General relativity, Lauricella's hypergeometric function F D and the theory of braids
aa r X i v : . [ g r- q c ] S e p General relativity, Lauricella’shypergeometric function F D and thetheory of braids G. V. Kraniotis ∗ Max Planck Institut f¨ur Physik,F¨ohringer Ring 6,D-80805 M¨unchen, Germany † November 8, 2018
Abstract
The exact (closed form) solutions of the equations of motion in the the-ory of general relativity that describe motion of test particle and photonin Kerr and Kerr-(anti) de Sitter spacetimes all involve the multivariablehypergeometric function of Lauricella F D : Kraniotis [Class. QuantumGrav. D n of the cor-responding function depends on the first integrals of motion associatedwith the isometries of the Kerr-(anti) de Sitter metric and Carter’s con-stant Q as well as on the cosmological constant Λ and the Kerr (rotation)parameter. In this work we discuss the topological properties of the do-main D n and in particular its fundamental connection with the theory ofbraids. An intrinsic relationship of general relativity with the pure braidsis established. ∗ [email protected] † MPP-2007-133, September 2007 Introduction
The exact (closed form) solutions of the equations of motion of the theory ofgeneral relativity (GTR) that describe orbits of test particle and photon in Kerrand Kerr-(anti) de Sitter spacetimes have yielded the following result [1],[2], [3]:All the physical amplitudes (measurable quantities) related to test particleorbits such as periapsis and gravitomagnetic (Lense-Thirring) precessions, or-bital periods as well as the bending of light by a rotating central mass (rotatingblack hole or rotating star), the gravitomagnetic precessions and orbital periodsof spherical photon orbits in Kerr spacetime with a cosmological constant havebeen elegantly expressed in terms of Lauricella’s multivariable hypergeomet-ric function F D ( α, β , β , · · · , β m , γ ; z , z , · · · , z m ). The domain of variables(moduli) of Lauricella’s function F D [4, 5] D n = ( z , z , · · · , z n ); z i = 0 , , (1 ≤ i ≤ n ) , z j = z k (1 ≤ j < k ≤ n ) (1)is related in the theory of General Relativity through the exact solutions ofthe geodesics system in Kerr-(anti) de Sitter spacetime to the first integrals ofmotion, as well as to the cosmological constant Λ and the rotation (Kerr) pa-rameter [1],[2],[3]. The generalised hypergeometric function of Appell-Lauricella F D is a very important function in Mathematical Analysis and as we shall seein this work it possesses very interesting topological properties. This then canlead to a fundamental relationship of General Relativity with topology. Theestablishment of such a relationship is the main theme and objective of thiswork.Indeed in pure Mathematics the domain of variables of F D has been stud-ied [6]; a main result of this investigation was the very interesting topologicalproperties of the domain D n . Essentially, the fundamental group of the domainunder discussion, π ( D n , a ), crudely speaking is the pure (or coloured) symme-try braid group [6]. Thus the results of [1],[2],[3] combined with the previousresult constitute a profound and intrinsic relation of the theory of General Rel-ativity with the field of algebraic topology and in particular with the theory ofbraids and links . The first , as a matter of fact, direct connection of a theoryof Physics with the theory of braids .The material of this paper is organized as follows: In section 2 we presentsome basic results of braid theory ( which are useful in understanding the mainresult); Namely the presentation theory of the braid and pure braid group aswell as their representation theory through the Burau and Gassner matricesrespectively (subsections 2.1.1 and 2.2). Having discussed the closure operationand the Markov moves in section 2.1, the topology invariants for knots and links As we shall see in the main body of the paper the closure of braids are knots and links.In particular, the closure of a coloured braid is a link. At this point we must mention that nice accounts of previous efforts and results connectingcertain models in physics with topological invariants can be found in [7, 8]. π ( D n , a ) and the hypergeometric representation of the pure or colouredgroup that it defines, an approach developed in [6]. Combining with the resultsin [1],[2],[3] the establishment of the fundamental connection of the theory ofGeneral Relativity with the coloured braids via the hypergeometric function ofLauricella F D is then achieved. Finally, section 5 is used for our conclusions. Braids (Z¨opfe) are very beautiful and profound mathematical entities. Theyhave been constructed by the German mathematician Emil Artin [9],[10] firstas an application to textile industry evolving into a central theme of topologywhere they currently serve as the fundamental theory of knots and links.In the space R , consider the points A i = ( i, ,
0) and B i = ( i, , i = 1 , , . . . , n . A polygonal line joining one of the points A i with one of thepoints B j is called ascending if in the motion of a point from A i to B j along theline its z-coordinate increases monotonically. A braid in n strands (or strings)is defined as a set of pairwise nonintersecting ascending polygonal lines (thestrands) joining the points A , . . . , A n to the points B , . . . , B n (in any order).One can also consider braids whose strands are ascending smooth lines (ratherthan polygonal ones); then it is natural to define equivalence as isotopy, i.e., asa smooth deformation in the class of braids. Examples of braids are given infig.1. y z x A1 A2 A3 A4B1 B2 B3 B4 A1 A2 A3 A4B1 B2 B3 B4 Figure 1: Examples of braids.Braids form a group and we now discuss its properties.We denote by σ i the braid which joins i to i + 1 by a path passing under thepath joining i + 1 to i (see Figure 2).The braid group B n is generated by the elements σ , σ , · · · σ n − with thefollowing presentation [9]: 3 i i+1 n Figure 2: The generator σ i . β Figure 3: Closure of a braidGenerators : σ , σ , · · · σ n − Relations : σ i σ i +1 σ i = σ i +1 σ i σ i +1 , ( i = 1 , · · · , n − σ i σ j = σ j σ i for | i − j | > far commutativity , because it saysthe generators commute pairwise when they are sufficiently far from each other,i.e., when their indices differ by two or more. In this section we discuss the relationship betweeen braids and links arising fromthe closure operation, which assigns a knot, link to each braid in a natural way.There is a canonical epimorphism ϕ : B n → S n of the braid group onto thepermutation group. In terms of relations, the group S n is obtained from B n byadding the relation σ i = 1. A link or knot β ( b ) is obtained by closing the braid b , i.e., tying the top end of each string (strand) to the same position at thebottom of the braid as shown in fig.3. The closure β ( b ) of the braid b is a knotif the permutation ϕ ( b ) associated to the braid generates the cyclic subgroup oforder n , Z /n Z , in the permutation group S n [8].Next we discuss an important theorem due to A. Markov which answers thequestion of when different braids can have isotopic closures, i.e. represent thesame knot, link. The first Markov move replaces b ∈ B n by aba − for a ∈ B n .4he second Markov move is the replacement b ↔ bσ ± n for b ∈ B n (note that σ n B n , so the notation bσ n makes sense algebraically only if we identify b withits image under the natural inclusion B n ֒ → B n +1 ) Then the theorem assertsthat the closures of two braids are isotopic if and only if one braid can be takento another by a finite sequence of Markov moves. A proof of this theorem canbe found in the book of J. S. Birman [11].Despite the difficulty of applying Markov’s theorem for studying knots viabraids, braids had first suggested themselves as a useful tool for investigatingfurther their relationship with links via the closure operation, after the discoveryby Werner Burau of a matrix representation of B n . This representation is thesubject of the following section If x is a non-zero complex number let M i , for 1 ≤ i ≤ n −
1, be the n × n matrix i − x xi + 1 1 0 1 . . .0 1 = M i where 1 − x is the i − i entry. One may easily check that M i M i +1 M i = M i +1 M i M i +1 and M i M j = M j M i if | i − j | ≥
2. Thus sending σ i to M i definesthe (non-reduced) Burau representation of B n [12]. Burau recognized that hisrepresentation was related to closed braids. More specifically if α ∈ B n and ψ is the reduced Burau representation then det(1 − ψ ( α )) is (1 + x + . . . + x n − )times the Alexander polynomial of the link ˆ α [13]. The kerner of the epimorphism ϕ defines the coloured braid symmetry group or pure braid group P n P n = Ker ϕ (3)We first define generators for P n .For any i < j , set A ij = A ji = σ j − σ j − · · · σ i +1 σ i σ − i +1 · · · σ − j − σ − j − .The generator A ij is depicted in Figure 4.Artin [10] gives the following presentation for P n As a matter of fact, the following short exact sequence is valid: 1 → P n ρ → B n ϕ → S n → One way to obtain a presentation for P n is using the Schreier-Reidemeister method. Thegroup P n is of index n ! in B n . One may choose as coset representatives for P n in B n any set i jj Figure 4: The generator A ij for the pure symmetry braid group.Generators : A ij ; 1 ≤ i < j < n Relations :1) A ǫrs A ik A − ǫrs = A ik if all indices are differentand if the pairs r, s and i, k do not separate each other2) A ǫrs A ir A − ǫrs = A − ǫis A ir A ǫis A ǫrs A is A − ǫrs = A − ǫis A − ǫir A is A ǫir A ǫis if finally the subscripts are all different andthe pairs r, s and i, k separate each other we get4) A ǫrs A ik A − ǫrs = A − ǫis A − ǫir A ǫis A ǫir .A ik .A − ǫir A − ǫis A ǫir A ǫis (4)and ǫ = ± A ij σ σ σ σ = A A = (5)The element ( A )( A A ) · · · ( A n A n · · · A n − n ) ∈ centre of P n .Let us give an example for n = 4, the symmetry group P has presentation of n ! words in the generators of B n whose images under ϕ range over all of S n . When thesecoset representatives form a Schreier set (i.e. any initial segment of a coset representative isagain a coset representative ) one can apply the Schreier-Reidemeister method [14]. A , A , A , A , A , A Relations : A A = A A A A = A A A A A − = A − A − A A A A − A − A A A A A − = A − A A A A A − = A − A A A A A − = A − A A A A A − = A − A A A A A − = A − A − A A A A A A − = A − A − A A A A A A − = A − A − A A A A A A − = A − A − A A A (6)It can easily be checked that the following matrices constitute a representa-tion of the pure braid group P , i.e. they satisfy the relations (6). A = − x + x x (1 − x ) x − x x A = − x + x x − x ) x − x )(1 − x ) 1 ( − x )(1 − x ) 01 − x x
00 0 0 1 A = − x + x x − x ) x (1 − x )(1 − x ) 1 0 ( − x )(1 − x )(1 − x )(1 − x ) 0 1 ( − x )(1 − x )1 − x x A = x ( − x ) (1 − x ) x
00 1 − x x
00 0 0 1 A = − x + x x − x ) x − x )(1 − x ) 1 ( − x )(1 − x )0 1 − x x = − x + x x (1 − x ) x − x x This matrix representation is the famous representation discovered by BettyJane Gassner in 1961 [15], generalizing the Burau representation.The element A A A A A A is represented by the matrix x ( − x x x ) − ( − x ) x − ( − x ) x x − ( − x ) x x x − x x (1 + x ( − x x )) − x ( − x ) x − x ( − x ) x x − x x − x x x x (1 + x ( − x )) − x x ( − x ) x − x x − x x − x x ( − x ) x x x More generally, denoting by A rs , ≤ r < s ≤ n , the generators of P n ,the (unreduced) Gassner representation is the homomorphism G n : P n → GL n ( Z [ x ± , · · · , x ± n ]) given by the formula G n ( A rs ) = I r − − x r + x r x s x r (1 − x r ) 00 ~u I s − r − ~v
00 1 − x s x r
00 0 0 0 I n − s where ~u = ((1 − x r +1 )(1 − x s ) · · · (1 − x s − )(1 − x s )) ⊤ (7)and ~v = ((1 − x r +1 )( x r − · · · (1 − x s − )( x r − ⊤ (8)and I k denotes the k × k identity matrix. Jones introduced his polynomial invariant for tame oriented links via certainrepresentations of the braid group [16] , exploiting the similarity of the Ocneanutrace in Hecke algebras with the Markov moves, first pointed out to him by JoanS Birman. There are two ways to introduce the Jones [16] and HOMFLY (orLYMPHTOFU) polynomials [17]. First through braids (each knot and linkexpressed as a word in the generators of the braid group) [16], and secondthrough the bracket polynomial due to L Kauffman [18]. We briefly discussboth approaches beginning with the definition and properties of the bracketpolynomial [18]. The faithfulness of G n for n ≥ Different from the Burau representation in section 2.1.1.
8o each nonoriented link diagram L a polynomial in the variables a, b, c isassigned, denoted by < L > which satisfies the following defining relations * + = a * + + b * + (9) < L ⊔ > = c < L > (10) < > = 1 (11)Here the little pictures in (9) denote three link diagrams L, L A , L B which areidentical outside a small disk and are as shown in the picture 5 inside it. L L A L B
Figure 5: Eliminating a crossing point.In this notation (9) may be rewritten as < L > = a < L A > + b < L B > .The arcs inside the small disks of the diagrams L A and L B are chosen in theregions A and B defined in Figure 6 AAB B
Figure 6: A and B regions near a crossing point.It turns out that the bracket polynomial is invariant under two of the Rei-demeister moves which imposes the constraints b = a − , c = − a − b .For instance for the knot 4 the bracket polynomial is calculated as follows * + = 1 + a − + a − a − a − (12)9hile for the link below we have the result * + = − a + 2 a − a + 2 a − − a − + a − (13)For the Kauffman polynomial one needs to consider oriented links, i.e. weassume that each component is supplied with an orientation (shown by arrowsin the figures). The writhe number is defined as follows ω ( L ) := X i ǫ i (14)where the sum is taken over all crossing points and the numbers ǫ i are equal to ± i th crossing point, which is defined in figure 7. ε=1 ε=−1 Figure 7: Positive and negative crossing points.Then the Kauffman polynomial X ( L ) on any oriented link diagram L isdefined by [18] X ( L ) := ( − a ) − ω ( L ) < | L | > (15)where the nonoriented diagram | L | is obtained from L by forgetting the orien-tation of all components. Now for the knot below the Kauffman polynomial iscalculated to be X = * + = 3 − a − − a + a − + a − a − − a (16)since ω ( L ) = 0. L+ L− L Figure 8: Orientation diagrams in the defining relation for X ( L ).10ubstituting a = q − / into X ( L ) one obrains V ( L ) the Jones polynomial ofthe oriented link L . For the reef knot in eq.(16) we have V = 3 − q − q − + q + q − − q − q − (17)The Jones polynomial satisfies the following relations [16] q − V ( L + ) − qV ( L − ) = ( q / − q − / ) V ( L ) (18) V ( L ⊔
0) = − ( q − / + q / ) V ( L ) (19) V (0) = 1 (20)Equation (18) is known as the skein relation , L + , L − , L are the three linkdiagrams exhibited in figure 8 . The condition L ⊔ L withan added circle that does not intersect L (and has no crossing points with L ).The last condition says that the Jones polynomial of the circle is 1.Using equations (18)-(20) one can calculate in an alternative way the poly-nomial V ( L ). Indeed, the skein relation for the figure 8 knot reads q − V δ+ ! − qV = ( q / − q − / ) V (21)or V δ+ ! = 1 + q + q − − q − q − (22)where we used the fact that V ( L ) for the Hopf link in equation (21) is equal to: − q − / − q − / . Our result using the skein relations agrees with the previouscalculation equation (12), where the rules for the bracket polynomial have beenapplied.Applying the skein relation for the Stevedore’s knot once we obtain − qV ! + q − V = ( q / − q − / ) V (23)The right hand side of equation (23) is a Hopf link with Jones polynonial: − q / − q / . Applying the skein relations to the second member of the lefthand side of (23) we obtain the unknot and a Hopf link. Eventually we obtainfor the Jones polynomial of the Stevedore’s (6 ) knot V ! = 2 − q − q − + q − + q − q − + q − (24)11s a final example the Jones polynomial for the link of the Borromean rings iscalculated to be: V = − q + 3 q − q + 4 − q − + 3 q − − q − (25) It is useful to express the links and knots we discussed in the previous section inwords in terms of the generators of the braid group. The braid word for the Hopflink, in terms of generators of the standard and pure braid group is: σ = A while for the Borromeo rings which appear in Eq. (25) the corresponding word isgiven by: σ σ − σ σ − σ σ − = A A − A − A On the other hand for the reefor square knot in eq.(16) the braid word is σ − σ , i.e. a braid with 3-strings.The two-variable HOMFLY polynomial of the oriented link L is defined asfollows [17],[19] X L ( x, λ ) := (cid:18) − − λx √ λ (1 − x ) (cid:19) n − ( √ λ ) e tr( π ( b )) (26)where b ∈ B n is any braid with β ( b ) = L , e being the exponent sum of b as aword on the σ i ’s and π the representation of B n in the Hecke algebra H ( x, n ), σ i → g i . The Jones one-variable polynomial of the previous section is then aspecial case of the HOMFLY polynomial [16],[19] V ( q ) = X L ( q, q ) (27)For instance for the Hopf link the HOMFLY polynomial is calculated to be X Hopf link ( x, λ ) = (cid:18) − − λx √ λ (1 − x ) (cid:19) − ( √ λ ) tr( g )= − − λx √ λ (1 − x ) λ tr (( x − g + x )= − − λx √ λ (1 − x ) λ (cid:18) ( x − − ) 1 − x − λx + x (cid:19) = − √ λ − x (cid:0) − x + x (1 − λ ) (cid:1) (28)and the Jones polynomial V ( q ) = X L ( q, q ) = − q / − q / which agrees withour calculation in the previous section using the skein relations or the bracketpolynomial. 12 The Fundamental group of D n The fundamental group of a topological space X can be introduced by makingthe homotopy equivalence classes of paths that start and end at a fixed pointin a space into a group. Indeed, for a point x in X , a loop at x is a paththat starts and ends at x . Then the fundamental group of X with base point x ,denoted π ( X, x ), is defined to be the set of equivalence classes of loops at x ,where the equivalence is by homotopy. Also the notions of covering spaces andfundamental groups are intimately related: coverings correspond to subgroupsof the fundamental group. There is a universal covering, from which all othercoverings can be constructed [20].In [6] the universal covering space ˜ D n of the domain D n of Appell-Lauricella’sfunction F D was determined. There it was shown that ˜ D n is isomorphic to E which is the space of the quotient of all simple, closed, rectilinear curves onRiemann’s sphere by a certain equivalence. Subsequently, a presentation for π ( D n , a ) was determined, where it was showed that π ( D n , a ) is isomorphicto P n +2 /Z n +2 . The normal subgroup Z n +2 of the pure braid group denotes itscentre.We follow the notation used in [6]. Consider the set N n = { , , . . . , n + 1 } where n is a non-negative integer. Being given a, C a on the Riemann’s sphere U and two different integers i, j ∈ N n , one considers a simple curve (path): u = u ij ( t ) , (0 ≤ t ≤
1) such that u ij (0) = a i , u ij (1) = a j , u ij ( t ) = u ji (1 − t )and that u ij ( t ) is contained in the domain U ( C a ) provided that 0 < t < h C a ,ij,s ( t ) of which the curve: u = h C a ,ij,s ( t ) is a lace incomparison with u ij ( s ) leaving a i . For I = { i α } ⊂ N n , one supposes always i α < i β if α < β . For each pair i, j ∈ N n , one denotes by A ij the element of π ( D n , a ) represented by the curve: z α = a α ( i = α ) , z i = h C a ,ij, ( t ) .Being given a group G n generated by { A ij ; i, j ∈ N n , i = j } , one poses, for I = { i α ; α ∈ N p } , A i i ··· i p ; i p +1 := A i i p +1 A i i p +1 · · · A i p i p +1 ,A I = A i i ··· i p +1 := A i ; i A i o i ; i · · · A i i ··· i p ; i p +1 (29)It is said that G n admits the relation R n if one has A ij = A ji ∀ i, j ∈ N n and that G n admits R nq ( I ) if one has, for all J = { j β ; β ∈ N q } with q ≤ p and J ⊂ I, A J ↔ A j α j β where ↔ signifies commutativity [6]. In addition, we saythat G n admits R nn if one has the relation A ··· n +1 = 1 the unity of G n . As it is remarked in [6] A ij can be regarded as an element of the coloured braid group.Indeed two sets ( z ) and ( z ′ ) of n + 2 complex numbers among whom no two can be equal weredefined to be equivalent if and only if z − z ′ = · · · = z n +1 − z ′ n +1 . Defining A ij by the curvewith the same formulae as above in this new space one can regard A ij as an element of the purebraid group. When i < j , using the elements σ α ( α = 0 , · · · , n ) of the braid group B n +2 , onecan write A ij = σ − i σ − i +1 · · · σ − j − σ j − σ j − · · · σ i and A ji = σ j − · · · σ i +1 σ i σ − i +1 · · · σ − j − . Using (29) the relation R nn is equivalent to: A A · · · A ··· n ; n +1 = A A A · · · A n +1 A n +1 · · · A nn +1 = 1. I ⊂ N n and a positive η one writes S I ( η ) = \ α ∈ N p { z = ( z , z , · · · , z n +1 ); z ∈ D n , | z i α − z i | < η sup {| z i − z i | ; i ∈ N n \ I }} (30)and ˚S I ( η ) is the set of the interior points.Then using a series of lemmas the author transports the paths and ho-motopies of D n to those of S I ( η ) and he proves that the fundamental group π ( D n , a ) is generated by [6]: { A ij ; i, j ∈ N n , i = j } and the relations among these elements are reduced to the set of relations: R n , R n ( N n ) , R n ( N n ) , R nn . Consequently one can chose ( n + 1)( n + 2) / − { A ij ; i, j ∈ N n , i = j } and the relations among the elements are reduced to those of the set of relations R n , R n ( N n ) , R n ( N n ) . In this work using and combining results from [1], [2], [3] and [6] we have estab-lished a direction connection of the theory of General Relativity with the theoryof the pure braid group. More specifically, the connection is established viathe generalised multivariable hypergeometric function of Lauricella F D throughwhich the exact solutions of the equations of motion of test and photon particlesin Kerr and Kerr-(anti) de Sitter spacetimes and of the corresponding physicalquantities such as periapsis and gravitomagnetic precessions, bending of lightand deflection angle were expressed [1], [2], [3] . As we discussed in the maintext the topological properties of the domain of variables D n are such that thefundamental group π ( D n , a ) is isomorphic to the quotient group P n +2 /Z n +2 [6]. The domain of variables D n of F D is related in the theory of General Rela-tivity to the first integrals of motion as well as to the cosmological constant andthe Kerr (spin) parameter of the rotating black hole or rotating central star.We also mentioned in the main body of the paper that the closure operationon pure braids lead to links.We believe that the link established in this work between General Relativitythe leading fundamental physical theory of gravity and low dimensional topol-ogy which involves the theory of coloured braids and links is very important. It14ay also provide us with hints and clues about the observed dimensionality ofspacetime and the topological origin of some physical quantities. On the topol-ogy side it might lead to new invariants for links through the hypergeometricrepresentation of the pure braid symmetry group. The first light from the dawnof a new era has reached us. This work is supported by a Max Planck research fellowship at the Max-Planck-Institute for Physics in Munich. At early stages it was partially supported bya fellowship at the Ludwig-Maximilians-Universit¨at in Munich. The author isgrateful to Dieter L¨ust for discussions.
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