The equivalence problem, namely the problem of deciding whether or not two given spacetime geometries are locally equivalent, is central to the area of exact solutions in general relativity. The classic resolution due to E. Cartan [1] involves computation of the tenth covariant derivative of the Riemann tensor, which, even with the aid of modern computer algebra systems, is prohibitive. The method of Cartan was expounded and simplified with specific reference to general relativity by C. H. Brans [2] and considerable progress in reducing the required order of differentiation was made by A. Karlhede [3], who devised an algorithmic method for classifying geometries.

The Karlhede algorithm starts by computing the Riemann tensor in a particular frame and then computes successively higher covariant derivatives, together with possible changes of frame, until a `complete' classification of the geometry has been obtained. If we denote the frame components of the Riemann tensor and its covariant derivatives up to the qth by R^q, then the algorithm essentially proceeds as follows :

1. Let q = 0.
2. Compute R^q.
3. Fix up the frame as much as possible by choosing a canonical form for R^q.
4. Find the invariance group H^q of the frame which leaves R^q invariant.
5. Find the number of functionally independent components t^q amongst the set R^q.
6. If t^q ¹ t^q-1 or dim(H^q) ¹ dim(H^q-1) then set q = q+1 and go to 2.
7. Otherwise the set {H^p, t^p, R^p}, p = 1,¼,q classifies the solution.

Then given two metrics g and g¢ which we wish to compare for equivalence, we start by completing the above classification for each metric. The rest of the procedure is contained in the following steps :

8. If the two sequences H⁰,t⁰; H¹,t¹;¼; H^q,t^q for g and g¢ differ, then so do the metrics.
9. If the set of simultaneous algebraic (but possibly transcendental) equations R⁰ = R¢⁰, R¹ = R¢¹,¼, R^q = R¢^q admit a coordinate transformation x¢ⁱ = x¢ⁱ(xⁱ), i = 1,¼,n as a solution then the metrics are equivalent, otherwise they are inequivalent.

Step 9 is not algorithmic, since there is no constructive procedure for solving simultaneous algebraic equations. The classification also provides a simple means of obtaining the dimension of the isometry group I of the metric through the result (see [1])

The theoretical bound on the highest order of covariant derivative required in the Karlhede approach is reduced to seven in the worst three cases (non-vacuum types N and D and vacuum type N) and five in all other cases (vacuum and non-vacuum types I, II, III and vacuum type D). Nonetheless, for any non-trivial solution, the complexity involved in the computation of all the terms up to such high derivatives is still beyond the capabilities of current computer algebra systems.

The issue of the required order of differentiation is of significance because a major effort is under way to use the Karlhede classification to construct a computer data base of all known exact solutions [4]. However, it turns out that in all the calculations for explicit solutions carried out to date in this work, computation of the third order derivatives has proved sufficient. This leads to two important questions:-

When (if ever) does one need to proceed beyond the third derivative?

Can one reduce the theoretical bound on the order of the required derivatives for the various cases?

In [4] and [5] it was shown that Karlhede's bound on the number of derivatives required for type D vacuum spacetimes could be reduced from 5 to 3 (the bound for the non-vacuum case was reduced to 4 in [6]). An important aspect of the approach taken in this work was the use of an extension of the NP formalism (Newman-Penrose formalism [7]) called GHP formalism (Geroch-Held-Penrose formalism [8]). The advantage of using this formalism results from the fact that type D spacetimes have a Weyl tensor which admits spin and boost transformations as its invariance group, and it is precisely these transformations that the GHP formalism respects. The key idea is to split the connection coefficients into two groups. Those which have simple scaling properties under spin and boost transformations and those that have more complicated behaviour. It turns out that the badly behaved spin coefficients may be combined with the directional derivatives to form new differential operators (called edth and thorn) which have a simple scaling under spin and boost transformations. These are examples of `invariant differential operators'.

For type N spacetimes the invariance group consists of null rotations rather than spin and boost transformations. In [9] a formalism involving totally symmetric spinors was introduced which is invariant under null rotations, and in [10] these ideas were combined with those of the GHP formalism to produce a spacetime calculus based on a single null direction. This was used in [11] to reduce the bound on the number of derivatives required to classify type N vacuum spacetimes from 7 to 5.

§3 Solutions of Einstein's Equations

Although the original motivation for introducing such invariant operators was to improve on the derivative bounds for the Karlhede classification it was also hoped that such a formalism would prove useful in finding solutions to Einstein's equations which are invariant under null rotations. The use of the GHP formalism to find exact solutions was pioneered by Held [12] and in the past few years has been applied to good effect by a number of authors including Edgar and Ludwig [13]. In particular they demonstrated how the GHP formalism could be used to obtain the complete class of conformally flat radiation metrics.

Another approach to the construction of exact solutions which was originally suggested by Karlhede and Lindström [14], is to apply the techniques used in classifying equivalent metrics in reverse and construct a geometry from a set of elements representing the Riemann tensor and some of its covariant derivatives. This is indeed possible provided certain integrability conditions are satisfied. Furthermore in a number of papers Bradley and Marklund have actually used the method to construct a class of locally rotationally symmetric perfect fluid spacetimes. In [15] Edgar and vickers combined the ideas used in these two approaches by performing the integration using a systematic application of the commutators of the invariant differential operators to the functional information obtained at each order of the Karlhede classification.

References

[1]

E. Cartan, Leçons sur la Geometrie des Espaces de Riemann (Gauthier-Villars, Paris, 1946).

[2]

C.H. Brans, J. Math. Phys. 6 (1965) 94.

[3]

A. Karlhede, G.R.G. 12 (1979) 693.

[4]

J.M. Collins, R.A. d'Inverno and J.A. Vickers, Class. Quantum Grav. 7 (1990) 2005.

[5]

J.M. Collins, R.A. d'Inverno and J.A. Vickers, Class. Quantum Grav. 8 (1991) L215.

[6]

J.M. Collins and R.A. d'Inverno Class. Quantum Grav. 10 (1993) 343.

[7]

E.T. Newman and R. Penrose J. Math. Phys. 3 (1962) 566.

[8]

R. Geroch, A. Held and R. Penrose J. Math. Phys. 14 (1973)874.

[9]

M.P. Machado Ramos and J.A. Vickers Proc. R. Soc. A 450(1996) 1.

[10]

M.P. Machado Ramos and J.A. Vickers Class. Quantum Grav. 13(1996) 1579.

[11]

M.P. Machado Ramos M P and J.A. Vickers Class. Quantum Grav. 13 (1996) 1589.

[12]

A. Held Comm. Math. Phys.37 (1974) 311.

[13]

S.B. Edgar and G. Ludwig G. R. G.29 (1997) 1309.

[14]

A. Karlhede and U. Lindström G. R. G.15 (1983) 597.

[15]

S.B. Edgar and J.A. Vickers Class. Quantum Grav. (in press)