Structural equations for Killing tensors of arbitrary rank^{∗}

Thomas Wolf School of Mathematical Sciences Queen Mary and Westfield College University of London London E1 4NS T.Wolf@maths.qmw.ac.uk

November 19, 2001

Abstract

An algorithm is given for bringing the equations of monomial first integrals of arbitrary degree of the geodesic motion in a Riemannian space V_{n} into the form (F_{A})_{;k} = ∑_{B} Γ_{kAB}F_{B}. The F_{A} are the components of a Killing tensor K_{i_{1}...i_{r}} of arbitrary rank r and its symmetrized covariant derivatives. Explicit formulas are given for rank 1,2 and 3. Killing tensor equations in structural form allow the formulation of algebraic integrability conditions and are supposed to be well suited for integration as it is demonstrated in the case of flat space. A method based on integrability conditions being algebraic is given to compute in a single generic point of spacetime numerically the number of non-trivial Killing tensors using numerical values for the Riemann tensor and its derivatives.

PACS numbers: 0420, 0240, 0490, 0290 By-line: Structural Equations for Killing Tensors

∗appeared in Comp. Phys. Comm., 115 (1998), 316 - 329

1 Introduction

As symmetries of a Riemannian space are associated with conservation laws of geodesic motion in this space, the investigation of Killing vectors (KV) and Killing tensors (KT) is a central task in General Relativity in classifying and understanding a spacetime.

A test particle moving in a Riemannian space V_{n} with velocity u^{i}, i = 1, …, n has as a constant of motion K_{i_{1}...i_{r}}u^{i_{1}} … u^{i_{r}} if K_{i_{1}...i_{r}} satisfies the conditions ([4])

K_{(i_{1}...i_{r};i_{(r+1)})} = 0  (1)

where () indicates complete symmetrization and [ ] in the following complete antisymmetrization.

But KT’s are not only linked to geodesic motion. When motion is not described by geodesics but when the corresponding Hamiltonian H has a potential function V

H = \frac{1}{2}g^{ij}(x^{k})p_{i}p_{j} + V (x^{k})

and when a first integral F is polynomial in the momentum variables p_{i}:

F = K_{s}(p)^{s} + K_{s−1}(p)^{(s−1)} + … + K_{0}

(indices suppressed) with symmetric contravariant tensors K_{i} of rank 0, 1, .., s then K_{s} and K_{s−1} have to be KT’s (see e.g. [20]) which are to be computed first as they enter the equations for the other K_{i}.

Another source of interest for KT’s comes from the fact that the separability of the Hamilton-Jacobi equation

g^{ij}∂_{x^{i}}W ∂_{x^{j}}W = E, g^{ij} = g^{ji}, 1 ≤ i, j ≤ n

is dependent on the existence of KT’s for the metric g_{ij} which was important for the integration of the geodesic equation for the Kerr metric by Carter and which is important in classical mechanics as well ([1], [2], [10], [27]).

Despite of their importance only few proper KT of rank two, very few of rank three and as far as the author is aware of, none of rank higher than three are known. The main problem in solving equations (1) lies in the high computational effort, as we will see below, especially when the rank is higher than two.

With the availability of computer algebra programs [23], [24],[25] for investigating and solving overdetermined PDE systems like (1) it is possible to try solving these equations directly and automatically. This is the approach pursuit by the package CLASSYM written by Guy Grebot [26] in the computer algebra system CLASSI. CLASSYM formulates the equations for KVs, conformal KVs and KTs of rank 2 including their integrability conditions. Then it uses the package CRACK to solve them. In this approach it is up to the intelligence of the program CRACK to apply the right integrability conditions at the right time to make the whole task feasible. Although methods used for computing a so-called differential Gröbner basis, for integrating total differentials and others, are algorithmic, there is much freedom left in these algorithms, e.g. the choice of the total ordering in computing the differential Gröbner basis. To use this freedom most efficiently is difficult - for humans as well as for programs.

In this paper a different approach is pursuit, one that is completely algorithmic, and not depending on a particular choice of total orderings of derivatives. It involves rewriting the KT equations of arbitrary rank in the form of structural equations

(F_{A})_{;k} = ∑_{B} Γ_{kAB}F_{B}  (2)

which are homogeneous linear first order PDEs expressing all first order derivatives of a set of functions F_{A} through expressions algebraic in these functions. Because all first order derivatives of all F_{A} are given in terms of F_{B}, the general solution can only contain free constants.

2 Advantage of tensorial structural equations

The advantage of structural equations is, that all integrability conditions of (2) are easily obtained by calculating (F_{A})_{;[kl]} at first from (2) with first order derivatives substituted through (2) and also using the RICCI identity (4) (below). The resulting algebraic linear integrability conditions can be differentiated again and first derivatives be substituted to obtain additional linear integrability conditions. This process can be continued until no new functionally independent conditions result [21].

An advantage of having structural equations in tensorial form, i.e. using the covariant derivative in (2), is that they can be easily contracted with the metric tensor or geometrically meaningful vector fields and scalar equations be computed, as there is a good chance that they may become relatively simple, i.e. short or integrable.

Although the computer algebra program CRACK ([23], [26]) for solving overdetermined PDEsystems such as (1), does apply integrability conditions, it does not know about the tensor structure of a PDE system. Therefore some advantage is expected to generate integrability conditions outside of CRACK in such a way that their tensorial character is preserved.

Why should it be advantageous to investigate structural equations and not to formulate integrability conditions, for example, by generating a differential Gröbner basis?

The equations of a differential Gröbner basis are generated on the basis of an ordering of variables and of functions. This ordering has to be chosen beforehand. The final form of the differential Gröbner basis is dependent on the ordering applied. As many orderings are possible, it is unlikely that, for example, the vanishing of one derivative of one component of the Killing tensor will show up as an equation in a differential Gröbner basis. But knowing about such equations is important in order to integrate the PDE-system.

On the other hand having a formulation of the PDE-system as structural equations available, such an equation would show up. The reason for this is that all first order derivatives of all Killing tensor components and of all their derivatives of order up to r are expressed through expressions in them. The vanishing of some of the derivatives will lead the way of integrations to be done to integrate the system step by step. The price of this order-independence is a large number of mostly long integrability conditions. The most simple case - flat space - as solved below indicates that once all integrability conditions are satisfied it is better to integrate the structural equations first and to impose the symmetries (13), (14) as linear algebraic conditions on the constants of integration afterwards. For Killing vectors the structural equations take the well known form

K_{i;j} = ω_{ij}
ω_{ij;k} = K_{m}R_{kji}^{m}

with the Killing vector K_{i} and the antisymmetric Killing bivector ω_{ij} = −ω_{ji} being the F_{A} of (2).

For the case of rank two Killing tensors Collinson [3] derived integrability conditions equivalent to the ones in this paper although the Killing tensor equations had not been rewritten as a first order system for some F_{A} to allow writing the integrability conditions as algebraic conditions for these F_{A}. Hauser and Malhiot [6],[7] introduced new tensors L, M to formulate structural equations for rank two Killing tensors. Their definition differs from ours and is especially well adapted to do investigations in a null tetrad [8]. Using definitions as those of Hauser and Malhiot, structural equations for conformal KT’s of rank 2 were given by Weir in [22]. The maximal number of KT’s which is attained if and only if V_{n} is of constant curvature has been found by G. Thompson (see [19],[20]) to be \frac{(n+r−1)!(n+r)!}{(n−1)!n!r!(r+1)!}. If K is a KT of rank r then all derivatives of K of order greater than r are expressible in terms of derivatives of K of order less than or equal to r through the solution of a linear algebraic system of equations [16],[20].

The aim of this paper is to give an algorithm for such a calculation in enough detail to be programmable and to give the concrete form of structural equations for lower ranks.

Applying our method the calculation of all rank r KT’s of flat space becomes very simple. It has been shown that KT’s of spaces with vanishing or constant curvature are reducible ([20]), i.e. they are linear combinations of symmetrized products of Killing vectors. The alternative constructive proof we give in appendix B will provide as a by-product the relation between our structural equations in terms of symmetrized covariant derivatives of the Killing tensor on one hand and the formalism of Hauser and Malhiot using tensors antisymmetric in pairs of indices on the other hand.

3 Notation

To keep formulae better readable we will use a bold letter K for the Killing tensor whenever possible which is understood throughout the paper to be of rank r, with r indices taking values 1, …, n in an n-dimensional Riemannian space. Covariant derivatives are denoted by D and a covariant derivative D_{i_{1}}D_{i_{2}} … D_{i_{s}} of order s is also written as D^{s}.

If a tensor is to be completely symmetrized over a number of indices then the symmetry operator S will be used, i.e. S_{i_{1}...i_{r}} will denote complete symmetrization over the indices i_{1}, …, i_{r}. Because the structural equations will be formulated in terms of completely symmetrized covariant derivatives of the Killing tensor K, we define tensors K^{(r)(s)} through

K_{i_{1}...i_{(r+s)}}^{(r)(s)} := K_{i_{1}...i_{r};(i_{(r+1)}...i_{(r+s)})}
= S_{i_{(r+1)}...i_{(r+s)}}K_{i_{1}...i_{r};i_{(r+1)}...i_{(r+s)}}, 0 ≤ s  (3)

with K^{(r)(0)} := K.

The sign of the Riemann tensor is fixed through the RICCI identity

2W_{i_{1}...i_{p};[cd]} = W_{bi_{2}...i_{p}}R_{i_{1}cd}^{b} + … + W_{i_{1}...i_{(p−1)}b}R_{i_{p}cd}^{b}.  (4)

Only for the purpose of proving the equivalence of Killing tensor equations to a system of structural equations we will drop terms from equations which come in only through the RICCI identity. If such terms are dropped we will use ‘≃’ instead of ‘=’. That means that with equalities of the form A ≃ B, where A, B are n’th order differential expressions in components of K we will express that

A = B + terms of lower than n’th order in K which

include the Riemann tensor and vanish in flat space.

4 Structural equations

The purpose of this section is to formulate structural equations that are equivalent to the Killing tensor equations (1) in terms of tensors K^{(r)(s)}.

This proceeds in two steps.

1. At first differential identities are derived expressing the first order derivative D K^{(r)(s)} in terms of K^{(r)(0)}, …, K^{(r)(s+1)} and the Riemann tensor.

2. In the second step the Killing equations and differential consequences of them are expressed as algebraic conditions for the K^{(r)(s)}. By proving K^{(r)(r+1)} ≃ 0 the resulting structural system is shown to have finitely many equations.

To do the first step the covariant derivatives of order s + 1 of the Killing tensor D^{(s+1)} K are expressed in terms of symmetrized derivatives K^{(r)(0)} … K^{(r)(s+1)} and the Riemann tensor using the RICCI identity (4) and its derivatives

2K_{i_{1}...;...i_{p}[cd]i_{(p+1)}...} = (K_{bi_{2}...;...i_{p}}R_{i_{1}cd}^{b} + … + K_{i_{1}...;...i_{(p−1)}b}R_{i_{p}cd}^{b})_{;i_{(p+1)}...}  (5)

to have

K_{i_{1}...i_{r};i_{(r+1)}...i_{(r+s+1)}} = K_{i_{1}...i_{r};(i_{(r+1)}...i_{(r+s+1)})}
+ antisymmetrized derivatives of K  (6)
= K_{i_{1}...i_{(r+s+1)}}^{(r)(s+1)} +
expressions in K^{(r)(0)}, …, K^{(r)(s−1)}.  (7)

In (6) antisymmetrized derivatives of K of order s + 1 have been reduced to derivatives of order s − 1 and each of them can be split in turn into a fully symmetrized s − 1 order derivative (a component of K^{(r)(s−1)}) and antisymmetrized derivatives of order s−3 and so on using the RICCI identity.

The first step is completed by a symmetrization over the indices i_{(r+1)}...i_{(r+s)} giving

K_{i_{1}...i_{r};(i_{(r+1)}...i_{(r+s)})i_{(r+s+1)}} = K_{i_{1}...i_{(r+s)};i_{(r+s+1)}}^{(r)(s)}
= K_{i_{1}...i_{(r+s+1)}}^{(r)(s+1)} +
expressions in K^{(r)(0)}, …, K^{(r)(s−1)}.  (8)

i.e. equations of the form

D K^{(r)(s)} ≃ K^{(r)(s+1)}, s = 0, 1, ….  (9)

Although the precise form of the right hand side (rhs) of (9) for arbitrary s is not given it can be easily calculated with the following procedure.

A tensor W with indices i_{1} … i_{p} is split into its symmetric part and antisymmetric parts by

W_{i_{1}...i_{p}} = \frac{1}{p!}(W_{i_{1}...i_{p}} + all permutations of \{i_{1} … i_{p}\})
+\frac{1}{p!}( (p! − 1)W_{i_{1}...i_{p}} − all permutations of \{i_{1} … i_{p}\})  (10)
= W_{(i_{1}...i_{p})} + ‘second term of (10)’.  (11)

Each of the (p! − 1) terms W_{i_{1}...i_{p}} in the second term on the rhs of (10) can be converted into any one of the (p! − 1) permutations to cancel it. This is done by repeatedly interchanging two neighboring indices such that the second bracket can be written as a sum of terms W_{...[jk]...} = (W_{...jk...} − W_{...kj...})/2. For example, with p = 3, a possible decomposition is

W_{abc} = W_{(abc)} + \frac{2}{3}W_{a[bc]} + \frac{2}{3}W_{b[ac]} + \frac{1}{3}W_{[ac]b} + \frac{1}{3}W_{[bc]a} + W_{[ab]c}  (12)

(which is not unique due to the Jacobi identity W_{[a[bc]]} +W_{[b[ca]]} +W_{[c[ab]]} ≡ 0). The decomposition of W_{(ab)c} as required in (8) would be identical to the rhs of (12) without the last term W_{[ab]c}.

This completes the first step. The system (9) has the form of structural equations (2) but so far with an unlimited number of functions and equations as s is not bounded.

In the second step the Killing equations and their differential consequences are combined with (9). The Killing equations read

0 = K_{(i_{1}...i_{r};i_{(r+1)})}
= K_{(i_{1}...i_{(r+1)})}^{(r)(1)}  (13)

and successive covariant differentiations and substitutions of first order derivatives of K^{(r)(s)} using (9) provides relations of the form

0 ≃ K_{(i_{1}...i_{(r+1)})i_{(r+2)}...i_{(r+s)}}^{(r)(s)}, s > 1.  (14)

Another identity needed below is

S_{a_{1}...a_{m}}S_{b_{1}...b_{(r+1−m)}}K_{a_{1}...a_{m}b_{1}...b_{(r−m)};b_{(r+1−m)}} =
− \frac{m}{(r + 1 − m)} S_{a_{1}...a_{m}}S_{b_{1}...b_{(r+1−m)}}K_{a_{1}...a_{(m−1)}b_{1}...b_{(r+1−m)};a_{m}},  (15)

for m = 0, …, r. This results from the Killing equations through

0 = (r + 1) K_{(a_{1}...a_{m}b_{1}...b_{(r−m)};b_{(r+1−m)})}
= S_{a_{1}...a_{m}}S_{b_{1}...b_{(r+1−m)}}(r + 1) K_{(a_{1}...a_{m}b_{1}...b_{(r−m)};b_{(r+1−m)})}
= S_{a_{1}...a_{m}}S_{b_{1}...b_{(r+1−m)}} (K_{a_{1}...a_{m}b_{1}...b_{(r−m)};b_{(r+1−m)}}+
K_{b_{(r+1−m)}a_{1}...a_{m}b_{1}...b_{(r−m−1)};b_{(r−m)}} +
…
K_{a_{2}...a_{m}b_{1}...b_{(r−m)}b_{(r+1−m)};a1})
= S_{a_{1}...a_{m}}S_{b_{1}...b_{(r+1−m)}}( (r + 1 − m) K_{a_{1}...a_{m}b_{1}...b_{(r−m)};b_{(r+1−m)}} +
m K_{a_{1}...a_{(m−1)}b_{1}...b_{(r+1−m)};a_{m}} )  (16)

and from (16) which is valid for m = 0, …, r+1, further (15). Identity (15) is used in the remainder of the second step to show that

K_{a_{1}...a_{r}b_{1}...b_{r}}^{(r)(r)} ≃ (−1)^{r} K_{b_{1}...b_{r}a_{1}...a_{r}}^{(r)(r)} and  (17)
K^{(r)(r+1)} ≃ 0.  (18)

This is done by using (15) and the RICCI identities repeatedly:

K_{a_{1}...a_{r}b_{1}...b_{r}}^{(r)(r)} = S_{a_{1}...a_{r}}S_{b_{1}...b_{r}}K_{a_{1}...a_{r};b_{1}...b_{r}}  {by def. of K^{(r)(s)}}
= −rS_{a_{1}...a_{r}}S_{b_{1}...b_{r}}K_{b_{1}a_{1}...a_{(r−1)};a_{r}b_{2}...b_{r}} {by (15) with m = r}
≃ −rS_{a_{1}...a_{r}}S_{b_{1}...b_{r}}K_{b_{1}a_{1}...a_{(r−1)};b_{2}...b_{r}a_{r}} {by RICCI ident. }
= (−1)^{2} \frac{r(r − 1)}{1 · 2} S_{a_{1}...a_{r}}S_{b_{1}...b_{r}}K_{b_{1}b_{2}a_{1}...a_{(r−2)};a_{(r−1)}b_{3}...b_{r}a_{r}}
≃ (−1)^{2} (^{r}_{2}) S_{a_{1}...a_{r}}S_{b_{1}...b_{r}}K_{b_{1}b_{2}a_{1}...a_{(r−2)};b_{3}...b_{r}a_{(r−1)}a_{r}}

…
= (−1)^{r} (^{r}_{r}) S_{a_{1}...a_{r}}S_{b_{1}...b_{r}}K_{b_{1}...b_{r};a_{1}...a_{r}}
= (−1)^{r} K_{b_{1}...b_{r}a_{1}...a_{r}}^{(r)(r)}.

Using this result we get further for K^{(r)(r+1)}

K_{a_{1}...a_{r}b_{1}...b_{(r+1)}}^{(r)(r+1)} = S_{a_{1}...a_{r}}S_{b_{1}...b_{(r+1)}}K_{a_{1}...a_{r};b_{1}...b_{(r+1)}}
≃ (−1)^{r}S_{a_{1}...a_{r}}S_{b_{1}...b_{(r+1)}}K_{b_{1}...b_{r};a_{1}...a_{r}b_{(r+1)}}
≃ (−1)^{r}S_{a_{1}...a_{r}}S_{b_{1}...b_{(r+1)}}K_{b_{1}...b_{r};b_{(r+1)}a_{1}...a_{r}}
= 0.

The possibility to express K^{(r)(r+1)} in terms of K^{(r)(0)}, …, K^{(r)(r−1)} makes the structural system (9) to be finite. To summarize:

Killing equations (1) for Killing tensors K of rank r are equivalent to structural equations (F_{A})_{;k} = ∑_{B} Γ_{kAB}F_{B} with F_{A} being the components K_{i_{1}...i_{(r+s)}}^{(r)(s)} of tensors K^{(r)(s)}, 0 ≤ s ≤ r and K = K^{(r)(0)} :

D K^{(r)(s)} ≃ (1 − δ_{r}^{s}) K^{(r)(s+1)}, s = 0, …, r  (19)

together with the algebraic relations (13) and (14):

0 = S_{i_{1}...i_{(r+1)}} K_{i_{1}...i_{(r+1)}}^{(r)(1)}  (20)
0 ≃ S_{i_{1}...i_{(r+1)}} K_{i_{1}...i_{(r+1)}i_{(r+2)}...i_{(r+s)}}^{(r)(s)}, 2 ≤ s ≤ r  (21)

and symmetries

K_{i_{1}...i_{(r+s)}}^{(r)(s)} = S_{i_{1}...i_{r}}S_{i_{(r+1)}...i_{(r+s)}} K_{i_{1}...i_{(r+s)}}^{(r)(s)}, 0 ≤ s ≤ r.  (22)

The exact form of (19) and (21) has to be worked out for each individual r by using the RICCI identity (5) in the derivation of these equations whenever covariant derivatives are interchanged (see appendix A. for r = 1, 2, 3). The equations (20) - (22) and (17) can be used to simplify the rhs of (19).

5 Flat space

Although it has been proven in [20] that all KT’s of flat space with rank ≥ 2 are fully reducible we give the explicit solution to demonstrate how easily these structural equations can be integrated.

In the special case of flat space all “≃” signs in (19),(13),(14) can be replaced by “=” signs and the equations (19) can be integrated in the order s = r, …, 0 directly to obtain

K_{i_{1}...i_{(2r)}}^{(r)(r)} = L_{i_{1}...i_{(2r)}}^{(r)(r)}

K_{i_{1}...i_{(2r−1)}}^{(r)(r−1)} = L_{i_{1}...i_{(2r)}}^{(r)(r)}x^{i_{(2r)}} + L_{i_{1}...i_{(2r−1)}}^{(r)(r−1)}
…
K_{i_{1}...i_{r}}^{(r)(0)} = ∑_{s=0}^{r} \frac{1}{s!} L_{i_{1}...i_{r}i_{(r+1)}...i_{(r+s)}}^{(r)(s)}x^{i_{(r+1)}} … x^{i_{(r+s)}}

where the L_{i_{1}...i_{(r+s)}}^{(r)(s)}, s = 0, …, r are constants of integration which satisfy the symmetry conditions

L_{i_{1}...i_{r}i_{(r+1)}...i_{(r+s)}}^{(r)(s)} = L_{(i_{1}...i_{r})(i_{(r+1)}...i_{(r+s)})}^{(r)(s)}  (23)
L_{(i_{1}...i_{r}i_{(r+1)})i_{(r+2)}...i_{(r+s)}}^{(r)(s)} = 0.  (24)

In the case of Killing vectors (r = 1) in space-time (n = 4, i_{k} = 0, … 3), the L_{i_{1}}^{(1)(0)} correspond to boosts, the L_{αβ}^{(1)(1)}, α, β = 1, 2, 3 correspond to Galilei rotations and the L_{0α}^{(1)(1)} to Lorentz rotations.

The example of flat space suggests that, if all integrability conditions are fulfilled then if possible, to integrate the structural equations first and only afterwards to solve the algebraic equations as conditions for the constants of integration.

6 Integrability conditions

Integrability conditions for the structural equations are obtained by calculating [DD] K^{(r)(r)} from (19) and again by using the RICCI identity. The resulting identity is a linear algebraic condition for the K^{(r)(0)} … K^{(r)(r)} with coefficients depending on the Riemann tensor and its derivatives of order up to r. The coefficients of the K^{(r)(s)} vanish in spaces of constant curvature. Further integrability conditions can be obtained through differentiation and substitution of derivatives of K^{(r)(s)} through the structural equations to obtain more equations algebraic in K^{(r)(s)}. This procedure can be repeated until such a differentiation and substitution step would only generate equations that are functionally dependent on the equations already known. Then neither would further steps generate new conditions as shown in [21].

In general, the integrability conditions resulting from the computations above will be algebraic expressions in all the K^{(r)(0)}, …, K^{(r)(r)} and will involve many very long equations. For computational purposes it would be interesting to know whether linear combinations of these equations always exist, that involve only a subset of the unknowns, for example no components K^{(r)(r)}. Such, probably shorter, relations could be used first in an attempt to solve the structural equations.

To delete K^{(r)(r)} from the integrability conditions one could use relations (17) and (21) for s = r:

• As the integrability conditions are derived from K_{i_{1},...,i_{(2r)};[i_{(2r+1)}i_{(2r+2)}]}^{(r)(r)}, a symmetrization of the integrability conditions wrt. S_{i_{1},...,ı_{(r+1)}} will yield relations equivalent to

(S_{i_{1},...,ı_{(r+1)}}K_{i_{1}...i_{(2r)}}^{(r)(r)})_{;[i_{(2r+1)}i_{(2r+2)}]}

which will not involve K^{(r)(r)} due to the symmetry (21)

S_{i_{1},...,ı_{(r+1)}}K_{i_{1}...i_{(2r)}}^{(r)(r)} ≃ 0.

• A similar idea is to apply (17) and to calculate

(K_{i_{1}...i_{(2r)}}^{(r)(r)} − (−1)^{r}K_{i_{(2r)}...i_{1}}^{(r)(r)})_{;[i_{(2r+1)}i_{(2r+2)}]},

i.e. to calculate the difference between the integrability conditions and (−1)^{r} times the integrability conditions with indices i_{1}, …, i_{2r} reversed.

The reason that this approach does not work and that only identities result this way, is that the integrability conditions for the structural equations D K^{(r)(s)} = …, s < r, are the equations D K^{(r)(s+1)} = … and are therefore identically satisfied. Only the equations D K^{(r)(r)} = … have non-trivial integrability conditions but they involve K^{(r)(r)}. A computer calculation for rank 2 confirms that only identities result.

Therefore, the only way to get length-reduced integrability conditions is to compute contractions of them not aiming at eliminating K^{(r)(r)}.

7 The combined system of structural equations and inte-

grability conditions

In order to solve the combined system of structural equations and integrability conditions, any single algebraic condition can be solved for one of the F_{A} and this F_{A} can be substitued in all other algebraic conditions and in the structural equations. The n structural equations with a first order derivative of F_{A} will become either identities 0=0 or new algebraic conditions after derivatives of other F_{B} resulting through this substitution have been replaced by other structural equations. This process is repeated until there is no algebraic condition left. If no unevaluated F_{A} is left then all F_{A} are zero and no KT of rank r exists. Otherwise the remaining equations still form a system of structural equations with all integrability conditions satisfied. If this involves k functions F_{A} then these are n × k equations with the general solution having k constants, i.e. this space has k KTs of rank r.

8 Non reducible Killing tensors and the rigidly rotating disk

of dust

In order to find the number of KT’s of rank r one can compute integrability conditions and substitute them in structural equations as described above. Another way, enabling numerical computations locally is the following.

With Z_{n}^{r} = \frac{1}{r+1} \binom{n+r−1}{r} \binom{n+r}{r} as the maximal number of KT’s of rank r ([20]) we can obtain the number of KT’s of a given space by subtracting the number of functionally independent algebraic integrability conditions from Z_{n}^{r}. The result is the total number of reducible and non-reducible rank r KT’s. If one knows all KV’s and KT’s of rank less than r (including the metric as a trivial KT of rank 2) then it is no problem to find the number of reducible KT’s of rank r by generating all linear independent fully symmetrized products of lower rank KT’s that give a tensor of rank r. The number of non-reducible KT’s is therefore the difference of both numbers. To complete this counting we needed the lower rank KT’s explicitly. It is not enough to know only the number of non-reducible lower rank KT’s. For example, in 4-dimensional flat space the 10 KV’s provide 10(10+1)/2=55 symmetrized products, but only 50 of them are linearly independent which is the maximal number of KT’s of rank 2.

To find only the number of non-trivial Killing tensors one can do the calculation in a single point because the integrability conditions are algebraic in the K^{(r)(s)}. This can become important like in the following problem.

In a number of publications ([12]-[17]), Meinel and Neugebauer gave the exact solution of Einstein’s Field Equations describing the rigidly rotating disk of dust.

In [15] it is shown that for fixed angular momentum there exists an upper limit for the total mass of the disk and at this limit a transition to the Kerr solution occurs which is known to have a rank 2 Killing tensor.

Another limiting procedure leads to the cosmological solution ([17])

ds^{2} = e^{2U} [e^{2k}(dr^{2} + r^{2}dϑ^{2}) + r^{2} sin^{2} ϑ dφ^{2}] − e^{2U}(dt + adϕ)^{2}

e^{2U} = Ω^{2}r^{2} · \frac{cos^{4} ϑ + 6 cos^{2} ϑ − 3}{cos^{2} ϑ + 1},

a = \frac{2}{Ω^{2}r} · \frac{cos^{2} ϑ − 1}{cos^{4} ϑ + 6 cos^{2} ϑ − 3}

e^{2k} = \frac{1}{4}(cos^{4} ϑ + 6 cos^{2} ϑ − 3).

Applying CLASSYM [26], 4 Killing vectors are found, i.e. two more than the dust disk has:

∂_{t}, ∂_{ϕ}, (\frac{t^{2}}{2} + \frac{1}{8r^{2}Ω^{2}} ) ∂_{t} − rt∂_{r} − \frac{1}{2rΩ}∂_{ϕ}, t∂_{t} − r∂_{r}.

Also in the Newtonian spherical symmetric limit, the Kepler potential, there is an additional vectorial conservation law quadratic in the momentum - the Runge-Lenz vector. It is the classical analog of a rank 2 Killing tensor.

Motivated through these extra conservation laws the question arises whether the dust disk has a rank 2 Killing tensor as well. Although the metric is known analytically, a direct investigation would be very difficult as it involves hyperelliptic integrals.

An alternative is:

• Compute the numerical value of the Riemann tensor and its first and second covariant derivatives to a high precision in a single point which is neither on the symmetry axis nor on the plane z = 0.

• Compute the coefficients of the 10 components K_{ij}^{(2)(0)}, the 20 K_{ijk}^{(2)(1)} satisfying (20) and 100 K_{ijkl}^{(2)(2)} satisfying (22) in the system of symmetry conditions (21) (or (29) below) and all integrability conditions (30).

• Compute the rank of the coefficient matrix of the 130 unknowns in these equations and to subtract it from 130, to find an upper bound for the number of Killing tensors.

• Subtract 4 from this number for the metric tensor and the 3 symmetrized products of the two Killing vectors in order to obtain an upper bound for the number of non-trivial KTs.

• If that number is non-zero then compute numerical values of third order derivatives of the Riemann tensor and differentiate the integrability conditions to compute the rank of the coefficient matrix of the increased set of equations. Repeat this process until the rank does not change or the number of non-trivial KTs is zero.

To perform this procedure for the dust disk, numerical values for the Riemann tensor and its first and second order derivatives had been provided by A. Kleinwächter [11] with an accuracy of 15 positions. After the first cycle at least a further cycle would have been necessary but the first set of integrability conditions was already too big to be differentiated again and the D K^{(r)(s)} to be substituted in a 24MB REDUCE session. Also, highly accurate numerical values for all third order derivatives of the Riemann tensor would have become necessary.

An alternative test is to check the Newtonian rigidly rotating dust disk for conservation laws quadratic in the momentum. The Hamiltonian describing motion of a test particle in this potential is ([17])

H = \frac{1}{2}G_{ij}p^{i}p^{j} + V,

G_{ij} = \frac{1}{mr^{2}} diag (\frac{1 + ξ^{2}}{ξ^{2} + η^{2}} , \frac{1 − η^{2}}{ξ^{2} + η^{2}} , \frac{1}{(1 + ξ^{2})(1 − η^{2})}) ,

V = −\frac{ω^{2}r^{2}}{π} (\frac{4}{3} arctan ξ + (ξ − (ξ^{2} + \frac{1}{3}) arctan ξ) (1 − 3η^{2})) .

In order to have conservation laws quadratic in p^{i}

const. = K_{ij}p^{i}p^{j} + L,

the symmetric coefficient matrix K_{ij} has to satisfy the Killing tensor equation as a necessary condition

K_{(ij;m)} = 0.  (25)

The covariant derivative is computed with G_{ij}. Using CLASSYM with the command (dimension 3), 10 equations (25) for 6 unknowns K_{ij} are generated. The general solution of this system found

by CRACK is

K_{ij} = c_{1}G_{ij} + c_{2}δ_{i}^{φ}δ_{j}^{φ} \frac{1}{(1 − η^{2})^{2}(1 + ξ^{2})^{2}}

with the second Killing tensor being the square of the Killing vector ∂_{ϕ} .

As the Newtonian dust disk does not have any non-trivial conservation laws quadratic in the momentum, it is highly unlikely, that the relativistic case has any non-trivial rank 2 Killing tensor.

Acknowledgment

The author wants to thank Guy Grebot for drawing attention to references [19] and [20] and further Hans Stephani for comments to the manuscript. The author also thanks the Relativity group in Jena for hospitality where part of this work was done as well as the Konrad Zuse Institut in Berlin where part of the programming was done.

Appendix A: Structural equations for Killing tensors of rank

1,2 and 3

The structural equations and integrability conditions provided below have been obtained with the help of a special purpose REDUCE computer algebra program [9], that performs tensor operations as described in section 3. In addition, it is able to factor out symmetry and antisymmetry operators S, A as is seen below in order to compactify expressions. The first upper index indicating the rank of K has been omitted as well as the ( ) of the second upper index to improve readability.

Rank one

For rank r = 1 the structural equations take the well-known form

K_{a;b}^{0} = K_{ab}^{1}  (26)
K_{ab;c}^{1} = K_{m}^{0}R_{cba}^{m}  (27)

with integrability conditions

0 = A_{ab} A_{cd} ( K_{m}^{0}R_{cba;d}^{m} + K_{mc}^{1}R_{dab}^{m} + K_{ma}^{1}R_{bcd}^{m})  (28)

where A is the antisymmetrization operator. The algebraic condition (13) takes the form 0 = S_{ab} K_{ab}^{1}.

Rank two

For rank r = 2 the structural equations take the form

K_{ab;c}^{0} = K_{abc}^{1}
K_{abc;d}^{1} = K_{abcd}^{2} + S_{ab} ( K_{am}^{0}R_{bcd}^{m})
K_{abcd;e}^{2} = 2 S_{ab} ( S_{cde} ( K_{cm}^{0}R_{bda;e}^{m} − K_{cm}^{0}R_{dea;b}^{m} +

2 K_{acm}^{1}R_{deb}^{m} − K_{cdm}^{1}R_{bea}^{m}) + S_{cd} ( \frac{1}{3} K_{am}^{0}R_{cdb;e}^{m} + \frac{2}{3} K_{am}^{0}R_{ecb;d}^{m} −

\frac{1}{3} K_{abm}^{1}R_{dec}^{m} + 2 K_{amc}^{1}R_{edb}^{m} + K_{ame}^{1}R_{cdb}^{m}) )

and algebraic conditions (13) (14) (17) become

0 = S_{abc} K_{abc}^{1}
0 = S_{abc} ( K_{abcd}^{2} + K_{am}^{0}R_{bcd}^{m})
K_{abcd}^{2} − K_{cdab}^{2} = 2 S_{ab} S_{cd} ( K_{am}^{0}R_{cdb}^{m} − K_{cm}^{0}R_{abd}^{m})  (29)

The integrability conditions are

0 = S_{ab} S_{cd} A_{ef} [K_{am}^{0}(2R_{ecb;df}^{m} − 3/2R_{nfe}^{m}R_{cdb}^{n} − R_{nfd}^{m}R_{ceb}^{n} − R_{bnc}^{m}R_{efd}^{n}

+R_{nfb}^{m}R_{dec}^{n} + R_{cnd}^{m}R_{efb}^{n} − 4R_{nfd}^{m}R_{ecb}^{n} − 2R_{ndb}^{m}R_{efc}^{n} + R_{fnd}^{m}R_{ceb}^{n} +R_{fnd}^{m}R_{ecb}^{n} − R_{fnb}^{m}R_{dec}^{n}) + K_{cm}^{0}(R_{bea;df}^{m} − R_{dea;bf}^{m} − R_{eda;bf}^{m} +R_{nfe}^{m}R_{bda}^{n} + R_{nfd}^{m}R_{bea}^{n} − R_{anb}^{m}R_{efd}^{n} − R_{nfb}^{m}R_{dea}^{n} + R_{and}^{m}R_{efb}^{n} +2R_{nfb}^{m}R_{eda}^{n} + 2R_{ndb}^{m}R_{efa}^{n} − R_{fnd}^{m}R_{bea}^{n} + R_{fnb}^{m}R_{dea}^{n} + 4R_{fnb}^{m}R_{eda}^{n} +3R_{dnb}^{m}R_{efa}^{n}) + K_{em}^{0}(R_{bca;df}^{m} − R_{cda;bf}^{m} + R_{nfd}^{m}R_{bca}^{n} − R_{nfb}^{m}R_{cda}^{n} −3R_{ndb}^{m}R_{fca}^{n} − R_{fnd}^{m}R_{bca}^{n} + R_{fnb}^{m}R_{cda}^{n} − 3R_{dnb}^{m}R_{fca}^{n}) − K_{abm}^{1}R_{dec;f}^{m} +2K_{acm}^{1}R_{deb;f}^{m} + 2K_{acm}^{1}R_{edb;f}^{m} + 2K_{aem}^{1}R_{cdb;f}^{m} + 6K_{amc}^{1}R_{edb;f}^{m} +2K_{ame}^{1}R_{cdb;f}^{m} − 2K_{ame}^{1}R_{fcb;d}^{m} − K_{cdm}^{1}R_{bea;f}^{m} − 2K_{cem}^{1}R_{bda;f}^{m} −K_{cme}^{1}R_{bda;f}^{m} − K_{cme}^{1}R_{bfa;d}^{m} + K_{cme}^{1}R_{dfa;b}^{m} + K_{cme}^{1}R_{fda;b}^{m}

+K_{emf}^{1}R_{bca;d}^{m} − K_{emf}^{1}R_{cda;b}^{m} + 3K_{abcm}^{2}R_{efd}^{m} + K_{abem}^{2}R_{dfc}^{m}
−6K_{acdm}^{2}R_{efb}^{m} − 2K_{acem}^{2}R_{dfb}^{m} + 4K_{acem}^{2}R_{fdb}^{m} + 6K_{aecm}^{2}R_{fdb}^{m}
+2K_{aefm}^{2}R_{cdb}^{m} + K_{cdem}^{2}R_{bfa}^{m} − 2K_{cefm}^{2}R_{bda}^{m} ]  (30)

Rank three

For rank r = 3 the structural equations take the form

K_{abc;d}^{0} = K_{abcd}^{1}
K_{abcd;e}^{1} = K_{abcde}^{2} + S_{abc} A_{de} 3K_{abm}^{0}R_{edc}^{m}
K_{abcde;f}^{2} = K_{abcdef}^{3} − S_{abc} S_{de} ( K_{abm}^{0}R_{cfd;e}^{m} + 2/3K_{abcm}^{1}R_{efd}^{m}
+3K_{abmd}^{1}R_{cfe}^{m})
K_{abcdef;g}^{3} = S_{abc} S_{def} ( about 200 terms involving K^{0}, K^{1}, K^{2})

Symmetries of K^{1}, …, K^{3} as a consequence of the Killing equations (13) and their derivatives (14) are

0 = S_{abcd} K_{abcd}^{1}
0 = S_{abcd} ( 2K_{abcde}^{2} − 3K_{abm}^{0}R_{dec}^{m})
0 = S_{abcd} ( 2K_{abcdef}^{3} − K_{abm}^{0} (3R_{dec;f}^{m} + R_{dfc;e}^{m} + R_{cfe;d}^{m})
−2/3K_{abcm}^{1} (R_{efd}^{m} + R_{dfe}^{m})
−3K_{abmc}^{1}R_{dfe}^{m} − 3K_{abme}^{1}R_{dfc}^{m} − 3K_{abmf}^{1}R_{dec}^{m})

Integrability conditions for rank r=3 are very large in size which may be discouraging. But products vanish as soon as only one factor vanishes and for a given metric with many vanishing components of the curvature tensor expressions might shrink drastically. Also, for a given metric to be investigated it would be enough to compute only one integrability condition and to use that to reduce the system of structural equations as described in section 7. Another simplification is given in dimensions lower than 4. In three dimensions the Riemann tensor can be expressed through the Ricci tensor and in two dimensions through the Ricci scalar.

Appendix B: Reducibility of KT’s of flat space

In this appendix we prove that the general rank r Killing tensor of flat space

K_{a_{1}...a_{r}} = ∑_{s=0}^{r} \frac{1}{s!} L_{a_{1}...a_{r}b_{1}...b_{s}}^{(r)(s)}x^{b_{1}} … x^{b_{s}}  (31)

is fully reducible, i.e. it can be written as a sum of fully symmetrized products of Killing vectors.

Although this reducibility was already proven before in [20], we included this direct version for two reasons.

• As a byproduct we will get a general one-to-one correspondence between any tensors L_{a_{1}...a_{r}b_{1}...b_{s}} and C_{a_{1}...a_{r}b_{1}...b_{s}} both of rank r+s, r ≥ s where L has symmetries L_{a_{1}...a_{r}b_{1}...b_{s}} = L_{(a_{1}...a_{r})(b_{1}...b_{s})}, L_{(a_{1}...a_{r}b_{1})(b_{2}...b_{s})} = 0 and C is antisymmetric in each pair a_{i}, b_{i} and is furthermore symmetric in a_{s+1} … a_{r} and symmetric w.r.t. exchanges of pairs a_{i}b_{i} ↔ a_{k}b_{k}.

• In the non-flat case an analogous calculation, starting from (34) but only using K^{(r)(s)} instead of L^{(r)(s)} and the RICCI-identity whenever covariant derivatives are interchanged, shows an equivalence between tensors K^{(r)(s)} and C^{(r)(s)}. In the work of Hauser & Malhiot [6, 7] structural equations for rank 2 Killing tensors are formulated using tensors L, M. These tensors are identical to C^{(2)(1)}, C^{(2)(2)} apart from a constant factor and an interchange of indices C_{abc}^{1} ↔ L_{acb}, C_{abcd}^{2} ↔ M_{acbd}. As the constructive method to express C^{(r)(s)}, K^{(r)(s)} in terms of each other does work for arbitrary rank, we achieved as well a generalization of the formalism of Hauser & Malhiot [6, 7] to arbitrary dimension. This formulation of structural equations using L, M was prefered by Hauser & Malhiot when a null tetrad formulation is to be used to formulate the structural equations for a given spacetime.

Proof: An n-dimensional flat space has the Killing vectors ξ^{A}, A = 1, …, n with components ξ_{i}^{A} = δ_{i}^{A} and the Killing vectors ζ^{α}, α = 1, …, n(n − 1)/2 with components ζ_{i}^{α} = ϵ_{ij}^{α}x^{j}, ϵ_{ij}^{α} = −ϵ_{ji}^{α} ∈ (−1, 0, 1). To generate the term L_{a_{1}...a_{r}b_{1}...b_{s}}^{(r)(s)} x^{b_{1}} … x^{b_{s}} in (31) from these Killing vectors, we have to consider a linear combination of products symmetrized over the indices a_{1}, …, a_{r} of s vectors ζ^{α} (due to the power s of x^{i}) and r − s vectors ξ^{A}:

S_{a_{1}...a_{r}} C_{α_{1}...α_{s}A_{1}...A_{(r−s)}}^{(r)(s)}ϵ_{a_{1}b_{1}}^{α_{1}}x^{b_{1}} … ϵ_{a_{s}b_{s}}^{α_{s}}x^{b_{s}}δ_{a_{(s+1)}}^{A_{1}} … δ_{a_{r}}^{A_{(r−s)}}  (32)

where summation over identical Latin indices goes from 1 to n and summation over Greek indices goes from 1 to n(n − 1)/2. We have to show that coefficients C^{(r)(s)} can be chosen such that any L^{(r)(s)} with its symmetries (23) can be expressed through

L_{a_{1}...a_{r}b_{1}...b_{s}}^{(r)(s)} = S_{a_{1}...a_{r}}S_{b_{1}...b_{s}} C_{α_{1}...α_{s}a_{(s+1)}...a_{r}}^{(r)(s)}ϵ_{a_{1}b_{1}}^{α_{1}} … ϵ_{a_{s}b_{s}}^{α_{s}}.  (33)

with S_{b_{1}...b_{s}} taking over the symmetrizing effect of x^{b_{1}} … x^{b_{s}} or, equivalently,

L_{a_{1}...a_{r}b_{1}...b_{s}}^{(r)(s)} = S_{a_{1}...a_{r}}S_{b_{1}...b_{s}} C_{a_{1}...a_{r}b_{1}...b_{s}}^{(r)(s)} , 0 ≤ s ≤ r  (34)

with coefficients C_{a_{1}...a_{r}b_{1}...b_{s}}^{(r)(s)} that are antisymmetric in each pair a_{m}b_{m}, 1 ≤ m ≤ s.

The symmetry (24) of L^{(r)(s)}, i.e. the vanishing of L^{(r)(s)} symmetrized over the first r+1 indices is also satisfied by the rhs of (34). Such a symmetrization would involve for each term of the sum given by the rhs of (34) at least one pair of indices a_{m}b_{m} and would therefore vanish.

In the following we will give a constructive proof of the existence of C^{(r)(s)} by showing how to calculate them from given L^{(r)(s)}. A pair of indices p_{i}q_{i} will stand for a pair a_{i}b_{i} with either p_{i} = a_{i}, q_{i} = b_{i} or p_{i} = b_{i}, q_{i} = a_{i}. The calculation is done by a sequence of antisymmetrizations successively over the index pairs a_{1}b_{1}, a_{2}b_{2}, …, a_{s}b_{s}. A single antisymmetrization step is done through manipulating each term by the following step

S_{a_{1}...a_{r}}S_{b_{1}...b_{s}} L_{p_{1}...p_{(i−1)}a_{i}p_{(i+1)}...p_{r}q_{1}...q_{(i−1)}b_{i}q_{(i+1)}...q_{s}}^{(r)(s)}

= S_{a_{1}...a_{r}}S_{b_{1}...b_{s}} L_{p_{1}...p_{(i−1)}a_{i}p_{(i+1)}...p_{r}b_{i}q_{1}...q_{(i−1)}q_{(i+1)}...q_{s}}^{(r)(s)}

(It is assumed that among the first r indices of L^{(r)(s)} there are m a-indices and (r − m) b-indices. The identity (16) then gives:)

= S_{a_{1}...a_{r}}S_{b_{1}...b_{s}}(L_{p_{1}...p_{(i−1)}a_{i}p_{(i+1)}...p_{r}b_{i}q_{1}...q_{(i−1)}q_{(i+1)}...q_{s}}^{(r)(s)} −

\frac{1}{(r + 1)}((r + 1 − m) L_{p_{1}...p_{(i−1)}a_{i}p_{(i+1)}...p_{r}b_{i}q_{1}...q_{(i−1)}q_{(i+1)}...q_{s}}^{(r)(s)} +

m L_{p_{1}...p_{(i−1)}b_{i}p_{(i+1)}...p_{r}a_{i}q_{1}...q_{(i−1)}q_{(i+1)}...q_{s}}^{(r)(s)}))
= S_{a_{1}...a_{r}}S_{b_{1}...b_{s}} \frac{m}{(r + 1)}(L_{p_{1}...p_{(i−1)}a_{i}p_{(i+1)}...p_{r}b_{i}q_{1}...q_{(i−1)}q_{(i+1)}...q_{s}}^{(r)(s)} −
L_{p_{1}...p_{(i−1)}b_{i}p_{(i+1)}...p_{r}a_{i}q_{1}...q_{(i−1)}q_{(i+1)}...q_{s}}^{(r)(s)})
= S_{a_{1}...a_{r}}S_{b_{1}...b_{s}} \frac{m}{(r + 1)}(L_{p_{1}...p_{(i−1)}a_{i}p_{(i+1)}...p_{r}q_{1}...q_{(i−1)}b_{i}q_{(i+1)}...q_{s}}^{(r)(s)} −
L_{p_{1}...p_{(i−1)}b_{i}p_{(i+1)}...p_{r}q_{1}...q_{(i−1)}a_{i}q_{(i+1)}...q_{s}}^{(r)(s)})

i.e. if m of the first r indices of L^{(r)(s)} are a-indices then

S_{a_{1}...a_{r}}S_{b_{1}...b_{s}} L_{...a_{i}...b_{i}...}^{(r)(s)}
= S_{a_{1}...a_{r}}S_{b_{1}...b_{s}} \frac{m}{(r + 1)}(L_{...a_{i}...b_{i}...}^{(r)(s)} − L_{...b_{i}...a_{i}...}^{(r)(s)}).  (35)

This antisymmetrization step wrt. pairs a_{i}b_{i}, i = 1, …, s is applied successively to each term that arose in the previous step when a_{(i−1)}, b_{(i−1)} were antisymmetrized:

L_{a_{1}...a_{r}b_{1}...b_{s}}^{(r)(s)}
= S_{a_{1}...a_{r}}S_{b_{1}...b_{s}} L_{a_{1}...a_{r}b_{1}...b_{s}}^{(r)(s)}
= S_{a_{1}...a_{r}}S_{b_{1}...b_{s}} \frac{r}{(r + 1)} (L_{a_{1}...a_{r}b_{1}...b_{s}}^{(r)(s)} − L_{b_{1}a_{2}...a_{r}a_{1}b_{2}...b_{s}}^{(r)(s)})
= S_{a_{1}...a_{r}}S_{b_{1}...b_{s}} \frac{r}{(r + 1)} (\frac{r}{(r + 1)} (L_{a_{1}...a_{r}b_{1}...b_{s}}^{(r)(s)} − L_{a_{1}b_{2}a_{3}...a_{r}b_{1}a_{2}b_{3}...b_{s}}^{(r)(s)})

−\frac{(r − 1)}{(r + 1)} (L_{b_{1}a_{2}...a_{r}a_{1}b_{2}...b_{s}}^{(r)(s)} − L_{b_{1}b_{2}a_{3}...a_{r}a_{1}a_{2}b_{3}...b_{s}}^{(r)(s)}))

…

= S_{a_{1}...a_{r}}S_{b_{1}...b_{s}} C_{a_{1}...a_{r}b_{1}...b_{s}}^{(r)(s)}.

The expressions in L^{(r)(s)} obtained finally are identified with C^{(r)(s)}.

For example, for r = 2 the method yields

C_{a_{1}a_{2}b_{1}}^{(2)(1)} = \frac{2}{3} ( L_{a_{1}a_{2}b_{1}}^{(2)(1)} − L_{b_{1}a_{2}a_{1}}^{(2)(1)})
C_{a_{1}a_{2}b_{1}b_{2}}^{(2)(2)} = \frac{2}{9} ( 2 L_{a_{1}a_{2}b_{1}b_{2}}^{(2)(2)} − 2 L_{a_{1}b_{2}b_{1}a_{2}}^{(2)(2)}
− L_{b_{1}a_{2}a_{1}b_{2}}^{(2)(2)} + L_{b_{1}b_{2}a_{1}a_{2}}^{(2)(2)}).

This algorithm for computing C^{(r)(s)} completes the proof of reducibility of all Killing tensors of flat space to linear combinations of symmetrized products of Killing vectors.

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