INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY Class. Quantum Grav. 18 (2001) 2297–2304 www.iop.org/Journals/cq PII: S0264-9381(01)22408-3

Existence of Lanczos potentials and superpotentials for the Weyl spinor/tensor

Fredrik Andersson and S Brian Edgar

Department of Mathematics, Linkoping¨ University, S581 83 Linkoping,¨ Sweden

E-mail: [email protected] and [email protected]

Received 2 March 2001

Abstract A new and concise proof of existence—emphasizing the very natural and simple structure—is given for the Lanczos spinor potential LABCA of an A arbitrary symmetric spinor WABCD defined by WABCD = 2∇(A LBCD)A ; this proof is easily translated into tensors in such a way that it is valid in four- dimensional spaces of any signature. In particular, this means that the Weyl spinor ABCD has Lanczos potentials in all spacetimes, and furthermore that the Weyl tensor has Lanczos potentials on all four-dimensional spaces, irrespective of signature. In addition, two superpotentials for WABCD are identified: the D first TABCD (= T(ABC)D) is given by LABCA =∇A TABCD, while the second HABAB (= H(AB)(AB)) (which is restricted to Einstein spacetimes) is given by B LABCA =∇(A HBC)AB . The superpotential TABCD is used to describe the gauge freedom in the Lanczos potential.

PACS numbers: 0420, 0240

1. Introduction

The fact that there exists a potential (called the Lanczos potential [1]) for the Weyl tensor/spinor in analogy to the well known electromagnetic potential for the electromagnetic field does not seem to be well known. Even among those aware of its existence it does not seem to be fully appreciated just how close is the analogy with the electromagnetic case, nor how simple and natural is the structure of the potential for the gravitational case. These facts are particularly striking in spinor notation where the Lanczos spinor potential LABCA of the Weyl spinor ABCD is defined by A ABCD = 2∇(A LBCD)A (1) is an obvious generalization of the familiar electromagnetic case, A ϕAB =∇(A AB)A . The renewed interest began with the work of Bampi and Caviglia [2], who showed that Lanczos’ original proof of existence [1] was flawed; however, they themselves supplied a rigorous but

0264-9381/01/122297+08$30.00 © 2001 IOP Publishing Ltd Printed in the UK 2297 2298 F Andersson and S B Edgar

1 complicated proof of local existence for the Lanczos tensor potential Labc of an arbitrary Weyl candidate in four-dimensional analytic spaces, independent of signature. Illge [5] has supplied a more conventional proof of existence for the Lanczos spinor LABCA (as part of a Cauchy problem analysis) without requiring any assumption on analyticity; since he uses spinors his result is restricted to spacetimes (four-dimensional spaces with a metric of Lorentz signature), and it seems difficult to translate his proof into tensor notation in a form valid for four-dimensional spaces of any signature. The work by Illge clearly demonstrates the superiority of the spinor formalism for studying the Lanczos potential in spacetimes (see [4] for a recent summary of properties of the Lanczos spinor/tensor potential). Some results which suggest that the Lanczos potential has a more important role in general relativity, in the future, are the following.

• In vacuum, Illge [5] has shown that a Lanczos potential of the Weyl spinor obeys the simple homogeneous wave equation

LABCA = 0

AA in spacetimes, in the Lanczos differential gauge ∇ LABCA = 0 (see below); and it is shown in [4, 6] that this remarkably simple wave equation is a feature of all four- dimensional spaces, irrespective of signature. In a vacuum spacetime it can also be shown [4] that generically this wave equation, together with the Lanczos differential gauge condition, is a sufficient condition for LABCA to be a Lanczos potential of the Weyl spinor. • In a large class of the Kerr–Schild spacetimes [7] it has been shown that Lanczos potentials can be used to define curvature-free asymmetric connections [8, 9]. Crucial to this construction is the existence of a symmetric potential HABAB (= H(AB)(AB)) for the Lanczos potential, i.e. a superpotential for the Weyl spinor. In the Kerr spacetime, Bergqvist and Ludvigsen [10] have used this superpotential to construct an expression for quasi-local momentum, while, in the Kerr–Schild class, Harnett [11] has used the superpotential as a link to twistor concepts. These results have since been generalized considerably [12]. • In general, the existence of a Lanczos potential tensor is restricted to four dimensions [13]; this is what Lanczos originally claimed [1], but with no attempt at a proof. This generic result given in [13] does not rule out the possibility, in higher dimensions, of the existence of Lanczos potential tensors of Weyl tensors for some special classes of spaces; but if we consider this possibility via the definition (8) below, we find that the tensor version of the simple wave equation given above becomes much more complicated, with the addition of extra second-order derivative terms and also nonlinear terms involving the first derivative [4]. The non-existence of the Lanczos potential in higher dimensions is a significant difference from the electromagnetic case, and this must increase interest in its role as a ‘gravitational potential’ for classical general relativity. The purpose of this paper is to highlight the very natural and simple structure of the Lanczos spinor potential, by a new and concise proof of existence. Like Illge’s proof [5], it uses spinors, and although it lacks the uniqueness element of his proof, it is more direct. In addition, the proof emphasizes new aspects of the analogy with the electromagnetic case, as well as introducing an (asymmetric) potential TABCD (= T(ABC)D) for the Lanczos potential; this new potential may have important properties in its own right. This spinor proof can, without too much effort, be translated into a tensor proof—and hence is valid for Weyl tensors

1 The standard notation of [3] is used; for Lanczos spinors and tensors we follow the notation of [4]. Lanczos potentials and superpotentials 2299 in all signatures in four-dimensional analytic spaces. The superpotential TABCD is used in section 4 to express the gauge freedom of the Lanczos potential. In addition, we report the existence of another superpotential HABAB (= H(AB)(AB)) for the Weyl spinor (this time, a symmetric potential of the Lanczos potential) for all Einstein spacetimes [14].

2. Existence of Lanczos spinor potentials

Given symmetric spinor fields WABCD and ζBC we wish to find a symmetric spinor field LABCA such that A AA 2∇(A LBCD)A = WABCD, ∇ LABCA = ζBC. (2) These equations are easily seen to be equivalent to the single equation

A 3 ∇A LBCDA = WABCD − εA(B ζCD). 2 2 (3) We will restrict ourselves to look for solutions of the form D LABCA =∇A TABCD where TABCD = T(ABC)D. Hence, we want to find TABCD such that

A E 3 ∇A ∇A TBCDE = WABCD − εA(B ζCD). 2 2 However, by using the commutators, we can rewrite the left-hand side of the previous equation as A E EG 2∇A ∇A TBCDE =−
TBCDA +6 A(B TCD)GE
 E −6 εA(B TCD)E + TA(BCD) + TBCDA . (4) After these preliminary considerations we are ready to prove our main result:

Theorem 2.1. Given symmetric spinors WABCD (= W(ABCD)) and ζAB = ζ(AB) there exists a symmetric spinor LABCA (= L(ABC)A ) satisfying the equation

A 3 ∇A LBCDA = WABCD − εA(B ζCD) 2 2 (5) D locally, such that LABCA =∇A TABCD for some spinor TABCD (= T(ABC)D).

Proof. Consider the equation
 EG E 0 =
TBCDA − 6 A(B TCD)GE +6 εA(B TCD)E + TA(BCD) + TBCDA

3 WABCD − εA(B ζCD) + 2 (6) which is a linear second-order hyperbolic system. By the standard theory for differential D equations [15] it has a local solution TABCD = T(ABC)D. Put LABCA =∇A TABCD. From the above calculations it follows that A A E EG 2∇A LBCDA = 2∇A ∇A TBCDE =−
TBCDA +6 A(B TCD)GE
 E −6 εA(B TCD)E + TA(BCD) + TBCDA

3 = WABCD − εA(B ζCD) 2 where the last equality is a consequence of TABCD being a solution of (6).
For completeness, we mention three generalizations of this result. 2300 F Andersson and S B Edgar

• (n, ) W (n − The obvious generalization to symmetric 0 -spinors A1A2...An with symmetric , ) L A 1 1 -spinor potentials A1A2...An−1 is trivial. B B ...B • (n, m) W 1 2 m A generalization to symmetric -spinors A1A2...An is also quite straightforward, provided that in such cases the (n − 1,m +1)-spinor potential B B ...B L 1 2 m+1 A1A2...An−1 is not required to be completely symmetric. • Alternatively a generalization to symmetric (n, m)-spinors with asymmetric (n+1,m−1)- B B ...B L 1 2 m−1 spinor potentials A1A2...An+1 is straightforward. Turning to specific applications, we note that this theorem (in common with the theorems of Bampi and Caviglia [2] and of Illge [5]) is for Weyl candidates, i.e. arbitrary symmetric 4-spinors, but of course the most interesting application is when we choose WABCD = ABCD as the Weyl curvature spinor. It is then usual to call ζAB the differential gauge and when the choice ζAB = 0 is made it is called the Lanczos differential gauge. The following corollary is an immediate consequence of the previous theorem.

Corollary 2.2. Given the symmetric Weyl spinor ABCD and the differential gauge ζBC there exists a symmetric spinor LABCA (= L(ABC)A ) satisfying the equations

A AA ABCD = 2∇(A LBCD)A and ζBC =∇ LABCA D locally. Also LABCA =∇A TABCD for some spinor TABCD (= T(ABC)D) given by (6). A The equation ABCD = 2∇(A LBCD)A has come to be known as the Weyl–Lanczos equation. The proof of the above theorem, for the symmetric 4-spinor WABCD, modifies very easily to any symmetric 2-spinor FAB , and in particular to the Maxwell spinor, ϕAB for the electromagnetic field.

Corollary 2.3. Given the symmetric Maxwell spinor ϕAB and the differential gauge α, there A AA exists a spinor AAA satisfying the equations ∇(A AB)A = ϕAB and ∇ AAA = α locally. D Also AAA =∇A TAD for some spinor TAD. It is interesting to note that the existence of an electromagnetic potential is usually presented as a consequence of the second of Maxwell’s equations via Poincare’s´ lemma. Here we see that the existence of a (complex) potential AAA is independent of Maxwell’s equations; the role of Maxwell’s equations is to ensure that this potential can be chosen as Hermitian [5]. The superpotential TAB is linked to the familiar flat-space Hertz potential [14]. We also wish to point out that if the differential gauge ζBC is not specified, then we can D always find a symmetric solution TABCD = T(ABCD) of (6) and LABCA =∇A TABCD will still be a Lanczos potential of WABCD. It is easily checked that in this case the differential gauge has been fixed by CDE ζAB = 2 CDE(ATB) . We note that this does not prove that every Lanczos candidate has a symmetric T -potential, but simply that every Weyl candidate has a Lanczos potential with a symmetric T -potential, i.e. every Weyl candidate has a symmetric superpotential TABCD.

3. Existence of Lanczos tensor potentials in four dimensions

The Lanczos tensor Labc corresponding to LABCA has the properties that b Labc = L[ab]c,L[abc] = 0 and Lab = 0 (7) Lanczos potentials and superpotentials 2301 and the (four-dimensional) tensor version of (1), as given by Lanczos [1], is W = L L − ∗L∗ − ∗L∗ abcd ab[c;d] + cd[a;b] ab[c;d] cd[a;b] (8) in the differential gauge c ζab = Lab ;c. (9) The n-dimensional tensor equation [2, 4]

2 e e 2 e e Wabcd = 2Lab c;d +2Lcd a;b − ga c(L|b| d ;e + Ld b;e) + gb c(L|a| d ;e + Ld a;e) [ ] [ ] n − 2 [ ] ] n − 2 [ ] ] (10) is equivalent to (8) in four dimensions (it is this form which provides a possible Lanczos potential in n>4 dimensions). To obtain a tensor version of the theorem for n = 4, we use C the fact that TABCD = T(ABC)D can be decomposed into UABCD = T(ABCD) and VAB = TABC . The wave equation (6) then splits into two parts, whose respective tensor versions are

2 1 ef ef e f e f ∇ Uabcd − (Cab Ucdef Ccd Uabef ) − Cc a Ub fde Cd a Ub fce 2 + 2 [ ] +2 [ ] R 3 e e − (Cab c Vd e + Ccd a Vb e) + Uabcd =−Wabcd 2 [ ] [ ] 2 (11) R 2 def 1 de ∇ Vbc +4C bUc def + Cbc Vde − Vbc =−2ζbc [ ] 2 6 where the tensors Uabcd (a Weyl candidate, having Weyl symmetry) and Vab (a 2-form) correspond, respectively, to the spinors UABCD and VAB ; the 2-form ζab corresponds to ζAB . 2 a Note that ∇ =∇ ∇a is not necessarily a wave operator since the space is of arbitrary signature. Now, if the space is (real) analytic, and both Wabcd and ζbc are (real) analytic then, by the Cauchy–Kovalevskaya theorem, this system of equations always has a local solution and by translating equation (1) into tensors we can construct a Lanczos tensor potential Labc of Wabcd in the differential gauge ζbc from the solutions of (11) by

d 1 1 d Labc =−Uabc ;d − (Vab;c − Vc a;b ) − gc aVb ;d . 2 [ ] 2 [ ] (12) So we have the following result.

Corollary 3.1. Given the Weyl candidate tensor Wabcd and the 2-form ζab, there exists a tensor Labc with the properties (7), satisfying equations (8) and (9) locally. Also Labc is given by (12) for some Weyl candidate Uabcd and 2-form Vab given by (11). Hence, we have shown that Lanczos potentials exist in four-dimensional analytic spacetimes of arbitrary signature in agreement with the result of Bampi and Caviglia [2]. The specialization to the Weyl curvature tensor in four dimensions follows trivially.

4. The gauge freedom in the Lanczos potential

In applications it is of interest to know what gauge freedom we have in choosing a Lanczos potential. We already know that we are allowed to choose the differential gauge ζAB arbitrarily, but this does not exhaust the gauge freedom. The question is then: what remaining gauge freedom is there after the differential gauge has been fixed? Illge [5] gives an answer to this question: the values of LABCA on a spacelike past-compact hypersurface can be arbitrarily assigned. However, not all spacetimes have enough structure for such a hypersurface to be picked out in a natural way. Therefore, we want to express this remaining gauge freedom in a 2302 F Andersson and S B Edgar way that does not rely on a hypersurface being singled out. As we will see in this section, the theorems of section 2 enable us to express the remaining gauge freedom in such a way. We note that there are no overwhelming reasons for choosing a particular gauge in general, yet the choice of Lanczos differential gauge does give the very attractive wave equation
LABCA = 0. ˜ Let WABCD and ζBC be given symmetric spinors. Let LABCA and LABCA be two Lanczos potentials of WABCD in the same differential gauge ζBC, i.e. A A ˜ WABCD = 2∇(A LBCD)A = 2∇(A LBCD)A (13) AA AA ˜ ζBC =∇ LABCA =∇ LABCA . ˜ Put MABCA = LABCA − LABCA so that A AA 2∇(A MBCD)A = 0, ∇ MABCA = 0 which is equivalent to A ∇A MBCDA = 0. (14)

Note that this does not mean that MABCA is a Lanczos potential for the Weyl spinor of a conformally flat spacetime as ∇AA is not the Levi-Civita connection of such a spacetime. Now, according to the generalizations of theorem 2.1 there exists a spinor TABCD = D T(ABC)D such that MABCA =∇A TABCD in the special case WABCD = 0, ζAB = 0, and so we retain the same notation as in section 2). The same calculations as in the previous section tells us that TABCD must satisfy the following wave equation (we note that this wave equation is precisely equation (6)): EG E 0 =
TBCDA − 6 A(B TCD)GE +6(εA(B TCD)E + TA(BCD) + TBCDA) (15) or equivalently EG G 0 =
UABCD − 6 (AB UCD)GE +3 (ABC VD)G +12UABCD DEF DE (16) 0 =
VBC − 4 (B UC)DEF + BC VDE − 4VBC D where UABCD = T(ABCD) and VBC = TBCD . Since these equations are coupled we see that in general we need both UABCD and VBC to be non-zero to obtain a proper gauge transformation. An important exception is when M is conformally flat (i.e. ABCD = 0) where the equations decouple, and so we could obtain gauge transformations where one of UABCD and VBC is zero, but not the other. Thus, we have completely characterized the gauge transformations of the Lanczos potential that leaves the differential gauge intact. We summarize our findings in the following theorem.

Theorem 4.1. Let WABCD and ζBC be given symmetric spinors. Let LABCA be a Lanczos ˜ potential of WABCD in the differential gauge ζBC. Then the symmetric spinor LABCA is ˜ also a Lanczos potential of WABCD in the differential gauge ζBC if and only if LABCA = LABCA + MABCA where D MABCA =∇A TABCD and TABCD = T(ABC)D is a solution of (15). The tensor version of this result follows easily from (11). It is, however, worth noting that if we allow a change in the differential gauge, i.e. we only require that the first of equations (16) are satisfied, then we can take VBC = 0 and thus obtain a gauge transformation with just a symmetric UABCD, i.e. to any Lanczos potential LABCA D we can add a gauge term MABCA =∇A UABCD, where UABCD = U(ABCD) is a solution of ˜ the first of equations (16). The resulting sum LABCA = LABCA + MABCA is still a Lanczos potential of WABCD, but the differential gauge will have changed. Lanczos potentials and superpotentials 2303

5. Discussion

The remarkably simple structure and the corresponding proof for the existence of symmetric L A W potentials A1A2...An−1 for symmetric spinors A1A2...An raises the question as to whether, for all symmetric (n, m)-spinors, such symmetric potentials exist, and can be confirmed by this method. The method of our theorem can indeed be used to prove the existence of asymmetric spinor potentials for all symmetric spinors; however, we cannot, in general, prove the existence of symmetric spinor potentials for all symmetric spinors by this method. Indeed, it has been shown that such symmetric potentials for all symmetric spinors cannot exist in every curved spacetime because of algebraic inconsistencies [5]. However, there may well be significant classes of curved spaces where significant classes of spinors do have symmetric potentials. In particular, it is of interest to know whether the Lanczos potential LABCA of the Weyl spinor has a symmetric potential of the form HABAB (= H(AB)(AB)) given by

B LABCA =∇(A HBC)AB , (17) i.e. a symmetric superpotential for the Weyl spinor (we remember that for H-spaces in complex relativity, such a superpotential of the Weyl spinor exists and plays a very important role in the H-space programme [16]). In fact, the existence of a symmetric superpotential HABAB for ABCD given by (17) has recently been confirmed for a very large class of spacetimes (including all Einstein spaces and many algebraically special non-vacuum spacetimes [12, 14]). Actually, the result in [14] is significantly more general; it is shown that in an Einstein spacetime, any symmetric spinor LABCA has a completely symmetric potential as defined in (17). The knowledge that explicit superpotentials for the Weyl tensor exist for such a large class of spacetimes means that applications such as curvature-free connections, definitions of quasi- local angular momentum and links to twistor concepts may be developed in a wider context. In particular, it has recently been shown that an asymmetric curvature-free connection can be constructed from such superpotentials for a large class of algebraically special spacetimes [12].

Acknowledgment

BE gratefully acknowledges the financial support by the Natural Science Research Council of Sweden.

References

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