K lyenko - Null strings in 4d minkowski space-time - страница 1



Серія фізична. 2001. Вип.34. С.338-343

VISNYK LV1V UNIV. Ser.Physic. 2001. № 34. Р.338-343

УДК 539.12

PACS number(s): ll.25.-w, 11.25.Mj



Institute for Radiophysics and Electronics, NAS of Ukraine, Department of Theoretical Physics, 12 Ak Proskura Street, Kharkiv 61085, Ukraine

We present a twistor action functional for null strings (null 2-surfaces) in 4D Minkowski space-time. The proposed formulation is reparametrization invariant and free of algebraic and differential constraints. Proposed approach results in derivation of evolution equations for the null strings. It is shown that non-geodesic null strings are contained in the presented formalism. A discussion of the problem of minimality for -surfaces with degenerate induced metric is given. We also speculate on the possible description of strings (time­like 2-surfaces) and conventional (space-like) 2-surfaces.

Keywords: tensionless (null) string, degenerate induced metric, null bivector, twistor variational principle."-'

The notion of null 2-surface was put forward by A. Schild as a tensionless (null) string, i.e. a 2-dimensional, ruled by a one-parameter family of light-like (null) geodesies, submanifold of 4D Minkowski or curved space-times. The induced metric on such submanifolds is degenerate:

(дтх)2(дах)2-(дтх-дах)2=0. (1) Here x" (r, cr) is the world-sheet coordinate, the dots and primes denote differentiation with respect to the parameters r and a, dzx-dax stands for dtx°daxa. It should be emphasised that the null property (1) is manifestly reparametrization invariant while the original Schild's variational principle lacked this feature [1].

A different approach was developed in the articles [2,3]. In the paper [2] J. Stachel chose a null bivector (a 2-form obeying algebraic constraints

PabM*Pa"(x) = pab(x)p'"'(x) = 0, (2) where the star denotes the duality operation) as the Lagrangian density and showed that Schild's null strings can be described in this way. He also imposed the integrability condition

М*)^/>&]« = о (3)

for the null bivector (the square brackets denote antisymmetrization, Va stands for the covariant derivative). It stems from the requirements of differential geometry and was adopted from the book [4]. Recently, O.E. Gusev and A.A. Zheltukhin [3] have solved the algebraic constraints in the physical dimension of Minkowski space-time using a fundamental result of the spinor calculus of Cartan-Penrose [5,Vol. 1]. According to that



©IlyenkoK., 2001

In the present contribution we show that the variational principle (4) admits a natural twistor generalization. Additionally to the solutions already found in [3], the corresponding Euler-Lagrange equations contain as their solutions generic, i.e. not necessarily ruled by null geodesies of the ambient 4D Minkowski space-time (referred to here as non-geodesic), null 2-surfaces (null strings). We derive a non-linear evolution equation governing propagation of a generic non-geodesic null string. A heuristic argument in favour of defining the minimal null 2-surfaces as those ruled by null geodesies of 4D Minkowski space-time is presented. We also draw attention of the reader to the possibility and advantages of using the proposed formalism for expressing variational principles for strings (time-like 2-surfaces) and conventional (space-like) 2-surfaces of 4D Minkowski space-time (cf. [3]). To switch between vector and spinor indices the conventions of [5,Vols. 1&2] are used.

Action principle. By definition, the null bivector, pab, obeys the pair of above stated algebraic constraints. This fact leads to the existence of a pair of 1-forms with components ua(x) and vb(x) such thatраъ = (1/2!) (uavb - ubva) = u[avb). Moreover, it follows that without loss of generality one has uau" = uav" = 0, vav" < 0, and the particular value of the Lorentz norm of v,, is irrelevant for the variational problem in question [4].

Let the spinor fields nA{x) and rjA(x) to constitute a normalized Newman-Penrose dyad (лАт]л = 1) in 4D Minkowski space-time and additionally assume the spinor field JcA to be chosen in such a way as to represent the coincident principal null directions of the null bivector pab. Then, one can write ua = тїАкА, and v0 = nAr)A. + 7iA,rjA. This representation is, up to an overall functional multiplier, the general solution of the algebraic constraints for the null bivector pab(x) (see, for example, [5]).

If one assembles a pair of 1-forms uadx" = лАпА4хлл and vadx" = (nAr]A. + nArjA.)dxM', introduces the null twistors = {ал,лл.) and W = (%л,г}А.), expresses the 1-forms in terms of the null twistors and substitutes the results in the formula for pab(x)dx" л dx° then (s)he obtains the following twistor action for a null string in 4D Minkowski space-time:


Here the spinor fields coA and £ * are given by the usual definitions: coA - ixM nA, and %A = ix^ rj^. Za and Wa are the conjugate null twistors. The null property of the twistors Za and W corresponds to the Hermitian property of xM' and leads to the identities ZaZa = WaW = ZaW = WaZa = 0. It reflects the reality condition imposed

on the points of 4D Minkowski space-time. The 2-form in (5) is understood to be restricted to a 2-dimensional submanifold of 4D Minkowski space-time parametrized by r and a.

The Lagrangian density of the twistor action functional (5) is multiplied by a factor of q* under the gauge transformations of the form

Z" -+qZa,   Г->?-'Г+ pZa. (6)

Here q(r,cr) is a nowhere vanishing real-valued function and р(т,ст) is an arbitrary complex-valued function. This is an admissible freedom for a differential form representing a surface [4]. It gives rise to invariance of the Euler-Lagrange equations under the above mentioned transformations. The invariance corresponds to the possibility of rescaling with real multiples of the extent of the null direction tangent to the null string world-sheet WA -> qnA and to an addition of real multiples of the null direction to the space—like direction tangent to the world-sheet fjA —> q"rfA + рлА. Thus the gauge freedom of the null string world-sheet comprises the null- and boost-rotations [5].

Evolution equations. The Euler-Lagrange equations for the variational principle (5) were obtained in the author's doctoral thesis [6]. After some tedious but straightforward algebra they lead to the property (1) (cf. the article [3]). In addition, one can show that the differential constraint does not introduce new equations to those obtained by the variational procedure from the action (5). The latter can be proved by expressing the differential constraint in terms of the spinor fields nA and rjA [6].

It follows that the twistor action functional (5) describes the null string as a 2-dimensional submanifold of 4D Minkowski space-time with the degenerate induced metric. It is also remarkable that this formulation is reparametrization invariant and free of additional algebraic and differential constraints present in the space-time description [2] and of the two auxiliary world-sheet quantities pM artificially introduced in an earlier formulation by LA. Bandos and A.A. Zheltukhin [7].

One can choose orthogonal gauge for the world-sheet of a null string so that (dTxy = 0 and (1) implies {dtx-dax) = 0. In spinor terms this means that д,хм, = глАлА. and дахм, = i{gfjAnA, - длАт]А,), where r(j,a) is the real-valued flagpole extent and g(r,a) is, in general, a complex function. The equations of motion for the variational principle (5) also imply that (dznA)(g-g~) = 0. This means either that дглА vanishes or the function g is real-valued.

The first opportunity was spotted by O.E. Gusev and A.A. Zheltukhin in the article [3]. It immediately leads to the equation d\xa <x dtx", which states that the integral curves of the vector field dTx" are (null) geodesies in the sense of the ambient 4D Minkowski space-time. If the affine parametrization for the null geodesies is chosen then one can write an evolution equation for geodesic null strings in the form:

d\xa = 0. (7)

The second opportunity allows one to take the so-called natural parametrization, г~\т,а) = к з їїАїї'л' Ч ввлл (for the geodesic case the spin-coefficient к vanishes identically), and results in the following complete set of equations of motion for the non-geodesic null strings:

Зтдґ"' = кхлАлА\  дахм' = ід(їїАцА' - Щлжл'),

к\ялдаял +тсл'далА.) = &гл'дттіА, -лАдттіА), (8)


The spin-coefficient к reflects the existence of interaction between the null string and external fields. Such interactions preserve the null character of the string world-sheet but violate its geodesic property. For an example the interested reader is referred to the article [8]. It should be pointed out that the set of equations (8) coincides with the system derived in that paper for the non-geodesic (interacting) null string. This proves an equivalence of the both approaches on the classical level.

In the natural parametrization one finds the next identities ткА)пл = 1, and it follows that ffA - -dznA . These results can be used to show that there exists a non­linear evolution equation for non-geodesic null strings:

д]х[(д„х?дІх°-(д^-дІх^х'Уід.хУідІх-дІлду =o (9)

The non-geodesic null string evolution equation is invariant under a subgroup of diffeomorphisms of the null string world-sheet which preserves the orthogonal gauge.

Minimal null 2-surfaces. The results of L.P. Hughston and W.T. Shaw [9] on the connection between non-interacting (free) strings and minimal time-like 2-surfaces in 4D Minkowski space-time provide an impetus for attempts of finding a similar correspondence between geodesic (i.e. non-interacting, cf. [8]) null strings and minimal null 2-surfaces. The task of formulating the conditions of minimality for a null 2-surface in 4D Minkowski space-time seems to be a rather difficult one. The standard approach of the classical geometry of surfaces in the Riemannian space, which uses a suitable variational principle, fails in this case. The problem lies in the degenerate property of the induced metric (the first fundamental form) of such surfaces and, therefore, one cannot easily construct an analogue of the area element like in the case of time-like 2-surfaces.

Nevertheless, it is possible to formulate the minimality conditions for a null two-surface in 4D Minkowski space-time. In order to find them, a limiting procedure which takes the tangent space to a space-like 2-surface element to that of a null 2-surface element was built in [6].

The conditions of minimality for a space-like 2-surface in 4D Minkowski space-time can be formulated in the same way as those for a 2-surface in the ordinary Riemannian geometry. This has its origin in the non-degenerate property of the first fundamental form of a space-like 2-surface. In particular, the conditions of minimality can be given by the requirement of vanishing of the relevant mean curvatures. The mean curvatures are calculated by taking a trace of the corresponding second fundamental forms. Taking the limit of the well-defined minimality conditions for the space-like 2-surfaces with the aid of that procedure, one finds that minimal 2-surfaces admit a one parameter family of null geodesies.

Hit geometry of the minimal null 2-surfaces depends on whether the corresponding line of striction is a null or space-like curve. In the former case the minimal null2-surface is (locally) developable and the null geodesies of the congruence are strongly incident; in the latter case the null generators of the 2-surface present an example of weakly incident light rays as has been discussed recently by R. Penrose [10].

Discussion. The method employed for obtaining the action functional (5) of a null string can be in principle used for designing the variational principles for strings and space-like 2-surfaces. The idea is to take a simple bivector (pab *p"° = 0) and impose one of the conditions рлЬр°" = 1 or -1. The first condition would single out the string

while the second correspond to a space-like 2-surface. It is easy to see that such a procedure uniquely fixes the symmetric second rank spin-tensor фАВ(х) in the standard decomposition of an antisymmetric 4D Minkowski space-time tensor


Then, the variational principle


defines a 2-surface subject to the differential constraint (3). Now, one hopes that the use of spinor decomposition for pab{x) consistent with the either pair of the above stated algebraic constraints would provide equations of motion, which automatically incorporate the differential constraint. This assertion is supported by the success of this procedure for the null bivectors. It may also be possible to derive the analogues of the evolution equation (9) for generic (interacting) strings in 4D Minkowski space-time and curved space-times of general relativity, where exist explicit spinor constructions.

In the same way one could build twistor action functionals in the both cases and find corresponding objects on the null twistor space for generic time-like and space-like 2-surfaces of 4D Minkowski space-time. This would accomplish the task of finding a twistor description for 2-surfaces.

1.     Schild A. Classical null strings /I Physical review D. 1977. Vol.16. N6. P. 1722-1726.

2.     Stachel J. Thickening the string II. The null-string dust // Physical review D. 1980. Vol. 21. N8. P. 2182-2184.

3.     Gusev O.E., Zheltukhin A.A. Twistor description of world surfaces and the action integral of strings //JET? letters. 1996. Vol. 64. N7. P. 487-494.

4.     Schouten J.A. Ricci-Calculus. - Berlin "Springer-Verlag". 1954. P. 617.

5.     Penrose R., Rindler W. Spinors and space-time, Vol. 1. - Cambridge "Cambridge University Press". 1984. P. 528 .

6.     Ilyenko K. Doctor of philosophy thesis: Twistor description of null strings. -University of Oxford. 1999. P. 96.

7.     Bandos I.A., Zheltukhin A.A. Covariant quantization of null-super-membranes in four-dimensional space-time // Theoretical and mathematical physics. 1992. Vol. 88. N3. P. 925-937.

8.     Ilienko K., Zheltukhin A.A. Tensionless string in the notoph background //Classical and quantum gravity. 1999. Vol. 16, N2. P. 383-393.

9.     Hughston L.P., Shaw W.T. Twistors and strings // LMS lecture notes. 1990. Vol. 156. P.218-245.

10. Penrose R. Twistor geometry of light rays // Classical and quantum gravity. 1997. Vol. 14.N1.P.A299-A323.



Інститут радіофізики та електроніки НАН України, відділ теоретичної фізики, вул. Ак. Проскури, 12, 61085 Харків, Україна

Побудовано твісторний функціонал дії для нуль струн (двовимірних ізотропних поверхонь) чотиривимірного простору-часу Мінковського. Цей підхід є репараметризаційно інваріантним і вільним від штучних алгебричних та диференціальних умов, що є в працях попередників. Отримано еволюційне рівняння для нуль струн. Негеодезичні нуль струни також можна описати за допомогою цього формалізму. Обговорено проблему визначення мінімальних двовимірних поверхонь із виродженою індукованою метрикою. Ми припускаємо аналогічну можливість опису струн (часоподібних двовимірних) і звичайних (простороподібних) двовимірних поверхонь.

Ключові слова: нуль струна, вироджена індукована метрика, нуль бівектор, твісторний варіаційний принцип.

Стаття надійшла до редколегії       15.05.2001 Прийнята до друку 26.09.2001


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K lyenko - Null strings in 4d minkowski space-time