Past and future gauge in numerical relativity

Maurice H P M van Putten

MIT, Cambridge, MA 02139-4307, USA

Received 25 October 2001
Published 6 March 2002

PACS numbers: 0425D, 0460N

1. Introduction

The initial value problem for general relativity is receiving much attention in the prediction of waveforms from candidate sources for the upcoming gravitational wave detectors LIGO/VIRGO [1, 2]. The structure of gravitational waves has recently been elucidated in new hyperbolic formulations of general relativity (see, e.g., [3] for references), which holds promise for accurate integration schemes. Long-time integrations also require accurate conservation of the associated elliptic constraints, representing energy and momentum conservation [4]. While analytically these constraints are conserved under dynamical evolution, the nonlinear nature of the equation typically tends to introduce some numerical departure from exact conservation.

Gauge choice in numerical relativity involves slicing spacetime into space-like hypersurfaces. Data from one hypersurface are evolved numerically onto the next, for example using a hyperbolic formulation. Algebraic slicing is particularly illustrative, which considers prescribed lapse and shift functions N_a = g_ta for the metric tensor g_ab.

Here, we propose a new approach for numerical relativity with preservation of constraints using a choice of gauge in the future and a dynamical gauge in the past. With the prescribed gauge and dynamical space-like components of the metric in the future, this obtains a complete system of ten equations in ten unknowns.

The approach presented recognizes two types of constraints in Hamiltonian approaches within the context of the underlying four-covariant theory of general relativity. Recall that Hamiltonian formulations assign the 3-metric h_ij(t) and its canonical momentum π_ij(t) as dynamical variables to a hypersurface of constant coordinate time t. This implicitly carries a geometric constraint, in having h_ij(t) as exact projections of the four-covariant metric g_ab onto the hypersurfaces of constant t. In addition, general relativity obeys energy-momentum conservations by the Bianchi identity, which translates into the Gauss-Codacci relations on the same hypersurfaces. These observations may be contrasted with the nonlinear wave equations of [6], which describe the four-covariant hyperbolic evolution of the Riemann-Cartan connections ω_μab first, with subsequent slicing of spacetime for the purpose of numerical implementation. In the continuum limit, both the projections and energy-momentum constraints are exact and consistent. This need not carry over to the discrete case, given the nonlinear nature of the Einstein equations. This may be illustrated by the observed violations of energy-momentum constraints in present numerical simulations [4].

A discretization of the Einstein equations introduces finite deviations from the continuum limit. For the purpose of numerical relativity, these deviations are acceptable, provided they permit stable evolution and convergence upon refinement of the discretization. The question therefore is: where are discretizations allowed to introduce deviations from the continuum theory? The above suggests one should consider seeking a trade-off between the geometric constraint of exact projections and energy-momentum conservation. This points towards a numerical scheme in which the past slicing is adjusted within the chosen discretization level, so as to satisfy energy-momentum conservation identically for a given definition of the discretized Einstein tensor. This gives a complete system of evolution equations, as follows from simple counting: the contracted Bianchi identity shows that the Einstein equations provide six constraints on the second time derivatives of the 3-metric h_ij and four constraints on the first time derivatives of the lapse and shift functions N_a = g_at. The first tells us that the discretized Einstein equations live on three time slices, whereas the latter indicates that the lapse and shift functions are constraints on a pair of them, e.g. the first (past) and the third (future) time-slice. Thus, choosing a gauge for one of these leaves the gauge on the other determined self-consistently by the Einstein equations. It seems natural to maintain control over the future gauge, i.e. the future gauge is the choice of the user. This leaves the past gauge as a dynamical variable, to be determined implicitly during numerical evolution. This can be achieved using Newton's method.

Non-exact projections naturally permit an uncertainty between the 3-metric and its canonical momentum within the underlying context of a four-covariant theory, i.e. also in regards to the association with the hypersurface at hand. In the covariant approach of [6], this would thus reflect an uncertainty in the tetrad elements, which define the projection, and their connections. This points towards a potential connection to quantum gravity. Indeed, soon after this work was proposed [5], the author learned of a very interesting independent discussion on the problem of consistent discretizations in this context [7].

2. A discretized initial value problem

We illustrate our present approach on the vacuum Einstein equations, described by the vanishing Ricci tensor

The Ricci tensor R_ab is a second-order expression in the metric g_ab. Hence, (1) defines a relationship between metric data on a triple of time slices :

Here, is derived from the Riemann tensor expressed in coordinate form in terms of the Christoffel symbols

This expression can be discretized by finite differencing on a triple of time slices with preservation of the quasi-linear second-order structure of R_ab.

Algebraic gauge-fixing takes the form of specifying the components N_a = g_ta in coordinates with t = x¹ time like. A gauge-choice on a triple of time-slices, therefore, amounts to a choice of . Recall that this gauge-choice in the metric arises explicitly in the Gauss-Codacci relations for energy-momentum conservation. The components h_ij = g_ij, where i, j =  2, 3, 4 refer to projections of the metric into the time-slice t =  const, which describe the dynamical part of the metric. The combination (h_ij, N_a) reflects the mixed hyperbolic-elliptic structure in numerical relativity and (1) represents ten evolution equations in these variables on a triple of time slices.

In algebraic gauge-fixing, we prescribe as a future gauge in computing on a future hypersurface t = t_n + 1 from the data at present and past hypersurfaces t = t_n - 1 and t = t_n. We propose closure by reintroducing as dynamical gauge in the past, leaving fixed. Combined, this defines an advanced hyperbolic-retarded elliptic evolution of the metric. The paritioning of the metric in past and future variables as

thus obtains ten dynamical variables in the ten equations

Here the dots refer to the remaining data , which are kept fixed while solving for . Thus, equation (5) takes into account all ten Einstein equations with no reduction of variables. Time-stepping by (5) evolves the metric into the future with dynamical gauge in the past, in an effort to satisfy energy-momentum conservation within the definition of the discretized Ricci tensor. Because (5) comprises derivatives of N_a only to first order in time, numerically through the data and , we anticipate that the evolution of N_a is of first order in the t-discretization Δt. This introduces non-exactness in as projections of g_ab on t - Δt to within the same order of accuracy. It may result in a first order drift in the t-labelling of the hypersurfaces - permitted by coordinate invariance.

The approach presented can be illustrated on a polarized Gowdy wave. Gowdy cosmologies are an extensively-studied class of universes with compact space-like hypersurfaces with two Killing vectors ∂_σ and ∂_δ. With cyclic boundary conditions, the space-like hypersurfaces are homeomorphic to the 3-torus as a model universe collapsing into a singularity. The associated line-element is (see, e.g. [8])

where λ = λ(τ, θ) and dΣ denotes the surface element in the space supported by the two Killing vectors. Polarized Gowdy waves form a special case which permits a reduction to

Here P satisfies a linear wave equation ; a long wavelength solution is

where Y₀ is the Bessel function of the second kind of order zero. This leaves

A spectrally accurate numerical integration is described in [9].

The implicit equation (5) for the dynamical variables has been implemented numerically. We have done so by solving for the all ten components using Newton iterations on these variables. This procedure uses a numerical evaluation of the Jacobian

where the capital indices A, B =  1, 2,  ..., 10 refer to the labelling

The Ricci tensor (3) has been implemented by second-order finite differencing, such that it remains quasi-linear in the second derivatives. In particular, the Christoffel symbol

is obtained by symmetric finite differencing on the metric components, and is itself differentiated by the product rule following individual numerical differentiations of g^ab and (g_cb,a + g_ac,b - g_ab,c). The choice of future gauge is provided by the components

of the analytical line element (6)-(9), which facilitates error analysis by direct comparison of the numerical results with the analytic expression for the line element (6). It will be appreciated that, in principle, other choices of can be made.

Figure 1 shows numerical results for the evolution of initial data on 0 ≤τ≤ 4. The results show that all Einstein equations in the form of R_ab =  0 are satisfied with arbitrary precision, while the metric components are solved accurately to within 1%. The asymptotic behaviour of the implicit corrections to the lapse functions are shown in figure 2. Note that these corrections are finite to first order in Δt - a testimonial to the appearance of the lapse function in the Einstein equations to first order in time.

Figure 2. The self-consistent corrections on the slicing t = tn + 1, introduced by the difference between the past gauge to the hypersurface t = tn + 2 and the earlier future gauge to the hypersurface t = tn. The three curves refer to different discretizations m1 =  16, 32 and 64 points with, respectively, m2 =  256, 512 and 1024 time steps. These similar results for various discretizations indicate asymptotic behaviour consistent with the first order appearance of the lapse function in the Einstein equations. A first-order accuracy in lapse introduces a commensurate offset in slicing or, equivalently, an offset in the coordinate t.

In summary, a dynamical gauge in the past gives a complete number of ten degrees of freedom in evolving to a future hypersurface, permitting all ten Einstein equations to be satisfied with arbitrary precision. The Einstein equations are hereby satisfied numerically on a triple of hypersurfaces of past, present and future within the definition of the discretized Riemann tensor. The simulation of a nonlinear one-dimensional Gowdy wave by implicit time-stepping according to the ten discretized vacuum Einstein equations (5), serves to illustrate a numerical implementation. The approach presented ensures conservation of energy and momentum, within the definition of the discretization used. Satisfying these constraints is generally a necessary condition for stability in long-time integrations. It is presently not known if satisfying these constraints is a sufficient condition for stability. It would be of interest to study this method in evolving Schwarzschild black holes with singularity-avoiding slicings of spacetime, and its behaviour in the presence of outgoing radiative boundary conditions. More generally, it would be of interest to consider this approach in higher dimensions, including a self-consistent integration of any of the modern hyperbolic formulations and efficient elliptic solvers.

Acknowledgments

This research is partially supported by NASA Grant 5-7012 and an MIT C E Reed Award. The author thanks P Hoefflich for drawing attention to the asymptotic behaviour of slicing and L Wen for stimulating discussions.

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