7.1 Motivations

7.1.1 How do the twistors emerge?

We saw in Section 3.1.1 that in the Newtonian theory of gravity the mass of the source in D can be expressed as the flux integral of the gravitational field strength on the boundary 𝒮 := ∂D. Similarly, in the weak field (linear) approximation of general relativity on Minkowski spacetime the source of the gravitational field (i.e., the linearized energy-momentum tensor) is still analogous to charge. In fact, the total energy-momentum and angular momentum of the source can be expressed as appropriate two-surface integrals of the curvature at infinity [481]. Thus, it is natural to expect that the energy-momentum and angular momentum of the source in a finite three-volume Σ, given by Eq. (2.5View Equation), can also be expressed as the charge integral of the curvature on the two-surface 𝒮. However, the curvature tensor can be integrated on 𝒮 only if at least one pair of its indices is annihilated by some tensor via contraction, i.e., according to Eq. (3.14View Equation) if some ωab = ω[ab] is chosen and μab = 𝜀ab. To simplify the subsequent analysis, ωab will be chosen to be anti-self-dual: ωab = 𝜀A′B′ωAB with ωAB = ω (AB ).11 Thus, our goal is to find an appropriate spinor field AB ω on 𝒮 such that

∫ ∮ Q [K ] := K Tab 1-𝜀 = --1-- ωABR =: A [ω ]. (7.1 ) 𝒮 Σ a 3! bcde 8πG 𝒮 ABcd 𝒮
Since the dual of the exterior derivative of the integrand on the right, and, by Einstein’s equations, the dual of the 8πG times the integrand on the left, respectively, is
ecdf AB F F′(A BC ) F ′ E′F AB F′ FA 𝜀 ∇e(ω RABcd ) = − 2iψ A′BC′ ∇ ω + 2′ ϕABE ′ i∇ ω + 4Λi∇ A ω , (7.2 ) − 8 πGKaT af = 2ϕF AF A KAA ′ + 6ΛKF F . (7.3 )
expressions (7.2View Equation) and (7.3View Equation) are equal if AB ω satisfies
A ′A BC A(B C )A′ ∇ ω = − i𝜀 K . (7.4 )
This equation in its symmetrized form, ∇A ′(A ωBC ) = 0, is the valence 2 twistor equation, a specific example for the general twistor equation A ′(A BC...E ) ∇ ω = 0 for BC...E (BC...E) ω = ω. Thus, as could be expected, ωAB depends on the Killing vector Ka, and, in fact, Ka can be recovered from ωAB as ′ ′ KA A = 23i∇AB ωAB. Thus, ωAB plays the role of a potential for the Killing vector ′ KA A. However, as a consequence of Eq. (7.4View Equation), Ka is a self-dual Killing 1-form in the sense that its derivative is a self-dual (s.d.) 2-form: In fact, the general solution of Eq. (7.4View Equation) and the corresponding Killing vector are
′ ′ ′ ωAB = − ixAA xBB M¯A ′B′ + ix (AA ′T B)A + ΩAB, AA′ AA′ AB′ A′ (7.5 ) K = T + 2x M¯B′,
where M¯ ′ ′ A B, T AA′, and ΩAB are constant spinors, and using the notation xAA′ := xaσAA-′ℰ A¯â„°A ′ a A- A′, where {ℰAA-} is a constant spin frame (the ‘Cartesian spin frame’) and ′ σAaA- are the standard SL (2,ℂ) Pauli matrices (divided by √ -- 2). These yield that Ka is, in fact, self-dual, ∇AA ′KBB ′ = 𝜀ABM¯A ′B ′, T AA′ is a translation and M¯ ′ ′ A B generates self-dual rotations. Then Q [K ] = T ′P AA′ 𝒮 AA ¯ ¯A ′B′ +2 MA ′B′J, implying that the charges corresponding to AB Ω are vanishing, the four components of the quasi-local energy-momentum correspond to the real ′ T AAs, and the spatial angular momentum and center-of-mass are combined into the three complex components of the self-dual angular momentum J¯A′B ′, generated by M¯A ′B′.

7.1.2 Twistor space and the kinematical twistor

Recall that the space of the contravariant valence-one twistors of Minkowski spacetime is the set of the pairs α A Z := (λ ,πA ′) of spinor fields, which solve the valence-one–twistor equation A ′A B AB A′ ∇ λ = − i𝜀 π. If Zα is a solution of this equation, then Zˆα := (λA,πA ′ + iϒA ′AλA ) is a solution of the corresponding equation in the conformally-rescaled spacetime, where ϒa := Ω− 1∇aΩ and Ω is the conformal factor. In general, the twistor equation has only the trivial solution, but in the (conformal) Minkowski spacetime it has a four complex-parameter family of solutions. Its general solution in the Minkowski spacetime is A A AA ′ λ = Λ − ix πA ′, where A Λ and πA′ are constant spinors. Thus, the space Tα of valence-one twistors, called the twistor space, is four–complex-dimensional, and hence, has the structure Tα = SA ⊕ ¯SA ′. T α admits a natural Hermitian scalar product: if W β = (ωB, σ ′) B is another twistor, then H ′Z αW¯ β′ := λAσ¯ + π ′¯ωA ′ αβ A A. Its signature is (+, +, − ,− ), it is conformally invariant, α ¯β′ α β′ Hαβ′Zˆ Wˆ = H αβ′Z W¯, and it is constant on Minkowski spacetime. The metric H αβ′ defines a natural isomorphism between the complex conjugate twistor space, ¯Tα′, and the dual twistor space, T β := SB ⊕ S¯B ′, by (¯λA ′,¯π ) ↦→ (¯π ,¯λA′) A A. This makes it possible to use only twistors with unprimed indices. In particular, the complex conjugate ¯ A α′β′ of the covariant valence-two twistor A αβ can be represented by the conjugate twistor ′ ′ ¯A αβ := A¯α′β′H α αH ββ. We should mention two special, higher-valence twistors. The first is the infinity twistor. This and its conjugate are given explicitly by

( 𝜀AB 0) ′′ ( 0 0 ) Iαβ := , Iαβ := ¯Iαβ H α′αH β′β = A′B ′ . (7.6 ) 0 0 0 𝜀
The other is the completely anti-symmetric twistor 𝜀αβγδ, whose component 𝜀0123 in an H ′ αβ-orthonormal basis is required to be one. The only nonvanishing spinor parts of 𝜀 αβγδ are those with two primed and two unprimed spinor indices: A′B ′ A′B′ 𝜀 CD = 𝜀 𝜀CD, A′ C ′ A ′C′ 𝜀 B D = − 𝜀 𝜀BD, C′D′ C ′D ′ 𝜀AB = 𝜀AB𝜀, …. Thus, for any four twistors α A i Z i = (λi ,π A′), i = 1,...,4, the determinant of the 4 × 4 matrix, whose i-th column is (λ0i,λ1i,πi0′,πi1′), where the λ0i, …, πi1′ are the components of the spinors λAi and πi ′ A in some spin frame, is
( 0 0 0 0 ) λ 1 λ2 λ 3 λ4 || λ1 λ1 λ1 λ1 || ν := det | 1 2 3 4 | = 1𝜖ijklλAi λBj πkA′πlB ′𝜀AB 𝜀A′B′ = 1𝜀αβγδZα1 Zβ2Z γ3Z δ4, (7.7 ) |( π10′ π20′ π30′ π40′|) 4 4 1 2 3 4 π1′ π1′ π1′ π 1′
where 𝜖ij kl is the totally antisymmetric Levi-Civita symbol. Then Iαβ and I α β are dual to each other in the sense that α β 1 αβγδ I = 2𝜀 Iγδ, and by the simplicity of αβ I one has αβ γδ 𝜀αβγδI I = 0.

The solution AB ω of the valence-two twistor equation, given by Eq. (7.5View Equation), can always be written as a linear combination of the symmetrized product λ(AωB ) of the solutions λA and ωA of the valence-one twistor equation. ωAB uniquely defines a symmetric twistor ω αβ (see, e.g., [426Jump To The Next Citation Point]). Its spinor parts are

( ) ωAB − 1KAB ′ ω αβ = 2 . − 12KA ′B − iM ¯A ′B′

However, Eq. (7.1View Equation) can be interpreted as a ℂ-linear mapping of ω αβ into ℂ, i.e., Eq. (7.1View Equation) defines a dual twistor, the (symmetric) kinematical twistor Aαβ, which therefore has the structure

( ′ ) 0 PAB Aα β = A ′ ¯A′B ′ . (7.8 ) P B 2iJ
Thus, the quasi-local energy-momentum and self-dual angular momentum of the source are certain spinor parts of the kinematical twistor. In contrast to the ten complex components of a general symmetric twistor, it has only ten real components as a consequence of its structure (its spinor part AAB is identically zero) and the reality of P AA′. These properties can be reformulated by the infinity twistor and the Hermitian metric as conditions on Aαβ: the vanishing of the spinor part AAB is equivalent to αγ βδ A αβI I = 0 and the energy momentum is the ′ A αβZαI βγH γγ′Z¯γ part of the kinematical twistor, while the whole reality condition (ensuring both AAB = 0 and the reality of the energy-momentum) is equivalent to
βγ β′γ′ A αβI H γδ′ = A¯δ′β′¯I H γ′α. (7.9 )
Using the conjugate twistors, this can be rewritten (and, in fact, usually is written) as βγ γα′ ¯ β′δ A αβI = (H Aα′β′H ) ¯δ′γ′ ¯γδ (Hδδ′I Hγ′α) = A Iδα. The quasi-local mass can also be expressed by the kinematical twistor as its Hermitian norm [420Jump To The Next Citation Point] or as its determinant [510Jump To The Next Citation Point]:
m2 = − PAA ′P AA′ = − 1A αβ ¯Aα′β′H αα′H ββ′ = − 1A αβ ¯Aαβ, (7.10 ) 4 1 αβ2γδ μνρσ 2 m = 4det Aαβ = 3!𝜀 𝜀 A αμAβνA γρAδσ. (7.11 )
Similarly, as Helfer shows [264Jump To The Next Citation Point], the various components of the Pauli–Lubanski spin vector 1 b cd Sa := 2𝜀abcdP J can also be expressed by the kinematic and infinity twistors and by certain special null twistors: if ′ Z α = (− ixAB πB′,πA′) and ′ W α = (− ixAB σB ′,σA′) are two different (null) twistors such that A αβZ αZ β = 0 and A αβW αW β = 0, then
( ) ( A αβZ αW β ) (2P eπE ′π¯EP fσF′¯σF )−1¯πAπA ′¯σB ¯σB ′ SaP b − SbP a = − ℜ -----γ---δ- . (7.12 ) IγδZ W
(ℜ on the right means ‘real part’.)

Thus, to summarize, the various spinor parts of the kinematical twistor A αβ are the energy-momentum and s.d. angular momentum. However, additional structures, namely the infinity twistor and the Hermitian scalar product, are needed to be able to ‘isolate’ its energy-momentum and angular momentum parts, and, in particular, to define the mass and express the Pauli–Lubanski spin. Furthermore, the Hermiticity condition ensuring that Aαβ has the correct number of components (ten reals) is also formulated in terms of these additional structures.

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