Essay Final Hand In

An Introduction toLanczos Potential Theory

May 2, 2014

Overview

In this essay, we present a brief review of the development of the theory of the Lanczospotential, a three index tensor which arises as a Lagrange multiplier when consideringaction principles for four dimensional Riemannian and pseudo-Riemannian geometries.

The essay is composed of five chapters, the structure of which we will now givean overview. The first chapter will give a brief introduction to various geometricobjects which will be important in our later discussions. To supplement this any texton differential geometry or general relativity, such as Nakahara [10] or Hawking andEllis [4], should suffice.

We then proceed, in the second chapter, to investigate the motivations and his-torical development of the Lanczos tensor, following the work of Cornelius Lanczosin geometric action principles from the 1930s up to his landmark paper in 1962. Thesource material for this chapter can be found in papers by Lanczos [5–9] as well assections 4.1-3 of O’Donnell [11].

In chapter 3, the algebra and analysis of two component spinors is introduced. Mostimportantly, the embedding of the tensor algebra within the spinor algebra, and thecorrespondence between a number of tensor and spinor operations is discussed. Thischapter follows material found in sections 2.3-5 and 3.3-4 of Penrose and Rindler [13],as well as sections 2.2-8 and 3.1-2 of O’Donnell [11].

Applying the two-component spinor formalism, we obtain the spinor equivalent ofthe Lanczos tensor in the fourth chapter. We continue to find the spinor equivalent ofthe Weyl-Lanczos equations, finding that they simplify significantly. Here, we followsections 4.3-4 of O’Donnell [11].

In the final chapter, we investigate the potential physical significance of the Lanczostensor. Considering the Jordan form of general relativity brings forth many similari-ties between general relativity and electromagnetism, and suggests that the Lanczostensor may play a similar role in gravity as the electromagnetic vector potential doesin electrodynamics. We follow an investigation by Zund [18] which considers transfor-

iv

mations akin to the U(1) gauge transformations of electrodynamics. We also commenton work by Roberts [15] which suggests that an effect similar to the Aharonov-Bohmeffect of electrodynamics may come into play due to the Lanczos potential.

Contents

Overview iii

Contents v

1 Introduction 11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Covariant derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Riemann curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Weyl tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Historical formulation of the Lanczos tensor 72.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Early work in action principles of geometric quantities . . . . . . . . . 82.3 Anti-self-dual and self-dual splitting of the Riemann tensor . . . . . . . 122.4 Lanczos’ canonical Lagrangian . . . . . . . . . . . . . . . . . . . . . . . 132.5 Lanczos’ gauge and normalisation conditions . . . . . . . . . . . . . . . 162.6 The fundamental nature of Habc . . . . . . . . . . . . . . . . . . . . . . 172.7 The Weyl-Lanczos equations . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Spinor algebra and analysis 213.1 Spinor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Spinor analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3 Spin coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 The Lanczos spinor 394.1 Lanczos spinor and decomposition . . . . . . . . . . . . . . . . . . . . . 394.2 Spinor form of Weyl-Lanczos equations . . . . . . . . . . . . . . . . . . 43

vi Contents

5 Interpretation of the Lanczos tensor 455.1 Jordan form of general relativity . . . . . . . . . . . . . . . . . . . . . . 455.2 Gauge tensor candidates . . . . . . . . . . . . . . . . . . . . . . . . . . 475.3 Aharonov-Bohm-esque effects . . . . . . . . . . . . . . . . . . . . . . . 48

Conclusion 49

References 51

Chapter 1Introduction

Before examining the history and theory of the Lanczos potential, we discuss a numberof preliminary topics. In this chapter, we define the fundamental geometric quantitieswhich are manifest in physical theories, namely the Riemann curvature tensor andrelated quantities. In addition, a number of conventions to be used throughout thispaper are defined and justified.

1.1 Preliminaries

1.1.1 Units

We will assume natural gravitational units unless otherwise stated. That is

G = c = 1. (1.1)

1.1.2 Conventions

Differentiation shorthand

Ordinary partial differentiation of a tensor with respect to a basis vector will berepresented by an ordinary comma (e.g. Rabcd,e is shorthand for ∂

∂xeRabcd). Similarly,covariant differentiation with respect to a basis vector will be represented by a semi-colon (e.g Rabcd;e is shorthand for ∇eRabcd).

2 Introduction

(Anti-)Symmetrisation shorthand

Symmetrisation over a set of indices will be represented by ordinary round bracketsover the set of indices to be symmetrised (e.g. T(ab) = 1

2 (Tab + Tba )). Similarly, anti-symmetrisation will be represented by square brackets over the set of indices to beanti-symmetrised (e.g. T[ab] = 1

2 (Tab − Tba )). If a set of indices within the (anti-)symmetrisation is to be omitted from the operation, it shall be separated by verticalbars (e.g. T(a|b|c) = 1

2 (Tabc + Tcba )).

Duals of Rabcd

The Riemann tensor, due to its anti-symmetries (1.10) and (1.11), can be dualisedover the first two, the last two, or both pairs of indices. In the literature during theearly and middle of the 20th century, on which the majority of this review is based, asingle asterisk, *Rabcd, was often used to indicate the dual over both pairs of indices.We shall employ a different convention in which the dual over the first pair of indicesshall be represented by *Rabcd, over the second pair as R*abcd, and over both pairs as*R*abcd.

1.1.3 Abstract index notation

This essay will be employing the abstract index notation described in section 2.1 ofO’Donnell [11] for describing tensor and spinor equations in a basis invariant formal-ism, while still being able to use the conveniences of Einstein notation. For a morerigorous description of abstract index notation, chapter 2 of Penrose and Rindler [13]gives a complete and formal formulation.

Briefly, unbolded Roman indices (a, b, c, d, . . . ) shall represent basis independentindices, while bold Roman indices (a, b, c,d, . . . ) shall represent basis dependent in-dices. If an equation contains only unbolded indices, then the equation is valid inall bases. On the other hand if an expression contains uncontracted bold indices, itis only valid in a specific basis (often the case if we move to local inertial frame oremploy some symmetry of a system).

In general we shall adopt the notation that lower-case letters represent tenso-rial indices, while unprimed upper-case letters represent spinorial indices and primedupper-case letters represent conjugate spinorial indices.

1.2 Covariant derivative 3

1.2 Covariant derivative

The gradient of a scalar function on a manifold M is given a unique invariant meaningthrough the differential structure of M (see [13], eq. (4.1.32)). However, the definitionof a gradient of any arbitrary valence tensor1 requires an additional structure on M,known as a connection which defines the notion of the covariant derivative.

A covariant derivative operator ∇a is defined as a linear map from the (p; q) tensorsto the (p; q + 1) tensors which, given an arbitrary valence (p; q) tensor T b1b2...bp

a1a2...aq

and (r; s) tensor U b1b2...bra1a2...as

, satisfies the following properties:

1. Leibniz Rule:

∇a

(T b1b2...bp

a1a2...aqU b1b2...br

a1a2...as

)= T b1b2...bp

a1a2...aq∇aU

b1b2...bra1a2...as

+ U b1b2...bra1a2...as

∇aTb1b2...bp

a1a2...aq. (1.2)

2. Commutation with index substitution.

3. Commutation with contraction:

∇aTb1b2...c...bp

a1a2...c...aq= g d

c ∇aTb1b2...c...bp

a1a2...d...aq

= ∇agd

c Tb1b2...c...bp

a1a2...d...aq. (1.3)

4. The metric tensor gab is covariantly constant:

∇agbc = 0 and ∇agbc = 0. (1.4)

5. The torsion of the connection vanishes2:

(∇a∇b − ∇b∇a)λ = 0, (1.5)

for any scalar function λ.1Besides, of course, a valence (0; 0) tensor, i.e. a scalar.2This is not a strict axiom of a general covariant derivative in Riemannian geometry, however

we will henceforth make the assumption that the connection is torsion-free, and is therefore theLevi-Civita connection.

4 Introduction

6. The covariant derivative acts as an ordinary partial coordinate derivative onscalars:

∇aλ = ∂aλ, (1.6)

for any scalar function λ.

1.3 Riemann curvature tensor

The concept of curvature arises naturally if we question whether (1.5) holds for ageneral tensor rather than simply scalar functions. We shall see that in the case wherethe covariant derivative does not commute with itself, then the manifold has non-zerocurvature. The curvature is measured by a quantity known as the Riemann curvaturetensor Ra

bcd, which we shall define below.

1.3.1 Definition

We define the Riemann curvature tensor to be the mapping of an arbitrary co-vectorfield Aa under the commutator of a covariant derivative, as

RabcdAa = [∇c,∇d]Ab = Ab;cd − Ab;dc. (1.7)

In terms of the Christoffel symbols, one can write the Riemann curvature tensor ex-plicitly as

Rabcd = ∂cΓa

db − ∂dΓacb + Γa

ceΓedb − Γa

deΓecb. (1.8)

Lowering the first index with the metric tensor yields the Riemann curvature tensorwith all covariant indices:

Rabcd = gaeRebcd. (1.9)

1.3.2 Symmetries

From the definition of the Riemann tensor Rabcd in terms of the Christoffel sym-bols (1.8) combined with the lowering of the first index (1.9), a number of symmetriesare evident:

1. Anti-symmetry in a and b:

Rabcd = −Rbacd. (1.10)

1.3 Riemann curvature tensor 5

2. Anti-symmetry in c and d:

Rabcd = −Rabdc. (1.11)

3. Symmetry in the exchange of the index pair ab with the index pair cd:

Rabcd = Rcdab. (1.12)

4. First Bianchi identity (often called cyclic identity):

Rabcd +Racdb +Radbc = 0, (1.13)

or equivalentlyRa[bcd] = 0. (1.14)

5. Second Bianchi identity (often simply referred to as the Bianchi identity):

Rabcd;e +Rabde;c +Rabec;d = 0, (1.15)

or equivalentlyRab[cd;e] = 0. (1.16)

1.3.3 Ricci tensor and scalar

We define the Ricci tensor Rab, often known as the contracted curvature tensor,through a contraction of the Riemann tensor as

Rab = Rcacb. (1.17)

As a consequence of the first Bianchi identity (1.14), the Ricci tensor is symmetric:

Rab = Rba. (1.18)

It is interesting to note that due to the symmetries of the Riemann curvature ten-sor, the Ricci tensor (up to a sign) is the only possible non-trivial tensor which onecan achieve through a single contraction of any two indices of the Riemann tensor.

6 Introduction

Performing a second contraction yields a quantity

R = R aa (1.19)

which is known as the Ricci scalar and is the trace of the Ricci tensor. The Einsteintensor Gab of general relativity can be constructed through a trace reversal of the Riccitensor as

Gab = Rab − 12Rgab. (1.20)

1.4 Weyl tensor

The Weyl tensor Cabcd can be thought of as the traceless part of the Riemann curvaturetensor. In general relativity, it is the only part of the Riemann tensor which is leftlocally undetermined by the Einstein field equations.

1.4.1 Definition

We define the Weyl tensor Cabcd through a decomposition of the Riemann tensor intoits various traces and the Weyl tensor, known as the Ricci decomposition, as

Rabcd = Cabcd − 12 (Racgbd +Rbdgac −Radgbc −Rbcgad)

+ 16R (gacgbd − gadgbc) (1.21)

(see [11], eq. 4.29). The definition is such that tracing over any pair of indices of Cabcd

vanishes.

1.4.2 Symmetries

The Weyl tensor shares symmetries (1)−(4) of the Riemann tensor in section 1.3.2 dueto its definition (1.21). In addition, we may express its traceless property concisely as

Cabac = 0. (1.22)

Chapter 2Historical formulation of the Lanczos tensor

In this chapter we will present the historical development of the Lanczos potential.In 1962 Lanczos released a paper entitled The Splitting of the Riemann Tensor [9] inwhich the Lanczos tensor was derived though an action principle. This groundbreak-ing paper was motivated heavily by the work of Einstein in his famous 1916 paperconnecting Riemannian geometry and physics [1], and in his less famous 1928 paperon how the classical theories of electromagnetism and general relativity might be uni-fied under action principles of purely geometric quantities [3]. Lanczos preceded his1962 paper by his exploration of action principles in geometric quantities in the early1930s [5–8].

2.1 Motivations

2.1.1 Einstein’s theory of general relativity

In 1916 Einstein released his review on the theory of general relativity [1], assertingthat space-time may not be simply flat Minkowski space, but rather a curved mani-fold for which the contracted Riemann tensor is sourced by the stress-energy tensordescribing the matter present in space-time. This was not the first theory of physicsto introduce geometry as the source of gravitation; Nordström’s 1912 and 1913 the-ories were predecessors of general relativity but were experimentally verified to beincorrect. In his 1962 paper [9], Lanczos observes that since Einstein’s theory was sorevolutionary and supported the “fundamental significance of the contracted curva-ture tensor for the description of the geometrical properties of the physical universe”,it eclipsed the more general aspects of Riemannian geometry for physicists. Lanczoseven mentions that when searching for a unified geometric theory of electromagnetism

8 Historical formulation of the Lanczos tensor

and gravitation, Einstein “was compelled to drop the classical Riemannian geometry”in favour of other mathematical tools due to the inability of the Ricci tensor to containany information which could be “correlated to electric quantities.”

2.1.2 Unified theories of classical gravity and classical elec-tromagnetism

After the onset of Einstein’s theory of general relativity, geometry became somewhatof a ‘fad’ in developing new theories of physics. Most notable of this trend are theoriessuch as Einstein’s theory of “distant parallelism”, Veblen’s, Hoffman’s and Pauli’stheory of “projective reality”, as well as the famous five-dimensional Kaluza-Kleintheory. These theories all had in common the desire to unify the classical field theoriesof electromagnetism and gravity, and express all of known physics in terms of space-time structure.

2.2 Early work in action principles of geometricquantities

Lanczos was highly motivated to find a theory which explained both electromagneticand gravitational phenomena through purely geometric means, essentially encodingthe information about either the field strength tensor or the vector potential into thecurvature coefficients, alongside general relativity. Lanczos released several interestingarticles on Riemannian geometry, particularly concerning the determination of four-dimensional Riemannian geometries through extremal action principles, for severaldecades before writing his 1962 paper highlighting the Lanczos tensor as a fundamentalgeometric quantity.

In 1938 Lanczos investigated the potential types of scalar invariants which mightappear in a geometric Lagrangian in A Remarkable Property of the Riemann-ChristoffelTensor in Four Dimensions [6]. The paper considers the properties of a scale inde-pendent1 action principle in the components of the Riemann tensor. Lanczos assumes

1Weyl [17] developed the notion of a scale independent geometry which requires a "gauge-symmetry" of metric scaling invariance. That is the transformation g′

ab = λgab where λ is anarbitrary function of the coordinates, leaving physical quantities on the manifold invariant. Thisinvariance is motivated as it has the effect of “changing the scale of calibration of the infinitesimalyard-stick.” That is, we are only changing the units by which we are measuring other invariants ofthe theory.

2.2 Early work in action principles of geometric quantities 9

that the fundamental independent quantities of the theory were the components of themetric tensor and the curvature tensor. This scale-invariance insists that the scalarsfrom which the Lagrangian is built are quadratic functions of Rabcd. Lanczos con-tinues by analysing all possible quadratic scalars and finds that any scale-invariantLagrangian may be constructed from the scalars

I1 = RabRab, (2.1)

I2 = R2 (2.2)

(see [6], eq. 6.5) with all other quadratic scalar terms either being reducible to somelinear combination of (2.1) and (2.2), contributing only as boundary terms, or van-ishing identically. This work confirms the importance of the Ricci tensor in physicaltheories of geometry asserted by Einstein, assuming that all physical theories can bederived from some extremal action principle.

The generalised Lagrangian L determining a geometry is thus given by

L = RabRab + cR2 (2.3)

(see [7], eq. 2.1), where c is an undetermined quantity following the investigationin [6]. Lanczos continued this investigation in 1942 [7] by considering a tensor

Sabdef= Rab − 1

4κRgab (2.4)

(see [7], eq. 2.2) for which κ is undetermined. The generalised Lagrangian (2.3) cannow be rewritten as

L = SabSab (2.5)

(see [7], eq. 2.3) under the condition:

κ(2 − κ) + 4c = 0 (2.6)

(see [7], eq. 2.4). For any c ≥ 14 , real solutions for κ can be found. Since the Lagrangian

only contains a term quadratic in Sab , essentially treating it as a free-field, it is clearthat

Sab = 0 (2.7)

is a potential solution. Lanczos argues that this is the most stable solution as it


minimizes the action integral to zero. Comparing the solution (2.7) with (2.4) we find

Rab − 14κRgab = 0 (2.8)

(see [7], eq. 2.5) and, after contracting through with gab:

(1 − κ)R = 0 (2.9)

(see [7], eq. 2.6). Here, R = 0 or κ = 1. The former case, together with with (2.8),gives the trivial field equations Rab = 0. More interesting is the latter case, κ = 1,which when combined with (2.6) gives c = −1

4 and simplifies (2.4) to

Sab = Rab − 14Rgab (2.10)

(see [7], eq. 2.12). This is precisely the quantity suggested by Einstein in 1919 [2]to replace the stress-energy tensor in his model of the electron given below in (2.22).The 1942 article [7] continues by suggesting the stable solution (2.7) was a “metricalplateau” over which a high-frequency superposition of carrier fields could be added asa perturbation. Such a construction would result in a microscopic scale average curva-ture radius, potentially allowing for the explanation of fundamental particles throughcurvature. The basic field equations of this theory were fourth order in gab, whichtroubled Lanczos as the equations of nature “seem to offer themselves as differentialequations of first and second order.”

In 1949 Lanczos revisited the problem in Lagrangian Multipliers and RiemannianSpaces [8] and considered the metric components gab and the curvature componentsRabcd, together with Lagrangian multipliers to impose the Bianchi identity, as indepen-dent metrical quantities in order to reduce the order of the fundamental field equationspresented in [7] to second order. The Bianchi identity is more naturally expressed interms of the dual of the Riemann tensor. We define the dual of the Riemann tensoras

*R*abcd = Rijklϵijabϵklcd (2.11)

(see [8], eq. II.1) where ϵabcd is the totally anti-symmetric Levi-Civita tensor. TheBianchi identity is then expressed as

*R*abcd;a = 0 (2.12)

2.2 Early work in action principles of geometric quantities 11

(see [8], eq. II.2). Lanczos argues that instead of the scalar given in equation (2.1),one may define a different scalar as

I ′1 = *R*abcd*R*abcd (2.13)

(see [8], eq. III.1) through a linear combination of (2.1) and (2.2) without losinginformation. He then states that for the case of “infinitesimal fields” the contributionof the scalar (2.2) is an “infinitesimal of the second order and thus negligible for ourpresent purposes.” This leaves us with an action integral of the following form

12

∫(*R*abcd)2 dτ (2.14)

(see [8], eq. III.3). The Bianchi identity (2.12) is incorporated into the action (2.14)through a Lagrangian multiplier Ha

bc , which is anti-symmetric in b and c due tothe properties of the Bianchi identity, but with the symmetry of the first index leftunspecified [12]. This constraint is added to the action integral as

−12

∫Ha

bc;d *R*abcd (2.15)

(see [8], eq. III.2), after performing an integration by parts and discarding boundaryterms. After performing the variation to extremize the action, Lanczos found thefollowing expression for the dual of the Riemann tensor in terms of the Lagrangianmultiplier:

*R*abcd = Ha

bc;d −H adbc; +H a

b d;c −H ac d;b (2.16)

(see [12], eq. 2; [8], eq. III.5). This is immediately analogous to the definition ofthe Riemann tensor given in terms of the Christoffel symbols, however Ha

bc and Γabc

have significantly different properties as noted in [12]. Firstly, Γabc transforms as a

connection while Habc transforms as a tensor. Secondly, Ha

bc is anti-symmetric in b

and c while Γabc, being defined as the torsion-free Levi-Civita connection, is symmetric

in b and c.Lanczos finally derives field equations for Ha

bc by substituting the form found forthe dual Riemann tensor in terms of Ha

bc (2.16) into the Bianchi identity (2.12),finding

∆Habc −Hd a

bc;d +Ha db ;dc −Ha d

c ;db = 0 (2.17)


(see [8], eq. IV.1) where ∆ is the Laplacian operator2. Hence, using the action inte-gral (2.14) and the Bianchi identities, a system of 24 second order partial differentialequations for the determination of the 24 independent components of Ha

bc has beenfound.

Although the investigations of his 1949 (and earlier 1932) paper attempting tounify electricity and magnetism through geometrical quantities in space-time were notfruitful as a valid theory, the insights Lanczos made into defining geometries via actionprinciples was an important precursor to the realization of the Lanczos tensor as afundamental quantity in four-dimensional Riemannian geometry.

The tensor Habc was revisited by Lanczos in his 1962 paper, albeit under a slightlydifferent, yet consistent, definition. It is the same quantity which is now commonlyreferred to as the Lanczos tensor. The historical development of this tensor’s propertieswill be the subject of the remainder of the chapter.

2.3 Anti-self-dual and self-dual splitting of the Rie-mann tensor

We define the anti-self-dual and self-dual parts of the Riemann tensor respectively as

Aabcddef= Rabcd − *R*abcd (2.18)

Sabcddef= Rabcd + *R*abcd (2.19)

so thatRabcd = 1

2 (Aabcd + Sabcd) (2.20)

(see [11], eq. 4.1-3). Clearly, taking the double dual of (2.18) reverses the sign andsimilarly leaves (2.19) invariant. This splitting was noted by Rainich in a short let-ter [14], in which the self-dual tensor (with 11 independent components) was attributedto gravitational phenomena while the anti-self-dual part (with 9 independent compo-nents) was attributed to electromagnetic phenomena.

The anti-self-dual tensor (2.18) was shown by Einstein to be reducible to the Ricci2Rather than �, as Lanczos was working in imaginary time.

2.4 Lanczos’ canonical Lagrangian 13

tensor through the following decomposition

Aabcd =(Rac − 1

4Rgac

)gbd +

(Rbd − 1

4Rgbd

)gac

−(Rad − 1

4Rgad

)gbc −

(Rbc − 1

4Rgbc

)gad (2.21)

(see [11], eq. 4.4), and is easily verified by moving to normal coordinates (where gµν

reduces to δµν ), however a fully covariant derivation of (2.21) can be found in section1 of [9]. Einstein used this identity to derive the following field equations in [2] in anattempt to model a stable electron:

Rab − 14Rgab = −KTab (2.22)

(see [11], eq. 4.5) where Tab is the energy-momentum tensor of Maxwell theory. Only9 of the 10 components of the energy-momentum tensor are determined by (2.22).This ambiguity in the field equations caused Einstein to abandon this analysis of theanti-self-dual component of the Riemann tensor.

Although Einstein was not fruitful in his development of the field equations (2.22),his analysis of the anti-self-dual tensor helped elucidate the nature of four-dimensionalgeometry. The structure of the self-dual tensor (2.19), however, was left unanalysed.Is there a “generating function” of lower order for the self-dual tensor, similar tothe Ricci tensor in the case of the anti-self-dual tensor? Lanczos [9] makes the boldstatement that without this analysis “we cannot claim to have fully understood thestructure of four-dimensional Riemannian geometry.”

2.4 Lanczos’ canonical Lagrangian

Lanczos, taking the investigation of the anti-self-dual curvature tensor as motivation,revisited action principles in Riemannian geometry by considering, in analogy to (2.14)and (2.15), the Lagrangian

L′ = L (*R*abcd, gab) +Habc *R*abcd;d + P ab

c

(Γc

ab −{c

ab

})+ ρab

(Rab + Γc

bc,a − Γcab,c + Γc

adΓdbc − Γc

abΓddc

)(2.23)


(see [11], eq. 4.9), where again Habc is a Lagrangian multiplier ensuring the Bianchiidentity (2.12) is enforced, P ab

c is a Lagrangian multiplier enforcing the equality be-tween the torsion-free affine connection and the Christoffel symbols

Γcab −

{c

ab

}= 0 (2.24)

(see [11], eq. 4.7), and finally ρab is the Lagrangian multiplier enforcing the definitionof the Ricci tensor3 in terms of the Christoffel symbols

Rab + Γcbc,a − Γc

ab,c + ΓcadΓd

bc − ΓcabΓd

dc = 0 (2.25)

(see [11], eq. 4.8). The Lagrangian (2.23) has canonical variables gab, Γcab, and *R*abcd,

with conjugate variables ρab, P abc , and Habc respectively. The Lagrangian multipliers

P abc and ρab are symmetric in a and b, however Habc is anti-symmetric in a and b4.Lanczos proceeds with his investigation of the Lagrangian multipliers by preparing

to perform the variation with respect to the canonical variables. First he notes thatHabc , the conjugate of the dual Riemann tensor (which has 20 independent compo-nents), with its anti-symmetry in the first two indices has 24 independent components.He therefore adds the additional condition, without loss of generality,

*Hada = 1

2Habc ϵabcd = 0 (2.26)

(see [9], eq. 2.9), in order to restrict the number of components to 20, which isequivalent to imposing the cyclic identity:

Habc +Hbca +Hcab = 0 (2.27)

(see [9], eq. 2.10). Also in preparation of the variation, he splits ρab into its trace andits trace-free parts by defining a scalar q as

qdef= 1

4ρabgab (2.28)

3In [6], Lanczos shows that the full Riemann curvature tensor does not enter into the action prin-ciple, so here only the definition of the contracted curvature tensor must be enforced as a constraint.

4This definition of the multiplier Habc differs from that in Lanczos’ 1949 paper, however they areequivalent, and we shall henceforth use the multiplier with these symmetry properties.

2.4 Lanczos’ canonical Lagrangian 15

(see [9], eq. 2.12) and a tensor Qab as

Qab def= ρab − qgab (2.29)

so thatρab = Qab + qgab (2.30)

(see [9], eq. 2.11). Finally, we rewrite the second term of (2.23) using integration byparts and neglecting the boundary term (as its variation vanishes by construction)

Habc *R*abcd;d = −Habc;d *R*abcd (2.31)

(see [11], eq. 4.17). We can now rewrite the canonical Lagrangian as

L′ = L (*R*abcd, gab) −Habc;d *R*abcd + P abc

(Γc

ab −{c

ab

})+(Qab + qgab

) (Rab + Γc

bc,a − Γcab,c + Γc

adΓdbc − Γc

abΓddc

). (2.32)

The variation with respect to Γcab can be performed to obtain a relation for P ab

c

without specifying L (*R*abcd, gab) as it does not depend on Γcab explicitly. Lanczos

does this (see [9], eq. 2.14) and finds that P abc can be written in terms of other

variables and thus is not a fundamental quantity.Next, Lanczos performed the variation with respect to gab to obtain the following

relation5

∂L∂gab

+ 12Lgab = Qij*R*iajb +Qai*R*b

i +Qbi*R*ai

− 12RQ

ab − q*R*ab − 12Rqg

ab

− 12(P aib

;i + P bia;i − P abi

;i

)(2.33)

(see [9], eq. 2.15).Finally when performing the variation with respect to *R*abcd we notice that the

terms of the Lagrangian with Christoffel symbols drop out, and the variation δ*R*abcd

becomes∂L

∂*R*abcd −Habc;d +Qacgbd − qgacgbd (2.34)

5Here we have made use of that fact that *R* ca bc = *R*ab = −

(Rab − 1

2 Rgab

)and *R* a

a =*R* = R.


(see [9], eq. 2.16). Due to symmetries of *R*abcd, we must impose a symmetrisation(represented by square brackets)6 before allowing the variation to vanish. The resultingequations from the variation are thus given by[

∂L∂*R*abcd

]=[Habc;d − (Qac − qgac) gbd

](2.35)

(see [9], eq. 2.19).It is important to note that the preceding analysis is completely general and de-

termines the form of the Lagrangian multipliers for a geometry corresponding to theform of L, and not only for infinitesimal geometries as previously considered in the1942 and 1949 papers [7, 8].

2.5 Lanczos’ gauge and normalisation conditions

Lanczos noticed that (2.35) possessed a particular gauge invariance through the fol-lowing transformations

Habc → Habc − Φbgac + Φagbc (2.36)

Qab → Qab + Φa;b + Φb;a − 12Φc

;cgab (2.37)

q → q − 12Φa

;a (2.38)

(see [9], eq. 2.20). Since this invariance leaves the content of (2.35) physically equiv-alent, we have the freedom to choose a gauge to further restrict (or “normalise”) theform of Habc through choosing the form of Φa. If we choose

Φb = −13Habcg

ac (2.39)

(see [9], eq. 2.21), then the resulting transformation leads to the following identity,now known as the trace-free condition,

H ba b = 0 (2.40)

6Lanczos defines the bracketed version of a four-index tensor Aabcd as:[Aabcd] = 1

2 (Aabcd + Acdab + Abadc + Adcba − Abacd − Acdba − Aabdc − Adcab) . Note that this spe-cial symmetrisation leaves the Riemann tensor and the double dual Riemann tensor invariant (up toa constant of proportionality).

2.6 The fundamental nature of Habc 17

(see [9], eq. 2.23) which reduces the number of independent components of Habc from20 to 16.

We can further restrict the form of Habc by introducing the “divergence-free gaugecondition”

H cab ;c = 0 (2.41)

(see [9], eq. 3.15). These six additional gauge conditions do not respect the generalform of the variation equations (2.33) and (2.35), and reduce the number of indepen-dent components from 16 to 10. However, for certain explicit forms of L (2.35) isoverdetermined, and we can use (2.41) if we are only concerned with solving (2.35)and disregard (2.33).

2.6 The fundamental nature of Habc

In his exploration of the self-dual tensor, Lanczos revisits the scalar invariant K =Rabcd*R*abcd, which he had previously shown in [6] to not generate field equations inaction principles for four-dimensional geometries as its variation vanished identically.However, Lanczos makes the realization that it is “exactly for this reason we havehere a variational property which characterizes all Riemannian geometries of fourdimensions.”

We can thus make the choice L = 18Rabcd*R*abcd without placing constraints on

the resultant geometry. We find

∂L∂*R*abcd = 1

4Rabcd (2.42)

(see [9], eq. 3.2-3). Inserting this into (2.35), and using symmetries of the Riemanntensor, we obtain

Rabcd =[Habc;d − (Qac − qgac) gbd

](2.43)

(see [9], eq. 3.4). Multiplying (2.43) through by gad and performing the contractiongives the Ricci-Lanczos equations:

Rbc = Habc;a −Ha

ba;c +H ac a;b −H a

c b;a − 2Qbc + 6qgbc (2.44)

(see [11], eq. 4.26), and after contracting through again by gbc we find

R = 4Habb;a + 24q (2.45)


(see [11], eq. 4.27). Rearranging (2.44) and using (2.45) we arrive at

Qbc − qgbc = H i(bc);i +H i

(c|i| ;b) − 13H

ijj;igbc − 1

2Rbc + 112Rgbc. (2.46)

We can substitute (2.46) into (2.43) to find the following expression for the Riemanntensor, also known as the Riemann-Lanczos equations:

Rabcd = Habc;d +Hcda;b +Hbad;c +Hdcb;a

+(H i

(ac);i +H i(a|i| ;c)

)gbd

+(H i

(bd);i +H i(b|i| ;d)

)gac

−(H i

(ad);i +H i(a|i| ;d)

)gbc

−(H i

(bc);i +H i(b|i| ;c)

)gad

− 23H

ijj;i (gacgbd − gadgbc)

− 12 (Racgbd +Rbdgac −Radgbc −Rbcgad)


(see [11], eq. 4.28). Employing the trace-free and divergence-free gauge conditions (2.40)and (2.41) simplifies these equations to the following form:

Rabcd = Habc;d +Hcda;b +Hbad;c +Hdcb;a

+H iac;igbd +H i

bd;igac

−H iad;igbc −H i

bc;igad

− 12 (Racgbd +Rbdgac −Radgbc −Rbcgad)


(see [11], eq. 4.35). Lanczos revisited the splitting of the Riemann tensor (2.20),balancing the splitting by defining

Uabcddef= Aabcd + 1

6R (gacgbd − gadgbc) (2.49)

Vabcddef= Sabcd − 1

6R (gacgbd − gadgbc) (2.50)

so that Rabcd = Uabcd + Vabcd , with each Uabcd and Vabcd having 10 independent com-

2.7 The Weyl-Lanczos equations 19

ponents. A similar expression to (2.21) can be found for Uabcd by simply plugging thedefinition (2.49) into (2.21) to obtain

Uabcd =(Rac − 1

6Rgac

)gbd +

(Rbd − 1

6Rgbd

)gac

−(Rad − 1

6Rgad

)gbc −

(Rbc − 1

6Rgbc

)gad (2.51)

(see [9], eq. 3.12). Multiplying (2.47) by a factor of two and using the fact that2Rabcd = Uabcd + Vabcd and subtracting (2.51), we can find an expression for Vabcd :

Vabcd = 2(Habc;d +Hcda;b +Hbad;c +Hdcb;a

+(H i


)gbd

+(H i


)gac

−(H i


)gbc

−(H i


)gad

− 23H

ijj;i (gacgbd − gadgbc)). (2.52)

Although Habc had appeared before in Lanczos’ 1949 investigation of a theory ofinfinitesimal fields, he shows here that it holds fundamental significance to all Rie-mannian geometries of four dimensions. Lanczos found that Habc is a “generatingfunction” of Vabcd . Lanczos made the realization that “the tensor Vabcd contains ex-actly those components of the full Riemann tensor which are not reducible to thecontracted tensor Rab.” This highlights the fundamental nature of Habc as it con-tains the exact information necessary to complete the description of the curvaturecomponents of four-dimensional geometries provided gab and Rab.

2.7 The Weyl-Lanczos equations

Although Lanczos realized the fundamental significance of (2.52), it wasn’t untilTakeno investigated further in 1964 [16] that it was realized that Vabcd was in factproportional to the Weyl tensor, Cabcd . We use the Ricci decomposition (1.21) of the


Riemann tensor along with (2.47) to obtain an expression for the Weyl tensor,

Cabcd = Habc;d +Hcda;b +Hbad;c +Hdcb;a

+(H i


)gbd

+(H i


)gac

−(H i


)gbc

−(H i


)gad

− 23H

ijj;i (gacgbd − gadgbc) (2.53)

(see [11], eq. 4.30), known as the Weyl-Lanczos equations. From these it is clearthat Vabcd = 2Cabcd . If we again employ the trace-free and divergence-free gaugeconditions (2.40) and (2.41), the Weyl-Lanczos equations simplify to

Cabcd = Habc;d +Hcda;b +Hbad;c +Hdcb;a

+H iac;igbd +H i

bd;igac

−H iad;igbc −H i

bc;igad (2.54)

(see [11], eq. 4.36).

Chapter 3Spinor algebra and analysis

In the context of theoretical physics, spin-vectors are often first defined as a represen-tation of an orthogonal group (such as the rotations or the Lorentz group) such thatthey extend the properties of vectors in a natural way. This is traditionally motivatedby the fact that the properties arising from this representation naturally occur in thedescription of physical phenomena, such as Dirac fermions. In this chapter a verybrief introduction to the algebra and analysis of two-component spinors is given. Fora more rigorous and complete description of the two-spinor formalism, one should lookat Penrose and Rindler [13] and O’Donnell [11].

3.1 Spinor algebra

3.1.1 Spin-vectors and spin-space

A real valence (r; s) tensor Aa1...arb1...bs

at a point p in a manifold is often thought of asa multi-linear map from r vectors in Tp (Minkowski space for a Lorentzian manifold)and s co-vectors in T ∗

p to R. We can construct algebras analogous to the tensor algebraby changing the vector space to which the vectors belong, rather than simply choosingit to be Tp.

We define spin space S as a two-dimensional vector space over C which possessesa bi-linear, skew-symmetric and non-degenerate inner-product (see [11], section 1.5).Elements of S are known as spin-vectors. If we consider spin-vectors ζ,η,θ,φ ∈ S

and λ ∈ C, then the properties of the inner product [ , ] directly imply the following

22 Spinor algebra and analysis

relations:

[ζ,η] = − [η, ζ] , (3.1)

λ [ζ,η] = [λζ,η] , (3.2)

[ζ + η,φ + θ] = [ζ,φ] + [ζ,θ] + [η,φ] + [η,θ] . (3.3)

Consider two spin-vectors ζ,η ∈ S which are linearly related by complex factor λ,that is ζ = λη. It is clear from (3.1) and (3.2) that

[ζ,η] = 0. (3.4)

Let (o, ι) be a spin-basis for S where o and ι are arbitrary spin-vectors, under thecondition

[o, ι] = 1 (3.5)

(see [11], eq. 1.42). We can expand any spin-vector η on this basis

η = η0o + η1ι (3.6)

where η0 = [η, ι] and η1 = [η,o] are the components of η in this spin basis. In termsof components, the inner product takes the form

[ζ,η] = ζ0η1 − ζ1η0 (3.7)

(see [11], eq. 1.46).

3.1.2 Spin transformations

We consider now general linear transformations of the components of a spin-vector

ζ =ζ0

ζ1

:

ζ0 → ζ0 = aζ0 + bζ1

ζ1 → ζ1 = cζ0 + dζ1 (3.8)

(see [11], eq. 1.29), where a, b, c, d ∈ C and ad − bc = 1 (so that the inner product ispreserved). These transformations are known as the spin transformations and can be

3.1 Spinor algebra 23

written in matrix form as ζ0

ζ1

=a b

c d

ζ0

ζ1

. (3.9)

These matrices form a faithful representation of the group SL(2,C), and the spin-vectors can be thought of as a representation space of this transformation group.

3.1.3 General valence spinors

We now introduce the dual spin-space S∗. For every spin-vector η ∈ S we can con-struct the map [η, ] ∈ S∗, which we call spin-co-vectors. Similarly to how tensors ofvalence (p; q) are constructed as maps from p co-vectors and q vectors to the scalars,we can construct valence (p; 0; q; 0) spinors as bilinear maps from p spin-co-vectorsand q spin-vectors to C.

We now introduce the operation of complex conjugation of a spinor index. Werepresent the complex conjugate of a spin-vector η ∈ S by η ∈ S. Similarly, we canconjugate spin-co-vectors [η, ] ∈ S∗ simply by dualising the corresponding element inS, that is [η, ] = [η, ].

With complex conjugation defined, we can define the most general valence (p, q; r, s)spinor as a bilinear map from p spin-co-vectors, q conjugate spin-co-vectors, r spin-vectors, and s conjugate spin-vectors to the complex numbers. We introduce abstractindex notation for spinors analogously as we did for tensors, however rather thanlower-case letters, upper-case letters represent spinor indices. If an index is primed, itrepresents a conjugate index. Hence a (p, q; r, s) spinor, M , can be represented by thenotation M

A1...ApB1′...Bq′

C1...CrD1′...Ds′ . In the case of two-spinors, all indices run over

the set {0, 1}.

3.1.4 Levi-Civita (ϵ−) spinors

The properties of the inner product (3.1), (3.2), and (3.3) ensure the existence of aspinor ϵAB, called the Levi-Civita spinor, such that for any two spin-vectors ζ,η ∈ S

we have[ζ,η] = ϵABζ

AηB (3.10)

[η, ζ] = −ϵABζBηA (3.11)

(see [11], eq. 2.13-4), from which it is obvious that ϵAB = −ϵBA.


The Levi-Civita spinor plays an analogous role to the metric tensor gab. We canuse ϵAB to lower indices as is apparent from (3.10) or (3.11) so that ϵABζ

A is the dualof ζB and thus

ζB = ϵABζA (3.12)

(see [11], eq. 2.16). We can thus write the inner product [ζ,η] in index form as ζAηA,

or as ζ0η0 + ζ1η

1 in an explicit spin basis. The Levi-Civita spinor with indices upstairsis similarly defined and raises indices, for example

ζA = ϵABζB (3.13)

(see [11], eq. 2.20). If we insert (3.12) into (3.13) we find

ζA = ϵCBϵABζC = δA

CζC (3.14)

where δAC is the usual Kronecker delta, and so we arrive at the following properties

ϵABϵAC = δ BC = ϵ B

C , (3.15)

−ϵCBϵBA = δCA = −ϵC

A (3.16)

(see [11], eq. 2.21-2). We can thus replace the Kroneker delta δ BA = δB

A with theLevi-Civita spinor ϵ B

A , if we respect the anti-symmetry properties

ϵ BA = −ϵB

A, (3.17)

ϵAB = −ϵBA (3.18)

(see [11], eq. 2.23-4).

3.1.5 Spinor dyad bases

We introduced the idea of a spin basis above in (3.5), however the relationship betweenthe basis spin-vectors can now be written equivalently using (3.10) as

oAιA = 1, ιAo

A = −1,

oAoA = 0, ιAι

A = 0, (3.19)


(see [13], eq. 2.5.39-41). We say that the two basis spinors, oA, ιA form a dyad in spin-space in analogy to how the four basis vectors ta, xa, ya, za form a tetrad in Minkowskispace. We can write the dyad oA, ιA collectively as ϵ A

A , where

ϵ A0 = oA, ϵ A

1 = ιA (3.20)

(see [13], eq. 2.5.44). The dual basis, ϵ AA , must satisfy the condition

ϵ AA ϵ B

A = ϵ BA =

1 00 1

(3.21)

(see [13], eq. 2.5.47). The components of a spinor with respect to a dyad can be foundusing ϵ A

A , for example the components of ϵAB are

ϵAB = ϵABϵA

A ϵ BB =

0 oAιA

oAιA 0

(3.22)

(see [13], eq. 2.5.45).

3.1.6 Spinor representation of tensors

The algebra of tensors is embedded in the spinor algebra we have introduced. Thisis directly related to the fact that there exists a homomorphism between the lineartransformation group of spin-vectors, SL(2,C), and that of vectors, L(4), althoughbecause this is a homomorphism (and not an isomorphism) there is not always atensorial equivalent of a spinorial object (although the spinor equivalent of a tensorcan always be found). If we represent tensors in terms of their equivalent spinors, weshall find that many complicated tensorial expressions are simplified by the additionalstructure present in the spinor algebra.

We introduce the Infeld-van der Waerden symbols, σaAB′ , as the objects which

connect tensorial indices to their spinor counterparts. Here the index a is a tensorialindex which runs over the tetrad in the tangent space, and A,B′ are spinor indiceswhich run over the spinor dyad. Leaving the spinor indices free, each tensorial com-ponent is defined as a 2 × 2 Hermitian matrix which in a normalised spin-basis (asdefined in (3.5)) are the identity matrix and the three Pauli matrices (up to a factor


1√2):

σ0AB′ = 1√

2

1 00 1

, σ1AB′ = 1√

2

0 11 0

,σ2

AB′ = 1√2

0 i

−i 0

, σ3AB′ = 1√

2

1 00 −1

(3.23)

(see [11], eq. 2.69). The Infeld-van der Waerden symbols are an explicit description ofthe group homomorphism described above. From their form it is clear that a vectorcan be represented by a spin-vector and a conjugate spin-vector. Similarly, any tensorindex is matched to an unprimed and primed spinor index.

We now give an example of how the relationship between tensors and spinors isrealized explicitly. If T b

a is a valence (1; 1) tensor, then its spinor-equivalent is

T ba = T BB′

AA′ = T ba σa

AA′σ BB′

b (3.24)

where we have used the Infeld-van der Waerden symbols with spinorial indices raisedusing the Levi-Civita spinor, and tensorial indices lowered using the metric tensor.

Through explicit calculation, one can verify the following equations:

σ AA′

a σbAA′ = δ b

a , (3.25)

σaAA′σ BB′

a = ϵ BA ϵ B′

A′ (3.26)

(see [11], 2.74-5). The first property (3.25) is just the statement that the Pauli matrices(and identity) form an orthonormal basis for the faithful representation of SL(2,C).The second property (3.26) is that the four world-vectors obtained by keeping A andA′ fixed and allowing the tensorial index to vary, form an orthonormal basis for thefaithful representation of L(4).

We define the metric tensor in terms of the Infeld-van der Waerden symbols as

gab = ϵABϵA′B′σ AA′

a σ BB′

b (3.27)

(see [11], 2.73), from which the explicit relationship between the Levi-Civita spinorand the metric tensor can be found by applying (3.25) and (3.26) to (3.27), obtaining

gabσa

AA′σbBB′ = ϵABϵA′B′ (3.28)


(see [11], 2.76).From this point forward omission of explicit Infeld-van der Waerden symbols is

standard and it is understood when tensorial indices are replaced by spinor indices (orvice-versa) that Infeld-van der Waerden symbols have been invoked.

3.1.7 Symmetry operations

One of the chief reasons which cause spinor equivalents of tensorial expressions tobe more elegant or natural algebraically is due to the symmetry properties of two-dimensional spin-space. Due to these symmetries, any spinor which is anti-symmetricin three or more primed or unprimed indices vanishes identically. That is, for anyspinor of the form AAIJK or BAI′J ′K′ , where the index A represents an arbitrary setof spinorial indices, we have

AA[IJK] = 0, BA[I′J ′K′] = 0 (3.29)

(see [13], eq. 3.3.24). This property is due to the fact that spin-space is two-dimensional; hence in any set of three anti-symmetrised indices at least two of theindices must refer to the same dyad component. Let us consider the particular caseof (3.29) for the Levi-Civita spinor

ϵA[BϵCD] = 0 (3.30)

(see [11], eq. 2.54), which we can rewrite as

ϵABϵCD + ϵACϵDB + ϵADϵBC = 0 (3.31)

(see [11], eq. 2.55). After raising the indices C and D and rearranging we have

ϵ CA ϵ D

B − ϵ CB ϵ D

A = 2ϵ C[A ϵ D

B] = ϵABϵCD (3.32)

(see [11], eq. 2.56).Another convenient property of spinors is that they can, in a sense, be reduced to

symmetric spinors. Contracting (3.32) through with an arbitrary spinor ΦACD, gives

2ΦA[AB] = Φ CAC ϵAB (3.33)


(see [13], eq. 2.5.24). We can then write the spinor ΦAAB as

ΦAAB = ΦA(AB) + ΦA[AB] = ΦA(AB) + 12Φ C

AC ϵAB (3.34)

(see [11], eq. 2.59-60).We define the equivalence relation ∼ between two spinors if their difference is an

outer-product of Levi-Civita spinors and symmetric spinors of lower valence than theoriginals. We first show that

ΦAA...Z ∼ ΦA(A...Z) (3.35)

holds for each ΦAA...Z . We express the symmetrisation of ΦA(A...Z) as

ΦA(A...Z) = 1n

(ΦAA(BC...Z) + ΦAB(AC...Z) + · · · + ΦAZ(AB...Y )

)(3.36)

(see [13], eq. 3.3.50). If we consider the difference between the first term and any otherterm in the bracketed expression on the right-hand-side of (3.36), and using (3.34),we find

ΦAC(AB...Z) = ΦAA(BC...Z) + ϵACΦ XA (BX...Z) (3.37)

(see [13], eq. 3.3.51). If we substitute this form of ΦAC(AB...Z) into (3.36) and repeatfor all other terms on the right-hand-side, we establish the equivalence

ΦA(AB...Z) ∼ ΦAA(B...Z) (3.38)

(see [13], eq. 3.3.52). We can now absorb the index A on the right-hand-side of (3.38)into the index set A and repeat the argument to obtain

ΦA(AB...Z) ∼ ΦAA(B...Z) ∼ ΦAAB(C...Z) ∼ · · · ∼ ΦAABC...Z (3.39)

establishing the desired result. This argument is equally valid for sets of primedindices, and if we have an arbitrary spinor we have the general result

ΦA′B′C′...F ′P QR...Z ∼ Φ(A′B′C′...F ′)(P QR...Z). (3.40)

Furthermore, although the algorithm above was done on a spinor with all indicesdownstairs, the argument is general as we may simply lower all the indices of a mixed


valence spinor and perform the algorithm.The decomposition above is actually the decomposition of general spinors into the

direct sum of irreducible representations of SL(2,C), which are in fact the spacesof symmetric spinors. Another way of expressing this irreducibility of completelysymmetric spinors is by realizing that if we impose any additional (anti-)symmetry,we either find the symmetry is redundant or we destroy the spinor.

We give an example of the decomposition to help elucidate the idea of the reductionto symmetric spinors. Consider the spinor ΦABA′B′ . We can decompose it as follows

ΦABA′B′ = Φ(AB)(A′B′)

− 12ϵABΦC

C(A′B′)

− 12ϵA′B′Φ C′

(AB) C′

+ 14ϵABϵA′B′ΦC C′

C C′ (3.41)

(see [13], eq. 3.3.56).

3.1.8 Tensor equivalents of spinor operations

It is clear from the way we have constructed the connection between tensors and spinorsthat every algebraic tensor operation has a spinor analogue. However if we have aspinor associated with some tensor, the richer spinor algebra allows certain tensorialoperations to be represented elegantly. The spinorial operations of exchanging twounprimed or primed indices do not have a simple tensorial counterpart at first glance,and we find that such operations arise much more naturally when working in the spinorframework.

Trace reversal over symmetric indices

Consider an arbitrary symmetric tensor of valence (0; 2), Tab = Tba . We can convertthis to the spinor formalism as

TAA′BB′ = TBB′AA′ (3.42)


(see [13], eq. 3.4.2). Adding and subtracting TABBA from the right-hand-side andrearranging1 gives

TABA′B′ = 12 (TABA′B′ + TABB′A′ ) + 1

2 (TBAB′A′ − TABB′A′ ) (3.43)

(see [13], eq. 3.4.3). The first term on the right-hand-side is symmetric in A and B

and in A′ and B′ through (3.42), and can be written as T(AB)(A′B′) . The second termon the right-hand-side is anti-symmetric in A and B and in A′ and B′ by (3.42), andcan be written as T[AB][A′B′] . Using (3.33) twice on each set of antisymmetric indicesin the second term, we get2

Tab = TABA′B′ = SABA′B′ + ϵABϵA′B′τ (3.44)

(see [13], eq. 3.4.4), where SABA′B′def= T(AB)(A′B′) and τ

def= 14T

CC′CC′ = 1

4Tc

c . We canrewrite (3.44) in tensor form as

Tab = Sab + gabτ (3.45)

(see [13], eq. 3.4.7), which is the canonical decomposition of a tensor into its trace-freepart Sab and its trace τ .

We perform the operation of the trace reversal to Tab , effectively leaving the trace-free part invariant and negating the trace, by

Tabdef= Tab − 1

2Tc

c gab (3.46)

(see [13], eq. 3.4.10). In spinor form we have

Tab = TAA′BB′ = SABA′B′ − ϵABϵA′B′τ (3.47)

(see [13], eq. 3.4.12) which upon comparison with (3.44), and recalling the anti-symmetry of the Levi-Civita spinor gives the simple relation

TAA′BB′ = TBAA′B′ = TABB′A′ (3.48)1Note that primed and unprimed indices may pass through each other with no effect. Only the

exchange of (un)primed indices with other (un)primed indices has a non-trivial effect.2Note that this decomposition is equivalent to the one presented in (3.41) with additional sym-

metry.


(see [13], eq. 3.4.13). We thus see that trace reversal applied to a pair of symmetrictensorial indices a and b is realized in the spinor formalism simply by interchangingthe spinor indices A and B or equivalently interchanging A′ and B′.

Dualisation over anti-symmetric indices

Consider now an arbitrary anti-symmetric tensor of valence (0; 2), Fab = −Fba , oftencalled a bivector. We can convert this to the spinor formalism as

FAA′BB′ = −FBB′AA′ (3.49)

(see [13], eq. 3.4.15). Adding and subtracting FABB′A′ from the right-hand-side andrearranging gives

FABA′B′ = 12 (FABA′B′ − FABB′A′ ) + 1

2 (FABB′A′ − FBAB′A′ ) (3.50)

(see [13], eq. 3.4.16). The first term on the right-hand-side is anti-symmetric in A′ andB′ and so we can apply (3.33) to rewrite it as 1

2FC′

ABC′ ϵA′B′ . Similarly, the secondterm on the right-hand-side is anti-symmetric in A and B and through applying (3.33)we can rewrite it as 1

2FC

C A′B′ ϵAB. Rewriting (3.50) in terms of these decompositions,we have

FABA′B′ = 12F

C′

ABC′ ϵA′B′ + 12F

CC A′B′ ϵAB (3.51)

(see [13], eq. 3.4.17).The Hodge dual of a bivector Fab is given by

*Fabdef= 1

2ϵcd

ab Fcd (3.52)

(see [13], eq. 3.4.21), where ϵ cdab is the alternating tensor which has spinor form

ϵ cdab = iϵ C

A ϵ DB ϵ D′

A′ ϵ C′

B′ − iϵ DA ϵ C

B ϵ C′

A′ ϵ D′

B′ (3.53)

(see [13], eq. 3.3.44). Applying (3.52) and (3.53) to (3.51) gives

*Fab = *FABA′B′ = −i12FC′

ABC′ ϵA′B′ + i12F

CC A′B′ ϵAB (3.54)


(see [13], eq. 3.4.22), from which it is clear upon comparison with (3.51) that

*FABA′B′ = iFABB′A′ = −iFBAA′B′ (3.55)

(see [13], eq. 3.4.23). Hence we see that dualisation of a pair of anti-symmetrictensorial indices a and b is realized in the spinor formalism simply by interchangingthe spinor indices A and B and multiplication by −i or equivalently interchanging A′

and B′ and multiplication by i.

Procedure over a general pair of indices

Now suppose we have an arbitrary world-tensor of valence (0; 2), Gab,with no symmetryin a and b. We can decompose Gab into its symmetric and anti-symmetric parts:

GAA′BB′ = G(ab) +G[ab]. (3.56)

Using (3.48) and (3.55) on the symmetric part and antisymmetric part of Gab respec-tively, we arrive at

GBA′AB′ = G(ab) + i*G[ab], (3.57)

GAB′BA′ = G(ab) − i*G[ab] (3.58)

(see [13], eq. 3.4.51-2). If we write these expressions out explicitly in tensorial form,we have

GBA′AB′ = 12(Gab +Gba −G c

c gab + iϵabcdGcd), (3.59)

GAB′BA′ = 12(Gab +Gba −G c

c gab − iϵabcdGcd)

(3.60)

(see [13], eq. 3.4.53-4). The tensorial complexity which arises from the simplestspinor operation, namely the exchange of two indices, is remarkable. Moreover, theoperations of trace reversal and dualisation appear with high frequency in physics,often allowing the spinor formalism to be a more ‘natural’ setting to derive and expressphysical expressions.

3.2 Spinor analysis 33

3.2 Spinor analysis

So far we have introduced the algebra of spinors at a point on a manifold. If we wantto express geometric theories, such as general relativity, in a spinor formalism we mustdefine some sort of covariant derivative on the manifold by which spinors at differentpoints can be related.

3.2.1 Spinor covariant derivative

We define the spinor covariant derivative ∇a = ∇AA′ , in analogy to the tensor covariantderivative in section 1.2, by a linear map from the (p, q; r, s) spinors to the (p, q; r +1, s + 1) spinors which satisfies the following properties, given an arbitrary valence(p, q; r, s) spinor T C1...CpD1′...Dq

′

A1...ArB1′...Bs′ :

1. Leibniz rule as in (1.2).

2. Commutation with index substitution.

3. Commutation with contraction as in (1.3), however noting that in the spinorcase the contraction may be over a single pair of spinor indices and is performedwith the Levi-Civita spinor ϵ D

C or ϵ D′C′ (rather than the metric tensor).

4. Commutation with complex conjugation:

∇aTC1...CpD1′...Dq

′

A1...ArB1′...Bs′

= ∇aTC1...CpD1′...Dq

′

A1...ArB1′...Bs′ (3.61)

(see [11], eq. 3.7).

5. The Levi-Civita spinor (rather than the metric tensor in (1.4)) is covariantlyconstant:

∇aϵBC = 0 and ∇aϵBC = 0 (3.62)

(see [11], eq. 3.9).

6. The torsion of the connection vanishes as in (1.5).

7. The spinor covariant derivative acts as an ordinary partial coordinate derivativeon scalars as in (1.6).


3.2.2 The curvature spinors

Consider the spinor equivalent of the Riemann tensor:

RAA′BB′CC′DD′ = Rabcd (3.63)

(see [11], eq. 3.15). Recalling the anti-symmetry of the Riemann tensor in a and b

in (1.10) and using the splitting in (3.50), we have

Rabcd = 12 (RABA′B′CDC′D′ −RABB′A′CDC′D′)

+ 12 (RABB′A′CDC′D′ −RBAB′A′CDC′D′)

= RAB[A′B′]CDC′D′ +R[AB]B′A′CDC′D′ (3.64)

(see [11], eq. 3.16). Using the anti-symmetry in the second set of indices c and d

in (1.11) and performing the same procedure to each term in (3.64) leads to

Rabcd = RAB[A′B′]CD[C′D′] +RAB[A′B′][CD]C′D′

+R[AB]B′A′CD[C′D′] +R[AB]B′A′[CD]C′D′ (3.65)

(see [11], eq. 3.19). We can then apply (3.33) twice to each term in (3.65) to get

Rabcd = XABCD ϵA′B′ϵC′D′ + ΦABC′D′ϵA′B′ϵCD

+ ΦA′B′CDϵABϵC′D′ +XA′B′C′D′ϵABϵCD (3.66)

(see [11], eq. 3.20), where XABCD and ΦABC′D′ are defined as

XABCDdef= 1

4RE′ F ′

ABE′ CDF ′ , (3.67)

ΦABC′D′def= 1

4RE′ F

ABE′ F C′D′ (3.68)

(see [11], eq. 3.21-2). XABCD and ΦABC′D′ are referred to as curvature spinors andcontain the complete information of the original Riemann curvature tensor.

3.2 Spinor analysis 35

3.2.3 The Weyl spinor

Let us now investigate the properties of XABCD further. Under simultaneous exchangeof A and B as well as A′ and B′3, we have anti-symmetry in the Riemann tensor asin (1.10); contracting over one pair of these indices forces symmetry in the other. Like-wise, exchange symmetry in the index sets AA′BB′ and CC ′DD′ translates throughto the curvature spinors directly.

Hence, XABCD has the symmetries:

XABCD = X(AB)(CD) , (3.69)

XABCD = XCDAB (3.70)

(see [11], eq. 3.23, 3.25). Investigating the manifestation of the cyclic symmetry (1.14)in the curvature spinor XABCD yields the following property4:

X BABC = 3ΛϵAC (3.71)

(see [11], eq. 3.36), where Λ def= 16X

ABAB is proportional to the trace of XABCD . Hence,

after contracting over the second and fourth indices, we are left with a completely anti-symmetric spinor5.

In an attempt to decompose the anti-symmetric parts of XABCD , we expand usingthe following identity6:

XABCD = X(ABCD) +X[ABCD]

+ 1112XABCD − 1

12(XACDB +XADBC +XBCAD +XBADC

+XBDCA +XCDAB +XCABD +XDACB

+XDCBA +XDBAC ) (3.72)

(see [11], eq. 3.54). Clearly, X[ABCD] = 0 by (3.69), as anti-symmetrising a pair of3This also holds when exchanging both C and D as well as C ′ and D′.4To derive this one should express the cyclic identity in terms of the right dual of the Riemann

tensor as R* cbab = 0 and find the corresponding spinor equation in terms of the curvature spinors.

One finds that ΦABC′D′ drops out of the expression due to its symmetries (see [11], sec. 3.2.1).5Note that this is not violating the rule of decomposition of anti-symmetric spinors into symmetric

spinors as here we have the direct product of the completely symmetric zero valence spinor Λ withϵAC .

6One can verify this identity by simply expanding the terms X(ABCD) and X[ABCD] .


symmetric indices annihilates the object. Similarly, (3.69) gives X(ABCD) = XA(BCD) ,as symmetrising an index set with a member of a symmetric index set is equivalentto simply symmetrising over the union of both index sets. With this, (3.72) can besimplified to

XABCD = XA(BCD) + 13(XABCD −XACBD ) + 1

3(XABCD −XADCB )

= XA(BCD) + 13XA[BC]D + 1

3XA[B|C|D] (3.73)

(see [11], eq. 3.56), which is now in a form where we can apply (3.33) to arrive at

XABCD = XA(BCD) + 13ϵBCX

EAE D + 1

3ϵBDXE

AEC (3.74)

(see [11], eq. 3.57). Applying (3.71) to (3.74), we can complete the decomposition as

XABCD = ΨABCD + Λ (ϵBCϵAD + ϵBDϵAC) (3.75)

(see [11], eq. 3.58), where ΨABCD is defined as

ΨABCDdef= XA(BCD) (3.76)

and is often called the Weyl spinor or gravitational spinor.Rewriting (3.66) using the Weyl spinor to yields

Rabcd = ΨABCDϵA′B′ϵC′D′ + ΨA′B′C′D′ϵABϵCD

+ ΦABC′D′ϵA′B′ϵCD + ΦA′B′CDϵABϵC′D′

+ 2Λ (ϵACϵBDϵA′C′ϵB′D′ − ϵADϵBCϵA′D′ϵB′C′) . (3.77)

This decomposition is the spinor equivalent of the Ricci decomposition (1.21), andthrough further analysis it is possible to show the equivalence of the Weyl tensor andthe Weyl spinor as

Cabcd = ΨABCDϵA′B′ϵC′D′ + ΨA′B′C′D′ϵABϵCD (3.78)

(see [11], eq. 3.61).

3.3 Spin coefficients 37

3.3 Spin coefficients

The spin-coefficient formalism, also known as the Newman-Penrose formalism, is anotation equivalent to the spinor formalism often used numerical relativity. Althoughsomewhat beyond the scope of this essay, we shall provide a brief summary of theformalism here.

A set of four null vectors is chosen: two real null vectors and two complex-conjugates. The idea is that the information in tensors in the theory are projectedonto this null tetrad. In essence, the spin-coefficient formalism gives the relationshipbetween:

1. Twelve complex spin-coefficients which store information about the directionalcovariant derivatives along the tetrad vectors.

2. Five complex functions which store information about the Weyl tensor in thetetrad basis.

3. Four real functions and three complex functions (and their conjugates) whichstore information about the Ricci tensor in the tetrad basis.

Although seemingly complicated and arduous, working with a specific null tetradto exploit the symmetries of the space-time can cause several of the spin-coefficientsto be trivial. Working in the spin-coefficient formalism is often a preferred method innumerical relativity. The reader is suggestion to consult section 3.5-6 of [11] for moreinformation on the spin-coefficient formalism.

Chapter 4The Lanczos spinor

We saw in the Lanczosian development of Habc in chapter 2 that the Lanczos tensor isa fundamental quantity in four-dimensional Riemannian geometry. Shortly after thisrealization was made, it was shown that in fact the Lanczos tensor was a potentialfor the Weyl tensor through the Weyl-Lanczos equations (2.54). However, these ex-pressions are quite complicated and it can be hard to work with them analytically.Furthermore, suppose we wish to solve these equations numerically for the quantityHabc ; the Weyl-Lanczos equations are 16 non-linear equations in 16 unknowns in aconvoluted form, and are thus computationally expensive.

In this chapter we make use of the properties of the 2-spinor formalism introducedin chapter 3 to express the theory of the Lanczos potential in spinor form, closelyfollowing the procedure in O’Donnell [11].

4.1 Lanczos spinor and decomposition

We convert the Lanczos tensor Habc to spinor form

HAA′BB′CC′ = Habc (4.1)

in the usual way, omitting the explicit invocation of the Infeld-van der Waerden sym-bols. The Lanczos tensor has an anti-symmetry in the a and b index, which is expressedin the spinor formalism as

HAA′BB′CC′ = −HBB′AA′CC′ (4.2)

40 The Lanczos spinor

(see [11], eq. 4.43). We split (4.2) using (3.50) as

Habc = HABA′B′CC′

= 12 (HABA′B′CC′ −HABB′A′CC′ )

+ 12 (HABB′A′CC′ −HBAB′A′CC′ ) (4.3)

(see [11], eq. 4.44). The first term on the right-hand-side is anti-symmetric in A′ andB′, which when combined with (4.2) indicates symmetry in A and B. The secondterm on the right-hand-side is anti-symmetric in A and B, which similarly indicatessymmetry in A′ and B′. We can thus write Habc as

Habc = HABA′B′CC′ = H(AB)[A′B′]CC′ +H[AB](B′A′)CC′ (4.4)

(see [11], eq. 4.45). Recalling the identity (3.33) for a pair of anti-symmetric spinorindices, we can rewrite each term in (4.4) as the direct product of a Levi-Civita spinorand a spinor of lower valence, and hence (4.4) can be rewritten as

Habc = HABA′B′CC′ = 12ϵA′B′H D′

(AB)D′ CC′ + 12ϵABH

DD (B′A′)CC′ (4.5)

(see [11], eq. 4.47).We now define the spinors HABCC′ and φA′B′CC′ for convenience as

HABCC′ = H(AB)CC′def= 1

2HD′

(AB)D′ CC′ , (4.6)

φA′B′CC′ = φ(A′B′)CC′def= 1

2HD

D (B′A′)CC′ (4.7)

(see [11], eq.4.48), allowing us to write (4.5) as

Habc = HABA′B′CC′ = ϵA′B′HABCC′ + ϵABφA′B′CC′ (4.8)

(see [11], eq. 4.49). Taking the conjugate of (4.8) and noting the fact that theconjugate of a real tensor is itself gives

Habc = ϵABHA′B′CC′ + ϵA′B′φABCC′ (4.9)

(see [11], eq. 4.50), which when equating with (4.8) gives the consistent relations

4.1 Lanczos spinor and decomposition 41

φA′B′CC′ = HA′B′CC′ and φABCC′ = HABCC′ . Thus, we may write (4.8) in terms ofHABCC′ and its conjugate HA′B′CC′ alone:

Habc = HABA′B′CC′ = ϵA′B′HABCC′ + ϵABHA′B′CC′ (4.10)

(see [11], eq. 4.51). The complete information contained in the Lanczos tensor is alsocontained in the quantity HABCC′ , which we henceforth will refer to as the Lanczosspinor.

So far, we have only imposed the anti-symmetry of the Lanczos tensor in its firsttwo indices, resulting in the following symmetry of the Lanczos spinor:

HABCC′ = H(AB)CC′ . (4.11)

Without imposing additional symmetry, the Lanczos spinor as it stands has 12 inde-pendent complex components, containing the same information as the 24 real inde-pendent components of the Lanczos tensor before imposing (2.26), (2.40), and (2.41).We proceed to find the form of these additional symmetries for the Lanczos spinor.First we investigate the trace-free gauge condition (2.40),

H ba b = Habcg

bc = HAA′BB′CC′ ϵBCϵB′C′ = 0, (4.12)

which after substituting the form of HAA′BB′CC′ in (4.10) gives

H ba b =

(ϵA′B′HABCC′ + ϵABHA′B′CC′

)ϵBCϵB′C′

= −HABCA′ ϵBC −HA′B′C′AϵB′C′ = 0, (4.13)

or, equivalently,−HABCA′ ϵBC = HA′B′C′Aϵ

B′C′ (4.14)

(see [11], eq. 4.53).Now we impose (2.26), which is equivalent to the cyclic property (2.27), in spinor

form as

*Habcgbc =

(−iϵA′B′HABCC′ + iϵABHA′B′CC′

)ϵBCϵB′C′

= −HABCA′ ϵBC +HA′B′C′AϵB′C′ = 0, (4.15)


or, equivalently,HABCA′ ϵBC = HA′B′C′Aϵ

B′C′ (4.16)

(see [11], eq. 4.54) by recalling (3.55). We see that the right-hand-side of (4.14)and (4.16) are equal, and thus equating the left-hand-sides implies

H DAD A′ = 0 (4.17)

(see [11], eq. 4.55). Once again calling upon the identity (3.33), although this time inreverse, we rewrite (4.17) as

12ϵBCH

DAD A′ = HA[BC]A′ = 0. (4.18)

Since, when anti-symmetrised on indices B and C, HABCA′ vanishes identically, itmust instead be symmetric in B and C, that is

HABCC′ = HA(BC)C′ (4.19)

(see [11], eq. 4.56), which when combined with the original symmetry of the Lanczosspinor (4.11) can be written as

HABCC′ = H(ABC)C′ (4.20)

(see [11], eq. 4.57). At this point, there are eight remaining independent complexcomponents of the Lanczos spinor which match the 16 real components of the Lanczostensor after imposing the cyclic and trace-free conditions. We do not yet imposethe divergence-free condition, but note that it has the following form in the spinorformalism1:

H cab ;c = ∇CC′

(H CC′

AB ϵA′B′ +H C′CA′B′ ϵAB

)= ϵA′B′∇CC′H CC′

AB + ϵAB∇CC′H C′CA′B′

= 0 (4.21)

(see [11], eq. 4.59). Contracting through with ϵA′B′ and recalling the symmetry of the1The second expression follows since we have assumed the Levi-Civita spinor is covariantly con-

stant in (3.62).

4.2 Spinor form of Weyl-Lanczos equations 43

Lanczos spinor and its conjugate in the first two indices in (4.6) gives

∇CC′H CC′

AB = 0 (4.22)

(see [11], eq. 4.60). Contracting (4.21) through with ϵAB, we find a similar expressionfor the conjugate Lanczos spinor:

∇CC′H C′CA′B′ = 0. (4.23)

4.2 Spinor form of Weyl-Lanczos equations

Assuming the divergence-free gauge condition, however not invoking it explicitly andsimply holding (4.22) and (4.23) as auxiliary conditions, we investigate the form ofthe Weyl-Lanczos equations (2.54) in the spinor formalism, recalling the decomposi-tion (3.78) of the Weyl tensor into the Weyl spinor and its conjugate, as

Cabcd = ΨABCDϵA′B′ϵC′D′ + ΨA′B′C′D′ϵABϵCD

= ∇DD′

(HABCC′ ϵA′B′ +HA′B′C′CϵAB

)− ∇CC′

(HABDD′ ϵA′B′ +HA′B′D′DϵAB

)+ ∇BB′

(HCDAA′ ϵC′D′ +HC′D′A′AϵCD

)− ∇AA′

(HCDBB′ ϵC′D′ +HC′D′B′BϵCD

)− ∇EE′

(H E

A CC′ ϵ E′

A′ +H E′

A′ C′CϵE

A

)ϵBDϵB′D′

− ∇EE′

(H E

B DD′ ϵ E′

B′ +H E′

B′ D′DϵE

B

)ϵACϵA′C′

+ ∇EE′

(H E

A DD′ ϵ E′

A′ +H E′

A′ D′DϵE

A

)ϵBCϵB′C′

+ ∇EE′

(H E

B CC′ ϵ E′

B′ +H E′

B′ C′CϵE

B

)ϵADϵA′D′ . (4.24)

Contracting through with ϵA′B′ϵC′D′ , recalling that the Weyl spinor and its conju-

gate inherit symmetry in their first and last index pairs from the curvature spinorXABCD and that the Levi-Civita symbol is covariantly constant, leads to the followingsimplification:

2ΨABCD = ∇ E′

D HABCE′ + ∇ E′

C HABDE′

+ ∇ E′

B HCDAE′ + ∇ E′

A HCDBE′ (4.25)


(see [11], eq. 4.58). These are the Weyl-Lanczos equations expressed in the spinorformalism, together with the divergence-free gauge condition (4.22).

We can express the divergence-free condition in a way which can be directly incor-porated into (4.25). We apply (3.33) in reverse to (4.22) to write the gauge conditionas

12ϵDE∇CC′

HABCC′ = ∇ C′

[E H|AB|D]C′ = 0, (4.26)

or equivalently∇ C′

E HABDC′ = ∇ C′

D HABEC′ (4.27)

(see [11], eq. 4.61). Finally, we can use two applications of (4.27) with different indexpermutations to simplify (4.25) to its final form, with both gauge conditions invoked:

ΨABCD = 2∇ E′

D HABCE′ (4.28)

(see [11], eq. 4.62).Although elegant, the final spinor form of the Weyl-Lanczos equations (4.28) are

still not easy to work with. A common approach to solving for the Lanczos spinor ina given space-time is by applying the spin-coefficient formalism, briefly described insection 3.3, to find the Lanczos coefficients. For more information, one should consultsection 4.4-12 in [11].

Chapter 5Interpretation of the Lanczos tensor

We have shown the fundamental significance of the Lanczos tensor in describing four-dimensional Riemannian geometries, however its physical interpretation is often over-looked. In a 2010 review of Lanczos potential theory [12], O’Donnell and Pye statethat the interpretation of the Lanczos potential “remains largely uninvestigated andis also arguably the most important factor of the theory that requires delineation.”

In this chapter, we first explore Zund’s 1975 investigation [18] of the ramificationsof the Weyl-Lanczos equations in the Jordan form of general relativity. We thenbriefly discuss the ideas behind Robert’s 1995 investigation of the possibility of aneffect analogous to the Aharonov-Bohm effect arising in the quantum realm of gravity.

5.1 Jordan form of general relativity

The field equations of general relativity are most commonly expressed as a local de-termination of the Ricci tensor Rab in terms of the stress-energy tensor Tab presentin the space-time. In particular we have a proportionality between the trace-reversedRicci tensor and the stress-energy tensor, known as the Einstein field equations:

Rab − 12gabR = 8πTab . (5.1)

These equations are a set of 10 coupled non-linear partial differential equations in themetric and its first and second derivatives1.

As can be seen from the Ricci decomposition of the Riemann tensor (1.21), the1Note, however, that the contracted Bianchi identity and the fact that the metric is covariantly

constant implies that the Einstein equations are divergenceless, and thus are equivalent to 6 inde-pendent differential equations.

46 Interpretation of the Lanczos tensor

Ricci tensor and Weyl tensor are algebraically independent parts of the curvature, andthus the Weyl tensor is not determined locally by the Einstein equations. However,applying the second Bianchi identity (1.16) to the Ricci decomposition (1.21) leads toa relation between the first derivatives of the Weyl tensor and the first derivatives ofthe Ricci tensor. Hence, the global form of the Weyl tensor is determined indirectlyby the matter distribution.

A once contracted form of the second Bianchi identity can be written using theWeyl tensor as

Cabcd;d = Jabc, (5.2)

where we defineJabc def= Rc[a;b] + 1

6gc[bR;a] (5.3)

(see [4], eq. 4.28-9). Since the Ricci tensor is completely determined locally by thestress-energy tensor, it is possible to rewrite (5.3) fully using Tab and its trace T .Therefore, we can think of (5.2) as the field equations of general relativity encoded asfirst-order differential equations for the components of the Weyl tensor, with sourceJabc. This form of general relativity bears striking resemblance to the first Maxwellequations in terms of the field tensor:

F ab;b = Ja. (5.4)

The second Maxwell equation in terms of the dual field tensor,

*F ab;b = 0, (5.5)

has an analogous expression corresponding to the left dual2 of the Weyl tensor, byrewriting the traceless property (1.22) as

*Cabcd;a = 0, (5.6)

(see [18], eq. 14).The field tensor Fab is generated differentially in terms of the vector potential Aa

asFab = ∇[aAb]. (5.7)

In a similar way, the Weyl-Lanczos equations (2.53) show that the Weyl tensor is2Or, equivalently due to the symmetries of the Weyl tensor, the right dual.

5.2 Gauge tensor candidates 47

generated differentially in terms of the Lanczos tensor Habc .

5.2 Gauge tensor candidates

In a 1975 article entitled The Theory of the Lanczos Spinor [18], Zund observes theabove correspondences between general relativity and electromagnetism and asks theobvious question of whether there also exists a gauge group under which the equationsof general relativity, as they are posed in (5.2), remain invariant; similar to howMaxwell’s equations are invariant under the U(1) transformation:

Aa → A′a = Aa + ∂aφ (5.8)

which leaves the field tensor invariant. More precisely, Zund asks whether there existsa tensor Labc, such that the transformation

Habc → H ′abc = Habc + Labc (5.9)

(see [18], eq. 18) leave the Weyl tensor invariant:

C ′abcd = Cabcd . (5.10)

In his investigation Zund sought forms of Labc that leave the Weyl tensor invariant.One particular form he considered, as the closest analogy to electromagnetism, was

Labc = ∇cψab (5.11)

(see [18], eq. 19). ψab must be a bivector as the Lanczos tensor is anti-symmetric inits first two indices. Furthermore, the cyclic property and trace-free condition become

*ψab;c = 0 and ψab;a = 0, (5.12)

which are equivalent to the source-free Maxwell equations, implying that ψab is asingular bivector. Unfortunately, the divergence-free condition causes difficulties, andthus we make the further assumption that ψab is recurrent, that is

ψab;c = χcψab, (5.13)

48 Interpretation of the Lanczos tensor

(see [18], eq. 20). With this assumption, the conditions (5.12) require the co-vectorfield χc to be a principal null direction of ψab. The divergence free equation thenbecomes, χa

;a = 0. Finally Zund found that C ′abcd = Cabcd if

ψ c[b χa];c = 0, (5.14)

which is satisfied if χa is a parallel null vector field. Hence, if χa is both a parallel nullvector field and principally null with respect to ψab, which itself is a singular recurrentbivector with recurrence vector χa, then (5.11) is a possible form of the gauge field.

There are certainly other forms of Labc which leave the Weyl tensor invariant.Zund concluded, however, that “only a deeper physical study of the gravitational fieldwill indicate what kind of gauge tensor is appropriate.” It will certainly be fruitfulto consider the algebraic structure of the full group of gauge transformations, andconsider possibly restricting it to a subgroup which can be physically motivated bygeneral relativity. Furthermore, considering the Weyl-Lanczos equations in spinorform may lead to simplifications in this analysis.

5.3 Aharonov-Bohm-esque effects

In his paper, entitled The physical interpretation of the Lanczos tensor [15], Robertsgives a potential effect of the presence of the Lanczos potential, analogous to theAharonov-Bohm effect of electrodynamics, occurring in the quantum realm of gravity.The Aharonov-Bohm effect arises from the fact that Maxwell’s equations are cast asa gauge theory, and thus we can replace partial derivatives with a gauge covariantderivative with respect to the gauge group of electrodynamics. Replacing the partialderivatives by gauge covariant derivatives in the Schroedinger equation gives rise toseveral factors of the vector potential, which affects charged particles in the systemdirectly. Consider, for example, a space-time with vanishing field tensor, but non-trivial vector potential. Classically such a scenario would not have any electromagneticeffect on charged particles in the space-time. However, in the quantum realm this isnot the case and the vector potential is shown to be, in a sense, ‘physical’.

Roberts constructed many potential gauge covariant derivatives for the Jordanform of general relativity in terms of the Lanczos potential. In the end, the results ofthe study were inconclusive as to whether the Lanczos tensor may produce an effectsimilar to the Aharonov-Bohm effect.

Conclusion

Although Lanczos failed to realize his intentions of producing a unified geometric the-ory of electromagnetism and gravity, his work in geometric action principles helped elu-cidate a fundamental quantity in four-dimensional Riemannian and pseudo-Riemanniangeometry. The Weyl-Lanczos equations highlight that we should not consider the Weyltensor as being fundamental, but rather the Lanczos tensor should be considered afundamental constituent of four-dimensional Riemannian geometry. Moving to thespinor formalism, we see that the complex relationship of the Weyl-Lanczos equationssimplifies incredibly in terms of the Weyl spinor and the Lanczos spinor, highlightingthe power of simple operations in the spinor formalism in replicating complex tensorialoperations.

It seems that the Lanczos tensor is currently largely unnoticed, possibly in partdue to our knowledge of classical gravity being regarded as somewhat complete whileother fields often recognized as more fundamental or applicable take the stage. How-ever, there is still significant work to be done regarding the Lanczos tensor, such asfurther investigating its role as a potential in the Jordan form of general relativity, orgeneralizing solutions for the Lanczos tensor to arbitrary space-times.

References

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[2] Einstein, A. (1919). Spielen Gravitationsfelder im Aufbau der materiellen Ele-mentarteilchen eine wesentliche Rolle? Preussische Akademie der Wissenshchaften,(1):349–356.

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[10] Nakahara, M. (2003). Geometry, Topology, and Physics. IOP Publishing.

[11] O’Donnell, P. (2003). Introduction to 2-Spinors in General Relativity. WorldScientific.

[12] O’Donnell, P. and Pye, H. (2010). A brief historical review of the importantdevelopments in Lanczos potential theory. Electronic Journal of Theoretical Physics,7(24):327–350.

[13] Penrose, R. and Rindler, W. (1984). Spinors and Space-time; Volume 1: Two-spinor calculus and relativistic fields. Cambridge University Press.

[14] Rainich, G. (1925). Electricity in curved space-time. Nature, 115(2892):498.

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[15] Roberts, M. (1995). The physical interpretation of the Lanczos tensor. Il NuovoCimento B, 110(10):1165–1176.

[16] Takeno, H. (1964). On the spintensor of Lanczos. Tensor, 104:103–119.

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[18] Zund, J. (1975). The theory of the Lanczos spinor. Annali di Matematica Puraed Applicata, 15(1):239–268.

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