These two methods are separately explained below because they give different formulas. We refer to [208, 209, 210] for the excision approach and to [202, 203, 106] for the puncture one. For a more detailed discussion of the decomposition of Einstein’s equation and the formalism, we refer to [29, 231, 44], and for the hydrostatics [80, 214, 219].
In the excision approach, Einstein’s equations in the conformal thin-sandwich formalism are solved for constructing quasi-equilibrium configurations . The line element in the 3+1 form is written as[80, 78]).
The spatial metric is further decomposed into a conformal factor and a background spatial metric as
The extrinsic curvature is defined by[229, 145], it is split into the trace and the traceless part as
Equations (25), (26), and (27) yield
The matter terms in the right-hand side of Equations (28), (30), and (31) are derived from the projections of the stress-energy tensor into the spatial hypersurface , defined by
The set of Equations (28) – (31) has four functions that can be chosen freely; , , , and . For computing quasi-equilibrium states, one usually assumes the presence of a helical Killing vector, , and the absence of gravitational waves in the wave zone. Under these assumptions, it is natural to choose the time direction so as to satisfy (in the comoving frame), and to set and . For the choice of and , two ways have been proposed. The first one is to choose a maximal slicing and to adopt a flat metric , for simplicity. The other choice is to use the Kerr–Schild metric for and in the vicinity of the BH or in the whole computational space. One of the advantages of choosing maximal slicing and the flat metric background is that the source term becomes simple and falls off rapidly for enough to obtain accurate results. The disadvantage is that it is not possible to construct the Kerr BH even with a distant orbit, and moreover, the set of Equations (28) – (31) may have non-unique solutions [159, 17, 224]. In particular, the spin of a BH has two values for the same spin parameter (see Equation (49) for the definition) . The lower branch of the spin as a function of , which should be physically reasonable because it approaches to the Schwarzschild BH as , can reach only , much less than the maximum spin of a Kerr BH. The advantage of using the Kerr–Schild metric for and is that one can calculate a spinning BH with nearly maximum spin, . The disadvantage is that the source term becomes complicated and falls off slowly for . Because of this situation, it is not easy to derive the results as accurately as those with the conformal three metric, if one adopts this background metric in the whole computational space . To recover the accuracy, a modification of the Kerr–Schild metric seems to be necessary; the metric is chosen to be nearly the Kerr–Schild one in the vicinity of the BH, whereas away from the BH, the metric should approach exponentially a conformally-flat metric and a maximal slicing [75, 132].
In the following, we restrict ourselves to the case of maximal slicing and flat spatial background metric for simplicity. Then, Equations (28), (30), and (31) can be written as
To solve the gravitational field equations (40), (41), and (42), it is necessary to impose appropriate boundary conditions on two different boundaries in the excision approach: outer boundaries at spatial infinity and inner boundaries on the BH horizons. Assuming asymptotic flatness, the boundary conditions at spatial infinity are written as (see also [40, 45] as well as the related isolated horizon formalism, e.g., [10, 81]). The boundary condition for the conformal factor is [55, 40] ( described the original formulation and  reported a numerical result for BH-BH binaries based on the formulation of [55, 47]), or by a new alternative method for finding approximate Killing vectors of closed 2-spheres [48, 132]. is iteratively determined until the quasi-local spin angular momentum of BH relaxes to a required value .
According to , the boundary condition on the lapse function can be chosen freely. For example, we can choose a Neumann boundary condition
A “puncture” method  was proposed by Brandt and Brügmann to describe multiple BH with arbitrary linear momenta and spin angular momenta, extending the original work by Brill and Lindquist . A “moving-puncture approach” [39, 15] was revealed to be quite useful in dynamic simulations. Here, we describe the puncture approach for quasi-equilibrium in the context of BH-NS binaries, which was originally developed by Shibata and Uryū [202, 203] and subsequently modified by Kyutoku et al. . The puncture approach employs a mixture of the conformal thin-sandwich decomposition and the conformal transverse-traceless decomposition of Einstein’s equations. The trace part of the extrinsic curvature is set to zero (maximal slicing) and the three metric is assumed to be conformally flat in the work so far.
The basic equations for the gravitational field are Equations. (40), (41), and (42) as in the excision approach. In the puncture approach, the metric quantities, which appear in Equations (40) and (42), and , are decomposed into an analytic singular part and a regular part. The former part denotes the contribution from a BH and the latter one is obtained by numerically solving the basic equations. Assuming that the puncture is located at , and are given by
denotes an auxiliary three-dimensional function and . The equation for is derived by substituting Equation (56) into the momentum constraint equation. Because the total linear momentum of the binary system should vanish1, the linear momentum of the BH, , is related to that of the companion NS as
The field equations to be solved are summarized as follows:
Equations (59) – (62) are elliptic in type, and hence, appropriate boundary conditions have to be imposed at spatial infinity. Because of the asymptotic flatness, the boundary conditions at spatial infinity are written as
In contrast to the case in which the excision approach is adopted, the inner boundary conditions do not have to be imposed in the puncture approach. This could be a drawback in this approach, because one cannot impose physical boundary conditions (e.g., Killing horizon boundary conditions) for the BH. However, this could also be an advantage, because we do not have to impose a special condition for the geometric variables, and as a result, flexibility for adjusting a quasi-equilibrium state to a desired state is preserved.
The hydrostatic equations governing quasi-equilibrium states are the Euler and continuity equations. The matter in the NS interior needs to satisfy those equations. There are several versions for deriving the hydrostatic equations [26, 9, 192, 215, 80, 204]. In this section, we review the version in [204, 219].
The equation of motion is written as
Equation (67) can be further modified. Defining a spatial velocity in the comoving frame, the 4-velocity is written as
It is interesting to note that if we use the Cartan identity,, is obtained. Because of the helical symmetry and the irrotational flow , the Cartan identity yields
The next task is to derive an equation for the velocity potential . The term in the left-hand side of the equation of continuity (68) is rewritten as
Finally, we comment on the prescription for determining the constant in the right-hand side of Equation (77). For this task, the central value of the quantities in the left-hand side is usually used. Here the center of the NS is defined as the location of the maximum baryon rest-mass density. Equation (77) includes one more constant, which should be determined for each quasi-equilibrium figure; the orbital angular velocity as found from Equations (24), (69), (74), and (82). The method for calculating it will be explained in the next Section 2.1.4.
The first integral of the Euler equation (77) includes information of the orbital angular velocity, , through Equations. (24), (69), (74), and (82). should be determined from the condition that a binary system is in a quasi-equilibrium circular orbit. In the following, we describe two typical methods referring to the rotation axis of the binary system as the Z-axis and to the axis connecting the BH’s and NS’s centers as the X-axis.
In one of two typical methods, a force balance along the X-axis is required. The force balance equation is derived from the condition that the central value of the enthalpy gradient for the NS is zero,
In the other method, is determined by requiring the enthalpy at two points on the NS’s surface along the X-axis to be equal to on the surface. At these two points, the pressure is absent. Namely, the sum of the gravitational force from the BH, self-gravitational force from the NS, and the centrifugal force associated with the orbital motion is balanced. These two conditions may be regarded as the conditions that determine and , and thus, can be determined for a given set of gravitational field variables. The work by Taniguchi et al. [208, 209, 210, 214] confirmed that in both methods, an accurate numerical result can be computed with a reasonable number of iterations, and that the results by these two methods coincide within the convergence level of the enthalpy. Therefore, both methods work well.
Equation (84) also depends on the location of the center of mass, because the rotating shift includes , which is the radial coordinate measured from the center of mass of the binary system. To determine the location of the center of mass of a binary system, in the framework of the excision approach, Taniguchi et al. [208, 209, 210, 214] and Grandclément [82, 83] require that the linear momentum of the system vanishesZ-axis.
In the puncture approach, the situation is totally different from the above, because Condition (85) has already been used to calculate the linear momentum of the BH; see Equation (58) in Section 2.1.2. In this framework, there is no known natural, physical condition for determining the center of mass of the system. Until now, three methods have been employed to determine the center of mass. In the first method, the dipole part of at spatial infinity is required to be zero [202, 203]. However, it was found that in this condition, the angular momentum derived for a close orbit of is 2% smaller than that derived by the 3PN approximation for . Because the 3PN approximation should be an excellent approximation of general relativity for a fairly distant orbit, as such , the obtained initial data deviates from the true quasi-circular state, and hence, the initial orbit would be eccentric.
In the second method, the azimuthal component of the shift vector at the location of the puncture is required to be equal to ; a corotating gauge condition at the location of the puncture is imposed . This method gives a slightly better result than that of the first method. However, the angular momentum derived for a close orbit of is also 2% smaller than that derived by the 3PN relation for a larger mass ratio . The disagreement is larger for the larger mass ratio. Such initial conditions are likely to deviate from the true quasi-circular state and hence the orbital eccentricity is large as well.
In the last method, the center of mass is determined in a phenomenological manner: One imposes a condition that the total angular momentum of the binary system for a given value of agrees with that derived by the 3PN approximation . This condition can be achieved by appropriately choosing the position of the center of mass. With this method, the drawback in the previous two methods, i.e., the angular momentum becoming smaller than the expected value, is overcome. Recent numerical simulations by the KT group have been performed employing initial conditions obtained by this method, and showed that the binary orbit is not very eccentric with these initial conditions (cf. Section 3.1.1).
A wide variety of EOS has been adopted for the study of quasi-equilibrium states of BH-NS binaries, which are employed as initial conditions of numerical simulations (see Section 3). However, the only EOS used for the study of quasi-equilibrium sequences has been the polytrope,
For the polytropic EOS, we have the following natural units, i.e., polytropic units, to normalize the length, mass, and time scales:
Even though the EOS used for constructing sequences is only the polytrope, a lot of quasi-equilibria with several EOS have been derived as initial data for the merger simulations. We will summarize those initial data in Section 3.1.1.
The several key quantities that are necessary in the quantitative analysis of quasi-equilibrium sequences are summarized in this section; the irreducible mass and spin of BH, the baryon rest mass of NS, the ADM mass, the Komar mass, and the total angular momentum of the system. It is reasonable to consider that the irreducible mass and spin of the BH and the baryon rest mass of the NS are conserved during the inspiral of BH-NS binaries. In addition, temperature and entropy of the NS may be assumed to be approximately zero because the thermal effects of not-young NS are negligible to their structure, i.e., it is reasonable to use a fixed cold EOS throughout the sequence. For such a sequence, the ADM mass, the Komar mass, and the total angular momentum of the system vary with the decrease of the orbital separation. These global quantities characterize the quasi-equilibrium sequence.
We classify the study by seven parameters and summarize in Table 1:
In the framework of the excision approach, corresponds to the excision surface. On the other hand, it is necessary to determine the apparent horizon in the framework of the puncture approach (although it is not a difficult task).
In addition, the definition of the linear momentum (which is usually set to be zero) is the same as Equation (85),
Then, the binding energy of the binary system is often defined by
In order to measure a global error in the numerical results, the virial error is often defined as the fractional difference between the ADM and Komar masses,[76, 204]).
To determine the orbit at the onset of mass shedding of a NS, Gourgoulhon et al. [80, 211, 212] defined a “sensitive mass-shedding indicator” (in the context of NS-NS binaries) as[208, 209, 210] analyze this parameter for identifying the mass-shedding limit of BH-NS binaries.
Living Rev. Relativity 14, (2011), 6
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