Space-time models with dust and cosmological constant, that allow - PDF document

Space-time models with dust and cosmological constant, that allow integrating the Hamilton-Jacobi test particle equation by separation of variables method. Konstantin E. Osetrin Tomsk State Pedagogical University

Space-times models with dust and radiation T ij = Λ g ij + ρu i u j + εl i l j , g ij u i u j = 1 , g ij l i l j = 0 , i, j = 1 , ...n signature is (+ , − , − , − ) Λ – cosmological constant, ρ – mass density of dust matter, u i – velocity of dust matter, ε – energy density of radiation, l i – wave vector of radiation. Equation of test particle in Hamilton-Jacobi form g ij S ,i S ,j = m 2 i, j = 1 , ...n

ST¨ ACKEL SPACES Definition 1 Let V n be a n –dimensional Rie- mannian space with metric tensor g ij . The Hamilton – Jacobi equation g ij S ,i S ,j = m 2 i, j = 1 , ...n (1 . 1) can be integrated by complete separation of variables method if co-ordinate set { u i } exists for which complete integral can be presented in the form: n φ i ( u i , λ ) � S = (1 . 2) i =1 where λ 1 ...λ n – is the essential parameter. Definition 2 V n is called the St¨ ackel space if the Hamilton–Jacobi equation (1.1) can be in- tegrated by complete separation of variables method.

Theorem 1 Let V n be the St¨ ackel space. Then g ij in privileged co-ordinate set can be shown in the form g ij = (Φ − 1 ) ν n G ij � ν , ν G ij ν = G ij ν ( u ν ) , Φ ν µ = Φ ν µ ( u µ ) , (2 . 1) G ij ν = δ i ν δ j ν ε ν ( u ν )+ ( δ i ν δ j p + δ j ν δ i p ) G νp δ i p δ j q G pq � � ν + ν , p p,q (no summation over ν ) i, j = 1 , ...n, p, q = 1 , ...N, ν, µ = N + 1 , ...n. 1 ≤ N < n where Φ ν µ ( u µ ) – is called the St¨ ackel matrix.

Geodesic equations of St¨ ackel spaces admit the first integrals that commutes pairwise with respect to the Poisson bracket µ = (Φ − 1 ) ν µ ( ε ν p 2 ν + 2 G νp ν p p p ν + h pq ν p p p q ) , X i p i , p = Y (2 . 2) Y p p, q = 1 , ...N ; ν, µ = N + 1 , ..., n. Φ ν µ ( u µ ) – is called the St¨ ackel matrix, functions ε ν , G νp ν , h pq depends only from u ν , ν p i – momentum. If we write the functions X ν , Y p in the form: ij p i p j , i p i ν = X p = Y (2 . 3) X Y ν p then ν ( ij ; k ) = Y p ( i ; j ) = 0 X (the semicolon denotes the covariant deriva- tive and the brackets denote symmetrization). Therefore Y p i , X ν ij are the components of vector and tensor Killing fields respectively.

Definition 3 Pairwais commuting Killing vec- tors Y p i , where p = 1 , ...N and Killing tensors X ν ij , where ν = N + 1 , ...n form a so called complete set of the type ( N.N 0 ) , where i Y N 0 = N − rank || Y q i || p Theorem 2 A necessary and sufficient geo- metrical criterion of a St¨ ackel space is the pres- ence of a complete set of the type ( N.N 0 ). Then the Hamilton - Jacobi equation can be integrated by the complete separation of vari- ables method if and only if the complete set of the first integrals exists. Definition 4 Space - time is called the St¨ ackel one of the type ( N.N 0 ) if the complete set of the type ( N.N 0 ) exists.

Let us consider the Hamilton–Jacobi equation for the charged particle g ij ( S ,i + A i )( S ,j + A j ) = m 2 . (2 . 8) Theorem 3 If eq.(2.8) admits complete sepa- rations of variables then g ij is the metric tensor of the St¨ ackel space type ( N.N 0 ) . Using this theorem one can show that the sep- aration takes place for the same privileged co- ordinate set and A i = (Φ − 1 ) ν A i A i = (Φ − 1 ) ν n h i ν ( u ν ) , n h ν ( u ν ) . (2 . 9) The last condition can be regarded as an additional functional equation. St¨ ackel space-time is called the special one if (2.8) admits complete separation of variables.

Separation of variables for the Klein – Gordon – Fock equation. Let us consider the Klein–Gordon–Fock equa- tion for the Riemannian space. H − m 2 ) ψ ≡ [ − g kl ∇ k ∇ l − m 2 ] ψ = 0 ( ˆ (3 . 1) ∇ i - is covariant derivative, ψ – scalar function. Definition 5 Eq.(3.1) admits complete sepa- ration of variables if co-ordinate set { u i } exists for which the complete solution can be pre- sented in the form n det || ∂ 2 ψ φ i ( u i , λ ) , � ψ = || � = 0 . (3 . 2) ∂u i ∂λ j i =1

Theorem 4 Let Klein–Gordon–Fock equation admits complete separation of variables. Then g ij is a metric tensor of a St¨ ackel space. More- over the separation of variables takes place at the same privileged co - ordinate set. It was proved that in the special St¨ ackel elec- trovac spacetimes Klein–Gordon–Fock equa- tion can be integrated by the complete sep- aration of variables method. (Recall that St¨ ackel space - time is called the special one if HJ-equation with A i admits com- plete separation of variables)

St¨ ackel spaces and field equations of the theories of gravity. The metrics of the St¨ ackel spaces can be used for integrating the field equations of General Relativity and other metric theories of Gravity. Note that such famous GR exact solutions as Schwarcshild, Kerr, NUT, Friedman belong to the class of St¨ ackel spaces. At the moment all St¨ ackel spaces satisfying the Einstein–Maxwell equations have been found in our papers (Bagrov, Obukhov, Shapovalov, Osetrin) .

Classification of Stackel space-times for the following theories have been considered in our papers: 1. Brans–Dicke scalar-tensor theory. The field equations have the form R ij − 1 2 g ij R = 8 π φ T ij − ω φ 2 ( φ ; i φ ; j − − 1 2 g ij φ ; k φ ; k ) − 1 φ ( φ ; ij − g ij ✷ φ ) (5 . 1) 8 π 3 + 2 ωT ii , ✷ = g ij ∇ i ∇ j , ✷ φ = ω = const. 2. The classification problem for the Einstein– Vaidya equations when the stress–energy tensor have the form: l i l i = 0 T ij = a ( x ) l i l j , (5 . 2)

Conformally St¨ ackel spaces. Let us consider the Hamilton–Jacobi equation for a massless particle g ij S ,i S ,j = 0 (6 . 1) Obviously this equation admits complete sep- aration of variables for a St¨ ackel space. Yet one can verify that if g ij has the form g ij = ˜ g ij ( x ) exp 2 ω ( x ) (6 . 2) where ˜ g ij is a metric tensor of the St¨ ackel space, then eq.(6.1) can be solved by complete separation of variables method too.

Note that conformally St¨ ackel spaces play im- portant role when massless quantum equations are considered (f.e. conformal invariant Chernikov–Penrose equation, Weyl’s equation etc.). The problem of classification of conformally St¨ ackel spaces satisfying the Einstein equation R ij = Λ g ij , Λ = const (6 . 3) g ij = ˜ g ij ( x ) exp 2 ω ( x ) is more difficult than appropriate problem for the St¨ ackel spaces. We obtained the following form of integrability conditions: � C δαβγ exp ( n − 3) ω � ˜ ˜ ∇ δ = 0 . (6 . 4) If dimension of the space V n equals to 4, eq. (6.4) has the form ∇ δ � � ˜ ˜ C δαβγ exp ω = 0 . (6 . 5)

Using integrability conditions we have proved the following theorem Theorem 5 Let g ij be the metric tensor of the St¨ ackel space of type (N.1). Then Einstein space conformal to ˜ V 4 admits the same Killing vectors as V 4 . Moreover one can prove the following state- ment. Theorem 6 Let ˜ V n is conformally St¨ ackel space of type (N.1) ( N ≥ 2 ) satisfying the Einstein equation (6.3). Then Hamilton–Jacobi equa- tion (1.1) admits the complete separation of variables. In other words nontrivial null conformally St¨ ackel solutions of the Einstein equations belong only to (1.1)-type Stackel spaces.

HOMOGENEOUS ST¨ ACKEL SPACES Let us consider the problem of classification of space-homogeneous models of space-times which admit a complete separation of variables in Hamilton-Jacobi equation. The most interesting models for cosmology are space-homogeneous models, which admit a 3- parametrical transitive group of motions with space-like orbits. On the other hand, the St¨ ackel space ( N.N 0 ) type admit N Killing vectors. Thus, there is a problem of finding a subclass of homogeneous space-times admitting com- plete sets of integrals of motion. In other words, a space-time with a complete set must admit a 3-parametrical transitive group of motions with space-like orbits.

Let us consider Stackel space of type (3.1). This type of Stackel space-times is rather in- teresting in this context, because its metric depend only on such variable of privileged co- ordinate set, which corresponds to null (wave) hypersurface of the Einstein equation. In other words, the Stackel space-times of this type are common to spaces filled of radiation (gravitational, electromagnetic etc.). In a privileged coordinate set metric of (3.1) type has the form  b 2 ( x 0 ) b 3 ( x 0 )  0 1 1 0 0 0   g ij =   , b 2 ( x 0 ) a 22 ( x 0 ) a 23 ( x 0 )   0     b 3 ( x 0 ) a 23 ( x 0 ) a 33 ( x 0 ) 0 where x 0 is the wave-like null variable.

Space admit 3 commuting Killing vectors X 1 , X 2 , X 3 ; [ X p , X q ] = 0 , p, q, r = 1 , 2 , 3 with components X pi = δ i p . The metric projection on orbits of this group of motions is degenerated. Thus, we need an additional Killing vector X i 4 = ξ i . The commutative relations of group X 1 — X 4 have the form [ X m , X 4 ] = α m X 4 + β mn X n [ X 1 , X 4 ] = α 1 X 4 + β 1 p X p , p, q = 1 , 2 , 3 [ X p , X q ] = 0 The Jacobi identities for the structure con- stants have the form β 11 α m = 0 α 2 β 3 n = α 3 β 2 n α m β 1 n = α 1 β mn