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3.5 3D characteristic evolution

There has been rapid progress in 3D characteristic evolution. There are now two independent 3D codes, one developed at Canberra and the other at Pittsburgh (the PITT code), which have the capability to study gravitational waves in single black hole spacetimes, at a level still not mastered by Cauchy codes. Several years ago the Pittsburgh group established robust stability and second order accuracy of a fully nonlinear 3D code which calculates waveforms at null infinity [42Jump To The Next Citation Point31Jump To The Next Citation Point] and also tracks a dynamical black hole and excises its internal singularity from the computational grid [105Jump To The Next Citation Point102Jump To The Next Citation Point]. The Canberra group has implemented an independent nonlinear 3D code which accurately evolves the exterior region of a Schwarzschild black hole. Both codes pose data on an initial null hypersurface and on a worldtube boundary, and evolve the exterior spacetime out to a compactified version of null infinity, where the waveform is computed. However, there are essential differences in the underlying geometrical formalisms and numerical techniques used in the two codes and in their success in evolving generic black hole spacetimes.

3.5.1 Geometrical formalism

The PITT code uses a standard Bondi-Sachs null coordinate system,

( ) 2 2bV 2 A B 2 2b 2 B A 2 A B ds = - e -- - r hAB U U du - 2e du dr- 2r hAB U dudx + r hAB dx dx , (21) r
where det(hAB) = det(qAB) for some standard choice qAB of the unit sphere metric. This generalizes Equation (13View Equation) to the three-dimensional case. The hypersurface equations derive from the Gmn\ ~/ nu components of the Einstein tensor. They take the explicit form
1 b,r = ---rhAC hBD hAB,r hCD,r, (22) ( ) 16 ( ) r4e- 2bhABU B,r ,r = 2r4 r -2b,A ,r - r2hBC DC (hAB,r) (23) (( ) ) 2e-2bV,r = R - 2DADAb - 2(DAb)DAb + r- 2e- 2bDA r4U A ,r - 1r4e- 4bhAB U AU B, (24) 2 ,r ,r
where DA is the covariant derivative and R the curvature scalar of the conformal 2-metric hAB of the r = const. surfaces, and capital Latin indices are raised and lowered with h AB. Given the null data hAB on an outgoing null hypersurface, this hierarchy of equations can be integrated radially in order to determine b, UA and V on the hypersurface in terms of integration constants on an inner boundary. The evolution equations for the u-derivative of the null data derive from the trace-free part of the angular components of the Einstein tensor, i.e. the components mAmBGAB where hAB = 2m(AmB). They take the explicit form
( mAmB (rhAB,u),r- 1-(rV hAB,r),r - 2ebDADBeb + rhAC DB(U C) 2r r ,r r3 - 2b C D r- C - 2 e hAC hBD U ,r U,r + 2DA UB + 2 hAB,r DC U ) +rU CDC (hAB,r) + rhAD,r hCD(DBUC - DC UB) = 0. (25)
The Canberra code employs a null quasi-spherical (NQS) gauge (not to be confused with the quasi-spherical approximation in which quadratically aspherical terms are ignored [42Jump To The Next Citation Point]). The NQS gauge takes advantage of the possibility of mapping the angular part of the Bondi metric conformally onto a unit sphere metric, so that hAB --> qAB. The required transformation A A A x --> y (u, r,x ) is in general dependent upon u and r so that the NQS angular coordinates yA are not constant along the outgoing null rays, unlike the Bondi-Sachs angular coordinates. Instead the coordinates A y display the analogue of a shift on the null hypersurfaces u = const. In addition, the NQS spheres (u,r) = const. are not the same as the Bondi spheres. The radiation content of the metric is contained in a shear vector describing this shift. This results in the description of the radiation in terms of a spin-weight 1 field, rather than the spin-weight 2 field associated with hAB in the Bondi-Sachs formalism. In both the Bondi-Sachs and NQS gauges, the independent gravitational data on a null hypersurface is the conformal part of its degenerate 3-metric. The Bondi-Sachs null data consists of hAB, which determines the intrinsic conformal metric of the null hypersurface. In the NQS case, hAB = qAB and the shear vector comprises the only non-trivial part of the conformal 3-metric. Both the Bondi-Sachs and NQS gauges can be arranged to coincide in the special case of shear-free Robinson-Trautman metrics [7120Jump To The Next Citation Point].

The formulation of Einstein’s equations in the NQS gauge is presented in [19], and the associated gauge freedom arising from (u,r) dependent rotation and boosts of the unit sphere is discussed in [20]. As in the PITT code, the main equations involve integrating a hierarchy of hypersurface equations along the radial null geodesics extending from the inner boundary to null infinity. In the NQS gauge the source terms for these radial ODE’s are rather simple when the unknowns are chosen to be the connection coefficients. However, as a price to pay for this simplicity, after the radial integrations are performed on each null hypersurface a first order elliptic equation must be solved on each r = const. cross-section to reconstruct the underlying metric.

3.5.2 Numerical methodology

The PITT code is an explicit second order finite difference evolution algorithm based upon retarded time steps on a uniform three-dimensional null coordinate grid. The straightforward numerical approach and the second order convergence of the finite difference equations has facilitated code development. The Canberra code uses an assortment of novel and elegant numerical methods. Most of these involve smoothing or filtering and have obvious advantage for removing short wavelength noise but would be unsuitable for modeling shocks.

Both codes require the ability to handle tensor fields and their derivatives on the sphere. Spherical coordinates and spherical harmonics are natural analytic tools for the description of radiation, but their implementation in computational work requires dealing with the impossibility of smoothly covering the sphere with a single coordinate grid. Polar coordinate singularities in axisymmetric systems can be regularized by standard tricks. In the absence of symmetry, these techniques do not generalize and would be especially prohibitive to develop for tensor fields.

A crucial ingredient of the PITT code is the eth-module [104] which incorporates a computational version of the Newman-Penrose eth-formalism [159]. The eth-module covers the sphere with two overlapping stereographic coordinate grids (North and South). It provides everywhere regular, second order accurate, finite difference expressions for tensor fields on the sphere and their covariant derivatives. The eth-calculus simplifies the underlying equations, avoids spurious coordinate singularities and allows accurate differentiation of tensor fields on the sphere in a computationally efficient and clean way. Its main weakness is the numerical noise introduced by interpolations (fourth order accurate) between the North and South patches. For parabolic or elliptic equations on the sphere, the finite difference approach of the eth-calculus would be less efficient than a spectral approach, but no parabolic or elliptic equations appear in the Bondi-Sachs evolution scheme.

The Canberra code handles fields on the sphere by means of a 3-fold representation: (i) as discretized functions on a spherical grid uniformly spaced in standard (h,f) coordinates, (ii) as fast-Fourier transforms with respect to (h,f) (based upon a smooth map of the torus onto the sphere), and (iii) as a spectral decomposition of scalar, vector, and tensor fields in terms of spin-weighted spherical harmonics. The grid values are used in carrying out nonlinear algebraic operations; the Fourier representation is used to calculate (h,f)-derivatives; and the spherical harmonic representation is used to solve global problems, such as the solution of the first order elliptic equation for the reconstruction of the metric, whose unique solution requires pinning down the l = 1 gauge freedom. The sizes of the grid and of the Fourier and spherical harmonic representations are coordinated. In practice, the spherical harmonic expansion is carried out to 15th order in l, but the resulting coefficients must then be projected into the l < 10 subspace in order to avoid inconsistencies between the spherical harmonic, grid, and Fourier representations.

The Canberra code solves the null hypersurface equations by combining an 8th order Runge-Kutta integration with a convolution spline to interpolate field values. The radial grid points are dynamically positioned to approximate ingoing null geodesics, a technique originally due to Goldwirth and Piran [95] to avoid the problems with a uniform r-grid near a horizon which arise from the degeneracy of an areal coordinate on a stationary horizon. The time evolution uses the method of lines with a fourth order Runge-Kutta integrator, which introduces further high frequency filtering.

3.5.3 Stability

PITT code
 
Analytic stability analysis of the finite difference equations has been crucial in the development of a stable evolution algorithm, subject to the standard Courant-Friedrichs-Lewy (CFL) condition for an explicit code. Linear stability analysis on Minkowski and Schwarzschild backgrounds showed that certain field variables must be represented on the half-grid [106Jump To The Next Citation Point42Jump To The Next Citation Point]. Nonlinear stability analysis was essential in revealing and curing a mode coupling instability that was not present in the original axisymmetric version of the code [31Jump To The Next Citation Point142Jump To The Next Citation Point]. This has led to a code whose stability persists even in the regime that the u-direction, along which the grid flows, becomes spacelike, such as outside the velocity of light cone in a rotating coordinate system. Severe tests were used to verify stability. In the linear regime, robust stability was established by imposing random initial data on the initial characteristic hypersurface and random constraint violating boundary data on an inner worldtube. (Robust stability was later adopted as one of the standardized tests for Cauchy codes [5Jump To The Next Citation Point].) The code ran stably for 10,000 grid crossing times under these conditions [42Jump To The Next Citation Point]. The PITT code was the first 3D general relativistic code to pass this robust stability test. The use of random data is only possible in sufficiently weak cases where effective energy terms quadratic in the field gradients are not dominant. Stability in the highly nonlinear regime was tested in two ways. Runs for a time of 60,000 M were carried out for a moving, distorted Schwarzschild black hole (of mass M), with the marginally trapped surface at the inner boundary tracked and its interior excised from the computational grid [102Jump To The Next Citation Point103]. This remains one of the longest simulations of a dynamic black hole carried out to date. Furthermore, the scattering of a gravitational wave off a Schwarzschild black hole was successfully carried out in the extreme nonlinear regime where the backscattered Bondi news was as large as N = 400 (in dimensionless geometric units) [31Jump To The Next Citation Point], showing that the code can cope with the enormous power output N 2c5/G ~~ 1060 W in conventional units. This exceeds the power that would be produced if, in 1 second, the entire galaxy were converted to gravitational radiation.

Canberra code
 
Analytic stability analysis of the underlying finite difference equations is impractical because of the extensive mix of spectral techniques, higher order methods, and splines. Although there is no clear-cut CFL limit on the code, stability tests show that there is a limit on the time step. The damping of high frequency modes due to the implicit filtering would be expected to suppress numerical instability, but the stability of the Canberra code is nevertheless subject to two qualifications [222324]: (i) At late times (less than 100 M), the evolution terminates as it approaches an event horizon, apparently because of a breakdown of the NQS gauge condition, although an analysis of how and why this should occur has not yet been given. (ii) Numerical instabilities arise from dynamic inner boundary conditions and restrict the inner boundary to a fixed Schwarzschild horizon. Tests in the extreme nonlinear regime were not reported.

3.5.4 Accuracy

PITT code
 
Second order accuracy has been verified in an extensive number of testbeds [42Jump To The Next Citation Point31Jump To The Next Citation Point102Jump To The Next Citation Point228Jump To The Next Citation Point229Jump To The Next Citation Point], including new exact solutions specifically constructed in null coordinates for the purpose of convergence tests:

Canberra code
 
The complexity of the algorithm and NQS gauge makes it problematic to establish accuracy by direct means. Exact solutions do not provide an effective convergence check, because the Schwarzschild solution is trivial in the NQS gauge and other known solutions in this gauge require dynamic inner boundary conditions which destabilize the present version of the code. Convergence to linearized solutions is a possible check but has not yet been performed. Instead indirect tests by means of geometric consistency and partial convergence tests are used to calibrate accuracy. The consistency tests were based on the constraint equations, which are not enforced during null evolution except at the inner boundary. The balance between mass loss and radiation flux through I+ is a global consequence of these constraints. No appreciable growth of the constraints was noticeable until within 5 M of the final breakdown of the code. In weak field tests where angular resolution does not dominate the error, partial convergence tests based upon varying the radial grid size verify the 8th order convergence in the shear expected from the Runge-Kutta integration and splines. When the radial source of error is small, reduced error with smaller time step can also be discerned.

In practical runs, the major source of inaccuracy is the spherical harmonic resolution, which was restricted to l < 15 by hardware limitations. Truncation of the spherical harmonic expansion has the effect of modifying the equations to a system for which the constraints are no longer satisfied. The relative error in the constraints is 10- 3 for strong field simulations [21Jump To The Next Citation Point].

3.5.5 First versus second differential order

The PITT code was originally formulated in the second differential form of Equations (22View Equation, 23View Equation, 24View Equation, 25View Equation), which in the spin-weighted version leads to an economical number of 2 real and 2 complex variables. Subsequently, the variable

QA = r2e- 2bhAB UB,r (26)
was introduced to reduce Equation (23View Equation) to two first order radial equations, which simplified the startup procedure at the initial boundary. Although the resulting code has been verified to be stable and second order accurate, its application to increasingly difficult problems involving strong fields, and gradients have led to numerical errors that make important physical effects hard to measure. In particular, in initial attempts to simulate a white hole fission, Gómez [96Jump To The Next Citation Point] encountered an oscillatory error pattern in the angular directions near the time of fission. The origin of the problem was tracked to numerical error of an oscillatory nature introduced by o2 terms in the hypersurface and evolution equations. Gómez’ solution was to remove the offending second angular derivatives by introducing additional variables and reducing the system to first differential order in the angular directions. This suppressed the oscillatory mode and subsequently improved performance in the simulation of the white hole fission problem [98Jump To The Next Citation Point] (see Section 3.7.2).

This success opens the issue of whether a completely first differential order code might perform even better, as has been proposed by Gómez and Frittelli [97]. They gave a first order quasi-linear formulation of the Bondi system which, at the analytic level, obeys a standard uniqueness and existence theorem (extending previous work for the linearized case [89]); and they point out, at the numerical level, that a first order code could also benefit from the applicability of standard numerical techniques. This is an important issue which is not simple to resolve without code comparison. The part of the code in which the o2 operator introduced the oscillatory error mode was not identified in [96], i.e. whether it originated in the inner boundary treatment or in the interpolations between stereographic patches where second derivatives might be troublesome. There are other possible ways to remove the oscillatory angular modes, such as adding angular dissipation or using more accurate methods of patching the sphere. The current finite difference algorithm only introduces numerical dissipation in the radial direction [142]. The economy of variables in the original Bondi scheme should not be abandoned without further tests and investigation.

3.5.6 Nonlinear scattering off a Schwarzschild black hole

A natural physical application of a characteristic evolution code is the nonlinear version of the classic problem of scattering off a Schwarzschild black hole, first solved perturbatively by Price [175Jump To The Next Citation Point]. Here the inner worldtube for the characteristic initial value problem consists of the ingoing branch of the r = 2m hypersurface (the past horizon), where Schwarzschild data are prescribed. The nonlinear problem of a gravitational wave scattering off a Schwarzschild black hole is then posed in terms of data on an outgoing null cone which describe an incoming pulse with compact support. Part of the energy of this pulse falls into the black hole and part is backscattered to I+. This problem has been investigated using both the PITT and Canberra codes.

The Pittsburgh group studied the backscattered waveform (described by the Bondi news function) as a function of incoming pulse amplitude. The computational eth-module smoothly handled the complicated time dependent transformation between the non-inertial computational frame at I+ and the inertial (Bondi) frame necessary to obtain the standard “plus” and “cross” polarization modes. In the perturbative regime, the news corresponds to the backscattering of the incoming pulse off the effective Schwarzschild potential. When the energy of the pulse is no larger than the central Schwarzschild mass, the backscattered waveform still depends roughly linearly on the amplitude of the incoming pulse. However, for very high amplitudes the waveform behaves quite differently. Its amplitude is greater than that predicted by linear scaling and its shape drastically changes and exhibits extra oscillations. In this very high amplitude case, the mass of the system is completely dominated by the incoming pulse, which essentially backscatters off itself in a nonlinear way.

The Canberra code was used to study the change in Bondi mass due to the radiation [21]. The Hawking mass MH(u, r) was calculated as a function of radius and retarded time, with the Bondi mass MB(u) then obtained by taking the limit r-- > oo. The limit had good numerical behavior. For a strong initial pulse with l = 4 angular dependence, in a run from u = 0 to u = 70 (in units where the interior Schwarzschild mass is 1), the Bondi mass dropped from 1.8 to 1.00002, showing that almost half of the initial energy of the system was backscattered and that a surprisingly negligible amount of energy fell into the black hole. A possible explanation is that the truncation of the spherical harmonic expansion cuts off wavelengths small enough to effectively penetrate the horizon. The Bondi mass decreased monotonically in time, as necessary theoretically, but its rate of change exhibited an interesting pulsing behavior whose time scale could not be obviously explained in terms of quasinormal oscillations. The Bondi mass loss formula was confirmed with relative error of less than 10-3. This is impressive accuracy considering the potential sources of numerical error introduced by taking the limit of the Hawking mass with limited resolution. The code was also used to study the appearance of logarithmic terms in the asymptotic expansion of the Weyl tensor [25]. In addition, the Canberra group studied the effect of the initial pulse amplitude on the waveform of the backscattered radiation, but did not extend their study to the very high amplitude regime in which qualitatively interesting nonlinear effects occur.

3.5.7 Black hole in a box

The PITT code has also been implemented to evolve along an advanced time foliation by ingoing null cones, with data given on a worldtube at their outer boundary and on the initial ingoing null cone. The code was used to evolve a black hole in the region interior to the worldtube by implementing a horizon finder to locate the marginally trapped surface (MTS) on the ingoing cones and excising its singular interior [105Jump To The Next Citation Point]. The code tracks the motion of the MTS and measures its area during the evolution. It was used to simulate a distorted “black hole in a box” [102Jump To The Next Citation Point]. Data at the outer worldtube was induced from a Schwarzschild or Kerr spacetime but the worldtube was allowed to move relative to the stationary trajectories; i.e. with respect to the grid the worldtube is fixed but the black hole moves inside it. The initial null data consisted of a pulse of radiation which subsequently travels outward to the worldtube where it reflects back toward the black hole. The approach of the system to equilibrium was monitored by the area of the MTS, which also equals its Hawking mass. When the worldtube is stationary (static or rotating in place), the distorted black hole inside evolved to equilibrium with the boundary. A boost or other motion of the worldtube with respect to the black hole did not affect this result. The marginally trapped surface always reached equilibrium with the outer boundary, confirming that the motion of the boundary was “pure gauge”.

The code runs “forever” even when the worldtube wobbles with respect to the black hole to produce artificial periodic time dependence. An initially distorted, wobbling black hole was evolved for a time of 60,000 M, longer by orders of magnitude than permitted by the stability of other existing 3D black hole codes at the time. This exceptional performance opens a promising new approach to handle the inner boundary condition for Cauchy evolution of black holes by the matching methods reviewed in Section 4.

Note that setting the pulse to zero is equivalent to prescribing shear free data on the initial null cone. Combined with Schwarzschild boundary data on the outer world tube, this would be complete data for a Schwarzschild space time. However, the evolution of such shear free null data combined with Kerr boundary data would have an initial transient phase before settling down to a Kerr black hole. This is because the twist of the shear-free Kerr null congruence implies that Kerr data specified on a null hypersurface are not generally shear free. The event horizon is an exception but Kerr null data on other null hypersurfaces have not been cast in explicit analytic form. This makes the Kerr spacetime an awkward testbed for characteristic codes. (Curiously, Kerr data on a null hypersurface with a conical type singularity do take a simple analytic form, although unsuitable for numerical evolution [79].) Using some intermediate analytic results of Israel and Pretorius [173], Venter and Bishop [219] have recently constructed a numerical algorithm for transforming the Kerr solution into Bondi coordinates and in that way provide the necessary null data numerically.


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