3+1 formalism and bases of numerical relativity - LUTh ...

114 Asymptotic flatness and global quantities
turns out to be true in practice, as we shall see on the specific example of Kerr spacetime
in Sec. 7.6.3.
7.5.2 The “cure”
In view of the above coordinate dependence problem, one may define the angular momentum
as a quantity which remains invariant only with respect to a subclass of the coordinate changes
(7.7). This is made by imposing decay conditions stronger than (7.1)-(7.4). For instance, York
[276] has proposed the following conditions 3 on the flat divergence of the conformal metric and
the trace of the extrinsic curvature:
∂˜γij
∂x j = O(r−3 ), (7.64)
K = O(r −3 ). (7.65)
Clearly these conditions are stronger than respectively (7.35) and (7.3). Actually they are so
severe that they exclude some well known coordinates that one would like to use to describe
asymptotically flat spacetimes, for instance the standard Schwarzschild coordinates (7.16) for the
Schwarzschild solution. For this reason, conditions (7.64) and (7.65) are considered as asymptotic
gauge conditions, i.e. conditions restricting the choice of coordinates, rather than conditions on
the nature of spacetime at spatial infinity. Condition (7.64) is called the quasi-isotropic gauge.
The isotropic coordinates (6.24) of the Schwarzschild solution trivially belong to this gauge (since
˜γij = fij for them). Condition (7.65) is called the asymptotically maximal gauge, since for
maximal hypersurfaces K vanishes identically. York has shown that in the gauge (7.64)-(7.65),
the angular momentum as defined by the integral (7.63) is carried by the O(r −3 ) piece of K
(the O(r −2 ) piece carrying the linear momentum Pi) and is invariant (i.e. behaves as a vector)
for any coordinate change within this gauge.
Alternative decay requirements have been proposed by other authors to fix the ambiguities
in the angular momentum definition (see e.g. [92] and references therein). For instance, Regge
and Teitelboim [209] impose a specific form and some parity conditions on the coefficient of the
O(r −1 ) term in Eq. (7.1) and on the coefficient of the O(r −2 ) term in Eq. (7.3) (cf. also M.
Henneaux’ lecture [157]).
As we shall see in Sec. 7.6.3, in the particular case of an axisymmetric spacetime, there exists
a unique definition of the angular momentum, which is independent of any coordinate system.
Remark : In the literature, there is often mention of the “ADM angular momentum”, on the
same footing as the ADM mass and ADM linear momentum. But as discussed above,
there is no such thing as the “ADM angular momentum”. One has to specify a gauge first
and define the angular momentum within that gauge. In particular, there is no mention
whatsoever of angular momentum in the original ADM article [23].
3 Actually the first condition proposed by York, Eq. (90) of Ref. [276], is not exactly (7.64) but can be shown
to be equivalent to it; see also Sec. V of Ref. [246].