Spacetime encodings. IV. The relationship between Weyl curvature and Killing tensors in stationary axisymmetric vacuum spacetimes

Spacetime encodings. IV. The relationship between Weyl curvature and Killing tensorsin stationary axisymmetric vacuum spacetimes

Jeandrew Brink

Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91103, USA(Received 4 November 2009; published 21 January 2010)

The problem of obtaining an explicit representation for the fourth invariant of geodesic motion

(generalized Carter constant) of an arbitrary stationary axisymmetric vacuum spacetime generated

from an Ernst potential is considered. The coupling between the nonlocal curvature content of the

spacetime as encoded in the Weyl tensor, and the existence of a Killing tensor is explored and a

constructive, algebraic test for a fourth-order Killing tensor suggested. The approach used exploits the

variables defined for the Backlund transformations to clarify the relationship between Weyl curvature,

constants of geodesic motion, expressed as Killing tensors, and the solution-generation techniques. A new

symmetric noncovariant formulation of the Killing equations is given. This formulation transforms the

problem of looking for fourth-order Killing tensors in 4D into one of looking for four interlocking two-

manifolds admitting fourth-order Killing tensors in 2D.

DOI: 10.1103/PhysRevD.81.022002 PACS numbers: 04.80.Cc, 04.20.Jb, 04.30.Db

I. INTRODUCTION

Very little is understood about the implications that thecurvature of a spacetime manifold has for particle motionwithin the spacetime. In the context of extreme mass ratioinspiral (EMRI) gravitational-wave observations carefulknowledge of particle motion around compact objectscould lead to a spacetime mapping algorithm [1,2]. Thispaper provides a framework in which the relationshipbetween orbital invariants and the curvature expressed bytheWeyl tensor can be explored. In particular, it formulatesa constructive algebraic test to see whether a particularstationary axisymmetric vacuum (SAV) spacetime admitsan additional invariant, a generalized Carter constant, as-sumed to result from a Killing tensor. Only SAV space-times that have two commuting Killing vectors @t and @�and thus can be generated from a complex Ernst potentialare considered.

The existence of a totally symmetric tensor Tð�1��mÞ oforderm, which obeys the Killing equations Tð�1��m;�Þ ¼ 0

implies that the quantity

Q ¼ Tð�1��mÞp�1� � �p�m

(1)

is constant along a geodesic, and thus provides a constantof motion. In Eq. (1), p� indicates the particle momentum.

It was shown in [3] that the condition that a SAV space-time admits a second-order Killing tensor places directrestrictions on the components of the Weyl tensor. Inparticular, it limits the Petrov type to D [4,5]. The approachto the problem of checking whether a particular SAVspacetime admits a Killing tensor of rankm is conceptuallysimple: all that is required is to formulate the condition thateach component of the Killing tensor exists. These inte-grability conditions result in a number of conditions on the

Weyl tensor. Numerical experiments [3,6] suggest that alarge number of SAV spacetimes may possess a fourth-order Killing tensor. Now, a totally symmetric tensor oforder four in four dimensions has 35 possible independentcomponents and the Killing equations impose 56 condi-tions on the gradients of these components. In the directcheck suggested, the Killing equations have to be satisfiedin conjunction with the 10 vacuum field equations.Furthermore, by writing out integrability conditions,many more equations and unknown fields are generated.The magnitude of the calculation may make the notion ofthe practical implementation of this idea seem absurd.Possibly for this reason, no literature on and no examplesof fourth-order Killing tensors in general relativity seemsto exist [7,8]. This paper demonstrates how, with somefinesse, it is possible to check whether SAV spacetimesadmit a fourth-order Killing tensor, and construct itscomponents.A number of ideas lead to the problem becoming trac-

table. Adopt the point of view that a set of constants ofgeodesic motion are equivalent to a coordinate systemideally suited to describing the motions of free fallingparticles in spacetime. Suppose now that an observerwith his/her own inertial frame is conducting ballistic testsand keeping track of the velocity and positions of a seriesof projectiles, in an attempt to discover experimentallywhat this special coordinate system is. Suppose furtherthat the observer planned to one day communicate theresults of the experiment to someone else traveling throughthe same SAV spacetime, in an unambiguous manner. He isconcerned about how to orient the coordinate system of theexperiment to do so with the greatest ease. A mathemati-cian may suggest that both observer and the traveler ori-entate their coordinate systems according to the geometryof the spacetime, for instance along the principle null

PHYSICAL REVIEW D 81, 022002 (2010)

1550-7998=2010=81(2)=022002(16) 022002-1 � 2010 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.81.022002

directions of the Weyl tensor, which in most spacetimeswill be unique [7], or by selecting a transverse frame.Whether this advice is experimentally feasible is irrelevantfor the rest of the paper, but a well-chosen tetrad is intro-duced in Sec. III and all quantities expressed with refer-ence to it.

The investigation [1–3] into understanding the relation-ship between curvature and geodesics was begun to ex-plore the possibility of exploiting the algebraic propertiesof the solution-generation techniques to provide a methodof mapping spacetimes and cataloging their propertiesobservationally. One of the simplest methods of mappingone spacetime onto the next, and one that is directly relatedto the underlying SL(2,R) symmetry of the SAV fieldequations, is the discovery of a Backlund transformation.Developed by Harrison [9] and Neugebauer [10], theBacklund transformations (BT) are used to generate anew solution from an existing SAV solution using a func-tion known as a pseudopotential (or in recent mathematicalliterature, an ‘‘integral extension’’) to carry out the map-ping. The choice of variables used in these papers for thecurvature quantities makes the field equations particularlytransparent and easy to program using algebraic computersystems such as MATHEMATICA. These variables areadopted in this paper. Ultimately, we can ask the questionof what additional conditions, if any, should be imposed onthe pseudopotential so that for a given BT both the curva-ture components and the Killing tensor components aremapped to the next solution.

It can be observed that the Killing equations for SAVspacetimes decouple in such a manner that a subgroup ofthe equations is equivalent to the Killing equations for atwo-manifold in the (�, z) plane. This allows us to use thegeometric picture, derived from the theory of dynamicalsystems, of what a Killing tensor on a two-manifold ac-tually represents [2], and to exploit the accompanyingsymmetries and analytic structure. The remaining Killingequations couple onto the two-dimensional decoupled sys-tem in a ‘‘treelike structure’’. Found by inspection, thisproperty allows a large number of Killing tensor compo-nents to be sequentially eliminated by well-chosen inte-grability conditions. The remaining 10 components andtheir undetermined derivatives are arranged so that re-peated differentiation introduces a minimal number ofnew variables, so the number of integrability conditionsgrows much more rapidly than the new variables. Finallythe process terminates leading to a spectacularly overde-termined linear system for the components of the Killingtensor and certain of their higher-order derivatives, con-sidered to be independent potentials.

The coefficients of the potentials in the linear system arepolynomials in the field variables and their derivatives (thehighest derivative of the metric functions that is required isthe fifth). Inverting the linear system and writing down theconditions that a solution can be found result in the Killing

tensor components, given explicitly as rational functions ofthe field variables and their derivatives, up to a scalingfactor. The consistency conditions for the linear systemdetermine the conditions on the field variables that arerequired in order for a Killing tensor to exist. The directinversion of the overdetermined system will be termed thebrute forcemethod and is discussed in Sec. VI. I argue thatit is computationally feasible for existing computers to findthe analytic answer. The possible caveat being that oncethe condition on the field variables is written down, it maynot be easy to identify the physical implications of theresult. In practice, the main application of the brute forcemethod would be to construct the Killing tensor compo-nents for a given spacetime, once they are known to exist. Itshould also be noted that the analysis is local, so given aspacetime and enough local derivatives at a point theprocess described in Sec. VI would provide an immediatecheck of whether the Killing equations are consistent atthat point. A number of simplifications and insight-building special cases are suggested in VIIA, using thegeneral framework of the brute force method.A more elegant tack than the brute force method is to

ask, given a particular SAV spacetime solution that admitsa Killing tensor, which of the BT or other solution-generation techniques preserves this property. This ques-tion is formulated clearly mathematically but not answeredin Sec. VII B.The main effort in the calculation is to arrange quantities

so that it does not mushroom out of control. The actualarrangement can only be understood by reading throughthe following pages in detail. The emphasis is on beingable to characterize the relationship between curvature, thesolution-generation techniques and geodesic motion, so therole of the Weyl tensor components is made explicit wher-ever possible.In the derivation for second-order Killing tensors given

in [3], it was noted that the problem of seeking a second-order Killing tensor in a 4D spacetime was equivalent to 4interrelated problems of seeking second-order Killing ten-sors on 2D spacetimes. This allowed the techniques used intwo-degree-of-freedom dynamical systems [11] to be ap-plied in seeking a solution. This observation prompted thesearch for a similar formulation of the 4D, fourth-orderKilling equations considered in this paper. An equivalentsymmetric formulation of fourth-order Killing equationshaving a similar property was found and is presented inSec. VIII. In this alternative formulation, the equations arewritten in the form of the four fourth-order Killing equa-tions of a two-manifold with additional interlocking con-ditions. This formulation is particularly concise, andappears to offer the possibility of an explicit analyticsolution to the problem. Sec. IX concludes the paper witha brief summary of the main results, a few wild specula-tions, and a number of more serious comments about theoutlook of future work.

JEANDREW BRINK PHYSICAL REVIEW D 81, 022002 (2010)

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II. SAV METRIC AND FIELD EQUATIONS

Any SAV spacetime with two commuting Killing vec-tors, @t and @�, can be represented by means of the Lewis-

Papapetrou metric,

ds2 ¼ e�2c ½e2�ðd�2 þ dz2Þ þ R2d�2�� e2c ðdt�!d�Þ2: (2)

The metric functions can be determined entirely by solu-tions of the Ernst equation for the complex potential E,

<ðEÞ �r2E ¼ �rE � �rE; (3)

where �r2 ¼ @�� þ 1� @� þ @zz and

�r ¼ ð@�; @zÞ . The realpart of the Ernst potential is e2c ¼ <ðEÞ. Line integrals ofthe E potential determine the functions � and !. Thefunction R is any harmonic function obeying the equationR;zz þ R;�� ¼ 0, and represents a coordinate freedom in

this formulation of the metric. One choice that is often usedis to set R ¼ �.

Equivalent to the Ernst formulation of the SAV fieldequations is a formulation introduced by Harrison [9] andNeugebauer [10]. The variables introduced in [9,10] aremost suited to performing Backlund transformations andturn out to be directly related to the Ricci rotation coef-ficients computed for the tetrad, introduced in Sec. III. It isthe Harrison-Neugebauer notation (referred to as M vari-ables) that is adopted for the rest of the paper. This notationmakes explicit the relationship between the Backlundtransformations, the choice of tetrad (Sec. III), and ulti-mately the Killing equations (Sec. VI), and correspondingconstants of motion. Furthermore, the field equations (5)expressed in terms of these variables are very easy andefficient to program for symbolic manipulation. (The vari-able names differ from the original ones used.)

Introduce the complex variables � ¼ 1=2ð�þ izÞ and�� ¼ 1=2ð�� izÞ and define

M1 ¼ @ ��; M2 ¼@ ��R

R;

M3 ¼@ ��E

E þ �E; M4 ¼

@ ��E

E þ �E;

M�1 ¼ @��; M�

2 ¼@�R

R;

M�3 ¼

@�E

E þ �E; M�

4 ¼@� �E

E þ �E:

(4)

Note that the ‘‘*’’ operation is not simply complex con-jugation, in fact �M1 ¼ M�

1,�M2 ¼ M�

2, but�M3 ¼ M�

4 and�M4 ¼ M�

3.

In terms of the M variables, the field equations can beexpressed as

M1;� ¼ � 1

2ðM3M

�4 þM4M

�3Þ;

M2;� ¼ �M2M�2;

M3;� ¼ ��1

2ðM2M

�3 þM3M

�2Þ �M3M

�3 þM3M

�4

�;

M4;� ¼ ��1

2ðM2M

�4 þM4M

�2Þ þM4M

�3 �M4M

�4

�;

M2; �� ¼ �M22 þ 2ðM1M2 �M3M4Þ:

(5)

The remaining five field equations result from thecomplex-conjugate expressions of Eqs. (5), which arealso required to hold. It should be noted the field equationsdetermine only certain of the derivatives of the M varia-bles. In particular, M1; �� , M3; �� , M4; �� , and their complex

conjugates are left free and encode the nonlocal contentof the spacetime curvature. These quantities change as onesolution is mapped onto another one. How they enter in theWeyl scalars is shown in Sec. III.For reference sake, we give the derivatives of the re-

maining metric components,

2@ ��c ¼ M3 þM4; @ ��! ¼ RðM4 �M3Þe�2c : (6)

III. COMPUTATIONAL TETRAD AND WEYLCURVATURE

In order to understand the interrelationship betweenWeyl curvature, Petrov classification, and the Killing equa-tions, it is necessary to introduce a null tetrad. With theparticular tetrad constructed below it turns out that themetric functions can be completely eliminated, and theproblem written in terms of the M variables and the com-ponents of the Killing tensor on the tetrad. The first step isto define a local inertial frame or tetrad frame by the basisvectors

E1 ¼ �ec dtþ!ec d�; E2 ¼ e�cRd�;

E3 ¼ e��cd�; E4 ¼ e��c dz:(7)

The corresponding contravariant basis vectors are

E1 ¼ e�c @t; E2 ¼ !ec

R@t þ ec

R@�;

E3 ¼ ec��@�; E4 ¼ ec��@z:

(8)

Using this inertial frame, define the null tetrad to be

k ¼ 1ffiffiffi2

p ðE1 þ E2Þ; l ¼ 1ffiffiffi2

p ðE1 � E2Þ;

m ¼ 1ffiffiffi2

p ðE3 � iE4Þ; �m ¼ 1ffiffiffi2

p ðE3 þ iE4Þ:(9)

The Weyl tensor coefficients are

SPACETIME ENCODINGS. IV. THE RELATIONSHIP . . . PHYSICAL REVIEW D 81, 022002 (2010)

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�0 ¼ C��k�m�k�m�;

�1 ¼ C��k�l�k�m�;

�2 ¼ C��k�m� �m�l�;

�3 ¼ C��k�l� �m�l�;

�4 ¼ C�� m�l� �m�l�;

(10)

and when expressed on this tetrad in terms of the Ms theybecome (setting V ¼ e2��2c )

�0 ¼ 1

2Vð�2M�

1M�4 þM�

3M�4 þ 2M�2

4 þM�4;� Þ;

�2 ¼ 1

4VðM�

2M4 þM2M�4 � 2M4M

�4Þ;

�4 ¼ 1

2Vð�2M1M4 þM3M4 þ 2M2

4 þM4; �� Þ;

(11)

and �1 ¼ �3 ¼ 0. Note how the derivatives of the Mvariables that are not determined by the field equationsenter the Weyl scalars �4 and �0. The SAV solutions areof Petrov type D if theWeyl scalars expressed on this tetradobey the identity�0�4 ¼ 9�2

2. In Sec. V it is shown, as anexample of the general formalism how the existence of asecond-order Killing equation implies that this conditionholds.

The Killing equations treated in Sec. IV require knowl-edge of the Ricci rotation coefficients associated with thetetrad (k, l, m, �m). (See [5] for a definition of rotationcoefficients.) Given Eq. (12) below, these coefficients serveas dials in the calculation which can be adjusted to generatesimpler example problems, as discussed in Sec. VII A.

In terms of the M variables, the nonzero Ricci rotationcoefficients are (recall that the 24 rotation coefficients areantisymmetric in the first two indices �abc ¼ ��bac)

�123 ¼ M�3 �M�

4

2ffiffiffiffiffiffiffi2V

p ; �124 ¼ M4 �M3


p ;

�131 ¼ M�2 � 2M�

4


p ; �141 ¼ M2 � 2M3


p ;

�132 ¼ � M�2


p ; �142 ¼ � M2


p ;

�231 ¼ � M�2


p ; �241 ¼ � M2


p ;

�232 ¼ M�2 � 2M�

3


p ; �242 ¼ M2 � 2M4


p ;

�343 ¼ � M�B


p ; �344 ¼ MB


p ;

(12)

where MB ¼ @ �� lnV ¼ 2M1 �M3 �M4. Vanishing coef-

ficients are

�121 ¼ �122 ¼ �133 ¼ �134 ¼ �143 ¼ �144 ¼ 0;

�233 ¼ �234 ¼ �243 ¼ �244 ¼ �341 ¼ �342 ¼ 0:(13)

These rotation coefficients clarify the relationship betweenthe M variables used in the BT and the Newman-Penrosetetrad formalism. Furthermore, if the spacetime is staticthen �123 ¼ �124 ¼ 0 and if the gauge is chosen to be R ¼�, �132 ¼ �231 ¼ �142 ¼ �231.Finally, before moving onto the Killing equations, we

note that the constant matrix � that is used for raising andlowering indices on the null basis has four nonzero com-ponents �12 ¼ �21 ¼ �1, and �34 ¼ �43 ¼ 1.

IV. KILLING EQUATIONS

On the tetrad basis, the Killing equations for a totallysymmetric Killing tensor T of order m, or equivalently forthe vanishing of the totally symmetrized intrinsic deriva-tive Tða1��amjbÞ ¼ 0, can be expressed in terms of the direc-

tional derivatives and rotation coefficients as

Tða1��am;bÞ ¼ m�cd�cða1a2Ta3��ambÞd: (14)

In the case of SAV spacetimes, the absence of functionaldependence on the two coordinates t and�, and the choiceof tetrad, Eq. (9), imply that the directional derivatives inthe 1, 2 (or k, l) directions vanish. As a result only direc-tional derivatives in the 3, 4 (orm, �m) directions need to beconsidered. For the tetrad chosen in Eq. (9), these deriva-tives can be expressed as m� ¼ 1ffiffiffiffiffi

2Vp @� and �m� ¼ 1ffiffiffiffiffi

2Vp @ �� .

Both the directional derivatives and the rotation coeffi-

cients contain a factor 1=ffiffiffiffiffiffiffi2V

p. When programming the

check for the integrability conditions for the Killing com-ponents it is convenient to remove this last explicit depen-dence on the metric functions V ¼ e2��2c . To do so we

multiply Eq. (14) byffiffiffiffiffiffiffi2V

pand let T denote a vector in

which all the independent components of the Killing tensorT have been arranged, and we also let M and M� be four-dimensional vectors M ¼ ½M1;M2;M3;M4� and M� ¼½M�

1;M�2;M

�3;M

�4�.

Then the nKilling equations for a Killing tensor of orderm can be represented as the system of equations

CiAT � þ Ci

BT �� ¼ MCiDT þM�Ci

ET ; (15)

where i ¼ 1; � � � ; n and the matrices CiA, C

iB, C

iD and Ci

E�

contain only integer entries. This equation makes explicitthe coupling between the components of the Killing tensorand the curvature of the spacetime, expressed in terms ofMvariables.The Ci matrices contain a lot of structure inherited from

the Killing equations. We shall exploit this structure tolimit the number of computations that have to be per-formed to check whether a specific spacetime admits aKilling tensor of orderm. In particular, only the casesm ¼2 and m ¼ 4 have thus far been considered.It can always be shown that the Killing equations for the

components Tðk1��kmÞ with ki ¼ f3; 4g decouple from the

larger system, and are themth order Killing equations for atwo-manifold with conformal factor V. This observation


022002-4

allows us to use the geometric understanding obtainedfrom the field of dynamical systems about what theseKilling tensors represent [2] to identify appropriate varia-bles and possibly useful coordinate transformations.

The remaining components that contain the indices i ¼f1; 2g are ultimately coupled to this two-manifold. In them ¼ 2 case, by writing out the appropriate integrabilityconditions [3] they too can bewritten in a form of a second-order Killing equation on a two-manifold distinct from V.As is shown in Sec. VIII, this property persists for them ¼4 case, where four two-manifolds admitting a fourth-orderKilling tensor are in effect sought.

If a Killing tensor of even order is considered, it can beshown by explicitly writing out the Killing equations onthis tetrad that the equations for Killing tensor componentssuch as TAABk that admit an odd number of ki ¼ f3; 4g andA; B ¼ f1; 2g indices decouple from the rest, and can be setto zero, without loss of generality. A more subtle argumentusing the Jacobi metric [3] and the symmetries of theadditional invariant on the two-manifold yields the sameresult.

In the next two sections, the two examples for m ¼ 2and m ¼ 4 are used to illustrate the ideas described abovemore concretely. These examples are also used to demon-strate the reductions, in some cases empirical, that simplifythe problem to a computationally manageable size.

V. SECOND-ORDER KILLING TENSOR

In the second-order (m ¼ 2) case, there are six nonzerocomponents of the Killing tensor that have to be consideredand ten Killing (n ¼ 10) equations that limit the gradientsof these components. A summary of the structure of theKilling tensor components and the gradients determined bythe Killing equations is shown in Table I. To avoid anyconfusion in notation, and to illustrate the general method,the Killing equations are in some cases written out in detailfor this example. We begin with the second-order versionof Eq. (14), ffiffiffiffiffiffiffi

2Vp

Tðab;fÞ ¼ 2�nmffiffiffiffiffiffiffi2V

p�nðabTfÞm: (16)

The 10 Killing equations (16) can be divided into twogroups. Four equations for the three Killing tensor compo-nents Tðk1k2Þ, with ki ¼ f3; 4g decouple from the rest and are

the Killing equations for a two-manifold with conformalfactor V. After setting MB ¼ @ �� ðlnVÞ and M�

B ¼ @� ðlnVÞthese equations are

T33;� ¼ M�BT33; T34;� ¼ � 1

2ðT33; �� þMBT33Þ;

T44; �� ¼ MBT44; T34; �� ¼ � 1

2ðT44;� þM�

BT44Þ:(17)

The two equations on the left of (17) state that T33=V is ananalytic function of �� , and T44=V is an analytic function of� [2]. It is in principle always possible to choose a gauge orR function in (2) such that two functions are equal to theidentity, as was done in [3]. The equations on the right of(17) can be viewed as defining the gradient of the functionT34. However, the gradient can only be considered to bevalid if the cross-derivatives cancel, or equivalently if theintegrability condition

T33;�� þ ðMBT33Þ; �� ¼ T44;�� þ ðM�BT44Þ;� (18)

holds. If one differentiates Eq. (18) with respect to �� andsubstitute in the field equations (5) and Killing equations(17), then @3��T33 can expressed in terms of a linear function

of (@p�� T33, @p� T44), where p ¼ f0; 1; 2g and it is considered

to be determined by the integrability conditions. This isindicated in the fifth column of Table I. A similar argumentholds for @3�T44. The functions MB and M�

B and their

derivatives @q��MB and @q�M�B that are not defined by the

field equations (5) always enter as polynomial factors infront of the T variables.The remaining six Killing equations define the gradients

of the functions T11, T22 and T12 (the first row entry onTable I). Setting T ¼ ½T11; T12; T22; T33; T34; T44�t andS ¼ ½T11; T12; T22�t these equations have the general form

TABLE I. Details of the second-order Killing equations (K.E.). Integrability conditions abbreviated as IC. Indices k and i take on thevalues 3, 4 or equivalently indicate m, �m components. Indices A and B take on the values 1, 2 or indicate l, k components.

Properties of second-order Killing tensors in SAV spacetimes

m ¼ 2, n ¼ 10, K.E. involving 6 tensor components

Column 1 Column 2 Column 3 Column 4 Column 5 Column 6 Column 7

Tensor

components

Number of

components

Derivatives

fixed by K.E.

Number of

IC generated

Derivatives

fixed by IC

Resulting unknown

potentials to determine

Number

of potentials

TðABÞ 3 @� � @ �� 3

T34 1 @� � @ �� 1

T33 1 @� � @3�� @p�� T33, p ¼ f0; 1; 2g 3

T44 1 @ �� @3� � @p� T44, p ¼ f0; 1; 2g 3

Total 6 components 10 K.E. 4 IC L2 ¼ 6


022002-5

S i� ¼ MCi

DT þM�CiET ;

Si��¼ M ~Ci

DT þM� ~CiET ;

(19)

where C and ~C are constant matrices with integer entries.The integrability conditions of the function S are surpris-ingly simple. They are linear functions in onlyð@p�� T33; @

p� T44Þ, where p ¼ f0; 1g with coefficients polyno-

mial in the M variables (at most quadratic) and the firstderivatives of M�

3� , M4 �� (at most linear). No other compo-

nents of the Killing tensor enter the integrabilityconditions.

It can be shown [3] that appropriate linear combinationsof the integrability conditions for the functions TðABÞ,where fA; Bg take on the values f1; 2g, and of Eq. (18)result in three more equations similar to (18), but for ametric with conformal factor distinct from V. This createsthe geometric picture that one is looking for four two-manifolds that admit second-order Killing tensors.

By writing out the integrability conditions (henceforthIC) for the Killing tensor components whose gradients arefully determined by the Killing equations (the first two rowentries in Table I) and differentiating some of these ICequations to fix the higher-order undetermined derivativesof the remaining Killing components (third and fourthentries in Table I, column 5), one can generate a systemof linear equations for the remaining unknown functionsrepresented in column 6 of Table I. Taking additionalderivatives of the IC (column 4) and substituting in thefield equations (5) and Killing equations (17) as well as theexpressions for the derivatives fixed by the IC column 5further increases the number of equations while keepingthe number of unknowns L2 constant. In doing so we buildup a large overdetermined linear system. The number ofunknowns, and thus an indication of the size of the linearsystem is given in column 7 of Table I. The coefficients inthis system are polynomials in the M variables and theirderivatives. The field equations and Killing equations, aswritten in terms ofM variables and expressed on the tetradpresented in this paper, are very easy to program inMATHEMATICA.

The overdetermined system of equations so constructedcan be solved for the undetermined Killing tensor compo-nents (column 6) up to an overall scaling. (The tensorcomponents that have been eliminated by writing out theirIC can ultimately be constructed from a line integral.) Theconsistency conditions that ensure that a solution can befound provide polynomial conditions on the M variablesand their derivatives.

In the case of the second-order tensor we find, therequirement, amongst others, that

0 ¼ �0�4 � 9�22 (20)

or that the SAV spacetime is Petrov type D. The otherrequirements, namely, separability in a certain coordinate

system, and the explicit solution of the problem, are dis-cussed more fully in [3].The total number in the lower right-hand corner of the

tables in many ways represents the largest possible linearsystem that may be required if the coefficients of theundetermined potentials are to remain polynomial in theM variables, and one is working in a general gauge. Anumber of simplifications exist that decrease this numberand will be mentioned in Sec VIIA. In the case of thesecond-order Killing equations, we can always work in agauge in which the conditions on the M variables canimmediately be written down (effectively L2 ¼ 0). Amore careful treatment of the second rank Killing equa-tions is given in [3] and the coordinate systems, in whichthe metric functions are separable, are found and classifiedthere.The requirement that a second-order Killing tensor exist

on a SAV spacetime is very restrictive. Equation (15)makes explicit the coupling between the curvature of aSAV spacetime contained in the M variables on the right-hand side and the components of the Killing tensor. Onecan take the point of view that the n equations of (15) canbe used as a representation of the M variables. With thisperspective a rough counting argument immediately im-plies that it is impossible to represent the eight functionallyindependentM variables in terms of the sixT variables. Atthe very least, a larger representation, with more freefunctions in the T vector, is required, to encompass gen-eral spacetimes. It is also not known whether all possibleallowedM functions consistent with the field equations canbe written in the form (15). A practical method of checkingis proposed in the next section for the fourth-order case.

VI. FOURTH-ORDER KILLING TENSORS

The orbital crossing pattern of the numerical integrationof the geodesic orbits [2,6], the pole structure of the Ernstequation [12], the algebraic structure of the Weyl tensor[7], and the group structure of the Backlund transforma-tions [9], all indicate that an exploration into the existenceof a fourth-order Killing tensor on SAV spacetimes iswarranted. Possible experimental applications [1] provideurgency.The analysis is considerably more complicated and

computationally expensive than the second-order case. Itis also highly nonlinear, as opposed to the second-ordercase which can be linearized as in [3]. In this section weuse the technique outlined in Sec. V to make the problemof checking for and constructing a fourth-order Killingtensor on SAV spacetimes, vulnerable to a brute forceattack. It is argued that it lies within the range of currentcomputational capabilities to analytically compute the gra-dients of the Killing tensor components and the integra-bility conditions on the M variables representing thecurvature for all SAV spacetimes. In some sense an answeris assured, although it may be ugly. In subsequent sections,


022002-6

refinements of this approach are suggested that may yielddeeper insight into the field equations, and allow compu-tations of insightful special cases with less computerpower.

On SAV spacetimes, the 56 Killing equations for the 35independent components of a general fourth-order Killingtensor decouple into two groups. As mentioned in Sec. IVthe Killing equations for components such as TAABk andTAk1k2k3 , with an odd number of ki ¼ f3; 4g and A;B ¼f1; 2g indices, form an entirely separate group from thosewith even pairs of indices. The equations that result fromthe odd group have more restrictive conditions on thecurvature, and are trivially solved by setting them tozero. The more general case involving only the even groupis considered. This reduces the number of tensor compo-nents to 19, and the Killing equations by half. Table II liststhe derivative properties of the ‘‘even’’ Killing tensorcomponents.

The fourth-order Killing equations under considerationin this section are written out in full in Appendix A. Threegroups of equations can be identified. The decoupled groupgoverning the derivatives Tðk1k2k3k4Þ, with ki ¼ f3; 4g, arethe 6 fourth-order Killing equations for a two-manifoldwith conformal factor V, Eqs. (A4) and (A5). Two of theseEqs. (A4) can be combined to eliminate the T3344 compo-nent (Table II, third row). The second group consists of tenequations (A1) defining the gradients of the five TðA1A2A3A4Þcomponents where Ai ¼ f1; 2g. These components can beeliminated in favor of five integrability conditions[Table II, first row, and Eq. (B1)]. Finally there is a buffergroup of 12 equations (A2) and (A3), involving the gra-dients of the mixed terms TA1A2k1k2 , six of which [namely

Eqs. (A2)] can be eliminated in favor of the three integra-bility conditions for the TA1A234 terms [Table II, second

row, and Eq. (B2)]. In each case where an integrabilitycondition is written down for a potential mentioned above,it is absent in the resulting IC.The ‘‘buffer’’ group does not have its counterpart in the

second-order Killing equations. Now, a valid solution tothe fourth-order Killing equations can be constructed bytaking the exterior product of two second-order Killingtensors. This type of solution to the fourth-order Killingequations can be considered to be reducible. For irreduc-ible fourth-order Killing tensors, the buffer introduces afunctional freedom that will necessarily include a greaterset of SAV spacetimes than those admitting second-orderKilling tensors. The currently open question is how largethis set is.The nine IC in column 4 of Table II, [Eqs. (B1)–(B3)]

and the 34 unknown potentials in column 6 of Table II nowform a closed system to differentiation [provided the fieldequations (5), Killing equations represented in column 3,Table II or Eqs. (A3) and (A5) and the higher-order de-rivatives represented by column 5 are substituted whenevernecessary]. In particular if you consider a system of equa-tions generated by the nine IC, their first derivatives, alltheir second derivatives with respect to � and �� , and thederivatives @�� IC and @�� IC, a completely overdeter-

mined system for the L4 ¼ 34 potentials has been con-structed. This system may be excessive in that morederivatives of the IC than what is necessary may havebeen taken. However it gives one freedom in choosingthe pivots during the elimination of the Killing componentsas one moves toward finding the conditions on the Mvariables, so that the answer is assured. The coefficientsof the Killing components are polynomials in the M vari-ables and their derivatives. At most fourth-order deriva-tives of the M variables enter these equations.

TABLE II. Details of the fourth-order Killing equations (K.E.). Integrability conditions abbreviated as IC. Indices k and i take on thevalues 3, 4 or equivalently indicate m, �m components. Indices A, B, C, D take on the values 1, 2 and indicate l, k components.

Properties of fourth-order Killing tensors in SAV spacetimes

m ¼ 4, n ¼ 28, K.E. involving 19 tensor components

Column 1 Column 2 Column 3 Column 4 Column 5 Column 6 Column 7

Tensor

components

Number of

components

Derivatives

fixed by K.E.

Number of

IC generated

Derivatives

fixed by IC

Resulting unknown

potentials to determine.

Number

of potentials

TðABCDÞ 5 @� � @ �� 5

TðAB34Þ 3 @� � @ �� 3

Tð3344Þ 1 @� � @ �� 1

TðAB33Þ 3 @� � @3�� @p�� TðAB33Þ, p ¼ f0; 1; 2g 9

Tð3334Þ 1 @� � @3�� @p�� Tð3334Þ, p ¼ f0; 1; 2g 3

Tð3333Þ 1 @� � @5�� @p�� Tð3333Þ, p ¼ f0; 1; 2; 3; 4g 5

TðAB44Þ 3 @ �� @3�� @p� TðAB44Þ, p ¼ f0; 1; 2g 9

Tð3444Þ 1 @ �� @3� � @p� Tð3444Þ, p ¼ f0; 1; 2g 3

Tð4444Þ 1 @ �� @5� � @p� Tð4444Þ, p ¼ f0; 1; 2; 3; 4g 5

Total 19 components 28 K.E. 9 IC L4 ¼ 34


022002-7

The elimination process of the Killing components is atthis stage ad hoc, guided mainly by inspection and sym-metries in the tensor indices. It is essential to choose linearcombinations of the equations such that the pivots, whichbecome denominators in the subsequent manipulations,remain as simple as possible. Following this process it ispossible to eliminate all but the first derivatives of theT3333, T4444, T3444, and T4333 components and the ten basiccomponents themselves (the four remaining Tk1k2k2k2 and

the six remaining TA1A2kk components), before it is no

longer apparent how to choose the pivots without introduc-ing complicated denominators and the polynomials in thenumerator have become so large that MATHEMATICA hastrouble with memory. Thus currently one is in principle one14-by-14 full-matrix inversion away from an analytic ex-pression for the fourth-order Killing tensor components, interms of the M variables and their derivatives. The con-ditions that theM variables have to satisfy can be obtainedby factoring the resulting consistency conditions after thematrix inversion is complete. The polynomials in M andtheir derivatives are very large. However both the matrixinversion and the subsequent factoring of integrabilityconditions lie within the range of current computationalcapacity. They have to be performed only once to getexplicit expressions for the Killing tensor components forall SAV spacetimes. The only caveat is that once the finalfactoring step has been performed, the resulting conditionson theM variables may not be in an easily recognizable orcompact form. In the following sections we discuss a seriesof examples that exploit the framework just given but seekways to simplify the computations.

VII. SIMPLIFICATION OF THE GENERALFRAMEWORK

A. General considerations

Currently the greatest obstacle that hampers the furtherreduction of the linear system is the complexity of thepolynomial coefficients that multiply the tensor compo-nents. Since the eight M variables and their derivatives upto the fourth enter the coefficients, any reduction in thenumber of M variables is welcome. One somewhat inef-fective reduction is the choice of gauge R ¼ � whichimplies that M2 ¼ M�

2. In static spacetimes, we find M3 ¼M4, M�

3 ¼ M�4 and T11kk ¼ T22kk, T1111 ¼ T2222 and

T1112 ¼ T1222 by a symmetry argument. This reduces thenumber of integrability conditions by three, and unknownsby six and makes the brute force approach more accessible,at the cost of generality.

The analysis performed in the previous section is local,so the existence of a fourth-order Killing tensor can bedisproved by showing that the consistency conditions failat a particular point in space. The large overdeterminedsystem generated in the previous section can thus be usedto show that there is no consistent solution to the Killingequations given a particular spacetime.

Another simplification that can be imposed is the choiceof a gauge in which T4444=V

2 ¼ T3333=V2 ¼ 1, which I

shall call the Killing gauge. This choice reduces the num-ber of unknown potentials by ten, however it does nothingto reduce the complexity of the coefficients of the remain-ing components. Choosing this gauge removes the powerof the formalism to check for the existence of a Killingtensor given a metric, but it may be useful in understandingthe structure of the equations. This was the case in second-order Killing equations [3], where an analogous gaugechoice was made.However the most powerful application of the technique

appears to be, given a metric that is integrable and admits afourth-order (possibly reducible) invariant, and a parame-terized family of metrics associated with it, to checkwhich of these metrics retain the property of integrability.The next subsection suggests an approach that uses thisidea, and that could in principle be used to check theintegrability of all SAV metrics. This method alsogives more insight into the choice of the M variables.

B. Checking Backlund tranformations, a tale of twomanifolds

This section suggests an elegant method that exploitsthe framework outlined in Sec VI and couples it to ourknowledge of the solution-generation techniques, in orderto ascertain which SAV spacetimes admit a fourth-order invariant. It was in large part the elegance of thesolution-generation techniques, and, in particular, theBacklund transformations that originally motivated thechoice of the M variables. This choice makes pro-gramming the check proposed in the previous sectionscomputationally possible. The approach given here canbe viewed as an alternative to the brute force com-putation.BT map a valid solution of the SAV field equations onto

a another with a different Weyl tensor. The mapping iscarried out with a so-called pseudopotential, and for agiven starting spacetime only 16 of these transformationsexist [9]. The basic idea is to start with a metric such asKerr, which is known to admit a fourth-order Killingtensor, and to ask which of the 16 possible BT [9] maintainthe property that the new spacetime also admits a fourth-order Killing tensor. The main advantage of working off anexisting solution is that the derivatives of the M functionsthat are not fixed by the field equations are known and thusthe complexity of the polynomials entering the overdeter-mined linear problem is reduced.A more mathematical statement of this approach is the

following. Consider two manifolds ðM; gÞ and ð ~M; ~gÞwith metrics g and ~g, that obey the SAV vacuum fieldequations. Suppose that M admits a fourth-order Killing

tensor and that M and ~M are related by a Backlundtransformation. Namely, ~N ¼ BN and ~N� ¼ B�N�, whereN ¼ ðM3;M4;M2Þ and B and B� are the 3� 3 transforma-


022002-8

tion matrices. The entries of these transformation matricesare very simple rational functions of the pseudopotentials qand . The 16 possibilities are collated in [9], wherehowever the notation may vary slightly from that usedhere. Furthermore ~M1 is fixed by the last equation of (5).

The potential is chosen to be ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðil� ��Þ=ðilþ �Þp,

and as a result

d ¼ 1

2½ð2 � 1ÞM�

2d� þ �1ð2 � 1ÞM2d ��: (21)

The pseudopotential q lies within the prolongation struc-ture and obeys the Ricatti equation [9]

dq ¼�qð1þ qÞM�

3 � ðqþ ÞM�4 þ

1

2ð1� q2ÞM�

2

�d�

þ �1

�qðqþ ÞM4 � ð1þ qÞM3

þ 1

2ð1� q2ÞM2Þ

�d ��: (22)

The gradients of q and are thus determined in terms ofthe M variables of the known solution. Furthermore, thederivatives of theM variables that are not fixed by the fieldequations are also explicitly known. An overdeterminedlinear system of Killing equations such as that suggested inTable II can then be built and used to check the conditionson q that are required to maintain fourth-order integra-bility. This approach has not been implemented in full,however the result would allow us to very accuratelyquantify the size of the subgroup of SAV spacetimes thatadmit a fourth-order Killing tensor and would lead to adeeper understanding about the relationship between thesolution-generation techniques and the geodesic structureof the resulting spacetimes.

VIII. ALTERNATE FORMULATION OF FOURTH-ORDER KILLING EQUATIONS

The geometric origin of the variables used in the pre-vious sections is obvious by construction: namely, theyconform to the components of the Killing tensor expressedon an orthonormal tetrad basis adapted to the underlyinggeometry of the metric. In other words, the tetrad is chosento respect the symmetry imposed by the Killing vectors ofthe SAV spacetime, and turns out to correspond to thetransverse frame of that spacetime. The other spacetime-dependent quantities that enter the equations are the rota-tion coefficients associated with the tetrad.

In this section I present an alternate formulation of theKilling equations, which was found by inspection. Whilethe system of equations presented here and those ofAppendix A are entirely equivalent, the former has sym-metry properties that highlight the difference betweenstatic and stationary spacetimes, allowing easy simplifica-tion. The formulation presented here has the additionalfeature that the spacetime-dependent quantities that enterthe equations are functions of the metric variables and their

derivatives rather than the M variables previously used.The relative simplicity of the resulting equations alsoallows insight into some of the features of the solutionsthat are sought. Looking at the structure of these equationsled to an ansatz for the Killing tensor components on theequatorial plane of static spacetimes. This ansatz appearsto be correct, as shown in [13]. The new variables, denotedby Phi:ji, are linear combinations of the Killing tensor

components on the transverse tetrad and of functions enter-ing the metric. While there is some vaguely systematic wayof searching for the new variables, they are best motivatedby the fact they have been shown to work to aid explicitcalculation in practice. The original intent was to find a setof variables that cast the Killing equations in as symmetricas possible a form.Based the derivation of the second-order Killing tensors

given in [3], I suspected that one could find a formulationin which the Killing equations would take on the form offour ‘‘interlocking’’ fourth-order problems for a two-manifold. The formulation given here is the result of thatsearch. A clearly defined algorithm for finding these coor-dinates does not exist, except to say grope roughly in thisdirection and if it is the right way, the answer should havethe form suggested in [3] for fourth-order Killing tensorson a two-manifold, if not continue groping. In what followsI thus resort to the annoying approach of giving the anzatzfor the transformation without fully being able to disclosehow it was obtained, or what the quantities refer to, exceptto say that the resulting equations are easier to solve andconform to some intuitive picture I had of their existencebefore they were found.All the properties of the Killing equations previously

discussed in Secs. IV and VI are inherited by the newsystem. The naming convention of the new variables dis-plays some of this structure. For the variable name Phi:ji,the index j gives some indication of the differentiabilityproperties of the variable. The designation j ¼ 0, for ex-ample, indicates that the Killing equations fully determineits gradient, and that it can be removed from this system toyield an integrability condition. There are nine of thesevariables, Phi:0i with i 2 f�4; � � � ;�1; 1; � � � ; 5g. The in-

dices j ¼ 3 and j ¼ 4 indicate that the Killing equationsfix derivatives with respect to � and �� , respectively. Theindex i labels different variables with a fixed derivativestructure. The final linear system is built up of the tenvariables Phi:ji with i 2 f1 � � � 5g and j 2 f3; 4g, and of

higher derivatives.Without further hesitation, make the following ansatz:

define the set of variables that are already decoupled as thefourth-order Killing equation of a two-manifold as

T3444 ¼ Ph1:4ie2��2c ; T3334 ¼ Ph1:3ie2��2c ;

T3333 ¼ Ph5:3ie4��4c ; T4444 ¼ Ph5:4ie4��4c ;

T3344 ¼ Ph�1:0i:

(23)


022002-9

Furthermore, set the mixed components TABij equal to

T1233 ¼ 1

12e2��4c ð�4e2cPh1:3i þ e4c ð�3!2Ph2:3i þ 6!Ph3:3i þ 6Ph4:3iÞ þ 3R2Ph2:3iÞ;

T1244 ¼ 1

12e2��4c ð�4e2cPh1:4i þ e4c ð�3!2Ph2:4i þ 6!Ph3:4i þ 6Ph4:4iÞ þ 3R2Ph2:4iÞ;

T1133 ¼ � 1

4e2��4c ðe4c ð!2Ph2:3i � 2!Ph3:3i � 2Ph4:3iÞ þ 2Re2c ð!Ph2:3i � Ph3:3iÞ þ R2Ph2:3iÞ;

T1144 ¼ � 1

4e2��4c ðe4c ð!2Ph2:4i � 2!Ph3:4i � 2Ph4:4iÞ þ 2Re2c ð!Ph2:4i � Ph3:4iÞ þ R2Ph2:4iÞ;

T2233 ¼ � 1

4e2��4c ðe4c ð!2Ph2:3i � 2!Ph3:3i � 2Ph4:3iÞ � 2Re2c ð!Ph2:3i � Ph3:3iÞ þ R2Ph2:3iÞ;

T2244 ¼ � 1

4e2��4c ðe4c ð!2Ph2:4i � 2!Ph3:4i � 2Ph4:4iÞ � 2Re2c ð!Ph2:4i � Ph3:4iÞ þ R2Ph2:4iÞ;

T1234 ¼ 1

8e�2c ð�4e2cPh�1:0i þ e4c ð!ð2Ph�3:0i �!Ph�2:0iÞ þ 2Ph�4:0iÞ þ R2Ph�2:0iÞ;

T1134 ¼ 1

8e�2c ðe4c ð!ð2Ph�3:0i �!Ph�2:0iÞ þ 2Ph�4:0iÞ þ 2Re2c ðPh�3:0i �!Ph�2:0iÞ þ R2ð�Ph�2:0iÞÞ;

T2234 ¼ 1

8e�2c ðe4c ð!ð2Ph�3:0i �!Ph�2:0iÞ þ 2Ph�4:0iÞ � 2Re2c ðPh�3:0i �!Ph�2:0iÞ þ R2ð�Ph�2:0iÞÞ:

(24)

Finally, set the components of the form TðA1A2A3A4Þ, Ai ¼ f1; 2g equal to

T1111 ¼ 9

64½e�4cPh1:0iðRþ e2c!Þ4 þ 4e�2cPh2:0iðRþ e2c!Þ3 þ 8Ph3:0iðRþ e2c!Þ2�

þ 9

16½e2cPh4:0iðRþ e2c!Þ þ e4cPh5:0i�;

T2222 ¼ 9

64½e�4cPh1:0iðR� e2c!Þ4 þ 4e�2cPh2:0iðe2c!� RÞ3 þ 8Ph3:0iðR� e2c!Þ2�

þ 9

16½e2cPh4:0iðe2c!� RÞ þ e4cPh5:0i�;

T1112 ¼�9

64e�4cPh1:0iðe4c!2 � R2Þ þ 3

8e�2cPh�2:0i � 9

32e�2cPh2:0iðR� 2e2c!Þ

�ðRþ e2c!Þ2

þ�9

8e2c!Ph3:0i � 3

4Ph�3:0i

�ðRþ e2c!Þ � 3

4e2cPh�4:0i þ 9

32e2cPh4:0iðRþ 2e2c!Þ þ 9

16e4cPh5:0i;

T1222 ¼�9

64e�4cPh1:0iðe4c!2 � R2Þ þ 3

8e�2cPh�2:0i þ 9

32e�2cPh2:0iðRþ 2e2c!Þ

�ðR� e2c!Þ2

þ�9

8e2c!Ph3:0i � 3

4Ph�3:0i

�ðe2c!� RÞ � 3

4e2cPh�4:0i þ 9

32e2cPh4:0ið2e2c!� RÞ þ 9

16e4cPh5:0i;

T1122 ¼ Ph�1:0i þ 9

64e�4cPh1:0iðR2 � e4c!2Þ2 þ

�1

2e�2cPh�2:0i þ 9

16!Ph2:0i

�ðe4c!2 � R2Þ � e2c!Ph�3:0i

� 3

8Ph3:0iðR2 � 3e4c!2Þ � e2cPh�4:0i þ 9

16e4c!Ph4:0i þ 9

16e4cPh5:0i:

(25)

After this horror of an ansatz, the resulting Killing equa-tions are much more amicable and concise. Substituting theansatz in Eqs. (23)–(25), into the Killing equations (A1)–(A5), yields, after simplification, the following equivalentsystem of Killing equations; the terms whose gradientsare not fully defined and that enter the linear systemused to construct the integrability conditions are givenbelow,

Phi:3i;� ¼ �fi; ��Ph5:3i � 1

4fiPh5:3i; �� i 2 f1 � � � 4g;

Phi:4i; �� ¼ �fi;�Ph5:4i � 1

4fiPh5:4i;� i 2 f1 � � � 4g;

Ph5:3i;� ¼ Ph5:4i; �� ¼ 0: (26)

The functions fi entering these equations are defined interms of the metric functions as


022002-10

f1 ¼ e2��2c ¼ V; f2 ¼ 2e2�

3R2;

f3 ¼ 2e2�!

3R2; f4 ¼ e2�ðR2e�4c �!2Þ

3R2:

(27)

Note that in Eqs. (26) Ph5:4i is an analytic function of � andindicates a gauge freedom still present in the metric.Without loss of generality one can set Ph5:4i ¼ Ph5:3i ¼ 1,

with vanishing derivatives further simplifying the expres-sions (26) to

Phi:3i;� ¼ �fi; �� ; Phi:4i; �� ¼ �fi;� i 2 f1 � � � 4g:(28)

The equations governing the field components whosegradients are fully described are now given by

Ph�1:0i;� ¼ � 4

3Ph1:3if1; ��

2

3f1Ph1:3i; �� ;

Ph�1:0i; �� ¼ � 4

3Ph1:4if1;� � 2

3f1Ph1:4i;� ;

Ph�2:0i;� ¼ �2Ph1:3if2; �� f2Ph1:3i; �� 2f1; ��Ph2:3i � f1Ph2:3i; �� ;

Ph�2:0i; �� ¼ �2Ph1:4if2;� � f2Ph1:4i;� � 2f1;�Ph2:4i � f1Ph2:4i;� ;





Ph1:0i;� ¼ �8Ph2:3if2; �� 4f2Ph2:3i; �� ;

Ph1:0i; �� ¼ �8Ph2:4if2;� � 4f2Ph2:4i;� ;

Ph2:0i;� ¼ þ4Ph2:3if3; �� þ 2f3Ph2:3i; �� þ 4f2; ��Ph3:3i þ 2f2Ph3:3i; �� ;

Ph2:0i; �� ¼ þ4Ph2:4if3;� þ 2f3Ph2:4i;� þ 4f2;�Ph3:4i þ 2f2Ph3:4i;� ;

Ph3:0i;� ¼ þ2Ph2:3if4; �� þ f4Ph2:3i; �� þ 2f2; ��Ph4:3i � 4Ph3:3if3; �� 2f3Ph3:3i; �� þ f2Ph4:3i; �� ;

Ph3:0i; �� ¼ þ2Ph2:4if4;� þ f4Ph2:4i;� þ 2f2;�Ph4:4i � 4Ph3:4if3;� � 2f3Ph3:4i;� þ f2Ph4:4i;� ;

Ph4:0i;� ¼ �8Ph3:3if4; �� 4f4Ph3:3i; �� 8f3; ��Ph4:3i � 4f3Ph4:3i; �� ;

Ph4:0i; �� ¼ �8Ph3:4if4;� � 4f4Ph3:4i;� � 8f3;�Ph4:4i � 4f3Ph4:4i;� ;

Ph5:0i;� ¼ �8Ph4:3if4; �� 4f4Ph4:3i; �� ;

Ph5:0i; �� ¼ �8Ph4:4if4;� � 4f4Ph4:4i;� :

(29)

An additional advantage of this formulation is that theterms on the right-hand side of the gradients do not containany of the Phi:0i variables whose gradients are being de-fined. Thus, unlike the case treated in Appendix A, it isimmediately obvious that the IC contain only the tenvariables Phi:3i and Phi:4i.

In the event that the spacetime is static rather than sta-tionary, f3 ¼ 0, and the following Phi;ji variables can be setto zero:

Ph3:3i ¼ Ph3:4i ¼ Ph2:0i ¼ Ph4:0i ¼ Ph�3:0i ¼ 0: (30)

In the general SAV case, the integrablilty conditions forthis formulation can be generated by cross-derivatives ofEq. (29). Before continuing, however, it is useful to definethe following differential operators, which illuminates the

structure in these equations. Let

FP ði; jÞ ¼ 2fi;��Phj:4i þ 3fi;�Phj:4i;� þ fiPhj:4i;��

FP ði; jÞ ¼ 2fi;��Phj:3i þ 3fi; ��Phj:3i; �� þ fiPhj:3i;�� : (31)

The IC for the fields fPh�1:0i; Ph1:0i; Ph5:0ig can be written as

F P ði; iÞ ¼ FP ði; iÞ; (32)

where the index i takes on the values i 2 f1; 2; 4g. Note thatthese IC are equivalent to the IC on the conformal factor ofa two-metric in a two-dimensional spacetime that admits afourth-order invariant [2]. Thus in an abstract sense one islooking for at least three distinct interrelated two-metricsthat admit fourth-order invariants in addition to satisfying


022002-11

the IC generated by the remaining fields. The IC for thefields fPh�2:0i; Ph�3:0i; Ph�4:0i; Ph2:0i; Ph4:0ig take the form

F P ði; jÞ þFP ðj; iÞ ¼ FP ði; jÞ þFP ðj; iÞ; (33)

where the index pair ði; jÞ takes on the values ði; jÞ ¼fð1; 2Þ; ð1; 3Þ; ð1; 4Þ; ð2; 3Þ; ð3; 4Þg. The remaining IC forPh3:0i takes the form

2FP ð3; 3Þ �FP ð2; 4Þ �FP ð4; 2Þ¼ 2FP ð3; 3Þ �FP ð2; 4Þ �FP ð4; 2Þ; (34)

which slightly breaks the symmetry of the previous twosets of expressions (32) and (33). However, if in addition tothe Killing equations, f3 and Ph3:4i obey the integrability

conditions of a two-manifold with a fourth-order invariant,namely, Eq. (32) with i ¼ 3, the symmetry is beautifullyrestored. Equation (34) provides the missing pair ði; jÞ ¼ð2; 4Þ in (33). The existence of a fourth-order invariant on aSAV spacetime then becomes synonmous with four inter-locking two-manifolds admitting a fourth-order invariant[Eqs. (32)] with additional conditions [Eqs. (33)].

These IC coupled with Eqs. (26), differentiated a fewmore times can also be used to build a large linear systemto perform a brute force check. It does however turn outthat they are of greater use as a framework for guessing theapproximations of the invariant, whether or not it actuallyexists. A means of doing this systematically is explored ingreater detail in [13,14].

IX. CONCLUSION

In many ways the geodesics of a spacetime can beviewed as one of its most fundamental descriptions. Thenature of the self-contained paths through space that mas-sive particles favor gives us a direct observational charac-terization of the inertial field and of the mass distributionwithin the spacetime. The field equations of SAV space-times are completely integrable. If the integrability prop-erties are inherited by the geodesics of the manifold andthat relationship can be made explicit; the practical appli-cation of a nonperturbative description of geodesic motionin SAV spacetimes can possibly be implemented in anexperimental environment.

This paper provides a framework by which the relation-ship between constants of motion, expressed as higher-order Killing tensors, and the curvature content of thespacetime can be quantified. To do so most elegantly it issuggested that the solution-generation techniques them-selves be exploited to simplify the calculation. This willalso increase our understanding of what these transforma-tions imply physically for the spacetimes they are map-ping. If the ability to maintain the existence of a fourth-

order Killing tensor for certain of the Backlund transfor-mations of a given spacetime can be quantified, it will givesome insight into the size of the subgroup of SAV space-times that admit a fourth-order Killing tensor.The main sticking point at the moment is the lack of a

good method of eliminating all the Killing tensor compo-nents to obtain the conditions for the existence of thefourth-order Killing tensor on the M variables withoutbuilding up excessively large polynomials. It is hopedthat an elegant method of inverting the linear integrabilitymatrix can be developed instead of the current ad hocapproach for choosing pivots. Failing that, however, thebrute force method will eventually yield a result.If all SAV spacetimes admit a fourth-order Killing ten-

sor, then there is a direct relationship between the geodesicstructure contained in the Killing equations and the alge-braic structure of the Weyl tensor. It further implies that thepole structure of the Ernst equation can be understood interms of the poles of the analytic functions hidden in theKilling equations. The field of two-degree-of-freedom dy-namical systems, with its paucity of examples of systemsadmitting a fourth-order invariant will then gain thesolution-generation techniques for Ernst’s equations andthe potential of a bi-infinite series of examples generatedby the HKX transformations and other solution-generationtechniques. Moser presented a geometric picture of howintegrability on the Jacobi ellipsoids arises, and of therelationship to quadrics, by means of a very simple geo-metric construction. Neugebauer [10] pointed out that eachaxisymmetric stationary vacuum field corresponds to aminimal surface on a hyperbolic paraboloid xþ yþ u2 þv2 � w2 ¼ 0 embedded in a five-dimensional pseudo-Euclidean space. He further ventured that the Backlundtransformations were mapping minimal surfaces into mini-mal surfaces. It is possible that Moser’s picture could begeneralized to quartics which have their origin in theLorentz boosts and that using Neugebauer’s observation asimilar geometric picture could be built. This paragraph isspeculative; it could list a lot of wishful thinking. In factnumerical evidence of chaotic behavior exists in someregions of phase space for the SAV spacetimes [6]. Thispaper provides an algebraic test for the validity of theseideas.The alternative symmetric formulation of the Killing

equations given in Sec. VIII has several advantages, de-spite its currently obscure mathematical origin. By con-firming the hunch that the fourth-order Killing equations in4D spacetime can be written in the form of four fourth-order Killing equations of a two-manifold with additionalinterlocking conditions, it gives greater insight into thestructure of the Killing equations in 4D spacetimes. Thehigh degree of symmetry of these equations and theircompactness eases the way for further analytic exploration,increasing the possibility for their analytic solution, whichwill be attempted in [14]. Furthermore, the simplifications


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that result from the additional assumption that the space-time is static become immediately apparent, eliminating alarge number of variables from the problem. This fact,coupled with an additional assumption of equatorial sym-metry, allows one to obtain an approximation for thePoincare map on the equatorial plane of these spacetimes.This program has been carried out for the Zipoy-Voorheesmetric [13] and the results agree very well with numericalsimulations of the orbits. Finding a more systematic deri-vation of the variables used in this formulation beyond justbeing the product of a fortuitous postulate should be aninteresting exercise in itself.

When gravitational-wave observatories such as LIGOand LISA mature and allow us to probe the nether regionsof spacetime around compact objects, we stand to learn agreat deal. It is hoped that this and the preceding papers inthis series, [1–3], will provide a mathematical frameworkin which to discuss practical algorithms for mappingspacetimes. So doing to facilitate optimally decodingthe information gleaned from gravitational-wave ob-servatories.

ACKNOWLEDGMENTS

My sincere thanks to Frank Estabrook for many usefuldiscussions. I am also indebted to Tanja Hinderer andMichele Vallisneri for their insightful comments on themanuscript. I gratefully acknowledge support from NSFGrants No. PHY-0653653, No. PHY-0601459, NASAGrant No. NNX07AH06G, and the Brinson Foundationand the David and Barbara Groce startup fund at Caltech.

APPENDIX A: FOURTH-ORDER KILLINGEQUATIONS IN ATRANSVERSE TETRAD

Killing tensor components confined to the directions setby the SAV Killing vectors have their gradients fullydefined by the Killing equations. These equations, ex-pressed in terms of the directional derivatives and rotationcoefficients are given below,

T1111;3 ¼ 2T1111�131 � 4T1112�131 � 12T1134�131 � 4T1111�132 � 12T1133�141 � 2T1111�232;

T1111;4 ¼�12T1144�131 þ 2T1111�141 � 4T1112�141 � 12T1134�141 � 4T1111�142 � 2T1111�242;

T2222;3 ¼�2T2222�131 � 4T2222�132 � 4T1222�232 þ 2T2222�232 � 12T2234�232 � 12T2233�242;

T2222;4 ¼�2T2222�141 � 4T2222�142 � 12T2244�232 � 4T1222�242 þ 2T2222�242 � 12T2234�242;

T1112;3 ¼ ðT1112 � 3T1122� 6T1234Þ�131 �ð4T1112 þ 6T1134Þ�132 � 6T1233�141 � 6T1133�142 �ðT1111þT1112Þ�232;

T1112;4 ¼�6T1244�131 � 6T1144�132 þðT1112� 3T1122 � 6T1234Þ�141 �ð4T1112þ 6T1134Þ�142 �ðT1111 þT1112Þ�242;

T1222;3 ¼�ðT1222þT2222Þ�131 �ð4T1222 þ 6T2234Þ�132 � 6T2233�142 þð�3T1122 þT1222� 6T1234Þ�232 � 6T1233�242;

T1222;4 ¼�6T2244�132 �ðT1222þT2222Þ�141 �ð4T1222 þ 6T2234Þ�142 � 6T1244�232 þð�3T1122 þT1222� 6T1234Þ�242;

T1122;3 ¼�2ðT1222þT2234Þ�131 � 4ðT1122 þ 2T1234Þ�132 � 2T2233�141 � 8T1233�142 � 2ðT1112þT1134Þ�232 � 2T1133�242;

T1122;4 ¼�2T2244�131 � 8T1244�132 � 2ðT1222þT2234Þ�141 � 4ðT1122 þ 2T1234Þ�142 � 2T1144�232 � 2ðT1112þT1134Þ�242:

(A1)

The equations governing Killing tensor components with mixed indices TA1A2k1k2 that have fully defined gradients are


022002-13

T1134;3 ¼ 1

2ð�T1133;4 � 2T1134ð2�132 þ �232 � �131Þ � T1133ð2�142 þ �242 � �141 þ 2�344ÞÞ � ð2T1234 þ T3344Þ�131

� ðT1233 þ T3334Þ�141;

T1134;4 ¼ 1

2ð�T1144;3 � 2T1134ð��141 þ 2�142 þ �242Þ þ T1144ð�131 � 2�132 � �232 þ 2�343ÞÞ � ðT1244 þ T3444Þ�131

� ð2T1234 þ T3344Þ�141;

T2234;3 ¼ 1

2ð�T2233;4 � 2T2234ð�131 þ 2�132 � �232Þ � T2233ð�141 þ 2�142 � �242 þ 2�344ÞÞ � ð2T1234 þ T3344Þ�232

� ðT1233 þ T3334Þ�242;

T2234;4 ¼ 1

2ð�T2244;3 � 2ðþT2234ð�141 þ 2�142 � �242ÞþÞ � T2244ð�131 þ 2�132 � �232 � 2�343ÞÞ � ðT1244 þ T3444Þ�232

� ð2T1234 þ T3344Þ�242;

T1234;3 ¼ 1

2ð�T1233;4 � 2T2234�131 � T2233�141 � 2T1134�232 � T1133�242 � 2T1233�344Þ � ð2T1234 þ T3344Þ�132

� ðT1233 þ T3334Þ�142;

T1234;4 ¼ 1

2ð�T1244;3 � T2244�131 � 2T2234�141 � T1144�232 � 2T1134�242 þ 2T1244�343Þ � ðT1244 þ T3444Þ�132

� ð2T1234 þ T3344Þ�142: (A2)

Mixed components that have partially defined gradients are governed by the following equations:

T1133;3 ¼ T1133ð�131 � 2�132 � �232 � 2�343Þ � 2

3ð3T1233�131 þ T3334�131 þ T3333�141Þ;

T2233;3 ¼ � 2

3ð3T1233�232 þ T3334�232 þ T3333�242Þ � T2233ð�131 þ 2�132 � �232 þ 2�343Þ;

T1233;3 ¼ 1

3ð�3T2233�131 � 2T3334�132 � 2T3333�142 � 3T1133�232 � 6T1233ð�132 þ �343ÞÞ;

T1144;4 ¼ � 2

3ðT4444�131 þ ð3T1244 þ T3444Þ�141Þ þ T1144ð�141 � 2�142 � �242 þ 2�344Þ;

T2244;4 ¼ � 2

3ððT4444�232 þ ð3T1244 þ T3444Þ�242ÞÞ � T2244ð�141 þ 2�142 � �242 � 2�344Þ;

T1244;4 ¼ 1

3ð�2T4444�132 � 3T2244�141 � 6T1244�142 � 2T3444�142 � 3T1144�242 þ 6T1244�344Þ:

(A3)

The components orthogonal to the Killing directionsTðk1k2k3k4Þ, ki ¼ f3; 4g constitute a subgroup of equationsthat represent the fourth-order Killing equations of a two-manifold. Within this subgroup components with gradientsfully defined are

T3344;3 ¼ � 2

3ðT3334;4 þ 2T3334�344Þ;

T3344;4 ¼ � 2

3ðT3444;3 � 2T3444�343Þ:

(A4)

The remaining components with partially defined gradientsare governed by the equations

T3334;3 ¼ � 1

4T3333;4 � T3333�344 � 2T3334�343;

T3444;4 ¼ � 1

4T4444;3 þ T4444�343 þ 2T3444�344;

T3333;3 ¼ �4T3333�343; T4444;4 ¼ 4T4444�344:

(A5)

APPENDIX B: INTEGRABILITY CONDITIONSTHE GENERAL FOURTH-ORDER CASE

A large number of Killing tensor components can beremoved by writing down the integrability conditions forall Killing tensor components whose gradients have beenfully defined, namely, Eqs. (A1), (A2), and (A4). In par-


022002-14

ticular the five tensor components T1111, T1112, T1122, T1222, T2222 can be completely removed from consideration usingcross-derivatives of Eqs. (A1) to yield the integrability conditions:

ð6T1244 þ 2T3444Þ�2131 þ 3T1144;3�131 þ 2T1144�131;3

þT1144ð�5�2131 þ ð14�132 þ 5�232 � 4�343Þ�131Þ

!¼ ð6T1233 þ 2T3334Þ�2

141 þ 3T1133;4�141 þ 2T1133�141;4

�T1133ð5�2141 � ð14�142 þ 5�242 þ 4�344Þ�141Þ

!;

ð6T1244 þ 2T3444Þ�2232 þ 3T2244;3�232 þ 2T2244�232;3

þT2244ðð�5�232 þ 5�131 þ 14�132 � 4�343Þ�232Þ

!¼ ð6T1233 þ 2T3334Þ�2

242 þ 3T2233;4�242 þ 2T2233�242;4

�T2233ðð5�242 � 5�141 � 14�142 � 4�344Þ�242Þ

!;

þ3T1244;3�131 þ 3T1144;3�132 þ 2T1244�131;3

þ3T2244�2131 þ 4T3444�132�131 þ 9T1144�131�232

þ3T1144�132ð4�132 þ �232 � 2�343 � �131Þþ2T1244ð��2

131 þ ð10�132 þ �232 � 2�343Þ�131Þ

0BBBBB@

1CCCCCA ¼

þ3T1233;4�141 þ 3T1133;4�142 þ 2T1233�141;4

þ3T2233�2141 þ 4T3334�142�141 þ 9T1133�141�242

þ3T1133ð�142ð4�142 þ �242 � �141 þ 2�344ÞÞ�2T1233ð�2

141 � ð10�142 þ �242 þ 2�344Þ�141Þ

0BBBBB@

1CCCCCA;

þ3T1244;3�232 þ 3T2244;3 þ 2T1244�232;3�132

þ3T1144�2232 þ 4T3444�132�232 þ 9T2244�131�232

þ3T2244ð�132ð�131 þ 4�132 � �232 � 2�343ÞÞþ2T1244ðð��232 þ �131 þ 10�132 � 2�343Þ�232Þ

0BBBBB@

1CCCCCA ¼

þ3T1233;4�242 þ 3T2233;4�142 þ 2T1233�242;4

þ3T1133�2242 þ 4T3334�142�242 þ 9T2233�141�242

þ3T2233ð�142ð�141 þ 4�142 � �242 þ 2�344ÞÞþ2T1233ðð��242 þ �141 þ 10�142 þ 2�344Þ�242Þ

0BBBBB@

1CCCCCA;

3T2244;3�131 þ 12T1244;3�132 þ 3T1144;3�232

þ2ðT2244�131;3 þ T1144�232;3ÞþT3444ð8�2

132 þ 4�131�232ÞþT2244ð�2

131 þ ð26�132 � �232 � 4�343Þ�131Þþ12T1244ð4�2

132 � 2�343�132 þ 3�131�232ÞþT1144ðð�232 � �131 þ 26�132 � 4�343Þ�232Þ

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA¼

3T2233;4�141 þ 12T1233;4�142 þ 3T1133;4�242

þ2ðT2233�141;4 þ T1133�242;4Þþ4T3334ð2�2

142 þ �141�242ÞþT2233ð�2

141 þ ð26�142 � �242 þ 4�344Þ�141Þþ12T1233ð4�2

142 þ 2�344�142 þ 3�141�242Þ�T1133ðð��242 þ �141 � 26�142 � 4�344Þ�242Þ

0BBBBBBBBBBB@

1CCCCCCCCCCCA: (B1)

Note that the five IC of Eq. (B1) only contain first derivatives of the Killing tensor components. The three tensorcomponents T1134, T2234, T1234 can be eliminated using cross-derivatives of Eqs. (A2) to yield the integrability conditions:

T3444ð�6�2131 þ 2ð12�132 þ 3�232 � 7�343Þ�131 þ 6�131;3Þ

þ3T1144ð�2131 þ ð�4�132 þ 2�232 þ 3�343Þ�131 þ 6�2

132 þ �2232 þ 2�2

343 � �131;3Þþ3T1144ð4�132�232 þ �232;3 � 8�132�343 � 3�232�343 � 2�343;3Þ

�6T1244ð�2131 � ð4�132 þ �232 � 3�343Þ�131 � �131;3Þ þ 6T2244�

2131

10T3444;3�131 þ T1144;3ð�6�131 þ 12�132 þ 6�232 � 9�343Þ þ 12T1244;3�131 þ 3T1144;33

0BBBBBBBB@

1CCCCCCCCA¼ Interchange

ind: 3 and 4

!;

2T3444ð�3�2232 þ 3�131�232 þ 12�132�232 � 7�343�232 þ 3�232;3Þ

þ3T2244ð�2131 þ ð4�132 þ 2�232 � 3�343Þ�131 þ 6�2

132 þ �2232 þ 2�2

343 þ �131;3Þþ3T2244ð�4�132�232 � �232;3 � 8�132�343 þ 3�232�343 � 2�343;3Þ

þ6T1244ð��2232 þ �131�232 þ 4�132�232 � 3�343�232 þ �232;3Þ þ 6T1144�

2232

þ10T3444;3�232 þ 12T1244;3�232 þ 3T2244;3ð2�131 þ 4�132 � 2�232 � 3�343Þ þ 3T2244;33

0BBBBBBBB@

1CCCCCCCCA¼ Interchange

ind: 3 and 4

!;

2T3444ð9�2132 � 10�343�132 þ 9�131�232Þ

þ3T1144ð�2232 � �131�232 þ 4�132�232 � 3�343�232 þ �232;3Þ

þ6T1244ð3�2132 � 4�343�132 þ �2

343 þ 3�131�232 � �343;3Þþ3T2244ð�2

131 þ ð4�132 � �232 � 3�343Þ�131 þ �131;3Þþ10T3444;3�132 þ 6T1144;3�232 þ 3T1244;3ð4�132 � 3�343Þ þ 6T2244;3�131 þ T1244;33

0BBBBBBBBB@

1CCCCCCCCCA¼ Interchange

ind: 3 and 4

!: (B2)

The structure of the three IC of Eq. (B2) differ slightly from that of Eqs. (B1) in that they contain second derivatives of theKilling tensor components. Finally the component T3344 can be eliminated using cross-derivatives of Eqs. (A4) to yield theIC:

2T3444ð�2343 � �343;3Þ � 3T3444;3�343 þ T3444;33 ¼ 2T3334ð�2

344 þ �344;4Þ þ 3T3334;4�344 þ T3334;44: (B3)


022002-15

Equations (B1)–(B3), constitute the nine basic integrability conditions on the remaining fields used to build up the linearsystem of equations for the Killing tensor components and their derivatives.

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(1970).[5] S. Chandrasekhar, The Mathematical Theory of Black

Holes (Clarendon, Oxford, 1983).[6] J. R. Gair, C. Li, and I. Mandel, Phys. Rev. D 77, 024035

(2008).[7] H. Stephani, D. Kramer, M. Maccallum, C. Hoenselaers,

and E. Herlt, Exact Solutions of Einstein’s Field Equations

(Cambridge University Press, Cambridge, England, 2003),2nd ed..

[8] T. Wolf, Comput. Phys. Commun. 316 (1998).[9] B. K. Harrison, J. Math. Phys. (N.Y.) 24, 2178 (1983).[10] G. Neugebauer, J. Phys. A 12, L67 (1979).[11] L. S. Hall, Physica D (Amsterdam) 8, 90 (1983).[12] C. Klein and O. Richter, Ernst Equation and Riemann

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[14] J. Brink, arXiv:0911.4161.


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Documents

Spacetime encodings. IV. The relationship between Weyl curvature and Killing tensors in stationary axisymmetric vacuum spacetimes