Killing Tensors as Space–Time Metrics

Annals of Physics 271, 2330 (1999) Article ID aphy.1998.5870, available online at http:www.idealibrary.com on

Killing Tensors as SpaceTime Metrics Franz Hinterleitner* , Department of Theoretical Physics and Astrophysics, Masaryk University, Kotlar ska 2, CZ 611 37 Brno, Czech Republic Received September 2, 1997; revised July 21, 1998

The symmetric Killing tensors of order two associated with orthogonal separable coordinates for the KleinGordon equation in flat 2+1-dimensional space-time are considered as metrics. So, as a by-product of variable separation in flat space-time new, generally curved, spaces are generated whose metric again admits Killing tensors. For each coordinate system there are distinguished Killing metrics whose curvature yields an energy-momentum tensor of a matter distribution which is comoving with the underlying separable coordinates. There are simple conformal relations between these particular Killing tensors of the different coordinate systems.  1999 Academic Press

1. INTRODUCTION Coordinate systems in n-dimensional spaces allowing for a separation of wave equations into ordinary differential equations by a product ansatz are connected with Stackel systems. A Stackel system is a system of n linearly independent symmetric Killing tensors of order two, including the metric g ik , characterized by the following generalization of the familiar Killing equation for vector fields, K (ik; l ) =0,

i, k, l=1, ..., n ,

(1)

the vanishing of the symmetrized covariant derivative. There exists a vast literature about orthogonal and non-orthogonal variable separation for the Hamilton Jacobi-, Helmholtz-, and scalar wave equations [13]. For all these equations the separability conditions and the connections with Killing tensors are very closely related. In the orthogonal case the connection between Stackel systems and separable coordinates is that the tangent vectors of the coordinate lines are the common eigenvectors of all the associated Killing tensors [4]. In this paper the Stackel systems of ``genuinely'' three-dimensional separable coorinates are consideredsystems that cannot be reduced by symmetry to translated or rotated separable systems in two dimensions. There are 31 such systems which may be arranged in 9 classes, according to 9 different expressions of the * Supported by the ``Jubilaumsfonds der Osterreichischen Nationalbank,'' project 5425. On leave from the Institut fur Theoretische Physik, Universitat Wien, Boltzmanngasse 5, A-1090 Wien, Austria.

23 0003-491699 30.00 Copyright  1999 by Academic Press All rights of reproduction in any form reserved.

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metric tensor in terms of the coordinates [5]. The three-dimensional case deserves interest as the simplest one where an energy-momentum tensor may be constructed via the Einstein equations.

2. SEPARABLE COORDINATE SYSTEMS, METRIC TENSORS Separable coordinates in three-dimensional Minkowski space label confocal surfaces of order two (or planes as their limits) [57]. In the following these curvilinear coordinates will be denoted by +, &, and \, each of them defined in one of the intervals (&, 0), (0, 1), (1, a), and (a, ) and +\&. Occasionally two of them or all three may lie in one interval. Transformed to +, &, \, the expressions for the flat metric tensor g ik for the corresponding nine classes take the following forms (x 0 =+, x 1 =&, x 2 =\). (1) Ellipsoidal and hyperboloidal coordinate systems, (1.a)

a#R

g (1.a) ik =diag (1.b)

_

(\&+)(\&&)

(+&&)(+&\) (&&+)(&&\) ( \&+)(\&&) . , , +(+&a)(+&b) &(&&a)(&&b) \(\&a)(\&b)

&

Hyperboloidal system in half-spaces, e.g., t>x g (2) ik =diag

(3)

(&&+)(&&\)

a, b complex conjugate

=diag g (1.b) ik (2)

(+&&)(+&\)

_+(+&1)(+&a) , &(&&1)(&&a) , \(\&1)( \&a)&

_

(+&&)(+&\) (&&+)(&&\) (\&+)(\&&) , , . +(+&1) 2 &(&&1) 2 \( \&1) 2

&

Hyperboloidal system in half-spaces g (3) ik =diag

_

(+&&)(+&\) (&&+)(&&\) (\&+)( \&&) , , . +3 &3 \3

&

(4) Paraboloidal coordinate systems, (4.a) g (4.a) ik =diag (4.b)

_

(+&&)(+&\) (&&+)(&&\) (\&+)( \&&) , , +(+&1) &(&&1) \(\&1)

&

a, b complex conjugate g (4.b) =diag ik

(+&&)(+&\) (&&+)(&&\) (\&+)( \&&)

_(+&a)(+&b) , (&&a)(&&b) , (\&a)(\&b) & .

25

KILLING TENSORS AS METRICS

(5)

Paraboloidal system in a half-space g (5) ik =diag

(6)

&

Paraboloidal system in a half-space g (6) ik =diag

(7)

_

(+&&)(+&\) (&&+)(&&\) (\&+)( \&&) , , . +2 &2 \2

_

(+&&)(+&\) (&&+)(&&\) (\&+)(\&&) . , , + & \

&

Paraboloidal system g (7) ik =diag [(+&&)(+&\), (&&+)(&&\), (\&+)(\&&)] .

All these metrics are flat metrics in parts of Minkowski spacethe domains of the separable coordinate systemsbounded by null horizons. For comparison with the extensive works on separable coordinates [5, 6], (1.a) corresponds to (B.1.ae) there, (1.b) to (B.1.f), (2) to (D.1), (3) to (F.1.a), (4.a) to (C.a,b), (4.b) to (C.d), (5) to (F.1.c), (6) to (E.1), and (7) to (G).

3. STACKEL SYSTEMS As the coordinate directions are given by common eigenvectors of g ik and two further Killing tensors, these tensors are diagonal too in terms of +, &, \. Here we begin with contravariant Killing tensors, given by K ii = f i g ii

(without summation).

(2)

The three functions f i(+, &, \) are solutions of the following partial differential equations [8]  i f j =( f i & f j )  i ln | g jj |.

(3)

Among the contravariant tensors there is the following relation: If K ij is an invertible Killing tensor in a space with metric g ij , then g ij is a Killing tensor in a space with the contravariant metric K ij (and whose covariant metric is the inverse of K ij ). The invertibility condition for K ij need not hold for systems with one coordinate along a Killing vector field v i. But such coordinate systems may be reduced to lower dimensional ones and are not considered here. In all the cases in this paper f 0 =&\,

f 1 =+\,

f 2 =+&

(4)

f 2 =++&

(5)

and f0 =&+\,

f 1 =++\,

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FRANZ HINTERLEITNER

are two linearly independent solutions of (3). Any linear combination of the Killing tensors K ij(1) constructed with (4), K ij(2) arising from (5) and the metric is a Killing tensor again. The Stackel system S is the linear space spanned by these three tensors.

4. KILLING METRICS Now, more generally, the inverse of any (invertible) linear combination k ij =:K ij(1) +;K ij(2) +# g ij # S

(6)

may be taken as a covariant space-time metric. (This is not a covariant Killing tensor w.r. to g ij !) For an arbitrary k ij the off-diagonal elements of the resulting Ricci- and Einstein-tensors are proportional to :#&; 2.

(7)

:=1,

;=#=0,

(8)

:=1,

;=&1,

#=1

(9)

:=1,

;=&a,

#=a 2

(10)

For the particular choices

and

the Killing metrics are conformally flat (conformally equivalent to another form of the flat metric given in Section 2), except the cases (4.b) and (7). In the case (1.a) we obtain with (10) = k (1.a) ij

1 g (4.a) , ( +&a)(&&a)( \&a) ij

(11)

with (8) = k (1.a) ij

(+&&)(+&\) (&&+)(&&\) (\&+)(\&&) 1 diag , , +&\ (+&1)(+&a) (&&1)(&&a) (\&1)(\&a)

_

&

(12)

and with (9) = k (1.a) ij

1 (+&&)(+&\) (&&+)(&&\) ( \&+)( \&&) . , , (+&1)(&&1)( \&1) +(+&a) &(&&a) \(\&a)

_

&

(13)

In (11) the conformal relation to the flat metric g (4.a) is obvious, in (12) and (13) the diagonal metric can be transformed to g (4.a) by trivial shift and scaling

KILLING TENSORS AS METRICS

FIG. 1.

27

The conformal relations between Killing metrics.

transformations of the variables. In the same manner one obtains further special, conformally flat Killing metrics 1 (4.b) , g +&\ ij

(14)

k (2) ij =

1 g (4.a), (+&1)(&&1)(\&1)

(15)

k (3) ij =

1 (5) g , +&\ ij

(16)

1 g (6) , (+&1)(&&1)(\&1) ij

(17)

k (5) ij =

1 (6) g , +&\ ij

(18)

k (6) ij =

1 (7) g . +&\ ij

(19)

k (1.b) = ij

= k (4.a) ij

Figure 1 shows the conformal relationships between the conformally flat Killing metrics and the original flat ones expressed in terms of separable coordinates.

5. CURVATURE, PHYSICAL SIGNIFICANCE Calculating the curvature quantities resulting from the above Killing metrics shows that in all cases the Einstein tensors are diagonal and so, by Einstein's equations, are the candidates for the energy-momentum tensors of possible material sources of curvature. The conventions are chosen so that the Einstein tensor and the energy-momentum tensor have the same sign.

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FRANZ HINTERLEITNER

In the case (1.b) the version (12) for k ij yields the simplest expression for the Einstein-tensor, G 00 =

(a+b) +&\&ab(2+&+2+\+&\) k 00 , +&\

G 11 =

(a+b) +&\&ab(2+&++\+2&\) k 11 , +&\

G 22 =

(a+b) +&\&ab(+&+2+\+2&\) k 22 . +&\

(20)

Case (1.a) is obtained from (1.b) by taking a real and b=1. In cases (2), (5) and (6) the curvature is constant and G ij B k ij , in case (3), k ij is locally flat. The metric k (4.b) yields the Einstein-tensor ij G 00 =&

+&\++&++\&3+&2&&2\+4 k 00 , (+&1)(&&1)(\&1)

G 11 =&

+&\++&+&\&2+&3&&2\+4 k 11 , (+&1)(&&1)( \&1)

G 22 =&

+&\++\+&\&2+&2&&3\+4 k 22 . (+&1)(&&1)(\&1)

(21)

In the remaining cases (4.b) and (7), which represent the endpoints in the figure, the Killing tensors constructed according to (8)(10) are not conformally flat, nevertheless their Einstein-tensors are diagonal as in all the other cases. The case of the locally flat Killing metric deserves special interest: Here the conformal factor +&\ is equal to &(t&x) 2 in terms of cartesian coordinates. So k ij is equivalent to the Minkowski metric everywhere except on the null plane t=x, which is part of the coordinate horizon.

6. SUMMARY It was shown by Eisenhart [4] that a diagonal Ricci-tensor is a necessary condition for orthogonal separation of the Helmholtz equation g &12  i (g 12 g ij  j )=E

(22)

(for a positive definite g ij ). Here symmetric Killing tensors of order two associated with the separation of the KleinGordon equation were tentatively considered as metrics. So from the (nonsingular) Stackel system of any considered coordinate system an n&1-parameter family of (curved) metrics (6) can be generated. All the Killing tensors, including

KILLING TENSORS AS METRICS

29

the original flat metric, are in involution; that means the quantities K :=k ijp i p j constructed with the tangent covectors p i along the geodesics of space-time have vanishing poisson brackets. In the language of Hamiltonian mechanics the K 's are constants of geodesic motion. Like the Helmholtz equation, the wave scalar equation ((22) with a Lorentzian metric and E=0) is separable with respect to the coordinates +, &, and \ for the Killing metrics with diagonal Ricci tensors, distinguished by (7). The physical interest in Killing metrics arises from the comparison with Killing vector fields. There is a conjecture due to Horsky and Mitskievich [9], stating that from an empty space metric with a Killing vector field a solution of the Einstein Maxwell equations with the same symmetry and the electromagnetic vector potential along the Killing vector may be generated. Separable curvilinear coordinates in flat space are arbitrary objects, but for the Killing metrics taken from the associated Stackel system the coordinate lines gain some physical significance. These Killing metrics generate massive solutions of the Einstein equations with the source of curvature at rest with respect to the coordinates. Case (3), the expression for the flat metric in a half-space, is a parallel to the conformal vacuum in a half-space considered in [10]. Concerning the nature of the ``matter'' in these models, it can be classical matter at most in parts of space-time, because it does not fulfill the positive and the dominant energy conditions everywhere. Let's take, as an example, the Einstein-tensor of case (1.a). The timelike component, being proportional to the energy density, is G 22 =G \\ , G \\ =

a(+&+2+\+2&\)&(a+1) +&\ ( \&+)( \&&) , (+&\) 2 ( \&1)( \&a)

G ++ =

a(2+&+2+\+&\)&(a+1) +&\ (+&&)(+&\) , (+&\) 2 (+&1)(+&a)

G && =

a(2+&++\+2&\)&(a+1) +&\ (&&+)(&&\) . (+&\) 2 (&&1)(&&a)

(23)

From these expressions it can be seen that the dominant energy condition |G \\ |  max( |G ++ |, |G && | ) holds (with or without a cosmic term 4k ij ) if \ is close enough to the boundary values 1 and a, and + and & are not. For an examination of the positive energy condition G \\ 0 the subcase &< &<0<\<1, a<+ (an example of hyperbolic coordinates) is taken. The condition is fulfilled under the restrictions \
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FRANZ HINTERLEITNER

closest to physical acceptability. The main significance lies in the fact that among the considered Stackel systems the flat metric is distinguished among all the Killing tensors in that way that its (vanishing) energy-momentum tensor is the only one to fulfill globally and trivially all classical energy conditions.

ACKNOWLEDGMENT The author thanks the relativity group of the Department of Theoretical Physics and Astrophysics at the Masaryk University in Brno, CZ, for hospitality and helpful discussion.

REFERENCES E. G. Kalnins and W. Miller, SIAM J. Math. Anal. 11 (1980), 1011. E. G. Kalnins and W. Miller, SIAM J. Math. Anal. 12 (1981), 617. E. G. Kalnins, S. Benenti, and W. Miller, J. Math. Phys. 38 (1997), 2345. L. P. Eisenhart, Ann. Math. 35 (1934), 284. F. Hinterleitner, submitted for publication. E. G. Kalnins and W. Miller, SIAM J. Math. Anal. 6, No. 2 (1975), 340. E. G. Kalnins and W. Miller, J. Math. Phys. 17 (1976), 331. S. Benenti and M. Francaviglia, in ``General Relativity and Gravitation'' (A. Held, Ed.), p. 393, Plenum, New York, 1980. 9. J. Horsky and N. Mitskievich, Czechoslovak J. Phys. B 39 (1989), 957. 10. M. R. Brown, A. C. Ottewill, and S. T. C. Siklos, Phys. Rev. D 26 (1982), 1881. 1. 2. 3. 4. 5. 6. 7. 8.