Approaches Based on the Nester

8 Approaches Based on the Nester–Witten Two-Form

We saw in Section 3.2 that

both the ADM and Bondi–Sachs energy-momenta can be re-expressed by the integral of the Nester–Witten two-form $u (λ, ¯μ)ab$ ,
the proof of the positivity of the ADM and Bondi—Sachs masses is relatively simple in terms of the two-component spinors.

Thus, from a pragmatic point of view, it seems natural to search for the quasi-local energy-momentum in the form of the integral of the Nester–Witten two-form. Now we will show that

the integral of Møller’s tetrad superpotential for the energy-momentum, coming from his tetrad Lagrangian (3.1 ), is just the integral of $A- ¯B′ u(λ ,λ )ab$ , where $A- {λA }$ is a normalized spinor dyad.

Hence, all the quasi-local energy-momenta based on the integral of the Nester–Witten two-form have a natural Lagrangian interpretation in the sense that they are charge integrals of the canonical Noether current derived from Møller’s first-order tetrad Lagrangian. Update

If $𝒮$ is any closed, orientable spacelike two-surface and at minimum an open neighborhood of $𝒮$ is time and space orientable, then an open neighborhood of $𝒮$ is always a trivialization domain of both the orthonormal and the spin frame bundles [461 ]. Therefore, the orthonormal frame ${Eaa}$ can be chosen to be globally defined on $𝒮$ , and the integral of the dual of Møller’s superpotential, $1Ke ∨ ab1𝜀 2 e 2 abcd$ , appearing on the right-hand side of the superpotential equation (3.3 ), is well defined. If $a a (t ,v )$ is a pair of globally-defined normals of $𝒮$ in the spacetime, then in terms of the geometric objects introduced in Section 4.1, this integral takes the form

∮ Q [K ] : =--1-- 1Ke ∨eab 1𝜀abcd 8πG 𝒮 2 2 1 ∮ e( ⊥ ba ⊥ b ba a a ab b ) = ----- K − 𝜀eaQb − Ae − 𝜀ea(δbE b)η-E a + δe(taE a)η--Ebvb d𝒮. (8.1 ) 8πG 𝒮

The first term on the right is just the dual mean curvature vector of $𝒮$ , the second is the connection one-form on the normal bundle, while the remaining terms are explicitly $SO (1,3)$ gauge dependent. On the other hand, this is boost gauge invariant (the boost gauge dependence of the second term is compensated by the last one), and depends on the tetrad field and the vector field $a K$ given only on $𝒮$ , but is independent in the way in which they are extended off the surface. As we will see, the general form of other quasi-local energy-momentum expressions show some resemblance to Equation (8.1

Then, suppose that the orthonormal basis is built from a normalized spinor dyad, i.e., $a AB-′ A¯A ′ E a = σa ℰ AℰB-′$ , where $′ σAaB-$ are the $SL (2,ℂ)$ Pauli matrices (divided by $√-- 2$ ) and ${ℰAA-}$ , $A- = 0,1$ , is a normalized spinor basis. A straightforward calculation yields the following remarkable expression for the dual of Møller’s superpotential:

where the overline denotes complex conjugation. Thus, the real part of the Nester–Witten two-form, and hence, by Equation (3.5

), apart from an exact two-form, the Nester–Witten two-form itself, built from the spinors of a normalized spinor basis, is just the superpotential two-form derived from Møller’s first-order tetrad Lagrangian [461 ].

Next we will discuss some general properties of the integral of $u(λ, ¯μ )ab$ , where $λA$ and $μA$ are arbitrary spinor fields on $𝒮$ . Then, in the integral $H [λ,μ¯] 𝒮$ , defined by Equation (7.15 ), only the tangential derivative of $λA$ appears. ( $μA$ is involved in $H𝒮 [λ, ¯μ]$ algebraically.) Thus, by Equation (3.5 ), $H 𝒮 : C ∞ (𝒮,SA ) × C∞ (𝒮, SA) → ℂ$ is a Hermitian scalar product on the (infinite-dimensional complex) vector space of smooth spinor fields on $𝒮$ . Thus, in particular, the spinor fields in $H𝒮 [λ,μ¯]$ need be defined only on $𝒮$ , and $-------- H𝒮 [λ, ¯μ] = H𝒮 [μ, ¯λ]$ holds. A remarkable property of $H 𝒮$ is that if $λA$ is a constant spinor field on $𝒮$ with respect to the covariant derivative $Δe$ , then $H 𝒮[λ, ¯μ ] = 0$ for any smooth spinor field $μA$ on $𝒮$ . Furthermore, if $A- 0 1 λ A = (λA,λ A)$ is any pair of smooth spinor fields on $𝒮$ , then for any constant $SL (2,ℂ )$ matrix $ΛAB--$ one has $H 𝒮[λC-ΛCA, ¯λD′¯ΛD-′B′] = H 𝒮[λC,λ¯D-′]ΛC-A-¯ΛD-′B′$ , i.e., the integrals $H 𝒮[λA,-¯λB-′]$ transform as the spinor components of a real Lorentz vector over the two–complex-dimensional space spanned by $λ0 A$ and $λ1 A$ . Therefore, to have a well-defined quasi-local energy-momentum vector we have to specify some two-dimensional subspace $A S --$ of the infinite-dimensional space $∞ C (𝒮, SA )$ and a symplectic metric $𝜀AB--$ thereon. Thus, underlined capital Roman indices will be referring to this space. The elements of this subspace would be interpreted as the spinor constituents of the ‘quasi-translations’ of the surface $𝒮$ . Note, however, that in general the symplectic metric $𝜀 AB--$ need not be related to the pointwise symplectic metric $𝜀AB$ on the spinor spaces, i.e., the spinor fields $0 λA$ and $1 λA$ that span $A S --$ are not expected to form a normalized spin frame on $𝒮$ . Since, in Møller’s tetrad approach it is natural to choose the orthonormal vector basis to be a basis in which the translations have constant components (just like the constant orthonormal bases in Minkowski spacetime, which are bases in the space of translations), the spinor fields $λA- A$ could also be interpreted as the spinor basis that should be used to construct the orthonormal vector basis in Møller’s superpotential (3.2 ). In this sense the choice of the subspace $A S--$ and the metric $𝜀AB-$ is just a gauge reduction, or a choice for the ‘reference configuration’ of Section 3.3.3.

Once the spin space $(SA, 𝜀 ) AB-$ is chosen, the quasi-local energy-momentum is defined to be $AB-′ ′ P 𝒮 := H 𝒮[λA,-¯λB-]$ and the corresponding quasi-local mass $m 𝒮$ is $AA-′ BB-′ m2𝒮 := 𝜀AB-𝜀A′B-′P 𝒮 P 𝒮$ . In particular, if one of the spinor fields $λA- A$ , e.g., $λ0 A$ , is constant on $𝒮$ (which means that the geometry of $𝒮$ is considerably restricted), then $P00′= P01′= P10′= 0 𝒮 𝒮 𝒮$ , and hence, the corresponding mass $m 𝒮$ is zero. If both $0 λA$ and $1 λA$ are constant (in particular, when they are the restrictions to $𝒮$ of the two constant spinor fields in the Minkowski spacetime), then $′ P𝒮AB-$ itself is vanishing.

Therefore, to summarize, the only thing that needs to be specified is the spin space $(SA,-𝜀 ) AB-$ , and the various suggestions for the quasi-local energy-momentum based on the integral of the Nester–Witten two-form correspond to the various choices for this spin space.

8.1 The Ludvigsen–Vickers construction
  8.1.1 The definition
  8.1.2 Remarks on the validity of the construction
  8.1.3 Monotonicity, mass-positivity and the various limits
8.2 The Dougan–Mason constructions
  8.2.1 Holomorphic/antiholomorphic spinor fields
  8.2.2 The genericity of the generic two-surfaces
  8.2.3 Positivity properties
  8.2.4 The various limits
8.3 A specific construction for the Kerr spacetime