Absence of Differential Correlations Between the Wave Equations for Upper-Lower One-Index Twistor Fields Borne by the Infeld-van der Waerden Spinor Formalisms for General Relativity
aa r X i v : . [ m a t h - ph ] N ov Absence of Differential Correlations Between theWave Equations for Upper-Lower One-IndexTwistor Fields Borne by the Infeld-van derWaerden Spinor Formalisms for GeneralRelativity
Karla WeberDepartment of PhysicsCentre for Technological Sciences-UDESCJoinville 89223-100, Santa Catarina, Brazil.e-mail: karlaweberfi[email protected]. G. CardosoDepartment of MathematicsCentre for Technological Sciences-UDESCJoinville 89223-100, Santa Catarina,Brazil.e-mail: [email protected] numbers:04.20.Gz, 03.65.Pm, 04.20.Cv, 04.90.+eKEY WORDS: wave equations; twistor fields;Infeld-van der Waerden spinor formalisms.
Abstract
It is pointed out that the wave equations for any upper-lower one-index twistor fields which take place in the frameworks of the Infeld-vander Waerden γε -formalisms must be formally the same. The only reasonfor the occurrence of this result seems to be directly related to the fact thatthe spinor translation of the traditional conformal Killing equation yieldstwistor equations of the same form. It thus appears that the conventionaltorsionless devices for keeping track in the γ -formalism of valences ofspinor differential configurations turn out not to be useful for sorting outthe typical patterns of the equations at issue. Introduction
Certain calculational techniques were utilized in an earlier paper [1] for workingout the twistor equation for contravariant one-index fields in curved spacetimes.The main aim associated to the completion of the relevant procedures was to de-rive one of the simplest sets of wave equations for conformally invariant spinorfields that should presumably take place in the frameworks of the Infeld-vander Waerden γε -formalisms [2-4]. A striking feature of these wave equations isthat they involve no couplings between the twistor fields and wave functionsfor gravitons [5-7]. In actuality, the only coupling configurations brought aboutby the techniques allowed for thereabout take up appropriate outer productscarrying the fields themselves along with some electromagnetic wave functionsfor the γ -formalism [4, 5]. Loosely speaking, the non-occurrence of ε -formalismcouplings stems even in the case of charged fields from the applicability of apeculiar property of partially contracted second-order covariant derivatives ofspin-tensor densities which carry only one type of indices as well as suitablegeometric attributes [8-10]. Indeed, the electromagnetic curvature contribu-tions that normally enter such derivative expansions really cancel out wheneverthe non-vanishing entries of the valences of the differentiated densities are ade-quately related to the respective weights and antiweights [4].The present paper just brings forward the result that the above-mentionedwave equations possess the same form as the ones for the corresponding lower-index fields. It shall become clear that the legitimacy of this result rests uponthe fact that the spinor translation of the classical conformal Killing equationleads to twistor equations which must be formally the same. Consequently, theconventional covariant devices for keeping track of valences of spinor differentialconfigurations in the γ -formalism [4, 6], turn out not to be useful as regardsthe attainment of the full specification of the formal patterns for the field andwave equations being considered. We mention, in passing, that such deviceshad originally been built up in connection with the derivation of a system ofsourceless gravitational and electromagnetic wave equations [5], with the perti-nent construction having crucially been based upon the implementation of thetraditional eigenvalue equations for the γ -formalism metric spinors [2, 3]. Itmay be said that the motivations for elaborating upon the situation entertainedherein rely on our interest in completing the work of Ref. [1], thereby making upappropriately the set of γε -wave equations which emerge from the curved-spaceversion of twistor equations for one-index fields.The paper has been outlined as follows. In Section 2, we exhibit the twistorfield equations which are of immediate relevance to us at this stage. We lookat the twistor wave equations in Section 3, but the key remarks concerning thelack of differential correlations between them shall be made in Section 4. It willbe convenient to employ the world-spin index notation adhered to in Ref. [11].In particular, the action on an index block of the symmetry operator will beindicated by surrounding the indices singled out with round brackets. Withoutany risk of confusion, we will utilize a torsion-free operator ∇ a upon takingaccount of covariant derivatives in each formalism. Likewise, the D’Alembertian2perator for either ∇ a will be written as (cid:3) . A horizontal bar will be used oncein Section 4 to denote the ordinary operation of complex conjugation. Einstein’sequations should thus be taken as2Ξ ab = κ ( T ab − T g ab ) , T + T ab g ab , where T ab amounts to the world version of the energy-momentum tensor of somesources, g ab denotes a covariant spacetime metric tensor and κ stands for theEinstein gravitational constant. By definition, the quantity ( − ab is identifiedwith the trace-free part of the Ricci tensor R ab for the Christoffel connexion of g ab . The cosmological constant λ will be allowed for implicitly through thewell-known trace relation R = 4 λ + κT, R + R ab g ab . Our choice of sign convention for R ab coincides with the one made in Ref. [11],namely, R ab + R ahbh , with R abcd being the corresponding Riemann tensor. We will henceforth assumethat the local world-metric signature is (+ − −− ). The calculational techniquesreferred to before shall be taken for granted at the outset. The differential patterns borne by the original formulation of twistor equationsin a curved spacetime [12-14] may be thought of as arising in either formalismfrom ∇ ( AA ′ K BB ′ ) = 14 ( ∇ CC ′ K CC ′ ) M AB M A ′ B ′ , (1)and ∇ ( AA ′ K BB ′ ) = 14 ( ∇ CC ′ K CC ′ ) M AB M A ′ B ′ , (2)where the K -objects amount to nothing else but the Hermitian spinor versionsof a null conformal Killing vector, and the kernel letter M accordingly standsfor either γ or ε .It should be emphatically observed that the genuineness of (1) and (2) asa system of equivalent field equations lies behind a general covariant-constancyproperty of the Hermitian connecting objects for both formalisms [2, 3]. Thus,these equations can be obtained from one another on the basis of the metric-compatibility requirements ∇ a ( M AB M A ′ B ′ ) = 0 ⇔ ∇ a ( M AB M A ′ B ′ ) = 0 . (3) The symmetry operation involved in Eqs. (1) and (2) must be applied to the index pairs. K AA ′ = ξ A ξ A ′ , (4)along with its lower-index version, after accounting for some manipulations, weget the statements ∇ A ′ ( A ξ B ) = 0 , ∇ A ′ ( A ξ B ) = 0 , (5)which, when combined together with their complex conjugates, bring out thetypical form of twistor equations. We stress that solutions to twistor equationsare generally subject to strong consistency conditions (see, for instance, Ref.[1]).Either ξ -field of (5) bears conformal invariance [13, 14], regardless of whetherthe underlying spacetime background bears conformal flatness. In the γ -formalism,the entries of the pair ( ξ A , ξ A ), and their complex conjugates, come into playas spin vectors under the action of the Weyl gauge group of general relativity[15], whereas their ε -formalism counterparts appear as spin-vector densities ofweights (+1 / , − /
2) and antiweights (+1 / , − / In the γ -formalism, ξ A shows up [1] as a solution to the wave equation( (cid:3) − R
12 ) ξ A = 2 i φ AB ξ B , (6)with φ AB denoting a wave function for Infeld-van der Waerden photons [16-18]. In order to derive in a manifestly transparent manner the γ -formalismwave equation for the lower-index field ξ A , we initially recast the second of thestatements (5) into 2 ∇ A ′ A ξ B = γ AB γ LM ∇ A ′ L ξ M , (7)and then operate on (7) with ∇ A ′ C . It follows that, calling upon the splitting [5] ∇ A ′ C ∇ AA ′ = 12 γ AC (cid:3) − ∆ AC , (8)together with the definition ∆ AC + −∇ A ′ ( A ∇ C ) A ′ , (9)and the property [4] ∇ a ( γ AB γ LM ) = 0 , (10)we arrive at (cid:3) ξ A −
23 ∆ AB ξ B = 0 . (11)The explicit calculation of the ∆-derivative of (11) gives∆ AB ξ B = R ξ A + iφ AB ξ B , (12)4hence, fitting pieces together suitably, yields( (cid:3) − R
12 ) ξ A = 2 i φ AB ξ B . (13)It should be evident that the equality (11) remains formally valid in the ε -formalism as well. Therefore, since the ε -formalism field ξ A is a covariant one-index spin-vector density of weight − /
2, the ε -counterpart of the derivative(12) has to be expressed as the purely gravitational contribution ∆ AB ξ B = R ξ A . (14)Hence, the ε -formalism statement corresponding to (13) must be spelt out as( (cid:3) − R
12 ) ξ A = 0 . (15) The formulae shown in Section 3 supply the entire set of wave equations forone-index conformal Killing spinors that should be tied in with the context ofthe γε -frameworks. It is worth pointing out that the common overall sign on theright-hand sides of (6) and (13), is due to the γ -formalism metric relationshipbetween the differential configuration (12) and∆ AB ξ B = − ( R ξ A + iφ AB ξ B ) , with the aforesaid relationship actually coming about when we invoke the well-known derivatives [4]∆ AB γ CD = 2 iφ AB γ CD , ∆ AB γ CD = − iφ AB γ CD . What happens with regard to it is, in effect, that the pieces of those contracted∆ ξ -derivatives somehow compensate for each other while producing the formalcommonness feature of the apposite couplings through∆ AB ξ B + ∆ AB ξ B = 2 iφ AB ξ B . At first sight, one might think that a set of differential correlations betweenthe γ -formalism wave equations for ξ A and ξ A could at once arise out of utilizingthe devices [4, 5] (cid:3) ξ A = γ AB (cid:3) ξ B + ( (cid:3) γ AB ) ξ B + 2( ∇ h γ AB ) ∇ h ξ B , and (cid:3) ξ A = γ BA (cid:3) ξ B + ( (cid:3) γ BA ) ξ B + 2( ∇ h γ BA ) ∇ h ξ B , For a similar reason, the ε -formalism version of (6) reads ( (cid:3) − R ) ξ A = 0. It will becomemanifest later in Section 4 that the relation (14) is compatible with this assertion.
5n conjunction with the eigenvalue equations [2-4] ∇ a γ AB = iβ a γ AB , ∇ a γ AB = ( − iβ a ) γ AB , and (cid:3) γ AB = − Θ γ AB , (cid:3) γ AB = − Θ γ AB , where Θ + β h β h + i ∇ h β h , and β a is a gauge-invariant real world vector. If any such raising-loweringdevice were implemented in a straightforward way, then a considerable amountof ”strange” information would thereupon be brought into the picture whilstsome of the contributions involved in the intermediate steps of the calculationsthat give rise to the characteristic statements ∇ ( A ( A ′ K B ′ ) B ) = 0 , ∇ ( A ( A ′ K B ′ ) B ) = 0 , would eventually be ruled out. We can conclude that any attempt at making useof a metric prescription to recover either of (6) and (13) from the other, wouldvisibly carry a serious inadequacy in that the twistor equations (5) could not bebrought forth simultaneously. It is obvious that the property we have deducedultimately reflects the absence of index contractions from twistor equations.It would be worthwhile to derive the γε -wave equations for twistor fields ofarbitrary valences. This issue will probably be considered further elsewhere. ACKNOWLEDGEMENT:
One of us (KW) should like to acknowledge the Brazilian agency CAPES forfinancial support.
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