8 Approaches Based on the Nester–Witten 2-Form

We saw in Section 3.2 that

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 2-form. Now we will show that

Hence, all the quasi-local energy-momenta based on the integral of the Nester–Witten 2-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.

If 𝒮 is any closed, orientable spacelike two-surface and 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 [500Jump To The Next Citation Point]. Therefore, the orthonormal frame a {E a} can be chosen to be globally defined on 𝒮, and the integral of the dual of Møller’s superpotential, 12Ke ∨eab12𝜀abcd, appearing on the right-hand side of the superpotential Eq. (3.7View Equation), is well defined. If (ta,va) 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

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

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

------------- 1σa Ee ∨ ab1-𝜀 = u (ℰ , ¯â„° ′) + u(ℰ , ¯â„° ′) , (8.2 ) 4 AB-′ a e 2 abcd A- B- cd B- A- cd
where the overline denotes complex conjugation. Thus, the real part of the Nester–Witten 2-form, and hence, by Eq. (3.11View Equation), apart from an exact 2-form, the Nester–Witten 2-form itself, built from the spinors of a normalized spinor basis, is just the superpotential 2-form derived from Møller’s first-order tetrad Lagrangian [500].

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 𝒮[λ, ¯μ ], defined by Eq. (7.14View Equation), only the tangential derivative of λA appears. (μA is involved in H𝒮 [λ,μ¯] algebraically.) Thus, by Eq. (3.11View Equation), 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 H 𝒮 [λ,μ¯] need be defined only on 𝒮, and -------- H 𝒮[λ,μ¯] = H 𝒮[μ,¯λ ] holds. A remarkable property of H𝒮 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 λA-= (λ0 ,λ1 ) A A A is any pair of smooth spinor fields on 𝒮, then for any constant SL (2,ℂ ) matrix B- ΛA-- one has C- A-¯ D′¯ B′ C-¯D-′ A¯ B′ H 𝒮[λ ΛC--,λ ΛD-′ ] = H𝒮 [λ ,λ ]ΛC- ΛD-′, 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 λ0A and λ1A. Therefore, to have a well-defined quasi-local energy-momentum vector we have to specify some two-dimensional subspace SA- 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 𝜀AB- need not be related to the pointwise symplectic metric 𝜀 AB on the spinor spaces, i.e., the spinor fields λ0 A and λ1 A that span SA- 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 (3.6View Equation). In this sense the choice of the subspace A- S and the metric 𝜀AB- is just a gauge reduction (see Section 3.3.3).

Once the spin space (SA, 𝜀AB) is chosen, the quasi-local energy-momentum is defined to be AB-′ A-¯B-′ P 𝒮 := H 𝒮[λ ,λ ] and the corresponding quasi-local mass m 𝒮 is 2 AA-′ BB-′ m 𝒮 := 𝜀AB-𝜀A′B-′P 𝒮 P 𝒮. In particular, if one of the spinor fields A λA-, e.g., λ0A, is constant on 𝒮 (which means that the geometry of 𝒮 is considerably restricted), then P0𝒮0′= P0𝒮1′= P1𝒮0′= 0, and hence, the corresponding mass m 𝒮 is zero. If both λ0 A and λ1 A are constant (in particular, when they are the restrictions to 𝒮 of the two constant spinor fields in the Minkowski spacetime), then AB′ P𝒮 itself is vanishing.

Therefore, to summarize, the only thing that needs to be specified is the spin space (SA,-𝜀AB-), and the various suggestions for the quasi-local energy-momentum based on the integral of the Nester–Witten 2-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

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