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Gauge wave testbed

These tests look at the ability of formulations to handle gauge dynamics. This is done by considering flat Minkowski space in a slicing where the 3-metric $ h_{ij}$ is time dependent. Such gauge waves have been considered before, notably by [Alcubierre(1997)] and [Calabrese et al.(2002)Calabrese, Pullin, Sarbach, and Tiglio].

We have considered different profiles for gauge waves. For the purpose of this paper we focus on the case of a propagating gauge sine wave. This specific test was used in comparing systems with different hyperbolicity properties in [Calabrese et al.(2002)Calabrese, Pullin, Sarbach, and Tiglio].

The 4-metric is obtained from the Minkowski metric $ ds^2 =- d\hat t^2+d\hat x^2+d\hat y^2+d\hat z^2$ by the coordinate transformation

\begin{displaymath}\begin{array}[c]{r c l} \hat t&=& t + \frac {Ad}{4\pi}\cos \l... ...(x - t)}{d} \right), \\ \hat y&=& y, \\ \hat z&=& z \end{array}\end{displaymath} (1)

where $ d$ is the size of the evolution domain. This leads to the 4-metric

$\displaystyle ds^2=-H dt^2 +Hdx^2+dy^2+dz^2,$ (2)

where

$\displaystyle H = H(x-t)=1 + A \sin \left( \frac{2 \pi (x - t)}{d} \right),$ (3)

which describes a sinusoidal gauge wave of amplitude $ A$ propagating along the $ x$-axis. The extrinsic curvature is given by
$\displaystyle K_{xx}$ $\displaystyle =$ $\displaystyle -\frac{\pi A}{d} \frac{ \cos \left( \frac{2 \pi (x - t)}{d} \right) }{ \sqrt{1 + A \sin \left( \frac{2 \pi (x - t)}{d} \right) } },$ (4)
$\displaystyle K_{ij}$ $\displaystyle =$ $\displaystyle 0 \qquad \textrm{ otherwise}.$ (5)

Since this wave propagates along the $ x$-axis and all derivatives are zero in the $ y$ and $ z$ directions, the problem is essentially one dimensional and can simplify the system dramatically for certain formulations, as there is no finite-difference error in the orthogonal directions. A simple coordinate transformation causes the wave to propagate along a diagonal:

$\displaystyle x = \frac{1}{\sqrt{2}}(x^\prime - y^\prime), \qquad y = \frac{1}{\sqrt{2}}(x^\prime + y^\prime)$ (6)

The resulting metric is a function of

$\displaystyle \sin \left( \frac{2 \pi (x' - y' - t' \sqrt{2})}{d'} \right), \quad \textrm{where} \quad d' = d \sqrt{2} \, .$ (7)

Setting $ d'$ to the size of the evolution domain in the $ x'$ and $ y'$ directions gives periodicity along those directions. This test should be run in both axis-aligned and diagonal form.

As any evolution is a pure gauge effect, care must be taken in the choice of lapse and shift to allow a direct comparison between formulations. For example, the Bona-Masso family of gauges [Bona et al.(1995)Bona, Massó, Seidel, and Stela]

$\displaystyle \partial_t \alpha = - \alpha^2 f(\alpha) K$ (8)

will propagate a gauge wave in direction $ x^i$ with speed $ \alpha \sqrt{f g^{ii}}$, which can be varied. In contrast, maximal slicing would freeze any gauge pulse, stopping it from propagating.

Note that the time coordinate $ t$ in the metric (2) is harmonic, which corresponds to

$\displaystyle f(\alpha) = 1.$ (9)

This gauge condition can easily be integrated to $ \alpha = h(x^i) \sqrt(\det g)$. Also, in this case the gauge speed is simply the speed of light. To ensure that we can directly compare as many formulations as possible we choose harmonic slicing to carry out the evolution for the gauge wave test. Different formulations and codes may demand different implementations of harmonic time slicing for optimal performance. The test should thus be implemented in a way that is analytically compatible with the metric (2), which still leaves significant freedom.

We run the gauge wave with amplitudes $ A=10^{-1}$ and $ A=10^{-2}$. We have found that smaller amplitudes are quite simple for all codes whilst larger amplitudes can cause numerical error to trigger gauge pathologies, such as the formation of coordinate singularities, very quickly.

The specified wave has wavelength $ d=1$ in the 1D simulation and wavelength $ d'=\sqrt{2}$ in the diagonal simulation. We find that 50 grid points are sufficient to resolve the profile and therefore make the following choices for the computational grid:

  • Simulation domain:
    1D: $ \quad x \in [-0.5; +0.5],$ $ \quad y = 0,$ $ \quad z = 0,$ $ \quad d=1$
    diagonal: $ \quad x \in [-0.5; +0.5],$ $ \quad y \in [-0.5; +0.5], $ $ \quad z = 0,$ $ \quad d'=\sqrt{2}$
  • Grid: $ x^i = -0.5 + (n-\frac{1}{2}) dx, \quad n=1\ldots 50\rho, \quad dx=1/(50\rho), \quad \rho \in \mathbb{Z}$
  • Time step: $ dt = dx/4 = 0.005 / \rho$
The 1D evolution is carried out for $ T=1000$ crossing times, i.e.  $ 2\times10^5\rho$ time steps (or until the code crashes), with output every 10 crossing times. The 2D diagonal runs are carried out for $ T=100$, with output every crossing time. We run using $ \rho=1,2,4$.

We output the $ L_{2}$, the maxima and minima, and profiles along the $ x$-axis through the center of the grid of $ g_{xx}$, $ \alpha $, $ tr(K)$, the Hamiltonian constraint and any other independent constraints that arise in a nontrivial way in a particular formulation. We also calculate the $ L_{2}$-norm of the difference from the exact solution for $ g_{xx}$ and calculate the convergence factor. Figure 1 provides examples of test output obtained with a standard ADM code. These results do not show a problem with ADM, but illustrate the expansion of the numerical space-time. This is due to the reasons mentioned in section III, stating that any non trivial space-time with $ T^3$ topology must have a singularity either in the past or in the future. The behavior of the $ L_2$-norm of $ g_{xx}$ indicates how the volume element of the space behaves. It is also seen that the evolution is convergent for a long time, nevertheless the higher order terms cause the deviation from convergence.

Figure 1: Results for the 1D gauge wave using the standard ADM formulation. The left hand plot shows $ \vert\vert g_{xx}\vert\vert _2$; the central plot shows the maximum of the lapse $ \alpha $, and the right hand plot shows the convergence factor of $ g_{xx}$ calculated using the three resolutions. A value of $ 4$ would mark exact second order convergence.
\includegraphics[width=14pc]{gaugewave_1D_gxx_nm2_no-noise.eps} \includegraphics[width=14pc]{gaugewave_1D_lapse_max_no-noise.eps} \includegraphics[width=14pc]{gaugewave_1D_gxx_nm2_convergence.eps}

Bibliography

Alcubierre(1997)
M. Alcubierre, Phys. Rev. D 55, 5981 (1997), gr-qc/9609015.

Calabrese et al.(2002)Calabrese, Pullin, Sarbach, and Tiglio
G. Calabrese, J. Pullin, O. Sarbach, and M. Tiglio, Phys. Rev. D 66, 041501 (2002), gr-qc/0207018.

Bona et al.(1995)Bona, Massó, Seidel, and Stela
C. Bona, J. Massó, E. Seidel, and J. Stela, Phys. Rev. Lett. 75, 600 (1995), gr-qc/9412071.

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$Date: 2004/01/08 17:45:06 $