Improved constrained scheme for the Einstein equations: An approach to the uniqueness issue

Uniqueness problems in the elliptic sector of constrained formulations of Einstein equations have a dramatic effect on the physical validity of some numerical solutions, for instance, when calculating the spacetime of very compact stars or nascent black holes. The fully constrained formulation (FCF) proposed by Bonazzola, Gourgoulhon, Grandclément, and Novak is one of these formulations. It contains, as a particular case, the approximation of the conformal flatness condition (CFC) which, in the last ten years, has been used in many astrophysical applications. The elliptic part of the FCF basically shares the same differential operators as the elliptic equations in the CFC scheme. We present here a reformulation of the elliptic sector of the CFC that has the fundamental property of overcoming the local uniqueness problems. The correct behavior of our new formulation is confirmed by means of a battery of numerical simulations. Finally, we extend these ideas to the FCF, complementing the mathematical analysis carried out in previous studies.

Figure 1

Time evolution of the central lapse $N_{\mathrm{c}}$ (top panel) and the ADM mass $M_{\mathrm{ADM}}$ (bottom panel) for a collapsing packet of gravitational waves, using the integration scheme proposed in 23. The unit of $t$ is given by the initial width of the wave packet.

Figure 2

Time evolution of the central rest-mass density ${\rho }_{\mathrm{c}}$ (top panel) and the central lapse $N_{\mathrm{c}}$ (bottom panel) for the migration of the unstable neutron star model, SU, to the stable branch, with either a polytropic (solid lines) or an ideal gas (dashed lines) equation of state. The dotted horizontal lines mark the values of ${\rho }_{\mathrm{c}}$ and $N_{\mathrm{c}}$ for the SS equilibrium configuration from the stable branch with the same baryon mass $M_{\mathrm{b}}$ as the SU model, while the dash-dotted lines are obtained from a series of equilibrium models where mass shedding, like in the migration model with an ideal gas equation of state, is taken into account. In the inset the baryon mass $M_{\mathrm{b}}$ versus the ${\rho }_{\mathrm{c}}$ relation for this model setup is displayed. The SU (the initial model) and SS (the final state for a polytropic equation of state) models as well as the final state for an ideal gas equation of state are marked. The arrows symbolize the respective migration paths.

Figure 3

Collapse to a black hole for the spherical model SU, and the rotating models D1 and D4. The top panel shows the time evolution of the central lapse $N_{\mathrm{c}}$ (thin lines) and the central rest-mass density ${\rho }_{\mathrm{c}}$ relative to the initial value ${\rho }_{\mathrm{c},0}$ (thick lines). The bottom panel shows the time evolution of the apparent horizon radius $r_{\mathrm{AH},\mathrm{e}}$ in the equatorial plane (thin lines) and the rest mass $M_{\mathrm{outside}\mathrm{AH}}$ remaining outside the apparent horizon relative to the total rest mass $M$ (thick lines). The dashed vertical lines mark the time when the apparent horizon first appears. If the axes of the lower panel were exchanged, the resulting plot would resemble the typical spacetime diagram of a star collapsing to a black hole.

Figure 4

Isocontours of the rest-mass density for model D4 after the apparent horizon first appears at $t=129.9$. The dashed line shows the location of the apparent horizon.

Figure 5

Consistency of the approximation for rotating neutron star models. In the top panel $\max |{\widehat{A}}_{\mathrm{TT}}^{ij}/{\widehat{A}}^{ij}|$ for the FCF (solid line) and the CFC (dashed line) as well as the maximum deviation from conformal flatness $\max |h^{ij}|$ for the FCF (dash-dotted line) are plotted against the ellipticity $e$. The bottom panel shows the absolute difference $|N_{\mathrm{c},\mathrm{CFC}}-N_{\mathrm{c}}|$ in the central value of the lapse between the CFC and FCF (solid line) and the absolute difference $|N_{\mathrm{c},\mathrm{CFC}}-N_{\mathrm{c},{\mathrm{CFC}}^{\prime }}|$ between the regular CFC and the CFC neglecting ${\widehat{A}}_{\mathrm{TT}}^{ij}$ in Eq. (29) (dashed line). The Kepler limit is marked by vertical dotted lines, while the slanted dotted lines represent the order of accuracy with respect to powers of $e$.