where we've omitted the terms that will vanish in the limit as a result of requiring
.I-noi-nd ic flc atness. Similarly, we define angular momentum by
J = lim (Kb-K b bS,62)
where we have introduced the extrinsic curvature, Kab, of E, and the rotational K~illingf
vector a". For a generic metric perturbation of the K~err spacetime, we have
1 r2xr rx
6J =lim r 7sin 86,4 r2 Sin 8 r 4 ded#. (6-23)
Though these definitions provide the most general prescription for computing the mass
and angular momentum, for stationary and axially symmetric spacetimes (those containing
both timelike and axial K~illing vectors), the K~omar formulae [73] evaluated at infinity
allow us to compute the value of the perturbationS2 of M~ and J. though not the entire
perturbation in the interior and exterior spacetime. The formulae are given by
6M = (Lb- 9a)?a b 3x,~ (6-24)
6J Lb 9b)na b 3x (6-25)
where E is spacelike hypersurface that extends to infinity, n" is the unit normal to it, to
and #" are the timelike and axial K~illing vectors and 2/7d3Z is the volume element on E.
Because our stress-energy tensor is confined to a spacelike hypersurface, Ep, at r = ro, to
compute the ADM mass we must take the limit as To oo. In this limit, with the source
given by Equations 6-7-6-12, the K~omar formulae give (for the K~err spacetime)
sM = pE, (6-26)
6J = p-L. (6-27)
2 We thank John Friedman for -II--- _t h-r;! the use of the K~omar formulae.