Generalized exterior forms, geometry and space-time

Transcription

1 Generalized exterior forms, geometry and sace-time P Nurowski Instytut Fizyki Teoretycznej Uniwersytet Warszawski ul. Hoza 69, Warszawa nurowski@fuw.edu.l D C Robinson Mathematics Deartment King s College London Strand, London WC2R 2LS United Kingdom david.c.robinson@kcl.ac.uk November 8, 2005 Abstract The roerties of generalised -forms, first introduced by Sarling, are discussed and develoed. Generalised Cartan structure equations for generalised affine connections are introduced. A new reresentation of Einstein s equations, using genneralised forms is given. 1

2 In this letter a develoment of a generalized exterior algebras and calculi of -forms, will be resented. This tye of extension of the ordinary calculus and algebra of differential forms was first introduced by Sarling in order to associate an abstract twistor structure with any real analytic Einstein sacetime, [1-4]. However it is clear that it is a tool which can be emloyed in more general hysical and geometrical contexts. Here the aim is to show how the formalism can be further develoed by constructing certain generalized connections and using them to rovide a formulation of Einstein s vacuum equations. A generalized -form, a, is defined to be an ordered air of ordinary - and +1-forms, that is a ( α, +1 α ) Λ Λ +1, (1) where Λ denotes the module of -forms on a differentiable manifold M of dimension n. By defining a minus one-form to be an ordered air 1 a = (0, 0 α), (2) where α 0 is a function on M, the range of can be taken to be 1 n. The manifold and forms may be real or comlex but here n is taken to be the real dimension of M. The module of generalised -forms will be denoted by Λ G and the formal sum 1 n Λ G will be denoted by Λ G. The letters over the forms indicate the degrees of the forms. Whenever these degrees are obvious they will be omitted. In the following bold Latin letters will be used for generalized forms and normal Greek letters for ordinary forms. A generalized form given by a air ( α, 0) will be identified with the ordinary -form α. Hence, for examle, a function on M will be identified with the generalized 0-form ( α, 0 0) while the air (0, α) 0 defines a generalized minus one-form. If a ( α, +1 α ) and b q ( β, q q+1 β ), then the (left) generalized exterior roduct, : Λ G Λq G Λ+q G, and the (left) generalized exterior derivative, d : Λ G Λ+1, are defined to be: G and a b q ( α β, q α q+1 β + ( 1) q+1 α β) q (3) d a (d α + ( 1) +1 k +1 α,d +1 α ), (4) 2

3 where k is a constant which, in the following, is assumed to be non-zero. It should be noted that it follows that 1 a 1 b = (0, 0). (5) These exterior roducts and derivatives of generalised forms can easily be shown to satisfy the standard rules of exterior algebra and calculus. This exterior roduct is associative and distributive. If a and b are, resectively generalized -forms and q-forms then a b = ( 1) q b a. The exterior derivative is an anti-derivation from -forms to (+1)-forms, that is d(a b) = da b + ( 1) a db. Furthermore, d 2 = 0. The above rules for the exterior algebra and calculus follow immediately if, following Sarling in references [1-3], the existence of a form ς, of degree minus one, is assumed to exist and satisfy all the basic rules of exterior algebra and calculus, together with the condition that dς = k. Tthe rules for exterior multilication and exterior differentiation, given above, follow when the generalized -form a = ( α, +1 α ), is identified with α + +1 α ς and then added, multilied and differentiated according to the ordinary rules of exterior algebra and calculus. 1 The following generalized Poincaré lemma holds. Let a ( α, +1 α ) be a closed generalized -form, so that da = 0. Then, (a) d 1 a = 0 if and only if 1 a = 0; (b) in any simly connected neighbourhood of any oint of M, (i) da 0 = 0 if and only if there exist ordinary 0-forms β 0 such that a 0 = ( β 0,k 1 dβ) 0 or a 0 = d 1 b, where 1 b = (0,k 1 β). 0 (ii) da = 0, 1 n, if and only if there exist ordinary (-1)- and -forms 1 β and β such that a = (d 1 β + ( 1) kβ,d β). Hence a = d 1 b, where 1 b = ( 1 β, β). Next consider Lie grous and Lie algebras and let G= Gl(n) or one of its sub-grous. In the resent context it is natural to associate with G the 1 Similarly rules for right exterior multilication and right exterior derivatives can be obtained by identifying a generalised -form a is identified with α+ς +1 α. The resulting right algebra and calculus is isomorhic to the left exterior algebra and calculus and it is the latter which is always used in this aer. 3

4 semi-direct roduct of G and the Lie algebra of G (viewed as an additive abelian grou). Define the (associated) Lie grou G by G = {a a = α(1,a)}, (6) α(1,a) (α, 0) (1,A) = (α,αa), where a is a generalized zero form, α belongs to the Lie grou G (GL(n) or a subgrou), with identity 1, and A is an ordinary 1-form with values in the Lie algebra of G. The roduct of two elements of G, a = α(1,a) and b = β(1,b) is given by the above rules for left exterior multilication, and is ab = αβ(1,b + β 1 Aβ). The inverse of a is a 1 = α 1 (1, αaα 1 ) and the identity is (1, 0), where 0 the zero 1-form. Right fundamental 1-forms r are formally defined to be forms of the tye da a 1 = (dα α 1 kαaα 1,α(dA + ka A)α 1 ), (7) and satisfy the Maurer-Cartan equation dr r r =0. (8) Similarly left fundamental 1-forms l are formally defined by a 1 da. When k is non-zero l can be neatly written in the form l = (λ, k 1 (dλ + λ λ), and l satisfies the Maurer-Cartan equation λ = α 1 dα ka, (9) dl + l l =0. (10) Next, in order to construct generalised Cartan structure equations for generalized connection and curvature forms on M, a generalised moving coframe of 1-forms, e a = (θ a, Θ a ), and a generalized Lie algebra valued 1-form Γ a b = (ωa b, Ωa b ) are introduced. When the aim is to identify the latter as a generalized affine connection, as it will be initially, the Lie algebra corresonding to the generalized structure grou, G, is gl(n) or a sub-algebra and the lower case Latin indices range and sum over 1 to n. It will be convenient to use covariant exterior derivatives and the generalized covariant exterior derivative is denoted here by D. The first generalized Cartan structure equation is given by T a = De a de a e b Γ a b. (11) 4

5 The generalized torsion, the 2-form T a, is in fact given by T a = (dθ a θ b ω a b kθ a, DΘ a + θ b Ω a b), where DΘ a = dθ a + Θ b ω a b. (12) Here D denotes an ordinary exterior covariant derivative. The second generalized Cartan structure equation is given by and F a b is the generalized curvature of Γa b. Here F a b = dγ a b + Γ a c Γ c b, (13) F a b = (dω a b + ω a c ω c b kω a b, DΩ a b), where DΩ a b = dω a b + Ω c b ω a c Ω a c ω c b. (14) When θ a is a co-frame on M, and ωb a are connection 1-forms with torsion kθ a and curvature kω a b, then the ordinary Cartan structure equations and the ordinary Bianchi identities dθ a θ b ω a b = kθ a, dω a b + ω a c ω c b = kω a b, (15) DΘ a = θ b Ω a b, DΩ a b = 0, (16) are satisfied if and only if the generalized affine connection is flat. That is equations (15) and (16) are equivalent to De a = 0, F a b = 0. (17) Hence when the ordinary Cartan structure equations are satisfied e a = (a 1 ) a b dxb, Γ a b = (a 1 ) a c d(a)c b, (18) where a a b = αa c(δb c,aa b ) is a generalized 0-form with values in the Lie grou G. Here αb a has values in GL(n) (or the aroriate sub-grou) and Aa b has values in the corresonding Lie algebra. 5

6 Gauge transformations are determined by generalized functions on M with values in the Lie grou G, as above. The gauge transformations determined by an element of G, again written a a b = αa c(δb c,aa b ), are given by e a (a 1 ) a b eb, Γ a b (a 1 ) a c dac b + (a 1 ) a c Γc da d b, De a (a 1 ) a b Deb, F a b (a 1 ) a c Fc da d b (19) These transformations are equivalent to the following θ a (α 1 ) a bθ b, ω a b (α 1 ) a cdα c b + (α 1 ) a cω c dα d b ka a b ϖ a b ka a b, Θ a (α 1 ) a b[θ b α b ca c d(α 1 ) d e θ e ], Ω a b (α 1 ) a cω c dα d b D ϖ A a b + ka a c A c b, (20) where D ϖ denotes the covariant exterior derivative with resect to ϖb a. These formulae encode the affine structure and the formalism rovides a unifying framework for different connections. A simle examle of this is rovided by the following result. Proosition : Let a metric on M have line element ds 2 = η ab θ a θ b, (21) where η ab =η ba, are the comonents of the metric with resect to the coframe θ a, are constant. Let ωb a = ωa bc θc be a general connection with torsion Θ a = 1 2 Θa bc θb θ c, Θ a bc = Θa cb, and let A ab = A abc θ c. Then if in the above generalized gauge transformations αb a = δa b and (i) if A (ab) = k 1 ω (ab) the transformed connection is metric; (ii) if, furthermore, A [ab] = {1/2[Θ cab Θ bca Θ abc ]+k 1 ω (ac)b k 1 ω (bc)a }θ c, the transformed connection is also torsion free and hence is the Levi-Civita connection of the metric. Similar results can be derived for generalised connections on rincial and associated bundles. Rather than ursue that line here, an illustration of the use of formalism above will be given by emloying it to rovide a simle formulation of Einstein s vacuum field equations in four dimensions. In this examle it is assumed that k is non-zero and not equal to one. Uer 6

7 case Latin indices sum and range over 0-1 and are two-comonent sinor indices,[5]. Consider the 4-metric on a four dimensional manifold M given by ds 2 = α A β A + β A α A, (22) where, α A and β A are sinor-valued 1-forms on M. Define the two generalized sinor valued 1-forms r A (α A, k 1 µ A ), and the generalized connection 1-form s A (β A,k 1 υ A ), (23) Γ A B = (ω A B, Ω A B), where Ω A B = dω A B + ω A C ω C B, (24) and ωb A is a sl(2,c)-valued 1-form. Then it can be seen directly from reference [6], (see also [7]), that the metric is Ricci flat if and only if the generalized sinor valued 1-forms have vanishing exterior derivatives and their generalized exterior roduct is an ordinary 2-form. That is the metric is Ricci flat if and only if r (A s B) are ordinary 2-forms and Dr A = 0, Ds A = 0, (25) where D is the generalized exterior covariant derivative determined by Γ A B. These conditions encode the Ricci flatness of the metric and ensure that ω A B is the anti-self dual art of the Levi-Civita sin connection. Equations (25) are formally similar to the first order equations used by Plebanski, [8], in his analysis of half-flat geometries. In conclusion it should be noted that the concet of a generalized - form discussed above is a secial case of a broader generalisation in which generalized -forms are reresented by n+1-tules of ordinary forms. Using the tye of notation introduced by Sarling and mentioned above on an n dimensional manifold, assume that n minus one-forms ς a exist and satisfy the conditions, ζ 1 ζ 2... ζ n 0, dζ a = k a, k a constants,a = 1...n. (26) 7

8 Now define a generalized -form to be a= α + +1 α a 1 ζa n α a 1a 2...a n ς a1 ς a2... ς an (27) a 1...a k ] α or, including zeros, the equivalent (n+1) tule. Here α k a 1...a k k[ =, k = to n, are ordinary k-forms (all sub-scrited indices ranging and summing over 1 to n). The rules for exterior multilication and exterior derivative can be comuted immediately from the last two equations. Extensions to include suer-symmetry and the infinite dimensional case aear to ose no major formal roblems. Further develoments of the above formalisms, including the definitions of Lie derivatives, general generalized connection and metric geometries, generalized Hodge duality, co-differentials or adjoints, inner roducts, Lalacians, co-homology, and hysical alications, will be resented elsewhere. 8

9 REFERENCES [1] Sarling G A J 1998 Abstract/virtual/reality/comlexity, in The Geometric Universe: Science, Geometry and the Work of Roger Penrose, Eds. S. A. Huggett et al. (Oxford University Press, Oxford) [2] Sarling G A J 1997 The abstract twistor theory of a rototye self-dual sace-time, U. of Pittsburgh rerint [3] Perjes Z & Sarling G A J 1998 The Abstract Twistor Sace of the Schwarzschild Sace-Time Prerint E S I Vienna number 520 [4] Sarling G A J 1999 Gen. Rel. Grav [5] Penrose R & Rindler W 1984 Sinors and Sace-time (C.U.P., Cambridge) [6] Robinson D C 1996 Class. & Quantum Grav. 13, 307 [7] Robinson D C 1998 Int. J. Theor. Phys. 37, 2067 [8] Plebanski J 1977 J.Math.Phys. 18,