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Spherically symmetric scalar field collapse in any dimension

We describe a formalism and numerical approach for studying spherically symmetric scalar field collapse for arbitrary spacetime dimension d and cosmological constant Lambda. The presciption uses a double null formalism, and is based on field redefinitions first used to simplify the field equations in generic two-dimensional dilaton gravity. The formalism is used to construct code in which d and Lambda are input parameters. The code reproduces known results in d = 4 and d = 6 with Lambda = 0. We present new results for d = 5 with zero and negative Lambda.


I. INTRODUCTION
It is an interesting fact that spherically symmetric gravitational collapse exhibits critical behaviour [1]. This is a classical effect in phase space: there are one parameter families of initial data sets, for a variety of matter fields, such that as the parameter is tuned, a transition from reflection of infalling matter to black hole formation is observed numerically. Two types of behaviour are observed close to this transition point. Depending on the matter type, black holes form with zero or non-zero initial mass, and the matter field exhibits discrete or continuous self-similarity. A variety of matter fields have been studied since the initial seminal work by Choptuik. There have also been some extensions beyond spherical symmetry, as well as a semi-analytic perturbation theory understanding of the critical exponent. (Recent reviews may be found in Refs. [2,3].) If an exact time dependent solution were available with the appropriate boundary conditions, this critical behaviour would be manifested in the solution of the apparent horizon condition, which is a transcendental equation [4]. However, to date only one time dependent (and non-self similar solution) is known, but this does not have the required asymptotically flat boundary conditions [5]. Thus the full PDE problem must be studied numerically.
Most numerical studies of spherically symmetric collapse have been in four spacetime dimensions. The only exceptions are the massless minimally coupled scalar field by Garfinkle [7] in six spacetime dimensions, and by Pretorius and Choptuik [8], and Husain and Olivier [9] in three dimensions with negative cosmological constant. In addition, there exist two papers that study this problem in the much simpler case of the self-similarity ansatz, an ODE problem, for any spacetime dimension [10,11].
The purpose of this work is to present a formalism and numerical method for studying the gravitational collapse of a spherically symmetric scalar field for any value of spacetime dimension and cosmological constant. The approach reproduces and supplements known results in four and six dimensions with zero cosmological constant, and gives new results in five spacetime dimensions with zero and negative cosmological constant.

II. FIELD EQUATIONS
The basic idea for obtaining field equations valid for any dimension is to reduce the d−dimensional Einstein-scalar field equations by imposing spherical symmetry, and then use a field redefinition originally motivated by generic 2dimensional dilaton gravity [12]. This allows the Einstein equations for d−dimensional, spherically symmetric scalar field collapse to be put into a form that can be managed numerically by a single code. In actuality, the formalism and code we describe below is applicable to a more general class of models that includes non-minimal scalar field coupling. This class of theories is quite broad, and contains as a sub-class the d−dimensional spherically symmetric case.
Einstein gravity with cosmological constant in d spacetime dimensions is given by the action The corresponding action for a minimally coupled scalar field is To impose spherical symmetry, we write the d−dimensional metric g µν as where dΩ (d−2) is the metric on S d−2 and α, β = 1, 2. This gives the reduced action 1 where n = d − 2, V (n) is the volume of the unit n-sphere, and The key simplification in our formalism is achieved by the following conformal reparametrization of the metric, which eliminates the kinetic term forφ from the action [12]. Let where and C is an arbitrary constant. Now define a dimensionless "dilaton" field Note that φ is proportional to the area of the n-sphere at radius r. With these redefintions the reduced action takes the simpler form where and the overall factor of V (n) has been dropped. For arbitrary functions V (φ) and H(φ), the action (12) is that of generic dilaton gravity theory coupled to a scalar field in two spacetime dimensions. This theory has been studied in great detail [12]. The vacuum equations (χ = 0) can be solved exactly. By choosing an adapted coordinate system in which the the dilaton φ plays the role of the spatial coordinate (i.e. x = lφ), the vacuum solution for the metric is where M plays the role of mass, and Note that we have dropped the superscript n denoting spacetime dimension, since the above solution applies to the generic case. For the specific case of d−dimensional spherically symmetric gravity It can easily be verified by making the appropriate substitutions and conformal reparametrization (8) that the physical line element ds 2 (d) corresponding to (15) is precisely that of a d dimensional deSitter/anti-deSitter black hole with mass M . 2 It is important to note that the metric (15) is singular at φ = 0 even when M = 0. Up to numerical constants, j goes to zero as near φ = 0. This is not a physical singularity since the physical metric g is indeed the Minkowski metric when M and Λ are zero. Nonetheless, the vanishing of j(φ) will affect the choice of boundary conditions in our numerical method. We now examine the field equations that derive from (12) in double null coordinates, for which the metric may be parametrized as where the prime denotes partial differentiation with respect to the null coordinate v. (Recall that this is just the u − v part of the physical metric.) The corresponding field equations arė In the above, the dot refers to differentiation with respect to u, which is treated like the "time" coordinate for the purposes of the following numerical integration. Remarkably, for arbitrary n Eqs. (20-22) are virtually identical in form to those studied in [9] in the context of 2+1 dimensional AdS gravity. 3 However, the boundary conditions are special in the case of n = 1 (d = 3), so we will not consider this case further. Except where explicitly stated, we henceforth restrict consideration to n ≥ 2 (which means spacetime dimension d ≥ 4). The evolution equations may be put in a form more useful for numerical solution by defining the variable This effectively replaces the scalar field χ by h. The evolution equations arė and χ is now to be considered a functional of h and g given by The integration constant K 3 (u) must be zero because the definition of h requires h = χ at φ = 0. 4 The function g is a functional of h and φ, obtained by integrating the constraint (21): where K 1 (u) is again an integration constant (i.e. independent of the "spatial" coordinate v). We consider the case of a spherically symmetric, collapsing shell of matter, with no black hole in the interior, initially. Thus, our boundary condition should be such as to guarantee that the metric at r = 0 (which translates to φ = 0) goes over to the vacuum solution (15). By transforming the vacuum (M = 0) metric (15) to double null coordinates (19) we obtain a metric of the form where u = t − φ * and v = t + φ * , with the generalized "tortoise coordinate" φ * defined by With these definitions, φ = 0 corresponds to the surface v = u. Moreover, it follows that for the vacuum solution Comparing metric (29) to our general form (19) we see that g = 1 for the vacuum solution. Since we would like the numerical solution to approach the vacuum at φ = 0, the above analysis determines the required boundary conditions. In particular the integration constant K 1 (u) = 1, anḋ which vanishes at φ = 0 in agreement with the expression (26).

III. NUMERICAL METHOD
The numerical scheme uses a v ('space') discretization to obtain a set of coupled ODEs: where i = 0, · · · , N specifies the v grid. Initial data for these two functions is prescribed on a constant v slice, from which the functions g(u, v),g(u, v) are constructed. Evolution in the 'time' variable u is performed using the 4th. order Runge-Kutta method. The general scheme is similar to that used in [13], together with some refinements used in [6]. This procedure was also used for the 3−dimensional collapse calculations in [9]. The initial scalar field configuration χ(φ, u = 0) is most conveniently specified as a function of φ rather than r. (Recall that φ ∝ r n .) This together with the initial arrangement of the radial points φ(v, u = 0) fixes all other functions. We used the initial specification φ(0, v) = v.
We consider two types of initial scalar field configurations: the Gaussian and "tanh" functions and These choices permit us to test "universality," which is defined to be the independence of the details of the collapse, such as the critical exponent, on the choice of initial data shapes and parameters. Although universality may be tested for a large variety of shapes and parameters, the emphasis here is on different spacetime dimensions, so we have restricted attention to varying the amplitude parameter for these two shapes.
The initial values of the other functions are determined in terms of the above by computing the integrals for g n andg n using Simpson's rule. In all cases, we used values of φ 0 = 1 and σ = 0.3 for the Gaussian initial data.
The boundary conditions at fixed u are φ k = 0,g k = 0, g k = 1.
where k is the index corresponding to the position of the origin φ = 0. (In the algorithm used, all grid points 0 ≤ i ≤ k − 1 correspond to ingoing rays that have reached the origin and are dropped from the grid; see below). These conditions are equivalent to r(u, u) = 0, g| r=0 = g(u, u) = 1, and guarantee regularity of the metric at r = 0. Notice that for our initial data, φ k and hence h k are initially zero, and therefore remain zero at the origin because of Eqn. (25). As evolution proceeds via the Runge-Kutta procedure, the entries in the φ i array sequentially reach 0, at which point they are dropped from the grid. Thus the radial grid loses points with evolution. This is similar to the procedure used in [13] and [6].
At each u step, a check is made to see if an apparent horizon has formed by observing the function whose vanishing signals the formation of an apparent horizon. 5 For each run of the code with fixed amplitude a, this function is scanned from larger to smaller radial values after each Runge-Kutta iteration, and evolution is terminated if the value of this function reaches 10 −3 . The corresponding radial coordinate value is recorded as R ah . In the subcritical case, it is expected that all the radial grid points reach zero without detection of an apparent horizon. This is the signal of pulse reflection. The results (a, R ah ) are collated as in [1], by seeking a relationship of the form where a * is the critical amplitude which separates the black hole and reflection solutions. R ah is the basic dimensional scale if a black hole forms, and is linearly related to black hole mass in four spacetime dimensions. To improve numerical accuracy near φ = 0 we follow a procedure similar to [6], where all functions on a constant u surface are expanded in power series in φ at φ = 0, and the first three values of the constraint integrals are derived using the respective power series. We write and calculate the parameters h 0 , h 1 using the linear least squares fit for the first fifteen points in h i (u). 6 From this the expressions for χ, g andg follow. The remaining N − 3 values of these functions are computed from their integral using Simpson's rule for equally spaced points. This linear fit near the φ = 0 is necessary because it elegantly handles the problematic 1/φ factor in the h evolution equation, which would persist even if a finer mesh were used. The v−derivatives of functions (needed for computing g andg) are calculated using Further comments on the procedure are the following. The number of v grid points decreases as ingoing null geodesics cross r = 0, and so a reflected pulse cannot be followed back out toward infinity. 7 Also, again due to the loss of grid points, and hence resolution, we are not able to observe the detailed behaviour of the scalar field very near criticality. An additional numerical adjustment concerns the enforcement of boundary conditions at the origin: there is the gravitational tendency for the matter to pile up at the origin as the collapse proceeds. However the formalism has the competing implicit condition h(u, u) = 0 (ie. at φ = 0). This can lead to a shift of this boundary condition under evolution. It is rectified by adjusting the scalar field function at each time step by adding a constant shift at all points. This shift is of order 10 −5 or less at each time step, and therefore there is a minor loss of accuracy, but a corresponding gain in stability.
The code was tested for grid sizes ranging from 2000 to 6000 points, and with the u and v step sizes ranging from 10 −2 to 10 −4 , for the two types of initial data used, as well as the vacuum case of vanishing scalar field. These tests established that the code converges. Further tests of the code is the reproduction of the known results in four and six dimensions, which also demonstrates the accuracy of the results we obtain. All the results presented below were for a grid size of 6000 points, with u and v step sizes of 10 −3 .
Finally, we point out that this procedure allows a more accurate analysis of the supercritical case than the subcritical one, again because of the loss of grid points as the evolution proceeds. This is less of an issue in the supercritical case because the number of points lost depends on the initial pulse amplitude, and our termination condition is such that typically one quarter of the original number of points are still present at termination, for the closest approach to criticality. We could of course monitor the subcritical case up to this point as well in an attempt to observe the self-similarity of the scalar field, but the very nature of this behaviour requires a way to replace lost grid points, as observed in [6]. For this reason we focus on supercritical evolutions.

IV. RESULTS AND CONCLUSIONS
The code was first tested to recalculate known results in four and six dimensions. It was then run for the five dimensional case with zero, positive and negative cosmological constant. All the calculations were performed for amplitudes above the threshold for black hole formation, and for initial data specified in both Gaussian and tanh forms, parametrized by amplitude a. The figures below show the scaling law Eqn. (38). The squares represent the points (a, R ah ) and the lines are the least squares fit to these points.
The 4−dimensional results for the Gaussian initial data are illustrated in Figure 1. The least squares fit gives a slope of γ = 0.36, in good agreement (∼ 4%) with earlier studies [1][2][3]. The figure also shows the oscillation about the fit line, again in accord with earlier work. The results for the four and six dimensional calculations for the tanh initial data are γ = 0.35 and γ = 0.41 respectively. This provides a further check of our code, and further evidence of the insensitivity of the critical exponent to the shape of the initial data ("universality") in both four and six dimensions. Note that in six dimensions this is the first evidence for universality since Ref. [7] contains results only for a specific gaussian form of initial data, different from the one used here. Note also that the agreement of our apparent horizon radius scaling results with those of the earlier works cited, shows that the apparent horizon appears to be a fairly good approximation to the event horizon of the long time static limit, (in sofar as these earlier studies actually find this limit).
These tests of our formalism and code establish the consistency of our results with the earlier works mentioned above, and set the stage for new calculations for arbitrary values of the cosmological constant. Although our code allows calculations for any dimension, we focus on the 5−dimensional case mainly because results already exist for 3, 4 and 6 dimensions.
With Λ = 0, for the tanh initial data, we find a critical exponent of γ = 0.41 (Figure 3). This value falls between the 4 and 6 dimensional cases as conjectured in [7]. With Λ = −1, and Gaussian initial data, we find a critical exponent of γ = 0.49 ( Figure 4). All of the graphs show an oscillation about the least squares fit line. This is a known feature for zero cosmological constant, and is concomitant with discrete self-similarity of the critical solution. Our results for negative comological constant also show this feature, which indicates that the critical solution for this case also has discrete self-similarity.
We find that in five dimensions the critical exponent appears not to be universal, at least in the supercritical approach to computing it. The Gaussian initial data yielded γ = 0.52 for Λ = 0, in comparison to γ = 0.41 for the tanh initial data ( Figure 5). The reason for this is not clear to us and it would be worthwhile calculating the exponents using the subcritical approach suggested in Ref. [15], where the Ricci scalar at the origin is calculated near criticality from below. It is also worth noting that a similar lack of universality is manifested in the 3−dimensional AdS case using the supercritical apparent horizon method of computing γ [16].
We also carried out a preliminary investigation of the positive cosmological constant case in five dimensions. This is an interesting case because of the presence of a cosmological horizon in addition to the potential apparent horizon. Figure 6 shows graphs of the scalar field and apparent horizon functions h and ah in the left and right columns respectively, as functions of φ, prior to and at the onset of apparent horizon formation in the two successive rows. Note the location of the cosmological horizon in the right hand column near φ = 5.8. We find that the function φ(u, v) evolved such that instead of the radial grid contracting as for the zero and negative Λ cases, it expanded as the scalar field moved towards the origin. This feature is visible in Figure 6: the range of φ in the lower graphes has expanded to 8 from 6. In fact the closer is the onset of apparent horizon formation, the larger the range of the φ variable (and hence the radial grid). This prevented us from extracting accurate apparent horizon radii since the interesting features became confined to an ever shrinking part of the grid. We hope to study this in detail in future work. In summary, we have described a formalism and code for studying spherically symmetric gravitational collapse of a scalar field for any d and Λ, presented new results in five dimensions, and given evidence for universality in six dimensions. In future work we will present results of a systematic analysis of the critical exponent as a function of both spacetime dimension and cosmological constant.

V. ACKNOWLEDGEMENTS
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. G.K. would like to thank J. Gegenberg for helpful conversations.