Initial data for Einstein's equations with superposed gravitational waves

A method is presented to construct initial data for Einstein's equations as a superposition of a gravitational wave perturbation on an arbitrary stationary background spacetime. The method combines the conformal thin sandwich formalism with linear gravitational waves, and allows detailed control over characteristics of the superposed gravitational wave like shape, location and propagation direction. It is furthermore fully covariant with respect to spatial coordinate changes and allows for very large amplitude of the gravitational wave.

Figure 1

Domain decomposition in $R^3$. A cube covers the central region which is not covered by the spherical shells.

Figure 2

Constraint violation of linear gravitational wave in flat background prior to solving the constraints.

Figure 3

Convergence of the elliptic solver for different amplitudes $A$. Plotted is the residual in the Hamiltonian constraint (root mean square) versus the number of radial basis functions in the middle spherical shell.

Figure 4

Convergence of the elliptic solver for different amplitudes $A$. Plotted is the difference of the ADM energy to the next higher resolution solution versus the number of radial basis functions in the middle spherical shell.

Figure 5

ADM energy of an ingoing Gaussian pulse in flat space. The dashed line indicates the low-amplitude quadratic behavior.

Figure 6

Cuts through the equatorial plane of the $A=0.3$ data set of Fig. 5. The large plot shows lapse and conformal factor, the inset shows the scalar curvature of the 3-metric.

Figure 7

Black hole with superposed gravitational wave: Residual of the Hamiltonian constraint versus radial number of basis functions in middle spherical shell.

Figure 8

Black hole with superposed gravitational wave.

Figure 9

Apparent horizon mass during an evolution of a perturbed black hole spacetime. The dashed line indicates $E_{\mathrm{A}\mathrm{D}\mathrm{M}}$ as computed from the initial data set.