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7.1 Motivations

7.1.1 How do the twistors emerge?

In the Newtonian theory of gravity the mass contained in some finite 3-volume S can be expressed as the flux integral of the gravitational field strength on the boundary S := @S:

gf mS = --1-- va(Daf) dS, (42) 4pG S
where f is the gravitational potential and va is the outward directed unit normal to S. If S is deformed in S through a source-free region, then the mass does not change. Thus the mass mS 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 2-surface integrals of the curvature at infinity [349]. Thus it is natural to expect that the energy-momentum and angular momentum of the source in a finite 3-volume S, given by Equation (5View Equation), can also be expressed as the charge integral of the curvature on the 2-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 (15View Equation) if some ab [ab] w = w is chosen and ab ab m = e. To simplify the subsequent analysis ab w will be chosen to be anti-self-dual: ab A'B' AB w = e w with AB (AB) w = w 11. Thus our claim is to find an appropriate spinor field wAB on S such that

integral gf ab 1 --1-- AB QS [K] := S KaT 3!ebcde = 8pG S w RABcd =: AS [w] . (43)
Since the dual of the exterior derivative of the integrand on the right, and, by Einstein’s equations, the dual of the 8pG times the integrand on the left, respectively, is
' ' ' ' eecdf \~/ e(wABRABcd) = - 2iyFABC \~/ F (AwBC) + 2fABE'F i \~/ E F wAB + 4/\i \~/ FA wFA, (44) af FAF'A' ' FF' -8pGKaT = 2f KAA + 6/\K , (45)
Equations (44View Equation) and (45View Equation) are equal if wAB satisfies
A'A BC A(B C)A' \~/ w = -ie K . (46)
This equation in its symmetrized form, \~/ A'(AwBC) = 0, is the valence 2 twistor equation, a specific example for the general twistor equation \~/ A'(AwBC...E) = 0 for wBC...E = w(BC...E). Thus, as could be expected, AB w depends on the Killing vector a K, and, in fact, a K can be recovered from AB w as A'A 2 A' AB K = 3i \~/ B w. Thus AB w plays the role of a potential for the Killing vector A'A K. However, as a consequence of Equation (46View Equation), Ka is a self-dual Killing 1-form in the sense that its derivative is a self-dual (or s.d.) 2-form: In fact, the general solution of equation (46View Equation) and the corresponding Killing vector are
wAB = - ixAA'xBB'MA'B'+ ix(AA'T B)A'+ _O_AB, ' ' ' ' (47) KAA = T AA + 2xAB M AB',
where MA'B', AA' T, and AB _O_ are constant spinors, using the notation AA' a AA-' A A' x := x sa E AEA-', where {E AA} is a constant spin frame (the ‘Cartesian spin frame’) and ' sAaA- are the standard SL(2, C) Pauli matrices (divided by V~ 2-). These yield that K a is, in fact, self-dual, \~/ 'K '= e M '' AA BB AB A B, and TAA' is a translation and MA'B' generates self-dual rotations. Then AA' A'B' QS [K] = TAA'P + 2MA'B'J, implying that the charges corresponding to _O_AB are vanishing; the four components of the quasi-local energy-momentum correspond to the real ' T AAs, and the spatial angular momentum and centre-of-mass are combined into the three complex components of the self-dual angular momentum JA'B', generated by '' MA B.

7.1.2 Twistor space and the kinematical twistor

Recall that the space of the contravariant valence 1 twistors of Minkowski spacetime is the set of the pairs Za := (cA, pA') of spinor fields, which solve the so-called valence 1 twistor equation \~/ A'AcB = - ieABpA'. If Za is a solution of this equation, then ^Za := (cA, p '+ iU ' cA) A A A is a solution of the corresponding equation in the conformally rescaled spacetime, where -1 Ua := _O_ \~/ a_O_ and _O_ 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 cA = /\A - ixAA'pA', where /\A and pA' are constant spinors. Thus the space Ta of valence 1 twistors, the so-called twistor-space, is 4-complex-dimensional, and hence has the structure a A T = S o+ SA'. a T admits a natural Hermitian scalar product: If b B W = (w ,sB') is another twistor, then a b' A A' Hab'Z W := c sA + pA'w. Its signature is (+, +, -,- ), it is conformally invariant, Hab'^Za W^b'= Hab'ZaW b', and it is constant on Minkowski spacetime. The metric H ' ab defines a natural isomorphism between the complex conjugate twistor space, a' T, and the dual twistor space, B' Tb := SB o+ S, by ' ' (cA ,pA) '--> (pA, cA ). This makes it possible to use only twistors with unprimed indices. In particular, the complex conjugate Aa'b' of the covariant valence 2 twistor Aab can be represented by the so-called conjugate twistor Aab := Aa'b'Ha'aHb'b. We should mention two special, higher valence twistors. The first is the so-called infinity twistor. This and its conjugate are given explicitly by

( AB ) ( ) Iab := e 0 , Iab := Ia'b'Ha'aHb'b = 0 0A'B' . (48) 0 0 0 e
The other is the completely anti-symmetric twistor eabgd, whose component e0123 in an Hab'-orthonormal basis is required to be one. The only non-vanishing spinor parts of eabgd are those with two primed and two unprimed spinor indices: ' ' ' ' eAB CD = eA B eCD, ' ' ' ' eA BC D'= - eA C eBD, eABC'D'= eABeC'D', …. Thus for any four twistors Zai = (cAi ,piA'), i = 1,...,4, the determinant of the 4 × 4 matrix whose i-th column is (c0,c1,pi',pi') i i 0 1, where the c0 i, …, pi' 1 are the components of the spinors A ci and i pA' in some spin frame, is
( ) c01 c02 c03 c04 1 1 1 1 c 1 c2 c 3 c4 1 ij A B k l A'B' 1 a b g d n := det 1 2 3 4 = 4e klci c j pA'pB'eABe = 4eabgdZ1 Z2Z 3Z 4, (49) p0'p 0'p0'p 0' p11'p21'p31'p41'
where eijkl is the totally antisymmetric Levi-Civita symbol. Then Iab and Iab are dual to each other in the sense that Iab = 12eabgdIgd, and by the simplicity of Iab one has eabgdIabIgd = 0.

The solution wAB of the valence 2 twistor equation, given by Equation (47View Equation), can always be written as a linear combination of the symmetrized product (A B) c w of the solutions A c and A w of the valence 1 twistor equation. AB w defines uniquely a symmetric twistor ab w (see for example [313Jump To The Next Citation Point]). Its spinor parts are

( AB 1 A ) ab w - 2K B' w = 1 'B '' . - 2KA - iMA B

However, Equation (43View Equation) can be interpreted as a C-linear mapping of wab into C, i.e. Equation (43View Equation) defines a dual twistor, the (symmetric) kinematical twistor Aab, which therefore has the structure

( ' ) 0 PAB Aab = A' A'B' . (50) 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 Aab: The vanishing of the spinor part AAB is equivalent to AabIagIbd = 0 and the energy momentum is the a bg ' g' AabZ I Hgg Z part of the kinematical twistor, while the whole reality condition (ensuring both AAB = 0 and the reality of the energy-momentum) is equivalent to
AabIbgHgd'= Ad'b'Ib'g'Hg'a. (51)
Using the conjugate twistors this can be rewritten (and, in fact, usually it is written) as AabIbg = (Hga'Aa'b'Hb'd) (Hdd'Id'g'Hg'a) = AgdIda. Finally, the quasi-local mass can also be expressed by the kinematical twistor as its Hermitian norm [307Jump To The Next Citation Point] or as its determinant [371Jump To The Next Citation Point]:
2 A' A 1 aa' bb' 1 ab m = -PA P A'= - 2AabAa'b'H H = - 2AabA , (52) m4 = 4det Aab = 1eabgdemnrsAamAbnAgrAds. (53) 3!
Thus, to summarize, the various spinor parts of the kinematical twistor Aab are the energy-momentum and s.d. angular momentum. However, additional structures, namely the infinity twistor and the Hermitian scalar product, were needed to be able to ‘isolate’ its energy-momentum and angular momentum parts, and, in particular, to define the mass. Furthermore, the Hermiticity condition ensuring A ab to have the correct number of components (ten reals) were also formulated in terms of these additional structures.
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