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, e.g., [441Jump To The Next Citation Point, 419]). The basic idea is that we foliate an open subset U of the spacetime by a two-parameter family of (e.g., closed) spacelike two-surfaces. If 𝒮 is the typical two-surface, then this foliation is defined by a smooth embedding Ī• : 𝒮 × (− 𝜖,𝜖) × (− 𝜖,𝜖) → U : (p, ν+,ν− ) â†Ļ→ Ī•(p,ν+, ν− ). Then, keeping ν+ fixed and varying ν−, or keeping ν− fixed and varying ν+, : 𝒮ν+,ν− = Ī•(𝒮,ν+, ν− ) defines two one-parameter families of hypersurfaces Σν+ and Σ ν− respectively. Requiring one (or both) of the hypersurfaces Σν± to be null, we get a null (or double-null, respectively) foliation of U. (In Section 4.1.8 we require the hypersurfaces Σν± to be null only for the special value ν = 0 ± 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 ν+ and p ∈ 𝒮 the prescription ν− â†Ļ→ Ī•(p,ν+,ν− ) defines a curve through Ī• (p,ν+,0) ∈ 𝒮ν+,0 in Σ ν+, and hence a vector field ξa+ := (∂∕∂ν − )a tangent everywhere to Σ+ on U. The Lie bracket of ξa+ and the analogously-defined ξa− are zero. There are several inequivalent ways of introducing coordinates or rigid frame fields on U, which are fit naturally to the null or double null foliation {𝒮 ν+,ν− }, in which the (vacuum) Einstein equations and Bianchi identities take a relatively simple form [441, 209, 160Jump To The Next Citation Point, 480, 522Jump To The Next Citation Point, 245Jump To The Next Citation Point, 225, 105, 254Jump To The Next Citation Point].

Defining the ‘time derivative’ to be the Lie derivative, for example, along the vector field ξa+, 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 been made. (Otherwise, the variation of the action with respect to the less-than-ten nontrivial components of the metric would not yield all ten 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 [160, 522, 245Jump To The Next Citation Point, 254Jump To The Next Citation Point]. 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 [245, 248Jump To The Next Citation Point] 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 [282], starting with the idea of the ‘freely falling two-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 [247] 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., 𝒩u, but the other was timelike, e.g., Bv. The former defined a foliation of the latter in terms of the spacelike two-surfaces 𝒮u,v := 𝒩u ∩ Bv. Yoon found [567Jump To The Next Citation Point, 568Jump To The Next Citation Point] a certain two-surface integral on 𝒮u,v, denoted by &tidle;E(u,v ), for which the difference E&tidle;(u2,v) − &tidle;E (u1,v), u1 < u2, could be expressed as a flux integral on the portion of the timelike hypersurface B v between 𝒮 u1,v and 𝒮 u2,v. In general this flux does not have a definite sign, but Yoon showed that asymptotically, when Bv is ‘pushed out to null infinity’ (i.e., in the v → ∞ limit in an asymptotically flat spacetime), it becomes negative definite. In fact, ‘renormalizing’ &tidle;E (u,v) by a subtraction term, ∘ -------------------- E (u,v) := &tidle;E(u, v) − Area(𝒮0,v)∕(16πG2 ) tends to the Bondi energy, and the flux integral tends to the Bondi mass-loss between the cuts u = u1 and u = u2 [567, 568]. These investigations were extended for other integrals in [569, 570, 571], which are analogous to spatial momentum and angular momentum. However, all these integrals, including &tidle; E (u,v ) above, depend not only on the geometry of the spacelike two-surface 𝒮u,v but on the 2 + 2 foliation on an open neighborhood of 𝒮u,v as well.


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