Constrained scheme for the Einstein equations based on the Dirac gauge and spherical coordinates

We propose a new formulation for $3+1$ numerical relativity, based on a constrained scheme and a generalization of Dirac gauge to spherical coordinates. This is made possible thanks to the introduction of a flat 3-metric on the spatial hypersurfaces $t=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$, which corresponds to the asymptotic structure of the physical 3-metric induced by the spacetime metric. Thanks to the joint use of Dirac gauge, maximal slicing and spherical components of tensor fields, the ten Einstein equations are reduced to a system of five quasilinear elliptic equations (including the Hamiltonian and momentum constraints) coupled to two quasilinear scalar wave equations. The remaining 3 degrees of freedom are fixed by the Dirac gauge. Indeed this gauge allows a direct computation of the spherical components of the conformal metric from the two scalar potentials which obey the wave equations. We present some numerical evolution of 3D gravitational wave spacetimes which demonstrates the stability of the proposed scheme.

Figure 1

Evolution of the $h^{\phi \phi }$ component of $\mathit{h}$ in the plane $\theta =\pi /2$, between $t=0$ (upper left) and $t=8r_0$ (lower right). The various snapshots are separated by a constant time interval $\Delta t=r_0$. The size of the depicted square is $16r_0$, so that the wave extraction surface at $R_{\mathrm{e}\mathrm{x}\mathrm{t}}=8r_0$ is given by the circle inscribed in this square.

Figure 2

Evolution of the ADM mass for three different computational settings, corresponding to different values of the time step $\delta t$ and the wave extraction radius $R_{\mathrm{e}\mathrm{x}\mathrm{t}}$.

Figure 3

Evolution of the ADM mass (logarithmic scale, contrary to Fig. 2) for two different values of the wave extraction radius $R_{\mathrm{e}\mathrm{x}\mathrm{t}}$.

Figure 4

Relative error $\epsilon$ [Eq. (175)] on the time derivative of the conformal factor $\Psi$ in the central domain ($r\le r_0$).

Figure 5

Evolution of the maximum of absolute value of the potential $\chi$ [Eq. (140)] for the long term run.