spacetimes that become flat near infinity. The most precise definition of .l-i-inia..'l~e
flatness requires a detailed analysis of the conformal structure of spacetime [72], but for
our purposes it will suffice to simply consider the .I- i-n ng .)i oi falloff of the components
of the metric. More precisely, for a set of coordinates (x, y, x) in a metric, gAb, and
r = .1.2 2 2,,~ we required
lim gab 0ab(1
r-oo r(6-19)
These conditions are satisfied hv the Schwarzschild and K~err spacetimes we wish
to consider, but we must he careful to choose an appropriate gauge for the metric
perturbation to ensure that Equations 6-19 are satisfied. Assuming an .I-i-mptotically
flat spacetime, the ADM mass is defined by
Af= lim (bb-Der S (6-20)
16xr stoo
where the symbols need a bit of explanation: we denote the hypersurface of constant t
by E, and its boundary by S. The three-metric on Et is Yub. Then Kub = ub y, b, with
yOb being the metric of flat spacetime (in the same coordinates aS Yub) and a~ = ab ,0 byO"
Additionally, D, is the covariant derivative compatible with ,b, ra is the unit normal
to S, and dS is the surface element on S. For an arbitrary metric perturbation, hub, this
evaluates to
,M = ,itli 2r~rOJ~Z ir sin O, d2d (6-21)
1 In general, having a well-defined angular momentum actually requires a faster falloff
than that given helow. However, because we're restricting our attention to spacetimes with
axial killing vectors, the falloff required for .I-i-inia..'lc flatness is sufficient.