hep-th/ v2 20 Aug 1997

Transcription

1 Newman-Penrose dyads and the null string motion equations in a background of antisymmetric elds hep-th/ v2 20 Aug 1997 K Ilienko Balliol College and Mathematical Institute, University of Oxford, 24{29 St Giles', Oxford OX1 3LB, UK A A Zheltukhin y Kharkiv Institute for Physics and Technology, 1 Akademichna Str, Kharkiv , Ukraine Abstract An application of the Newman-Penrose dyad formalism for description of tensionless (null) string dynamics in 4D Minkowski spacetime is studied in a background of antisymmetric elds. We present a special class of background antisymmetric elds which admits exact solutions of the null string motion equations and formulate a sucient condition of the absence of interaction of a null string with such elds. address: ilienko@math.ox.ac.uk y address: zheltukhin@kipt.kharkov.ua 1

2 1 Introduction Recently, there has been considerable interest in investigation of the null string motion equations in dierent backgrounds of external elds and in curved spacetimes (e.g. [1, 2, 3] and references therein). Finding of the exact solutions for the equations of motion in such systems is a rather dicult task, mainly due to the nonlinear character of the motion equations, so, it seems interesting to study those situations in which these equations possess exact solutions. It is known that the motion equations for strings in 4D Minkowski spacetime, null strings in some backgrounds and for particular types of curved spacetimes can be solved exactly [4, 5, 6]. Being a zero tension limit of strings [7], null strings possess simpler equations of motion than those of strings and can be considered as a zero approximation with the string tension as the perturbation parameter [8, 9, 10]. This explains our current interest in this problem. Geometrically world sheets of null strings are lightlike (null) 2-surfaces which generalise world lines of massless particles [11, 12, 13]. By convention, null string interactions with various elds can be analysed into two types. To the rst type we attribute the interactions which violate the lightlike character of the world sheets and, therefore, lead to generation of non-zero tension. All other interactions fall into the second type. In particular, null string interactions with antisymmetric background tensor elds are of the second type. As shown in [14], 2-component spinors of 3D Minkowski spacetime provide particularly convenient framework for studying the general solutions of null string motion equations in arbitrary antisymmetric background elds T mn (x). On the other hand, convenience of adopting 2-spinor description for spacelike and timelike strings in 4D Minkowski spacetime was demonstrated in [15] and for free null (super) strings and p-branes in [16]. Therefore, it is interesting to investigate the possibility of application of such a formalism in the form of the Newman-Penrose dyad calculus in 4D Minkowski spacetime for the description of null strings in background antisymmetric elds which is one of the main goals of the present article. In addition, we nd a special class of background antisymmetric elds which gives rise to exact solutions of the null string motion equations and formulate a sucient condition for the absence of interaction of a null string with background antisymmetric elds. 2 Equations of motion Let us consider the action of a null string in an external antisymmetric eld T mn (x). In the spinor form it can be rewritten as follows S = Z x AB0 o A o B 0? " 1 1 x A 1B 0 2 x A 2B 0 2 T A1 A 2 B 0 1B 0 2(x AB0 ) i d 2 ; (1) where T A1 A 2 B 0 1B 0 2, x AB0 and d 2 represent eld T mn, coordinates of the null string x m (; ) and the area element dd of the world sheet. We assume that = (; ) is a smooth parametrisation of the null-string world sheet, choose " =?" = 2

3 ?1, ij = diag(+???) and use The coordinates on the world sheet are dimensionless and we assume the world sheet vector density () to have the dimensions of inverse length. ensures invariance of action (1) under arbitrary non-degenerate reparametrisations of the null string world sheet. Interaction constant has the dimensions of inverse length. Spinor elds o A and A form a basis for 2-dimensional complex vector space and obey the normalisation condition o A A = (o B 0 B0 ) = ; (2) where (x AB0 ) represents a possibility of rescaling of the Newman-Penrose dyad element A at each spacetime point x AB0. The relation T A1 B 0 1A 2 B 0 2 = T A1 A 2 B 0 1B 0 2 =?T A2 A 1 B 0 2B 0 1 (3) holds for antisymmetric tensor eld T mn and we AB AB0, _x AB0 x AB0, x AB0 x AB0 and 3@ [AB 0T A1 A 2 B 0 1B 0 2] AB 0T A1 A 2 B 0 1B 0 2 A1 B 0 1 T A 2 AB 0 2B 0 A 2 B 0 2 T AA 1 B 0 B 0 1 : (4) Variation of action (1) results in the equations describing the null string x AB0 o A = x AB0 o A o B 0 = ( o A o B 0) + 3" 1 1 x A 1B 0 2 x A 2B 0 [AB 0T A1 A 2 B 0 1B 0 2] = 0; (5) and complex conjugate of them. The rst equation in (5) and its complex conjugate x AB0 = eo A o B0 ; (6) where e() is an arbitrary real-valued function with transformation properties of a scalar world sheet density. Assuming that is a nowhere zero function we rewrite this equation as _x AB0 = e oa o B0? Taking into account the second equation in (5) we obtain xab0 : (7) x AB0 o A o B 0 = 0: (8) This equation yields representation for spin-tensor x AB0 in the form 1 x AB0 = o A r B0 + r A o B0 ; (9) where condition o A r A 6= 0 is imposed on the spinor eld r A. Action (1) is invariant with respect to the following transformations ~o A = o A ; ~ A = A + uo A ; (10) 1 Representations (7) and (9) imply _x 2 = ( = ) 2 x 2 and _xx =?( = )x 2. Hence, the determinant of the induced metric on the null string world sheet vanishes identically ( _x 2 x 2? ( _xx) 2 = 0). This veries that the action principle (1) provides a description of a null string world sheet. 3

4 and ~o A = e v o A ; ~ A = e?v A ; ~ = e?(v+v) (11) with complex-valued functions u() and v(). Expanding spinor eld r A in the basis (o A, A ) as r A = po A + q A, where functions p() and q() are complex-valued, representing q in the form q = jqj exp(i') and using (9) we get x AB0 = (p + p)o A o B0 + jqj(e?i' o A B0 + e i' A o B0 ): (12) Carrying out successive transformations (10) and (11) with parameters u = jqj?1 p exp(?i') and v =?[ln( =e) + i']=2 we obtain _x AB0 = o A o B0? xab0 ; x AB0 = jqj(o A B0 + A o B0 ): (13) Here we omitted the tildes over the basis spinors, because motion equations (5) are invariant with respect to transformations (10) and (11). Using the possibility of rescaling the dyad element A ~o A = o A ; ~ A = A ; ~ = ; (14) we incorporate factor jqj into A and write (13) in the form _x AB0 = o A o B0? xab0 ; x AB0 = o A B0 + A o B0 : (15) Using (15) we represent the last equation in (5) ( o A o B 0) = i(q AD 0o D0 o B 0? o A Q CB 0o C ); (16) where spin-tensor Q AB 0 corresponds to vector Q k = " [l T mn] with 3!@ [l T mn] = " lmnp " q T rs and we take " 0123 =?" 0123 = 1. Invariance of action (1) under arbitrary reparametrisations of the world sheet implies (via the second Noether's theorem [17, 18]) that the solutions of the motion equations depend upon two arbitrary real-valued functions. We x one of them using a condition = 0: (17) Then, the null string motion equations take the form _x AB0 = o A o B0 ; x AB0 = o A B0 + A o B0 ; ( o A o B 0) : = i(q AD 0 o D0 o B 0? o A o C Q CB 0); (18) and the rst two of them result in the Virasoro constraints appearing in the standard theory of null strings. _x 2 = 0 and _xx = 0 (19) 4

5 3 Absence of interaction with antisymmetric eld Projecting equation (16) on the spin-tensor basis element o A B0 we obtain o o A = io A Q AB 0o B0 : (20) When the external antisymmetric eld vanishes Q AB 0 is equal to zero and (20) gives a reparametrization invariant equation of a null string world sheet which is generated by null geodesics (cf. [7]). Thus, setting Q AB 0 to zero we get under gauge condition (17) equation _o A o A = 0 (21) as a sucient condition for a null string world sheet to be a geodesic one. We note that this condition, together with the null string motion equations and the Virasoro constraints, is invariant under non-degenerate transformations of the form (see [14]) ~ = ~(; ) and ~ = ~(): (22) Equation (21) under gauge condition (17) shows that the sucient condition of the absence of the null string interaction with an antisymmetric external eld is Q AB 0o A o B0 = 0; (23) or Q k _x k = 0 in vector form. Since vector Q k is dual to the eld strength [l T mn] we observe that condition (23) is equivalent to the impossibility of constructing a non-zero 4-volume out of the eld strength tensor and vector _x k. Thus, the measure of this volume can be taken as a measure of interaction. This case corresponds to system (40) with = 0 (cf. equation (21)). The solution for spinors o A and A is given by and (2) implies where o A = A ()e m 2 ; A = [ A () + Z o (o A + io A )e m 2 d]e? m 2 (24) A A + A A R = e?m?m 2 ; (25) R = Z 0 e m+m 2 d; (26) as the normalisation condition. Representation (24) allows to write (18) in the form _x AB0 = A B0 _R; x AB0 = A B 0 + A B0 + ( A B0 R) 0 ; R + _ R _ = i R[( _ A + A R)Q AB 0 B0? A Q AB 0( B0 + B0 R)]: Equation (23) and the rst two equations in (27) ensure that the restriction of the background antisymmetric eld T mn (x) to the null string world sheet given by spintensor Q AB 0 depends on only through R. Hence making use of (24) in (23) we obtain Q AB 0 = A w B 0(R; ) + w A (R; ) B 0: (28) 5 (27)

6 Imposing the second gauge condition in a form = 1; (29) analogues to [14], we are able to nd solution for R of the last equation in system (27) in implicit form = Z R 0 Z y where the function F (x; ) is given by 0 F (x; )dx?1dy; (30) F (x; ) = i[( A A + x A A ) B0 w B 0(x; )? A w A (x; )( B 0 B 0 + x B 0 B0 )]: (31) Taking into account invariance of the null string motion equations under transformations (22) we may choose new local coordinates by ~ = R(; ) and ~ =. Then, the rst equation in system (27) results in the equation of motion for a free null string ~x AB0 (~ ; ~) = 0 (32) (or ~x m = 0 in the vector form), where the dierentiation is performed with respect to ~. This equation possesses exact solution and can be easily integrated, thus we nd that the null string motion equations in background antisymmetric elds which obey conditions (23) can also be solved exactly. 4 Conclusion In this paper null string motion equations in arbitrary background of antisymmetric elds are considered. We nd a special class of the elds, singled out by condition (23), which can be eliminated by a particular choice of parametrisation for the null string world sheet, such that the null string motion equations are reduced to those of a free null string and, therefore, admit exact solution. At this point, it seems interesting to generalise the above mentioned results to the case of tensionless p- branes in antisymmetric external elds and for d-dimensional spacetimes. 5 Acknowledgements This work is supported, in part, by Dutch Government and INTAS Grants No 94{ 2317, No ext and No ext. The rst author would like to acknowledge the Dervorguilla Graduate Scholarship awarded by Balliol College. He is also grateful to Profs R Penrose and Yu P Stepanovsky for fruitful discussions. The second author wish to acknowledge nancial support by the Government Fund for Fundamental Research of the Ministry of Research and Technology of Ukraine and a High Energy Physics Grant awarded by the ministry. After submitting this article the authors learnt about recent paper [19] which contains an interesting approach to the description of interaction of tensionless strings with other elds and has an interesting intersection with our consideration. We are grateful to Dr B Jensen who informed us about this work. 6

7 A Analysis of consistency relations Let us analyse consistency relations for the representations of _x AB0 ( _x AB0 ) 0 is equal to (x AB0 ) : the spinors o A and A obey the relation and x AB0. Since o A o B0 + o A o B0 = o A _ B0 + _o A B0 + A _o B0 + _ A o B0 : (33) Multiplying both sides of equation (33) by o A o B 0 we get which yields _o A o A + _o B0 o B 0 = 0; (34) _o A o A + i = 0; (35) where () is an arbitrary real-valued function. Allowing for (2), one can write _o A = 1 2 _moa? i A : (36) Here m() is an arbitrary complex-valued function. Substituting (36) into (33) and projecting the obtained equation on o A B 0 we nd (_ A _ m A? o A )o A = 0: (37) From (37) it follows that _ A _ m A? o A = ~o A : (38) Substituting again (36) and (38) into (33) we obtain ~ = i; (39) where () is an arbitrary real-valued function. Thus, consistency relation (33) results in a system of equations for the basis spinors o A and A _o A = 1 2 _moa? i A ; _ A = io A? 1 2 _ m A + o A : (40) 7

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