A local potential for the Weyl tensor in all dimensions

S Brian Edgar¹ and José M M Senovilla²

¹Matematiska Institutionen, Linköpings Universitet, Linköping, S-581 83, Sweden
²Física Teórica, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain

Email: bredg@mai.liu.se and wtpmasej@lg.ehu.es

Received 29 August 2004, in final form 31 August 2004
Published 29 October 2004

PACS numbers: 02.40.Ky, 04.20.Cv

In n dimensions, the existence of a 1-form potential A_a for the 2-form electromagnetic field F_ab enables the electromagnetic field equations to be written as a wave equation (in Lorentz signature) for the potential, which is particularly simple in the differential gauge^Note1A^a_{; a}  =  0 [12]:

In four dimensions, Lanczos [11] proposed the existence of a double (2, 1)-form potential H^ab_c  =  H^[ab]_c for the double (2, 2)-form Weyl tensor C^ab_cd (see e.g. [14] for the definition and properties of r-fold forms), and this result was confirmed in [2] (for any double (2, 2)-form with the algebraic properties of the Weyl conformal curvature tensor) and equivalently for any symmetric spinor _ABCD in [10]; see also [1, 5].

It is straightforward to conjecture a direct n-dimensional analogue for the Lanczos-Weyl equation (see (13) below) but, unfortunately, it has been shown that such a potential cannot exist, in general, in dimensions n > 4 [6]. As a consequence, interest in the existence of potentials for the Weyl tensor in dimensions n > 4 has diminished.

However, there are a number of reasons for a continuing interest in a potential for the Weyl tensor, especially one which is defined in arbitrary dimensions. In particular, one attractive property of any possible potential for the Weyl tensor is that it has units L ^- 1, the same as the connection (or the first derivative of the metric). This means that any of its squares (such as its superenergy tensor [14]) would have units L ^- 2 which are precisely the units we would expect for quantities related to gravitational `energies', in contrast to the familiar Bel-Robinson superenergy tensor [3, 14] whose units are L <sup>- 4</sup> (Roberts [13] pointed this out for the Lanczos potential). As a matter of fact, another related attractive property of these potentials is that they are at a level similar to the connection. As the potential is an `integral' of the Weyl tensor, which itself is a second derivative of the metric, the potential and the connection are necessarily connected. The advantage is that the potential is a tensorial object. Then, of course, the impossibility of constructing tensors from the metric and its first derivatives manifests itself - as it must - in the lack of uniqueness of the potentials. Thus, already at this stage we realize that any potential must be affected by a gauge freedom.

In this letter we shall show, in arbitrary dimension and signature, that all (2, 2)-forms with the algebraic properties of the Weyl tensor locally have a double (2, 3)-form potential P^ab_cde  =  P^[ab]_[cde]. We will further show that, in four dimensions, the double Hodge dual of the new potential, which is a double (2, 1)-form, is exactly the classical Lanczos potential. Hence a very natural and very general potential for the Weyl tensor is this new double (2, 3)-form potential.

We begin by considering an arbitrary n-dimensional pseudo-Riemannian manifold with metric g_ab of any signature. We call a Weyl candidate any double (2, 2)-form X_abcd  =  X_[ab]cd  =  X_ab[cd] with the algebraic properties of the Weyl conformal curvature tensor:

so that X_abcd is a traceless and symmetric double (2, 2)-form. We will now exploit these properties and rearrange as follows,

and using the Ricci identity on the second and fourth terms, we obtain

where R_ab is the Ricci tensor. Using the shorthand notation {R ⊗ X}_abcd for all the terms linear in the curvature tensors, that is to say, the second and third line of (2), we can rewrite by using (1) repeatedly

Now define the double (2, 3)-form

which will inherit from X_abcd the properties

Immediate consequences from the first of these are the following useful properties:

Hence (3) can be rewritten as

It is easily confirmed that this expression constructed from P_abcde, ignoring the {R ⊗ X}_abcd terms, has the necessary Weyl candidate index symmetries.

Now consider any Weyl candidate W_abcd. We can always find a Weyl candidate `superpotential' X_abcd locally for W_abcd by appealing to the Cauchy-Kowalewsky theorem [4] which guarantees a local solution of the linear second-order equation

in a given analytic background space. From the superpotential X_abcd we can then construct the potential P_abcde using (4), and obtain our main result.

Theorem 1. Any Weyl candidate tensor field W_abcd has a double (2, 3)-form local potential P_abcde with the properties (5) such that

The potential itself can be given in terms of a Weyl candidate superpotential X_abcd by (4) so that W_abcd is given in terms of this superpotential by (8).

Of course the above theorem has a direct application to the Weyl tensor C^ab_cd of any pseudo-Riemannian manifold.

The number of independent components of the potential can be computed easily by using the properties (5) and the result is (n  +  2)n(n  -  3)(n²  -  n  +  4)/24 (16 if n  =  4, 70 if n  =  5). This is larger than the number of independent components of a Weyl candidate, which is known to be (n  +  2)(n  +  1)n(n  -  3)/12 (that is, 10 if n  =  4, 35 if n  =  5). It is also larger (less, in the case n  =  4) than the number of independent Ricci rotation coefficients, or of independent components of the connection in a given basis. Although this large number of components may seem unsatisfactory, one must bear in mind that this number can be substantially reduced in any particular case by means of the gauge differential freedom as we show in [7].

The choice of P_abcde is not unique. As is usual with potentials - see, e.g., [12] - one can redefine significant parts of P_{abcde; f} without altering the combination in (9) which produces a particular W_abcd. From known results about p-forms, and standard arguments on the independence of the exterior differential versus the divergence, we can easily identify and exploit some of the gauge freedom for the new potential. A detailed discussion on gauge, with explicit formulae for the gauge redefinition of P_abcde, will be given in [7].

Consider now the special case of n  =  4. It has already been noted that if n  =  4 the Weyl tensor and more generally all Weyl candidates have a so-called Lanzcos potential [1, 2, 10, 11]. This is a double (2, 1)-form H^ab_c  =  H^[ab]_c with the extra properties

and all Weyl candidates have this type of potentials such that

What is the relation, if any, between the new potential P_abcde and the Lanczos potential?

To answer this question, we first remark that a double (2, 3)-form is equivalent, via dualization with the Hodge * operator, to a double (2, 1)-form in n  =  4. For the formulae and conventions about the Hodge dual operator, we refer the reader to [14] - allowing for an extra sign depending on the signature of the space. In particular, for any traceless double (2, 2)-form, and denoting the canonical volume element 4-form by η_abcd  =  η_[abcd], we have [14]

where  =  ±1 =  sign(det(g_ab)) is a sign depending on the signature (  =  1 in positive-definite metrics, and  =   - 1 in the Lorentzian case). Similarly, for any double (2, 3)-form P_abcde we can write [14]

Observe that the properties (5) translate for the double dual into, respectively,

which are the Lanczos potential properties (10) exactly. Hence, by taking the double dual of (9), using the previous formulae and after a little bit of algebra we can prove that, in four dimensions,

which coincides with (11) by identifying

For completeness, we also give the inverse formula: P_abcde  =  (*H*)_abcde.

Therefore, we have recovered the classical Lanczos potential for the Weyl tensor in four dimensions as the double dual of the new potential P_abcde. Note that the familiar differential gauge of the Lanczos potential H_abc^{; c} [1, 2, 10, 11] becomes H_abc^{; c}  =  (*P*)_abc^{; c}, and so the differential gauge for the new potential resides in P^ab_{[cde; f]}, via double dualization.

In dimensions greater than 4, a natural generalization of (11) to arbitrary dimension is to keep the double (2, 1)-form H_abc with properties (10) and consider the appropriate formula analogous to (11) which keeps the Weyl candidate symmetries. This formula is unique and reads

However, as already noted, this Lanczos potential exists exclusively in n  =  4 dimensions [6]. On the other hand, we now know that there is a different counterpart to the Lanczos potential in n > 4 dimensions; it is the double dual of P^ab_cde, i.e., a double (n  -  2, n  -  3)-form , and an expression - giving any Weyl candidate in terms of such a potential - can be obtained in any dimension by the same method that led to (12). Clearly such a dimensionally dependent result is much less natural and convenient than our defining formula (9), which is independent of dimension. Therefore, in particular, we have proved that the natural and proper way to consider a potential for a traceless and symmetric double (2, 2)-form is as a double (2, 3)-form; this version carries over to all dimensions by means of (9). The traditional double (2, 1)-form Lanczos potential H^ab_c in n  =  4 dimensions is nothing but the double dual of the general potential P^ab_cde.

In electromagnetic theory the local existence of the potential A_a for the field tensor F_ab is a direct consequence of applying Poincaré's lemma to one of Maxwell's equations, F_{[ab; c]}  =  0. However, it is important to note that the local existence of the new potential P_abcde for the Weyl tensor does not require any such `field equations'. Our result is actually analogous to the result that any 2-form always has a pair of local potentials, a 1-form A_a and a 3-form B_abc such that

If the 2-form F_ab is closed, then B_abc can be chosen to be zero, while if it is divergence free, then A_a can be set to zero [10, 12]. Our result for Weyl candidates is the generalization of this fact but, given the special structure of traceless and symmetric double (2, 2)-forms, we achieve dealing with just one potential. A detailed discussion will be presented in [7].

Finally we emphasize that all our results are local and depend on the analyticity of the pseudo-Riemannian metric [4]. However, as is usually the case, we expect that these results can be generalized, by using appropriate techniques of existence and uniqueness of solutions to differential equations, to the smooth case and even to spaces of low differentiability; from the point of view of general relativity, we can appeal to stronger theorems [8, 9] when we specialize to spaces with Lorentz signature, and the second-order differential equations in the theorems become wave equations.

Acknowledgments

JMMS gratefully acknowledges financial support from the Wenner-Gren Foundations, Sweden, and from grants BFM2000-0018 of the Spanish CICyT and 9/UPV 00172.310-14456/2002 of the University of the Basque Country. JMMS thanks the Applied Mathematics Department at Linköping University, where this work was carried out, for hospitality.

[1] 

Andersson F and Edgar S B 2001 Existence of Lanczos potentials and superpotentials for the Weyl spinor/tensor Class. Quantum Grav. 18 2297 
IOPscience

[2] 

Bampi F and Caviglia G 1983 Third-order tensor potentials for the Riemann and Weyl tensors Gen. Rel. Grav. 15 375 
CrossRef

[03a1] 

Bel L 1958 C.R. Acad. Sci. Paris 247 1094 

[03a2] 

Bel L 1962 Les états de radiation et le problème de l'énergie en relativité générale Cah. Phys. 16 59-80 

[03a3] 

Bel L 2000 Gen. Rel. Grav. 32 2047 (Engl. Transl.) 
CrossRef
Robinson I 1958  King's College Lectures Unpublished

Robinson I 1997 On the Bel-Robinson tensor Class. Quantum Grav. 14 A331-3 
IOPscience

[4] 

Courant R and Hilbert D 1989 Methods of Mathematical Physics Vol II (New York: Interscience)  

[5] 

Edgar S B and Höglund A 1997 The Lanczos potential for the Weyl curvature tensor: existence, wave equations and algorithms Proc. R. Soc. A 453 835 
CrossRef

[6] 

Edgar S B and Höglund A 2000 The Lanczos potential for Weyl candidates exists only in four dimensions Gen. Rel. Grav. 32 2307 
CrossRef
Edgar S B and Höglund A 2002 The non-existence of a Lanczos potential for a Weyl curvature tensor in dimensions n ≥ 7 Gen. Rel. Grav. 34 2149-53 
CrossRef

[7] 

Edgar S B and Senovilla J M M Gauge freedom for general potentials and new potentials for curvature tensors  in preparation

[8] 

Friedlander F G 1975 The Wave Equation on a Curved Spacetime (Cambridge: Cambridge University Press)  

[9] 

Hawking S W and Ellis G F R 1973 The Large-Scale Structure of Space-Time (Cambridge Monographs on Mathematical Physics, No. 1) (London: Cambridge University Press)  

[10] 

Illge R 1988 On potentials for several classes of spinor and tensor fields in curved space-times Gen. Rel. Grav. 20 551-64 
CrossRef

[11] 

Lanczos C 1962 The splitting of the Riemann tensor Rev. Mod. Phys. 34 379 
CrossRef

[12] 

Misner C W, Thorne K S and Wheeler J A 1973 Gravitation (San Francisco: Freeman)  

[13] 

Roberts M D 1988 A new gravitational energy tensor Gen. Rel. Grav. 20 775-92 
CrossRef

[14] 

Senovilla J M M 2000 Super-energy tensors Class. Quantum Grav. 17 2799-841 
IOPscience

Notes

Note1

 A semicolon indicates covariant derivative with respect to the canonical connection. As usual, we use round and square brackets to denote symmetrization and antisymmetrization of indices, respectively. Our convention for the Riemann tensor follows from the Ricci identity: 2v_{a; [bc]}  =  R^d_abcv_d.