Comparing Full General Relativity with the Conformally Flat Approximation in Rotating Supernova Core Collapse
C.D. Ott, H. Dimmelmeier
Introduction:
The task of numerical relativity is to find numerical solutions of the Einstein equations, which are the equations governing the influence of gravity on an astrophysical system like ordinary stars, galaxies, neutron stars, black holes, or the entire universe. On the one hand, these equations are mathematically very complex, but additionally their numerical solution is also often hampered by instability problems. Therefore, in the past decades several approximation schemes have been introduced to both reduce the complexity of the Einstein equations and to circumvent the problems related to numerical instabilities.
One such approximative approach is the so-called conformal flatness condition (CFC) or Isenberg-Wilson-Mathews approximation [Wilson, et al., 1996], which we have utilized in numerical simulations of general relativistic rotating core collapse with a simple equation of state [Dimmelmeier, et al., 2001, Dimmelmeier, et al., 2002a, Dimmelmeier, et al., 2002b] and also advanced microphysics [Ott, et al., 2006b, Dimmelmeier, et al., 2007], as well as an investigation of pulsations of rotating neutron stars [Dimmelmeier, et al., 2006].
In CFC the Einstein equations are simplified by assuming conformal flatness for the three-metric γ_{i j}:
γ_{i j}^{CFC} = ^{φ 4} γ_{i j}^{flat},
where φ is the conformal factor and γ_{i j}^{flat} is the flat three-metric. Apart from reducing the complexity of the hydrodynamic and metric equations, this approach also exhibits numerical stability even for long evolution times, as it solves all constraint equations and thus cannot violate them by definition.
The up-to-now most successful formulation of the exact Einstein equations for numerical relativity is the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation, also known as Nakamura-Oohara-Kojima (NOK) formalism. This way of reformulating the Einstein equations has been known for several years, but only recently new developments for gauge conditions paved the way for its application to simulations of rotating supernova core collapse.
In order to quantitatively estimate how accurate the CFC approximation is compared to other approximation schemes or to solving the exact Einstein equations (e.g. by using the BSSN formulation), several attempts have been made. An early study [Shibata and Sekiguchi, 2004], which was limited to only a few models of rotating supernova core collapse, has demonstrated that the results using either CFC or BSSN agree rather well in this astrophysical scenario. A different approach was undertaken by comparing the CFC approximation to an extension called CFC+ [Cerda, et al., 2005], which again showed the outstanding quality of CFC, this time also for rapidly rotating cold neutron stars, which can attain a higher compactness (and are thus subject to stronger gravity) than the proto-neutron stars in supernova core collapse.
Simulations:
To unambiguously establish the excellent performance of the CFC approximation, we have now made a direct comparison of two codes with different grid setups, coordinate choices, gauge conditions, and mesh refinement prescriptions, with one of the codes using the CFC approxiations for gravity (CoCoNuT), and the other one utilizing the BSSN scheme (Cactus-Whisky) [Ott, et al., 2006a, Ott, et al., 2006b, Ott, et al., 2006c].
The astrophysical models we choose are collapsing stellar cores in a supernova, which lead to the formation of a rotating proto-neutron star. For the description of the matter we have either used a simple analytic (so-called hybrid) equation of state, or a more realistic treatment of microphysics based on a tabulated non-zero temperature equation of state combined with a simple but effective method to account for the influence of neutrinos during the collapse.
When comparing results from the fully general relativistic BSSN and approximate CFC collapse calculations, we find no significant deviations that could be attributed to systematic deficiencies of the CFC approximation. Both quantities relevant for the dynamic evolution of the collapsing core and the gravitational radiation waveforms agree very well in both approaches. This study is the by far most advanced and general demonstration of the how well CFC actually approximates the fully general relativistic Einstein equations.
Additionally, by means of the Cotton-York tensor,
Y^{ i j} = ε^{i l m} ∇_{l} (R_{m}^{j} - 1/4 δ_{m}^{j} R)
(which is zero in a situation where the spacetime is conformally flat, i.e. if CFC is an exact representation of the Einstein equations), we are able to quantitatively analyze the deviation of CFC from BSSN. Indeed we find that the (density-weighted volume integral of the) Cotton-York tensor is always below 2% during the entire collapse, even for rapidly rotating models.
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