A Note on Clifford Algebras in Gravitational Physics. Rot.pdf · Clifford Algebras One of the main objects we study in this paper are clifford bundles over Riemannian manifolds. This

A Note on Clifford Algebras in Gravitational Physics.

Thomas Rot

October 21, 2009

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Abstract

Clifford algebras are widely used in quantum mechanics. The gamma matrices, found in the Diracequation, are a representation of the four dimensional Clifford algebra with Lorentzian signature. Inother areas of physics the Clifford structure is not used often. We investigate the use of these algebrasin gravitational physics. A review of Clifford algebras are given and the algebraic structure is liftedto the tangent bundle for any (pseudo)-Riemannian manifold. We discover that certain sections of theClifford bundle can be interpreted as Lorentz transformations. The Dirac operator is defined, whichis a differential operator subordinate to the Clifford structure. Einstein’s field equations are writtenin terms of the Dirac operator. This formulation makes it clear that the field equations are waveequations. For this no approximation is made, all results are exact. As an example the field equationsfor diagonal metrics are explicitly computed in this formalism. We conclude with some suggestionsfor further research.

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Contents

1 Introduction 5

2 Clifford Algebras 82.1 Construction of the exterior algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 The dot and wedge product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 The scalar product and the norm of multivectors . . . . . . . . . . . . . . . . . . . . 13

3 The Clifford Structure and the Dirac Operator 153.1 The Clifford structure of the exterior bundle . . . . . . . . . . . . . . . . . . . . . . 153.2 The connection on the exterior bundle . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 The Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 The geodesic equations in the Clifford formalism . . . . . . . . . . . . . . . . . . . 213.5 A Critique on Vector Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Curvature Identities of the Dirac Operator 23

5 Einstein’s Equations 285.1 The field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 The physical interpretation of the field equations . . . . . . . . . . . . . . . . . . . . 295.3 Solution techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6 The Fiducial Frame and the Spin Connection 31

7 Examples of Curvature Calculations 357.1 Diagonal Metrics in vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.2 The Schwarzschild metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

8 Lorentz Transformations and Bivector fields 418.1 Reflections and rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.2 Lorentz transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9 Outlook 479.1 Fundamental geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.2 The field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.3 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.4 Spinors and quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3

4 CONTENTS

9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

10 Appendix Bundles 4910.1 Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.2 Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.3 Pseudo-Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Chapter 1

Introduction

General Relativity (GR) is a difficult theory. The simplest non trivial example, that of the Schwarzschildmetric, is already computationally hard. We will try to remove some of the computational difficultiesby an attempt to formulate GR in a new formalism. The formalism GR is conventionally formulatedin is that of tensor calculus on manifolds. A main object of study, the Riemann tensor, consists of a 4indexed tensor. That means it has 4 × 4 × 4 × 4 = 256 components. A feature of this tensor is thehigh degree of symmetry. Many indices of the tensor can be related to each other via the switching ofindices. Thus the Riemann tensor stores superfluous information. One might wonder if the calcula-tions can be simplified if one does not carry along all the redundant information in calculations.

We will study a formulation of GR, not as a theory on the tensor bundle, but on a construction wewill call a clifford bundle. This clifford bundle is essentially a bundle of Clifford algebras, defined oneach tangent space at each point of the original Riemannian manifold. The algebraıc properties of theclifford bundle we will try to exploit, hopefully resolving computational problems in standard GR,and gaining new physical insight in the theory.

The origins of this approach can be traced to Hestenes and Sobczyk [6]. Hestenes’ aim is to for-mulate all modern physics and mathematics in terms of clifford algebras1. He argues the language isextremely potent and simple, able to describe all physical theories, and most of mathematics [4].

Hestenes, in his book [6], tries to generalize the Clifford algebra to manifolds. He calls the objectsvector manifolds. He never clearly defines this notion, making the claim of a mathematical correcttheory strange. His construction is outlined in section 3.5. While he does not define the concept of avector manifold rigorously, it can be done I believe. This definition uses a generalization of Nash’sembedding theorem[9] by Clarke[3]. The construction should be isometrically equivalent to the con-struction we will coin in chapter 3. We will not prove this claim. Our construction is constructive, wedo not merely prove existence, which is an added bonus.

After we have constructed the clifford bundle, we will study standard curvature objects, and in-terpret them in the new formalism. The Einstein equations will turn out to be a generalization of thewave equation. This interpretation is in the standard formulation of GR only available in the weakfield limit.

1Hestenes calls Clifford algebra by the name geometric algebra. This name originally was coined by Clifford, who firstdescribed the construction. Most of the literature speaks of Clifford algebras, hence we have stuck with this name.

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6 CHAPTER 1. INTRODUCTION

We will also study active and passive Lorentz transformations in the formalism. We will showthat the clifford algebra contains a representation of the Lorentz group, and we will find a beautifulformula describing lorentz transformations in terms of a bivector fields. This is done in chapter 8.

We will finally compute the Schwarzschild metric in the new formalism in chapter 7, as an exam-ple how curvature calculations can be done in this formalism.

It is interesting to note that in recent years Doran, Lasenby, and Gull [7] have used the formalismof clifford bundles to phrase GR as a gauge theory. This is a remarkable success, since all other fun-damental quantum theories are formulated as gauge theories.

We finally state some possible research directions for the use of clifford algebras in gravitationalphysics.

Notations and conventions

Some of the conventions used in this thesis are not standard, or are non-uniform in the literature. Forreference we list them here.

Convention 1.1. We will uphold the following conventions regarding notations, and regarding thephysical theory of general relativity.

1. We assume the standard formulation of general relativity is correct; i.e. gravitational physics istorsion free.

2. General Relativity is a theory on a pseudo-Riemannian manifold. Officially we should speakof the pseudo-metric, or bilinear form. Writing pseudo for every instance of metric reduces thereadability. Therefore we follow the physical convention that we will call the pseudo-metricby the name metric. This will not cause confusion since we still make a distinction between aRiemannian manifold and a pseudo-Riemannian manifold, from this the essence of the metricwill be clear.

3. We assume all maps between manifolds to be smooth unless explicitly stated that they are not.

4. We use Einstein’s summation convention, repeated indices are summed over, unless explicitlystated. We use a metric of signature {−1,+1,+1,+1}.

5. It is standard to view vector fields as differential operators in differential geometry. One writes

X(f) = df(X) for f ∈ C∞(M) and X ∈ Γ (TM) . (1.1)

it is then customary to denote the basis of vector fields generated by the chart functions ki aspartial derivatives ∂i. This habit will cause confusion with our convention to write the cliffordproduct by juxtaposition. It is not clear if ∂i∂j should be understood as a clifford multiplicationof ∂i and ∂j or if we should differentiate twice. We will therefore not follow the general conven-tion. The basis vectors generated by the chart functions ki will be denoted by γi. Differentiatingwith respect to a vector field will be written as

∂X(f) := df(X). (1.2)

We also write, for a coordinate basis γµ,

∂µ(f) := df(γµ), (1.3)

if it is clear that there is only one coordinate base used.

7

Chapter 2

Clifford Algebras

One of the main objects we study in this paper are clifford bundles over Riemannian manifolds. Thissection is a basic review of clifford algebras, the algebra over which the bundle is defined. For thosefamiliar with the theory of clifford algebras this might serve as an introduction in the notation used inthis thesis. This part is highly inspired by the introduction to tensor algebras by Looijenga[8].

In this thesis we only consider clifford algebras defined on vector spaces over the field of the realsR. Most of the theorems in this section carry over to general fields K. Only in characteristic1 2 someproblems arise, due to the triviality of bilinear forms in that characteristic. We will not be studyingalgebra in this thesis, therefore we do not aim to formulate the most general theorems on cliffordalgebras possible, a wealth of information is available in the literature.

We will view the Clifford algebra as a multiplicative product on the exterior algebra of a vectorspace when the underlying vector space is endowed with an bilinear form. This is in contrast withmost approaches to clifford algebras. It is customary to define the algebra as a quotient of the tensoralgebra, with an ideal generated by products of the form v⊗ v− 〈v, v〉. This construction is harder tolift to a Riemannian manifold. It is however not impossible; the final constructions are equivalent.

2.1 Construction of the exterior algebra

In this subsection we construct the exterior algebra as a quotient algebra of the tensor algebra. Withoutfurther ado we define the tensor algebra.

Definition 2.1. Let V be a finite dimensional vector space over R. The tensor algebra T (V ) is definedas

T (V ) :=⊕k≥0

V ⊗k. (2.1)

This is clearly an infinite dimensional vector space. It is a ring if we take ⊗ as the product. Wewill now construct out of this algebra a quotient algebra known as the exterior algebra. This will be

1the characteristic of a field is the smallest prime number of times that one has to repeat adding the multiplicative identityelement 1 to obtain the additive identity element 0. For fields like R or C this never happens, and we say that these fieldshave characteristic 0. The finite fields Zp have characteristic p. In our application we will always work with the field ofreals R so this remark should not trouble us.

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2.1. CONSTRUCTION OF THE EXTERIOR ALGEBRA 9

an algebra with dimension dim(∧V ) = 2 dim(V ). We will proceed by defining a new multiplicative

product on this space resulting in the clifford algebra.

Let E(V ) be the two sided ideal generated in T (V ) by all elements of the form v⊗ v with v ∈ V .Thus for x ∈ E(V ) we have a representation

x =∑i

ai ⊗ (vi ⊗ vi)⊗ bi where ai, bi ∈ T (V ) and vi ∈ V. (2.2)

We have all the tools available to define the exterior algebra.

Definition 2.2. We define the exterior algebra∧V of a vector space V as∧

V = T (V )/E(V ). (2.3)

The product in∧V is the image of the tensor product, and is denoted by the symbol ∧. We will also

write∧V for the exterior algebra.

The following lemma is folklore.

Lemma 2.3. Let V be a finite dimensional vector space, and let (e1, ..., en) be a basis. Then thefollowing statements hold.

1. For any v1, v2, · · · , vk ∈ V we have

vσ(1) ∧ vσ(2) ∧ · · · ∧ vσ(k) = sgn(σ)v1 ∧ v2 ∧ · · · ∧ vk (2.4)

Here σ is an element of the symmetry group with k elements.

2. The set

B = {ei1i2···ik := ei1 ∧ ei2 ∧ · · · ∧ eik | with 1 ≤ i1 < i2 < · · · < ik ≤ n}⋃{1}

(2.5)is a basis for

∧V . In particular the exterior algebra is an 2n dimensional vector space.

3. We have the decomposition∧V =

⊕0≤k≤n

∧kV with

∧kV = span {ei1i2···ik} . (2.6)

Proof. (1) The proof of the first statement is not hard. Let v, v′ ∈ V . Then

(v + v′) ∧ (v + v′) = v ∧ v + v ∧ v′ + v′ ∧ v + v′ ∧ v′. (2.7)

For any w ∈ V we have w ∧ w = 0, since this is an element of the ideal K(V ). We havev ∧ v′ + v′ ∧ v = 0, yielding v ∧ v′ = −v′ ∧ v. Using induction we now see that the first state-ment in the lemma is true.

(2) It is clear thatB generates∧V . The tensors ei1⊗· · ·⊗eik generate T (V ), hence ei1∧· · ·∧eik

generate∧V . The latter formula is zero unless all ei are distinct. We can always arrange the order

at the cost of a minus sign, by part (1) of this proof. Therefore B generates∧V . We compute

10 CHAPTER 2. CLIFFORD ALGEBRAS

the cardinality of the set B. This is a basic combinatorial problem. For each k we have to pick kelements out of a choice of n elements. We disregard the order of the elements, because afterwardswe rearrange it in increasing order at the cost of some minus signs. It is well known that there are

n!k!(n−k)! possibilities to do so. Now we have to sum for all k ∈ {0, 1, ..., n}.. It is easy to verify that

n∑k=0

n!k!(n− k)!

= 2n. (2.8)

Hence the cardinality of the set B is 2n.

(3) This part directly follows from (2) with the conclusion that the space is finite dimensional.

2.2 Clifford algebras

We have opted to introduce the clifford algebra as an additional structure on the exterior algebraof vector spaces on which a bilinear form 〈 · , · 〉 is defined. This approach nicely generalizes to(pseudo)-Riemannian manifolds. We first need some additional mappings defined on the exterioralgebra. The reversion map is not needed for the construction of the clifford algebra, but is needed forrotor descriptions of rotations cf. 8.2.

Definition 2.4. The grading mapping grad assigns to pure elements u ∈∧l V the grade l. It is not

defined for none-pure elements. The grade involute is the mapping α :∧V 7→

∧V : u 7→ u∗ defined

byu∗ = (−1)grad(u) u, (2.9)

for a pure element and extended to the whole of∧V by imposing linearity. The reversion map is the

mapping β :∧V 7→

∧V : u 7→ u defined by

u = (−1)grad(u)(grad(u)−1)

2 u, (2.10)

for a pure element, and is extended to the whole of∧V by linearity.

Remark 2.5. Both mappings are involutions on∧V , i.e. they square to the identity. The first involu-

tion generates a Z2 grading.

Lemma 2.6. The grade involute generates a Z2 grading. The ±1 eigenspaces of the grade involuteare the gradation. We denote these by

∧± V . The Z2 grading implies that for any x±, y± ∈∧± V

we have the relations

x+ ∧ y+ ∈∧+

V x+ ∧ y− ∈∧−

V x− ∧ y− ∈∧+

V. (2.11)

and we have the decomposition∧+V =

⊕k≥0

∧2kV

∧−V =

⊕k≥0

∧2k+1V. (2.12)

Proof. The lemma is immediate.

2.2. CLIFFORD ALGEBRAS 11

An explanation of the nomenclature of the reversion map might be in order. Consider the expan-sion of a pure element in a basis γi

u = λγk1 ∧ · · · ∧ γkn (2.13)

Then a simple computation yieldsu = λγkn ∧ · · · ∧ γk1 . (2.14)

Thus the order of the elements is reversed.

The above tools allow us to define the left contraction y. The left contraction is only used in thedefinition of the clifford algebra, it will not be used after the construction of the algebra.

Definition 2.7. Let x, y ∈∧1 V , u, v ∈

∧V , and λ ∈

∧0 V . The left contraction is the mappingy :∧V ×

∧V 7→

∧V defined by the equations

1.x y y = 〈x, y〉 (2.15)

2.x y(u ∧ v) = (x yu) ∧ v + u∗ ∧ (x y v) (2.16)

3.(u ∧ v) yw = u y(v yw) (2.17)

4.λ yu = 0 (2.18)

and the demand that y is bilinear.

It is clear that this defines a unique mapping. The first statement tells us that contraction on vectorsequals taking the inner product. The second and third statement allows us to reduce the applicationof y to any element of the exterior algebra. The left contraction lowers the grade of the element it isworking on. Note that this application is not associative.

Lemma 2.8. We have, for u ∈∧k V , v ∈

∧l V , the property

u y v ∈∧l−k

V (2.19)

in particular u y v = 0 if k > l.

Proof. If k = l = 1 the statement is exactly the first axiom of definition 2.7. We will first applyinduction over l. Suppose the statement holds for l. Then we compute

x y(y ∧ u) =(x y y) ∧ u+ y ∧ (x yu)=〈x, y〉u+ y ∧ (x yu)

(2.20)

But the right hand side is an element of∧l V . Since left contraction is linear we conclude that

x yw ∈∧gradw−1(V ). Now we apply induction over k. Suppose u y v ∈

∧grad(v)−grad(u)(V ), then

(x ∧ u) y v =x y(u y v) (2.21)

by our hypothesis u y v ∈∧grad v−gradu(V ), and the preceding yields x y(u y v) ∈

∧grad v−gradu−1(V ).By linearity we conclude that u y v ∈

∧grad v−gradu. The last statement of the lemma directly followsfrom the fourth axiom of the left contraction.


We finally are able to define the clifford product.

Definition 2.9. The clifford product µ :∧V ×

∧V 7→

∧V : (u, v) 7→ u v is defined as

xu = x yu+ x ∧ u if x ∈∧1

V and u ∈∧V. (2.22)

and this is used to generate the clifford product on any two elements of the exterior algebra by impos-ing associativity and linearity of the product.

Definition 2.10. Let V be a finite dimensional vector space with an (pseudo)-inner product. TheClifford Algebra is the vector space

∧V equipped with the clifford product µ.

Remark 2.11. The grade involute also induces a Z2 grading with respect to the clifford product.Lemma 2.6 also holds for the clifford product.

Lemma 2.12. The grade involute generates a Z2 grading. The ±1 eigenspaces of the grade involuteare the gradation. We denote these by

∧± V . The Z2 grading implies that for any x±, y± ∈∧± V

we have the relations

x+y+ ∈∧+

V x+y− ∈∧−

V x−y− ∈∧+

V. (2.23)

and we have the decomposition∧+V =

⊕k≥0

∧2kV

∧−V =

⊕k≥0

∧2k+1V. (2.24)

2.3 The dot and wedge product

The clifford product has a grade lowering and a grade raising part. It is sometimes useful to considerthese operations separately. The grade raising part is the exterior product already familiar of course.The grade lowering part is similar to the left contraction, but yet undefined.

Definition 2.13. The projection map πk maps an element u ∈∧V to its k-th pure part. The dot

product is defined byu · v = π| grad(u)−grad(v)|(u v) (2.25)

for pure elements. For non-pure elements we extend the definition by demanding that · is bilinear.Another expression for the wedge product can be given by

u ∧ v = π| gradu+grad v|(u v). (2.26)

Convention 2.14. The three products thus defined, i.e. the clifford, dot, and wedge product, arenot associative with respect to each other. We need a convention for the order of application of theseoperators, to reduce excessive cluttering of expressions with unnecessary parentheses. The conventionwe use is the following

• The highest priority has the dot product.

• Then comes the wedge product.

• The clifford product has the lowest priority

Example 2.15. We put in the parentheses in the following expression

u · w ∧ xy · z = ((u · w) ∧ x)(y · z), (2.27)

to clarify our convention.

2.4. THE SCALAR PRODUCT AND THE NORM OF MULTIVECTORS 13

2.4 The scalar product and the norm of multivectors

We have defined several products on the clifford algebra. We cannot speak of the length, or the anglebetween different multivectors. We will need these operations. Once we have defined the scalarproduct and the norm, it will turn out that the clifford product is a Banach algebra. An interestingproperty to say the least. The apparatus constructed here is a bit cumbersome. This is basicallybecause we do not assume that the inner product on V is positive definite, it is assumed to be non-degenerate however. The scalar product is the positive definite cousin of the inner product on theunderlying vector space.

Definition 2.16. Let ek be a orthogonal basis of the vector space V . That is 〈ek, el〉 = ηkl where η isthe signature indicator. Define the parity operation :

∧V →

∧V on the basis vectors ek by

ek = ηkkek (no sum) (2.28)

and extend this to the whole of∧V by demanding that this operation is linear and commutes with the

clifford product. The scalar product

? :∧V ×

∧V → R : (u, v) 7→ u ? v (2.29)

is defined byu ? v := π0(u v). (2.30)

Remark 2.17. The scalar product is only used to prove some functional analytic results concerning theclifford exponential map 8.2. We will therefore not use the parity operation in the forthcoming much.We will consider the reciprocal frame, defined in chapter 6, which is a basis of the tangent space dualto the coordinate basis. This is basically the coordinate basis, acted upon with the parity operation.

Lemma 2.18. The scalar product is an inner product.

Proof. We have to verify three properties. First we do symmetry. Let u, v ∈∧V , then

u ? v = π0(uv) = π0(uv), (2.31)

since for scalars the parity equals the identity. Since π0 only projects, we can continue

π0(uv) = π0(uv) = π0(vu) = v ? u. (2.32)

Hence we have verified the symmetry property u ? v = v ? u for all multivectors u, v ∈∧V .

We also can easily verify linearity in the first argument. Let u, v, w ∈∧V , α ∈ R, then

(αu+ v) ? w =π0 ((αu+ v)w) = π0 (αuw) + π0 (vw)=απ0 (uw) + π0 (vw) = αu ? w + v ? w.

(2.33)

The linearity and symmetry combined yield that the scalar product is linear in the second argument aswell.

The last property is usually the hardest to verify for inner products. This is the positive definitenessproperty u ? u ≥ 0 for all u ∈

∧V and is only zero if u = 0. We can check that all basis elements

ei1ei2 · · · eik are orthogonal and have a positive norm ||u|| := (u?u)12 . These span the clifford algebra∧

V . Hence we have proven that the scalar product is an inner product.


Remark 2.19. All∧k V are orthogonal hyperplanes with respect to this inner product. The inner

product preserves the direct sum decomposition.

Theorem 2.20. The clifford algebra, endowed with the norm defined above, is a unital Banach alge-bra. In particular

||u v|| ≤ ||u|| ||v|| for all u, v ∈∧V. (2.34)

Proof. The underlying vectorspace∧V is finite dimensional, hence Banach. To prove that

∧V is a

Banach algebra, we only need to verify the identity. We compute the norm of the product of u and v.

||u v|| = π0(u vu v) = π0(u vvu). (2.35)

We can decomposevv = π0(vv) + π⊥0 (vv). (2.36)

since the clifford algebra has a direct sum decomposition. With π⊥0 we mean the projection outside∧0 V , that is projection on⊕

i≥1

∧i V . We apply this

||u v|| =π0(u (π0(vv) + π⊥0 (vv))u)

=π0(uu)π0(v v) + π0(uπ⊥0 (vv)u)≤ ||u|| ||v||

(2.37)

since the second term is obviously non-negative. It is trivial to see that the norm of the unit 1 is 1,

||1|| = π0(11) = π0(1) = 1. (2.38)

Thus the clifford algebra is a unital Banach algebra.

Remark 2.21. A caution on nomenclature is needed. We have defined several products, with namescommonly associated with inner products. These are the metric, left contraction, the dot product,and the scalar product. This is confusing. Here we attempt to reduce the confusion by listing thedifferences and similarity between these products.

1. The metric is defined on the vector space V from which we construct the clifford algebra. Thismetric does not have to be positive definite2. It measures angles and lengths of vectors. Lengthsof vectors do not have to be positive. It is not an inner product generically!

2. The left contraction is used to construct the clifford algebra. It is not needed in the rest ofthe thesis, and we will consider it devoid of any physical meaning. It is merely mathematicalapparatus needed for the construction.

3. The dot product is a projection operator. The expression u · v measures the volume of u con-tained in v and projects it to the orthogonal complement in the span of v.

4. The scalar product is the only product that is actually inner. We will not really give this aphysical meaning. It is used in chapter 8.2 to prove some functional analytic facts about theclifford exponential map.

2In general relativity it is not

Chapter 3

The Clifford Structure and the DiracOperator

In this section we endow the exterior bundle of a pseudo-Riemannian manifold with a clifford product.The clifford product is defined in chapter 2. The exterior bundle is briefly recalled in appendix 10.The clifford product will be smooth in the same sense the metric is smooth.

We proceed by lifting basic constructions on TM to the whole of the exterior bundle∧TM . The

Levi-Civita connection is a prime example. The lifted Levi-Civita connection does not respond to theclifford structure. That is why we introduce the Dirac operator. This operator is defined for sectionson all of the exterior bundle. The Dirac operator is a generalization of the curl, the divergence andgradient operators encountered in basic vector calculus. This operator has many interesting analyticand algebraic properties. It also allows a study of the topology of spacetime, but we will not pursuethis direction.

3.1 The Clifford structure of the exterior bundle

Let M be a pseudo-Riemannian manifold with metric g. The exterior bundle∧TM and the exterior

cobundle∧T ∗M are manifolds as well due to corollary 10.3. The mappings + and ∧ are smooth.

We can introduce the clifford product as explained in 2 on each exterior space∧TpM , by using as

a symmetric bilinear form the metric gp on the tangent space. Since this construction is given ascompositions of the maps + , gp , and ∧, which are smooth in the basepoint p, the clifford product µpdepends smoothly on p ∈M . Let’s formalize this

Definition 3.1. Let M be a pseudo-Riemannian manifold with metric g. Endow the exterior algebraof each tangent space, denoted

∧TpM with a clifford structure µp. Let u, v ∈ Γ (

∧TM) be smooth

sections of the exterior bundle. Then the mapping µ : Γ (∧TM)×Γ (

∧TM)→ Γ (

∧TM) defined

by the equationµ(u, v)(p) = (p, µp(u(p), v(p))) (3.1)

maps pairs of smooth sections to smooth sections. It is called the clifford product. We usually omitthe symbol µ, and write the clifford product by juxtaposition, i.e.

uv := µ(u, v). (3.2)

15

16 CHAPTER 3. THE CLIFFORD STRUCTURE AND THE DIRAC OPERATOR

Proof. For each p ∈ M the clifford structure on∧TpM is given as a composition of scalar mul-

tiplication, summation, the wedge product and the metric. These operations are all smooth in p.Composition of smooth maps is smooth, hence the clifford product is smooth.

3.2 The connection on the exterior bundle

We want to construct a differential operator on the exterior bundle∧TM which respects the clifford

product in some sense. We first lift the Levi-Civita connection to the exterior bundle. In a sense itdoes not know of the algebraic structure of the exterior and clifford products. First recall the notionof the Levi-Civita connection. The proof that this connection is unique can be found in any book onRiemannian geometry, so we do not repeat this here.

Definition 3.2. There exists a unique connection, called the Levi-Civita connection ∇ : Γ (TM) ×Γ (TM) → Γ (TM) on the tangent bundle TM which satisfies the following two identities for anythree vector fields X,Y, Z ∈ Γ (TM)1

•∇XY −∇YX − [X,Y ] = 0 (3.3)

•∂Xg(Y, Z) = g(∇XY, Z) + g(Y,∇XZ) (3.4)

The first property is known as the symmetry property. The second property encapsulates the flatness ofthe metric with respect to the Levi-Civita connection, and is therefore known as the flatness property.

This connection can be lifted to the exterior bundle.

Lemma 3.3. Let M be a pseudo-Riemannian manifold with metric g. Let ∇ be its Levi-Civita con-nection on TM . The operator ∇ : Γ (TM)× Γ (

∧TM) 7→ Γ (

∧TM) defined by the equations

∇Xf =X(f)

∇XY =∇XY∇Xu ∧ v =∇Xu ∧ v + u ∧ ∇Xv∇X(u+ v) =∇Xu+ ∇Xv

(3.5)

is a connection on∧TM . We took f ∈ Γ(

∧0 TM), X,Y ∈ Γ(∧1 TM), and u, v ∈ Γ (

∧TM).

Since ∇X = ∇X on Γ(∧1 TM) we denote the connection ∇ on Γ (

∧TM) with the same symbol as

the Levi-Civita connection∇ = ∇ without confusion.

Proof. We need to verify three properties to establish that this is indeed a connection. These are:

• Local nature. If we restrict both X and u to an open neighborhood U ⊂ M , the connectionshould remain invariant under this restriction, i.e.

(∇Xu)|U = ∇X|Uu|U. (3.6)

1We write ∇Xu := ∇(X, u), as this is common practice

3.3. THE DIRAC OPERATOR 17

• R-Linearity. The identity ∇X(u+λv) = ∇Xu+λ∇X(v) should hold, for allX ∈ Γ(∧1 TM),

and all u, v ∈ Γ (∧TM), and λ ∈ R.

• Variance in X. For all functions f the equality

∇fX = f∇X (3.7)

holds.

• The Leibniz Rule. For all functions f we should have

∇Xfu = ∂X(f)u+ f∇Xu. (3.8)

The local nature of the operator ∇X is clear, since it is completely defined in terms of the localoperators ∇ and X . It also is R linear in the second argument, because both the exterior product andthe Levi-Civita connection are. We can easily verify that the variance in X is satisfied, it inheritsfrom the Levi-Civita connection. If we apply ∇fX on Y ∧u we get by induction, assuming ∇f Xu =f∇Xu,

∇fXY ∧ u =(∇fXY ) ∧ u+ Y ∧ ∇fXu=(f∇XY ) ∧ u+ Y ∧ (f∇Xu)

=f∇XY ∧ u.(3.9)

The last axiom of a connection we need to check is the Leibnitz rule. The third axiom for our operator∇ yields for f ∈ Γ(

∧0 TM)

∇Xf u = ∇Xf ∧ u = X(f) ∧ u+ f ∧ ∇Xu = X(f)u+ f∇X(u) (3.10)

Hence ∇ is a connection on∧TM .

3.3 The Dirac operator

It is customary to define, next to the tangent bundle TM the cotangent bundle T ∗M . Sections ofthis bundle are interpreted as 1-forms, assigning numbers to vectors. The metric product on thetangent bundle induces a metric on the cotangent bundle. We can analogously to

∧TM define a

clifford structure on the bundle∧T ∗M . The metric, being non-degenerate, allows us to identify both

bundles. Hence we have no need for the cotangent bundle. We now study the notion of a reciprocalframe, a notion usually associated to the dual bundle. This reciprocal frame allows the introduction ofthe Dirac operator in a constructive fashion.

Definition 3.4. Let γµ be a coordinate frame. The reciprocal frame γµ is defined via

γµ := gµνγν . (3.11)

where the inverse metric gµν is defined as the inverse of the matrix gµν := γµ · γν . We have thereciprocal relation

γµ · γν = δµν . (3.12)


Proof. The claim in the definition is a short computation.

γµ · γν = gµαγα · γν = gµαgαν = δµν . (3.13)

We disregard the convention that upper indices are reserved for forms, and lower indices are re-served for vectors. Both the frame and the reciprocal frame are sections of

∧TM , the same space.

There is some trouble with this. One might expect that if one computes the coordinate frame, in dif-ferent coordinates, that the reciprocal frame would transform similarly. This is not true, it transformsas the dual.

Convention 3.5. In the rest of the paper we will use γµ for the basis vectors arriving from a coordinatechart. In general these can only be defined locally. All theorems and lemmas are therefore localstatements. These local statements can usually be patched together to form global statements. Somecare should be taken though. Even though the Dirac operator, defined below, is defined globally, theformula defining it is a local formula. In general a manifold cannot be described in one chart. Foreach chart the definition is correct. If two charts overlap, the formula is the same for both coordinatesystems.

The Levi-Civita connection extended to the exterior bundle knows nothing of the clifford structurewe have placed upon it. In the following we define a derivative operator, known as the Dirac operator,which does know of this structure. It is both a generalization of the divergence and the curl operators,known from classical analysis.

Remark 3.6. On flat space, the Dirac operator is an invertible mapping, where the curl and the di-vergence operators are not invertible, which is shown in [6]. I currently do not know if this can begeneralized to generic (pseudo)-Riemannian manifolds.

Definition 3.7. The Dirac Operator is the mapping 6∇ : Γ(∧

TM)→ Γ

(∧TM

)defined in local

coordinates via the equation6∇u = γµ∇γµu := µ(γµ,∇γµ u). (3.14)

The clifford product gave rise to the dot product and the exterior product, the grade loweringand grade raising part of the clifford product, see definition 2.13. The differential analogue of thisconstruction is the divergence and the curl, which are the grade lowering and grade raising parts ofthe Dirac operator. Both these operators satisfy the Leibniz rule, and are thus differential operators.

Definition 3.8. The curl operator 6∇∧ : Γ(∧

TM)→ Γ

(∧TM

)is the operator defined by the

equation6∇ ∧ u := γµ ∧∇γµu (3.15)

The curl operator acting on functions is known as the gradient. Analogously the divergence operator6∇· : Γ

(∧TM

)→ Γ

(∧TM

)is defined by

6∇ · u := γµ · ∇γµu. (3.16)

Remark 3.9. Like the wedge and dot products, these operators have a graded structure. The curloperator raises the grade of a pure multivector, while the divergence operator lowers it. By linearity wesee that for non-pure multivectors, we can see the raising and lowering componentwise respectively.

3.3. THE DIRAC OPERATOR 19

Convention 3.10. The operators defined above are not associative with respect to each other. Thereforewe need to establish some rules, in which order we apply the operators if we do not use parentheses.This is done to reduce excessive cluttering of parentheses. The convention is thus

• The divergence operator has the highest priority,

• The curl operator has the second priority,

• The Dirac operator has the lowest priority.

This convention completely agrees with the multiplication convention for the exterior, interior andclifford product on a vector space, cf. convention 2.14. It is important to remember this convention toreduce confusion. After minus signs, associativity problems create the most calculational errors.

Example 3.11. As an example for the use of our convention we write down the correct parentheses inthe formula below, for any u ∈

∧TM

6∇ ∧ 6∇ · 6∇6∇u = (6∇ ∧ ( 6∇ · 6∇))6∇u. (3.17)

This is not hard, and in fact will make our life easier. It is important to remember this convention.

Some basic facts about the Dirac operator should be stated

Lemma 3.12. The Dirac operator satisfies the following properties

1. The Dirac operator is independent of the coordinate chart we expressed it in.

2. The Dirac operator can be decomposed in the gradient and the curl

6∇u = 6∇ · u+ 6∇ ∧ u. (3.18)

3. The Dirac operator is associative, i.e.

6∇(6∇u) = (6∇6∇)u for any section u ∈∧TM, (3.19)

hence we can write without ambiguity 6∇6∇u.

Proof. (1) Suppose γµ are basis vectors of another coordinate chart (with coordinate charts xµ). Thenthese satisfy the transformation rule:

γµ =∂xα

∂xµγα, (3.20)

while the reciprocal frame has transformation rule

γµ =∂xµ

∂xαγα. (3.21)

Using that ∂xα

∂xµ is the inverse matrix of ∂xµ

∂xα yields

6∇ = γµ∇γν = ∂xµ

∂xα γα∇ ∂xβ

∂xµγβ

= ∂xµ

∂xα∂xβ

∂xµ γα∇γβ = δβαγ

α∇γβ= γα∇γα ,

(3.22)


establishing the required independence of the coordinate chart. In the second line we applied the vari-ance property of the connection.

(2) The reciprocal frame consist of vectors. The clifford product of a vector with a multivector isparticularly simple. That is X u = X ·u+X ∧u for any X ∈

∧1 TM and u ∈∧TM by definition.

This yields the decomposition

6∇u = γµ∇γµu = (γµ ·+γµ∧)∇γµu = 6∇ · u+ 6∇ ∧ u (3.23)

(3) The proof of the last statement follows from a straightforward computation by applying theassociativity of the clifford product

6∇(6∇u) =γµ∇γµ(γν∇γνu) = γµ∇γµγν(∇γνu)=γµ(∇γµγν)(∇γνu) + γµγν∇γµ(∇γνu).

(3.24)

on the other hand, we can evaluate

(6∇6∇)u =(γµ∇γµγν∇γν )u=(γµ(∇γµγν)∇γν )u+ (γµγν∇γµ∇γν )u=γµ(∇γµγν)(∇γνu) + γµγν∇γµ(∇γνu).

(3.25)

proving the associativity of the Dirac operator and thereby all statements of the lemma.

Note that the divergence operators is not associative, in contrast with the Dirac and curl operators.The Dirac operator has interesting properties. On flat space the Dirac operator is the root of thelaplacian. See example 3.13. That is 6∇2 = ∆. In curved space the Dirac operator also stores thescalar curvature. This is known as Weitzenbock’s theorem. The Ricci curvature is the curvature weare interested in physically. Via some tricks we can compute the Ricci curvature (and hence theEinstein tensor) from the Dirac operator acting on sections of our manifold. We proceed to state theEinstein equations (the equations governing the dynamics of general relativity) in terms of the Diracoperator. We will interpret the resulting formula as a wave equation. This is fundamentally differentfrom newtonian physics. The corresponding equation is the Poisson equation. This is completelystatic. Gravitational information travels with infinite speed in newtonian physics.

Example 3.13. As an example let’s calculate the square of the Dirac operator acting on a function inflat space. The signature is determined by the metric ηµν . Let f : Rn → R then

6∇6∇f =γµ∇γµγν∇γνf=(∂µ∂νf)γνγµ

(3.26)

We used that all the Christoffel symbols vanish, hence∇γµγν = 0, see lemma 4.1, hence the covariantderivatives become ordinary derivatives. We now split the clifford product γµγν into a function and abivector part.

6∇6∇f = (∂µ∂νf)ηµν + (∂µ∂νf)γµ ∧ γν . (3.27)

In this equation we sum over µ and ν. The ordinary derivatives commute ∂µ∂νf = ∂ν∂µf , and theexterior product is anticommutative, hence the second term drops out of the equations. Therefore

6∇6∇f = sgn(ηµµ)(∂2µf) (3.28)

3.4. THE GEODESIC EQUATIONS IN THE CLIFFORD FORMALISM 21

In the last expression it is understood that there is a sum over µ. This is one of those expressions whereEinstein’s summation convention fails. We see that in Euclidean 3−space (sgn ηµµ = (+1,+1,+1))the square of the Dirac operator is the laplacian. In Minkowski space (sgn ηµµ = (−1,+1,+1,+1)the square of the Dirac operator is the wave operator if we act on functions. If we act on vectors,we would find the Laplacian, or wave operator, acting on the coefficients of these vectors. The samephenomenon happens for bivectors and higher multivectors.

3.4 The geodesic equations in the Clifford formalism

The geodesic equations describe motions of freely falling particles, i.e. they are unaccelerated. Theyare derived by imposing the condition

∇x x = 0 (3.29)

on a path I →M , for some interval I ⊂ R. Writing this out in a coordinate basis x = xλγλ yields

(xλ + Γλµν xµxν)γλ = 0. (3.30)

We applied the chain rule in the derivation.

Lemma 3.14. The geodesic equations are equivalent to

(x · 6∇)x = 0 (3.31)

Proof. No deep analysis is needed for the derivation. We start with the definition of the Dirac operator,and we expand the left vector x = xµγµ in a coordinate basis.

(xµγµ · γν∇γν )x =0(xµδνµ∇γν )x =0

∇xx =0.

(3.32)

In the last step we used variance of the connection in the first argument, i.e. xν∇γν = ∇x.

3.5 A Critique on Vector Manifolds

As mentioned in the introduction, this research originated in trying to understand the notion of a vectormanifold. Hestenes[6] does not give a rigorous definition of a vector manifold. His construction is inthe following spirit2.

Definition 3.15. Let M be a (pseudo)-Riemannian manifold. There exists an isometric embeddingφ : M ↪→ Rp,q into a real space with signature (p, q). Construct the Clifford algebra of Rp,q, followingthe construction outlined in chapter 2. The manifold M can be canonically embedded in

∧Rp,q, via

an extension φ : M ↪→∧

Rp,q. Each tangent space at a point of the embedded space φ(M) generatesa subalgebra of

∧Rp,q, call this the tangent algebra. The bundle of these tangent algebras is known

as a vector manifold.

2Note that we try to formalize the definition of a vector manifold outlined in chapter 4 of [6]; his description is morevague.


This definition has some problems. The first problem is existence. It is a well known result byNash[9], that any Riemannian manifold M can be isometrically embedded in some Euclidean spaceRn. He finds some dimensional bounds

• M is compact: n ≤ dim(M)2 (3 dim(M) + 11)

• M is not compact: n ≤ dim(M)2 (dim(M) + 1)(3 dim(M) + 11)

for smooth embeddings. Nash does not speak of pseudo-Riemannian manifolds. The first thoughtthat comes to mind that no such result is possible. This inspired the search for another construction,combining clifford algebras and curved spacetime, to be able to formulate general relativity in thislanguage. This construction is formulated in the next section. Later we found the paper by Clarke[3],in which he generalizes the construction of Nash for the pseudo-Riemannian case. He founds muchless strict bounds, i.e. the embedding space is much larger. His main theorem is

Theorem 3.16. Any m− dimensional manifold M with Ck pseudo-Riemannian metric, with k ≥ 3,of rank3 r and signature (l, s) can be embedded in Ep,p+q provided that

p ≥ m− 12

(r + s) + 1 and q ≥ 12m(3m+ 11), (3.33)

if M is compact. If M is not compact the bounds are

p ≥ m− 12

(r + s) + 1 and q ≥ 16m(2m2 + 37) +

52m2 + 1. (3.34)

Thus we see this theorem solves the first problem, that of existence. This does introduce a secondproblem, that we introduce many superfluous, i.e. unphysical, dimensions. The theorem amountsfor spacetime (a pseudo-Riemannian manifold of signature (3, 1)) to an embedding space R2,48 forcompact spacetimes, and an embedding space R2,89 if spacetime turns out to be noncompact4. Thenumber of extra dimensions required for Hestenes’ construction of a vector algebra is enormous.These extra dimensions are redundant. They do not contain any physics, and should be eradicated fromthe theory. This is why we have constructed the clifford algebra at each tangent space intrinsically inthis chapter. One might also think that there is a third problem. The construction is not unique. Ifwe found an embedding, we can always find a new one in a higher dimensional space, by just addingextra dimensions to the embedding space. This is true, however we must think about what we considerto be unique. The construction is unique up to diffeomorphic isometries, i.e. it is unique in the senseof Riemannian manifolds. Hence the third problem is not really a problem.

3The rank of a manifold is a measure of how many dimensions are flat. We will not define this notion properly, since wewill not use it in the rest of the thesis. The bounds hold if we pick r = 0, but the bounds are not the strictest possible ones.

4Of course these bounds might turn out to be non-optimal, we might find a smaller dimension in which we can embedgeneral spacetimes.

Chapter 4

Curvature Identities of the DiracOperator

In flat space the square of the Dirac operator equals the wave operator. On generic Riemannian man-ifolds the square of the Dirac operator stores the scalar curvature as well. By studying combinationsof the Dirac operator, the divergence, and the curl one can find nice identities relating combinationsof these operators with the curvature tensors (Riemann, Ricci, and Einstein) of Riemannian geometry.These identities are used in the next section to formulate Einstein’s equations as a wave equation forsections of

∧1 TM . These types of identities are known as Weitzenbock formulas. The proofs ofthese identities are straightforward, once you know what combinations of derivatives to look for.

We can express covariant derivatives on the reciprocal frame via the Christoffel symbols.

Lemma 4.1. The identity∇γαγµ = −Γµασγ

σ. (4.1)

holds.

Proof. We apply ∇γα to equation 3.12

∇γα(γµ · γν) = ∇γαδµν = ∂γαδµν = 0. (4.2)

The Leibnitz rule now yields that

∇γα(γµ · γν) = (∇γαγµ) · γν + γµ · (∇γαγν) = 0. (4.3)

We can express the second term in the Christoffel symbols, i.e.

γµ · (∇γαγν) = γµ · Γσανγσ = −Γµαν , (4.4)

which results in(∇γαγµ) · γν = −Γµαν . (4.5)

Another fact is that the reciprocal frame is curlless.

23

24 CHAPTER 4. CURVATURE IDENTITIES OF THE DIRAC OPERATOR

Lemma 4.2. The reciprocal frame is curlless. i.e.

6∇ ∧ γµ = 0. (4.6)

Therefore the Dirac operator of a scalar function is curlless

6∇ ∧ 6∇φ = 0 for any φ ∈ Γ(∧0

TM). (4.7)

Proof. We compute

6∇ ∧ γµ =γν ∧∇γνγµ

=γν ∧ (−Γµνα)γα

=− 12

Γµνα(γνγα − γαγν)

=− 12

(Γµνα − Γµαν)γνγα = 0.

(4.8)

In the last step we used that the Christoffel symbols are symmetric in the lower indices. Now wecompute the curl of the gradient of φ. We apply the previous result to obtain the third expression.

6∇ ∧ 6∇φ =6∇ ∧ (γµ∂µφ)=(6∇ ∧ γµ)∂µφ+ (γν ∧ γµ)∇γν∂µφ=(γν ∧ γµ)∂ν∂µφ = 0.

(4.9)

The last expression is zero, for the same reason the contraction of the Christoffel symbols with γµ∧γνwas zero; γµ ∧ γν is antisymmetric in µ and ν while ∂µ∂ν is symmetric in these indices.

Lemma 4.3. We have the identity

6∇ ∧ 6∇γρ =12Rie ρσµν(γν ∧ γµ)γσ. (4.10)

whereRie is the Riemann tensor.

Proof. The proof does not contain any deep idea, it is a straightforward computation. The problem iskeeping track of all the minus signs. We compute

∇γµ∇γνγρ =−∇γµΓρνσγσ

=(−∂µΓρνσ + ΓρνλΓλµσ)γσ.(4.11)

We applied the Leibnitz rule, and we renamed the dummy index σ to λ in the second term. We madeuse of lemma 4.1 twice. We can now compute

(∇γν∇γµ −∇γµ∇γν )γρ = (−∂νΓρµσ + ΓρµλΓλνσ − (−∂µΓρνσ + ΓρνλΓλµσ))γσ. (4.12)

The right hand side is exactly the Riemann curvature tensor applied on γσ expressed in local coordi-nates. Hence

(∇γν∇γµ −∇γµ∇γν )γρ = Rie ρσµνγσ. (4.13)

25

From this the following computation follows

6∇ ∧ 6∇γρ =γν∇γν ∧ γµ∇γµγρ

=γν ∧ γµ∇γν∇γµγρ, since the reciprocal frame is curlless

=12

(γνγµ)(∇γν∇γµ −∇γµ∇γν )γρ

=12

(γνγµ)Rie ρσµνγσ.

(4.14)

Since the Riemann tensor is antisymmetric in the last two indices, and we sum over these, the sym-metric part of γνγµ vanishes, hence we can write

6∇ ∧ 6∇γρ =12

(γν ∧ γµ)Rie ρσµνγσ, (4.15)

proving the lemma.

Remark 4.4. The identity is due to Hestenes [5], but there are some errors in minus signs in hisderivation. This is fixed here.

Lemma 4.5. Let f ∈ C∞(M) and u ∈ Γ (∧TM). The operator 6∇ ∧ 6∇ is a C∞(M)-linear map on

Γ (∧TM) i.e. the identity

6∇ ∧ 6∇f u = f 6∇ ∧ 6∇u (4.16)

holds. Hence 6∇ ∧ 6∇ is not a differential operator, i.e. it does not depend on derivatives of f .

Proof. We compute

( 6∇ ∧ 6∇)f u =6∇ ∧ γµ(∂µf)u+ 6∇ ∧ γµ f∇γµu=γν ∧ γµ

((∂ν∂µf)u+ (∂µf)∇γνu+ (∂νf)∇γµu+ f∇γν∇γµu

).

(4.17)

The first term vanishes because we contract a symmetric, i.e. (∂ν∂µf)u, and an antisymmetric, i.e.γν ∧ γµ object. We rewrite the third term

γν ∧ γµ(∂νf)∇γµu = γµ ∧ γν(∂µf)∇γνu = −γν ∧ γµ(∂µf)∇γνu (4.18)

and see that it cancels the second term in equation (4.17). Therefore we can conclude that

6∇ ∧ 6∇f u = f 6∇ ∧ 6∇u. (4.19)

because all terms with derivatives of f drop out of the expression, hence we have proven the lemma.

Lemma 4.6. For any X ∈ Γ(∧1 TM

)we have that 6∇ ∧ 6∇X ∈ Γ

(∧1TM

). The expression

does not have a trivector part. This is equivalent to the statement that the Ricci (or the first Bianchi)identity holds, i.e.

Rie ρσµνγν ∧ γµ ∧ γσ = 0, (4.20)

or in traditional tensor notation

Rie ρσµν +Rie ρµνσ +Rie ρνσµ = 0. (4.21)

26 CHAPTER 4. CURVATURE IDENTITIES OF THE DIRAC OPERATOR

Proof. Recall that we can decompose the Dirac operator in the curl and the divergence, cf. lemma3.12. We do this for the expression 6∇6∇γµ for the first and second Dirac operator separately, whichwe can do because the Dirac operator is associative as well

(6∇ ·+6∇∧) 6∇γµ = 6∇ (6∇ ·+6∇∧) γµ. (4.22)

The reciprocal frame is curlless, in accord with lemma 4.2, thus the second term on the right hand sideis zero. Rearranging the terms one finds

6∇ ∧ 6∇γµ = 6∇(6∇ · γµ)− (6∇ · 6∇)γµ. (4.23)

The right hand side is obviously vectorial. We conclude that the trivector part on the left hand sidemust vanish. Lemma 4.3 therefore yields the Ricci identity

Rie ρσµνγν ∧ γµ ∧ γσ = 0. (4.24)

If we dot this expression with γα ∧ γβ ∧ γε we find a more familiar form

0 =Rie ρσµν(γν ∧ γµ ∧ γσ) · (γα ∧ γβ ∧ γε)=Rie ραβε −Rie ραεβ −Rie ρβαε +Rie ρβεα +Rie ρεαβ −Rie ρεβα=Rie ραβε +Rie ρβεα +Rie ρεαβ

(4.25)

In the third line we applied the antisymmetry of the Riemann tensor in the last two indices, and dividedthe expression by a factor of two.

Theorem 4.7. We have the following Weitzenbock formula

6∇ ∧ 6∇X = 6∇( 6∇ ·X) + 6∇ · (6∇ ∧X)− (6∇ · 6∇)X = Ric (X) for any X ∈ Γ(∧1

TM).

(4.26)HereRic is the Ricci tensor i.e. the nontrivial contraction of the Riemann tensorRie .

Proof. The plan of attack is now to compute the expression in the theorem for the special case X =γµ. The C∞(M)-linearity, cf. lemma 4.5, makes that the theorem holds for any X ∈ Γ

(∧1 TM)

.Since the operator 6∇∧ 6∇maps vector fields to vector fields, lemma 4.3 yields that the clifford productwith γσ in fact reduces to the dot product of that vector. That is

6∇ ∧ 6∇γρ =12Rie ρσµν(γν ∧ γµ) · γσ. (4.27)

Which we can work out

6∇ ∧ 6∇γρ =12Rie ρσσνγν −

12Rie ρσµσγµ = Ric ρνγν . (4.28)

The Ricci tensor is the only non-trivial contraction of the Riemann tensor (up to minus signs).

To obtain the inner identity, we have another look at equation (4.22). This identity also holds forany vector field X . Then one finds after rearranging the terms

6∇ ∧ 6∇X = 6∇( 6∇ ·X) + 6∇( 6∇ ∧X)− ( 6∇ · 6∇)X. (4.29)

27

The second term on the right hand side cannot have a trivector part. The left hand side does not, cf.lemma 4.6, hence right hand side cannot have a trivector part. The second term is the only possibleway of arriving at a trivectoral contribution, therefore it must vanish. Thus we can expand the aboveequation to

6∇ ∧ 6∇X = 6∇(6∇ ·X) + 6∇ · ( 6∇ ∧X)− (6∇ · 6∇)X. (4.30)

Chapter 5

Einstein’s Equations

5.1 The field equations

With the apparatus developed in section 4 we can express the Einstein equation using the Dirac, curland divergence operators. The wave nature of the gravitational field will become much more apparentthan in conventional treatments of the subject. It is only after one makes a limit, where one considersgravitational perturbations of a fixed background, that one discovers wave behavior. We will see thatthe Einstein equations are the wave equations, where the energy momentum tensor acts as a sourcefor the waves. We employ the following convention

Convention 5.1. The units in which we study physical systems are the so-called geometrized units.That is we set G = c = 1. This is the natural scale in which to study gravitational phenomena.

Theorem 5.2. The Einstein equations can be written in the clifford formalism as the equivalence

6∇ ∧ 6∇X = 8π(T − 1

2Tr T

)X, (5.1)

or the equivalence

6∇ ( 6∇ ·X) + 6∇ · (6∇ ∧X)− (6∇ · 6∇)X = 8π(T − 1

2Tr T

)X, (5.2)

for all vector fields X ∈ Γ(∧1 TM

).

Proof. We will not “derive” the field equations from first principles. There exist much better physicaltreatments on this subject. We refer the interested reader to the physics books of Carroll [1] and Bruhat[2]. These treatments make it plausible, reasoning from first principles, that the field equations are

Ric µν = 8π(Tµν −

12

Tr T gµν). (5.3)

For those not familiar with this formula, this is called the trace-reversed form of the field equations.The tensor Tµν is known as the energy-momentum tensor. It contains the mass, energy and momen-tum content of the physical system we want to describe.

28

5.2. THE PHYSICAL INTERPRETATION OF THE FIELD EQUATIONS 29

The tensors on both sides of trace reversed equation (5.3) can be applied to a vector field X .Hence, in any coordinate frame we will have the compact formula

RicX = 8π(T − TrT )X. (5.4)

Applying the Weitzenbock formula, i.e. theorem 4.7, now yields the required identities.

One might wonder why we bother with the description (5.2) at all, since formula (5.1) is morecompact. The beauty of the second equation is that it enlightens the wave nature of the gravitationalfield.

5.2 The physical interpretation of the field equations

We have promised a wave interpretation of the field equations. For this we first go to vacuum, i.e. theright hand side of (5.2) vanishes. Gradient fields, for example the reciprocal frame, have the propertythat 6∇ ∧ X = 0. The second term in the equation vanishes. The equation on these type of fields invacuum reduces to

6∇ (6∇ ·X)− ( 6∇ · 6∇)X = 0. (5.5)

Now demanding that the field is divergenceless, i.e. 6∇ · X = 0 or at least constant, shows that Xsatisfies the wave equation

( 6∇ · 6∇)X = 0. (5.6)

Of course this might seem like an obvious statement in flat space. The wave operator is defined to bethe divergence of a gradient. Still, with source terms this does not seem very obvious. We then arrive,after imposing the conditions of being a gradient and being divergence free, at the equation

− ( 6∇ · 6∇)X = 8π(T − 1

2Tr T

)X. (5.7)

Where we see a discrepancy between being a solution to the wave equation and having the divergenceof the gradient being zero.

5.3 Solution techniques

It might seem that the preceding discussion also yields a possible solution technique for solving thefield equations. Pick a suitable “gauge” and solve the equation in this gauge. By a gauge we mean acondition like vanishing of the divergence, or vanishing of the curl. One basically reduces the singlesystem of differential equations of order 2 to two systems of differential equations of order 1. Thisis something one can always do. The interesting this is that we can expect to pick physical gauges inthis manner. Gauges respecting certain symmetries of the system under consideration.

However this thinking is not correct. One does not really have a gauge to fix. One gauge mightimpose conditions on the metric, but we have to impose all gauges, forming a class of conditions. Allthese conditions together will be equivalent to the condition that equation (5.1) holds for all vectorfields.

For clarification we restate this. A gauge corresponds to a symmetry of nature. Nature does notcare in which gauge one considers the equations in, all physical phenomena resulting from different

30 CHAPTER 5. EINSTEIN’S EQUATIONS

gauges are the same. Here the situation is different. One can simplify the formulas by imposing cer-tain conditions on the vector fields. But nature needs to know that these equations hold for all vectorfields. Computing the equations on a basis of vector fields will suffice.

Chapter 6

The Fiducial Frame and the SpinConnection

It is not hard to compute the field equations for some toy models using theorem 5.2. If we expandthe Dirac and curl operator, we will find the formula for the Ricci tensor. This approach has beenstudied exhaustively. It has been argued[5] that it is natural to look at orthonormal frames instead ofthe coordinate frames. The coordinatization we choose to describe spacetime does not have a deepphysical significance; it is merely a means to label events. An orthonormal frame does have a physicalsignificance however. The orthonormal frame is a local apparatus to measure distance and direction.This observation motivates Hestenes [5] to coin the term fiducial frame for an orthonormal frame.This frame is not unique. The time direction however can be specified uniquely for a massive testparticle. It is the proper velocity of this particle. The spacelike directions can now be transformedusing a rotation, i.e. a SO(3) symmetry.

Now that we agree that it might be worthwhile to study gravitation via an orthonormal frame wewill prove some more curvature lemmas. First a definition.

Definition 6.1. Let γµ be a local coordinate frame, and ea a local orthonormal frame, defined in thesame chart. Since both are bases for the tangent space at each point they can be expressed in terms ofeach other. This is done via the fiducial tensor haµ

ea = haµγµ.

and some conventions will be necessary

Convention 6.2. Indices of the coordinate frame will be denoted by Greek letters {µ, ν, ...}. Indicesof orthonormal components will be denoted by Roman letters a, b, .... This will also be done forcomponents. Hence a vector field can be expressed in four equivalent ways

X = Xµγµ = Xµγµ = Xaea = Xae

a.

Hence lowering and raising of coordinate components will be done via the metric components Xµ =gµνXµ and raising and lowering of fiducial components will be done with the Minkowski metricXa = ηabXb.

If we both use numerals for the coordinate and the fiducial components errors will occur. Hencewe will only use numerals for the fiducial components, and will express the coordinate indices using

31

32 CHAPTER 6. THE FIDUCIAL FRAME AND THE SPIN CONNECTION

the names of the charts (variables). For example in the Schwarzschild metric, we will use t, r, φ, θ asnames for the coordinate components.

The following lemma is immediately clear.

Lemma 6.3. The fiducial tensor is invertible, its inverse denoted by haµ. We can compute the fiducialtensor via the expression

haµ = ea · γµ. (6.1)

It should be remarked that the fiducial tensor is the root of the metric tensor. This might seemstrange, but the proof of this statement is actually incredibly simple. We formulate this result

Theorem 6.4. The inverse of the fiducial tensor is the root of the metric tensor. That is

gµν = haµhbνηab (6.2)

Proof. Recall that the metric tensor components can be computed via

gµν = γµ · γν , (6.3)

writing the right hand side in terms of the fiducial frame yields the equality

gµν = haµea · hbνeb = haµhbνηab (6.4)

We can compute the Dirac operator using an orthonormal frame.

Lemma 6.5.

The Dirac operator can be expressed in terms of the fiducial frame as

6∇u = ea∇eau (6.5)

Proof. We use lemma 6.3 to compute

6∇ = gµνγµ∇γν = gµνhaνeahbµ∇eb . (6.6)

by the variance of the connection. Since the fiducial frame is the root of the metric tensor, taking theinverse of the equation in theorem 6.4, we have

gµνhaνhbµ = ηab. (6.7)

Hence we find that6∇ = ea∇ea . (6.8)

It is natural to consider the spin connection when dealing with fiducial components. The spinconnection is defined thus

Definition 6.6. The spin connection Scab is defined via the equation

Scab := ec · ∇eaeb. (6.9)

33

The spin connection obeys some identities, comparable to those of the Christoffel symbols.

Lemma 6.7. We can compute the components of the spin connection via the formula

Scab =(ha

µ∂µhbν + ha

µ hbλΓνµλ

)hcν (6.10)

The spin connection satisfies the identity

Scab = −Sbac (6.11)

Proof. These statements are a mere computation, without deep thought. The first statement amountsto the calculation below

Scab =ec · ∇eaeb=ec ·

(ha

µ∇γµhbνγν)

=ec ·(ha

µ(∂µhbν)γν + haµhb

λΓνµλγν)

=(ha

µ∂µhbν + ha

µ hbλΓνµλ

)hcν .

(6.12)

The second statement follows from the computation of

∇eaeb = ∇eaηbcec = ηbcSdaced = Sdabed, (6.13)

on one hand. On the other hand

∇ea(eb · ec) = ∇ea(δbc) = 0. (6.14)

Let’s apply the Leibnitz rule on the left hand side. This yields

(∇eaeb) · ed = −eb∇eaed = −Sbad (6.15)

Thus we have(∇eaeb) = −Sbaded = Sba

ded. (6.16)

Comparing this with equation (6.13) yields the required identity, after we have renamed some dummyindices.

We can express the field equations in the spin connection. This is not a deep insight. This formulawill come in handy for the explicit examples of the next section. We will only consider the vacuumequations

Lemma 6.8. The vacuum field equations are equivalent to the statement that, for any a and b

ScfbSfca + ∂ebS

ffa + ScfaS

fbc = ScbfS

fca + ∂efS

fba + ScbaS

ffc (6.17)

Proof. This directly follows from expressing 6∇ ∧ 6∇ea = 0, cf. theorem 5.2, in the fiducial frame.

6∇ ∧ 6∇ea =eb ∧∇ebec∇ecea

=eb ∧(−Scbded + ec∇eb

)Sf caef

(6.18)

34 CHAPTER 6. THE FIDUCIAL FRAME AND THE SPIN CONNECTION

Here we made use of the spin connection identities. If we relabel the indices and work out the covari-ant derivative on the right hand side we find

eb ∧ edef(−ScbdSf ca + ∂ebS

fda + ScdaS

fbc

)(6.19)

The Bianchi identity holds, that is the trivector part of the last equation vanishes, cf. lemma 4.6. Usingthe easily verifiable identity

π1(eb ∧ edef ) = ebδdf − edδbf , (6.20)

we arrive at the claim in the lemma, where we renamed some dummy indices.

Chapter 7

Examples of Curvature Calculations

7.1 Diagonal Metrics in vacuum

Not all metrics are globally diagonalizable, and not all physical systems are in vacuum. It is stillinteresting to study these simplified toy models. Two of the most important gravitational systems doobey this criterium. These are the Schwarzschild solution, describing both the exterior field of stars,as well as black holes, and the Friedmann−Lemaıtre−Robertson−Walker (FLRW) metric, describinghomogeneous and isotropic universes.

It is much easier to calculate with diagonal metrics, than it is with non-diagonal ones. A basicfeature is that the fiducial tensor is diagonal as well, removing many of the summations necessary inthe non-diagonal case.

Convention 7.1. Let (M, g) be a pseudo-Riemannian manifold. Assume that the metric g is diagonal-izable, i.e. it admits a cover of charts in which the metric tensor gµν = γµ · γν is diagonal. Then wecan easily normalize the coordinate frame to form a fiducial frame. That is, we define

ea := h(a)γa no sum (7.1)

such that ea · eb = ηab. We can easily compute that the fiducial functions h(a) obey h(a) =√

1γa·γa .

The fiducial functions are actually the diagonal components of the fiducial tensor, defined in chapter6. That is the fiducial tensor can be expressed in the fiducial functions via

haµ = h(a)δaµ. (7.2)

Note that we do not sum over a in the right hand side. The parameter a in h(a) is understood not tobe summed over. We introduce this convention to remove a huge amount of unnecessary summationsin the case of diagonal metrics. In the same manner we define the signature functions η(a) = ηaa

Diagonal metrics have surprisingly simple spin connection components

Lemma 7.2. Let a 6= b 6= c. We have some formulas for the spin connection. The indices are

35

36 CHAPTER 7. EXAMPLES OF CURVATURE CALCULATIONS

understood not to be summed over in the formulas below

Scab =Scac = Sccc = 0.

Scca =− h(a)h(c)

∂ah(c)

Sacc =h(a)h(c)

η(c)η(a)∂ah(c) = −η(c)η(a)Scca

(7.3)

Proof. Simple bookkeeping will yield the identities above. If we apply the simplification (7.2) to theroot equation (6.2) we find the root equation in terms of the fiducial functions

gµν =1

h(µ)h(ν)ηµν (7.4)

and the reciprocal root equationgµν = h(µ)h(ν)ηµν (7.5)

These relations can be used to compute the Christoffel symbols in terms of the fiducial functions

Γcab = ηabηcµ h(c)2

h(a)2∂µh(a)− 1

h(c)∂ah(c)δcb −

1h(c)

∂bh(c)δca. (7.6)

Hence, by application of formula (6.10), we find that the spin components obey

Scab =h(c)h(a)

ηabηcd∂dh(a)− h(b)

h(c)∂bh(c)δca. (7.7)

Filling in explicit combinations of the components, we find the formulas of the lemma.

We have the following simplification of lemma 6.8 for diagonal metrics.

Lemma 7.3. Assume that the metric is diagonalizable. Let a 6= b then the field equations are equiva-lent to the statements1 ∑

c 6=b , c 6=a

(h(b)∂bScca + Sccb

(Scca − Sbba

))= 0, (7.8)

which are basically the diagonal components of the Ricci tensor multiplied with a function, and forthe diagonal components we have the equations

∑c

(h(a)∂aScca + η(a)η(c)h(c)∂cSaac + SccaS

cca +

∑d

η(a)η(d)SaadSccd

)= 0. (7.9)

Proof. We will use the expression for the field equations, as formulated in lemma 6.8, and try tosimplify them using the identities of 7.2. The off diagonal components simplify enormously, but thediagonal components unfortunately do not.

We will try to simplify all the terms of lemma 6.8, bringing all spin connection components tothe form Sf fg which we can do by lemma 7.2. In the formulas that come, we explicitly sum over all

1We do not use the summation convention in the formulas here

7.1. DIAGONAL METRICS IN VACUUM 37

indices not a or b, even if they come in quadruples, or if they are not “upper” and “lower” pairs. Westart with the off-diagonal components, assuming a 6= b. Then

ScfbSfca =SabbSbaa + SccbS

cca

=SbbaSaab + SccbScca.

(7.10)

We used the identity SabbSbaa = SbbaSaab which holds by lemma 7.2. We continue

∂ebSffa = h(b)∂bScca, (7.11)

and we haveScfaS

fbc = SbaaS

abb = SaabS

bba, (7.12)

which completes the three terms on the lefthand side of the equation in lemma 6.8. The right handside computations are similar. We list them here

ScbfSfca =ScbbSbca = SabbS

baa = SbbaS

aab.

∂efSfba =h(b)∂bSbba.

ScbaSffc =SbbaSccb.

(7.13)

Combining these results finds the equation

h(b)∂b(Scca − Sbba) + Sbba (Saab − Sccb) + SccbScca = 0. (7.14)

We see that we can simplify this if we do not sum over all values of c, but only the values c 6= a andc 6= b. This yields the equation∑

c 6=b , c 6=a

(h(b)∂bScca + Sccb

(Scca − Sbba

))= 0. (7.15)

We have mentioned before that the diagonal components do not simplify as much. We compute themhere (without much remarks, all computations are rather straightforward). For the left hand side ofthe equation in lemma 6.8 with a = b we find the simplifications

Sf caSfca =SccaScca

∂eaSffa =h(a)∂aScca

Sf caSfac =ScaaSaac = −η(a)η(c)SaacSaac.

(7.16)

And the terms on the right hand side we find

ScafSfca =SaafSf aa = −η(a)η(c)SaacSaac

∂efSfaa =− η(a)η(c)h(c)∂cSaac

ScaaSffc =η(a)η(c)SaacSddc.

(7.17)

Summing these up, we find

∑c

(h(a)∂aScca + η(a)η(c)h(c)∂cSaac + SccaS

cca +

∑d

η(a)η(d)SaadSccd

)= 0. (7.18)


Remark 7.4. In principle we have 10 different equations here. 6 off-diagonal 2ones a < b and 4diagonal ones a = b. We have assumed diagonality, which is a symmetry, thus we actually expect alower amount of independent equations. We have an amount of redundancy, but we do not attempt toremove this redundancy.

7.2 The Schwarzschild metric

The previous result is about as far as we can get without assuming more about spacetime. We couldexpand the previous results in terms of the fiducial functions. The result is not a simplification how-ever. Almost no terms cancel. That is why we will work out the celebrated Schwarzschild metric. Thismetric describes the exterior of non-rotating pointmass sources. We assume a spacetime of signature(3, 1) which can be foliated by 3-spheres. After some work we realize that the line element can bewritten in the form

ds2 = e2Φ(t,r)dt2 + e2Λ(t,r)dr2 + r2dθ2 + r2 sin2(θ)dφ2. (7.19)

For reference, see for example Carroll [1] why the line element has this structure. Many of the tricksused to arrive at the final line element are also inspired by this book. To find the field equations using7.3 we have to compute all spin connection components of the form Scca. From here on we do notwrite down explicitly the arguments of Φ and Λ. These unknown functions depend on t and r untilwe state otherwise.

Lemma 7.5. The fiducial functions for the Schwarzschild metric are

h(t) = e−Φ(t,r), h(r) = e−Λ(t,r), h(θ) =1r, and h(φ) =

1r sin(θ)

. (7.20)

And the spin components of the form Saab are computed to be

Sttr =e−Λ∂rΦ

Srrt =e−Φ∂tΛ

Sθθr =e−Λ

r

Sφφr =e−Λ

r

Sφφθ =cos θ

r sin(θ).

(7.21)

Proof. The fiducial functions can be directly read off from equation 7.19. These fiducial functionscan be plugged in lemma 7.2. This yields the formulas of the lemma.

The natural next step would be to write down all field equations and solve them. Let’s do so forthe off diagonal components.

Lemma 7.6. From all off diagonal components we can conclude that Λ is a function of r only. HenceSrrt = 0.

2We can of course have a > b but these are equivalent to a < b, thus we do not consider them as different.

7.2. THE SCHWARZSCHILD METRIC 39

Proof. We compute equation (7.8) for a = t and b = r. That is∑c 6=t,r

(h(r)∂rScct + Sccr (Scct − Srrt)) = 0. (7.22)

We directly see that the first term vanishes since Scct is only nonzero if c = r but we explicitly do notsum over c = r. We continue∑

c 6=t,rSccr (Scct − Srrt) = Sθθr

(Sθθt − Srrt

)+ Sφφr

(Sφφt − Srrt

). (7.23)

But we have Sθθt = Sφφt = 0, plugging in the remaining components we find

− 2e−Λ−Φ∂tΛ

r= 0. (7.24)

The expression −2 e−Λ−Φ

r is strictly negative, thus we conclude that

∂tΛ = 0. (7.25)

If we compute all the remaining off diagonal components, we discover that they are identically zero,thus we cannot determine the functions Λ and Φ fully from the off-diagonal components. We have tolook at the diagonal ones.

Now we turn to the diagonal equations

Lemma 7.7. The diagonal equations are, for a = t

e−Λ

r

(−2∂rΦ + r ∂rΦ ∂rΛ− r (∂rΦ)2 − r ∂2

rΦ)

= 0, (7.26)

for a = r

e−Λ

r

(−2∂rΛ− r ∂rΛ∂rΦ + r (∂rΦ)2 + r ∂2

rΦ)

= 0. (7.27)

For a = θ and a = φ we have the same equation.

− e−2Λ

r2

(−1 + e2Λ + r ∂rΛ− r ∂rΦ

)= 0. (7.28)

Proof. This is straightforward computation, similar to the last lemma. We omit this computation. Justplug in the spin connection components in formula (7.9).

We now only have to solve the previous partial differential equations to find the Schwarzschildmetric. We make a physical assumption regarding an integration constant in the proof.

Theorem 7.8. The Schwarzschild metric is

ds2 = −(

1− 2GMr

)dt2 +

(1− 2GM

r

)−1

dr2 + r2dθ2 + r2 sin2(θ)dφ2. (7.29)


Proof. We solve the three differential equations found in the last lemma. A first remark is that theonly time dependent component in the third equation (7.28) is the last term, −r ∂rΦ. Thus to vanishidentically we have

∂rΦ = 0 (7.30)

Which induces a solutionΦ(t, r) = Φr(r) + Φt(t). (7.31)

But the t component can be transformed away by the transformation dt→ e−Φt(t)dt. Thus we assumewithout loss of generality that it vanishes. Hence we have that Φ and Λ are time independent. Let’sadd equations (7.26) and (7.27). We find

∂rΦ + ∂rΛ = 0 (7.32)

thereforeΦ = −Λ + constant (7.33)

which is a tremendous simplification. By a transformation rule, we can transform the constant away.We now only have to compute one of the unknown functions! The third equation (7.28) can be writtenas

− e−2Λ + 1 + 2r ∂rΛ e−2Λ = 0. (7.34)

Using the trivial identity∂r(r e−2Λ

)= e−2Λ − 2r e−2Λ ∂rΛ, (7.35)

we discover that∂r(r e−2Λ

)= 1, (7.36)

holds. Hencer e−2Λ = r + c, (7.37)

For some constant c. Thereforee2Λ =

(1 +

c

r

)−1. (7.38)

We have found that the line element equals

ds2 = −(

1 +c

r

)dt2 +

(1 +

c

r

)−1dr2 + r2dθ2 + r2 sin2(θ)dφ2. (7.39)

We expect that this metric describes point masses. We can therefore use the weak field3 limit to makea connection with Newtonian gravity theory and discover the value of the constant c. We see directlythat in the limit r →∞ we have

gtt →−(

1 +c

r

), and

grr →(

1− c

r

).

(7.40)

We compare this with the weak field limit equations, which Carroll[1] studies extensively

gtt =− (1 + 2V )grr = (1− 2V ) .

(7.41)

Here V is the Newtonian gravitational potential V = −GMr . If these results coincide we must have

c = −2GM . Hence we arrive at the result of the theorem.

3In this case the field is weak when r →∞

Chapter 8

Lorentz Transformations and Bivectorfields

8.1 Reflections and rotations

Consider Euclidean space V . That is a vector space endowed with a positive definite inner product〈·, ·〉. Endow this with the canonical clifford algebra, as explained in chapter 2. We can reflect a vectorv in the plane orthogonal to the unit vector n via the formula

rn(v) = v − 2〈v, n〉n.

This formula is easily seen to be correct by looking at figure 8.1. Writing the inner product in termsof the clifford product we find that

rn(v) = v − 212

(v n+ n v)n

rnHvL

n

v

<v,n>n

Figure 8.1: A reflection can be obtained by subtracting the projection twice

41

42 CHAPTER 8. LORENTZ TRANSFORMATIONS AND BIVECTOR FIELDS

Since n is normalized by assumption n2 = 1 therefore

rn(v) = −n v n.

If we reflect the vector v in another plane, this time orthogonal to m, also a unit vector we arrive at

rm(rn(v)) = mnv nm.

If m and n are orthogonal to each other this reduces to

rm(rn(v)) = m ∧ n v n ∧m.

It is well known that reflections generate rotations. Any even power of reflections equals a rotation.A stronger statement is that all rotations are generated this way. Thus generically bivectors generaterotations.

In this derivation we have assumed that the normal vector was normalized, that is 〈n, n〉 = 1. Ifwe want to reflect in any direction, say x, we have to normalize the vector x. Or we can define theinverse of the non-zero vector x via

x−1 :=x

xx. (8.1)

The inverse is well defined because xx = x ·x is scalar. We can easily verify xx−1 = x−1 x = 1.Then a reflection of v in the hyper-plane orthogonal to x can be computed via

rx(v) = x v x−1. (8.2)

This equation also works for the non-Euclidean case.

n

m

v

rn(v)

w

rn(w)

rm(rn(v))rm(rn(w))

m

n

Figure 8.2: We reflect the vectors v and w twice, in the planes orthogonal to m and n. The resulting transfor-mation is a rotation

8.2. LORENTZ TRANSFORMATIONS 43

8.2 Lorentz transformations

Bivector fields generate rotations, and in the Minkowski case Lorentz transformations at each pointvia the clifford exponential map defined below. Its definition requires a bit of functional analysis,which is not necessary for the use of this map. We study this to find a large class of sections on whichwe can exponentiate. Recall that all sections are assumed to be smooth cf. definition 10.4.

Definition 8.1. The supremum norm is the norm on sections Γ (∧TM) defined by the equation

||u||Γb(TM) := supp∈M||u(p)||∧TpM .

We define the bounded sections Γb(∧TM) ⊂ Γ (

∧TM). These are all sections u ∈ Γ (

∧TM) for

which the supremum norm is finite.

Remark 8.2. Note that we use the norm that comes from the scalar product, which is derived from themetric cf. 2.4. We need this scalar product, because the metric does not directly impose a sense ofcontinuity.

We will define the clifford exponential map. This clifford exponential map will be defined as alimit. The limiting sequence is Cauchy with respect to the norm defined above. To conclude that thelimit exists we need that the space of bounded sections is a Banach space. We will not prove this here.We outline a possible proof. If the space turns out to be non-Banach, which we do not conjecture, thelimit does exist pointwise. The final section does not need to be smooth if that is the case. But weclaim:

Claim 8.3. The space of bounded sections Γb(∧TM), endowed with the supremum norm

||u||Γb(TM) := supp∈M||u(p)||∧TpM

is a Banach space.

Proof. Let uk be a Cauchy sequence in Γb (∧TM). Since we basically want to proof that the limit

of smooth sections is smooth, we can assume without loss of generality that the Cauchy sequence isdefined on a open subset U ⊂M , such that the clifford bundle can be trivialized

∧TM ∼= U×

∧Rn.

We can now define a section

u(p) = limk→∞

uk(p), (8.3)

which clearly converges pointwise, since each∧

Rn is a Banach space. Now we only have toshow this section is smooth. We use a coordinatization for this. To simplify notation, we assume thatU ⊂ Rn. The chart1 is applied after the trivialization. Let k ∈ N, then

||u(x)− u(y)|| = ||u(x)− uk(x) + uk(x)− uk(y) + uk(y)− u(y)||≤ ||u(x)− uk(x)||+ ||uk(x)− uk(y)||+ ||uk(y)− u(y)|| .

(8.4)

Here we applied the triangle equality2. Thus we see that u is continuous as well. If we assume thatuk is C1 we have the estimate

1This has no problems with the norm, since all norms are equivalent on finite dimensional vector spaces, and a pullbackof a norm via a diffeomorphism is a norm itself

2we use the notation ||·|| for the induced norm on the charted space


||u(x)− u(x+ h)||||h||

≤ ||u(x)− uk(x)||||h||

+||uk(x)− uk(x+ h)||

||h||+||uk(x+ h)− u(x+ h)||

||h||,

(8.5)which goes to zero if ||h|| → 0. Thus we conclude that u is C1. By induction we now see that u hasat least the same smoothness degree as uk. Therefore Γb (

∧TM) is a Banach space.

Definition 8.4. Let u ∈ Γb (∧TM). Define the sequence of operatorsFk : Γb (

∧TM)→ Γb (

∧TM)

by the equation

Fk(u) =k∑

n=0

un

n!

This sequence is a Cauchy sequence. Hence the sequence has a limit, which we call the cliffordexponential map ��exp

��exp(u) := limk→∞

Fk(u).

Proof. Let k, l ∈ N and w.l.o.g. k > l. We compute

||Fk(u)−Fl(u)||Γb(∧TM) = supp∈M||Fk(u)(p)−Fl(u)(p)||∧TpM

= supp∈M

∣∣∣∣∣∣∣∣∣∣

k∑n=l+1

un(p)n!

∣∣∣∣∣∣∣∣∣∣∧

TpM

≤ supp∈M

k∑n=l+1

||un(p)||∧TpMn!

,

which is a straightforward application of the triangle inequality. The second term vanishes by as-sumption of a Cauchy sequence with respect to the supremum norm. At each point p ∈M the cliffordalgebra

∧TpM is a Banach algebra, see theorem 2.20. We can thus conclude

||Fk(u)−Fl(u)|| ≤ supp∈M

k∑n=l+1

||u(p)||n∧TpMn!

If we choose l big enough, we have the straightforward estimate ||u(p)||n+1

(n+1)! ≤ ||u(p)||nn! , for any u ∈

Γb (∧TM) and all n > l. Thus we can estimate the sum by the first term

≤ supp∈M

(k − l + 1) ||u(p)||l+1

(l + 1)!

But this goes to zero if we pick l big, since u(p) is bounded. Hence we conclude that the sequence isCauchy, and that there exists a limit, which we call the clifford exponential map.

This clifford exponential map has interesting properties. All local lorentz transformations can bewritten as the clifford exponential map of a bivector field. The bivector-field in every point defines theplane in which the boost should take place. This will be made clear by an example in flat space.

8.2. LORENTZ TRANSFORMATIONS 45

Example 8.5. Consider Euclidean three space R3. We can identify the tangent bundle with the spaceitself, hence we can view the clifford bundle as the exterior vector space

∧R3 endowed with the

clifford product defined in chapter 2. We compute the exponential map of −αγ1∧γ2

2 , with γi a basis ofR3 and α ∈ R.

R := ��exp(−αγ1 ∧ γ2

2)

First note, that since γ1∧γ2 = γ1γ2 we can drop the wedge in this equation. The identity γ1γ2γ1γ2 =−γ1γ1γ2γ2 = −1, gives

(γ1γ2)n =

{(−1)

n+12 γ1γ2 if n is even

(−1)n2 if n is odd

yields a splitting in even and odd parts in sum of the rotorR.

R :=��exp(−αγ1 ∧ γ2

2)

=∞∑n=0

(−αγ1γ2)n

n!

=∞∑n=0

(−1)n(−α)2n+1

(2n+ 1)!γ1γ2 +

∞∑n=0

(−1)n(−α)2n

(2n)!

=− sinα

2γ1γ2 + cos

α

2.

We have recognized the Taylor series of the sine and cosine in the last step, and the fact that the sine isodd, while the cosine is even. The rotor is a left-right representation of a rotation in the plane spannedby γ1 and γ2 through an angle α. That is the operation

R(u) := RuR

rotates the multivector u through an angle α in the plane spanned by γ1 and γ2. Lets compute this forthe three basis vectors γ1, γ2, γ3.

R(γ1) =(cos(α

2)− sin(

α

2)γ1γ2)γ1(cos(

α

2) + sin(

α

2)γ1γ2)

=(cos(α

2)− sin(

α

2)γ1γ2)(cos(

α

2)γ1 + sin(

α

2)γ2)

= cos(α)γ1 + sin(α)γ2,

for γ2

R(γ2) =(cos(α

2)− sin(

α

2)γ1γ2)γ2(cos(

α

2) + sin(

α

2)γ1γ2)

=(cos(α

2)− sin(

α

2)γ1γ2)(cos(

α

2)γ2 − sin(

α

2)γ1)

= cos(α)γ2 − sin(α)γ1,

and γ3

R(γ3) =(cos(α

2)− sin(

α

2)γ1γ2)γ3(cos(

α

2) + sin(

α

2)γ1γ2)

=(cos(α

2)− sin(

α

2)γ1γ2)(cos(

α

2) + sin(

α

2)γ1γ2)γ3

=γ3.


In the derivation we used the double angle formulas

cos2(α

2)− sin2(

α

2) = cos(α) (8.6)

and2 cos(

α

2) sin(

α

2) = sin(α). (8.7)

We recognize a rotation in the plane spanned by γ1 and γ2 through an angle α. We can compute thisoperation for multivectors as well. If we rotate a multivector u = γ1 ∧ γ3 we see that we can rotatethe vectors separately because

R(u) = Rγ1 ∧ γ3R = Rγ1γ3R = (Rγ1R)(Rγ3R).

using RR∗ = 1. Thus we now have a tool to rotate area and volume elements as well!

The factor two which arises in the rotation has a close connection with the spin group in threedimensions, which is a double cover of the rotation group in three dimensions. We will not study thisinteresting observation further.

Chapter 9

Outlook

We have seen some applications of the clifford formalism in gravitational physics. The thesis is just thetip of the iceberg, it is very probable that new research in this subject will yield interesting results. Welist some possible directions of research. Note that we only consider gravitational applications. Othersubjects in physics might benefit from a Clifford approach, but we do not suggest direct directions ofresearch.

9.1 Fundamental geometry

In this thesis we have not linked two important results. We have done some computations in the fidu-cial frame, cf. section 6. Here the fiducial tensor was introduced. We did not use the results of section8, which is more in the spirit of the clifford formalism. I would really enjoy reading some research inthis direction.

We also used the Riemann tensor in the derivation of many of the results. It is probable that theRiemann tensor can be viewed as some section of the clifford bundle. This is also a direction of re-search worthy of delving into.

It is mathematically interesting to generalize the exponential map1 to higher multivectors. Theexponential maps vectors in the tangent space to points on the base manifold, bivectors can probablybe locally diffeomorphically mapped to areas on the manifold. Similarly for higher multivectors. Thismight simplify integration theory.

9.2 The field equations

We have derived the beautiful field equations in terms of the Dirac operator, showing the mathemati-cal equivalence to the old formulation. Afterward we have interpreted the resulting equations as waveequations. It would do the theory good if we could derive the equations from basic physical principles.Hestenes [5] notes this can be done by imposing that the formulation of physics should be the samein all possible fiducial frames.

1We mean the exponential map of classical Riemannian geometry, not the Clifford exponential map defined in chapter 8

47

48 CHAPTER 9. OUTLOOK

Closely related to the latter point is an interpretation of gravity in terms of a gauge field. Lasenbyet al.[7] state that gravitational theory can be formulated with the use of a gauge field. Their formula-tion strongly uses clifford algebra, but not on manifolds. They state that gravitation can be describedby fields living on flat space. It seems improbable that gravitational systems with non-trivial topology2

can be described in this theory. It is worthy of note that it is not known if the universe is topologicallytrivial or not. Locally it is as far as we know, thus this research direction is worthy of pursue.

9.3 Topology

There is a rich branch of mathematics concerned with the clifford structure of Riemannian manifoldsand the topology of the manifold. Many of the results are not known to the physics community. It isworthwhile to study if these results will have physical implications.

9.4 Spinors and quantum mechanics

It is well known that spinors play an important role in physics, especially in quantum mechanics.Since spinors are defined in terms of clifford algebra we get these for free in the clifford formalism.This might pave the way to quantum gravity.

9.5 Conclusion

We have seen some applications of clifford algebras in gravitational physics. We have understoodhow we can interpret the field equations as wave equations. The clifford formalism also shows thatwe can recognize Lorentz transformations as elements of the clifford bundle. Hopefully it is clearfrom this thesis that clifford algebras are not quantum objects, they arise naturally in the setting ofclassical physics.

We expected computational simplifications by introducing the extra algebraic structure. This di-rection of research seemed to have negative results. The curvature calculations do not seem to simplifythat much. This might be because we basically did a semi-tensorial approach in chapter 7. It wouldbe a good idea to try to formulate all quantities in terms of sections of the clifford bundle.

9.6 Acknowledgements

I’d like to thank various persons, outside of my supervisors, who I had interesting discussions withregarding this thesis. These include, but are not limited to, Erik van den Ban, Jan Jitse Venselaarand most of all Sweitse van Leeuwen. I’m thankful for both my supervisors, Frank Witte and RenateLoll, for taking their valuable time supervising me. Finally I’d like to indebted both my parents forsupporting me through a difficult time writing this thesis.

2Such as closed FLRW universe

Chapter 10

Appendix Bundles

10.1 Vector bundles

We have chosen to construct the clifford bundle as the exterior bundle, where we define a new producton the latter bundle. It is useful to recall some facts about vector bundles, since both bundles are. Avector bundle of rank r over a manifoldM can be thought of as a manifoldE, which is constructed byattaching a vector space of dimension r at each point p ∈M in a smooth manner. The exterior bundle,hence the clifford bundle, is a vector bundle of rank 2dimM . The tangent bundle is a vector bundleof rank dimM . In this appendix we will not prove any results. All serious texts about manifoldswill have these standard results. For reference one can look at [8]. Lets define the notion of a vectorbundle.

Definition 10.1. A vector bundle of rank r over a manifoldM is a mapping of manifolds π : E →Msuch that Ep := π−1({p}) has the structure of an r dimensional vector space over R. This structuredepends smoothly on p in the sense that there exists an open covering Uα of M and smooth mappingsρ : EUα = π−1(Uα)→ Rr with the following properties

1. The restriction of ρ to any fiber Ep, ρ|Ep → Rr is an isomorphism of vector spaces.

2. The map ρ = (ρ, π|EUα ) : EUα → Rr × U is a diffeomorphism.

The manifold E is called the total space, and the manifold M is known as the base space. π is alsoknown as the projection map. A pair (Uα, ρ|α) is called a local trivialization.

The following lemma let’s us construct vector bundles from collections of linear spaces indexedby a manifold, with a certain smoothness condition. This allows us to construct a many vector bundlesfrom more primitive ones.

Lemma 10.2. Let {Ep}p∈M be a collection of vector spaces of dimension r indexed by a manifold.Suppose we are given an open covering Uα of M and for every α and p ∈ Uα an isomorphism ofvector spaces ρα,p : Ep → Rr, for which the transition maps

hα,β : Uα ∩ Uβ → Gl(r,R), with hα,β = ρβ,p ◦ ρ−1α,p, (10.1)

are smooth. Then the disjoint union E :=∐p∈M Ep → M has a unique vector bundle structure of

rank r for which

(Uα, ρα : EUα :=∐p∈Uα

Ep → Rr, where ρα(p, x) = ρα,p(x)) (10.2)

49

50 CHAPTER 10. APPENDIX BUNDLES

make up an atlas.

This lemma has the extremely useful corollary

Corollary 10.3. Let ξ : E → M and ξ′ : E′ → M be two vector bundles of rank r and r′. Then thedisjoint union over p ∈ M of the vector spaces E∗p ,

∧k Ep ,∧Ep , Ep ⊕ E′p , and Ep ⊗ E′p have in

a natural manner a vector bundle structure denoted by ξ∗ ,∧k ξ ,

∧ξ , ξ ⊕ ξ′ , and ξ ⊗ ξ′.

10.2 Sections

We will be studying sections a lot in the main text. The notion of a section is defined here.

Definition 10.4. A section X of a vector bundle ξ : E → M is a smooth map M → E such that theprojection ξ : M →M equals the identity on M . The set of all smooth sections is denoted Γ(E).

10.3 Pseudo-Riemannian manifolds

It might be worthwhile to review the definition of a (pseudo)-Riemannian manifold.

Definition 10.5. A pseudo-Riemannian metric g on a manifold M is a family of non-degeneratesymmetric bilinear forms

gp : TpM × TpM → R p ∈M (10.3)

which is differentiable with respect to p in the sense that

gp(X(p), Y (p)) (10.4)

is a smooth mapping for all vector fields X,Y ∈ Γ (TM). A manifold endowed with a pseudo-Riemannian metric is called a pseudo-Riemannian manifold.

In the case the family of non-degenerate symmetric bilinear forms is of signature (dim(M), 0)a manifold is known as a Riemannian manifold (and Riemannian metric). If the signature equals(1, dim(M)−1), or (dim(M)−1, 1), we speak of a Lorentzian manifold, respectively an Lorentzianmetric. The Lorentzian manifold of dimension 4 of general relativity is known as space-time.

Thus a (pseudo)-Riemannian manifold is a manifold, for which we have a metric at each tangentspace, which depends smoothly on the base point. The structure of these is strictly richer than that ofmanifold theory. Each manifold can be endowed with a Riemannian structure, but this is in no senseunique.

Bibliography

[1] S. M. Carroll. Spacetime and Geometry: An Introduction to General Relativity. Benjamin Cum-mings, 2003.

[2] Y. Choquet-Bruhat. General Relativity and the Einstein Equations. Oxford Mathematical Mono-graphs. Oxford University Press, 2009.

[3] C. J. S. Clarke. On the Global Isometric Embedding of Pseudo-Riemannian Manifolds. Proceed-ings of the Royal Society of London. A. Mathematical and Physical Sciences, 314(1518):417–428,1970.

[4] D. Hestenes. Oersted medal lecture 2002: Reforming the mathematical language of physics.American Journal of Physics, 71(2):104–121, 2003.

[5] D. Hestenes. Spacetime physics with geometric algebra. American Journal of Physics, 71:691–714, July 2003.

[6] D. Hestenes and G. Sobczyk. Clifford algebra to geometric calculus.

[7] A. Lasenby, C. Doran, and S. Gull. Gravity, gauge theories and geometric algebra. PhilisophicalTransactions A, pages 487–582, 1998.

[8] E. J. N. Looijenga. Smooth manifolds. 2008.

[9] J. Nash. The imbedding problem for Riemannian manifolds. Annals Math., 63:20–63, 1956.

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A Note on Clifford Algebras in Gravitational Physics. Rot.pdf · Clifford Algebras One of the main objects we study in this paper are clifford bundles over Riemannian manifolds. This