Motivations

7.1 Motivations

7.1.1 How do the twistors emerge?

In the Newtonian theory of gravity the mass contained in some finite three-volume $Σ$ can be expressed as the flux integral of the gravitational field strength on the boundary $𝒮 := ∂Σ$ :

where $ϕ$ is the gravitational potential and $va$ is the outward-directed unit normal to $𝒮$ . If $𝒮$ is deformed in $Σ$ through a source-free region, then the mass does not change. Thus, the mass $m Σ$ is analogous to charge in electrostatics.

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 [442 ]. Thus, it is natural to expect that the energy-momentum and angular momentum of the source in a finite three-volume $Σ$ , given by Equation (2.5 ), 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 Equation (3.7 ) if some $ab [ab] ω = ω$ is chosen and $ab ab μ = 𝜀$ . To simplify the subsequent analysis, $ab ω$ will be chosen to be anti-self-dual: $ab A′B′ AB ω = 𝜀 ω$ with $AB (AB ) ω = ω$ ¹⁰ . Thus, our goal is to find an appropriate spinor field $ωAB$ on $𝒮$ such that

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

Expressions (7.3

) and (7.4

) are equal if $ωAB$ satisfies

This equation in its symmetrized form, $∇A ′(A ωBC ) = 0$ , is the valence 2 twistor equation, a specific example for the general twistor equation $∇A ′(AωBC...E ) = 0$ for $ωBC...E = ω(BC...E)$ . Thus, as could be expected, $AB ω$ depends on the Killing vector $a K$ , and, in fact, $a K$ can be recovered from $AB ω$ as $A ′A 2 A′ AB K = 3i∇ B ω$ . Thus, $AB ω$ plays the role of a potential for the Killing vector $A′A K$ . However, as a consequence of Equation (7.5

), $Ka$ is a self-dual Killing 1-form in the sense that its derivative is a self-dual (s.d.) two-form: In fact, the general solution of Equation (7.5

) and the corresponding Killing vector are

where $¯ MA ′B′$ , $AA′ T$ , and $AB Ω$ are constant spinors, and using the notation $AA′ a AA-′ A¯A ′ x := x σa ℰA-ℰA′$ , 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 $K a$ is, in fact, self-dual, $∇ ′K ′ = 𝜀 M¯ ′ ′ AA BB AB AB$ , $AA′ T$ is a translation and $¯ MA ′B′$ generates self-dual rotations. Then $AA′ Q𝒮 [K ] = TAA ′P$ $′′ +2 M¯A ′B′ ¯JA B$ , implying that the charges corresponding to $ΩAB$ are vanishing; the four components of the quasi-local energy-momentum correspond to the real $′ T AA$ s, and the spatial angular momentum and center-of-mass are combined into the three complex components of the self-dual angular momentum $¯A′B ′ J$ , generated by $¯ ′′ MA B$ .

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 $Z α := (λA,πA ′)$ of spinor fields, which solve the valence-one–twistorequation $∇A ′A λB = − i𝜀ABπA ′$ . If $Zα$ is a solution of this equation, then $Zˆα := (λA,π ′ + iϒ ′λA ) A A A$ is a solution of the corresponding equation in the conformally-rescaled spacetime, where $− 1 ϒa := Ω ∇aΩ$ 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 $λA = ΛA − ixAA ′πA ′$ , where $ΛA$ 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 $α A ¯ T = S ⊕ SA ′$ . $α T$ admits a natural Hermitian scalar product: if $β B W = (ω ,σB ′)$ is another twistor, then $α β′ A A ′ H αβ′Z W¯ := λ σ¯A + πA′¯ω$ . Its signature is $(+, +, − ,− )$ , it is conformally invariant, $Hαβ′ZˆαW¯ˆβ′ = H αβ′Z αW¯ β′$ , and it is constant on Minkowski spacetime. The metric $H ′ αβ$ defines a natural isomorphism between the complex conjugate twistor space, $¯α′ T$ , and the dual twistor space, $¯B′ T β := SB ⊕ S$ , by $′ ′ (¯λA ,¯πA ) ↦→ (¯πA, ¯λA )$ . This makes it possible to use only twistors with unprimed indices. In particular, the complex conjugate $A¯α′β′$ of the covariant valence 2 twistor $A αβ$ can be represented by the conjugate twistor $¯A αβ := A¯α′β′H α′αH β′β$ . We should mention two special, higher-valence twistors. The first is the infinity twistor. This and its conjugate are given explicitly by

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 $𝜀αβγδ$ are those with two primed and two unprimed spinor indices: $′ ′ ′ ′ 𝜀AB CD = 𝜀A B 𝜀CD$ , $′ ′ ′ ′ 𝜀A BC D ′ = − 𝜀A C 𝜀BD$ , $𝜀ABC ′D′ = 𝜀AB𝜀C ′D ′$ , …. Thus, for any four twistors $Z αi = (λAi ,πiA′)$ , $i = 1,...,4$ , the determinant of the $4 × 4$ matrix, whose $i$ -th column is $(λ0,λ1 ,πi′,πi ′) i i 0 1$ , where the $λ0 i$ , …, $πi ′ 1$ are the components of the spinors $A λ i$ and $i πA ′$ in some spin frame, is

( ) λ01 λ02 λ03 λ04 | 1 1 1 1 | || λ 1 λ2 λ 3 λ4 || 1 ij A B k l A′B′ 1 α β γ δ ν := det | 1 2 3 4 | = 4𝜖 klλi λ j πA′πB ′𝜀AB 𝜀 = 4𝜀αβγδZ1 Z2Z 3Z 4, (7.8 ) ( π0′ π0′ π0′ π 0′) π11′ π21′ π31′ π41′

where $𝜖ijkl$ is the totally antisymmetric Levi-Civita symbol. Then $Iαβ$ and $Iα β$ are dual to each other in the sense that $Iα β = 12𝜀αβγδIγδ$ , and by the simplicity of $Iαβ$ one has $𝜀αβγδI αβIγδ = 0$ .

The solution $ωAB$ of the valence 2 twistor equation, given by Equation (7.6 ), can always be written as a linear combination of the symmetrized product $(A B) λ ω$ of the solutions $A λ$ and $A ω$ of the valence-one twistor equation. $AB ω$ uniquely defines a symmetric twistor $αβ ω$ (see, for example, [392 ]). Its spinor parts are

( AB 1 A ) αβ ω − 2K B′ ω = 1 ′B ¯ ′′ . − 2KA − iMA B

However, Equation (7.2 ) can be interpreted as a $ℂ$ -linear mapping of $ωαβ$ into $ℂ$ , i.e., Equation (7.2 ) defines a dual twistor, the (symmetric) kinematical twistor $Aαβ$ , which therefore has the structure

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 $AA′ P$ . 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 $A αβIαγIβδ = 0$ 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

Using the conjugate twistors, this can be rewritten (and, in fact, usually is written) as $A αβIβγ = (H γα′ ¯Aα′β′H β′δ)$ $(Hδδ′¯Iδ′γ′Hγ′α) = A¯γδIδα$ . The quasi-local mass can also be expressed by the kinematical twistor as its Hermitian norm [386

] or as its determinant [469

Similarly, as Helfer shows [242

], the various components of the Pauli–Lubanski spin vector $Sa := 12𝜀abcdPbJ cd$ 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 $α β A αβZ Z = 0$ and $α β A αβW W = 0$ , then

( $ℜ$ 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 $A αβ$ has the correct number of components (ten reals) is also formulated in terms of these additional structures.