The decomposition of the spacetime in a 2 + 2 way with respect to two families of null hypersurfaces is as
old as the study of gravitational radiation and the concept of the characteristic initial value problem (see for
example [326, 306]). The basic idea is that we foliate an open subset
of the spacetime by a
2-parameter family of (e.g. closed) spacelike 2-surfaces: If
is the typical 2-surface, then this foliation is
defined by a smooth embedding
. Then, keeping
fixed and varying
, or keeping
fixed and varying
, respectively,
defines two 1-parameter families of hypersurfaces
and
. Requiring one (or both) of the
hypersurfaces
to be null, we get a so-called null (or double-null, respectively) foliation of
.
(In Section 4.1.8 we required the hypersurfaces
to be null only for the special value
of the parameters.) As is well known, because of the conjugate points, in the null or
double null cases the foliation can be well-defined only locally. For fixed
and
the
prescription
defines a curve through
in
, and hence a
vector field
tangent everywhere to
on
. The Lie bracket of
and the analogously defined
is zero. There are several inequivalent ways of introducing
coordinates or rigid frame fields on
, which are fit naturally to the null or double null foliation
, in which the (vacuum) Einstein equations and Bianchi identities take a relatively simple
form [326, 152, 123
, 348, 381
, 180
, 165, 82, 189
].
Defining the ‘time derivative’ to be the Lie derivative, for example, along the vector field , the
Hilbert action can be rewritten according to the 2 + 2 decomposition. Then the 2 + 2 form of the Einstein
equations can be derived from the corresponding action as the Euler-Lagrange equations provided the
fact that the foliation is null is imposed only after the variation has made. (Otherwise, the
variation of the action with respect to the less than ten nontrivial components of the metric
would not yield all the 10 Einstein equations.) One can form the corresponding Hamiltonian,
in which the null character of the foliation should appear as a constraint. Then the formal
Hamilton equations are just the Einstein equations in their 2 + 2 form [123, 381, 180
, 189
].
However, neither the boundary terms in this Hamiltonian nor the boundary conditions that
could ensure its functional differentiability were considered. Therefore, this Hamiltonian can be
‘correct’ only up to boundary terms. Such a Hamiltonian was used by Hayward [180, 183
] as the
basis of his quasi-local energy expression discussed already in Section 6.3. (A similar energy
expression was derived by Ikumi and Shiromizi [211], starting with the idea of the ‘freely falling
2-surfaces’.)
As we mentioned in Section 6.1.3, this double-null foliation was used by Hayward [182] to quasi-localize the
Bondi-Sachs mass-loss (and mass-gain) by using the Hawking energy. Thus we do not repeat the review of
his results here.
Update
Yoon investigated the vacuum field equations in a coordinate system based on a null 2 + 2 foliation.
Thus one family of hypersurfaces was (outgoing) null, e.g.
, but the other was timelike, say
. The
former defined a foliation of the latter in terms of the spacelike 2-surfaces
. Yoon
found [400
, 401
] a certain 2-surface integral on
, denoted by
, for which the difference
,
, could be expressed as a flux integral on the portion of the timelike
hypersurface
between
and
. In general this flux does not have a definite sign, but Yoon
showed that asymptotically, when
is ‘pushed out to null infinity’ (i.e. in the
limit in an
asymptotically flat spacetime), it becomes negative definite. In fact, ‘renormalizing’
by a
subtraction term,
tends to the Bondi energy, and the flux
integral tends to the Bondi mass-loss between the cuts
and
[400, 401]. These
investigations were extended for other integrals in [402, 403, 404], which are analogous to spatial
momentum and angular momentum. However, all these integrals, including
above, depend not
only on the geometry of the spacelike 2-surface
but on the 2 + 2 foliation on an open neighbourhood
of
too.
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