### 11.2 Approaches based on the double-null foliations

#### 11.2.1 The 2 + 2 decomposition

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 [326306]). 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 [32615212334838118016582189].

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 [123381180189]. 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 [180183] 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’.)

#### 11.2.2 The 2 + 2 quasi-localization of the Bondi-Sachs mass-loss

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. 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 [400401] 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  [400401]. These investigations were extended for other integrals in [402403404], 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.