown by Chruściel and Moncrief that strong cosmic censorship holds in AF electrovac solutions which admit a G₂ symmetric Cauchy surface [34]. The evolutions of these highly nonlinear equations are in fact non-singular.

2.1.3 Recent results

Garfinkle and Duncan [114] report preliminary results on the collapse of prolate configurations of Brill waves [60]. They use their axisymmetric code to explore the conjecture of Abrahams et al [2] that prolate configurations with no AH but large I in the initial slice will evolve to form naked singularities. Garfinkle and Duncan find that the configurations become less prolate as they evolve suggesting that black holes (rather than naked singularities) will form eventually from this type of initial data. Similar results have also been found by Hobill and Webster [149].

Pelath et al. [208] set out to generalize previous results [246, 241] that formation of a singularity in a slice with no AH did not indicate the absence of an EH. They looked specifically at trapped surfaces in two models of collapsing null dust, including the model considered by Barrabès et al. [10]. They indeed find a natural spacetime slicing in which the singularity forms at the poles before the outermost marginally trapped surface (OMTS) (which defines the AH) forms at the equator. Nonetheless, they also find that whether or not an OMTS forms in a slice closely (or at least more closely than one would expect if there were no relevance to the hoop conjecture) follows the predictions of the hoop conjecture.

2.1.4 Going further

Motivated by ST’s results [230], Echeverria [96] numerically studied the properties of the naked singularity that is known to form in the collapse of an infinite, cylindrical dust shell [239]. While the asymptotic state can be found analytically, the approach to it must be followed numerically. The analytic asymptotic solution can be matched to the numerical one (which cannot be followed all the way to the collapse) to show that the singularity is strong (an observer experiences infinite stretching parallel to the symmetry axis and squeezing perpendicular to the symmetry axis). A burst of gravitational radiation emitted just prior to the formation of the singularity stretches and squeezes in opposite directions to the singularity. This result for dust conflicts with rigorously nonsingular solutions for the electrovac case [34]. One wonders then if dust collapse gives any information about singularities of the gravitational field.

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One useful result from dust collapse has been the study of gravitational waves which might be associated with the formation of a naked singularity. Such a program has been carried out by Harada, Iguchi, and Nakao [160, 161, 198, 137, 138, 159].

Nakamura et al. (NSN) [197 ] conjectured that even if naked spindle singularities could exist, they would either disappear or become black holes. This demise of the naked singularity would be caused by the back reaction of the gravitational waves emitted by it. While NSN proposed a numerical test of their conjecture, they believed it to be beyond the current generation of computer technology.

Chiba [75] extended previous results [76] to search for AH’s in spacetimes without axisymmetry but with a discrete symmetry. The discrete symmetry is used to identify an analog of a symmetry axis to allow a prescription for an analog of the circumference. Given this construction, it is possible to formulate the hoop conjecture in this case and to explore its validity in numerically constructed momentarily static spacetimes. Explicit application was made to multiple black holes distributed along a ring. It was found that, if the quantity C defined as the circumference is less than approximately 1.168, a common apparent horizon surrounds the multi-black-hole configuration.

The results of all these searches for naked spindle singularities are controversial but could be resolved if the presence or absence of the EH could be determined. One could demonstrate numerically whether or not Wald’s view of ST’s results is correct by using existing EH finders [179, 184] in a relevant simulation. Of course, this could only be effective if the simulation covered enough of the spacetime to include (part of) the EH.

2.2.1 Gravitational collapse simulations

For a more detailed discussion of critical behavior see [126]. Since Gundlach’s Living Review article has appeared, the updates in this section will be restricted to results I find especially interesting.

Soon after this discovery, scaling and critical phenomena were found in a variety of contexts. Abrahams and Evans [1] discovered the same phenomenon in axisymmetric gravitational wave collapse with a different value of Δ and, to within numerical error, the same value of γ. (Note that the resealing of r with e^Δ ≈ 30 required Choptuik to use adaptive mesh refinement (AMR) to distinguish subsequent echoes. Abrahams and Evans’ smaller Δ (e^Δ ≈ 1.8) allowed them to see echoing with their 2 + 1 code without AMR.) Garfinkle [109] confirmed Choptuik’s results with a completely different algorithm that does not require AMR. He used Goldwirth and Piran’s [119] method of simulating Christodoulou’s [80] formulation of the spherically symmetric scalar field in null coordinates. This formulation allowed the grid to be automatically resealed by choosing the edge of the grid to be the null ray that just hits the central observer at the end of the critical evolution. (Missing points of null rays that cross the central observer’s world line are replaced by interpolation between those that remain.) Hamade and Stewart [135] have also repeated Choptuik’s calculation using null coordinates and AMR. They are able to achieve greater accuracy and find γ ≈ 0.374.

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2.2.2 Critical solutions as an eigenvalue problem

2.2.3 Recent results

Gundlach [129] completed his eigenvalue analysis of the Choptuik solution to find the growth rate of the unstable mode to be γ = 0.374 ± 0.001. He also predicted a periodic “wiggle” in the Choptuik mass scaling relation. This was later observed numerically by Hod and Piran [152]. Self-similar critical behavior has been seen in string theory related axion-dilaton models [95, 134] and in the nonlinear σ-model [146]. Garfinkle and Duncan have shown that subcritical collapse of a spherically symmetric scalar field yields a scaling relation for the maximum curvature observed by the central observer with critical parameters that would be expected on the basis of those found for supercritical collapse [113].

Choptuik et al. [79] have generalized the original Einstein-scalar field calculations to the Einstein-Yang-Mills (EYM) (for SU(2)) case. Here something new was found. Two types of behavior appeared depending on the initial data. In Type I, BH formation had a non-zero mass threshold. The critical solution is a regular, unstable solution to the EYM equations found previously by Bartnik and McKinnon [14]. In Type II collapse, the minimum BH mass is zero with the critical solution similar to that of Choptuik (with a different γ ≈ 0.20, Δ ≈ 0.74). Gundlach has also looked at this case with the same results [128]. The Type I behavior arises when the collapsing system has a metastable static solution in addition to the Choptuik critical one [132].

Brady, Chambers and Goncalves [71, 52] conjectured that addition of a mass to the scalar field of the original model would break scale invariance and might yield a distinct critical behavior. They found numerically the same Type I and II “phases” seen in the EYM case. The Type II solution can be understood as perturbations of Choptuik’s original model with a small scalar field mass μ. Here small means that λμ ≪ 1 where λ is the spatial extent of the original nonzero field region. (The scalar field is well within the Compton wavelength corresponding to μ.) On the other hand, λμ ≫ 1 yields Type I behavior. The minimum mass critical solution is an unstable soliton of the type found by Seidel and Suen [229]. The massive scalar field can be treated as collapsing dust to yield a criterion for BH formation [120].

The Choptuik solution has also been found to be the critical solution for charged scalar fields [132, 151]. As p → p*, Q/M → 0 for the black hole. Q obeys a power law scaling. Numerical study of the critical collapse of collision-less matter (Einstein-Vlasov equations) has yielded a non-zero minimum BH mass [217, 201]. Bizoń and Chmaj [47] have considered the critical collapse of skyrmions.

An astrophysical application of BH critical phenomena has been considered by Niemeyer and Jedamzik [199] and Yokoyama [252]. They consider its implications for primordial BH formation and suggest that it could be important.

2.2.4 Going further

The question is then why these critical phenomena should appear in so many collapsing gravitational systems. The discrete self-similarity of Choptuik’s solution may be understood as scaling of a limit cycle [136]. (The self-similarity of other systems may be understood as scaling of a limit point.) Garfinkle [110] originally conjectured that the scale invariance of Einstein’s equations might provide an underlying explanation for the self-similarity and discrete-self-similarity found in collapse and proposed a spacetime slicing which might manifestly show this. In fact, he later showed (with Meyer) [116] that, while the originally proposed slicing failed, a foliation based on maximal slicing did make the scaling manifest. These ideas formed the basis of a much more ambitious program by Garfinkle and Gundlach to use underlying actual or approximate symmetries to construct coordinate systems for numerical simulations [115].

An interesting “toy model” for general relativity in many contexts has been wave maps, also known as nonlinear σ models. One of these contexts is critical collapse [146]. Recently and independently, Bizoń et al. [48] and Liebling et al. [180] evolved wave maps from the base space of 3 + 1 Minkowski space to the target space S³. They found critical behavior separating singular and nonsingular solutions. For some families of initial data, the critical solution is self-similar and is an intermediate attractor. For others, a static solution separates the singular and nonsingular solutions. However, the static solution has several unstable modes and is therefore not a critical solution in the usual sense. Bizoń and Tabor [49] have studied Yang Mills fields in D + 1 dimensions and found that generic solutions with sufficiently large initial data blow up in a finite time and that the mechanism for blowup depends on D. Husa et al. [158] then considered the collapse of SU(2) nonlinear sigma models coupled to gravity and found a discretely self-similar critical solution for sufficiently large dimensionless coupling constant. They also observe that for sufficiently small coupling constant values, there is a continuously self-similar solution. Interestingly, there is an intermediate range of coupling constant where the discrete self-similarity is intermittent [238].

Until recently, only Abrahams and Evans [1] had ventured beyond spherical symmetry. The first additional departure has been made by Gundlach [131]. He considered spherical and non-spherical perturbations of P = 1/3ρ perfect fluid collapse. Only the original (spherical) growing mode survived.

Recently, critical phenomena have been explored in 2 + 1 gravity. Pretorius and Choptuik [215] numerically evolved circularly symmetric scalar field collapse in 2 + 1 anti-de Sitter space. They found a continuously self-similar critical solution at the threshold for black hole formation. The BH’s which form have BTZ [8] exteriors with strong curvature, spacelike singularities in the interior. Remarkably, Garfinkle obtained an analytic critical solution by assuming continuous self-similarity which agrees quite well with the one obtained numerically [112].

2.3.1 Overview

Unlike the simple singularity structure of the Schwarzschild solution, where the event horizon encloses a spacelike singularity at r = 0, charged and/or rotating BH’s have a much richer singularity structure. The extended spacetimes have an inner Cauchy horizon (CH) which is the boundary of predictability. To the future of the CH lies a timelike (ring) singularity [245]. Poisson and Israel [212, 213] began an analytic study of the effect of perturbations on the CH. Their goal was to check conjectures that the blue-shifted infalling radiation during collapse would convert the CH into a true singularity and thus prevent an observer’s passage into the rest of the extended regions. By including both ingoing and back-scattered outgoing radiation, they find for the Reissner-Nordström (RN) solution that the mass function (qualitatively R _αβγδ ∝ M/r³) diverges at the CH (mass inflation). However, Ori showed both for RN and Kerr [202, 203], that the metric perturbations are finite (even though R _μνρσ R ^μνρσ diverges) so that an observer would not be destroyed by tidal forces (the tidal distortion would be finite) and could survive passage through the CH. A numerical solution of the Einstein-Maxwell-scalar field equations would test these perturbative results.

For an excellent, brief review of the history of this topic see the introduction in [205].

2.3.2 Numerical studies

Gnedin and Gnedin [118] have numerically evolved the spherically symmetric Einstein-Maxwell with massless scalar field equations in a 2+2 formulation. The initial conditions place a scalar field on part of the RN event horizon (with zero field on the rest). An asymptotically null or spacelike singularity whose shape depends on the strength of the initial perturbation replaces the CH. For a sufficiently strong perturbation, the singularity is Schwarzschild-like. Although they claim to have found that the CH evolved to become a spacelike singularity, the diagrams in their paper show at least part of the final singularity to be null or asymptotically null in most cases.

Brady and Smith [56] used the Goldwirth-Piran formulation [119] to study the same problem. They assume the spacetime is RN for v > v₀. They follow the evolution of the CH into a null singularity, demonstrate mass inflation, and support (with observed exponential decay of the metric component g) the validity of previous analytic results [212, 213, 202, 203] including the “weak” nature of the singularity that forms. They find that the observer hits the null CH singularity before falling into the curvature singularity at r = 0. Whether or not these results are in conflict with Gnedin and Gnedin [118] is unclear [50]. However, it has become clear that Brady and Smith’s conclusions are correct. The collapse of a scalar field in a charged, spherically symmetric spacetime causes an initial RN CH to become a null singularity except at r = 0 where it is spacelike. The observer falling into the BH experiences (and potentially survives) the weak, null singularity [202, 203, 51] before the spacelike singularity forms. This has been confirmed by Droz [92] using a plane wave model of the interior and by Burko [63] using a collapsing scalar field. See also [65, 68].

Numerical studies of the interiors of non-Abelian black holes have been carried out by Breitenlohner et al. [57, 58] and by Gal’tsov et al. [91, 104, 105, 106] (see also [254]). Although there appear to be conflicts between the two groups, the differences can be attributed to misunderstandings of each other’s notation [59]. The main results include an interesting oscillatory behavior of the metric.

The current status of the topic of singularities within BH’s includes an apparent conflict between the belief [19] and numerical evidence [36] that the generic singularity is strong, oscillatory, and spacelike, and analytic evidence that the singularity inside a generic (rotating) BH is weak, oscillatory (but in a different way), and null. See the discussion at the end of [206].

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Various recent perturbative results reinforce the belief that the singularity within a “realistic” (i.e. one which results from collapse) black hole is of the weak, null type described by Ori [202, 203]. Brady et al. [54] performed an analysis in the spirit of Belinskii et al. [19] to argue that the singularity is of this type. They constructed an asymptotic expansion about the CH of a black hole formed in gravitational collapse without assuming any symmetry of the perturbed solution. To illustrate their techniques, they also considered a simplified “almost” plane symmetric model. Actual plane symmetric models with weak, null singularities were constructed by Ori [204].

The best numerical evidence for the nature of the singularity in realistic collapse is Hod and Piran’s simulation of the gravitational collapse of a spherically symmetric, charged scalar field [154, 153]. Rather than start with (part of) a RN spacetime which already has a singularity (as in, e.g., [56]), they begin with a regular spacetime and follow its dynamical evolution. They observe mass inflation, the formation of a null singularity, and the eventual formation of a spacelike singularity. Ori argues [206] that the rotating black hole case is different and that the spacelike singularity will never form. No numerical studies beyond perturbation theory have yet been made for the rotating BH.

Some insight into the conflict between the cosmological results and those from BH interiors may be found by comparing the approach to the singularity in Gowdy [121] spatially inhomogeneous cosmologies (see Section 3.4.2) with T³ [35] and S² × S¹ [111] spatial topologies. Early in the collapse, the boundary conditions associated with the S² × S¹ topology influence the gravitational waveforms. Eventually, however, the local behavior of the two spacetimes becomes qualitatively indistinguishable and characteristic of a (non-oscillatory in this case) spacelike singularity. This may be relevant because the BH environment imposes effective boundary conditions on the metric just as topology does. Unfortunately, no systematic study of the relationship between the cosmological and BH interior results yet exists.

2.3.3 Going further

Replacing the AF boundary conditions with Schwarzchild-de Sitter and RN-de Sitter BH’s was long believed to yield a counterexample to strong cosmic censorship. (See [185, 186;, 211, 70] and references therein for background and extended discussions.) The stability of the CH is related to the decay tails of the radiating scalar field. Numerical studies have determined these to be exponential [53, 70, 72] rather than power law as in AF spacetimes [67]. The decay tails of the radiation are necessary initial data for numerical study of CH stability [56] and are crucial to the development of the null singularity. Analytic studies had indicated that the CH is stable in RN-de Sitter BH’s for a restricted range of parameters near extremality. However, Brady et al. [55] have shown (using linear perturbation theory) that, if one includes the backscattering of outgoing modes which originate near the event horizon, the CH is always unstable for all ranges of parameters. Thus RN-de Sitter BH’s appear not to be a counterexample to strong cosmic censorship. Numerical studies are needed to demonstrate the existence of a null singularity at the CH in nonlinear evolution.

Extension of these studies to AF rotating BH’s has yielded the surprising result that the tails are not necessarily power law and differ for different fields. Frame dragging effects appear to be responsible [150].

As a potentially useful approach to the numerical study of singularities, we consider Hiibner’s [156, 157, 155] numerical scheme to evolve on a conformal compactified grid using Friedrich’s formalism [103]. He considers the spherically symmetric scalar field model in a 2 + 2 formulation. So far, this code has been used in this context to locate singularities. It was also used to search for Choptuik’s scaling [78] and failed to produce agreement with Choptuik’s results [156]. This was probably due to limitations of the code rather than inherent problems with the conformal method.

3.2.1 Symplectic methods

Symplectic numerical methods have proven useful in studies of the approach to the singularity in cosmological models [37]. Symplectic ODE and PDE [101, 189] methods comprise a type of operator splitting. An outline of the method (for one degree of freedom) follows. Details of the application to the Gowdy model (PDE’s in one space and one time direction) are given elsewhere [42].

3.2.2 Other methods

Symplectic methods can achieve convergence far beyond that of their formal accuracy if the full solution is very close to the exact solution from one of the sub-Hamiltonians. Examples where this is the case are given in [38, 35]. On the other hand, because symplectic algorithms are a type of operator splitting, suboperators can be subject to instabilities that are suppressed by the full operator. An example of this may be found in [41]. Other types of PDE solvers are more effective for such spacetimes. One currently popular method is iterative Crank-Nicholson (see [237] which is, in effect, an implicit method without matrix inversion. It was first applied to numerical cosmology by Garfinkle [111] and was recently used in this context to evolve T² symmetric cosmologies [41].

As pointed out in [42, 35, 41, 43], spiky features in collapsing inhomogeneous cosmologies will cause any fixed spatial resolution to become inadequate. Such evolutions are therefore candidates for adaptive mesh refinement such as that implemented by Hern and Stuart [143, 142].

3.3.1 Overview

Later, the BKL sensitivity to initial conditions was discussed in the language of chaos [11, 172]. An extended application of Bernoulli shifts and Farey trees was given by Rugh [224] and repeated by Cornish and Levin [85]. However, the chaotic nature of Mixmaster dynamics was questioned when numerical evolution of the Mixmaster equations yielded zero Lyapunov exponents (LE’s) [102, 62, 147]. (The LE measures the divergence of initially nearby trajectories. Only an exponential divergence, characteristic of a chaotic system, will yield a positive exponent.) Other numerical studies yielded positive LE’s [216]. This issue was resolved when the LE was shown numerically and analytically to depend on the choice of time variable [223, 27, 99]. Although MD itself is well-understood, its characterization as chaotic or not had been quite controversial [148].

LeBlanc et al. [178] have shown (analytically and numerically) that MD can arise in Bianchi VI₀ models with magnetic fields (see also [181]). In essence, the magnetic field provides the wall needed to close the potential in a way that yields the BKL map for u [29]. A similar study of magnetic Bianchi I has been given by LeBlanc [177]. Jantzen has discussed which vacuum and electromagnetic cosmologies could display MD [168].

Cornish and Levin (CL) [86] identified a coordinate invariant way to characterize MD. Sensitivity to initial conditions can lead to qualitatively distinct outcomes from initially nearby trajectories. While the LE measures the exponential divergence of the trajectories, one can also “color code” the regions of initial data space corresponding to particular outcomes. A chaotic system will exhibit a fractal pattern in the colors. CL defined the following set of discrete outcomes: During a numerical evolution, the BKL parameter u is evaluated from the trajectories. The first time u > 7 appears in an approximately Kasner epoch, the trajectory is examined to see which metric scale factor has the largest time derivative. This defines three outcomes and thus three colors for initial data space. However, one can easily invent prescriptions other than that given by Cornish and Levin [86] which would yield discrete outcomes. The fractal nature of initial data space should be common to all of them. It is not clear how the value of the fractal dimension as measured by Cornish and Levin would be affected. The CL prescription has been criticized because it requires only the early part of a trajectory for implementation [194]. Actually, this is the greatest strength of the prescription for numerical work. It replaces a single representative, infinitely long trajectory by (easier to compute arbitrarily many trajectory fragments.

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To study the CL fractal and ergodic properties of Mixmaster evolution [86], one could take advantage of a new numerical algorithm due to Berger, Garfinkle, and Strasser [38]. Symplectic methods are used to allow the known exact solution for a single Mixmaster bounce [236] to be used in the ODE solver. Standard ODE solvers [214] can take large time steps in the Kasner segments but must slow down at the bounce. The new algorithm patches together exactly solved bounces. Tens of orders of magnitude improvement in speed are obtained while the accuracy of machine precision is maintained. In [38], the new algorithm was used to distinguish Bianchi IX and magnetic Bianchi VI₀ bounces. This required an improvement of the BKL map (for parameters other than u) to take into account details of the exponential potential.

So far, most recent effort in MD has focused on diagonal (in the frame of the SU(2) 1-forms) Bianchi IX models. Long ago, Ryan [226] showed that off-diagonal metric components can contribute additional MSS potentials (e.g. a centrifugal wall). This has been further elaborated by Jantzen [169].

3.3.2 Recent developments

The most interesting recent developments in spatially homogeneous Mixmaster models have been mathematical. Despite the strong numerical evidence that Bianchi IX, etc. models are well-approximated by the BKL map sufficiently close to the singularity (see, e.g., [38]), there was very little rigorous information on the nature of these solutions. Recently, the existence of a strong singularity (curvature blowup) was proved for Bianchi VIII and IX collapse by Ringström [221, 222] and for magnetic Bianchi VI₀ by Weaver [248]. A remaining open question is how closely an actual Mixmaster evolution is approximated by a single BKL sequence [218, 222]. Since the Berger et al. algorithm [38] achieves machine level accuracy, it can be used to collect numerical evidence on this topic. For example, it has been shown that a given Mixmaster trajectory ceases to track the corresponding sequence of integers obtained from the BKL map (11) at the point where there have been enough era-ending (mixing) bounces to lose all the information encoded in finite precision initial data [38].

3.3.3 Going further

There are also numerical studies of Mixmaster dynamics in other theories of gravity. For example, Carretero-Gonzalez et al. [69] find evidence of chaotic behavior in Bianchi IX-Brans-Dicke solutions while Cotsakis et al. [87] have shown that Bianchi IX models in 4th order gravity theories have stable non-chaotic solutions. Barrow and Levin find evidence of chaos in Bianchi IX Einstein-Yang-Mills (EYM) cosmologies [12]. Their analysis may be applicable to the corresponding EYM black hole interior solutions. Bianchi I models with string-theory-inspired matter fields have been found by Damour and Henneau [89] to have an oscillatory singularity. This is interesting because many other examples exist where matter fields and/or higher dimensions suppress such oscillations (see, e.g., Recently, Coley has considered Bianchi IX brane-world models and found them not to be chaotic [84].

Finally, we remark on a successful application of numerical Regge calculus in 3 + 1 dimensions. Gentle and Miller have been able to evolve the Kasner solution [117].

3.4.1 Overview

BKL have conjectured that one should expect a generic singularity in spatially inhomogeneous cosmologies to be locally of the Mixmaster type (local Mixmaster dynamics (LMD)) For a review of homogeneous cosmologies, inhomogeneous cosmologies, and the relation between them, see [182]. The main difficulty with the acceptance of this conjecture has been the controversy over whether the required time slicing can be constructed globally [13, 122]. Montani [192], Belinskii [16], and Kirillov and Kochnev [175, 174] have pointed out that if the BKL conjecture is correct, the spatial structure of the singularity could become extremely complicated as bounces occur at different locations at different times. BKL seem to imply [22] that LMD should only be expected to occur in completely general spacetimes with no spatial symmetries. However, LMD is actually possible in any spatially inhomogeneous cosmology with a local MSS with a “closed” potential (in the sense of the standard triangular potentials of Bianchi VIII and IX). This closure may be provided by spatial curvature, matter fields, or rotation. A class of cosmological models which appear to have local MD are vacuum universes on T³ × R with a U(1) symmetry [190]. Even simpler plane symmetric Gowdy cosmologies [121, 26] have “open” local MSS potentials. However, these models are interesting in their own right since they have been conjectured to possess an AVTD singularity [123]. One way to obtain these Gowdy models is to allow spatial dependence in one direction in Bianchi I homogeneous cosmologies [26]. It is well-known that addition of matter terms to homogeneous Bianchi I, Bianchi VI₀, and other AVTD models can yield Mixmaster behavior [168, 178, 177]. Allowing spatial dependence in one direction in such models might then yield a spacetime with LMD. Application of this procedure to magnetic Bianchi VI₀ models yields magnetic Gowdy models [250, 247]. Of course, Gowdy cosmologies are not the most general T² symmetric vacuum spacetimes [121, 82, 40]. Restoring the “twists” introduces a centrifugal wall to close the MSS. Magnetic Gowdy and general T² symmetric models appear to admit LMD [247, 248, 41].

The past few years have seen the development of strong numerical evidence in support of the BKL claims [36]. The Method of Consistent Potentials (MCP) [123] has been used to organize the data obtained in simulations of spatially inhomogeneous cosmologies [42, 35, 250, 43, 36, 33, 41]. The main idea is to obtain a Kasner-like velocity term dominated (VTD) solution at every spatial point by solving Einstein’s equations truncated by removing all terms containing spatial derivatives. If the spacetime really is AVTD, all the neglected terms will be subdominant (exponentially small in variables where the VTD solution is linear in the time τ) when the VTD solution is substituted back into them. For the MCP to successfully predict whether or not the spacetime is AVTD, the dynamics of the full solution must be dominated at (almost) every spatial point by the VTD solution behavior. Surprisingly, MCP predictions have proved valid in numerical simulations of cosmological spacetimes with one [43] and two [42, 35, 41] spatial symmetries. In the case of U(1) symmetric models, a comparison between the observed behavior [43] and that in a vacuum, diagonal Blanch! IX model written in terms of U(1) variables provides strong support for LMD [45] since the phenomenology of the inhomogeneous cosmologies can be reproduced by this rewriting of the standard Blanch! IX MD.

Polarized plane symmetric cosmologies have been evolved numerically using standard techniques by Anninos, Centrella, and Matzner [5, 6]. The long-term project involving Berger, Garfinkle, and Moncrief and their students to study the generic cosmological singularity numerically has been reviewed in detail elsewhere [37, 36, 32] but will be discussed briefly here.

3.4.2 Gowdy cosmologies and their generalizations

The Gowdy model on T³ × R is described by gravitational wave amplitudes P(θ, τ) and Q(θ, τ) which propagate in a spatially inhomogeneous background universe described by λ(θ,τ). (We note that the physical behavior of a Gowdy spacetime can be computed from the effect of the metric evolution on a test cylinder [39].) We impose 0 ≤ θ ≤ 2π and periodic boundary conditions. The time variable τ measures the area in the symmetry plane with τ = ∞ being a curvature singularity.

For the special case of the polarized Gowdy model (Q = 0), P satisfies a linear wave equation whose exact solution is well-known [26]. For this case, it has been proven that the singularity is AVTD [165]. This has also been conjectured to be true for generic Gowdy models [123].

One striking property of the Gowdy models are the development of “spiky features” at isolated spatial points where the coefficient of a local “potential term” vanishes [42, 35]. Recently, Rendall and Weaver have shown analytically how to generate such spikes from a Gowdy solution without spikes [220].

Addition of a magnetic field to the vacuum Gowdy models (plus a topology change) which yields the inhomogeneous generalization of magnetic Bianchi VI₀ models provides an additional potential which grows exponentially if 0 > v > 1. Local Mixmaster behavior has recently been observed in these magnetic Gowdy models [250, 247].

Garfinkle has used a vacuum Gowdy model with S² × S¹ spatial topology to test an algorithm for axis regularity [111]. Along the way, he has shown that these models are also AVTD with behavior at generic spatial points that is eventually identical to that in the T³ case. Comparison of the two models illustrates that topology or other global or boundary conditions are important early in the simulation but become irrelevant as the singularity is approached.

Gowdy spacetimes are not the most general T² symmetric vacuum cosmologies. Certain off-diagonal metric components (the twists which are g _θσ , g _θδ in the notation of (12)) have been set to zero [121]. Restoring these terms (see [83, 40]) yields spacetimes that are not AVTD but rather appear to exhibit a novel type of LMD [41, 249]. The LMD in these models is an inhomogeneous generalization of non-diagonal Bianchi models with “centrifugal” MSS potential walls [227, 169] in addition to the usual curvature walls. In [41], remarkable quantitative agreement is found between predictions of the MCP and numerical simulation of the full Einstein equations. A version of the code with AMR has been developed [15]. (Asymptotic methods have been used to prove that the polarized version of these spacetimes have AVTD solutions [163].)

3.4.3 U(1) symmetric cosmologies

Current limitations of the U(1) code do not affect simulations for the polarized case since problematic spiky features do not develop. Polarized models have r = ω = 0. Grubišić and Moncrief [124] have conjectured that these polar ized models are AVTD. The numerical simulations provide strong support for this conjecture [37, 44]. Asymptotic methods have been used to prove that an open set of AVTD solutions exist for this case [164].

3.4.4 Going further

The MCP indicates that the term containing gradients of ω in (14) acts as a Mixmaster-like potential to drive the system away from AVTD behavior in generic U(1) models [24]. Numerical simulations provide support for this suggestion [37, 43]. Whether this potential term grows or decays depends on a function of the field momenta. This in turn is restricted by the Hamiltonian constraint. However, failure to enforce the constraints can cause an erroneous relationship among the momenta to yield qualitatively wrong behavior. There is numerical evidence that this error tends to suppress Mixmaster-like behavior leading to apparent AVTD behavior in extended spatial regions of U(1) symmetric cosmologies [30, 31]. In fact, it has been found that when the Hamiltonian constraint is enforced at every time step, the predicted local oscillatory behavior of the approach to the singularity is observed. (The momentum constraint is not enforced.) (Note that in a numerical study of vacuum Bianchi IX homogeneous cosmologies, Zardecki obtained a spurious enhancement of Mixmaster oscillations due to constraint violation [253, 147]. In this case, the constraint violation introduced negative energy.)

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Mixmaster simulations with the new algorithm [38] can easily evolve more than 250 bounces reaching |Ω|, ≈ 10⁶². This compares to earlier simulations yielding 30 or so bounces with |Ω| ≈ 10⁸. The larger number of bounces quickly reveals that it is necessary to enforce the Hamiltonian constraint. An explicitly constraint enforcing U(1) code was developed some years ago by Ove (see [207] and references therein).

It is well known [20] that a scalar field can suppress Mixmaster oscillations in homogeneous cosmologies. BKL argued that the suppression would also occur in spatially inhomogeneous models. This was demonstrated numerically for magnetic Gowdy and U(1) symmetric spacetimes Andersson and Rendall proved that completely general cosmological (spatially T³) spacetimes (no symmetries) with sufficiently strong scalar fields have generic AVTD solutions [3]. Garfinkle [107] has constructed a 3D harmonic code which, so far, has found AVTD solutions with a scalar field present. Work on generic vacuum models is in progress.

Cosmological models inspired by string theory contain higher derivative curvature terms and exotic matter fields. Damour and Henneaux have applied the BKL approach to such models and conclude that their approach to the singularity exhibits LMD [88].

Finally, there has been a study of the relationship between the “long wavelength approximation” and the BKL analyses by Deruelle and Langlois [90].