List of Footnotes

1		The situation is even more puzzling in the Einstein–Yang–Mills theory where the right side of Equation (1 ) acquires an additional term, $V δQ$ . In treatments based on stationary space-times, not only the Yang–Mills charge $Q$ , but also the potential $V$ (the analog of $Ω$ and $κ$ , is evaluated at infinity [176 ].
2		However, Condition (6 ) may be too strong in some problems, e.g., in the construction of quasi-equilibrium initial data sets, where the notion of WIH is more useful [125 ] (see Section 5.2).
3		Indeed, the situation is similar for black holes in equilibrium. While it is physically reasonable to restrict oneself to IHs, most results require only the WIH boundary conditions. The distinction can be important in certain applications, e.g., in finding boundary conditions on the quasi-equilibrium initial data at inner horizons.
4		Note that we could replace $¯T ab$ with $T ab$ because $g ^τaϕb = 0 ab$ . Thus the cosmological constant plays no role in this section.
5		This formula has a different sign from that given in [84 ] due to a difference in the sign convention in the definition of the extrinsic curvature.
6		Cook’s boundary condition on the conformal factor $ψ$ (Equation (82) in [70 ]) is equivalent to $Θ (ℓ) = 0$ which (in the co-rotating case, or more generally, when the 2-metric on $S$ is axi-symmetric) reduces to $ℒtψ = 0$ on $S$ . The Yo et al. boundary condition on $ψ$ (Equation (48) of [190 ]) is equivalent to $ℒ¯tψ = 0$ on $S$ , where, however, the evolution vector field $¯ta$ is obtained by a superposition of two Kerr–Schild data.
7		We are grateful to Alejandro Corichi for correspondence on the recent results in this area.