6.3 Approximate relativistic schemes

The first steps toward approximating the effects of GR included the use of 1PN dynamics or the CF approximation. Using a formalism derived by Blanchet, Damour, and Schäfer [48Jump To The Next Citation Point], the 1PN equations of motion require the solution of eight Poisson-like equations in the form
∇2ψ = f(⃗x ), (33 )
where the source terms f(⃗x) are compactly supported, and thus the fields ψ may be determined using the same techniques already in place to find the Newtonian potential. Adding in the lowest-order dissipative radiation reaction effects requires solution of a ninth Poisson equation for a reaction potential. The 1PN formalism was implemented in both grid-based [216] and SPH codes [10Jump To The Next Citation Point, 99Jump To The Next Citation Point, 101Jump To The Next Citation Point, 100Jump To The Next Citation Point]. Unfortunately, physically realistic NSs are difficult to model using a PN expansion, since the characteristic NS compactness 𝒞 ≳ 0.15, leads to first order “corrections” that often rival Newtonian terms in magnitude. To deal with this problem Ayal et al. [10] considered large (R ≈ 30 km), low-mass (< 1M ⊙) NSs, allowing them to study relativistic effects but making results more difficult to interpret for physically realistic mergers. In [99, 101Jump To The Next Citation Point, 100], a dual speed of light approach was used, in which all 1PN effects were scaled down by a constant factor to yield smaller quantities while Newtonian and radiation reaction terms were included at full-strength. Both SPH groups found that the GW signal in PN mergers is strongly modulated, whereas Newtonian merger calculations typically yielded smooth, either monotonically decreasing or nearly constant-amplitude ringdown signals. Even reduced 1PN effects were shown to suppress mass loss by a factor of 2 – 5 for initially synchronized cases, and disk formation was seen to be virtually non-existent for initially irrotational, equal-mass NSs with a stiff (Γ = 3 polytropic) EOS [101].

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. 18View Equation, 19View Equation, and 20View Equation, 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 [327Jump To The Next Citation Point, 328Jump To The Next Citation Point, 188Jump To The Next Citation Point, 187Jump To The Next Citation Point] 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., [313Jump To The Next Citation Point]). 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 [211Jump To The Next Citation Point] using a multigrid scheme and Faber, Grandclément, and Rasio [97Jump To The Next Citation Point] 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 ∼ 10−4 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.

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