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On the Geometry of Null Congruences in General Relativity* Proc. Indian Acad. Sci., Vol. 85 A, No. 6, 1977, pp. 546-551. On the geometry of null congruences in general relativity* ZAFAR AHSAN AND NIKHAT PARVEEN MALIK Department of Mathematics, Aligarh Muslim University, Aligarh 202001 MS received 16 August 1976; in revised form 5 January 1977 ABSTRACT Some theorems for the null conguences within the framework of general theory of relativity are given. These theorems are important in themselves as they illustrate the geometric meaning of the spin coeffi- cients. The newly developed Geroch-Held-Penrose (GHP) formalism has been used throughout the investigations. The salient features of GHP formalism that are necessary for the present work are given, and these techniques are applied to a pair of null congruences C (1) and C(n). 1. REVIEW OF THE TECHNIQUES SUPPOSE that two future pointing null directions are assigned at each point of the space-time. We can choose, as two of our tetrad vectors, two null vectors la, na (a, b,. — 1, 2, 3, 4) which point in these two directions and lava = 1 and the remaining tetrad vectors may be taken as Xa and Ya ortho- gonal to each la and na and to each other and also ma = 1/A/2 (Xa + iYa) and •Q = 1/x/2 (Xa — iYa), where a bar denotes the complex conjugation. If the scalar 7), with respect to the tetrad (la, ma, nza, na), undergoes the transformations _P aq (1) whenever la, na, ma and ma transform according as la ^ rla na . t.-1 l a ma -^ e^ 0 ma lilya-io ma (2) * Partially supported by a Junior Research Fellowship of Council of Scientific and Indu-- trial Research (India) under grant No. 7/112/358/73-GAU-I. 546 GEOMETRY OF NULL CONGRUENCES 547 where r and 8 are related by A 2 = re then we say that 71 is a spin- and boost-weighted scaler of type {p, q}. The spin-weight is 1/2 (p — q) and the boost-weight is 112 (p + q). Now suppose that oA and LA (A, B, ... = 1, 2) be a pair of spinor fields on the space-time normalized by OA LA = 1. (3) Such a pair of spinor fields is called a dyad or spin-frame. Also A oA', la = OA OA' , ma = oA 1A' , )7na = L L LA Z A' . (4) The twelve spin coefficients' in terms of dyad are K = OA OA' OB V AA OB ; Kr = — LA IA/ LB D AEI' La A 1A' Q =0 OBQdA'OB a = — LA O A' LB 0AA' LB; P = LA 0A' v V AA'OB ; P / = — OA L A' LB 0 AA' LB e (5) — A -A' B T L L O D AA' OB T = — 0A0A' LB VAA' LB; ^,q AI = V 6 LB 7 AN OB ^ F' ' = — LA OA' 0B7 AA 1 LB A' B ^7 E= O t ^/AA' Oa Er =— LA LA, OB VAA'LB. (6) The above notation differs from that of originally given by Newman-Pen- rose in that we have used only six different Greek letters rather than twelve. The relationship between the other six and that of ours is given by V = — K ' , A= —a, µ= —P, 7T —T, a=—/3', y=—e'. (7) The role of covariant derivative operator V a in Newman-Penrose formalism is given by A D = 0 0A' V AA' = 1a 7,, D' = LA 1A' V AA' = i1a V a, 8 = OA 1A' V AA' = ma V a, a, LA 0A' V AA' = in-a V a. (8) In the dyad or spin-frame the tetrad vectors la, na transform according as 0A -^ AO's, (9) and in terms of null tetrad, the transformation (9) has the form la --^ 1a la, ma AX 1 ma, ma -. A- 1 a tna, na -a A-' n 1 na. (10) Now we can say that a scaler q of type (p, q) is a quantity which transformed as AP Aq whenever the dyad (0A, LA) is changed according to (9) or the null tetrad is changed according to (10). A18—June 77 548 ZAFAR AIISAN AND NIKMAT PAP-VEEN MALIK The spin coefficients in (5) and (6) may now be divided into two classes according to whether or not they are proper spin- and boost-weighted quantities. Out of the twelve spin coefficients, only eight are found to be of good weight. These eight are given in the list (5). The role of the other remaining four spin coefficients appear in the definition of the derivative operators P (pronounced as thorn), P', z' (pronounced as edth) and defined for r9 of type {p, q} as: Pq=(D — pe — P'"1 =(D'+pe'+ga') , 7, qi)m " ^' ,1 = (S — pp + qq') , 71 = (S' -f- pR' — qP) 7]. (11) Uptill now this is just the standard GHP-formalism (we have given only that part of the formalism which is necessary for the present work, the details may be seen in the paper of Geroch, Held and Penrose 2). Using the for- malism just developed, we shall now obtain- certain results that will be used in section 2. With some calculations, we obtain the following basic equations. V a lb=—(P'—D ')la lb—Tla mb—Tla mb—(P—D)na lb Kna mb — Kna mb + (' — 8') ma lb + ama mb +Pmamb+(^' — 8)th a lb +Pmamb+amamb, (12) \anb =(P' — D')la nb —'' 1a mb —ic'la mb+(P — D)na nb T'na mb —T narnb — (^" — b')manb+Q mamb } p'ma mb -- (^' — 8) ma nb + P' ma mb + a ma mb, (13) i a mb=—is''a lb—Tla nb+(P'— D') /a mb—T' na1b — Kna nb „ +(P — D)namb+P'malb+Pmanb+(^ — 8') ma mb -1 a'ma lb + ama nb + (^' — 8) ma mb. (14) The equation for p a rnb is not written as it is just the complex conjugate of equation (14). Contracting equations (12), (13) and (14), we obtain Da la-- (p+P)—(P—D), (15) — Vana — (p' P')+(P' — D'), (16) V a ma=— (T'+T)—(^' -8). (17) The following simplifications shall also be useful in the later investigations: V[alb) = —2 (P — D)I(anb) — (? — "+ 5 ')I[amb] —(T—^'+8)I[a ^nb^ — Kn[ainbi — Kn[ainb^ f 2i(P — P)m[amba, (18) V[anb] = 2 (P — D)lfanb) — K'Ifamb1 —' 1[amb3 — (— z — b" + S') n[ a m b + (— T' — ^' + 8) n[ a m b] — 2i (p' — 0 m [ a fn b ., (19) GEOMETRY OF NULL CONGRUENCES 549 V ( amb) = — (T—T)lia nb J —(2i(P' — D')+P')lramb] —o' lra?nb] — (2i(P—D) +p) n[amb] ntamb]—G)' — 8 ) mcamb] (20) When GHP-derivatives (11) act on tetrad vectors, they give rise to (a) Pla = — Kma — K ma, (b) P' la = — T ma — T ma "1a = (c) '1a = — Pma — Q ma, (d) — vma — Pma, (21) (a) Pna = T'ma — T" ma, (b) P' n°" = — K' ma — K' ma, (c) 'na = — p' ma — a' ina, (d) b"na = —5'n — o' may (22) (a) Pma = — ^'la — Kna, (b) P'ma = — K la — Tn°, ma = — 'la — (23) (c) 'ma — _ v la — a na, (d) 2 p p71°, (a)Pma= —T la — icna, (b) P'ina=— K'1a —Tna, (c) 'ma = — a' la — a na, (d) „ ma = — p' la — p na. (24) 2. THEORY OF CONGRUENCES OF NULL CURVES In this section, we shall study the theory of congruences of null curves by giving certain theorems. These theorems originate in the works of Sachs 3 and later on was given by Newman and Penrose' in spin coefficients formalism. We shall mention here the statements of these theorems using the techniques of section 1, the proofs can easily be obtained. These theorems are important as they illustrate the geometrical meaning of the scalers in the list (5) and (6). THEOREM 1. The null congruence C(1) is geodesic if and only if K = 0 and by a suitable scaling, we may set P' — D' = 0. THEOREM 2 . The null congruence C (n) is geodesic if and only if K = 0 and by a suitable scaling, we may set P' — D' = 0. THEOREM 3. If we choose P — D = 0, then the tetrad (la, ma, Ana, na) undergoes parallel displacement along C (1) when K = - T ' = 0. THEOREM 4. If we choose P' — D' = 0, then the tetrad (la, ma, Ana, ra) undergoes parallel displacement along C (n) when — K' = r = 0. THEOREM 5. The null congruence C (1) is geodesic if and only if 1 [ bP1a ] =0. THEOREM 6 . The null congruence C (n) is geodesic if and only if n tbP'na ] — 0. 550 ZAFAR AHSAN AND NIKHAT PARVEEN MALIK We shall use the following lemma in our later investigations. LEMMA. Let C(1) be a null geodesic congruence, then 1 [a V b lc] = 0 is equivalent to p = P. According to Schouten4 the condition k[aV b = 0 is the necessary and sufl'ioient condition for the congruence C (k) to be hypersurface forming and thus by the lemma stated above, we have THEOREM 7. The null vector field l is hypersurface forming if and only if K=0 and p=p. THEOREM 8. The null vector field n is hypersurface forming if and only if K' = 0, and p' = p' . THEOREM 9 . If C (1) is an integrable hypersurface forming null con- gruence then T = — (i' — 8) and — (P — D) = 0. THEOREM 10. If C(n) is an integrable hypersurface forming null congruence then T = — (h" — 8') and — (P' — D') = 0. Now, calculating the dual of eq.
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