- Email: [email protected]
Structural equations for killing tensors of arbitrary rank
Computer Physics Communications ComputerPhysicsCommunicationsI15 ( 1998) 316-329
ELSNER
Structural equations for Killing tensors of arbitrary rank Thomas Wolf ’ School
of Mathematical
Sciences.
Queen Mary
and Westfield
Received
College,
27 February
University
of London,
London
El 4NS, UK
1998
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, into the form ( FA);I; = c, fk,uFB. The FA are the components of a Killing tensor Ki,,,,i, 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 space-time numerically the number of nontrivial Killing tensors using numerical values for the Riemann tensor and its derivatives. @ 1998 Elsevier ScienceB.V. PACS: 04.25.-g; 03.2O.+i; 02.40.K~ Keyword.~: Killing tensors;Conservation laws;Computeralgebra
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 space-time. A test particle moving in a Riemannian space V, with velocity u’, i = 1, . . . , n has as a constant of motion K;, ,,,;,& . . ui, if Ki, ,.,i, satisfies the conditions [ 41 K(ib..,,:i(r,lj)
= 0,
(1)
where ( ) indicates complete symmetrization and [ ] in the following indicates 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 = igii(Xk)pipj ’ E-mail:
+ V(nk) ,
[email protected]
OOlO-4655/98/$ - see front PIISOOIO-4655(98)00123-4
matter
@ 1998 Elsevier
Science
B.V. All rights
reserved.
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and when a first integral F is polynomial in the momentum variables pi, F = K,Jp)’
+ KS-, (p)‘“-I’
+ . . + K.
(indices suppressed) with symmetric contravariant tensors K; of rank 0, 1, .., s then K, and KS-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;. Another source of interest for KT’s comes from the fact that the separability of the Hamilton-Jacobi equation, g;j = 2 ,
gi’& Wa,, W = E ,
1
is dependent on the existence of KT’s for the metric g;j 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 [ 23-2.51 for investigating and solving overdetermined PDE systems like ( I), it is possible to try solving these equations directly and automatically. This is the approach pursuit by the package CLASSYMwritten by Guy Grebot [ 261 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 CRACKto 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 Grijbner 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 Grobner 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, (FA);k
= c
rk.utFB
(2)
3
B
which are homogeneous linear first order PDEs expressing all first order derivatives of a set of functions FA through expressions algebraic in these functions. Because all first order derivatives of all FA are given in terms of FB, 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 (FA) ;I k[) at first from (2) with first order derivatives substituted through (2) and also using the RICCJ 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 [ 2 11. 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 PDE-systems 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.
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Why should it be advantageous to investigate structural equations and not to formulate integrability conditions, for example, by generating a differential Griibner basis? The equations of a differential Griibner 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 Griibner 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 Grobner 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 ail 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 Ki;,j
@i,,;k
=
Wij
1
= KntR”;ji
>
with the Killing vector K; and the antisymmetric Killing bivector O;j = -Wji being the FA of (2). For the case of rank two Killing tensors, Collinson [ 31 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 FA to allow writing the integrability conditions as algebraic conditions for these FA. 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 [ 81. Using definitions as those of Hauser and Malhiot, structural equations for conformal KT’s of rank 2 were given by Weir in [ 221. The maximal number of KT’s which is attained if and only if V, is of constant curvature has been found by G. Thompson (see [19,20]) to be [(n+rl)!(n+r)!]/[(~~l)!n!r!(r+ I)!]. 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 [ 16,201. 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 [ 201, 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 I, with r indices taking values 1, . . . , rz in an n-dimensional Riemannian space. Covariant derivatives are denoted by D and a covariant derivative D;, Di, . . . DiT of order s is also written as D”. If a tensor is to be completely symmetrized over a number of indices then the symmetry operator S will be used, i.e. S,,...,r will denote complete symmetrization over the indices it, . . . , i,. Because the structural equations
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will be formulated in terms of completely symmetrized covariant derivatives of the Killing tensor K, we define tensors K(‘)(s) through Ky!“’ : = K.~~...i,:(i(,,l)...il,,,,) 11...4,,r, =Sil,~,l...ii,,,)Ki,...i,;i(,+,,...i,,+,I
7
0 I
(3)
St
with K(‘)(O) := K. The sign of the Riemann tensor is fixed through the RICCI identity 2~,,...,,,;lcdl
=
Wb.12,,;I,’R”.IId + . . . + W.,I...,,,a-,,tJb;,,cd.
(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 ‘N’ instead of ‘=‘. That means that with equalities of the form A 2 B, where A, B are nth order differential expressions in components of K, we will express that A=B+.
4. Structural
terms of lower than nth order in K which m&de the Riemann tensor and vanish in flat space.
equations
The purpose of this section is to formulate structural equations that are equivalent to the Killing tensor Eqs. ( 1) in terms of tensors KC’““‘. This proceeds in two steps. (i) At first differential identities are derived expressing the first order derivative D K(‘)(S) in terms of K”,(o), . . . , K(‘)(‘+I) and the Riemann tensor. (ii) In the second step the Killing equations and differential consequences of them are expressed as algebraic conditions for the K(‘)(r). By proving K(‘)(‘+‘) N 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 DC’+‘) K are expressed in terms of symmetrized derivatives K(‘)(O) . . . K(‘)(“+‘) and the Riemann tensor using the RICCI identity (4) and its derivatives, 2K;, .......i,,lcdli~,,,l~,..= Kb~~.,,;..,~,,Rb;,c~ f ” ’ + K~,,..;,..~ ,,,- ,,bRb;,,cd>
;il,,I,b...
,
(5)
to have + antisymmetrized derivatives of K K;I...i~;;,~,,l...,(,,,~,I, =K;,...i~:(i,~,,,...i(,,~,l)) = K~,,‘~‘~,“,f,‘,‘,, + expressions in KC’““,
(6)
. . . , K(‘)(“-‘)
(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(‘)(“-I)) 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 icr+i) . . . icr+$) giving K. (1. -~~~~Il,,ii...~~,.rl)~~,l~,I, . . .
= K!‘)(s) I,...ljr,S);ljr,,bll
= K:,T,!;,:Is’,‘,,+ expressions in K(‘)(O), . . . , K(“)(“-I)
.
(8)
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i.e., equations of the form D K(r)(“) N @‘)(“+I)
,
s=O,l,....
(9)
Although the precise form of the right-hand side (r.h.s.) of (9) for arbitrary s is not given it can be easily calculated with the following procedure. A tensor W with indices ii . . . i, is split into its symmetric part and antisymmetric parts by W;,,,,;,, = $ ( Wii,,,i,, + all permutations of {ii . . . ipI) +L((p!
- I)W;,...jP -all
P!
permutations of {ii . . . &)I
(10)
= WC!,...;,,) + ‘second term of (10)’
(11)
Each of the (p! - 1) terms W,,,,,,,Sin the second term on the r.h.s. 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...ljk]... = (W,,jk,,, - W..kj... I/2. For example, with p = 3, a possible decomposition is Wabc
= W(abc)
+
“;2Wn,bc,
$w bloc]
+
+
$‘&tc,b
+
$‘f$bc,u
+
(12)
W[abjc
(which is not unique due to the Jacobi identity w[,[b,ll + w[b[,,]l + W[c[ab]l f 0). The decomposition of W(&)c as required in (8) would be identical to the r.h.s. of (12) without the last term W[&]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
o=K(il...i,;;,,, I,) =K(r)(‘) (~l..Jlr1,,)
(13)
and successive covariant differentiations and substitutions of first order derivatives of K(r)(s) using (9) provides relations of the form 0 E &““’ (~l...~i~t,,);c,tz,...;~~i~, ’ s> 1.
t 14)
Another identity needed below is Su ,I.. *,,,sb
m I... b,,, I--n,, Ka I... a,,bl...b
,,-m, ;b,, ,I-“,,
= -
(,. +
1 _ m)
Sa,...u,Sb,...b,,,,-.,,K
al...o(.,-l,bi...b,,+I-,,,;cr,,,
7
(15)
for m = 0,. . . , 1. This results from the Killing equations through o=
tr =&I
+
1) K(o I... o,,b I... b( r-n, ,;b(,,,pn,))
I... ..,sb
= sa I... ..,sb
+f&...an,b =&,..,
..,sb
])K(
I... b(,,,-,,,,(T+ I._. b,,+,-,,,
I... b,,-.,,bc, I... b,,,,-.,,((r+
(Ka
[ll...n,,bi...br,-n,);b[~+l-“,,
I... amb I... b+n,j;b,r+
+I-.,,;a~
I--n,,
+
1 Kb ,r+ I - .,,al...a,bl...b~,-,,-l,:b~~-“,,,
) 1 -
m)K
al...~,,~bl...bi,-,,;b~,+~-~,~
+
mKa,...,,“,-,,b,...b,,,i-.,,;a,,)
t 16)
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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
@‘)(‘+I)
N 0.
This is done by using ( 15) and the RICCI identities repeatedly, f&H” = s a,...a,~b~...b,~~~,...a,;b,...h, 0I...n,hI...h, = 3
+-St,
p
(-l)*
=
(6l)‘
=
w%:.%,~
def.
. ..*. ~b~...b,~h,o,...o,,_,,;o,b~...h,
-&,...,Sh,
= (-1)
{by
..b,Kb,a
*r(r1) ------s 1.2
I.._ u ,,-, ‘I I... u,
(2’) scz j... u,&
of
K(‘)(“)}
{by
,;bs..b,n,
s h I... hrKh,b>o
(1.5)
with
m = r}
{by RICCI ident. } I... n,,-2,:al,-,,b~...b,tr,
I... b,Kblb2ol...ol,-2,;b,...b~~,,~~,~~,~
SC,,,,, a,%
I... b,Kb
I... b,;a I... t,r
I,,.. or .
Using this result we get further for K(‘)(‘+‘), K(‘)(‘+I)
LII... u,b ,... b,,,,,
= sa>...o,Sb
I... b,r,,,Ka
I... u,;b I... hi,+,,
=
(-1)‘s~
I... ,,,sb
I.,. b,,,,,&
2
(-l)‘&,
I... ,r,sb I... h,,+,iKb
/... h,;o I... o,b,,+,i j... b,;b,r+,,u
I... a,
-0.
The possibility to express K(‘)(‘+r ) in terms of K(‘)(O), . . . , K(‘)(‘-l) finite. To summarize:
makes the structural system (9) to be
Killing Eqs. (I) for Killing tensors K of rank r are equivalent to structural equations (r)(S), 0 < s < r and K = K(‘)(O) with FA being the components K!r)(s) t,,..i(r,r) of tensors K DK(‘)(“)
cv (1 - ,jF)K(‘)(.‘+‘) ,
s = 0,. . . , y,
( FA);~ = C, rkA~F~
(19)
together with the algebraic relations (13) and (14),
and symmetries (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). Eqs. (20)-(22) and (17) can be used to simplify the r.h.s. of (19).
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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 “e” signs in ( 19)) ( l3), ( 14) can be replaced by “=” signs and Eqs. ( 19) can be integrated in the order s = r, . . ,O directly to obtain
K!‘)(‘-1) = L!“)(,‘) Xi,2,, + L!‘)(‘-l) 1,...112,-1, Il...li?ri II --.lilr-I)
where the L,!,~~,~;‘,‘, (), s = 0, . . , r are constants of integration which satisfy the symmetry conditions
In the case of Killing vectors (r = 1) in space-time (n = 4, ik = 0,. . .3), the L,!,‘)(O) correspond to boosts, the Lhi”‘, LY,/I = 1,2,3 correspond to Galilei rotations and the Lh:“” to Lorentz rotations. The example of flat space suggests that, if all integrability conditions are fulfilled then if possible, to integrate the structural equations lirst 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] KCr)cr) from (19) and again by using the RICCI identity. The resulting identity is a linear algebraic condition for the K(‘)(O) . . K(‘~(‘~ with coefficients depending on the Riemann tensor and its derivatives of order up to r. The coefficients of the K(“)(“) 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 KC”)(“). 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 [ 2 I]. In general, the integrability conditions resulting from the computations above will be algebraic expressions in all the K(‘)(O), . . . , KC’)(‘) 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 (‘)(‘). Such, probably shorter, relations could be used first in an attempt to solve the structural equations. To delete KC’)(‘) from the integrability conditions one could use relations (17) and (21) for s = r: - As the integrability conditions are derived from KI(Ir~,(~~2r,.,jiZ,, ,,i,Zrt21,, a symmetrization of the integrability conditions w.r.t. St,,,,_,IL,,,, will yield relations equivalent to
323
which will not involve KC’“‘) due to the symmetry (21),
- A similar idea is to apply ( 17) and to calculate
i.e. to calculate the difference between the integrability conditions and ( - 1)’ times the integrability conditions with indices il , . , , izr 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”““’ = . ., s < r, are the equations D K”‘(“+‘) = . and are therefore identically satisfied. Only the equations D K (r)(‘) = . have nontrivial integrability conditions but they involve K(‘)(‘). 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 KC’)(‘).
7. The combined system of structural
equations and integrability
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 FA and this FA can be substituted in all other algebraic conditions and in the structural equations. The n structural equations with a first order derivative of FA will become either identities 0 = 0 or new algebraic conditions after derivatives of other FE 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,,+ is left then all FA 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,, then these are II x 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 Zi = -& (“+:I’) (“+lr) 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;. The result is the total number of reducible and non-reducible rank r KT’s. If one knows all KV’s 2nd 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 lO( 10 + 1)/2 = 55 symmetrized products, but only 50 of them are linearly independent which is the maxima1 number of KT’s of rank 2. To find only the number of nontrivial Killing tensors one can do the calculation in a single point because the integrability conditions are algebraic in the K (‘)(.‘). This can become important like in the following problem. In a number of publications [ l2-171, Meinel and Neugebauer gave the exact solution of Einstein’s Field Equations describing the rigidly rotating disk of dust.
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In [ 1.51 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 [ 171 ds* = e2” [e2k(dr2 e*’ = fi’r*
+ r2sin26dyo2]
+ r*d8*)
. ~os~i++6cos~~-3
cos28+
1
-e2U(dt+adc$)2,
'
2 cos* 6 - 1 a=~‘~os4~+6cos28-3’ e*“=~(~os~~++cos*6-3). Applying CLASSYM[26], 4 Killing vectors are found, i.e. two more than the dust disk has,
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/f)(‘), the 20 K!!)(‘) satisfying (20) and 100 Ki(i2k{(‘) 4 satisfying (22) in the system of symmetry conditions (21) (or (A.4) below) and all integrability conditions (AS). - 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 nontrivial KTs. - If that number is nonzero 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 nontrivial 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. Kleinwachter [ 1 l] 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 DK(‘)(“) to be substituted in a 24MB REDUCEsession. 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 [ 171 H = +G;,p’pj
G;, = -I--diag mr2
v=-
Eg(!
+ V,
-
1 +52
P++*’
I-72 ~ P+7q’
1 (I+p)(1-7+)
>
0.
arctan 5: + (5 - ([* + i) arctan 5) ( 1 - 377*
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In order to have conservation laws quadratic in p’, const. = Kijpipj + L , the symmetric coefficient matrix K;,i has to satisfy the Killing tensor equation as a necessary condition,
K(i,;n,)= 0.
(25)
The covariant derivative is computed with G;j. Using CLASSYM with the command (dimension 3), 10 equations (25) for 6 unknowns K;j are generated. The general solution of this system found by CRACK is 1 %I
= Cl%
+
c2Q7
(1
_ r12)2(,
+ 52)2
,
with the second Killing tensor being the square of the Killing vector 84 . As the Newtonian dust disk does not have any nontrivial conservation laws quadratic in the momentum, it is highly unlikely that the relativistic case has any nontrivial rank 2 Killing tensor.
Acknowledgements The author wants to thank Guy Grebot for drawing attention to Refs. [ 191 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
(A.11 (A.2) with integrability conditions (A.3) where A is the antisymmetrization
operator. The algebraic condition (13) takes the form
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Rank two
For rank r = 2 the structural equations take the form KZb:c= K!bc ’ K;,bc;d= K&cd + Strb( Kin, R’bcd1 9 2 Kt,,,cd;r
2&b
=
0
( &de > ( Ken, R"',,dn;r
+ Scd ( f K:,,,, R":.db;r
+
$ K:,,,
-
en
R":cb;d
R"L;b -
+
2Kf,C"Kkb
$!&;,,
+
-
K~dnFLI
2Kck,,cR":d,,
f
1
Khn$";d,,)
1
and algebraic conditions ( 13), ( 14), ( 17) become 0 = &c Kjbc , 0 = &rbc(f&.d Cbcd
-
+
K:"$"',,cJ
= 2&b&d
KZdab
( K:“,R”:db
-
K:“, R”:,
(A.4)
> .
The integrability conditions are 0 = &b&if&f.
- 3/2R”:,fpR”cdb - R”:,@“,,, - R”‘,,,,R”,fd -I- R”&Rndec [ K:“,(2R”:,,,,f +R”&dR”,fb - 4R”&dR”,,, - 2R”;,bR”,f, + R”lfndRncrb+ R”rS,dR”,cb - R”;nbi?ndrc) +K:n,
W’L~I~
-
fR”:,ndR”,f/, +K,q,,
+ -
-3h’“;,,,Rnfcc,
-
-&nR”;rc;
f
-
2R”:,g,R”,,,
( R”bca;df
-2Kin,,
R”;err;bf
R”bdo,,, t
R”:,d,;bf
+
R”lfndR”j,cc,
-
K:dnt
+
R”lfdu;b
+
K~n,~R”im;d
-2Kiiwn
R”;fb
+
4 Kkv,t
-
R”;,,bRncda
+
R”‘, fd R”,,,
R”;,,dR”bra
t
-
f
2KknR”tcdkf
-
2K:mR”bdo;f
-
K:“,eR”bdqf
Ke’n,fR”:da;b 6K:ec”,
R”:,nbR”,
R”‘fnbRndpu
fd
+
-
R”‘n fbR”dec,
4R”‘f,,bR”<>d,.,
f
3R”;,bt?‘,fa)
3R”;nbR”fce)
2K:mR”:db;f
+
-
R”;fbR”cdo
+
-
R”‘fdb
R”bdu -
R “i fd Rnbc”
R”bm,f
•t Kc&,
R”‘,fe
2R”LdbRnefo
-+ 2K~cn,R”;,,,~
R”ffcb;d
+
-+ %mn R”;dh
+
+ -
R”:
2 K:ef”,
fd
i-
R”:,db
6K~n,cR”~db;f K:“,pR”kfa;d
Kiw,, +
i+
R”;,-c
K:drn,
-
R”b fu
2K;,,,R”;d,;f
K!,,eR”;fa;i,
6K:c,,,,R”;p, -
2K,2,~“,R”‘,d,,
1 (A.51
Rank three
For rank r = 3 the structural equations take the form K:bc;d Ktbcd;r K:bcdr;f
=
Kfibcd =
3 +
Kixdr =
&bcAde3K:b,R”fedc
K:bcdef
-
S”bcSdc
K;bcdrf;R = S&S&f
9
t K:bnr
(about 200
R”:
fd;p
tWIllS
+
2/%ib,“,R”:fd
+
3K;b,dR”:y,
> 3
K”, K’, K2).
itlVOh’ing
Symmetries of K’, . . . , K” as a consequence of the Killing Eqs. (13) and their derivatives ( 14) are 0 = hbcd
Kf,bcd
0 = &cd
( 2Kfbcdr
0 = &bcd
=,“,d,
, f
3K:b”,R”;rc -
K:b”,(3R”;,c;f
) , +
R”;fc;,
+
R”&;d)
-
2/3Kfibc”,(
R”&d
+
R”drp)
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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,
(B.1) 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 [ 201, we included this direct version for two reasons. - As a byproduct we will get a general one-to-one correspondence between any tensors L,,, ...t,,~,,...,, and C‘I, LI,h,...b. both of rank r+s, r 2 s where L has symmetries L, ,...(,,h,...t,, = L(, ,...rr,)(b,...I,,), Lctl,...u,-h,)(/,z...t,,)= 0 and C is antisymmetric in each pair ai, b; and is furthermore symmetric in a,,+~ . . . a, and symmetric w.r.t. exchanges of pairs sib; ++ ukbk, 1 < i, k < s. - In the non-flat case an analogous calculation, starting from (B.4) but only using KC’)(‘) instead of L(‘)(s) and the RICCI-identity whenever covariant derivatives are interchanged, shows an equivalence between tensors K(“)(“) and C”“,” In the work of Hauser & Malhiot [ 671 structural equations for rank 2 Killing tensors are formulated using tensors L,M. These tensors are identical to C(*)(‘), C(2)(2) apart from a constant factor and an interchange of indices C& t-f Loch , C$,cd H Moct,d. As the constructive method to express C(‘)(,“, KC’)‘.‘) 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 rank. This formulation of structural equations using L, M was preferred by Hauser & Malhiot when a null tetrad formulation is to be used to formulate the structural equations for a given space-time. Prc~$ An n-dimensional flat space has the Killing vectors tA, A = 1,. . , n with components 6: = 8; and the Killing vectors l”, (Y= 1,. . .,n(n-1)/2withcomponents[,F=~G~‘,~P;=--E;E (-l,O,l).Togeneratethe term LS:“$ ,,,.,,( xhl . . . xbr in (B.l) from these Killing vectors, we have to consider a linear combination of products symmetrized over the indices al, . . . , a, of s vectors l” (due to the power s of x’) and r - s vectors SA?
where summation over identical Latin indices goes from 1 to n and summation over Greek indices goes from I to n( IZ- 1) /2. We have to show that coefficients Ct’)(“) can be chosen such that any LCrjCs) with its symmetries (23) can be expressed through
with &,, ...h. taking over the symmetrizing effect of xbl . . xb’ or, equivalently,
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with coefficients CJ,“l,$, ,,,,,,( that are antisymmetric in each pair a,,&,,, 1 5 m < s. The symmetry (24) of L”)(‘), i.e. the vanishing of Lcr)csf symmetrized over the first r + 1 indices, is also satisfied by the r.h.s. of (B.4). Such a symmetrization would involve for each term of the sum given by the r.h.s. of (B.4) at least one pair of indices a,,&, and would therefore vanish. In the following we will give a constructive proof of the existence of C (‘)(‘) by showing how to calculate them from given L(‘)(‘). A pair of indices p;qi will stand for a pair aibi with either p; = a;, qi = bi or p; = b;, qi = ai. The calculation is done by a sequence of antisymmetrizations successively over the index pairs asb,y. A single antisymmetrization step is done through manipulating each term by the following UI~I, ah,..., step:
(It is assumed that among the first r indices of L(‘)(s) there are m u-indices and (r - m) b-indices. The identity ( 16) then gives:)
i.e. if m of the first r indices of t(r)(s) are u-indices, then S~r~...cr,~b,...b,~...,,
(‘)(‘),.._ b ,,,, -- so,... u,Sb I... b
This antisymmetrization step w.r.t. pairs aibi, i = 1, . . . , s is applied successively to each term that arose in the previous step when a(/- t ), bci- 1) were antisymmetrized, LWW
(11...a.-b, ...b. = sa, . ..a.Sb I... b, Lj,;).:c;;b = sa, ...ar sb I... b, ~cr;
= sa I... ,,,sb
-~
(r
-
(r+
,,,, b, 1)
bjaz...a,a,b2...b,
_ L(‘)(S)
I... b, r
a,bza~...a,blazbl...h,
(r + 1)
1) 1)
L(r)~s) b,n2...o,o,b2...6,
= %, . ..a. sb, . ..b.C,(;.:,;)h,
_
L”““’ b,b~tn...n,o,azb~...b,
..,b, .
The expressions in L(‘)(“) obtained finally are identified with C(r)(s). For example, for r = 2 the method yields p(”a,crzb, c’2”2’ a,a2b,
= 3(2 p,‘;; b2
= 9( 2 2Lg;,‘;jb2
- ty,‘;;, - 2L3,‘h’j,,
- L;;j,,‘,T;b, + Lti2(;;n,
) .
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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. References [I] [2 1 [3 ] 14 1 [ 51 [ 61 [7 I 181 [9 1 1 101 III 1 [ 121 I 13 1 [ 141 [ 151 [ 16) 1171 [ 181 I 191 [201 I21 ] [ 22 ] [ 23 1 1241 12.5 1 1261 1271
S. Benenti, Intrinsic characterization of the variable separation in the Hamilton-Jacobi equation, J. Math. Phys. 38 ( 1997) 6578-6602. B. Carter, Global structure of the Kerr family of gravitational fields, Phys. Rev. D 174 ( 1968) 1559-1571. C.D. Collinson, Special quadratic first integrals of geodesics, J. Phys. A 4 ( 1971) 756-760. L.P. Eisenhart, Riemannian Geometry (Princeton Univ. Press, Princeton, NJ, 1966) pp. 128 ff. R.P. Geroch, Commun. Math. Phys. 13 ( 1969) 180. 1. Hauser, R.J. Malhiot, Structural equations for Killing tensors of order two I J. Math. Phys. 16 ( 1975) 150-152. 1. Hauser, R.J. Malhiot, Structural equations for Killing tensors of order two 11, J. Math. Phys. 16 ( 1975) 1625-1629. 1. Hauser, R.J. Malhiot, On space-time Killing tensors with a Segre characteristic I( II)( I I)], J. Math. Phys. 17 (1976) 1306-1312. A.C. Hearn, J.P. Fitch, REDUCE, User’s Manual 3.6 (Konrad-Zuse-Zentrum, Berlin, 1995). E.G. Kalnins, W.M. Miller, Killing tensors and variable separation for Hamilton-Jacobi and Helmholz equations, SIAM J. Math, Anal. I I (1980) pp. 1011 ff. A. Kleinwkhter, private communication. G. Neugebauer, R. Meinel, Astrophys. J. 414 (1993) L97. G. Neugebauer, R. Meinel, Phys. Rev. Lett. 73 ( 1994) 2166. G. Neugebauer, R. Meinel, Phys. Rev. Lett. 75 (1995) 3046. G. Neugebauer, A. Kleinwschter, R. Meinel, Helv. Phys. Acta 69 (1996) 472. A. Nijenhuis, Nederl. Akad. Wetensch. Proc. Ser. A 58 (1950) 390. R. Meinel. in: Proc. Second Mexican School on Gravitation and Mathematical Physics, A. Garcia, C. Liimmerzahl, A. Mac& T. Matos, D. Nutiez, eds., preprint gr-qc19703077. T.Y. Thomas. The fundamental theorem on quadratic first integrals, Proc. 32 (1946) IO-IS. G. Thompson, Polynomial constants of motion in flat space, J. Math. Phys. 25 ( 1984) 3474-3478. G. Thompson, Killing tensors in spaces of constant curvature, J. Math. Phys. 27 ( 1986) 2693-2699. 0. Veblen. T.Y. Thomas, The Geometry of Paths, Trans. Am. Math. Sot. 25 (1923) 551-608. G.J. Weir, Conformal Killing tensors in reducible spaces, J. Math. Phys. 18 ( 1977) 1782-1787. T. Wolf, A. Brand, The Computer Algebra Package CRACK for Investigating PDEs, Manual for the package CRACK distributed with REDUCE. T. Wolf. A. Brand, Examples of the investigation of differential equations with modularized programs, Math]. Comput. Modding 25 (1997) 133-139. T. Wolf, The program CRACK for solving PDEs in General Relativity, in: Relativity and Scientific Computing: Computer Algebra, Numerics, Visualization,F.W.Hehl, R.A. Puntigam, H. Ruder, eds. (Springer, Berlin, 1996) pp. 241-258. T. Wolf, 0. Grebot, Automatic symmetry investigation of space-time metrics, Int. J. Mod. Phys. D 3 (1994) 323-326. N.M.J. Woodhouse, Killing tensors and the separation of the Hamilton-Jacobi equation, Commun. Math. Phys. 44 (9) ( 1975).
ELSNER
Structural equations for Killing tensors of arbitrary rank Thomas Wolf ’ School
of Mathematical
Sciences.
Queen Mary
and Westfield
Received
College,
27 February
University
of London,
London
El 4NS, UK
1998
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, into the form ( FA);I; = c, fk,uFB. The FA are the components of a Killing tensor Ki,,,,i, 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 space-time numerically the number of nontrivial Killing tensors using numerical values for the Riemann tensor and its derivatives. @ 1998 Elsevier ScienceB.V. PACS: 04.25.-g; 03.2O.+i; 02.40.K~ Keyword.~: Killing tensors;Conservation laws;Computeralgebra
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 space-time. A test particle moving in a Riemannian space V, with velocity u’, i = 1, . . . , n has as a constant of motion K;, ,,,;,& . . ui, if Ki, ,.,i, satisfies the conditions [ 41 K(ib..,,:i(r,lj)
= 0,
(1)
where ( ) indicates complete symmetrization and [ ] in the following indicates 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 = igii(Xk)pipj ’ E-mail:
+ V(nk) ,
[email protected]
OOlO-4655/98/$ - see front PIISOOIO-4655(98)00123-4
matter
@ 1998 Elsevier
Science
B.V. All rights
reserved.
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and when a first integral F is polynomial in the momentum variables pi, F = K,Jp)’
+ KS-, (p)‘“-I’
+ . . + K.
(indices suppressed) with symmetric contravariant tensors K; of rank 0, 1, .., s then K, and KS-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;. Another source of interest for KT’s comes from the fact that the separability of the Hamilton-Jacobi equation, g;j = 2 ,
gi’& Wa,, W = E ,
1
is dependent on the existence of KT’s for the metric g;j 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 [ 23-2.51 for investigating and solving overdetermined PDE systems like ( I), it is possible to try solving these equations directly and automatically. This is the approach pursuit by the package CLASSYMwritten by Guy Grebot [ 261 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 CRACKto 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 Grijbner 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 Grobner 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, (FA);k
= c
rk.utFB
(2)
3
B
which are homogeneous linear first order PDEs expressing all first order derivatives of a set of functions FA through expressions algebraic in these functions. Because all first order derivatives of all FA are given in terms of FB, 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 (FA) ;I k[) at first from (2) with first order derivatives substituted through (2) and also using the RICCJ 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 [ 2 11. 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 PDE-systems 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.
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Why should it be advantageous to investigate structural equations and not to formulate integrability conditions, for example, by generating a differential Griibner basis? The equations of a differential Griibner 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 Griibner 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 Grobner 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 ail 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 Ki;,j
@i,,;k
=
Wij
1
= KntR”;ji
>
with the Killing vector K; and the antisymmetric Killing bivector O;j = -Wji being the FA of (2). For the case of rank two Killing tensors, Collinson [ 31 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 FA to allow writing the integrability conditions as algebraic conditions for these FA. 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 [ 81. Using definitions as those of Hauser and Malhiot, structural equations for conformal KT’s of rank 2 were given by Weir in [ 221. The maximal number of KT’s which is attained if and only if V, is of constant curvature has been found by G. Thompson (see [19,20]) to be [(n+rl)!(n+r)!]/[(~~l)!n!r!(r+ I)!]. 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 [ 16,201. 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 [ 201, 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 I, with r indices taking values 1, . . . , rz in an n-dimensional Riemannian space. Covariant derivatives are denoted by D and a covariant derivative D;, Di, . . . DiT of order s is also written as D”. If a tensor is to be completely symmetrized over a number of indices then the symmetry operator S will be used, i.e. S,,...,r will denote complete symmetrization over the indices it, . . . , i,. Because the structural equations
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will be formulated in terms of completely symmetrized covariant derivatives of the Killing tensor K, we define tensors K(‘)(s) through Ky!“’ : = K.~~...i,:(i(,,l)...il,,,,) 11...4,,r, =Sil,~,l...ii,,,)Ki,...i,;i(,+,,...i,,+,I
7
0 I
(3)
St
with K(‘)(O) := K. The sign of the Riemann tensor is fixed through the RICCI identity 2~,,...,,,;lcdl
=
Wb.12,,;I,’R”.IId + . . . + W.,I...,,,a-,,tJb;,,cd.
(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 ‘N’ instead of ‘=‘. That means that with equalities of the form A 2 B, where A, B are nth order differential expressions in components of K, we will express that A=B+.
4. Structural
terms of lower than nth order in K which m&de the Riemann tensor and vanish in flat space.
equations
The purpose of this section is to formulate structural equations that are equivalent to the Killing tensor Eqs. ( 1) in terms of tensors KC’““‘. This proceeds in two steps. (i) At first differential identities are derived expressing the first order derivative D K(‘)(S) in terms of K”,(o), . . . , K(‘)(‘+I) and the Riemann tensor. (ii) In the second step the Killing equations and differential consequences of them are expressed as algebraic conditions for the K(‘)(r). By proving K(‘)(‘+‘) N 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 DC’+‘) K are expressed in terms of symmetrized derivatives K(‘)(O) . . . K(‘)(“+‘) and the Riemann tensor using the RICCI identity (4) and its derivatives, 2K;, .......i,,lcdli~,,,l~,..= Kb~~.,,;..,~,,Rb;,c~ f ” ’ + K~,,..;,..~ ,,,- ,,bRb;,,cd>
;il,,I,b...
,
(5)
to have + antisymmetrized derivatives of K K;I...i~;;,~,,l...,(,,,~,I, =K;,...i~:(i,~,,,...i(,,~,l)) = K~,,‘~‘~,“,f,‘,‘,, + expressions in KC’““,
(6)
. . . , K(‘)(“-‘)
(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(‘)(“-I)) 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 icr+i) . . . icr+$) giving K. (1. -~~~~Il,,ii...~~,.rl)~~,l~,I, . . .
= K!‘)(s) I,...ljr,S);ljr,,bll
= K:,T,!;,:Is’,‘,,+ expressions in K(‘)(O), . . . , K(“)(“-I)
.
(8)
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i.e., equations of the form D K(r)(“) N @‘)(“+I)
,
s=O,l,....
(9)
Although the precise form of the right-hand side (r.h.s.) of (9) for arbitrary s is not given it can be easily calculated with the following procedure. A tensor W with indices ii . . . i, is split into its symmetric part and antisymmetric parts by W;,,,,;,, = $ ( Wii,,,i,, + all permutations of {ii . . . ipI) +L((p!
- I)W;,...jP -all
P!
permutations of {ii . . . &)I
(10)
= WC!,...;,,) + ‘second term of (10)’
(11)
Each of the (p! - 1) terms W,,,,,,,Sin the second term on the r.h.s. 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...ljk]... = (W,,jk,,, - W..kj... I/2. For example, with p = 3, a possible decomposition is Wabc
= W(abc)
+
“;2Wn,bc,
$w bloc]
+
+
$‘&tc,b
+
$‘f$bc,u
+
(12)
W[abjc
(which is not unique due to the Jacobi identity w[,[b,ll + w[b[,,]l + W[c[ab]l f 0). The decomposition of W(&)c as required in (8) would be identical to the r.h.s. of (12) without the last term W[&]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
o=K(il...i,;;,,, I,) =K(r)(‘) (~l..Jlr1,,)
(13)
and successive covariant differentiations and substitutions of first order derivatives of K(r)(s) using (9) provides relations of the form 0 E &““’ (~l...~i~t,,);c,tz,...;~~i~, ’ s> 1.
t 14)
Another identity needed below is Su ,I.. *,,,sb
m I... b,,, I--n,, Ka I... a,,bl...b
,,-m, ;b,, ,I-“,,
= -
(,. +
1 _ m)
Sa,...u,Sb,...b,,,,-.,,K
al...o(.,-l,bi...b,,+I-,,,;cr,,,
7
(15)
for m = 0,. . . , 1. This results from the Killing equations through o=
tr =&I
+
1) K(o I... o,,b I... b( r-n, ,;b(,,,pn,))
I... ..,sb
= sa I... ..,sb
+f&...an,b =&,..,
..,sb
])K(
I... b(,,,-,,,,(T+ I._. b,,+,-,,,
I... b,,-.,,bc, I... b,,,,-.,,((r+
(Ka
[ll...n,,bi...br,-n,);b[~+l-“,,
I... amb I... b+n,j;b,r+
+I-.,,;a~
I--n,,
+
1 Kb ,r+ I - .,,al...a,bl...b~,-,,-l,:b~~-“,,,
) 1 -
m)K
al...~,,~bl...bi,-,,;b~,+~-~,~
+
mKa,...,,“,-,,b,...b,,,i-.,,;a,,)
t 16)
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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
@‘)(‘+I)
N 0.
This is done by using ( 15) and the RICCI identities repeatedly, f&H” = s a,...a,~b~...b,~~~,...a,;b,...h, 0I...n,hI...h, = 3
+-St,
p
(-l)*
=
(6l)‘
=
w%:.%,~
def.
. ..*. ~b~...b,~h,o,...o,,_,,;o,b~...h,
-&,...,Sh,
= (-1)
{by
..b,Kb,a
*r(r1) ------s 1.2
I.._ u ,,-, ‘I I... u,
(2’) scz j... u,&
of
K(‘)(“)}
{by
,;bs..b,n,
s h I... hrKh,b>o
(1.5)
with
m = r}
{by RICCI ident. } I... n,,-2,:al,-,,b~...b,tr,
I... b,Kblb2ol...ol,-2,;b,...b~~,,~~,~~,~
SC,,,,, a,%
I... b,Kb
I... b,;a I... t,r
I,,.. or .
Using this result we get further for K(‘)(‘+‘), K(‘)(‘+I)
LII... u,b ,... b,,,,,
= sa>...o,Sb
I... b,r,,,Ka
I... u,;b I... hi,+,,
=
(-1)‘s~
I... ,,,sb
I.,. b,,,,,&
2
(-l)‘&,
I... ,r,sb I... h,,+,iKb
/... h,;o I... o,b,,+,i j... b,;b,r+,,u
I... a,
-0.
The possibility to express K(‘)(‘+r ) in terms of K(‘)(O), . . . , K(‘)(‘-l) finite. To summarize:
makes the structural system (9) to be
Killing Eqs. (I) for Killing tensors K of rank r are equivalent to structural equations (r)(S), 0 < s < r and K = K(‘)(O) with FA being the components K!r)(s) t,,..i(r,r) of tensors K DK(‘)(“)
cv (1 - ,jF)K(‘)(.‘+‘) ,
s = 0,. . . , y,
( FA);~ = C, rkA~F~
(19)
together with the algebraic relations (13) and (14),
and symmetries (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). Eqs. (20)-(22) and (17) can be used to simplify the r.h.s. of (19).
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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 “e” signs in ( 19)) ( l3), ( 14) can be replaced by “=” signs and Eqs. ( 19) can be integrated in the order s = r, . . ,O directly to obtain
K!‘)(‘-1) = L!“)(,‘) Xi,2,, + L!‘)(‘-l) 1,...112,-1, Il...li?ri II --.lilr-I)
where the L,!,~~,~;‘,‘, (), s = 0, . . , r are constants of integration which satisfy the symmetry conditions
In the case of Killing vectors (r = 1) in space-time (n = 4, ik = 0,. . .3), the L,!,‘)(O) correspond to boosts, the Lhi”‘, LY,/I = 1,2,3 correspond to Galilei rotations and the Lh:“” to Lorentz rotations. The example of flat space suggests that, if all integrability conditions are fulfilled then if possible, to integrate the structural equations lirst 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] KCr)cr) from (19) and again by using the RICCI identity. The resulting identity is a linear algebraic condition for the K(‘)(O) . . K(‘~(‘~ with coefficients depending on the Riemann tensor and its derivatives of order up to r. The coefficients of the K(“)(“) 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 KC”)(“). 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 [ 2 I]. In general, the integrability conditions resulting from the computations above will be algebraic expressions in all the K(‘)(O), . . . , KC’)(‘) 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 (‘)(‘). Such, probably shorter, relations could be used first in an attempt to solve the structural equations. To delete KC’)(‘) from the integrability conditions one could use relations (17) and (21) for s = r: - As the integrability conditions are derived from KI(Ir~,(~~2r,.,jiZ,, ,,i,Zrt21,, a symmetrization of the integrability conditions w.r.t. St,,,,_,IL,,,, will yield relations equivalent to
323
which will not involve KC’“‘) due to the symmetry (21),
- A similar idea is to apply ( 17) and to calculate
i.e. to calculate the difference between the integrability conditions and ( - 1)’ times the integrability conditions with indices il , . , , izr 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”““’ = . ., s < r, are the equations D K”‘(“+‘) = . and are therefore identically satisfied. Only the equations D K (r)(‘) = . have nontrivial integrability conditions but they involve K(‘)(‘). 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 KC’)(‘).
7. The combined system of structural
equations and integrability
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 FA and this FA can be substituted in all other algebraic conditions and in the structural equations. The n structural equations with a first order derivative of FA will become either identities 0 = 0 or new algebraic conditions after derivatives of other FE 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,,+ is left then all FA 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,, then these are II x 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 Zi = -& (“+:I’) (“+lr) 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;. The result is the total number of reducible and non-reducible rank r KT’s. If one knows all KV’s 2nd 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 lO( 10 + 1)/2 = 55 symmetrized products, but only 50 of them are linearly independent which is the maxima1 number of KT’s of rank 2. To find only the number of nontrivial Killing tensors one can do the calculation in a single point because the integrability conditions are algebraic in the K (‘)(.‘). This can become important like in the following problem. In a number of publications [ l2-171, Meinel and Neugebauer gave the exact solution of Einstein’s Field Equations describing the rigidly rotating disk of dust.
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In [ 1.51 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 [ 171 ds* = e2” [e2k(dr2 e*’ = fi’r*
+ r2sin26dyo2]
+ r*d8*)
. ~os~i++6cos~~-3
cos28+
1
-e2U(dt+adc$)2,
'
2 cos* 6 - 1 a=~‘~os4~+6cos28-3’ e*“=~(~os~~++cos*6-3). Applying CLASSYM[26], 4 Killing vectors are found, i.e. two more than the dust disk has,
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/f)(‘), the 20 K!!)(‘) satisfying (20) and 100 Ki(i2k{(‘) 4 satisfying (22) in the system of symmetry conditions (21) (or (A.4) below) and all integrability conditions (AS). - 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 nontrivial KTs. - If that number is nonzero 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 nontrivial 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. Kleinwachter [ 1 l] 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 DK(‘)(“) to be substituted in a 24MB REDUCEsession. 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 [ 171 H = +G;,p’pj
G;, = -I--diag mr2
v=-
Eg(!
+ V,
-
1 +52
P++*’
I-72 ~ P+7q’
1 (I+p)(1-7+)
>
0.
arctan 5: + (5 - ([* + i) arctan 5) ( 1 - 377*
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In order to have conservation laws quadratic in p’, const. = Kijpipj + L , the symmetric coefficient matrix K;,i has to satisfy the Killing tensor equation as a necessary condition,
K(i,;n,)= 0.
(25)
The covariant derivative is computed with G;j. Using CLASSYM with the command (dimension 3), 10 equations (25) for 6 unknowns K;j are generated. The general solution of this system found by CRACK is 1 %I
= Cl%
+
c2Q7
(1
_ r12)2(,
+ 52)2
,
with the second Killing tensor being the square of the Killing vector 84 . As the Newtonian dust disk does not have any nontrivial conservation laws quadratic in the momentum, it is highly unlikely that the relativistic case has any nontrivial rank 2 Killing tensor.
Acknowledgements The author wants to thank Guy Grebot for drawing attention to Refs. [ 191 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
(A.11 (A.2) with integrability conditions (A.3) where A is the antisymmetrization
operator. The algebraic condition (13) takes the form
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Rank two
For rank r = 2 the structural equations take the form KZb:c= K!bc ’ K;,bc;d= K&cd + Strb( Kin, R’bcd1 9 2 Kt,,,cd;r
2&b
=
0
( &de > ( Ken, R"',,dn;r
+ Scd ( f K:,,,, R":.db;r
+
$ K:,,,
-
en
R":cb;d
R"L;b -
+
2Kf,C"Kkb
$!&;,,
+
-
K~dnFLI
2Kck,,cR":d,,
f
1
Khn$";d,,)
1
and algebraic conditions ( 13), ( 14), ( 17) become 0 = &c Kjbc , 0 = &rbc(f&.d Cbcd
-
+
K:"$"',,cJ
= 2&b&d
KZdab
( K:“,R”:db
-
K:“, R”:,
(A.4)
> .
The integrability conditions are 0 = &b&if&f.
- 3/2R”:,fpR”cdb - R”:,@“,,, - R”‘,,,,R”,fd -I- R”&Rndec [ K:“,(2R”:,,,,f +R”&dR”,fb - 4R”&dR”,,, - 2R”;,bR”,f, + R”lfndRncrb+ R”rS,dR”,cb - R”;nbi?ndrc) +K:n,
W’L~I~
-
fR”:,ndR”,f/, +K,q,,
+ -
-3h’“;,,,Rnfcc,
-
-&nR”;rc;
f
-
2R”:,g,R”,,,
( R”bca;df
-2Kin,,
R”;err;bf
R”bdo,,, t
R”:,d,;bf
+
R”lfndR”j,cc,
-
K:dnt
+
R”lfdu;b
+
K~n,~R”im;d
-2Kiiwn
R”;fb
+
4 Kkv,t
-
R”;,,bRncda
+
R”‘, fd R”,,,
R”;,,dR”bra
t
-
f
2KknR”tcdkf
-
2K:mR”bdo;f
-
K:“,eR”bdqf
Ke’n,fR”:da;b 6K:ec”,
R”:,nbR”,
R”‘fnbRndpu
fd
+
-
R”‘n fbR”dec,
4R”‘f,,bR”<>d,.,
f
3R”;,bt?‘,fa)
3R”;nbR”fce)
2K:mR”:db;f
+
-
R”;fbR”cdo
+
-
R”‘fdb
R”bdu -
R “i fd Rnbc”
R”bm,f
•t Kc&,
R”‘,fe
2R”LdbRnefo
-+ 2K~cn,R”;,,,~
R”ffcb;d
+
-+ %mn R”;dh
+
+ -
R”:
2 K:ef”,
fd
i-
R”:,db
6K~n,cR”~db;f K:“,pR”kfa;d
Kiw,, +
i+
R”;,-c
K:drn,
-
R”b fu
2K;,,,R”;d,;f
K!,,eR”;fa;i,
6K:c,,,,R”;p, -
2K,2,~“,R”‘,d,,
1 (A.51
Rank three
For rank r = 3 the structural equations take the form K:bc;d Ktbcd;r K:bcdr;f
=
Kfibcd =
3 +
Kixdr =
&bcAde3K:b,R”fedc
K:bcdef
-
S”bcSdc
K;bcdrf;R = S&S&f
9
t K:bnr
(about 200
R”:
fd;p
tWIllS
+
2/%ib,“,R”:fd
+
3K;b,dR”:y,
> 3
K”, K’, K2).
itlVOh’ing
Symmetries of K’, . . . , K” as a consequence of the Killing Eqs. (13) and their derivatives ( 14) are 0 = hbcd
Kf,bcd
0 = &cd
( 2Kfbcdr
0 = &bcd
=,“,d,
, f
3K:b”,R”;rc -
K:b”,(3R”;,c;f
) , +
R”;fc;,
+
R”&;d)
-
2/3Kfibc”,(
R”&d
+
R”drp)
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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,
(B.1) 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 [ 201, we included this direct version for two reasons. - As a byproduct we will get a general one-to-one correspondence between any tensors L,,, ...t,,~,,...,, and C‘I, LI,h,...b. both of rank r+s, r 2 s where L has symmetries L, ,...(,,h,...t,, = L(, ,...rr,)(b,...I,,), Lctl,...u,-h,)(/,z...t,,)= 0 and C is antisymmetric in each pair ai, b; and is furthermore symmetric in a,,+~ . . . a, and symmetric w.r.t. exchanges of pairs sib; ++ ukbk, 1 < i, k < s. - In the non-flat case an analogous calculation, starting from (B.4) but only using KC’)(‘) instead of L(‘)(s) and the RICCI-identity whenever covariant derivatives are interchanged, shows an equivalence between tensors K(“)(“) and C”“,” In the work of Hauser & Malhiot [ 671 structural equations for rank 2 Killing tensors are formulated using tensors L,M. These tensors are identical to C(*)(‘), C(2)(2) apart from a constant factor and an interchange of indices C& t-f Loch , C$,cd H Moct,d. As the constructive method to express C(‘)(,“, KC’)‘.‘) 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 rank. This formulation of structural equations using L, M was preferred by Hauser & Malhiot when a null tetrad formulation is to be used to formulate the structural equations for a given space-time. Prc~$ An n-dimensional flat space has the Killing vectors tA, A = 1,. . , n with components 6: = 8; and the Killing vectors l”, (Y= 1,. . .,n(n-1)/2withcomponents[,F=~G~‘,~P;=--E;E (-l,O,l).Togeneratethe term LS:“$ ,,,.,,( xhl . . . xbr in (B.l) from these Killing vectors, we have to consider a linear combination of products symmetrized over the indices al, . . . , a, of s vectors l” (due to the power s of x’) and r - s vectors SA?
where summation over identical Latin indices goes from 1 to n and summation over Greek indices goes from I to n( IZ- 1) /2. We have to show that coefficients Ct’)(“) can be chosen such that any LCrjCs) with its symmetries (23) can be expressed through
with &,, ...h. taking over the symmetrizing effect of xbl . . xb’ or, equivalently,
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with coefficients CJ,“l,$, ,,,,,,( that are antisymmetric in each pair a,,&,,, 1 5 m < s. The symmetry (24) of L”)(‘), i.e. the vanishing of Lcr)csf symmetrized over the first r + 1 indices, is also satisfied by the r.h.s. of (B.4). Such a symmetrization would involve for each term of the sum given by the r.h.s. of (B.4) at least one pair of indices a,,&, and would therefore vanish. In the following we will give a constructive proof of the existence of C (‘)(‘) by showing how to calculate them from given L(‘)(‘). A pair of indices p;qi will stand for a pair aibi with either p; = a;, qi = bi or p; = b;, qi = ai. The calculation is done by a sequence of antisymmetrizations successively over the index pairs asb,y. A single antisymmetrization step is done through manipulating each term by the following UI~I, ah,..., step:
(It is assumed that among the first r indices of L(‘)(s) there are m u-indices and (r - m) b-indices. The identity ( 16) then gives:)
i.e. if m of the first r indices of t(r)(s) are u-indices, then S~r~...cr,~b,...b,~...,,
(‘)(‘),.._ b ,,,, -- so,... u,Sb I... b
This antisymmetrization step w.r.t. pairs aibi, i = 1, . . . , s is applied successively to each term that arose in the previous step when a(/- t ), bci- 1) were antisymmetrized, LWW
(11...a.-b, ...b. = sa, . ..a.Sb I... b, Lj,;).:c;;b = sa, ...ar sb I... b, ~cr;
= sa I... ,,,sb
-~
(r
-
(r+
,,,, b, 1)
bjaz...a,a,b2...b,
_ L(‘)(S)
I... b, r
a,bza~...a,blazbl...h,
(r + 1)
1) 1)
L(r)~s) b,n2...o,o,b2...6,
= %, . ..a. sb, . ..b.C,(;.:,;)h,
_
L”““’ b,b~tn...n,o,azb~...b,
..,b, .
The expressions in L(‘)(“) obtained finally are identified with C(r)(s). For example, for r = 2 the method yields p(”a,crzb, c’2”2’ a,a2b,
= 3(2 p,‘;; b2
= 9( 2 2Lg;,‘;jb2
- ty,‘;;, - 2L3,‘h’j,,
- L;;j,,‘,T;b, + Lti2(;;n,
) .
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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. References [I] [2 1 [3 ] 14 1 [ 51 [ 61 [7 I 181 [9 1 1 101 III 1 [ 121 I 13 1 [ 141 [ 151 [ 16) 1171 [ 181 I 191 [201 I21 ] [ 22 ] [ 23 1 1241 12.5 1 1261 1271
S. Benenti, Intrinsic characterization of the variable separation in the Hamilton-Jacobi equation, J. Math. Phys. 38 ( 1997) 6578-6602. B. Carter, Global structure of the Kerr family of gravitational fields, Phys. Rev. D 174 ( 1968) 1559-1571. C.D. Collinson, Special quadratic first integrals of geodesics, J. Phys. A 4 ( 1971) 756-760. L.P. Eisenhart, Riemannian Geometry (Princeton Univ. Press, Princeton, NJ, 1966) pp. 128 ff. R.P. Geroch, Commun. Math. Phys. 13 ( 1969) 180. 1. Hauser, R.J. Malhiot, Structural equations for Killing tensors of order two I J. Math. Phys. 16 ( 1975) 150-152. 1. Hauser, R.J. Malhiot, Structural equations for Killing tensors of order two 11, J. Math. Phys. 16 ( 1975) 1625-1629. 1. Hauser, R.J. Malhiot, On space-time Killing tensors with a Segre characteristic I( II)( I I)], J. Math. Phys. 17 (1976) 1306-1312. A.C. Hearn, J.P. Fitch, REDUCE, User’s Manual 3.6 (Konrad-Zuse-Zentrum, Berlin, 1995). E.G. Kalnins, W.M. Miller, Killing tensors and variable separation for Hamilton-Jacobi and Helmholz equations, SIAM J. Math, Anal. I I (1980) pp. 1011 ff. A. Kleinwkhter, private communication. G. Neugebauer, R. Meinel, Astrophys. J. 414 (1993) L97. G. Neugebauer, R. Meinel, Phys. Rev. Lett. 73 ( 1994) 2166. G. Neugebauer, R. Meinel, Phys. Rev. Lett. 75 (1995) 3046. G. Neugebauer, A. Kleinwschter, R. Meinel, Helv. Phys. Acta 69 (1996) 472. A. Nijenhuis, Nederl. Akad. Wetensch. Proc. Ser. A 58 (1950) 390. R. Meinel. in: Proc. Second Mexican School on Gravitation and Mathematical Physics, A. Garcia, C. Liimmerzahl, A. Mac& T. Matos, D. Nutiez, eds., preprint gr-qc19703077. T.Y. Thomas. The fundamental theorem on quadratic first integrals, Proc. 32 (1946) IO-IS. G. Thompson, Polynomial constants of motion in flat space, J. Math. Phys. 25 ( 1984) 3474-3478. G. Thompson, Killing tensors in spaces of constant curvature, J. Math. Phys. 27 ( 1986) 2693-2699. 0. Veblen. T.Y. Thomas, The Geometry of Paths, Trans. Am. Math. Sot. 25 (1923) 551-608. G.J. Weir, Conformal Killing tensors in reducible spaces, J. Math. Phys. 18 ( 1977) 1782-1787. T. Wolf, A. Brand, The Computer Algebra Package CRACK for Investigating PDEs, Manual for the package CRACK distributed with REDUCE. T. Wolf. A. Brand, Examples of the investigation of differential equations with modularized programs, Math]. Comput. Modding 25 (1997) 133-139. T. Wolf, The program CRACK for solving PDEs in General Relativity, in: Relativity and Scientific Computing: Computer Algebra, Numerics, Visualization,F.W.Hehl, R.A. Puntigam, H. Ruder, eds. (Springer, Berlin, 1996) pp. 241-258. T. Wolf, 0. Grebot, Automatic symmetry investigation of space-time metrics, Int. J. Mod. Phys. D 3 (1994) 323-326. N.M.J. Woodhouse, Killing tensors and the separation of the Hamilton-Jacobi equation, Commun. Math. Phys. 44 (9) ( 1975).