List of Footnotes

1 We could also formulate Einstein’s equations as a characteristic initial-value problem, but we will not pursue that approach in this paper.
2 A different sign choice for defining the extrinsic curvature is sometimes found in the literature.
3 The stress-energy tensor is decomposed as T = S + n j + n j + n n ρ μν μν μ ν ν μ μ ν or, equivalently, S ≡⊥ T ij ij, j ≡ − ⊥ nμT i iμ, and ρ ≡ nμnνT μν where ⊥ denotes the operation of projection onto the spatial hypersurface and it acts on all free indices. Note that the matter terms in Eqs. (12View Equation), (14View Equation), and (15View Equation) are defined with respect to a normal observer. This is in contrast to the usual definition of ρ with respect to the rest frame of the fluid in hydrodynamics.
4 Or, if we solve the momentum constraint in terms of a background momentum density &tidle;i 10 i j ≡ ψ j, the momentum constraint decouples with nonvanishing &tidle;i j.
5 Note that in (39View Equation) we have not used the usual conformal scaling of the extrinsic curvature given in equation (22View Equation).
6 Equations (43View Equation) and (45View Equation) take the place of defining &tidle;γ = 1 and the subsequent necessity of treating ψ and &tidle;γij as densities.
7 Spheres at constant areal-radial coordinate r have proper area 4πr2.
8 In Ref. [38Jump To The Next Citation Point], Eq. (B7), the line “1 for n = 1” should read “− 1 α for n = 1”.
9 Time-asymmetric solutions are also needed to represent time-independent solutions that cover the interior of a black hole.
10 If ξi(k) is a translational Killing vector, then (71View Equation) yields the linear momentum in the direction of that Killing vector. If ξi (k) is a rotational Killing vector, then (71View Equation) yields the corresponding angular momentum.
11 We note that for the preferred choice of ωstatic ≈ − 9, the gauge invariant parameters of the ISCO found in Ref. [47] do not match well with the same parameter determined via the effective-potential method in Ref. [39]. However, if 0 < ωstatic < 10, the agreement is much better.
12 For a barytropic fluid, the entropy per baryon and the fractional abundances of the different nuclear species are determined uniquely by the distribution of baryons. In this case, the total energy density 𝜀 can be expressed as a function of the pressure P (cf. Ref. [35Jump To The Next Citation Point]).
13 In spherical coordinates, ⃗ξ = ∂∕∂ϕ. In Cartesian coordinates, rotation about the z axis would be represented by ξμ = (0,− y,x,0).
14 For isentropic flow, the thermodynamic identity reduces to dh = 1∕ρ0 ⋅dP.
15 This boundary condition comes from the fact that the fluid motion at the surface of the star must be tangent to the surface, μ u ∇μρ0 = 0.