On the degrees of freedom in GR

Transcription

1 On the degrees of freedom in GR István Rácz Wigner RCP Budpest University of the Bsque Country Bilbo, 27 My, 2015 István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

2 Pln 1 The degrees of freedom 2 Folitions nd their use 3 Solving the constrints 4 Summry István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

3 The degrees of freedom Jnus-fced GR: The ren nd the phenomen : All the pre-gr physicl theories provide distinction between the ren in which physicl phenomen tke plce nd the phenomen themselves. ren: phenomen: clssicl mechnics phse spce: δ b dynmicl trjectories electrodynmics Minkowski spcetime: η b evolution of F b generl reltivity curved spcetime: g b evolution of g b Such cler distinction between the ren nd the phenomenon is simply not vilble in generl reltivity the metric plys both roles. GR is more thn merely field theoretic description of grvity. It is certin body of universl rules: modeling the spce of events by four-dimensionl differentible mnifold the use of tensor fields nd tensor equtions to describe physicl phenomen use of the (otherwise dynmicl) metric in mesuring of distnces, res, volumes, ngles... István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

4 The degrees of freedom The degrees of freedom in GR: Wht re the degrees of freedom? i theory possessing n initil vlue formultion: degrees of freedom is synonym of how mny distinct solutions of the equtions exist in ordinry prticle mechnics: the number of degrees of freedom is the number of quntities tht must be specified s initil dt divided by two The degrees of freedom in the linerized theory: Einstein (1916, 1918): the field equtions involve two degrees of freedom per spcetime point when studying linerized theory Is the full nonliner theory chrcterized by two degrees of freedom? Drmois (1927): probbly the erliest nswer in the confirmtory bsed on considertion of the Cuchy (or initil vlue) problem in GR How to identify these two degrees of freedom? They re supposed to be given in terms of components of the metric tensor nd its derivtives or such combintions of these s, e.g. the Riemnn tensor. István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

5 The degrees of freedom The degrees of freedom in GR: Wht re the min issues? initil dt: (h ij, K ij ), metric nd symmetric tensor on Σ 0 (3) R + ( K j j) 2 Kij K ij = 0 & D j K j i D i K j j = 0 D i denotes the covrint derivtive opertor ssocited with h ij. conforml method A. Lichnerowicz (1944) nd J.W. York (1972): the constrints re solved by trnsforming them into semiliner elliptic system by replcing the fields h ij nd K ij by φ 4 h ij nd φ 2 Kij no wy singles out precisely which functions (i.e., which of the 12 metric or extrinsic curvture components or functions of them) cn be freely specified, which functions re determined by the constrints, nd which functions correspond to guge trnsformtions. Indeed, one of the mjor obstcles to developing quntum theory of grvity is the inbility to single out the physicl degrees of freedom of the theory. R.M. Wld: Generl Reltivity, Univ. Chicgo Press, (1984) The min issue is not to find the only legitimte quntities representing the grvittionl degrees of freedom, rther, finding prticulrly convenient embodiment of this informtion (solving vrious problems). István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

6 The outline: The degrees of freedom Bsed on some recent ppers I. Rácz: Is the Binchi identity lwys hyperbolic?, Clss. Quntum Grv. 31 (2014) I. Rácz: Cuchy problem s two-surfce bsed geometrodynmics, Clss. Quntum Grv. 32 (2015) I. Rácz: Dynmicl determintion of the grvittionl degrees of freedom, submitted to Clss. Quntum Grv. I. Rácz nd J. Winicour: Blck hole initil dt without elliptic equtions, to pper in Phys. Rev. D The min messge: 1 Euclidend Lorentzin signture Einsteinin spces of n + 1-dimension (n 3), stisfying some mild topologicl ssumptions, will be considered. 2 the Binchi identity cn be used to explore reltions of vrious subsets of the bsic field equtions 3 new method in solving the constrints: s opposed to the conforml one by introducing some geometriclly distinguished vribles!!! regrdless whether the primry spce is Riemnnin or Lorentzin momentum constrint s first order symmetric hyperbolic system. the Hmiltonin constrint s prbolic or lgebric eqution 4 the conforml structure ppers to provide convenient embodiment of the degrees of freedom István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

7 Assumptions: The degrees of freedom The primry spce: (M, g b ) M : n + 1-dim. (n 3), smooth, prcompct, connected, orientble mnifold g b : smooth Lorentzin (,+,...,+) or Riemnnin (+,...,+) metric Einsteinin spce: Einstein s eqution restricting the geometry G b G b = 0 with source term G b hving vnishing divergence, G b = 0. or, i more conventionlly looking setup [R b 1 2 g b R] + Λ g b = 8π T b with mtter fields stisfying their field equtions with energy-momentum tensor T b nd with cosmologicl constnt Λ G b = 8π T b Λ g b István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

8 Folitions nd their use The primry 1 + n splitting: No restriction on the topology by Einstein s equtions! (locl PDEs) Assume: M is folited by one-prmeter fmily of homologous hypersurfces, i.e. M R Σ, for some codimension one mnifold Σ. known to hold for globlly hyperbolic spcetimes (Lorentzin cse) equivlent to the existence of smooth function σ : M R with non-vnishing grdient σ such tht the σ = const level surfces Σ σ = {σ} Σ comprise the one-prmeter folition of M. Σ σ István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

9 Projections: Folitions nd their use The projection opertor: the unit norm vector field tht is norml to the Σ σ level surfces = ɛ the sign of the norm of is not fixed. ɛ tkes the vlue 1 or +1 for Lorentzin or Riemnnin metric g b, respectively. the projection opertor to the level surfces of σ : M R. h b = δ b ɛ n b the induced metric on the σ = const level surfces h b = h e h f b g ef while D denotes the covrint derivtive opertor ssocited with h b. István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

10 Folitions nd their use Decompositions of vrious fields: Exmples: form field: L = δ e L e = (h e + ɛ n e ) L e = λ + L where λ = ɛ n e L e nd L = h e L e time evolution vector field σ : σ e e σ = 1 σ σ σ σ σ n σ σ = σ + σ = N n + N n σ σ σ where N nd N denotes the lps nd shift of σ = ( σ) : N = ɛ (σ e n e) nd N = h e σ e István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

11 Folitions nd their use Decompositions of vrious fields: Any symmetric tensor field P b cn be decomposed in terms of nd fields living on the σ = const level surfces s P b = π n b + [ p b + n b p ] + P b where π = n e n f P ef, p = ɛ h e n f P ef, P b = h e h f b P ef It is lso rewrding to inspect the decomposition of the contrction P b : ɛ ( P e ) n e = L n π + D e p e + [π (K e e) ɛ P ef K ef 2 ɛ ṅ e p e ] ( P e ) h e b = L n p b + D e P eb + [(K e e) p b + ṅ b π ɛ ṅ e P eb ] ṅ := n e e = ɛ D ln N bck István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

12 Folitions nd their use Decompositions of vrious fields: Exmples: the metric g b = ɛ n b + h b the source term G b = n b e + [ p b + n b p ] + S b where e = n e n f G ef, p = ɛ h e n f G ef, S b = h e h f b G ef the r.h.s. of our bsic field eqution E b = G b G b E b = n b E (H) + [ E (M) b + n b E (M) ] + (E (EVOL) b + h b E (H) ) E (H) = n e n f E ef, E (M) = ɛ h e n f E ef, E (EVOL) b = h e h f b E ef h b E (H) István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

13 Folitions nd their use Reltions between vrious prts of the bsic equtions: The decomposition of the covrint divergence E b = 0 of E b = G b G b : L n E (M) b L n E (H) + D e E (M) e + D (E (EVOL) b + [ E (H) (K e e) 2 ɛ (ṅ e E (M) e ) bck ɛ K e (E (EVOL) e + h e E (H) ) ] = 0 + h b E (H) ) + [ (K e e) E (M) b + E (H) ṅ b ɛ (E (EVOL) b + h b E (H) ) ṅ ] = 0 first order symmetric hyperbolic liner homogeneous system for (E (H), E (M) i ) T fosh Theorem Let (M, g b ) be s specified bove nd ssume tht the metric h b induced on the σ = const level surfces is Riemnnin. Then, regrdless whether g b is of Lorentzin or Eucliden signture, ny solution to the reduced equtions E (EVOL) b = 0 is lso solution to the full set of field equtions G b G b = 0 provided tht the constrint expressions E (H) nd E (M) vnish on one of the σ = const level surfces. bck István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

14 Folitions nd their use The secondry 1 + [n 1] splitting: Assume now tht on one of the σ = const level surfces sy on Σ 0 there exists smooth function ρ : Σ 0 R, with nowhere vnishing grdient such tht: the ρ = const level surfces S ρ re homologous to ech other nd such tht they re orientble compct without boundry in M. n i n i n i n i n i i n n i n i n i i n The metric h ij on Σ 0 cn be decomposed s Σ 0 h ij = ˆγ ij + ˆn iˆn j in terms of the positive definite metric ˆγ ij, induced on the S ρ hypersurfces, nd the unit norm field ˆn i = ˆN 1 [ ( ρ) i ˆN i ] norml to the S ρ hypersurfces on Σ 0, where ˆN nd ˆN i denotes the lps nd shift of n evolution vector field ρ i = ( ρ) i on Σ 0. István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

15 Folitions nd their use Secondry projections: The Lie trnsport of this folition of Σ 0 long the integrl curves of the vector field σ yields the two-prmeter folition S σ,ρ : σ σ σ σ σ σ n the fields ˆn i, ˆγ ij nd the ssocited projection op. ˆγ k l = h k l ˆn kˆn l to the codimension-two surfces S σ,ρ get to be well-defined throughout M. σ σ σ cn be put into the form Σ σ h e h f b E ef with some lgebr = E (EVOL) b + h b E (H) h e ih f j E ef = (n) E ij = (n) G ij (n) G ij István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

16 Folitions nd their use The integrbility condition for (n) G ij (n) G ij = 0 (n)e ij = Ê(H) ˆn iˆn j + [ˆn i Ê (M) j + ˆn j Ê (M) i ] + (Ê(EVOL) ij + ˆγ ij Ê (H) ) Ê (H) = ˆn eˆn f (n) E ef, Ê (M) i = ˆγ e j ˆn f (n) E ef, Ê (EVOL) ij = ˆγ e iˆγ f j (n)e ef ˆγ ij Ê (H) Lemm The integrbility condition D i [ (n) G ij ] = 0 holds on Σ σ if the momentum constrint expression E (M) b, long with its Lie derivtive L n E (M) b, vnishes there. Corollry Assume tht E (M) b = 0 oll the Σ σ level surfces, nd tht both Ê(H) nd Ê(M) vnish long world-tube W S in M. Theny solution to the secondry reduced equtions Ê(EVOL) = 0 is lso solution to the secondry equtions (n) ij G ij (n) G ij = 0. Corollry Theorem Assume, iddition, tht E (H) = 0 on Σ 0. Theny solution to the reduced equtions Ê(EVOL) ij = 0 is lso solution to the originl bsic field equtions G ij G ij = 0. [ E (M) b = 0 on Σ 0 = 0, {Ê(H) Ê(M) i = 0} on W S ] István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

17 Folitions nd their use The explicit forms: Expressions in the 1 + n decomposition: E (H) = n e n f E ef = 1 2 { ɛ (n) R + (K e e) 2 K ef K ef 2 e} E (M) E (EVOL) where = h e n f E ef = D e K e D K e e ɛ p b = (n) R b + ɛ { L n K b (K e e)k b + 2 K e K e b ɛ N 1 D D b N } ( ) + 1+ɛ (n 1) h b E (H) S b 1 n 1 h b [S ef h ef + ɛ e] e = n e n f G ef, p = ɛ h e n f G ef, S b = h e h f b G ef nd the extrinsic curvture K b which is defined s K b = h e e n b = 1 2 L nh b where L n stnds for the Lie derivtive with respect to István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

18 Folitions nd their use The explicit forms: Expressions in the 1 + [n 1] decomposition: Ê (H) = 1 2 { ˆR + ( ˆK l l) 2 ˆK kl ˆKkl 2 ê}, Ê (M) i = ˆD l ˆKli ˆD i ˆKl l ˆp i, Ê (EVOL) ij = ˆR ij Lˆn ˆKij ( ˆK l l) ˆK ij + 2 ˆK il ˆKl j ˆN 1 ˆDi ˆDj ˆN + ˆγ ij {Lˆn ˆKl l + ˆK kl ˆKkl + ˆN 1 ˆDl ˆDl ˆN} [ Ŝ ij ê ˆγ ij ] where ˆD i, ˆR ij nd ˆR denote the covrint derivtive opertor, the Ricci tensor nd the sclr curvture of ˆγ ij, respectively. The htted source terms ê, ˆp i nd Ŝ ij nd the extrinsic curvture ˆK ij re defined s nd ê = ˆn kˆn l (n) G kl, ˆp i = ˆγ k i ˆn l (n) G kl nd Ŝ ij = ˆγ k iˆγ l j (n) G kl ˆK ij = ˆγ l i D l ˆn j = 1 2 Lˆnˆγ ij István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

19 Solving the constrints The 1 + [n 1] decomposition of the extrinsic curvture: The Σ σ hypersurfces in both cses re spcelike: K ij = κ ˆn iˆn j + [ˆn i k j + ˆn j k i ] + K ij κ = ˆn kˆn l K kl = ˆn k (L nˆn k ) k i = ˆγ k iˆn l K kl = 1 2 ˆγk i (L nˆn k ) 1 2 ˆγ ki (L nˆn k ) K ij = ˆγ k iˆγ l j K kl = 1 2 ˆγk iˆγ l j (L nˆγ kl ) K l l = ˆγ kl K kl = 1 2 ˆγij (L nˆγ ij ) projection tking the trce free prts on the S σ,ρ surfces: Π kl ij = ˆγ k iˆγ l j 1 n 1 ˆγ ij ˆγ kl K ij = K ij 1 n 1 ˆγ ij(ˆγ ef K ef ) István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

20 Solving the constrints The 1 + n constrints The momentum constrint: E (M) = h e n f E ef = D e K e D K e e ɛ p = 0 div ( ˆK l l) k i + ˆD l K li + κ ˆn i + Lˆn k i ˆn l K li ˆD i κ n 2 n 1 ˆD i (K l l) ɛ p l ˆγ l i = 0 κ ( ˆK l l) + ˆD l k l K kl ˆKkl 2 ˆn l k l Lˆn (K l l) ɛ p l ˆn l = 0 where ˆnk = ˆn l D lˆn k = ˆD k (ln ˆN) With some lgebr in coordintes (ρ, x 3,..., x n+1 ) dopted to the folition S σ,ρ : n 1 (n 2) ˆN ˆγAB 0 (n 1) ˆN K ρ + (n 2) ˆN ˆγAB ˆγ AK k B 0 1 ˆγ BK ˆN K K K E + B A (k) = 0 B E (K) Is first order symmetric hyperbolic system for the vector vlued vrible (k B, K E E) T where the rdil coordinte ρ plys the role of time.... with chrcteristic cone (prt from the surfces S ρ with ˆn i ξ i = 0) [ˆγ ij (n 1) ˆn iˆn j ] ξ iξ j = 0 István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

21 Solving the constrints The 1 + n constrints The Hmiltonin constrint: E (H) = n e n f E ef = 1 2 { ɛ (n) R + (K e e) 2 K ef K ef 2 e} = 0 using { (n) R = ˆR 2 Lˆn ( ˆK l l) + ( ˆK l l) 2 + ˆK ˆKkl kl + 2 ˆN } 1 ˆDl ˆDl ˆN ɛ ˆR { + ɛ 2 Lˆn ( ˆK l l) +( ˆK l l) 2 + ˆK ˆKkl kl + 2 ˆN } 1 ˆDl ˆDl ˆN + 2 κ K l l + (K l l) 2 2 k l k l K kl K kl 2 e = 0 lgebric eqution for κ provided tht K l l does not vnish eliminting κ the momentum constrint becomes strongly hyperbolic system for (k i, K l l) T provided tht κ nd K l l re of opposite sign by choosing the free dt ( ˆN, ˆN i, ˆγ ij, K ij) on Σ 0 this cn be gurnteed loclly considering dt in Kerr-Schild form: g b = η b + 2Hl l b, (H smooth! on R 4, l is null with respect to both g b nd n implicit bckground Minkowski metric η b ) for ner Schwrzschild k A κ 0 pproximtions: Kl l κ 2(1+2H) 1+H everywhere! István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

22 Solving the constrints Conforml structure by splitting of the induced metric ˆγ ij : There exist smooth function Ω : Σ 0 R which does not vnish except t n origin where the folition S ρ smoothly reduces to point on the Σ 0 level surfces such tht the induced metric ˆγ ij cn be decomposed s ˆγ ij = Ω 2 γ ij where γ ij is such tht γ ij (L ρ γ ij ) = 0 throughout Σ 0 surfces. Wht does the second reltion men? In virtue of γ ij (L ρ γ ij ) = L ρ ln[det(γ ij )] the determinnt is independent of ρ but my depend on the ngulr coordintes. Does the desired smooth function Ω : Σ 0 R nd the metric γ ij exist? István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

23 Solving the constrints The conforml structure: γ ij = Ω 2 ˆγ ij The construction of Ω : Σ 0 R: for ny smooth distribution of the induced metric ˆγ ij one my integrte ˆγ ij (L ρˆγ ij ) = γ ij (L ρ γ ij ) + (n 1) L ρ (ln Ω 2 ) long the integrl curves of ρ on Σ 0, strting with smooth non-vnishing function Ω 0 = Ω 0 (x 3,..., x n+1 ) t S 0. Ω 2 = Ω 2 (ρ, x 3,..., x n+1 ) cn be gives [ Ω 2 = Ω 2 0 exp (ˆγ ij (L ρˆγ ij ) ) d ρ] 1 n 1 ρ 0 The conforml structure stisfying γ ij (L ρ γ ij ) = 0 cn be given thes: γ ij = Ω 2 ˆγ ij István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

24 Solving the constrints The other fces of the Hmiltonin constrint: ɛ ˆR { + ɛ 2 Lˆn ( ˆK l l) +( ˆK l l) 2 + ˆK ˆKkl kl + 2 ˆN } 1 ˆDl ˆDl ˆN + 2 κ K l l + (K l l) 2 2 k l k l K kl K kl 2 e = 0 ɛ = ±1 elliptic eqution for Ω: using ˆKl l = n 1 2 Lˆn ln Ω 2 ˆN 1 D k ˆN k nd ˆγ ij = Ω 2 γ ij = ˆR [ (γ)r { }] = Ω 2 (n 2) D l D l ln Ω 2 + (n 3) (D l ln Ω 2 )(D 4 l ln Ω 2 ) prbolic eqution for ˆN: ˆK l l = ˆN 1 [ n 1 2 L ρ ln Ω 2 ˆD k ˆN k ], Lˆn ( ˆK l l) = [...] Lˆn ˆN +... & ˆN 1 ˆDl ˆDl ˆN R. Brtnik (1993), R. Weinstein & B. Smith (2004) István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

25 Solving the constrints Sorting the components of (h ij, K ij ): The twelve independent components of the pir (h ij, K ij) my be represented by ( ˆN, ˆN i, Ω, γ ij; κ, k i, K l l, K ij) or by pplying κ = Ln ln ˆN nd k i = (2 ˆN) 1 ˆγ il (L n ˆN l ) K l l = n 1 2 L n ln Ω 2 nd K ij = 1 2 Ω2 γ k iγ l j (L nγ kl ) ( ˆN, ˆN i, Ω, γ ij; L n ˆN, Ln ˆN l, L nω, L nγ ij) The momentum constrint (stisfying hyperbolic system) clwys be solved s n initil vlue problem with initil dt specified t some S ρ Σ σ for the vribles L n ˆN l, L nω. The Hmiltonin constrint: ɛ = ±1 elliptic eqution for Ω: ill-posed together with the hyp.mom.constr. prbolic eqution for ˆN: freely specifible: ( ˆN i, Ω, γ ij ; L n ˆN, L nγ ij ) lgebric eqution for κ: freely specifible: ( ˆN, ˆN i, Ω, γ ij ; L nγ ij ) István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

26 Summry: Summry 1 Euclidend Lorentzin signture Einsteinin spces of n + 1-dimension (n 3) were considered. The topology of M ws restricted by ssuming: smoothly folited by one-prmeter fmily of homologous hypersurfces one of these level surfces is smoothly folited by one-prmeter fmily of codimension-two-surfces (orientble compct without boundry in M) 2 the Binchi identity nd pir of nested decompositions cn be used to explore reltions of vrious projections of the field equtions 3 solving the 1 + n constrints: by introducing some geometriclly distinguished vribles!!! regrdless whether the primry spce is Riemnnin or Lorentzin momentum constrint s first order symmetric hyperbolic system. the Hmiltonin constrint s prbolic or lgebric eqution 4 the conforml structure γ ij, defined on the foliting codimension-two surfces S ρ, ppers to provide convenient embodiment of the (n 1) n 2 1 degrees of freedom to vrious metric theories of grvity István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

27 Summry Thnks for your ttention István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

28 Summry First order symmetric hyperbolic liner homogeneous system for (E (H), E (M) i ) T : L n E (M) b L n E (H) + D e E (M) e + D (E (EVOL) b + [ E (H) (K e e) 2 ɛ (ṅ e E (M) e ) ɛ K e (E (EVOL) e + h e E (H) ) ] = 0 + h b E (H) ) + [ (K e e) E (M) b + E (H) ṅ b ɛ (E (EVOL) b + h b E (H) ) ṅ ] = 0 When writing them out explicitly in some locl coordintes (σ, x 1,..., x n ) dopted to the vector field σ = N + N : σ e eσ = 1 nd the folition {Σ σ}, red s {( 1 N N hij ) ( 1 σ + N k N h jk h ik 1 N N k h ij ) } ( ) E (H) k = E (M) i ( ) E E j where the source terms E nd E j re liner nd homogeneous in E (H) nd E (M) i. It is lso informtive to inspect the chrcteristic cone ssocited with the bove eqution which prt from the hypersurfces Σ σ with n i ξ i = 0 cn be gives (h ij n i n j ) ξ iξ j = 0 bck István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

29 Summry Reltions between vrious prts of the bsic equtions: Corollry If the constrint expressions E (H) surfces then the reltions nd E (M) vnish oll the σ = const level K b E (EVOL) b = 0 D E (EVOL) b ɛ ṅ E (EVOL) b = 0 hold for the evolutionry expression E (EVOL) b. István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30

30 Hving n origin Summry A world-line W ρ represents n origin in M: If the foliting codimension-two-surfces smoothly reduce to point on the Σ σ level surfces t the loction ρ = ρ. bck Note tht then Ω vnishes t ρ = ρ. = The existence of n origin on the individul Σ σ level surfces is signified by the limiting behvior ˆγ ij (L ρˆγ ij) ± while ρ ρ ±. σ σ σ n σ σ σ wρ * σ σ σ n To hve regulr origin in M: One needs to impose further conditions excluding the occurrence of vrious defects such s the existence of conicl singulrity. An origin W ρ will be referred s being regulr if there exist smooth functions ˆN (2), Ω (3) nd ˆN A (1) such tht, i neighborhood of the loction ρ = ρ on the Σ σ level surfces, the bsic vribles ˆN, Ω nd ˆN A cn be gives ˆN = 1 + (ρ ρ ) 2 ˆN (2), Ω = (ρ ρ ) + (ρ ρ ) 3 Ω (3), ˆN A = (ρ ρ ) ˆN A (1) István Rácz (Wigner RCP, Budpest) degrees of freedom 27 My, / 30