Moving beyond the linearized regime, several groups explored the CF approximation, which
incorporates many of the nonlinear effects of GR into an elliptic, rather than hyperbolic, evolution
scheme. While nonlinear elliptic solvers are expensive computationally, they typically yield stable
evolution schemes since field solutions are always calculated instantaneously from the given matter
configuration. Summarized quickly, the CF approach involves solving the CTS field equations,
Eqs. 18, 19
, and 20
, at every timestep, and evolving the matter configuration forward in time.
The metric fields act like potentials, with various gradients appearing in the Euler and energy
equations. While the CTS formalism remains the most widely used method to construct NS-NS
(and BH-NS) initial data, it does not provide a completely consistent dynamical solution to
the GR field equations. In particular, while it reproduces spherically symmetric configurations
like the Schwarzschild solution exactly, it cannot describe more complicated configurations,
including Kerr BHs. Moreover, because the CF approximation is time-symmetric, it also does not
allow one to consistently predict the GW signal from a merging configuration. As a result,
most dynamical calculations are performed by adding the lowest-order dissipative radiation
reaction terms, either in the quadrupole limit or via the radiation reaction potential introduced
in [48].
The CTS equations themselves were originally written down in essentially complete form by Isenberg in
the 1970s, but his paper was rejected and only published after a delay of nearly 30 years [137]. In the
intervening years, Wilson, Mathews, and Marronetti [327, 328
, 188
, 187
] independently re-derived the
entire formalism and used it to perform the first nonlinear calculations of NS-NS mergers (as a result, the
formalism is often referred to as the “Wilson–Mathews” or “Isenberg–Wilson–Mathews” formalism). The
key result in [327, 328, 188, 187] was the existence of a “collapse instability,” in which the deeper
gravitational wells experienced by the NSs as they approached each other prior to merger could
force one or both to collapse to BHs prior to the orbit itself becoming unstable. Unfortunately,
their results were affected by an error, pointed out in [104], which meant that much of the
observed compression was spurious. While their later calculations still found some increase in the
central density as the NSs approached each other [189], these results have been contradicted by
other QE sequence calculations (see, e.g., [313
]). Furthermore, using a “CF-like” formalism in
which the nonlinear source terms for the field equations are ignored, dynamical calculations
demonstrated the maximum allowed mass for a NS actually increases in response to the growing tidal
stress [273].
The CF approach was adapted into a Lagrangian scheme for SPH calculations by the same groups that
had investigated PN NS-NS mergers, with Oechslin, Rosswog, and Thielemann [211] using a multigrid
scheme and Faber, Grandclément, and Rasio [97
] a spectral solver based on the Lorene libraries [124].
The effects of nonlinear gravity were immediately evident in both sets of calculations. In [211], NS-NS
binaries consisting of initially synchronized NSs merged without appreciable mass loss, with no more that
of the total system mass ejected, strikingly different from previous Newtonian and PN
simulations. When evolving initially irrotational systems, [97] found no appreciable developments of “spiral
arms” whatsoever, indicating a complete lack of mass loss through the outer Lagrange points.
Both groups also found strong emission from remnants for a stiff EOS, as the triaxial merger
remnant produced an extended period of strong ringdown emission. Neither set of calculations
indicated that the remnant should collapse promptly to form a BH, but given the high spin of
the remnant it was noted that conformal flatness would have already broken down for those
systems.
http://www.livingreviews.org/lrr-2012-8 |
Living Rev. Relativity 15, (2012), 8
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