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 [231]. The line element in the 3+1 form is written as
where is the shift vector seen by a comoving observer. We note that the coordinates in the comoving observer frame are usually chosen for convenience in this field (e.g., [80, 78]).The spatial metric is further decomposed into a conformal factor and a background spatial metric as
Here, the condition, , is not always imposed. The shift vector seen by a comoving observer, , can be decomposed as where is the shift vector seen by an inertial observer and is a rotating shift. The rotating shift is written as , where is the orbital angular velocity vector of the binary system measured at infinity, and a coordinate vector for which the origin is located at the center of mass of the binary system.The extrinsic curvature is defined by
where is the Lie derivative along the unit normal to the hypersurface . Following [229, 145], it is split into the trace and the traceless part as The traceless part is rewritten as where . The Hamiltonian constraint then takes the form where denotes , the covariant derivative with respect to , and the scalar curvature with respect to .Equations (25), (26), and (27) yield
Inserting Equation (29) into the momentum constraint gives In addition to the Hamiltonian and momentum constraints, the trace of the evolution equation of the extrinsic curvature is often employed as one of the field equations: For a given condition for and , this equation reduces to an elliptic-type equation for , which possesses primary information of gravitational potential for an equilibrium or quasi-equilibrium configuration.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
For an ideal fluid for which the following relation holds: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) [132]. 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 [208]. 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
where and denote the flat Laplace operator and the partial derivative, and . Note here that the shift vector was replaced by in Equation (41), because the rotating shift does not contribute to the equation in the conformally-flat case. Equation (29) becomes where we have replaced by for the same reason as above.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
Inner boundary conditions arise from the excision of the BH interior. The assumption that the BH is in equilibrium leads to a set of boundary conditions for the conformal factor and the shift vector [47] (see also [40, 45] as well as the related isolated horizon formalism, e.g., [10, 81]). The boundary condition for the conformal factor is where is the outward pointing unit vector normal to the excision surface and is the induced metric on the excision surface, . The quantity is computed from the projection of the extrinsic curvature as . The boundary condition on the normal component of the shift vector is Note that the quantity is the the normal component of the shift vector in the comoving frame, . The tangential component must form a conformal Killing vector of the conformal metric on the excision surface. This can be achieved by choosing them to be Killing vectors of a 2-sphere, where is an arbitrary parameter related to the spin of the BH, and is derived by solving the conformal Killing equation on . Because we choose a conformally-flat spatial background, Equation (49) can be rewritten as Here is a freely specifiable vector, related to the BH spin, and is a Cartesian coordinate centered on the 2-sphere. The quasi-local spin angular momentum associated with an approximate Killing vector of is defined by where is the approximate Killing vector that is found by solving the Killing transport equations as described in [55, 40] ([55] described the original formulation and [40] 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 [40].According to [47], the boundary condition on the lapse function can be chosen freely. For example, we can choose a Neumann boundary condition
on the excision surface.
A “puncture” method [32] 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 [33]. 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. [106]. 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
where and are positive constants of mass dimension, and is a coordinate distance from the puncture. and denote the regular parts of and , respectively, and are determined by solving elliptic equations (see Equations (59) and (61) shown below). The quantity is an arbitrarily-chosen parameter called the puncture mass, while is determined by the condition that the ADM mass () and the Komar mass agree, Also, is decomposed into singular and regular parts as Here is the singular part, which denotes a weighted extrinsic curvature associated with the linear momentum, , and spin, , of the BH written by and is the completely anti-symmetric tensor on .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 vanish^{1}, the linear momentum of the BH, , is related to that of the companion NS as
where the right-hand side denotes the (minus) linear momentum of the NS.The field equations to be solved are summarized as follows:
We note that in the puncture approach is determined by solving Equation (56), not Equation (43), because is not straightforwardly defined for the location with when adopting Equation (43). In this approach, the elliptic equation for still has to be solved because is needed in solving hydrostatic equations.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
Here it is assumed that the equations are solved in the inertial frame.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
Assuming a perfect fluid, Equation (36), the term in the left-hand side of the equation of motion is written as where is the temperature, and is the entropy per baryon mass. To derive the second line of Equation (65), we use an equation of local thermodynamic equilibrium, Assuming that the entropy per baryon is constant everywhere inside the NS and using Equation (65), Equation. (64) yields where we use the equation of rest-mass conservation,Equation (67) can be further modified. Defining a spatial velocity in the comoving frame, the 4-velocity is written as
where is a helical Killing vector, and we have a relation, . denotes the time component of and is calculated through the normalization condition of , . Substituting Equation (69) into Equation (67) and using the relation, we obtain If the fluid motion in a NS is synchronized with the orbital motion, i.e., the corotating case, we have . Thus, Equation (71) can be integrated once to yield If the fluid motion in a NS is irrotational, we need to consider the vorticity of the fluid. The relativistic vorticity tensor is defined by For the irrotational flow, , there exists a scalar potential by which the term is expressed as Inserting Equation (74) into Equation (71), we have and thus, or, in another form,It is interesting to note that if we use the Cartan identity,
another form of the integrated Euler equation, e.g., Equation (33) in [80], is obtained. Because of the helical symmetry and the irrotational flow , the Cartan identity yields and thus, This equation is equivalent to Equation (77).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
Using Equations (69), (74), and the expression for the helical Killing vector, the equation of continuity (68) is rewritten as where Equation (81) is used.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,
where denotes the NS’s center at which the pressure gradient and self-gravitational force of the NS are absent. Thus, Equation (84) may be regarded as the condition that the gravitational force from the BH at the NS’s center is equal to the centrifugal force associated with the orbital motion, and hence, can be used to determine for a given set of gravitational field variables.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 vanishes
where the maximal slicing condition, , is assumed. The essence of this condition is as follows: for a hypothetical orbital angular velocity, , the total linear momentum of the system depends on the location of the center of mass, and hence, its location is determined by Condition (85). Once the location of the center of mass is determined in an iteration step, the positions of the BH and NS are moved, keeping the separation, in order for the center of mass of the binary system to locate on the Z-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 [197]. 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 [106]. 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,
where is a polytropic constant and is the adiabatic index. Hereafter, we review only the numerical results in this EOS.For the polytropic EOS, we have the following natural units, i.e., polytropic units, to normalize the length, mass, and time scales:
Because the geometric units with are adopted, the polytropic units normalize all of 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),
where the maximal slicing condition, , is assumed.Then, the binding energy of the binary system is often defined by
where is the ADM mass of the binary system at infinite orbital separation, (see Section 1.8 for the definition). For a non-spinning BH, coincides with the irreducible mass . For a spinning BH, the relation among , , and is given by [42]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,
Here, we note the presence of a theorem, which states that for the helical symmetric spacetime, the ADM mass and Komar mass are equal (e.g., [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
Here, the numerator of Equation (99), , is the radial derivative of the enthalpy in the equatorial plane at the surface along the direction toward the companion, and the denominator, , is that at the surface of the pole. The radial coordinate is measured from the center of the NS. For a spherical NS at infinite separation, , while indicates the formation of a cusp, and hence, the onset of mass shedding. Taniguchi et al. [208, 209, 210] analyze this parameter for identifying the mass-shedding limit of BH-NS binaries.
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