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. (12 ), (14 ), and (15 ) 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 (39 ) we have not used the usual conformal scaling of the extrinsic curvature given in equation (22 ).
6		Equations (43 ) and (45 ) 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. [38 ], 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 (71 ) yields the linear momentum in the direction of that Killing vector. If $ξi (k)$ is a rotational Killing vector, then (71 ) 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. [35 ]).
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, $μ u ∇μρ0 = 0$ .