Integrability condition for higher rank Killing-Stäckel tensors. Kentaro Tomoda

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1 2016/10/26 Osaka City Univ. Integrability condition for higher rank Killing-Stäckel tensors Kobe Univ. Institute of Cosmophysics, Kentaro Tomoda in collaboration with Tsuyoshi Houri (Kobe Univ.) and Yukinori Yasui (Setsunan Univ.)

2 2 / 22 A rank-r Killing-Stäckel tensor satisfies the Killing equation (b σ a 1 a r ) = 0 σ a 1 a r = σ (a 1 a r ) A Killing-Stäckel (KS) tensor is a symmetric tensor generalisation of the Killing vector The metric g is a trivial rank-2 KS tensor

3 3 / 22 A rank-r Killing-Stäckel tensor corresponds to a polynomial quantity Q = σ a 1 a r p a1 p ar p a a Q = 0 where p a are the canonical momenta To find Killing-Stäckel tensors To find polynomial conserved quantities for geodesic motion To understand the integrability of Hamiltonian systems on spacetimes (or curved spaces)

4 4 / 22 Research Purpose To count the number of linearly independent Killing- Stäckel tensors for a given metric Strategy: This can be accomplished by solving the Killing eq. But, in general the Killing eq. is not solvable We take up the task without solving the Killing eq.

5 Methodology Killing vector as an example 5 / 22

6 6 / 22 Consider the Killing equation (a σ b) = 0 If the dimension of a manifold is 4 The number of unknown variables are also 4 The number of equations are 10 The Killing equation forms an overdetermined system of PDEs

7 Rewrite the Killing equation (a σ b) = 0 as a σ b = ab ( ab = [a σ b] ) a bc = R cbad σ d These are called the prolonged equations This is a homogeneous system of linear equations The dimension of the solution space is bounded by initial values of σ a and ab (4 + 6 = 10) 7 / 22

8 8 / 22 The prolonged equations a σ b = ab ( ab = [a σ b] ) a bc = R cbad σ d Promoting the smoothness of solutions (from 1st line) [a b] σ c = R c[ba]d σ d = [a b]c R [abc]d σ d = 0 Bianchi id. (No information)

9 The prolonged equations a σ b = ab ( ab = [a σ b] ) a bc = R cbad σ d Promoting the smoothness of solutions (from 2nd line) [a b] cd = [a (R dc b]e σ e ) [a R cd b]e σ e R ab[c e d]e R cd[a e b]e = 0 This is the integrability condition The integrability condition is an algebraic equation 9 / 22

10 10 / 22 Demonstrations (1/5) = = = - - = = = = = - = = = = = = = = = = [ ] μν = = [[μ + ν + ]] {μ } {ν } [ ] - - [ ]

11 11 / 22 Demonstrations (2/5) Γ = Γρ = μν ρσ σ= μ νσ + ν - σ {μ } {ν } {ρ μσ μν = μνρ κ = = μν σ= = ν Γκ - μ Γκ μρ σ Γ σ - μν μ + Γ σ νρ μρ Γκ - σν νρ Γσ Γκ {μ } {ν σμ σ= Γ σ + Γ λ σν μν Γσ - λσ σν Γλ Γσ {μ } λμ σ= σ= λ= = ϵ = μνρκ μ νρκ ϵ + Γ ϵ σ - μσ νρκ Γσ ϵ - μν σρκ Γσ ϵ - μρ νσκ Γσ ϵ μκ νρσ σ= {κ } {ϵ } = ϕ = μνρκϵ μ ϕ + Γ ϕ σ - νρκϵ μσ νρκϵ Γσ ϕ - μν σρκϵ Γσ ϕ μρ νσκϵ σ= {ν } {ρ } {κ } {ϵ } {ϕ }

12 Demonstrations (3/5) 111 1σ11 σ = σ μ {μ } = {μ } {ν } μν σ = {μ } {ν } μν μν ω = σ { } { } { } = = σ = [][[μ + ν + σ σ σ σ σ 12 / 22

13 13 / 22 Demonstrations (4/5) = = μνρδκ = = - + = σ - σ - = σ - = σ - σ + κμρδν κνρδμ = - μνρ δκ σ + μνδ ρκ = = - - μρδν νρδμ κ = = {δ ρ + } {κ } κμνρ - - δ κμνδ ρ = κρδμ ν - κρδν = { } { } { + } { + } {μ } {ν μ + } {ρ } {δ ρ + } {κ } μνρδκ

14 Demonstrations (5/5) = [[{}] ] σ / / σ σ σ σ - ( - ) σ ( - ) σ σ - σ - σ - Possible variables σ 0, σ 2, σ 3, and 23 We know that there are four Killing vectors: t, ϕ, (sin ϕ θ + cot θ cos ϕ ϕ ) and (cos ϕ θ cot θ sin ϕ ϕ ) 14 / 22

15 15 / 22 Key Points The method can be applied to higher rank KSs But, the prolonged equations and the integrability conditions for higher rank KS tensors are not known In this work, we derived the prolonged equations and the integrability conditions for rank-2 and rank-3 KS tensors Here we present the results for rank-2 KS tensors

16 Main results for rank-2 Killing-Stäckel tensor 16 / 22

17 17 / 22 The Killing equation for rank-2 KS tensors New variables (a σ bc) = 0 abc = [a σ b]c ϕ abcd = ϕ [a [bc] d] ( ϕ abcd = (a b) σ cd ) Abbreviations (R σ) abc d = R abc mσ md ( R σ) a bcd e = a R bcd mσ me (R ) abc de = R abc m mde

18 18 / 22 The prolonged equations (a σ bc) = 0 a σ bc = P 0 [ abc ] a bcd = P 1 [(R σ) abc d ] + ϕ abcd a ϕ bcde = P 2 [( R σ) a bcd e + (R ) abc de ] where projection operators P 0, P 1 and P 2 are defined by P 0 [ abc ] = 4 3 a(bc) P 1 [(R σ) abc d ] = 5 8 (R σ)bca d 3 (R σ)bcd a (R σ)bad c 1 (R σ)bda c 4

19 19 / 22 The prolonged equations (a σ bc) = 0 a σ bc = P 0 [ abc ] a bcd = P 1 [(R σ) abc d ] + ϕ abcd a ϕ bcde = P 2 [( R σ) a bcd e + (R ) abc de ] where projection operators P 0, P 1 and P 2 are defined by P 2 [( R σ) a bcd e ] = 1 2 ( R σ)b cde a + 1 ( R σ)a bec d 4 ( R σ) b cda e + {[be] [cd]}

20 20 / 22 The prolonged equations (a σ bc) = 0 a σ bc = P 0 [ abc ] a bcd = P 1 [(R σ) abc d ] + ϕ abcd a ϕ bcde = P 2 [( R σ) a bcd e + (R ) abc de ] where projection operators P 0, P 1 and P 2 are defined by P 2 [(R ) abc de ] = 4(R ) abe cd 4 (R )abc ed (R )abc de + 1 (R )bec ad (R )bec da + {[be] [cd]}

21 The integrability condition I abcdef = [f a] ϕ bcde + P 2 [ [f ( R σ) a] bcd e + [f (R ) a]bc de ] = 0 Re-arranging gives I abcdef = ( R σ) ab ecd f + 2( R ) a fbe cd + 4 ( R )a fbc ed ( R )afbcde 2(R ϕ) afb ecd + 4 (R ϕ)abc def (R R σ)afb ec d 3 (R R σ)afb cd e (R R σ)abc de f 1 (R R σ)abc fe d 2 + {[af] [be] [cd]} 21 / 22

22 22 / 22 Summary This research aims to count the number of linearly independent Killing-Stäckel tensors for a given metric To this end, we need to derive the prolonged equations for rank-p KS tensors which leads to its integrability condition We derived the prolonged equations for rank-2 and rank-3 KS tensors Here we present the results for rank-2 KS tensors

23 Main result (Backup) for rank-3 Killing-Stäckel tensor 23 / 22

24 24 / 22 The Killing equation for rank-3 KS tensors New variables (a σ bcd) = 0 abcd = [a σ b]cd ϕ abcde = ϕ [a [bc] d]e ( ϕ abcd = (a b) σ cde ) η abcdef = η [a [b [cd] e] f] η abcdef = (a b c) σ def. Abbreviations (R R σ) abc de fg = R abc mr mde nσ nfg (R 2 σ) abc def g = R abc mr def nσ mng

25 The prolonged equations (a σ bcd) = 0 a σ bcd = P 0 [ abcd ] a bcde = P 1 [(R σ) abc de + ϕ abcde ] a ϕ bcdef = P 2 [( R σ) a bcd ef + (R ) abc def ] a η bcdefg +η abcdef = P 3 [( R σ) ab cde fg + ( R ) a bcd efg +(R R σ) abc de fg + (R 2 σ) abc def g ] where projection operators P 0, P 1 P 2 and P 3 are defined by P 0 [ abcd ] = 3 2 a(bcd) 25 / 22

26 + 1 6 ( R σ)bfceda + {[be] [cd]} 26 / 22 P 1 [ϕ abcde ] = 4 3 ϕabc(de) P 1 [(R σ) abc de ] =(R σ) abc (de) + 5 (R σ)ab(d e)c (R σ)a(d b c e) 1 (R σ)bc(d e)a 2 1 (R σ)b(de) ca 3 P 2 [( R σ) a bcd ef ] = 1 4 ( R σ)bcdeaf 1 ( R σ)bcafed ( R σ)bcfaed + 5 ( R σ)bacdfe ( R σ)bfcdae + 1 ( R σ)bacedf 3

27 27 / 22 P 2 [(R ) abc def ] = 5 8 (R )bec daf 3 (R )abf cde (R )fba cde 1 (R )abc fde (R )fbc ade + 1 (R )bec afd (R )bec fad (R )abc def (R )fbc dea 2 (R )abc edf (R )fbc eda 5 (R )abe cdf (R )fbe cda + {[be] [cd]}

28 28 / 22 P 3 [( R σ) ab cde fg ] = ( R σ) bc fde ag 1 ( R σ)bc dea fg ( R σ)ab cde fg + {[bg] [cf] [de]} P 3 [( R ) a bcd efg ] = 4( R ) b gca def + ( R ) b gac def 5 6 ( R )b acd feg ( R )b acd efg 1 ( R )b acd gef ( R )b gcd aef 2( R ) b gcf dea + 17 ( R )b gcd efa ( R )b gcd fea + {[bg] [cf] [de]}

29 29 / 22 P 3 [(R ϕ) abc defg ] = 2(R ϕ) abg cfde + 8 (R ϕ)abc defg (R ϕ)abc degf 7 6 (R ϕ)bgc defa + 1 (R ϕ)bgc deaf (R ϕ)bcd agef 16 9 (R ϕ)bcd agfe + 4 (R ϕ)bcd aefg 3 + {[bg] [cf] [de]} P 3 [(R R σ) abc de fg ] = 8 9 (R R σ)bcd ef ga 13 (R R σ)bcd ga ef (R R σ)bcf de ga 1 (R R σ)bgc df ea (R R σ)bgc fd ea + {[bg] [cf] [de]},

30 30 / 22 P 3 [(R 2 σ) abc def g ] = 1 24 (R2 σ) abc def g (R2 σ) abc deg f (R2 σ) abc dfg e 7 36 (R2 σ) abc dgf e (R2 σ) abg cfd e + {[bg] [cf] [de]}

31 The integrability condition I abcdefgh = [h a] η bcdefg + P 3 [ [h ( R σ) a]b cde fg + [h ( R ) a] bcd efg + [h (R ϕ) a]bc defg + [h (R R σ) a]bc de fg + [h (R 2 σ) a]bc def g ] = 0 Re-arranging gives I abcdefgh = 2 ( R ) ab cgh def 3 ( R )ab gcd efh 2 + ( R ) ab gcd feh + (On-going task) 31 / 22