1.8 Outline and notation

This review is organized as follows. In Section 2, we review the current status of the study of quasi-equilibrium states of BH-NS binaries in general relativity. First, the formalisms for a solution of quasi-equilibrium states are described and then the resulting solution, its sequence, and implications are summarized. In Section 3, we review the current status of the numerical-relativity study of BH-NS binaries. In Section 3.1, we summarize the formulations, procedures, and numerical implementations employed in numerical simulations. In Section 3.2, the parameter space (composed of masses of BH and NS, BH spin, and the EOS of NS) investigated so far is summarized. In Sections 3.33.7, the numerical results for the evolution process throughout the late inspiral to the merger, tidal disruption processes, properties of the remnant formed after the merger, and features of gravitational waveforms and gravitational-wave spectra are summarized. We recommend that readers who are interested only in these simulation results read Section 3.33.7.

In this article, we adopt the following notations:

gμν – the spacetime metric,
γμν ≡ gμν + nμn ν – the three metric on a three dimensional hypersurface Σt,
nμ – the timelike unit hypersurface normal,
Kij – the extrinsic curvature on Σt,
K – the trace of the extrinsic curvature,
α – the lapse function,
i β – the shift vector,
g – the determinant of gμν,
γ – the determinant of γij; √ --- √-- − g = α γ,
∇ μ – the covariant derivative associated with gμν,
Di – the covariant derivative associated with γij,
μν T – the stress energy momentum tensor,
ρ – the baryon rest-mass density,
𝜀 – the specific internal energy of the fluid,
P – the pressure of the fluid,
h ≡ 1 + 𝜀 + P- ρ – the specific enthalpy of the fluid,
κ – the polytropic constant,
Γ – the adiabatic index,
uμ – the four velocity of the fluid,
MNS and MBH – the gravitational masses of NS and BH in isolation,
MB – the baryon rest mass of NS,
m0 ≡ MNS + MBH – the total mass at infinite separation,
a ≡ -cJBH-- GM B2H – the non-dimensional spin of BH,
JBH – the spin angular momentum of BH,
GMNS--- 𝒞 ≡ c2R NS – the compactness of NS in isolation,
RNS – the circumferential radius of spherical NS,
Ω – the orbital angular velocity,
f – the frequency of gravitational waves,
λ – the wavelength of gravitational waves.

Latin and Greek indices denote spatial and spacetime components, respectively. t denotes the coordinate time. In Section 2 and the Appendix A, the geometrical units of c = G = 1 are used, whereas in other sections, c and G are recovered.

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