Lanczos Potential Theory

March 27, 2018 | Author: Dominic Reiss | Category: Tensor, General Relativity, Theoretical Physics, Differential Geometry, Mathematics


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An Introduction toLanczos Potential Theory Dominic Reiss Department of Applied Mathematics and Theoretical Physics University of Cambridge This essay is submitted as a part of Part III of the Mathematics Tripos Master of Advanced Study Fitzwilliam College May 2, 2014 Overview In this essay, we present a brief review of the development of the theory of the Lanczos potential, a three index tensor which arises as a Lagrange multiplier when considering action principles for four dimensional Riemannian and pseudo-Riemannian geometries. The essay is composed of five chapters, the structure of which we will now give an overview. The first chapter will give a brief introduction to various geometric objects which will be important in our later discussions. To supplement this any text on differential geometry or general relativity, such as Nakahara [10] or Hawking and Ellis [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 Lanczos in geometric action principles from the 1930s up to his landmark paper in 1962. The source material for this chapter can be found in papers by Lanczos [5–9] as well as sections 4.1-3 of O’Donnell [11]. In chapter 3, the algebra and analysis of two component spinors is introduced. Most importantly, the embedding of the tensor algebra within the spinor algebra, and the correspondence between a number of tensor and spinor operations is discussed. This chapter 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 of the Lanczos tensor in the fourth chapter. We continue to find the spinor equivalent of the Weyl-Lanczos equations, finding that they simplify significantly. Here, we follow sections 4.3-4 of O’Donnell [11]. In the final chapter, we investigate the potential physical significance of the Lanczos tensor. Considering the Jordan form of general relativity brings forth many similari- ties between general relativity and electromagnetism, and suggests that the Lanczos tensor may play a similar role in gravity as the electromagnetic vector potential does in electrodynamics. We follow an investigation by Zund [18] which considers transfor- iv mations akin to the U(1) gauge transformations of electrodynamics. We also comment on work by Roberts [15] which suggests that an effect similar to the Aharonov-Bohm effect of electrodynamics may come into play due to the Lanczos potential. Contents Overview iii Contents v 1 Introduction 1 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Covariant derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Riemann curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Weyl tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Historical formulation of the Lanczos tensor 7 2.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Early work in action principles of geometric quantities . . . . . . . . . 8 2.3 Anti-self-dual and self-dual splitting of the Riemann tensor . . . . . . . 12 2.4 Lanczos’ canonical Lagrangian . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Lanczos’ gauge and normalisation conditions . . . . . . . . . . . . . . . 16 2.6 The fundamental nature of H abc . . . . . . . . . . . . . . . . . . . . . . 17 2.7 The Weyl-Lanczos equations . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Spinor algebra and analysis 21 3.1 Spinor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Spinor analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Spin coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4 The Lanczos spinor 39 4.1 Lanczos spinor and decomposition . . . . . . . . . . . . . . . . . . . . . 39 4.2 Spinor form of Weyl-Lanczos equations . . . . . . . . . . . . . . . . . . 43 vi Contents 5 Interpretation of the Lanczos tensor 45 5.1 Jordan form of general relativity . . . . . . . . . . . . . . . . . . . . . . 45 5.2 Gauge tensor candidates . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.3 Aharonov-Bohm-esque effects . . . . . . . . . . . . . . . . . . . . . . . 48 Conclusion 49 References 51 Chapter 1 Introduction Before examining the history and theory of the Lanczos potential, we discuss a number of preliminary topics. In this chapter, we define the fundamental geometric quantities which are manifest in physical theories, namely the Riemann curvature tensor and related quantities. In addition, a number of conventions to be used throughout this paper 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 be represented by an ordinary comma (e.g. R abcd,e is shorthand for ∂ ∂x e R abcd ). Similarly, covariant differentiation with respect to a basis vector will be represented by a semi- colon (e.g R abcd;e is shorthand for ∇ e R abcd ). 2 Introduction (Anti-)Symmetrisation shorthand Symmetrisation over a set of indices will be represented by ordinary round brackets over the set of indices to be symmetrised (e.g. T (ab) = 1 2 (T ab +T ba )). Similarly, anti- symmetrisation will be represented by square brackets over the set of indices to be anti-symmetrised (e.g. T [ab] = 1 2 (T ab −T ba )). If a set of indices within the (anti-) symmetrisation is to be omitted from the operation, it shall be separated by vertical bars (e.g. T (a|b|c) = 1 2 (T abc +T cba )). Duals of R abcd The Riemann tensor, due to its anti-symmetries (1.10) and (1.11), can be dualised over the first two, the last two, or both pairs of indices. In the literature during the early and middle of the 20th century, on which the majority of this review is based, a single asterisk, *R abcd , 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 indices shall be represented by *R abcd , 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 of O’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 more rigorous 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 independent indices, 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 in all bases. On the other hand if an expression contains uncontracted bold indices, it is only valid in a specific basis (often the case if we move to local inertial frame or employ 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 primed upper-case letters represent conjugate spinorial indices. 1.2 Covariant derivative 3 1.2 Covariant derivative The gradient of a scalar function on a manifold Mis given a unique invariant meaning through the differential structure of M(see [13], eq. (4.1.32)). However, the definition of a gradient of any arbitrary valence tensor 1 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) tensors to the (p; q + 1) tensors which, given an arbitrary valence (p; q) tensor T b 1 b 2 ...b p a 1 a 2 ...a q and (r; s) tensor U b 1 b 2 ...b r a 1 a 2 ...a s , satisfies the following properties: 1. Leibniz Rule: ∇ a _ T b 1 b 2 ...b p a 1 a 2 ...a q U b 1 b 2 ...b r a 1 a 2 ...a s _ = T b 1 b 2 ...b p a 1 a 2 ...a q ∇ a U b 1 b 2 ...b r a 1 a 2 ...a s +U b 1 b 2 ...b r a 1 a 2 ...a s ∇ a T b 1 b 2 ...b p a 1 a 2 ...a q . (1.2) 2. Commutation with index substitution. 3. Commutation with contraction: ∇ a T b 1 b 2 ...c...b p a 1 a 2 ...c...a q = g d c ∇ a T b 1 b 2 ...c...b p a 1 a 2 ...d...a q = ∇ a g d c T b 1 b 2 ...c...b p a 1 a 2 ...d...a q . (1.3) 4. The metric tensor g ab is covariantly constant: ∇ a g bc = 0 and ∇ a g bc = 0. (1.4) 5. The torsion of the connection vanishes 2 : (∇ a ∇ b −∇ b ∇ a ) λ = 0, (1.5) for any scalar function λ. 1 Besides, of course, a valence (0; 0) tensor, i.e. a scalar. 2 This 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 the Levi-Civita connection. 4 Introduction 6. The covariant derivative acts as an ordinary partial coordinate derivative on scalars: ∇ a λ = ∂ a λ, (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 a general tensor rather than simply scalar functions. We shall see that in the case where the covariant derivative does not commute with itself, then the manifold has non-zero curvature. The curvature is measured by a quantity known as the Riemann curvature tensor R a bcd , which we shall define below. 1.3.1 Definition We define the Riemann curvature tensor to be the mapping of an arbitrary co-vector field A a under the commutator of a covariant derivative, as R a bcd A a = [∇ c , ∇ d ] A b = A b;cd −A b;dc . (1.7) In terms of the Christoffel symbols, one can write the Riemann curvature tensor ex- plicitly as R a bcd = ∂ c Γ a db −∂ d Γ a cb + Γ a ce Γ e db −Γ a de Γ e cb . (1.8) Lowering the first index with the metric tensor yields the Riemann curvature tensor with all covariant indices: R abcd = g ae R e bcd . (1.9) 1.3.2 Symmetries From the definition of the Riemann tensor R abcd in terms of the Christoffel sym- bols (1.8) combined with the lowering of the first index (1.9), a number of symmetries are evident: 1. Anti-symmetry in a and b: R abcd = −R bacd . (1.10) 1.3 Riemann curvature tensor 5 2. Anti-symmetry in c and d: R abcd = −R abdc . (1.11) 3. Symmetry in the exchange of the index pair ab with the index pair cd: R abcd = R cdab . (1.12) 4. First Bianchi identity (often called cyclic identity): R abcd +R acdb +R adbc = 0, (1.13) or equivalently R a[bcd] = 0. (1.14) 5. Second Bianchi identity (often simply referred to as the Bianchi identity): R abcd;e +R abde;c +R abec;d = 0, (1.15) or equivalently R ab[cd;e] = 0. (1.16) 1.3.3 Ricci tensor and scalar We define the Ricci tensor R ab , often known as the contracted curvature tensor, through a contraction of the Riemann tensor as R ab = R c acb . (1.17) As a consequence of the first Bianchi identity (1.14), the Ricci tensor is symmetric: R ab = R ba . (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 one can achieve through a single contraction of any two indices of the Riemann tensor. 6 Introduction Performing a second contraction yields a quantity R = R a a (1.19) which is known as the Ricci scalar and is the trace of the Ricci tensor. The Einstein tensor G ab of general relativity can be constructed through a trace reversal of the Ricci tensor as G ab = R ab − 1 2 Rg ab . (1.20) 1.4 Weyl tensor The Weyl tensor C abcd can be thought of as the traceless part of the Riemann curvature tensor. In general relativity, it is the only part of the Riemann tensor which is left locally undetermined by the Einstein field equations. 1.4.1 Definition We define the Weyl tensor C abcd through a decomposition of the Riemann tensor into its various traces and the Weyl tensor, known as the Ricci decomposition, as R abcd = C abcd − 1 2 (R ac g bd +R bd g ac −R ad g bc −R bc g ad ) + 1 6 R(g ac g bd −g ad g bc ) (1.21) (see [11], eq. 4.29). The definition is such that tracing over any pair of indices of C abcd vanishes. 1.4.2 Symmetries The Weyl tensor shares symmetries (1)−(4) of the Riemann tensor in section 1.3.2 due to its definition (1.21). In addition, we may express its traceless property concisely as C a bac = 0. (1.22) Chapter 2 Historical 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] in which 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 paper connecting Riemannian geometry and physics [1], and in his less famous 1928 paper on how the classical theories of electromagnetism and general relativity might be uni- fied under action principles of purely geometric quantities [3]. Lanczos preceded his 1962 paper by his exploration of action principles in geometric quantities in the early 1930s [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], asserting that 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 tensor describing the matter present in space-time. This was not the first theory of physics to introduce geometry as the source of gravitation; Nordström’s 1912 and 1913 the- ories were predecessors of general relativity but were experimentally verified to be incorrect. In his 1962 paper [9], Lanczos observes that since Einstein’s theory was so revolutionary 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. Lanczos even 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 contain any 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 somewhat of a ‘fad’ in developing new theories of physics. Most notable of this trend are theories such as Einstein’s theory of “distant parallelism”, Veblen’s, Hoffman’s and Pauli’s theory of “projective reality”, as well as the famous five-dimensional Kaluza-Klein theory. These theories all had in common the desire to unify the classical field theories of electromagnetism and gravity, and express all of known physics in terms of space- time structure. 2.2 Early work in action principles of geometric quantities Lanczos was highly motivated to find a theory which explained both electromagnetic and gravitational phenomena through purely geometric means, essentially encoding the information about either the field strength tensor or the vector potential into the curvature coefficients, alongside general relativity. Lanczos released several interesting articles on Riemannian geometry, particularly concerning the determination of four- dimensional Riemannian geometries through extremal action principles, for several decades before writing his 1962 paper highlighting the Lanczos tensor as a fundamental geometric quantity. In 1938 Lanczos investigated the potential types of scalar invariants which might appear in a geometric Lagrangian in A Remarkable Property of the Riemann-Christoffel Tensor in Four Dimensions [6]. The paper considers the properties of a scale inde- pendent 1 action principle in the components of the Riemann tensor. Lanczos assumes 1 Weyl [17] developed the notion of a scale independent geometry which requires a "gauge- symmetry" of metric scaling invariance. That is the transformation g ′ ab = λg ab where λ is an arbitrary function of the coordinates, leaving physical quantities on the manifold invariant. This invariance is motivated as it has the effect of “changing the scale of calibration of the infinitesimal yard-stick.” That is, we are only changing the units by which we are measuring other invariants of the theory. 2.2 Early work in action principles of geometric quantities 9 that the fundamental independent quantities of the theory were the components of the metric tensor and the curvature tensor. This scale-invariance insists that the scalars from which the Lagrangian is built are quadratic functions of R abcd . Lanczos con- tinues by analysing all possible quadratic scalars and finds that any scale-invariant Lagrangian may be constructed from the scalars I 1 = R ab R ab , (2.1) I 2 = R 2 (2.2) (see [6], eq. 6.5) with all other quadratic scalar terms either being reducible to some linear 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 physical theories of geometry asserted by Einstein, assuming that all physical theories can be derived from some extremal action principle. The generalised Lagrangian L determining a geometry is thus given by L = R ab R ab +cR 2 (2.3) (see [7], eq. 2.1), where c is an undetermined quantity following the investigation in [6]. Lanczos continued this investigation in 1942 [7] by considering a tensor S ab def = R ab − 1 4 κRg ab (2.4) (see [7], eq. 2.2) for which κ is undetermined. The generalised Lagrangian (2.3) can now be rewritten as L = S ab S ab (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 ≥ 1 4 , real solutions for κ can be found. Since the Lagrangian only contains a term quadratic in S ab , essentially treating it as a free-field, it is clear that S ab = 0 (2.7) is a potential solution. Lanczos argues that this is the most stable solution as it 10 Historical formulation of the Lanczos tensor minimizes the action integral to zero. Comparing the solution (2.7) with (2.4) we find R ab − 1 4 κRg ab = 0 (2.8) (see [7], eq. 2.5) and, after contracting through with g ab : (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 R ab = 0. More interesting is the latter case, κ = 1, which when combined with (2.6) gives c = − 1 4 and simplifies (2.4) to S ab = R ab − 1 4 Rg ab (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 “metrical plateau” over which a high-frequency superposition of carrier fields could be added as a perturbation. Such a construction would result in a microscopic scale average curva- ture radius, potentially allowing for the explanation of fundamental particles through curvature. The basic field equations of this theory were fourth order in g ab , which troubled Lanczos as the equations of nature “seem to offer themselves as differential equations of first and second order.” In 1949 Lanczos revisited the problem in Lagrangian Multipliers and Riemannian Spaces [8] and considered the metric components g ab and the curvature components R abcd , together with Lagrangian multipliers to impose the Bianchi identity, as indepen- dent metrical quantities in order to reduce the order of the fundamental field equations presented in [7] to second order. The Bianchi identity is more naturally expressed in terms of the dual of the Riemann tensor. We define the dual of the Riemann tensor as *R* abcd = R ijkl ϵ ijab ϵ klcd (2.11) (see [8], eq. II.1) where ϵ abcd is the totally anti-symmetric Levi-Civita tensor. The Bianchi identity is then expressed as *R* a bcd;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 losing information. He then states that for the case of “infinitesimal fields” the contribution of the scalar (2.2) is an “infinitesimal of the second order and thus negligible for our present purposes.” This leaves us with an action integral of the following form 1 2 _ (*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 H a bc , which is anti-symmetric in b and c due to the properties of the Bianchi identity, but with the symmetry of the first index left unspecified [12]. This constraint is added to the action integral as − 1 2 _ H a bc;d *R* abcd (2.15) (see [8], eq. III.2), after performing an integration by parts and discarding boundary terms. After performing the variation to extremize the action, Lanczos found the following expression for the dual of the Riemann tensor in terms of the Lagrangian multiplier: *R* a bcd = H a bc;d −H a dbc; +H a b d;c −H a c d;b (2.16) (see [12], eq. 2; [8], eq. III.5). This is immediately analogous to the definition of the Riemann tensor given in terms of the Christoffel symbols, however H a bc and Γ a bc have significantly different properties as noted in [12]. Firstly, Γ a bc transforms as a connection while H a bc transforms as a tensor. Secondly, H a bc is anti-symmetric in b and c while Γ a bc , being defined as the torsion-free Levi-Civita connection, is symmetric in b and c. Lanczos finally derives field equations for H a bc by substituting the form found for the dual Riemann tensor in terms of H a bc (2.16) into the Bianchi identity (2.12), finding ∆H a bc −H d a bc;d +H a d b ;dc −H a d c ;db = 0 (2.17) 12 Historical formulation of the Lanczos tensor (see [8], eq. IV.1) where ∆ is the Laplacian operator 2 . Hence, using the action inte- gral (2.14) and the Bianchi identities, a system of 24 second order partial differential equations for the determination of the 24 independent components of H a bc has been found. Although the investigations of his 1949 (and earlier 1932) paper attempting to unify electricity and magnetism through geometrical quantities in space-time were not fruitful as a valid theory, the insights Lanczos made into defining geometries via action principles was an important precursor to the realization of the Lanczos tensor as a fundamental quantity in four-dimensional Riemannian geometry. The tensor H abc was revisited by Lanczos in his 1962 paper, albeit under a slightly different, yet consistent, definition. It is the same quantity which is now commonly referred to as the Lanczos tensor. The historical development of this tensor’s properties will 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 A abcd def = R abcd −*R* abcd (2.18) S abcd def = R abcd + *R* abcd (2.19) so that R abcd = 1 2 (A abcd +S abcd ) (2.20) (see [11], eq. 4.1-3). Clearly, taking the double dual of (2.18) reverses the sign and similarly 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 attributed to 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 Ricci 2 Rather than , as Lanczos was working in imaginary time. 2.4 Lanczos’ canonical Lagrangian 13 tensor through the following decomposition A abcd = _ R ac − 1 4 Rg ac _ g bd + _ R bd − 1 4 Rg bd _ g ac − _ R ad − 1 4 Rg ad _ g bc − _ R bc − 1 4 Rg bc _ g ad (2.21) (see [11], eq. 4.4), and is easily verified by moving to normal coordinates (where g µν reduces to δ µν ), however a fully covariant derivation of (2.21) can be found in section 1 of [9]. Einstein used this identity to derive the following field equations in [2] in an attempt to model a stable electron: R ab − 1 4 Rg ab = −KT ab (2.22) (see [11], eq. 4.5) where T ab is the energy-momentum tensor of Maxwell theory. Only 9 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 the anti-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-dimensional geometry. 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 to the Ricci tensor in the case of the anti-self-dual tensor? Lanczos [9] makes the bold statement that without this analysis “we cannot claim to have fully understood the structure 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 , g ab ) + H abc *R* abcd ;d +P ab c _ Γ c ab − _ c ab __ +ρ ab _ R ab + Γ c bc,a −Γ c ab,c + Γ c ad Γ d bc −Γ c ab Γ d dc _ (2.23) 14 Historical formulation of the Lanczos tensor (see [11], eq. 4.9), where again H abc is a Lagrangian multiplier ensuring the Bianchi identity (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 Γ c ab − _ c ab _ = 0 (2.24) (see [11], eq. 4.7), and finally ρ ab is the Lagrangian multiplier enforcing the definition of the Ricci tensor 3 in terms of the Christoffel symbols R ab + Γ c bc,a −Γ c ab,c + Γ c ad Γ d bc −Γ c ab Γ d dc = 0 (2.25) (see [11], eq. 4.8). The Lagrangian (2.23) has canonical variables g ab , Γ c ab , and *R* abcd , with conjugate variables ρ ab , P ab c , and H abc respectively. The Lagrangian multipliers P ab c and ρ ab are symmetric in a and b, however H abc is anti-symmetric in a and b 4 . Lanczos proceeds with his investigation of the Lagrangian multipliers by preparing to perform the variation with respect to the canonical variables. First he notes that H abc , 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, *H ad a = 1 2 H abc ϵ abcd = 0 (2.26) (see [9], eq. 2.9), in order to restrict the number of components to 20, which is equivalent to imposing the cyclic identity: H abc +H bca +H cab = 0 (2.27) (see [9], eq. 2.10). Also in preparation of the variation, he splits ρ ab into its trace and its trace-free parts by defining a scalar q as q def = 1 4 ρ ab g ab (2.28) 3 In [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. 4 This definition of the multiplier H abc differs from that in Lanczos’ 1949 paper, however they are equivalent, 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 Q ab as Q ab def = ρ ab −qg ab (2.29) so that ρ ab = Q ab +qg ab (2.30) (see [9], eq. 2.11). Finally, we rewrite the second term of (2.23) using integration by parts and neglecting the boundary term (as its variation vanishes by construction) H abc *R* abcd ;d = −H abc;d *R* abcd (2.31) (see [11], eq. 4.17). We can now rewrite the canonical Lagrangian as L ′ = L(*R* abcd , g ab ) −H abc;d *R* abcd +P ab c _ Γ c ab − _ c ab __ + _ Q ab +qg ab _ _ R ab + Γ c bc,a −Γ c ab,c + Γ c ad Γ d bc −Γ c ab Γ d dc _ . (2.32) The variation with respect to Γ c ab can be performed to obtain a relation for P ab c without specifying L(*R* abcd , g ab ) as it does not depend on Γ c ab explicitly. Lanczos does this (see [9], eq. 2.14) and finds that P ab c can be written in terms of other variables and thus is not a fundamental quantity. Next, Lanczos performed the variation with respect to g ab to obtain the following relation 5 ∂L ∂g ab + 1 2 Lg ab = Q ij *R* iajb +Q ai *R* b i +Q bi *R* a i − 1 2 RQ ab −q*R* ab − 1 2 Rqg ab − 1 2 _ 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 −H abc;d +Q ac g bd −qg ac g bd (2.34) 5 Here we have made use of that fact that *R* c a bc = *R* ab = − _ R ab − 1 2 Rg ab _ and *R* a a = *R* = R. 16 Historical formulation of the Lanczos tensor (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 resulting equations from the variation are thus given by _ ∂L ∂*R* abcd _ = _ H abc;d −(Q ac −qg ac ) g bd _ (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 the form of L, and not only for infinitesimal geometries as previously considered in the 1942 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 H abc →H abc −Φ b g ac + Φ a g bc (2.36) Q ab →Q ab + Φ a;b + Φ b;a − 1 2 Φ c ;c g ab (2.37) q →q − 1 2 Φ a ;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”) the form of H abc through choosing the form of Φ a . If we choose Φ b = − 1 3 H abc g 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 b a b = 0 (2.40) 6 Lanczos defines the bracketed version of a four-index tensor A abcd as: [A abcd ] = 1 2 (A abcd +A cdab +A badc +A dcba −A bacd −A cdba −A abdc −A dcab ) . Note that this spe- cial symmetrisation leaves the Riemann tensor and the double dual Riemann tensor invariant (up to a constant of proportionality). 2.6 The fundamental nature of H abc 17 (see [9], eq. 2.23) which reduces the number of independent components of H abc from 20 to 16. We can further restrict the form of H abc by introducing the “divergence-free gauge condition” H c ab ;c = 0 (2.41) (see [9], eq. 3.15). These six additional gauge conditions do not respect the general form 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) is overdetermined, 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 H abc In his exploration of the self-dual tensor, Lanczos revisits the scalar invariant K = R abcd *R* abcd , which he had previously shown in [6] to not generate field equations in action principles for four-dimensional geometries as its variation vanished identically. However, Lanczos makes the realization that it is “exactly for this reason we have here a variational property which characterizes all Riemannian geometries of four dimensions.” We can thus make the choice L = 1 8 R abcd *R* abcd without placing constraints on the resultant geometry. We find ∂L ∂*R* abcd = 1 4 R abcd (2.42) (see [9], eq. 3.2-3). Inserting this into (2.35), and using symmetries of the Riemann tensor, we obtain R abcd = _ H abc;d −(Q ac −qg ac ) g bd _ (2.43) (see [9], eq. 3.4). Multiplying (2.43) through by g ad and performing the contraction gives the Ricci-Lanczos equations: R bc = H a bc;a −H a ba;c +H a c a;b −H a c b;a −2Q bc + 6qg bc (2.44) (see [11], eq. 4.26), and after contracting through again by g bc we find R = 4H ab b;a + 24q (2.45) 18 Historical formulation of the Lanczos tensor (see [11], eq. 4.27). Rearranging (2.44) and using (2.45) we arrive at Q bc −qg bc = H i (bc);i +H i (c|i| ;b) − 1 3 H ij j;i g bc − 1 2 R bc + 1 12 Rg bc . (2.46) We can substitute (2.46) into (2.43) to find the following expression for the Riemann tensor, also known as the Riemann-Lanczos equations: R abcd = H abc;d +H cda;b +H bad;c +H dcb;a + _ H i (ac);i +H i (a|i| ;c) _ g bd + _ H i (bd);i +H i (b|i| ;d) _ g ac − _ H i (ad);i +H i (a|i| ;d) _ g bc − _ H i (bc);i +H i (b|i| ;c) _ g ad − 2 3 H ij j;i (g ac g bd −g ad g bc ) − 1 2 (R ac g bd +R bd g ac −R ad g bc −R bc g ad ) + 1 6 R(g ac g bd −g ad g bc ) (2.47) (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: R abcd = H abc;d +H cda;b +H bad;c +H dcb;a +H i ac;i g bd +H i bd;i g ac −H i ad;i g bc −H i bc;i g ad − 1 2 (R ac g bd +R bd g ac −R ad g bc −R bc g ad ) + 1 6 R(g ac g bd −g ad g bc ) (2.48) (see [11], eq. 4.35). Lanczos revisited the splitting of the Riemann tensor (2.20), balancing the splitting by defining U abcd def = A abcd + 1 6 R(g ac g bd −g ad g bc ) (2.49) V abcd def = S abcd − 1 6 R(g ac g bd −g ad g bc ) (2.50) so that R abcd = U abcd + V abcd , with each U abcd and V abcd having 10 independent com- 2.7 The Weyl-Lanczos equations 19 ponents. A similar expression to (2.21) can be found for U abcd by simply plugging the definition (2.49) into (2.21) to obtain U abcd = _ R ac − 1 6 Rg ac _ g bd + _ R bd − 1 6 Rg bd _ g ac − _ R ad − 1 6 Rg ad _ g bc − _ R bc − 1 6 Rg bc _ g ad (2.51) (see [9], eq. 3.12). Multiplying (2.47) by a factor of two and using the fact that 2R abcd = U abcd +V abcd and subtracting (2.51), we can find an expression for V abcd : V abcd = 2(H abc;d +H cda;b +H bad;c +H dcb;a + _ H i (ac);i +H i (a|i| ;c) _ g bd + _ H i (bd);i +H i (b|i| ;d) _ g ac − _ H i (ad);i +H i (a|i| ;d) _ g bc − _ H i (bc);i +H i (b|i| ;c) _ g ad − 2 3 H ij j;i (g ac g bd −g ad g bc )). (2.52) Although H abc had appeared before in Lanczos’ 1949 investigation of a theory of infinitesimal fields, he shows here that it holds fundamental significance to all Rie- mannian geometries of four dimensions. Lanczos found that H abc is a “generating function” of V abcd . Lanczos made the realization that “the tensor V abcd contains ex- actly those components of the full Riemann tensor which are not reducible to the contracted tensor R ab .” This highlights the fundamental nature of H abc as it con- tains the exact information necessary to complete the description of the curvature components of four-dimensional geometries provided g ab and R ab . 2.7 The Weyl-Lanczos equations Although Lanczos realized the fundamental significance of (2.52), it wasn’t until Takeno investigated further in 1964 [16] that it was realized that V abcd was in fact proportional to the Weyl tensor, C abcd . We use the Ricci decomposition (1.21) of the 20 Historical formulation of the Lanczos tensor Riemann tensor along with (2.47) to obtain an expression for the Weyl tensor, C abcd = H abc;d +H cda;b +H bad;c +H dcb;a + _ H i (ac);i +H i (a|i| ;c) _ g bd + _ H i (bd);i +H i (b|i| ;d) _ g ac − _ H i (ad);i +H i (a|i| ;d) _ g bc − _ H i (bc);i +H i (b|i| ;c) _ g ad − 2 3 H ij j;i (g ac g bd −g ad g bc ) (2.53) (see [11], eq. 4.30), known as the Weyl-Lanczos equations. From these it is clear that V abcd = 2C abcd . If we again employ the trace-free and divergence-free gauge conditions (2.40) and (2.41), the Weyl-Lanczos equations simplify to C abcd = H abc;d +H cda;b +H bad;c +H dcb;a +H i ac;i g bd +H i bd;i g ac −H i ad;i g bc −H i bc;i g ad (2.54) (see [11], eq. 4.36). Chapter 3 Spinor 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 that they extend the properties of vectors in a natural way. This is traditionally motivated by the fact that the properties arising from this representation naturally occur in the description of physical phenomena, such as Dirac fermions. In this chapter a very brief introduction to the algebra and analysis of two-component spinors is given. For a more rigorous and complete description of the two-spinor formalism, one should look at 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 A a 1 ...a r b 1 ...b s at a point p in a manifold is often thought of as a multi-linear map from r vectors in T p (Minkowski space for a Lorentzian manifold) and s co-vectors in T ∗ p to R. We can construct algebras analogous to the tensor algebra by changing the vector space to which the vectors belong, rather than simply choosing it to be T p . We define spin space S as a two-dimensional vector space over C which possesses a 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 the condition [o, ι] = 1 (3.5) (see [11], eq. 1.42). We can expand any spin-vector η on this basis η = η 0 o +η 1 ι (3.6) where η 0 = [η, ι] and η 1 = [η, o] are the components of η in this spin basis. In terms of components, the inner product takes the form [ζ, η] = ζ 0 η 1 −ζ 1 η 0 (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 = _ _ ζ 0 ζ 1 _ _ : ζ 0 → ˆ ζ 0 = aζ 0 +bζ 1 ζ 1 → ˆ ζ 1 = cζ 0 +dζ 1 (3.8) (see [11], eq. 1.29), where a, b, c, d ∈ C and ad −bc = 1 (so that the inner product is preserved). These transformations are known as the spin transformations and can be 3.1 Spinor algebra 23 written in matrix form as _ _ ˆ ζ 0 ˆ ζ 1 _ _ = _ _ a b c d _ _ _ _ ζ 0 ζ 1 _ _ . (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 of valence (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-vectors and q spin-vectors to C. We now introduce the operation of complex conjugation of a spinor index. We represent the complex conjugate of a spin-vector η ∈ S by η ∈ S. Similarly, we can conjugate spin-co-vectors [η, ] ∈ S ∗ simply by dualising the corresponding element in S, 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 abstract index notation for spinors analogously as we did for tensors, however rather than lower-case letters, upper-case letters represent spinor indices. If an index is primed, it represents a conjugate index. Hence a (p, q; r, s) spinor, M, can be represented by the notation M A 1 ...A p B 1 ′ ...B q ′ C 1 ...C r D 1 ′ ...D s ′ . 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 a spinor ϵ AB , called the Levi-Civita spinor, such that for any two spin-vectors ζ, η ∈ S we have [ζ, η] = ϵ AB ζ A η B (3.10) [η, ζ] = −ϵ AB ζ B η A (3.11) (see [11], eq. 2.13-4), from which it is obvious that ϵ AB = −ϵ BA . 24 Spinor algebra and analysis The Levi-Civita spinor plays an analogous role to the metric tensor g ab . We can use ϵ AB to lower indices as is apparent from (3.10) or (3.11) so that ϵ AB ζ A is the dual of ζ B and thus ζ B = ϵ AB ζ A (3.12) (see [11], eq. 2.16). We can thus write the inner product [ζ, η] in index form as ζ A η A , or as ζ 0 η 0 +ζ 1 η 1 in an explicit spin basis. The Levi-Civita spinor with indices upstairs is similarly defined and raises indices, for example ζ A = ϵ AB ζ B (3.13) (see [11], eq. 2.20). If we insert (3.12) into (3.13) we find ζ A = ϵ CB ϵ AB ζ C = δ A C ζ C (3.14) where δ A C is the usual Kronecker delta, and so we arrive at the following properties ϵ AB ϵ AC = δ B C = ϵ B C , (3.15) −ϵ CB ϵ BA = δ C A = −ϵ C A (3.16) (see [11], eq. 2.21-2). We can thus replace the Kroneker delta δ B A = δ B A with the Levi-Civita spinor ϵ B A , if we respect the anti-symmetry properties ϵ B A = −ϵ B A , (3.17) ϵ AB = −ϵ BA (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 between the basis spin-vectors can now be written equivalently using (3.10) as o A ι A = 1, ι A o A = −1, o A o A = 0, ι A ι A = 0, (3.19) 3.1 Spinor algebra 25 (see [13], eq. 2.5.39-41). We say that the two basis spinors, o A , ι A form a dyad in spin- space in analogy to how the four basis vectors t a , x a , y a , z a form a tetrad in Minkowski space. We can write the dyad o A , ι A collectively as ϵ A A , where ϵ A 0 = o A , ϵ A 1 = ι A (3.20) (see [13], eq. 2.5.44). The dual basis, ϵ A A , must satisfy the condition ϵ A A ϵ B A = ϵ B A = _ _ 1 0 0 1 _ _ (3.21) (see [13], eq. 2.5.47). The components of a spinor with respect to a dyad can be found using ϵ A A , for example the components of ϵ AB are ϵ AB = ϵ AB ϵ A A ϵ B B = _ _ 0 o A ι A o A ι A 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. This is directly related to the fact that there exists a homomorphism between the linear transformation group of spin-vectors, SL(2, C), and that of vectors, L(4), although because this is a homomorphism (and not an isomorphism) there is not always a tensorial equivalent of a spinorial object (although the spinor equivalent of a tensor can always be found). If we represent tensors in terms of their equivalent spinors, we shall find that many complicated tensorial expressions are simplified by the additional structure present in the spinor algebra. We introduce the Infeld-van der Waerden symbols, σ a AB ′ , as the objects which connect tensorial indices to their spinor counterparts. Here the index a is a tensorial index which runs over the tetrad in the tangent space, and A, B ′ are spinor indices which 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 (as defined in (3.5)) are the identity matrix and the three Pauli matrices (up to a factor 26 Spinor algebra and analysis 1 √ 2 ): σ 0 AB ′ = 1 √ 2 _ _ 1 0 0 1 _ _ , σ 1 AB ′ = 1 √ 2 _ _ 0 1 1 0 _ _ , σ 2 AB ′ = 1 √ 2 _ _ 0 i −i 0 _ _ , σ 3 AB ′ = 1 √ 2 _ _ 1 0 0 −1 _ _ (3.23) (see [11], eq. 2.69). The Infeld-van der Waerden symbols are an explicit description of the group homomorphism described above. From their form it is clear that a vector can be represented by a spin-vector and a conjugate spin-vector. Similarly, any tensor index is matched to an unprimed and primed spinor index. We now give an example of how the relationship between tensors and spinors is realized explicitly. If T b a is a valence (1; 1) tensor, then its spinor-equivalent is T b a = T BB ′ AA ′ = T b a σ a AA ′ σ BB ′ b (3.24) where we have used the Infeld-van der Waerden symbols with spinorial indices raised using the Levi-Civita spinor, and tensorial indices lowered using the metric tensor. Through explicit calculation, one can verify the following equations: σ AA ′ a σ b AA ′ = δ b a , (3.25) σ a AA ′ σ BB ′ a = ϵ B A 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 and A ′ fixed and allowing the tensorial index to vary, form an orthonormal basis for the faithful representation of L(4). We define the metric tensor in terms of the Infeld-van der Waerden symbols as g ab = ϵ AB ϵ A ′ B ′ σ AA ′ a σ BB ′ b (3.27) (see [11], 2.73), from which the explicit relationship between the Levi-Civita spinor and the metric tensor can be found by applying (3.25) and (3.26) to (3.27), obtaining g ab σ a AA ′ σ b BB ′ = ϵ AB ϵ A ′ B ′ (3.28) 3.1 Spinor algebra 27 (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 (or vice-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 to be 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-symmetric in three or more primed or unprimed indices vanishes identically. That is, for any spinor of the form A AIJK or B AI ′ J ′ K ′ , where the index A represents an arbitrary set of spinorial indices, we have A A[IJK] = 0, B A[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 the indices must refer to the same dyad component. Let us consider the particular case of (3.29) for the Levi-Civita spinor ϵ A[B ϵ CD] = 0 (3.30) (see [11], eq. 2.54), which we can rewrite as ϵ AB ϵ CD +ϵ AC ϵ DB +ϵ AD ϵ BC = 0 (3.31) (see [11], eq. 2.55). After raising the indices C and D and rearranging we have ϵ C A ϵ D B −ϵ C B ϵ D A = 2ϵ C [A ϵ D B] = ϵ AB ϵ CD (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 Φ ACD , gives 2Φ A[AB] = Φ C AC ϵ AB (3.33) 28 Spinor algebra and analysis (see [13], eq. 2.5.24). We can then write the spinor Φ AAB as Φ AAB = Φ A(AB) + Φ A[AB] = Φ A(AB) + 1 2 Φ C AC ϵ AB (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 the originals. We first show that Φ AA...Z ∼ Φ A(A...Z) (3.35) holds for each Φ AA...Z . We express the symmetrisation of Φ A(A...Z) as Φ A(A...Z) = 1 n _ Φ AA(BC...Z) + Φ AB(AC...Z) +· · · + Φ AZ(AB...Y ) _ (3.36) (see [13], eq. 3.3.50). If we consider the difference between the first term and any other term in the bracketed expression on the right-hand-side of (3.36), and using (3.34), we find Φ AC(AB...Z) = Φ AA(BC...Z) +ϵ AC Φ X A (BX...Z) (3.37) (see [13], eq. 3.3.51). If we substitute this form of Φ AC(AB...Z) into (3.36) and repeat for all other terms on the right-hand-side, we establish the equivalence Φ A(AB...Z) ∼ Φ AA(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 Φ A(AB...Z) ∼ Φ AA(B...Z) ∼ Φ AAB(C...Z) ∼ · · · ∼ Φ AABC...Z (3.39) establishing the desired result. This argument is equally valid for sets of primed indices, and if we have an arbitrary spinor we have the general result Φ A ′ B ′ C ′ ...F ′ PQR...Z ∼ Φ (A ′ B ′ C ′ ...F ′ )(PQR...Z) . (3.40) Furthermore, although the algorithm above was done on a spinor with all indices downstairs, the argument is general as we may simply lower all the indices of a mixed 3.1 Spinor algebra 29 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 spaces of symmetric spinors. Another way of expressing this irreducibility of completely symmetric 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 reduction to symmetric spinors. Consider the spinor Φ ABA ′ B ′ . We can decompose it as follows Φ ABA ′ B ′ = Φ (AB)(A ′ B ′ ) − 1 2 ϵ AB Φ C C(A ′ B ′ ) − 1 2 ϵ A ′ B ′ Φ C ′ (AB) C ′ + 1 4 ϵ AB ϵ A ′ B ′ Φ C 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 spinors that every algebraic tensor operation has a spinor analogue. However if we have a spinor associated with some tensor, the richer spinor algebra allows certain tensorial operations to be represented elegantly. The spinorial operations of exchanging two unprimed 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 spinor framework. Trace reversal over symmetric indices Consider an arbitrary symmetric tensor of valence (0; 2), T ab = T ba . We can convert this to the spinor formalism as T AA ′ BB ′ = T BB ′ AA ′ (3.42) 30 Spinor algebra and analysis (see [13], eq. 3.4.2). Adding and subtracting T ABBA from the right-hand-side and rearranging 1 gives T ABA ′ B ′ = 1 2 (T ABA ′ B ′ +T ABB ′ A ′ ) + 1 2 (T BAB ′ A ′ −T ABB ′ 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 term on the right-hand-side is anti-symmetric in A and B and in A ′ and B ′ by (3.42), and can be written as T [AB][A ′ B ′ ] . Using (3.33) twice on each set of antisymmetric indices in the second term, we get 2 T ab = T ABA ′ B ′ = S ABA ′ B ′ +ϵ AB ϵ A ′ B ′ τ (3.44) (see [13], eq. 3.4.4), where S ABA ′ B ′ def = T (AB)(A ′ B ′ ) and τ def = 1 4 T CC ′ CC ′ = 1 4 T c c . We can rewrite (3.44) in tensor form as T ab = S ab +g ab τ (3.45) (see [13], eq. 3.4.7), which is the canonical decomposition of a tensor into its trace-free part S ab and its trace τ. We perform the operation of the trace reversal to T ab , effectively leaving the trace- free part invariant and negating the trace, by ˆ T ab def = T ab − 1 2 T c c g ab (3.46) (see [13], eq. 3.4.10). In spinor form we have ˆ T ab = ˆ T AA ′ BB ′ = S ABA ′ B ′ −ϵ AB ϵ A ′ 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 ˆ T AA ′ BB ′ = T BAA ′ B ′ = T ABB ′ A ′ (3.48) 1 Note 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. 2 Note that this decomposition is equivalent to the one presented in (3.41) with additional sym- metry. 3.1 Spinor algebra 31 (see [13], eq. 3.4.13). We thus see that trace reversal applied to a pair of symmetric tensorial indices a and b is realized in the spinor formalism simply by interchanging the 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), F ab = −F ba , often called a bivector. We can convert this to the spinor formalism as F AA ′ BB ′ = −F BB ′ AA ′ (3.49) (see [13], eq. 3.4.15). Adding and subtracting F ABB ′ A ′ from the right-hand-side and rearranging gives F ABA ′ B ′ = 1 2 (F ABA ′ B ′ −F ABB ′ A ′ ) + 1 2 (F ABB ′ A ′ −F BAB ′ A ′ ) (3.50) (see [13], eq. 3.4.16). The first term on the right-hand-side is anti-symmetric in A ′ and B ′ and so we can apply (3.33) to rewrite it as 1 2 F C ′ ABC ′ ϵ A ′ B ′ . Similarly, the second term on the right-hand-side is anti-symmetric in A and B and through applying (3.33) we can rewrite it as 1 2 F C C A ′ B ′ ϵ AB . Rewriting (3.50) in terms of these decompositions, we have F ABA ′ B ′ = 1 2 F C ′ ABC ′ ϵ A ′ B ′ + 1 2 F C C A ′ B ′ ϵ AB (3.51) (see [13], eq. 3.4.17). The Hodge dual of a bivector F ab is given by *F ab def = 1 2 ϵ cd ab F cd (3.52) (see [13], eq. 3.4.21), where ϵ cd ab is the alternating tensor which has spinor form ϵ cd ab = iϵ C A ϵ D B ϵ D ′ A ′ ϵ C ′ B ′ −iϵ D A 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 *F ab = *F ABA ′ B ′ = −i 1 2 F C ′ ABC ′ ϵ A ′ B ′ +i 1 2 F C C A ′ B ′ ϵ AB (3.54) 32 Spinor algebra and analysis (see [13], eq. 3.4.22), from which it is clear upon comparison with (3.51) that *F ABA ′ B ′ = iF ABB ′ A ′ = −iF BAA ′ B ′ (3.55) (see [13], eq. 3.4.23). Hence we see that dualisation of a pair of anti-symmetric tensorial indices a and b is realized in the spinor formalism simply by interchanging the 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), G ab ,with no symmetry in a and b. We can decompose G ab into its symmetric and anti-symmetric parts: G AA ′ BB ′ = G (ab) +G [ab] . (3.56) Using (3.48) and (3.55) on the symmetric part and antisymmetric part of G ab respec- tively, we arrive at G BA ′ AB ′ = ˆ G (ab) +i*G [ab] , (3.57) G AB ′ BA ′ = ˆ 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 G BA ′ AB ′ = 1 2 _ G ab +G ba −G c c g ab +iϵ abcd G cd _ , (3.59) G AB ′ BA ′ = 1 2 _ G ab +G ba −G c c g ab −iϵ abcd G cd _ (3.60) (see [13], eq. 3.4.53-4). The tensorial complexity which arises from the simplest spinor operation, namely the exchange of two indices, is remarkable. Moreover, the operations 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 express physical 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 want to express geometric theories, such as general relativity, in a spinor formalism we must define some sort of covariant derivative on the manifold by which spinors at different points can be related. 3.2.1 Spinor covariant derivative We define the spinor covariant derivative ∇ a = ∇ AA ′ , in analogy to the tensor covariant derivative 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 C 1 ...C p D 1 ′ ...D q ′ A 1 ...A r B 1 ′ ...B s ′ : 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 spinor case the contraction may be over a single pair of spinor indices and is performed with the Levi-Civita spinor ϵ D C or ϵ D ′ C ′ (rather than the metric tensor). 4. Commutation with complex conjugation: ∇ a T C 1 ...C p D 1 ′ ...D q ′ A 1 ...A r B 1 ′ ...B s ′ = ∇ a T C 1 ...C p D 1 ′ ...D q ′ A 1 ...A r B 1 ′ ...B s ′ (3.61) (see [11], eq. 3.7). 5. The Levi-Civita spinor (rather than the metric tensor in (1.4)) is covariantly constant: ∇ a ϵ BC = 0 and ∇ a ϵ BC = 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 derivative on scalars as in (1.6). 34 Spinor algebra and analysis 3.2.2 The curvature spinors Consider the spinor equivalent of the Riemann tensor: R AA ′ BB ′ CC ′ DD ′ = R abcd (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 R abcd = 1 2 (R ABA ′ B ′ CDC ′ D ′ −R ABB ′ A ′ CDC ′ D ′ ) + 1 2 (R ABB ′ A ′ CDC ′ D ′ −R BAB ′ A ′ CDC ′ D ′ ) = R AB[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 R abcd = R AB[A ′ B ′ ]CD[C ′ D ′ ] +R AB[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 R abcd = X ABCD ϵ A ′ B ′ ϵ C ′ D ′ + Φ ABC ′ D ′ ϵ A ′ B ′ ϵ CD + Φ A ′ B ′ CD ϵ AB ϵ C ′ D ′ +X A ′ B ′ C ′ D ′ ϵ AB ϵ CD (3.66) (see [11], eq. 3.20), where X ABCD and Φ ABC ′ D ′ are defined as X ABCD def = 1 4 R E ′ F ′ ABE ′ CDF ′ , (3.67) Φ ABC ′ D ′ def = 1 4 R E ′ F ABE ′ F C ′ D ′ (3.68) (see [11], eq. 3.21-2). X ABCD and Φ ABC ′ D ′ are referred to as curvature spinors and contain 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 X ABCD further. Under simultaneous exchange of A and B as well as A ′ and B ′3 , we have anti-symmetry in the Riemann tensor as in (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 through to the curvature spinors directly. Hence, X ABCD has the symmetries: X ABCD = X (AB)(CD) , (3.69) X ABCD = X CDAB (3.70) (see [11], eq. 3.23, 3.25). Investigating the manifestation of the cyclic symmetry (1.14) in the curvature spinor X ABCD yields the following property 4 : X B ABC = 3Λϵ AC (3.71) (see [11], eq. 3.36), where Λ def = 1 6 X AB AB is proportional to the trace of X ABCD . Hence, after contracting over the second and fourth indices, we are left with a completely anti- symmetric spinor 5 . In an attempt to decompose the anti-symmetric parts of X ABCD , we expand using the following identity 6 : X ABCD = X (ABCD) +X [ABCD] + 11 12 X ABCD − 1 12 (X ACDB +X ADBC +X BCAD +X BADC +X BDCA +X CDAB +X CABD +X DACB +X DCBA +X DBAC ) (3.72) (see [11], eq. 3.54). Clearly, X [ABCD] = 0 by (3.69), as anti-symmetrising a pair of 3 This also holds when exchanging both C and D as well as C ′ and D ′ . 4 To derive this one should express the cyclic identity in terms of the right dual of the Riemann tensor as R* cb ab = 0 and find the corresponding spinor equation in terms of the curvature spinors. One finds that Φ ABC ′ D ′ drops out of the expression due to its symmetries (see [11], sec. 3.2.1). 5 Note 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 with ϵ AC . 6 One can verify this identity by simply expanding the terms X (ABCD) and X [ABCD] . 36 Spinor algebra and analysis symmetric indices annihilates the object. Similarly, (3.69) gives X (ABCD) = X A(BCD) , as symmetrising an index set with a member of a symmetric index set is equivalent to simply symmetrising over the union of both index sets. With this, (3.72) can be simplified to X ABCD = X A(BCD) + 1 3 (X ABCD −X ACBD ) + 1 3 (X ABCD −X ADCB ) = X A(BCD) + 1 3 X A[BC]D + 1 3 X A[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 X ABCD = X A(BCD) + 1 3 ϵ BC X E AE D + 1 3 ϵ BD X E AEC (3.74) (see [11], eq. 3.57). Applying (3.71) to (3.74), we can complete the decomposition as X ABCD = Ψ ABCD + Λ(ϵ BC ϵ AD +ϵ BD ϵ AC ) (3.75) (see [11], eq. 3.58), where Ψ ABCD is defined as Ψ ABCD def = X A(BCD) (3.76) and is often called the Weyl spinor or gravitational spinor. Rewriting (3.66) using the Weyl spinor to yields R abcd = Ψ ABCD ϵ A ′ B ′ ϵ C ′ D ′ + Ψ A ′ B ′ C ′ D ′ ϵ AB ϵ CD + Φ ABC ′ D ′ ϵ A ′ B ′ ϵ CD + Φ A ′ B ′ CD ϵ AB ϵ C ′ D ′ + 2Λ(ϵ AC ϵ BD ϵ A ′ C ′ ϵ B ′ D ′ −ϵ AD ϵ BC ϵ A ′ D ′ ϵ B ′ C ′ ) . (3.77) This decomposition is the spinor equivalent of the Ricci decomposition (1.21), and through further analysis it is possible to show the equivalence of the Weyl tensor and the Weyl spinor as C abcd = Ψ ABCD ϵ A ′ B ′ ϵ C ′ D ′ + Ψ A ′ B ′ C ′ D ′ ϵ AB ϵ CD (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 a notation equivalent to the spinor formalism often used numerical relativity. Although somewhat beyond the scope of this essay, we shall provide a brief summary of the formalism 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 projected onto this null tetrad. In essence, the spin-coefficient formalism gives the relationship between: 1. Twelve complex spin-coefficients which store information about the directional covariant derivatives along the tetrad vectors. 2. Five complex functions which store information about the Weyl tensor in the tetrad basis. 3. Four real functions and three complex functions (and their conjugates) which store information about the Ricci tensor in the tetrad basis. Although seemingly complicated and arduous, working with a specific null tetrad to exploit the symmetries of the space-time can cause several of the spin-coefficients to be trivial. Working in the spin-coefficient formalism is often a preferred method in numerical relativity. The reader is suggestion to consult section 3.5-6 of [11] for more information on the spin-coefficient formalism. Chapter 4 The Lanczos spinor We saw in the Lanczosian development of H abc in chapter 2 that the Lanczos tensor is a fundamental quantity in four-dimensional Riemannian geometry. Shortly after this realization was made, it was shown that in fact the Lanczos tensor was a potential for 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 quantity H abc ; the Weyl-Lanczos equations are 16 non-linear equations in 16 unknowns in a convoluted form, and are thus computationally expensive. In this chapter we make use of the properties of the 2-spinor formalism introduced in chapter 3 to express the theory of the Lanczos potential in spinor form, closely following the procedure in O’Donnell [11]. 4.1 Lanczos spinor and decomposition We convert the Lanczos tensor H abc to spinor form H AA ′ BB ′ CC ′ = H abc (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 expressed in the spinor formalism as H AA ′ BB ′ CC ′ = −H BB ′ AA ′ CC ′ (4.2) 40 The Lanczos spinor (see [11], eq. 4.43). We split (4.2) using (3.50) as H abc = H ABA ′ B ′ CC ′ = 1 2 (H ABA ′ B ′ CC ′ −H ABB ′ A ′ CC ′ ) + 1 2 (H ABB ′ A ′ CC ′ −H BAB ′ A ′ CC ′ ) (4.3) (see [11], eq. 4.44). The first term on the right-hand-side is anti-symmetric in A ′ and B ′ , which when combined with (4.2) indicates symmetry in A and B. The second term on the right-hand-side is anti-symmetric in A and B, which similarly indicates symmetry in A ′ and B ′ . We can thus write H abc as H abc = H ABA ′ 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 spinor indices, we can rewrite each term in (4.4) as the direct product of a Levi-Civita spinor and a spinor of lower valence, and hence (4.4) can be rewritten as H abc = H ABA ′ B ′ CC ′ = 1 2 ϵ A ′ B ′ H D ′ (AB)D ′ CC ′ + 1 2 ϵ AB H D D (B ′ A ′ )CC ′ (4.5) (see [11], eq. 4.47). We now define the spinors H ABCC ′ and φ A ′ B ′ CC ′ for convenience as H ABCC ′ = H (AB)CC ′ def = 1 2 H D ′ (AB)D ′ CC ′ , (4.6) φ A ′ B ′ CC ′ = φ (A ′ B ′ )CC ′ def = 1 2 H D D (B ′ A ′ )CC ′ (4.7) (see [11], eq.4.48), allowing us to write (4.5) as H abc = H ABA ′ B ′ CC ′ = ϵ A ′ B ′ H ABCC ′ +ϵ AB φ A ′ B ′ CC ′ (4.8) (see [11], eq. 4.49). Taking the conjugate of (4.8) and noting the fact that the conjugate of a real tensor is itself gives H abc = ϵ AB H A ′ B ′ CC ′ +ϵ A ′ B ′ φ ABCC ′ (4.9) (see [11], eq. 4.50), which when equating with (4.8) gives the consistent relations 4.1 Lanczos spinor and decomposition 41 φ A ′ B ′ CC ′ = H A ′ B ′ CC ′ and φ ABCC ′ = H ABCC ′ . Thus, we may write (4.8) in terms of H ABCC ′ and its conjugate H A ′ B ′ CC ′ alone: H abc = H ABA ′ B ′ CC ′ = ϵ A ′ B ′ H ABCC ′ +ϵ AB H A ′ B ′ CC ′ (4.10) (see [11], eq. 4.51). The complete information contained in the Lanczos tensor is also contained in the quantity H ABCC ′ , which we henceforth will refer to as the Lanczos spinor. So far, we have only imposed the anti-symmetry of the Lanczos tensor in its first two indices, resulting in the following symmetry of the Lanczos spinor: H ABCC ′ = 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 b a b = H abc g bc = H AA ′ BB ′ CC ′ ϵ BC ϵ B ′ C ′ = 0, (4.12) which after substituting the form of H AA ′ BB ′ CC ′ in (4.10) gives H b a b = _ ϵ A ′ B ′ H ABCC ′ +ϵ AB H A ′ B ′ CC ′ _ ϵ BC ϵ B ′ C ′ = −H ABCA ′ ϵ BC −H A ′ B ′ C ′ A ϵ B ′ C ′ = 0, (4.13) or, equivalently, −H ABCA ′ ϵ BC = H A ′ B ′ C ′ A ϵ 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 *H abc g bc = _ −iϵ A ′ B ′ H ABCC ′ +iϵ AB H A ′ B ′ CC ′ _ ϵ BC ϵ B ′ C ′ = −H ABCA ′ ϵ BC +H A ′ B ′ C ′ A ϵ B ′ C ′ = 0, (4.15) 42 The Lanczos spinor or, equivalently, H ABCA ′ ϵ BC = H A ′ B ′ C ′ A ϵ 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 D AD A ′ = 0 (4.17) (see [11], eq. 4.55). Once again calling upon the identity (3.33), although this time in reverse, we rewrite (4.17) as 1 2 ϵ BC H D AD A ′ = H A[BC]A ′ = 0. (4.18) Since, when anti-symmetrised on indices B and C, H ABCA ′ vanishes identically, it must instead be symmetric in B and C, that is H ABCC ′ = H A(BC)C ′ (4.19) (see [11], eq. 4.56), which when combined with the original symmetry of the Lanczos spinor (4.11) can be written as H ABCC ′ = H (ABC)C ′ (4.20) (see [11], eq. 4.57). At this point, there are eight remaining independent complex components of the Lanczos spinor which match the 16 real components of the Lanczos tensor after imposing the cyclic and trace-free conditions. We do not yet impose the divergence-free condition, but note that it has the following form in the spinor formalism 1 : H c ab ;c = ∇ CC ′ _ H CC ′ AB ϵ A ′ B ′ +H C ′ C A ′ B ′ ϵ AB _ = ϵ A ′ B ′ ∇ CC ′ H CC ′ AB +ϵ AB ∇ CC ′ H C ′ C A ′ B ′ = 0 (4.21) (see [11], eq. 4.59). Contracting through with ϵ A ′ B ′ and recalling the symmetry of the 1 The 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 ϵ AB , we find a similar expression for the conjugate Lanczos spinor: ∇ CC ′ H C ′ C A ′ B ′ = 0. (4.23) 4.2 Spinor form of Weyl-Lanczos equations Assuming the divergence-free gauge condition, however not invoking it explicitly and simply holding (4.22) and (4.23) as auxiliary conditions, we investigate the form of the 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 C abcd = Ψ ABCD ϵ A ′ B ′ ϵ C ′ D ′ + Ψ A ′ B ′ C ′ D ′ ϵ AB ϵ CD = ∇ DD ′ _ H ABCC ′ ϵ A ′ B ′ +H A ′ B ′ C ′ C ϵ AB _ −∇ CC ′ _ H ABDD ′ ϵ A ′ B ′ +H A ′ B ′ D ′ D ϵ AB _ +∇ BB ′ _ H CDAA ′ ϵ C ′ D ′ +H C ′ D ′ A ′ A ϵ CD _ −∇ AA ′ _ H CDBB ′ ϵ C ′ D ′ +H C ′ D ′ B ′ B ϵ CD _ −∇ EE ′ _ H E A CC ′ ϵ E ′ A ′ +H E ′ A ′ C ′ C ϵ E A _ ϵ BD ϵ B ′ D ′ −∇ EE ′ _ H E B DD ′ ϵ E ′ B ′ +H E ′ B ′ D ′ D ϵ E B _ ϵ AC ϵ A ′ C ′ +∇ EE ′ _ H E A DD ′ ϵ E ′ A ′ +H E ′ A ′ D ′ D ϵ E A _ ϵ BC ϵ B ′ C ′ +∇ EE ′ _ H E B CC ′ ϵ E ′ B ′ +H E ′ B ′ C ′ C ϵ E B _ ϵ AD ϵ A ′ D ′ . (4.24) Contracting through with ϵ A ′ B ′ ϵ C ′ D ′ , recalling that the Weyl spinor and its conju- gate inherit symmetry in their first and last index pairs from the curvature spinor X ABCD and that the Levi-Civita symbol is covariantly constant, leads to the following simplification: 2Ψ ABCD = ∇ E ′ D H ABCE ′ +∇ E ′ C H ABDE ′ +∇ E ′ B H CDAE ′ +∇ E ′ A H CDBE ′ (4.25) 44 The Lanczos spinor (see [11], eq. 4.58). These are the Weyl-Lanczos equations expressed in the spinor formalism, 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 condition as 1 2 ϵ DE ∇ CC ′ H ABCC ′ = ∇ C ′ [E H |AB|D]C ′ = 0, (4.26) or equivalently ∇ C ′ E H ABDC ′ = ∇ C ′ D H ABEC ′ (4.27) (see [11], eq. 4.61). Finally, we can use two applications of (4.27) with different index permutations to simplify (4.25) to its final form, with both gauge conditions invoked: Ψ ABCD = 2∇ E ′ D H ABCE ′ (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 in a given space-time is by applying the spin-coefficient formalism, briefly described in section 3.3, to find the Lanczos coefficients. For more information, one should consult section 4.4-12 in [11]. Chapter 5 Interpretation 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 state that the interpretation of the Lanczos potential “remains largely uninvestigated and is 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 ramifications of the Weyl-Lanczos equations in the Jordan form of general relativity. We then briefly discuss the ideas behind Robert’s 1995 investigation of the possibility of an effect 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 R ab in terms of the stress-energy tensor T ab present in the space-time. In particular we have a proportionality between the trace-reversed Ricci tensor and the stress-energy tensor, known as the Einstein field equations: R ab − 1 2 g ab R = 8πT ab . (5.1) These equations are a set of 10 coupled non-linear partial differential equations in the metric and its first and second derivatives 1 . As can be seen from the Ricci decomposition of the Riemann tensor (1.21), the 1 Note, 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, and thus 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 to a relation between the first derivatives of the Weyl tensor and the first derivatives of the Ricci tensor. Hence, the global form of the Weyl tensor is determined indirectly by the matter distribution. A once contracted form of the second Bianchi identity can be written using the Weyl tensor as C abcd ;d = J abc , (5.2) where we define J abc def = R c[a;b] + 1 6 g c[b R ;a] (5.3) (see [4], eq. 4.28-9). Since the Ricci tensor is completely determined locally by the stress-energy tensor, it is possible to rewrite (5.3) fully using T ab and its trace T. Therefore, we can think of (5.2) as the field equations of general relativity encoded as first-order differential equations for the components of the Weyl tensor, with source J abc . This form of general relativity bears striking resemblance to the first Maxwell equations in terms of the field tensor: F ab ;b = J a . (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 dual 2 of the Weyl tensor, by rewriting the traceless property (1.22) as *C ab cd;a = 0, (5.6) (see [18], eq. 14). The field tensor F ab is generated differentially in terms of the vector potential A a as F ab = ∇ [a A b] . (5.7) In a similar way, the Weyl-Lanczos equations (2.53) show that the Weyl tensor is 2 Or, 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 H abc . 5.2 Gauge tensor candidates In a 1975 article entitled The Theory of the Lanczos Spinor [18], Zund observes the above correspondences between general relativity and electromagnetism and asks the obvious question of whether there also exists a gauge group under which the equations of general relativity, as they are posed in (5.2), remain invariant; similar to how Maxwell’s equations are invariant under the U(1) transformation: A a →A ′ a = A a +∂ a φ (5.8) which leaves the field tensor invariant. More precisely, Zund asks whether there exists a tensor L abc , such that the transformation H abc →H ′ abc = H abc +L abc (5.9) (see [18], eq. 18) leave the Weyl tensor invariant: C ′ abcd = C abcd . (5.10) In his investigation Zund sought forms of L abc that leave the Weyl tensor invariant. One particular form he considered, as the closest analogy to electromagnetism, was L abc = ∇ c ψ ab (5.11) (see [18], eq. 19). ψ ab must be a bivector as the Lanczos tensor is anti-symmetric in its first two indices. Furthermore, the cyclic property and trace-free condition become *ψ ab;c = 0 and ψ ab ;a = 0, (5.12) which are equivalent to the source-free Maxwell equations, implying that ψ ab is a singular bivector. Unfortunately, the divergence-free condition causes difficulties, and thus we make the further assumption that ψ ab is recurrent, that is ψ ab;c = χ c ψ ab , (5.13) 48 Interpretation of the Lanczos tensor (see [18], eq. 20). With this assumption, the conditions (5.12) require the co-vector field χ c to be a principal null direction of ψ ab . The divergence free equation then becomes, χ a ;a = 0. Finally Zund found that C ′ abcd = C abcd if ψ c [b χ a];c = 0, (5.14) which is satisfied if χ a is a parallel null vector field. Hence, if χ a is both a parallel null vector field and principally null with respect to ψ ab , which itself is a singular recurrent bivector with recurrence vector χ a , then (5.11) is a possible form of the gauge field. There are certainly other forms of L abc which leave the Weyl tensor invariant. Zund concluded, however, that “only a deeper physical study of the gravitational field will indicate what kind of gauge tensor is appropriate.” It will certainly be fruitful to consider the algebraic structure of the full group of gauge transformations, and consider possibly restricting it to a subgroup which can be physically motivated by general relativity. Furthermore, considering the Weyl-Lanczos equations in spinor form 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], Roberts gives a potential effect of the presence of the Lanczos potential, analogous to the Aharonov-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 as a gauge theory, and thus we can replace partial derivatives with a gauge covariant derivative with respect to the gauge group of electrodynamics. Replacing the partial derivatives by gauge covariant derivatives in the Schroedinger equation gives rise to several factors of the vector potential, which affects charged particles in the system directly. Consider, for example, a space-time with vanishing field tensor, but non- trivial vector potential. Classically such a scenario would not have any electromagnetic effect on charged particles in the space-time. However, in the quantum realm this is not the case and the vector potential is shown to be, in a sense, ‘physical’. Roberts constructed many potential gauge covariant derivatives for the Jordan form of general relativity in terms of the Lanczos potential. In the end, the results of the study were inconclusive as to whether the Lanczos tensor may produce an effect similar 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-Riemannian geometry. The Weyl-Lanczos equations highlight that we should not consider the Weyl tensor as being fundamental, but rather the Lanczos tensor should be considered a fundamental constituent of four-dimensional Riemannian geometry. Moving to the spinor formalism, we see that the complex relationship of the Weyl-Lanczos equations simplifies incredibly in terms of the Weyl spinor and the Lanczos spinor, highlighting the power of simple operations in the spinor formalism in replicating complex tensorial operations. It seems that the Lanczos tensor is currently largely unnoticed, possibly in part due to our knowledge of classical gravity being regarded as somewhat complete while other fields often recognized as more fundamental or applicable take the stage. 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