KFKI-1989-09/B
Z. PERJÉS
UNITARY SPINOR METHODS IN GENERAL RELATIVITY
Hungarian Academy of Sciences CENTRAL RESEARCH INSTITUTE FOR PHYSICS
BUDAPEST KFKI-1989-09/B PREPRINT
UNITARY SPINOR METHODS IN GENERAL RELATIVITY
Z. PERJES
Central Research Institute for Physics H 1526 Budapest 114, P.OB.49, Hungary
Mathematical Relativity miniconference, Australian National University. 1988.
HU ISSN 0368 5330 Z. Perjés: Unitary spinor methods in general relativity KFKI 1989 09/B
ABSTRACT A survey is given of the structure and applications of spinor fields in three dimensional (pseudo ) Riemannian manifolds. A systematic treatment, independent of the metric signature, is possible since there exists a fairly general structure, to be associated with unitary splnnors, which encompasses all but the reality properties. The discussion begins with the algebraic and analytic properties of unitary splnors, the Ricci identities and curvature spinor, followed by the spinor adjungation as space reflection, and 'he SUÍ2) and SU(1.1) spin coefficients with some applications. The rapidly increasing range of applications includes space times with Killing symmetries, the initial value formulation, positivity theorems on gravitational energy and topotogically massive gauge theories.
3. Перьеш: Унитарные спиномые методы в общей теории относительности. KFKI-1989-09/B
АННОТАЦИЯ
Рассматривается структура спинорных полей и их применение в трехмерных (псевдо-) римановских множествах. Независимо от сигнатуры метрики имеется воз можность систематического, независимого приближения, поскольку существует об щая структура, охватывающая, кроме действительности, все другие свойства. Об суждение начинается с рассмотрения алгебраических и аналитических свойств уни тарных спиноров, затем дается ознакомление с тождественностью Риччи и спинором кривизны, обратимостью спиноров, как пространственное отражение, а также со спиновыми коэффициентами SU(2) и SU(1,1). Количество применений быстро растет, в их числе: пространства-времена, допускающие симметрию Киллинга, задача с на чальными условиями, положительность гравитационной энергии и калибровочные по ля с топологической массой. I
Perjós Z.: Unitér spinormodszerek az általános relativitáselméletben. KFKI 1989 09/B KIVONAT Áttekintjük a splnormezők szerkezetét és alkalmazásait háromdimenziós (pszeudo ) Ríemann sokaságokban A metrika szignatúrájától független rendszeres megközelítés lehetséges, mert létezik a valósságon kívül minden más tulajdonságot felölelő általános struktúra. A tárgyalást a spinorok algebrai és analitikus tulajdonságaival kezdjük, majd a Ricci azonosságokat, a görbületi spinort, spinorok adjungáltját, mint térbeli tükrőzöttet, az SU(2) ós SU(1,1) splnegyütthatókat ismertetjük Az alkalmazások száma gyorsan nő, közöttük: a Killing szimmetriát megengedő téridők, a kezdőértékprobléma, a gravitációs energia pozitivitása ós a mértékterek topológiai tömeggel 1. A LITTLE HISTORY
One of the earliest successes of spinor techniques in general relativity is Witten's [l] version of the Petrov classification which was later perfected by Penrose [2] and complemented by the spin coefficient techniques of Newman and Penrose [3]. In the following years, the SL(2, C) spinors have gradually been accepted as a useful mathe matical working tool in relativity, even though they never ranked to the straightforward physical utility which spinors in particle physics enjoy. The prevailing view as to why spinors sail so well in curved space-time is that they are seen closely related to null and causal structures. Other insights of more physical nature have only recently been provided by the supersymmetric theories [4].
The applications of the spinors of unitary and pseudo-unitary subgroups of the Lorentz group escaped the attention of relativists until as late as 1970. Then, in an at tempt to establish a spinor approach to stationary space-times, the present author has worked out an SU(2) spinor formalism [5]. These technique* have been used for obtain ing exact solutions of the gravitational field equations [6]. A corresponding formalism for the non-compact group SU(1,1) has also been developed [7]. It hau been 2
shown possible to formulát» the stationary axisymmetric vacuum gravitational equations such that the complete description is carried by a set of SU(1,1) spin coefficients, and with no curvature quantities appearing.
In 1980, Sommers [8] introduced the notion of 'space spinors' in connection with SU(2) spinors relative to a space-like foliation of the space-time. This approach has been taken up by Sen [9] for quantizing the spin-3/2 massless field in a curved background.
Recently, the pace of applications has increased, due partly to the SU(2) spinorial nature of many supergravity models [4]. But SU(2) spinors enter Witten's expression for the gravitational energy [10], as well as the theory of topologically massive gauge fields [11]. A new spate of works followed Ashtekar's discovery of new canonical variables for the quantization of the gravitational degrees of freedom [12]. His configuration space is coordinatized essentially by the soldering forms of SU(2) spinors. Unitary spinors have been introduced on manifolds of arbitrary non-null congruences [13]. The ensuing description of space-time contains both the spinors in hypersurfaces and in the manifolds of Killing trajectories as special cases.
It has already been noted by Barut [14] that SU(2) and SU(1,1) spinors can be treated in a unified formalism in which one does not specify the signature of the 3- metric. The signature-independent properties, characterizing what will be called uni tary spinors, embrace all spinorial relations but those involving adjunction. In the present note we shall show that there exists a remarkably elaborate structure shared by both genres of spinor fields in (pseudo-) Riemannian З-врасев. Adjoining of spinors is relegated to Sec. 5. Applications in general relativity will be discussed at the end of •ach section.
s '{"* t
)~% 4É* 3
2. UNITARY SPINORS
The algebra of unitary spinors encompasses those properties of the SU(2) and SU(1,1) spinors which do not depend on the signature of the associated metric.
2.0 DEFINITION [15] Spin apace is a pair (E,c) where Б is a two-dimensional vector space over the field of complex (or real) numbers and с a symplectic structure on E.
Note that с provides an ismorphism
(2.1) £ : E -» E between E and the dual space E*. We shall use an index notation such that £д € E is
an element of the vector space E and £A € E* an element of the dual space. Capital Roman spinor indices A, B,... may take the values 0 and 1. Then (2.1) can be written
A д £ -» £в = £ «лв where {еАв) = ( « ft )• Note that cAB = -€вл- The inverse map employe the spinor tAB satisfying
(2.2) елв*св=6* where 6£ is the identity map on E.
2.1 DEFINITION A spinor nfd0- of p upstairs and q downstairs indices is a (p, q) tensor over E.
2.2 PROPOSITION The linear map L: E -» E preserving the spinor eAB has the determinant 1. The group of the maps L is SL(2,C).
The proof is all too easy.
2.3 DEFINITION The spinor
(2.3) QABCD — *л[с*о)в> 4 called the metric, yields a bilinear map V ® V —» С where V is the vector space
(2.4) V = {Ив № = 0}.
The inclusion of V in E ® E* defines the map a : E ® E* -»V rendering to each v* e E ® E* its trace-free part. This map is sometimes called a soldering form and has
the index structure o\ B where
(2.5) о\в=<ГВА.
Lower case Roman indices refer concisely to the vector space V.
2.4 DEFINITION Unitary spin space is the triplet (E,«, o) where a : E®E" -• V is the map satisfying
i A i C i A i C ii (2.6) °c ° B +<>c <>B =9 6E
Í ADCD andíT = The vector space V is oriented because Trvx (v3, v3] defines canonically a three-form e(v1,V2,v3) on V [16]. In the index notation we have tijk = е^ыу/д where eijk is the skew numerical Levi-Civita symbol and g = det[gik]. (Thus с<>* is imaginary when the signature of the metric is negative.) A 2.5 COROLLARY [5] The soldering forms oB satisfy the Lie product rule J (2.7) O~ÍACO% ~°ÍAO^ÍB — v2ieiiho AB. Here the right-hand side changes sign under reversal of the orientation of V. 2.6 PROPOSITION Let F<» = F{ih] be a bivector. Then the spinor FABCD = F**aABacD has the decomposition (2.8) FABOD — ЖвоФло -гСлсФво)- Proof. In three dimensions, the tensor Fik can be written equivalently ik (2.9) Fi = y/TfitiikF . 5 The (axial) vector Fi has the spinor components Флв = алвЪ- в в Using the identity <^ <т*в = -Í*, we have £ = -о? ФАв- Substitution of (2.9) in флв yields Флв=у/Що*лви,шР**. This contains the terms on the right hand side of Eq. (2.7), hence С (2.10) Флв=-%РАО в- Now adding the terms %(FADCB - FADCB) to FABCD, it can be written identically as FABCD = FA\B\C\D\ + F\A\D\C)B • For an arbitrary spinor 17 with a pair of skew indices R we have ibis -riBA = €ABnR . Thus FABCD = \(*BDFARC * + eAcFRD*B). Inserting here (2.10), the decomposition (2.8) follows. 2.7 DIGRESSION TO SL(2,C) The algebraic properties of unitary spinors can be derived [5] from the 5L(2, C) spinor algebra in the presence of a preferred non-null four-vector a. The soldering forms a of 5L(2, C) satisfy the defining relation [3] (211) a»AC'°vB +аиВС'°цА =9**еЛВ where Greek indices /1,1/,... range through the values 0, 1, 2 and 3 and primed spinor indices refer to the complex conjugate representation. A hat will signify that the entity refers to the 4-space-time whenever such an emphasis is necessary. Choose coordinates adapted to the non-null vector a such that the metric $M„ is independent of x° = t. Then, given a solution о at t = t0 of (2.11), this a will continue to be a solution for all possible values of t. We may choose the soldering form a to be independent of t. Let us denote the norm of the vector a by (2.12) / = a"aM=$00. Then the 4-metric has the decomposition (2.13) (M-('r\+/U'U' '?) with the inverse where g is the metric in the orthogonal 3-complement. We now decompose Eq. (2.11). From the mixed components with {p,v) = (t',0) we get: (2.14) Олегов' + *лс-°ов' = О- The spinor o0AC. — aAC> is invariantly defined. We now introduce the soldering forms in the 3-space by defining а (2.15) aAB = - -Г°ЛС'V*ÍcВ« • The multiplying factor here serves later convenience. Eq. (2.11) then expresses the symmetry (2.5) of the soldering forms in their spinor indices. The components of Eq. (2.11) with О*.") = (*.У) yield the anticommutation prop erty (2.6) of the soldering forms of unitary spinors. The product relation of soldering forms in the 4-space-time has the form [3] (2.16) ottADIo"CD, = iACeD>p<. By use of the 3+1 decomposition of scalar products 1 < fc (2.17) fi"6"=r («ofo-« f ffjfc) we obtain the metric (2.3) for the 3-space: (2.18) 3. SPINOR DERIVATIVES In this section we consider covariant derivatives of unitary spinor fieldso n (pseudo-) Riemannian 3-manifolds. 3.1 DEFINITION A unitary (p,g)- spinor field is a local section of a bundle of unitary (p, g)-spinors over a (pseudo-) Riemannian 3-manifold (M,g). The covariant derivative of a spinor with in reference to the metric g is defined in the contemporary literature such that the soldering forms and the fundamental spinor (.AB are covariantly constant. 3.2 DEFINITION The covariant derivative V of a (p, ?)-spinor field is a map into a (p + 1,<7 + l)-spinor field with the usual linearity and Leibnitz properties of derivatives. The map V coincides with the gradient map of scalars when (p,?) = (0,0). and has the further properties (3.1) Vko\P =0, Vi(AB=0. 3.3 DEFINITION The spinor affine connection TfA in the covariant derivative (3.2) VitA=b.i-Tiito can be expressed in the form [5] (3.3) r,J =-£,,-V,Ci4+o*X). 3.4 PROPOSITION The spinor Rieei identity has the form (3.4) Vp^V^ti =флвоо? +ЪЛ1вСо)Ь Proof. Eq. (3.4) can be derived from the Ricci identity (3.5) (VfcVy-VyV*)»* =RrijkVr 8 i i k by choosing V( to be the null vector v4 = of" £л£в. Transvecting with o BHa DQa cp and using (2.15) we get А (3.6) (VcpVpQ - VDQVCP){ÍBÍH) = -Яливысгоо£ Р where VAB = oABVi and the spinor structure of the Riemann tensor Rijkl is given by RAUBNCPDQ =0AuaBNaGP°,DQRan• The curvature tensor can be decomposed into irreducible parts by use of the scheme (2.8) for pairs of skew indices: RAUBNCPDQ = — *ЛВ*СоФм NPQ * *ЛВ*РяФмЫСО ~~ £*f Н^СРФАВРЯ — *U N*rQ + *ABePQ\*MCeND + tMD*AC) + «Л в fС I? (CJ/P СЛГ Q + с«0*ЛГ|»)] r where ^ABCJD = j(ßi* - \9ihR)oAB<^CD, R+k = Rirtk9 ' is the Ricci tensor and A = ß/24. Transvecting Eq. (3.6) with eFQijBriN where riN is an arbitary non- B N vanishing spinor field, and removing the overall factor 2{n £B )n , we obtain the spinor version (3.4) of the Ricci identity. 2.4 DIGRESSION TO SL(2,C) One can decompose the connection f in terms of Г. Introduce the complex gravitational vector [5] (3.8) *!7). This can be expressed in spinor terms as i B (3.9) GAB=o ABGi=$ A'aB)B,. Straightforward computation yields f"=-^°; (зло) is. = r,i - ^«./c5 - i-jpu&'I M 9 4. ADJUNCTION 'Che spinor aAC> establishes, in a natural way, a map between the primed and unprinvjd spin spaces. The 'unpriming' of spinor indices proceeds by contraction of aAB any primed spinor index with \l^\ • As an example, the unprimed version Ълв — Vi/i^^fl' of the vector v has the irreducible parts V{AB) — V*OiABi ^\AB\ — *ABVo- Thus the 3+1 decomposition of tensors becomes a symmetry operation over spinor indices. 4.0 DEFINITION [5] The adjoint spinor 4M is defined by (4.1) ?Л='\А-УВ'Ь: 4.1 PROPOSITION The double adjoint has the property Uo\ (ci\iA_l-ZA if f > 0 (time-like a), У > l? ' ~\£A if / < 0 (space-like a)0 . Proof. Contraction of Eq. (3.7) with a" a" yields the product rule for the aAB> spinors: (4.3) »4сИ»/' = {Я- 4.3 DEFINITION The norm of the spinor £ is defined by (4-4) ||С||в=,€м€д. 4.4 PROPOSITION The epinor norm (4.4) is real. Proof. The complex conjugation of scalar products proceeds as follows, (CAV) =L'fiA' = -,Ъ>*Ас'*ов'*1в> (4.5) / = ±íWA 10 where the upper and lower sign holds for / > 0 and / < 0, respectively. In particular, п-в) ey-iuV'^ni. 4.4 PROPOSITION The soldering forms have the Hermiticity property (4-7) *л'в=*"лв. Proof. Eq. (4.7) follows from the Hermiticity of the SL(2,C) soldering forms and from (2.12). Although the timelike case carries here a negative sign, this is the kind of behavior a Hermitian product aAB — CAtB\ exhibits. 4.5 PROPOSITION The covariant derivation commutes with adjunction. The Proof is straightforward and will be omitted here. When the four-vector at e-Iike, 3-spinors can be chosen real, but the adjunc tion will retain a direct geometrical interpretation, to be discussed in the next section. 5. TRIAD FORMALISM Consider a spinor 77 € E of positive norm. Such a spinor always exists and can be scaled to (6.1) VA4A=1- (In this section we shall adopt the tilde notation of Ref. [11] for adjointe.) Then (5.2) ^«vW/f is a real and self-adjoint unit vector: l* = f and (5.3) m - rfAo g tf 11 a null vector (real in the space-like case) with m'" the adjoint null vector. Adjunction may be expressed in geometric terms as a swop of the basis spinors n and ij, or as an exchange of the vectors m *.iid m while leaving the vector / intact. The triad of vectors (5.4) (VHOW.m'}, ? = ©,+,-, forms a normalized basis in the vector space V. The 3-metric acquires the form 1 0 0 0 0 1 (0 1 0 5.1 DEFINITION The Ricci rotation coefficient» are the invariants V (5-6) ЧРЧГ = **p<*q4*. The Ricci rotation coefficients will be given the individual notation [11] \mn —о +o + - о к к + P a —f Ъ P Table 1. The rotation coefficients. 5.2 DEFINITION The triad derivatives acting on scalar« are defined д„=гр'д<. 12 We also use the detailed notation (5.7) D = d0 5 = a+ « = d_. The triad derivatives have the commutators D6-6D=(p + c)6 + OŐ + KD (5.8) . . . 66 - 66 = H - r6 + (p - p)D. The Ricci identities have the detailed form 7 Da — 6к = тк+к + a(p + p + 2«) — 2$+ + 7 Dp — 6K = кк — кт •+• аЪ + p — ф00 (5.9) DT - It = kp - [к + т)д + е[к - т) + рт - 2ф0. 6а — 6р = 2ат + к{р — р) + 2ф0+ 6т + 6т =pp-aä + ">,тт - е(р - р) - 2ф+- + ф„„ where (5.10) фрч = | Rpq 5.3 DIGRESSION TO SL(2,C) One can decompose the null tetrad of Newman & Penrose [3] in terms of the triad. The null tetrad is fixed by the {oA,iA } basis in the A spin space, normalized by oAt = 1. This defines a tetrad AB Í" =oAo* 'öB> AB (5.11). th" =oAo» 'lB- MAB A" =lAc 'lB, The decomposition will depend on the relation of the o,i and the r),fj basis in the spin space. These spinor bases are are locally connected by an SL(2,C) rotation. It is sometimes convenient [5] to choose, for example, /2\Ч* (f\xl* i (5.12) °A = \j) Чл* ÍA = \V Чл' We then have 13 (5.13) т"=у/\Г\{т\-(т'ч)}. n-=i/í-+a- The spin coefficients of Newman and Penrose [3] have the decomposition 1=1(* + С.-в.) * = r"'(K-2G+) p = p + G0 2 a = a »—!/"•. (5.14) el,. 1 X=2fo a = 7/I/2(2r + G_ -G_) ,/2 fi=^f(P+G0) /? = --/ (2r + 3G++G+) *=\ri>(-k + 2G-) * = -J/(2*-SC.-ö.) and the Weyl curvature spinor is given by Ф0 = 2[SG+ - oG0 + TG+ + (2G+ + G+ )G+ ] l,9 *i = -f [DG+ - KG0 - eG+ + [2G0 + G0)G+ ] (5.15) Фа = i'^ + *G+ + KG- + (^ + G^G° "2G + G~ 1 ф 3/3 з = ^/ [£>G- - /cG0 + *G_ + (2G0 + G0)G_ j *« = =Л*С- - *Ge + TG. + (2G_ + G_ )G_). In a vacuum space-time, the Ricci tensor of the 3-space is determined by Einstein's gravitational equations as follows [5]: Apq +GpGq +GPG4 = 0. The triad components Gp of the gravitationa!' ..tor satisfy the vacuum field equations [5] DG0 + 6G+ + ÍG_ = {p + £)G0 - (л - r)G_ - (л - r)G+ + (G0 - Ge)G0 + (G+ - G+ )G_ + (G_ - G_ )G+ (5.16) £>G_ - $G0 = (/»- É)G_ + BG+ + KG0 - 60G. + G_ G„ DG+ - 6G0 = oG. + (p + «)G+ + /cG0 - G0G+ + G+ G0 6G- - ÄG+ = (p-/»)G0 - rG+ + fG. -G+G-+ G.G+. 14 The choice of the triad has so far been left arbitrary. A convenient orientation for the triad vector / is along the eigenrays of the gravitational field. These eigendirections are [5] the principal null directions of the gravitational spinor given by the canonical decomposition (5.17) GAB=X(A4B)- The eigenray condition can be written in triad terms as G+ = 0. Ai> eigenray congruence in the 3-space is geodesic if and only if the corresponding null congruence with tangent four-vector / is geodesic. Solutions of the vacuum gravitational equations with geodesic eigenrays have been found in Ref. [6]. REFERENCES [l] WITTEN, L., (1959) Invariants of general relativity and the classification of spaces. Phys. Rev. 113, 357-362. [2] PENROSE, R., (1960) A spinor approach to general relativity. Ann. Phys. 10, 171-201. [3] NEWMAN, E. T. and PENROSE, R., (1962) An approach to gravitational radiation by a method of spin coefficients coefficients. J. Math. Phys. 3, 566-578. [4j SCHWARZ, J. H., (1982) Superstring theory. Phys. Reports 89, 223-322. [5] PERJÉS, Z., (1970) Spinor treatment of stationary space-times. J. Math. Phys. 11, 3383-91. [6] KÓTA, J. and PERJÉS, Z., (1972) All stationary vacuum metrics with shear ing geodesic eigenrays. J. Math. Phys. 13,1695-99. [7] PERJÉS, Z., (1972) SU(1,1) spin coefficients. Acta Physic* Hung. 32, 207- 220. [8] SOMMERS, P., (1980) Space epinors. J. Math. Phys. 21, 2567-2571. [91 SEN, A., (1982) Quantum theory of spin-3/2 field in Einstein spaces. Int. J. Theor. Phys. 21,1-35. [10J HOROWITZ, G. T. and PERRY, M. J., (1981) Gravitational energy cannot become negative. Phys. Rev. Letters 81, 371-4. 15 [11] HALL, G. S., MORGAN, T. and PERJÉS, Z., (1987) Three-dimensional space-times. Gen. Rel. Grav. 10,1137-49. 112] ASHTEKAR, A., (1987) A new Hamiltonian formulation of general relativity. Phys. Rev. D36,1587-1602. [13] PERJÉS, Z., (1988) Parametric manifolJs. Proceedings of the Sth Marcel Grossm&nn Conference. Ed. R. Ruffini, World Scientific, to appear. [14] BARUT, A. O., (1967) Unified algebraic construction of representations of compact and non-compact Lie algebras and Lie groups. Mathematical Meth ods of Theoretical Physics, Eds. Brittin, W. E., Barut, A. O. and Guenin, M., Gordon and Breach, Lectures in Theoretical Physics 9,125-172. [15] WALD, R. M., (1984) General Relativity. University of Chicago Press 346. [16] FRIEDMAN, J. L. and SORKIN, R. D., (1980) A spin-statistics theorem for composites containing both electric and magnetic charges. Commun. Math. 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Title and classification of the issues published this year: KFKI-1989-01 /D G. Kocsis et al.: A possible method for ion temperature measurement by ion sen sitive probes KFKI-1989-02/G L. Perneczky et al: Using the pressurfzer spray line in order to minimize loop- seal effects (in Hungarian) KFKI-1989-03/E T Csiba et al.: Propagation of charge density wave voltage noise along a blue bronze, Rbo3Mo03 crystal KFKI-1989-04/G G Baranyai et al.: Experimental investigation of leakage of safety valves by means of acoustic emission detectors (in Hungarian) KFKI-1989-05/A Nguyen Ai Viet et al.: Can solitons exist in non linear models constructed by the non linear invariance principle? KFKI-1989-06/A Nguyen Ai Viet et al.: A non linearly invariant Skyrme type model KFKI-1989-07/A Nguyen Ai Vie* et al.: Static proparties of nuclejns in a modified Skyrme model KFKI-1989-08/B Z Perjés: Factor structure of the Tomimatsu Sato metrics KFKI-1989-09/B Z Perjés: Unitary spinor methods In general relativity Kiadja a Központi fizikai Kutató Intézet Felelős kiadó: Szegő Károly Szakmai lektor Rácz István Nyelvi lektor: Bencze Qyula Példányszám: 249 Törzsszám 69 67 Készült a KFKI sokszorosító üzemében Felelős vezető: Gonda Péter Budapest, 1989 február hó ! i !