PHYS451: MPhys Project Final Report Non-tensorial Properties of Higher Order Vectors & Their Combination with the Connection Alexander C Booth Project Supervisor - Dr. Jonathan Gratus April 24, 2015 Abstract The concept of combining the connection with higher order vectors on a manifold is intro- duced, demonstrating two different ways in which this can be done. Definitions in both index and coordinate free representations are suggested, then written in terms of useful geometric quantities such as the torsion and curvature. Large emphasis is given to the methods which have been developed to deal with the problem of taking the definitions from coordinate to coordinate free. Some possible applications are described, most notably rewriting an equation from general relativity and viewing higher order vectors as a new source of matter. 1
Transcript
PHYS451: MPhys Project Final Report
Non-tensorial Properties of Higher Order Vectors & Their
Combination with the Connection
Alexander C BoothProject Supervisor - Dr. Jonathan Gratus
April 24, 2015
Abstract
The concept of combining the connection with higher order vectors on a manifold is intro-duced, demonstrating two different ways in which this can be done. Definitions in both indexand coordinate free representations are suggested, then written in terms of useful geometricquantities such as the torsion and curvature. Large emphasis is given to the methods whichhave been developed to deal with the problem of taking the definitions from coordinateto coordinate free. Some possible applications are described, most notably rewriting anequation from general relativity and viewing higher order vectors as a new source of matter.
The connection is a highly useful geometric object which appears in many areas of physicsand mathematics. It is a idea that will be a familiar to many through its applications in gen-eral relativity and fluid physics, featuring in both the geodesic deviation and Navier-Stokesequations[10][12]. In these equations it acts as the covariant or directional derivative, providinga way of differentiating one vector field along another vector field on a manifold. A seemingunrelated concept at first is that of a higher order vector, introduced in a paper by Duval whichstudied differential operators on manifolds[5]. Their application to systems of ordinary differen-tial equations was then investigated by Aghasi et al in 2006, yet they remain a rather abstractconcept[1]. Although higher order vectors do not lend themselves to an intuitive introduction,a natural relationship exists between them and connection. This relationship becomes evidentwhen each of their transformation laws are calculated. It will be shown that both the connectionand higher order vectors are non-tensorial, that is to say in general they are dependent on thechoice of coordinate basis. With this shared property in mind, it can be asked whether the non-tensorial nature of the two objects can be exploited in such a way, that they can be combined toform an overall tensor. Tensors of course do not depend on the choice of basis, a property whichmakes them far more useful for constructing physical theories. This project began with nothingmore than the assumption that such tensorial objects should exist, at least when working withthe ‘lowest order,’ higher order vectors. The method then being to take products of the variousnon-tensorial objects in such a way that if searching for a vectorial component for example, onlyone free contravariant index is left. The transformation properties of this newly constructedobject are then worked out by direct computation, confirming whether or not a true vector hasbeen built. As far as we are aware, the combination of higher order vectors and the connectionin this way has not been seen before. Up until this point, research has been centred aroundsecond and third order vectors. It is believed however that an inductive definition, describinghow the connection and a vector of arbitrary order can be combined, does exist. This possibilitywill be explored in more detail in later sections.
Throughout the project, classical tensor calculus is the primary technique which is used. This isthe manipulation of tensorial and tensor-like objects using index notation. It is a very commonalgebraic method which features heavily at undergraduate level, in topics such as general rela-tivity. One of the main problems with this classical approach is that it requires reference to acoordinate system, which in turn means the introduction of a metric. From the project’s outset,the research has been focussed on defining in a coordinate free way, how higher order vectorscan be combined with the connection. At least at low orders, our research has found that fromthis viewpoint, the concepts of torsion and curvature are naturally introduced. These are twophysical quantities which play central roles in modern theories of nature. Curvature has longbeen considered in general relativity as the ‘source’ of gravity, whereas the possible significanceof torsion was only more recently recognised[12]. Potential areas of application which could ex-ploit this natural appearance of torsion, are discussed more closely toward the end of the report.
The description of objects and physical laws without reference to a basis is not a new idea. It isthe foundation of a field known as differential geometry, an extremely powerful tool in theoreticalphysics. In this language for example, all four of Maxwell’s equations can be reduced to just two,describing fully relativistically, the electro-magnetic fields in any spacetime[12]. Furthermore,a classical vector is no longer defined by its transformation properties, but by a set of basicalgebraic rules. It is believed that a coordinate free approach to higher order vectors has not yetbeen attempted. As well as the final definitions themselves, the report puts much emphasis onthe process by which the definitions evolve from coordinate, to coordinate free. During research,a number of tools were developed to do this effectively.
3
Many of the coordinate free manipulations and definitions which appear in the project involveconcepts which should be familiar. Basic knowledge of multivariable calculus along with covari-ant differentiation and tensors in index notation is assumed. However, to aid the reader whois unfamiliar with these ideas from the perspective of differential geometry, section 2 has beenincluded. This is an in depth discussion which covers all of the necessary background mathe-matics, restated in coordinate free language. Also, appendix A has been written to support theuse of exterior calculus seen briefly in section 6. With these two parts included, it is hoped thatthis document is completely self contained. That is to say, no reading beyond what is writtenherein should be required. Furthermore, there is wide use of both standard and non-standardnotation. Any notation which is not explained in the main body of the report can be found insection 8, a comprehensive glossary of all notation. Finally, the paper’s key results have beenhighlighted by borders for quick reference.
2 Preliminary Mathematics
2.1 Formal Treatment of Coordinate Systems & Taylor’s Theorem
Calculating transformation laws constitutes a large part of many sections in this report, it isimportant to discus therefore exactly what is meant by a coordinate system. A coordinatexa is simply a scalar function which takes a point p = (p1, · · · , pm) on an m-dimensionalmanifold M and maps it to a subset of R. It is assumed throughout that the manifoldM is m-dimensional. A coordinate system therefore is just a set of m of these functions,(x1, · · · , xm). An alternative coordinate system is given by a different set of scalar functions(y1(x1, · · · , xm), · · · , ym(x1, · · · , xm)
), which are all functions of the old coordinate functions.
The chain rule can therefore be used to relate an object O = O(x1, · · · , xm) in one coordinatesystem, to that same object O = O(y1, · · · , ym), in another coordinate system. This conventionof ‘hatted’ and ‘un-hatted’ frames will be used throughout. For further clarity, when working ina hatted frame, Greek indices will be used. When working in an un-hatted frame, Latin indiceswill be used.
Two incredibly useful relations that will be required when investigating transformation prop-erties will now be derived. Firstly, an expression relating the second order derivatives of frame(x1, · · · , xm) with respect to yα and second order derivatives of frame (y1, · · · , ym) with respectto xa.
Lemma 1. Given two coordinate frames (x1, · · · , xm) and (y1, · · · , ym), the following relationholds true.
∂2yα
∂xa∂xb∂xc
∂yα= −∂y
α
∂xa∂yβ
∂xb∂2xc
∂yα∂yβ(1)
Proof.
0 =∂
∂xaδcb =
∂
∂xa
(∂yα
∂xb∂xc
∂yα
)=
∂2yα
∂xa∂xb∂xc
∂yα+∂yβ
∂xb
(∂
∂xa∂xc
∂yβ
)=
∂2yα
∂xa∂xb∂xc
∂yα+∂yβ
∂xb∂yα
∂xa∂2xc
∂yα∂yβ
Rearranging the final line gives exactly (1).
Now a slightly more complicated expression is considered, relating the third order coordinatepartial derivatives.
4
2.1 Formal Treatment of Coordinate Systems & Taylor’s Theorem
Lemma 2. Given two coordinate frames (x1, · · · , xm) and (y1, · · · , ym), the following relationholds true.
∂3yε
∂xa∂xb∂xc= −
(∂yγ
∂xc∂yε
∂xd∂2yα
∂xa∂xb∂2xd
∂yα∂yγ+∂yε
∂xd∂yβ
∂xa∂2yα
∂xb∂xc∂2xd
∂yα∂yβ(2)
+∂yε
∂xd∂yα
∂xb∂2yβ
∂xa∂xc∂2xd
∂yα∂yβ+∂yγ
∂xc∂yε
∂xd∂yα
∂xb∂yβ
∂xa∂3xd
∂yα∂yβ∂yγ
)Proof. The result follows from partially differentiating each side of equation (1). Beginningwith the left hand side.
∂
∂yγ
(∂2yε
∂xa∂xb∂xc
∂yε
)=∂xd
∂yγ∂xc
∂yε∂3yε
∂xa∂xd∂xb+
∂2yε
∂xa∂xb∂2xc
∂yγ∂yε
Now the right hand side.
∂
∂yγ
(−∂y
α
∂xb∂yβ
∂xa∂2xc
∂yα∂yβ
)= −
(∂xd
∂yγ∂2yα
∂xb∂xd∂yβ
∂xa∂2xc
∂yα∂yβ+∂xd
∂yγ∂yα
∂xb∂2yβ
∂xa∂xd∂2xc
∂yα∂yβ
+∂yα
∂xb∂yβ
∂xa∂3xc
∂yα∂yβ∂yγ
)Rearranging and multiplying each side by ∂yγ
∂xf∂yε
∂xg gives
∂yγ
∂xf∂yε
∂xg∂xd
∂yγ∂xc
∂yε∂3yε
∂xa∂xd∂xb= −
(∂yγ
∂xf∂yε
∂xg∂2yα
∂xa∂xb∂2xc
∂yγ∂yα+∂yγ
∂xf∂yε
∂xg∂xd
∂yγ∂2yα
∂xb∂xd∂yβ
∂xa∂2xc
∂yα∂yβ
+∂yγ
∂xf∂yε
∂xg∂xd
∂yγ∂yα
∂xb∂2yβ
∂xa∂xd∂2xc
∂yα∂yβ+∂yγ
∂xf∂yε
∂xg∂yα
∂xb∂yβ
∂xa∂3xc
∂yα∂yβ∂yγ
)=⇒ δdfδ
cg
∂3yε
∂xa∂xd∂xb= −
(∂yγ
∂xf∂yε
∂xg∂2yα
∂xa∂xb∂2xc
∂yγ∂yα+ δdf
∂yε
∂xg∂2yα
∂xb∂xd∂yβ
∂xa∂2xc
∂yα∂yβ
+δdf∂yε
∂xg∂yα
∂xb∂2yβ
∂xa∂xd∂2xc
∂yα∂yβ+∂yγ
∂xf∂yε
∂xg∂yα
∂xb∂yβ
∂xa∂3xc
∂yα∂yβ∂yγ
)=⇒ ∂3yε
∂xa∂xf∂xb= −
(∂yγ
∂xf∂yε
∂xg∂2yα
∂xa∂xb∂2xg
∂yγ∂yα+∂yε
∂xg∂2yα
∂xb∂xf∂yβ
∂xa∂2xg
∂yα∂yβ
+∂yε
∂xg∂yα
∂xb∂2yβ
∂xa∂xf∂2xg
∂yα∂yβ+∂yγ
∂xf∂yε
∂xg∂yα
∂xb∂yβ
∂xa∂3xg
∂yα∂yβ∂yγ
)=⇒ ∂3yε
∂xa∂xb∂xc= −
(∂yγ
∂xc∂yε
∂xd∂2yα
∂xa∂xb∂2xd
∂yα∂yγ+∂yε
∂xd∂2yα
∂xb∂xc∂yβ
∂xa∂2xd
∂yα∂yβ
+∂yε
∂xd∂yα
∂xb∂2yβ
∂xa∂xc∂2xd
∂yα∂yβ+∂yγ
∂xc∂yε
∂xd∂yα
∂xb∂yβ
∂xa∂3xd
∂yα∂yβ∂yγ
)
This is exactly equation (2).
Since only the transformation properties of vectors up to and including third order are dealtwith in this report, there is no need for any higher order relationships.
In section 3, the most general basis of a third order vector is stated and proved. Central to thisproof is the following version of Taylor’s theorem[9].
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2.2 First Order Vectors & 1-Forms
Theorem 3. Given any function f ∈ ΓΛ0M that is differentiable at least q-times and describedby coordinates (x1, · · · , xm), it can be expressed about the point p = (0, · · · , 0) as
f(x1, · · · , xm) =∑|I|≤q
DIf
I!
∣∣∣∣p
xI +∑|I|=q
EI(x1, · · · , xm)xI (3)
Where E(x1, · · · , xm) is a finite error term with the property that it is continuous and
limxa→0
[EI(x1, · · · , xm)
]= 0 (4)
A full explanation of multi-index notation can be found in the glossary of notation, section 8.Equipped with this formal treatment of coordinate systems, vector fields are considered next.
2.2 First Order Vectors & 1-Forms
Before talking about higher order vectors, it is useful to introduce the coordinate free definitionof a ‘regular’ vector. Regular vectors refer to the type of vector usually dealt with in classicphysics, such as those in mechanics. That is to say, in index notation they are defined as allobjects u = ua ∂
∂xa , whose components ua obey the following transformation law[12].
uα =∂yα
∂xaua (5)
For the remainder of the document, these vectors will be known as first order vectors. Theclaim that all first order vectors can be written in the form u = ua ∂
∂xa will be covered by a moregeneral theorem in section 3.
In the language of differential geometry, a vector field v is defined as a function which takes ascalar field f and gives v〈f〉, a new scalar field[11]. Here angular brackets are used for clarity,avoiding any confusion between this type of action and simply listing a function and its variables.For example, g(x, y) is a scalar field in x and y. In order for this to be a full and completelyequivalent definition of a vector field, the function must satisfy two properties[11].
Definition 4. Given f, g ∈ ΓΛ0M, a vector field v ∈ ΓTM is a function v : ΓΛ0M→ ΓΛ0M,with v : f 7→ v〈f〉 such that it satisfies
v〈f + g〉 = v〈f〉+ v〈g〉 (6)
v〈fg〉 = fv〈g〉+ gv〈f〉 (7)
Equation (6) ensures that a vector acting upon a sum of scalar fields, gives a sum of the vectoracting on each scalar. This is known as plus linearity. Equation (7) says that a vector actingupon a product of scalars obeys the Leibniz rule.
Useful to keep in mind, yet far less important for the purposes of this project are 1-form fields.They are defined in a similar way to vector fields but instead of following a Leibniz rule, theyare ‘f-linear’[11].
Definition 5. Given f ∈ ΓΛ0M and v, w ∈ ΓTM, a 1-form field µ ∈ ΓΛ1M is a functionµ : ΓTM→ ΓΛ0M, with (v) 7→ µ : v such that it satisfies
µ : (v + w) = µ : v + µ : w (8)
µ : (fv) = fµ : v (9)
6
2.3 The Connection
Intuitively if a particular operation is f-linear, it means that a scalar field can be ‘pulled out’ ofthe operation. This is what is shown in equation (9). Exactly what is meant by f-linearity willbecome clear as the report moves forward. An alternative definition of a tensor for example isto view them as objects that are both plus and f-linear. Note the use of the colon, also seenlater in the project to represent a higher order vector combining with the connection.
In complete analogy with a first order vector field, given an m-dimensional manifold M withcoordinates (x1, · · · , xm), dxa for a = 1, · · · ,m denotes a 1-form basis on this manifold[12]. Itis possible to construct differential forms of arbitrary degree, the process by which this is doneis explained in section A. These higher order differential forms are a far more well establishedtool in mathematics and physics than higher order vectors.
2.3 The Connection
Of central importance to this project is the connection, ∇ appearing greatly in sections 5onwards. As previously explained, when dealing with vectors it is sometimes called the covariantderivative and represents differentiation of one vector field along another. This research onlyconsiders the combination of the connection with first and higher order vectors, although itsaction is defined on any tensor. Before considering the connection in a coordinate free way, itis useful to look at it using index notation. To do this, the following objects must be defined.
Definition 6. Given a general connection ∇ on M,
Γcab =(∇∂a∂b
)〈xc〉 (10)
Are the Christoffel Symbols of the second kind[11].
It can be shown that given a metric compatible and torsion free connection, the Christoffelsymbols are objects which can be written as a product of partial derivatives of the metric andthe inverse metric[11]. Metric compatibility describes the condition that the covariant derivativeof the metric is zero. In this project, the explicit form of these symbols is never required. Withdefinition 6 in mind and using classical tensor analysis, the covariant derivative of a vectorv ∈ ΓTM in the direction of a vector u ∈ ΓTM can be calculated.
(∇uv)c = ua∇∂a(vb∂b
)c= ua
(∇∂avb
)c∂b + uavb
(∇∂a∂b
)c(11)
The covariant derivative of vb, ∇∂avb is just the partial derivative of vb with respect to xa andusing equation (10) it follows that
(∇uv)c = ua∂vc
∂xa+ uavbΓcab = u〈vc〉+ uavbΓcab (12)
As with a first order vector, defining the connection in a coordinate free way involves viewingit as a function[11][12].
Definition 7. Given first order vector fields u, v, w ∈ ΓTM and f ∈ ΓΛ0M, a general con-nection ∇ on M is a function ∇ : ΓTM× ΓTM → ΓTM, with (u, v) 7→ ∇uv such that itsatisfies
∇u (v + w) = ∇uv +∇uw ∇u (fv) = u〈f〉v + f∇uv (13)
∇(u+w)v = ∇uv +∇wv ∇(fu)v = f∇uv (14)
The equations in (13) ensure that the connection is plus linear and Leibniz in the vector beingdifferentiated. The equations in (14) on the other hand ensure that it is plus linear in thedirection being differentiated in, but instead of being Leibniz in this argument it is f-linear.One further piece of notation featuring later in the text is ∇0, which is used to denote a torsionfree connection.
7
2.4 Torsion & Curvature
2.4 Torsion & Curvature
Torsion and curvature are both tensorial quantities which appear in differential geometry, pro-viding a way to quantify the warped nature of a particular manifold. Although Einstein’stheory of gravity assumes a Levi-Civita connection, that is to say a connection which is metriccompatible and torsion free, curvature plays a central role. The Riemann curvature tensor fea-tures explicitly not only in Einstein’s equation but also the geodesic deviation equation. Thisrelation quantifies the tidal forces between particles on neighbouring geodesics, a second or-der effect[12]. It would be reasonable to assume therefore that somewhere in their definitions,second derivatives and products of derivatives are involved.
Definition 8. Given first order vector fields u, v, w ∈ ΓTM, the curvature R of a connection∇ on M, is a function R : ΓTM× ΓTM× ΓTM → ΓTM, with (u, v, w) 7→ R(u, v)w suchthat
R(u, v)w = ∇u∇vw −∇v∇uw −∇[u,v]w (15)
It is plus and f-linear in all of its arguments.
This object is sometimes known as the curvature vector [11]. The equivalent coordinate expres-sion, viewing the curvature as a classical (1, 3) tensor is given by the following equation[12].
Rebac = ΓdabΓecd + ∂aΓ
ecb − ΓdcbΓ
ead − ∂cΓeab (16)
This report mostly deals with the coordinate free result.
Despite Einstein’s gravity only talking about the Levi-Civita connection, where possible in thereport, new objects are kept completely general. There are many alternative theories of gravitysuch as Einstein-Cartan theory, which do involve torsion[4]. As such, the torsion tensor is nowdefined[11][12].
Definition 9. Given first order vector fields u, v ∈ ΓTM, the torsion T of a connection ∇ onM is a function T : ΓTM× ΓTM→ ΓTM, with (u, v) 7→ T (u, v) such that
T (u, v) = ∇uv −∇vu− [u, v] (17)
It is plus and f-linear in all of its arguments.
As a classical tensor[12].T cab = Γcab − Γcba (18)
Both the torsion and the curvature will be seen again in section 5, where an equation relatingthe two will be required. An expression which does just this is Bianchi’s First Identity[8].
Ω(R(u, v)w
)= Ω
((∇uT )(v, w) + T (T (u, v), w)
)(19)
Here Ω denotes the cyclic sum over u, v and w. Most notably when working in the torsion freeregime, this immediately reduces rather nicely to the following[12].
R(u, v)w +R(w, u)v +R(v, w)u = 0 (20)
All of the tools which form the foundation of the report’s proofs and definitions have now beenintroduced. Next it is shown how higher order vectors are defined mathematically.
8
3 Introducing Higher Order Vectors
The main focus of this thesis is higher order vectors. There is a complete theory surroundingdifferential forms of arbitrary order, yet work on arbitrary order vectors rarely features in theliterature. As has been mentioned, the notion of a higher order operator was introduced byDuval in 1997 and their application to ordinary differential equations was recognised shortlyafter[1][5]. All of what are believed to be new results established in this project, involve secondand third order vector fields. In the first part of this section therefore, particular attention ispaid to these. The second part of this section, section 3.2, introduces how higher order vectorscan be defined in general. Such a definition would be necessary if our research were to beextended to arbitrary orders.
3.1 Second & Third Order Vectors
Beginning with the most simple extension to regular vector fields, second order vector fields.The space of all second order vector fields is denoted ΓT 2M. As with first order vectors, itis possible to define them in a coordinate free way by means of a plus linearity condition andLeibniz rule.
Definition 10. Given f, g ∈ ΓΛ0M, a second order vector field U ∈ ΓT 2M is a functionU : ΓΛ0M→ ΓΛ0M, with U : f 7→ U〈f〉 such that it satisfies
It is clear that this definition is similar to that of a first order vector field, equation (22) howeversays that second order vector fields do not obey the standard Leibniz rule. When acting upon aproduct of scalar fields there is the usual Leibniz part fU〈g〉+gU〈f〉 as would be expected, butthen an extra term U(1,1)〈f, g〉. This object belongs to the set Γ (TM⊗ TM) and is defined asa function U(1,1)〈−,−〉 : ΓΛ0M× ΓΛ0M→ ΓΛ0M. These two properties mean that it is itselfLeibniz in both arguments. That is to say given h ∈ ΓΛ0M also
Many will have written down a second order vector without realising. The Lie bracket of vectorsfor example
[u, v], itself a vector, when written in a coordinate free way expands as[
u, v]〈f〉 = u〈v〈f〉〉 − v〈u〈f〉〉 = u v〈f〉 − v u〈f〉 = (u v − v u) 〈f〉 (25)
Here the new notation u v is introduced, meaning ‘u operate v’. It is straightforward toshow that the object u v is a second order vector (see section 5.1). This simple example alsohighlights the fact that it is possible to write a first order vector as a linear combination ofsecond order vectors. Not only does this rule extend to higher order vectors but implies thatΓTM ⊂ ΓT 2M. When looking for a general basis for this new space, it should include termssimilar to those bases of a first order vector. It will be proven at third order, but for now simplystated in lemma 11, the most general form a second order vector field can take.
9
3.1 Second & Third Order Vectors
Lemma 11. Any second order vector field U ∈ ΓT 2M can be expressed
U = Ua∂
∂xa+Uab
2
∂2
∂xa∂xb(26)
WhereUa = U〈xa〉 , Uab = U(1,1)〈xa, xb〉 (27)
Proof. This result will follow immediately from lemma 13, since ΓT 2M ⊂ ΓT 3M. That is tosay, a second order vector is effectively a special case of a third order vector.
It is useful to note the symmetry Uab = U ba due to the equality of mixed partial derivatives.This observation will be a of great importance in later sections. Such a basis makes senseif second order vectors are viewed in analogy with differential forms. An example of a basiselement for a general 2-form is dxa ∧ dxb (see appendix A), ∂2ab can be written ∂a ∂b. Thetransformation and symmetry properties of second order vector fields can be exploited in such away, that they can be combined with the connection to give a new first order vector field. Thiswill be demonstrated in section 5. A third order vector field will now be defined, the extensionis not as obvious as perhaps would be expected.
Definition 12. Given f, g ∈ ΓΛ0M, a third order vector field V ∈ ΓT 3M is a functionV : ΓΛ0M→ ΓΛ0M, with V : f 7→ V 〈f〉 such that it satisfies
Defining V(1,2) and V(2,1) as belonging to sets Γ(TM⊗ T 2M
)and Γ
(T 2M⊗ TM
)respectively
means that V(1,2) is Leibniz in its first argument but not in its second and V(2,1) is Leibniz inits second but not in its first. This is best interpreted by introducing the following quantity.
This object is Leibniz in all of its arguments and is analogous to the additional term in equation(22). With this in mind and taking f, g, h ∈ ΓΛ0M, the Leibniz properties of V(1,2) can bewritten down less abstractly.
Similar equations apply for V(2,1). To see why such a definition may be reasonable, consider thespecific case that V ∈ ΓT 3M is such that for u ∈ ΓTM and U ∈ ΓT 2M, V = uU . It is shownlater in lemma 26 that u U is indeed a third order vector field. Simply using the definition
)Writing it in this way and comparing the two lines, it is clear to see why V(1,2)〈−,−〉 would beLeibniz in the first argument and V(2,1)〈−,−〉 Leibniz in the second argument. Definition 12can be used to show that third order vectors have the following basis in general.
Lemma 13. Any third order vector field V ∈ ΓT 3M can be expressed
V = V a ∂
∂xa+V ab
2
∂2
∂xa∂xb+V abc
6
∂3
∂xa∂xb∂xc(36)
Where
V a = V 〈xa〉 , V ab = V(1,2)〈xa, xb〉+ V(2,1)〈xa, xb〉 , V abc = V(1,1,1)〈xa, xb, xc〉 (37)
Proof. To prove this result requires Taylor theorem as stated in theorem 3, to express f ∈ ΓΛ0Mabout point p ∈ M. The action of a general third order vector on this scalar field will then beconsidered. It will be shown that the lemma holds for a third order vector, V ∈ T 3
pM at pointp = (0, · · · , 0). This is sufficient since there is always the freedom to chose the origin of thecoordinate system used. Furthermore the third order vector basis only involves derivatives upto third order, this means the error term can be introduced at this order. Hence,
f(x1, · · · , xm) = f
∣∣∣∣p
+∂f
∂xa
∣∣∣∣p
xa +1
2
∂2f
∂xa∂xb
∣∣∣∣p
xaxb +1
6
∂3f
∂xa∂xb∂xc
∣∣∣∣p
xaxbxc + Eabcxaxbxc (38)
Therefore
V〈f〉 = V
⟨f∣∣p
+ xa∂af∣∣p
+1
2xaxb∂2abf
∣∣p
+1
6xaxbxc∂3abcf
∣∣p
+ Eabcxaxbxc⟩
= V
⟨f∣∣p
⟩+ V
⟨xa∂af
∣∣p
⟩+ V
⟨1
2xaxb∂2abf
∣∣p
⟩+ V
⟨1
6xaxbxc∂3abcf
∣∣p
⟩+ V
⟨Eabcxaxbxc
⟩= 0 + xa
∣∣pV
⟨∂af
∣∣p
⟩+ ∂af
∣∣pV
⟨xa⟩
+ V(1,2)
⟨xa, ∂af
∣∣p
⟩+ V(2,1)
⟨xa, ∂af
∣∣p
⟩+
1
2
((xaxb)
∣∣pV
⟨∂2abf
∣∣p
⟩+ ∂2abf
∣∣pV
⟨xaxb
⟩+ V(1,2)
⟨xaxb, ∂af
∣∣p
⟩+ V(2,1)
⟨xaxb, ∂af
∣∣p
⟩)+
1
6
((xaxbxc)
∣∣pV
⟨∂3abcf
∣∣p
⟩+ ∂3abcf
∣∣pV
⟨xaxbxc
⟩+ V(1,2)
⟨xaxbxc, ∂af
∣∣p
⟩+ V(2,1)
⟨xaxbxc, ∂af
∣∣p
⟩)+ Eabc
∣∣pV
⟨xaxbxc
⟩+ (xaxbxc)
∣∣pV
⟨Eabc
⟩+ V(1,2)
⟨Eabc, xaxbxc
⟩+ V(2,1)
⟨Eabc, xaxbxc
⟩When a vector acts upon a constant, the result is 0. In addition recall that point p is in factthe origin, therefore all coordinate functions evaluated at p are zero. Finally, by definition ofthe error function in Taylor’s theorem (see theorem 3), it is zero in the limit that (x1, · · · , xm)tends to (0, · · · , 0). Applying all of these observations implies that
V〈f〉 = ∂af∣∣pV⟨xa⟩
+1
2∂2abf
∣∣pV⟨xaxb
⟩+
1
6∂3abcf
∣∣pV⟨xaxbxc
⟩+ V(1,2)
⟨Eabc, xaxbxc
⟩+ V(2,1)
⟨Eabc, xaxbxc
⟩11
3.2 Vectors of Arbitrary Order
= ∂af∣∣pV⟨xa⟩
+1
2∂2abf
∣∣p
(xa∣∣pV⟨xb⟩
+ xb∣∣pV⟨xa⟩
+ V(1,2)
⟨xa, xb
⟩+ V(2,1)
⟨xa, xb
⟩)+
1
6∂3abcf
∣∣p
((xbxc)
∣∣pV⟨xa⟩
+ xa∣∣pV⟨xbxc
⟩+ V(1,2)
⟨xa, xbxc
⟩+ V(2,1)
⟨xa, xbxc
⟩)+ xa
∣∣pV(1,2)
⟨Eabc, xbxc
⟩+ (xbxc)
∣∣pV(1,2)
⟨Eabc, xa
⟩+ V(1,1,1)
⟨Eabc, xa, xbxc
⟩+ xa
∣∣pV(2,1)
⟨Eabc, xbxc
⟩+ (xbxc)
∣∣pV(2,1)
⟨Eabc, xa
⟩= ∂af
∣∣pV⟨xa⟩
+1
2∂2abf
∣∣p
(V(1,2)
⟨xa, xb
⟩+ V(2,1)
⟨xa, xb
⟩)+
1
6∂3abcf
∣∣p
(V(1,2)
⟨xa, xbxc
⟩+ V(2,1)
⟨xa, xbxc
⟩)+ V(1,1,1)
⟨Eabc, xa, xbxc
⟩= ∂af
∣∣pV⟨xa⟩
+1
2∂2abf
∣∣p
(V(1,2)
⟨xa, xb
⟩+ V(2,1)
⟨xa, xb
⟩)+
1
6∂3abcf
∣∣p
(xb∣∣pV(1,2)
⟨xa, xc
⟩+ xc
∣∣pV(1,2)
⟨xa, xb
⟩+ V(1,1,1)
⟨xa, xb, xc
⟩+ xb
∣∣pV(2,1)
⟨xa, xc
⟩+ xc
∣∣pV(2,1)
⟨xa, xb
⟩)+ xb
∣∣pV(1,1,1)
⟨Eabc, xa, xc
⟩+ xc
∣∣pV(1,1,1)
⟨Eabc, xa, xb
⟩= ∂af
∣∣pV⟨xa⟩
+1
2∂2abf
∣∣p
(V(1,2)
⟨xa, xb
⟩+ V(2,1)
⟨xa, xb
⟩)+
1
6∂3abcf
∣∣pV(1,1,1)
⟨xa, xb, xc
⟩=
(V⟨xa⟩∂a +
1
2
(V(1,2)
⟨xa, xb
⟩+ V(2,1)
⟨xa, xb
⟩)∂2ab +
1
6V(1,1,1)
⟨xa, xb, xc
⟩∂3abc
)f∣∣p
=
(V a∣∣p∂a +
1
2V ab
∣∣p∂2ab +
1
6V abc
∣∣p∂3abc
)f∣∣p
Since this is true for all f
V = V a∣∣p∂a +
1
2V ab
∣∣p∂2ab +
1
6V abc
∣∣p∂3abc (39)
This expression is equation (36) evaluated at point p. A vector field is simply a collection ofvectors at points, therefore lemma 13 holds.
As with the basis of second order vector fields, this basis makes sense by analogy with threeform fields, whose basis is of the form dxa ∧ dxb ∧ dxc. Third order vector fields become veryimportant in section 5. Like second order vector fields, their specific transformation propertiesand natural symmetry of coefficients can be exploited. Note once again that V ab = V ba andV abc = V cba = V cab = · · · . They can be combined with the connection to construct bothvectorial and non-trivial scalar quantities. It will next be shown how a vector of nth order canbe defined.
3.2 Vectors of Arbitrary Order
Comparing the different bases of first, second and third order vector fields which have alreadybeen seen, there is a clear pattern emerging. Although this report will not explicitly use vectorsof fourth order and above, such a definition would be useful if research in this area were tobe taken any further. The definition of an nth order vector field will now be given in a formintroduced by Gratus, Banachek, Ross and Rose but is as yet unpublished. Definitions 10 and12 are of course specific cases of this more general definition.
12
3.3 Jet Spaces
Definition 14. Given f, g ∈ ΓΛ0M, an nth order vector field W ∈ ΓTnM is a functionW : ΓΛ0M→ ΓΛ0M, with W : f 7→W 〈f〉 such that it satisfies
In the case of a first order vector field n = 1, W(i,j) = 0 for all i and j since ΓT 0M is notexplicitly defined. The summation runs over all possible combinations of a and b such thata + b = n. It is clear to see that at large orders, things quickly become complicated. Take forexample W ∈ ΓT 5M, equation (41) will include a term of the form W(3,2) ∈ Γ
(T 3M⊗ T 2M
).
In order to do any meaningful calculations, W(3,2) must be broken down into terms which areLeibniz in most or all of their arguments, using a similar approach to that seen in the thirdorder case.
The most general basis of an nth order vector is as one would expect by extension of lemmas11 and 13. The proof of the exact expression is however beyond the level of the report and islargely irrelevant since our research involves vectors of order no higher than three.
3.3 Jet Spaces
It has been repeatedly highlighted that it is the specific transformation properties of higher ordervector components and the connection, which allows them to be combined in a meaningful way.The foundation of this ‘natural relationship’ is in prolongation and jet spaces. Here a briefoverview of these ideas is presented. Consider first of all a scalar function f :M→ R such thatf = f(x1, · · · , xm). The rth order jet space of f is denoted Jr(M→ R) and is best understoodby considering the first few values of r. The zero jet of f , J0(M→ R) is simply the set of allfunctions f : M → R and is the bundle R ×M over M[15]. It can be described thereforeby coordinates (x1, · · · , xm, f), meaning that the dimension of this jet space is m + 1. Higherorder jets can then be defined in a similar way, the table below shows the next three ordersof jets of f along with their corresponding coordinate system and dimension. The fractionswhich appear in the expressions for the dimension of each space, are there to account for thesymmetries fab = fba, fabc = fbca = fcab = · · · and so on.
Jets of f . Bundle. Coordinate System. a, b, c ∈ [1, · · · ,m] Dimension.
J0(M→ R) R×M (xa, f) m+ 1
J1(M→ R) R× T ∗M (xa, f, fa) 2m+ 1
J2(M→ R) - (xa, f, fa, fab)12m
2 + 2m+ 1
J3(M→ R) - (xa, f, fa, fab, fabc)16m
3 + 12m
2 + 2m+ 1
Table 1: Jets of f .
Here T ∗M refers to the dual space of TM. Now take for example the third order jet of f , J3fand consider the most general form of the third order vector V ∈ ΓT 3M shown in equation(36). Given an element of this jet space 3ϕ = (xa, ϕ, ϕa, ϕab, ϕabc) and the higher order vectorcomponents V a, V ab and V abc, they can be combined in the following way.
V : 3ϕ = V •f(3ϕ) + V afa(3ϕ) +
1
2V abfab(
3ϕ) +1
6V abcfabc(
3ϕ) (43)
13
3.3 Jet Spaces
Where V • is known as the secular component and is included in some definitions of higher ordervectors. Duval’s work on differential operators for example does include this term[5]. In thisreport however it was decided that the term be quotiented out of the higher order vector space.This is equivalent to taking V • = 0. An element of the third order jet space is said to be thethird prolongation of f , if all of the Latin subscripts correspond to partial differentiation. Thatis to say, fa(
3ϕ) = ∂aϕ, fab(3ϕ) = ∂2abϕ and so on. If it is assumed that in equation (43),
V • = 0 and it is the prolongation being dealt with then
V : 3ϕ = V a ∂ϕ
∂xa+
1
2V ab ∂2ϕ
∂xa∂xb+
1
6V abc ∂3ϕ
∂xa∂xb∂xc
= V 〈ϕ〉
That is to say, combining a third order vector with the third prolongation of f (secular termquotiented out), corresponds to our definition of a higher order vector acting upon a scalar field.The third order vector components therefore belong to the dual of jet J3f , denoted (J3f)∗.
It will later be shown in section 5 that taking combinations of higher order vector compo-nents and the connection, leads to the cancellation of non-tensorial terms. This is because theChristoffel symbols which ultimately define the connection, belong to the first order jet spaceon M. That is to say given a connection on M, Γ : M → J1M where J1M is the set of allfirst order jets on M[14].
14
4 Investigation of Transformation Properties
Equipped with the coordinate free definitions of second and third order vectors and havingshown what their most natural coordinate bases look like, their transformation properties canbe calculated. These transformation laws are the main motivation for this work, highlightingan intimate relationship between the Christoffel symbols and certain higher order vector com-ponents. This section requires the proper treatment of coordinates as in section 2.1, yet mostof the results are achieved simply by repeated application of the chain rule.
4.1 The Christoffel Symbols
It is well known that the Christoffel symbols are not tensorial, yet the derivation of this resultis usually done using the Levi-Civita expression for the symbols. That is to say, assuming thatthe connection is metric compatible and torsion free. Here the relationship is shown using justthe definition of Γcab in terms of the connection.
Lemma 15. Consider the Christoffel symbols of the second kind, Γcab on an m-dimensionalmanifold M in coordinate frame (x1, · · · , xm). In another coordinate frame (y1, · · · , ym), theChristoffel symbols are denoted Γγαβ. The symbols in each frame are related in the followingway.
Γγαβ =∂xa
∂yα∂xb
∂yβ∂yγ
∂xcΓcab +
∂yγ
∂xd∂2xd
∂yα∂yβ(44)
Proof.
Γγαβ =(∇∂α ∂β
)〈yγ〉 =
(∇ ∂
∂yα
∂
∂yβ
)〈yγ〉
=
[∇ ∂
∂yα
(∂xb
∂yβ∂
∂xb
)]〈yγ〉
=
[∂2xb
∂yα∂yβ∂
∂xb+∂xb
∂yβ∇ ∂xa
∂yα∂∂xa
(∂
∂xb
)]〈yγ〉
=∂2xb
∂yα∂yβ∂
∂xb〈yγ〉+
∂xa
∂yα∂xb
∂yβ
(∇ ∂
∂xa
∂
∂xb
)〈yγ〉
=∂2xb
∂yα∂yβ∂yγ
∂xb+∂xa
∂yα∂xb
∂yβ
(∇ ∂
∂xa
∂
∂xb
)∂yγ
∂xc〈xc〉
=∂yγ
∂xd∂2xd
∂yα∂yβ+∂xa
∂yα∂xb
∂yβ∂yγ
∂xcΓcab
The Christoffel symbols therefore transform into two terms. There is a tensorial term as wouldbe expected from a (1, 2) tensor and one extra term dependent on a second order partial deriva-tive. If tensorial expressions are to be formed from the Christoffel symbols and other objects,then these other objects must transform in a way such that this additional term is ‘cancelledout.’ It turns out that the higher order vector components are what is required.
It is straightforward to show that the Christoffel symbols of the first kind, defined in terms ofthe second kind and metric tensor as Γcab = gcdΓ
dab transform in a similar fashion.
Γγαβ =∂xa
∂yα∂xb
∂yβ∂xc
∂yγΓcab + gab
∂2xa
∂yα∂yβ
∂xb
∂yγ(45)
This relation will be useful when investigating combining a higher order vectors with the con-nection to obtain a scalar quantity.
15
4.2 Second & Third Order Vectors
4.2 Second & Third Order Vectors
It has been demonstrated that the Christoffel symbols transform as a (1, 2) tensor with theaddition of an extra term dependent on the second derivative. It will now be shown that theUa component of a second order vector shares a similar property. To do this, the invarianceof scalar fields under a change of coordinate system is used. By definition, for f ∈ ΓΛ0Mand U ∈ ΓTnM, U〈f〉 ∈ ΓΛ0M also. That is to say for two coordinate frames, hatted andun-hatted
U〈f〉 = U〈f〉 (46)
With this in mind, the following lemma is proposed.
Lemma 16. Consider a second order vector field U ∈ ΓT 2M with components Ua and Uab
in coordinate frame (x1, · · · , xm). In another coordinate frame (y1, · · · , ym), its componentsare Uα and Uαβ. The components in each frame are related in the following way.
Uα = Ua∂yα
∂xa+ Uab
1
2
∂2yα
∂xa∂xb, Uαβ = Uab
∂yα
∂xa∂yβ
∂xb(47)
Proof. For f ∈ ΓΛ0M
U〈f〉 = Uα∂f
∂yα+ Uαβ
1
2
∂2f
∂yα∂yβ
= Uα(∂xa
∂yα∂f
∂xa
)+ Uαβ
1
2
∂
∂yα
(∂xb
∂yβ∂f
∂xb
)=
(Uα
∂xa
∂yα
)∂f
∂xa+ Uαβ
1
2
(∂2xb
∂yα∂yβ∂f
∂xb+∂xa
∂yα∂xb
∂yβ∂2f
∂xa∂xb
)=
(Uα
∂xa
∂yα+ Uαβ
1
2
∂2xa
∂yα∂yβ
)∂f
∂xa+
(Uαβ
1
2
∂xa
∂yα∂xb
∂yβ
)∂2f
∂xa∂xb
The right hand side must be equal to U〈f〉 by (46), therefore(Uα
∂xa
∂yα+ Uαβ
1
2
∂2xa
∂yα∂yβ
)∂f
∂xa+
(Uαβ
1
2
∂xa
∂yα∂xb
∂yβ
)∂2f
∂xa∂xb= Ua
∂f
∂xa+ Uab
1
2
∂2f
∂xa∂xb(48)
Since the expression is true for all f , by comparing the coefficients of ∂af and ∂2abf , then usingthe freedom to relabel and interchange the hatted and un-hatted frame yields exactly (47).
Notice that the expression for Uα involves a tensorial term and a term dependent on a secondorder derivative of yα. The non-tensorial term yielded by the Christoffel symbols is a secondderivative of xa, however it was shown in section 2.1 that there is a simple expression, equation(1) relating the two.
The transformation laws for the components of a third order vector, V ∈ ΓT 3M are nowconsidered. Although more complex, a similar sort of pattern is followed.
16
4.2 Second & Third Order Vectors
Lemma 17. Consider a third order vector field V ∈ ΓT 3M with components V a, V ab
and V abc in coordinate frame (x1, · · · , xm). In another coordinate frame (y1, · · · , ym), itscomponents are V α, V αβ and V αβγ. The components in each frame are related in thefollowing way.
V α = V a∂yα
∂xa+ V ab 1
2
∂2yα
∂xa∂xb+ V abc 1
6
∂3yα
∂xa∂xb∂xcV αβγ = V abc∂y
α
∂xa∂yβ
∂xb∂yγ
∂xc(49)
V αβ = V ab∂yα
∂xa∂yβ
∂xb+ V abc 1
3
(∂yα
∂xa∂2yβ
∂xb∂xc+∂yα
∂xb∂2yβ
∂xa∂xc+∂yβ
∂xc∂2yα
∂xa∂xb
)(50)
Proof.
V 〈f〉 = V α ∂f
∂yα+ V αβ 1
2
∂2f
∂yα∂yβ+ V αβγ 1
6
∂3f
∂yα∂yβ∂yγ
= V α
(∂xa
∂yα∂f
∂xa
)+ V αβ 1
2
∂
∂yα
(∂xb
∂yβ∂f
∂xb
)+ V αβγ 1
6
∂
∂yα
(∂
∂yβ
(∂xc
∂yγ∂f
∂xc
))=
(V α ∂x
a
∂yα
)∂f
∂xa+ V αβ 1
2
(∂2xb
∂yα∂yβ∂f
∂xb+∂xa
∂yα∂xb
∂yβ∂2f
∂xa∂xb
)+ V αβγ 1
6
∂
∂yα
(∂2xc
∂yβ∂yγ∂f
∂xc+∂xb
∂yβ∂xc
∂yγ∂2f
∂xb∂xc
)=
(V α ∂x
a
∂yα
)∂f
∂xa+ V αβ 1
2
(∂2xb
∂yα∂yβ∂f
∂xb+∂xa
∂yα∂xb
∂yβ∂2f
∂xa∂xb
)+ V αβγ 1
6
(∂3xc
∂yα∂yβ∂yγ∂f
∂xc+
∂2xc
∂yβ∂yγ∂xa
∂yα∂2f
∂xa∂xc
+∂xa
∂yα∂xb
∂yβ∂xc
∂yγ∂3f
∂xa∂xb∂xc+∂xb
∂yβ∂2xc
∂yα∂yγ∂2f
∂xb∂xc+
∂2xb
∂yα∂yβ∂xc
∂yγ∂2f
∂xb∂xc
)=
(V α ∂x
a
∂yα+ V αβ 1
2
∂2xa
∂yα∂yβ+ V αβγ 1
6
∂3xa
∂yα∂yβ∂yγ
)∂f
∂xa
+
(V αβ ∂x
a
∂yα∂xb
∂yβ+ V αβγ 1
3
(∂2xa
∂yα∂yβ∂xb
∂yγ+∂xa
∂yβ∂2xb
∂yα∂yγ+
∂2xb
∂yβ∂yγ∂xa
∂yα
))1
2
∂2f
∂xa∂xb
+
(V αβγ ∂x
a
∂yα∂xb
∂yβ∂xc
∂yγ
)1
6
∂3f
∂xa∂xb∂xc
As with the second order vector transformation laws, by (46) the right hand side must be equalto V 〈f〉 for all f . The freedom to relabel and interchange the frames can be used again yieldingexactly equations (49) and (50) by comparing coefficients.
Taking U and V to be the higher order vectors used in lemmas 16 and 17, one can see that the‘largest order coefficients’ Uab and V abc both transform tensorially. Each of the other coefficientshave a tensorial piece and extra terms involving partial derivatives, whose maximum degreecorresponds to the order of the vector. The transformation of V a for example yields thirdorder partial derivatives of ya. This means that to form a vector quantity from V a and alinear combination of other objects, one of these other objects must involve a third order partialderivative of xa when transformed. It will later be seen that the derivative of a Christoffelsymbol provides such a term.
17
5 Combining Higher Order Vectors with the Connection
In the last section, it was shown that some of the components of second and third order vectorstransform in a similar way to the Christoffel symbols for a general connection. With thesetransformation properties in mind, one can ask the following question. Is it possible to builda tensorial object from a sum of terms, composed of these vector components and Christoffelsymbols? In this section it is shown that many of these combinations do indeed exist. Sinceit is transformation laws that are being dealt with, it is far more intuitive to start working inindex notation. Once a new object has been established, it is then a case of working backwardsto extract a sensible coordinate free definition. This is the method of approach used through-out this section of research. It is believed that all of the material presented in this section iscompletely new and absent from the literature.
It is sensible to demonstrate first of all, how a first order vector combines with the connection.This result is included here as it can almost be trivially defined. The combination is largelyuninteresting, however allows the introduction of the colon notation used throughout.
Definition 18. Given a first order vector field u ∈ ΓTM and a general connection ∇ onM,
u : ∇ = u (51)
This definition alone does not hold any new mathematics, but will be required later when it isextended to higher orders, becoming something more meaningful.
5.1 Second Order Vectors & the Connection
The most basic object with a non-tensorial transformation property is the Ua component of asecond order vector field U ∈ ΓT 2M, as shown in section 4.2. With this in mind the followingobject is defined.
Definition 19. Given a second order vector field U ∈ ΓT 2M such that U = Ua∂a + Uab
2 ∂2aband a general connection ∇ on M,
(U : ∇)c =Uab
2Γcab + U c (52)
The choice of notation, that is to say the use of ∇, will become clear when this object is definedin a coordinate free manner. It will now be shown that (U : ∇)c transforms as a bona fidevector.
Lemma 20. Given a second order vector U ∈ ΓT 2M, the object (U : ∇)c is a vectorquantity. That is to say (
U : ∇)γ
=∂yγ
∂xc(U : ∇)c (53)
18
5.1 Second Order Vectors & the Connection
Proof.(U : ∇
)γ=Uαβ
2Γγαβ + Uγ
=Uab
2
∂yα
∂xa∂yβ
∂xb
(∂yγ
∂xc∂xd
∂yα∂xe
∂yβΓcde +
∂yγ
∂xd∂2xd
∂yα∂yβ
)+ Ua
∂yγ
∂xa+ Uab
1
2
∂2yγ
∂xa∂xb
=Uab
2
(δdaδ
eb
∂yγ
∂xcΓcde +
∂yα
∂xa∂yβ
∂xb∂yγ
∂xd∂2xd
∂yα∂yβ+
∂2yγ
∂xa∂xb
)+ Ua
∂yγ
∂xa
=Uab
2
(∂yγ
∂xcΓcab −
∂2yγ
∂xa∂xb+
∂2yγ
∂xa∂xb
)+ Ua
∂yγ
∂xa
=∂yγ
∂xc
(Uab
2Γcab + U c
)=∂yγ
∂xc(U : ∇)c
The penultimate line is reached using lemma 1. In equation (25) it was reasoned that the Liebracket, although itself a first order vector, is a sum of second order vectors. These secondorder vectors were of the form ‘first order vector operate first order vector.’ Such a second ordervector is useful here, the following coordinate free object based on equation (52) is defined, forthe specific case that U ∈ ΓT 2M is such that U = v w.
Definition 21. Given v, w ∈ ΓTM and a general connection ∇ on M, (v w) : ∇ ∈ ΓTMis such that
(v w) : ∇ = ∇vw −1
2T (v, w) (54)
The motivation for this definition becomes clear when the next lemma is considered. For theproof, the definitions of the connection and torsion tensor introduced in sections 2.3 and 2.4respectively are required.
Lemma 22. Let a second order vector field U ∈ ΓT 2M have components given by
Ua = vd∂wa
∂xd, Uab = vawb + vbwa (55)
Then (Uab
2Γcab + U c
)∂
∂xc= ∇vw −
1
2T (v, w) (56)
Proof.(Uab
2Γcab + U c
)∂
∂xc=
(1
2
(vawb + vbwa
)Γcab + vd
∂wc
∂xd
)∂
∂xc
=
(vawbΓcab +
1
2
(vbwa − vawb
)Γcab + vd
∂wc
∂xd
)∂
∂xc
=
((vawbΓcab + vd
∂wc
∂xd
)+
1
2vbwaΓcab −
1
2vbwaΓcba
)∂
∂xc
=
((vawbΓcab + vd
∂wc
∂xd
)+
1
2vbwa (Γcab − Γcba)
)∂
∂xc
19
5.1 Second Order Vectors & the Connection
=
((∇vw)c +
1
2vbwaT cab
)∂
∂xc=
((∇vw)c − 1
2vbwaT cba
)∂
∂xc
=
((∇vw)c − 1
2T (v, w)c
)∂
∂xc= ∇vw −
1
2T (v, w)
To begin analysing this result, the choice of the second order vector components Ua and Uab
must be justified. As discussed in section 3.1, it is a straight forward exercise to prove that forv, w ∈ ΓTM, v w is a second order vector. This simple result will now be shown.
Lemma 23. Given v, w ∈ ΓTM, then U ∈ ΓT 2M if
U = v w (57)
Furthermore in index notation this may be written
U = va∂wb
∂xa∂
∂xb+
(vbwa + vawb
)2
∂2
∂xa∂xb(58)
Proof. This proof begins using definition 4 of a first order vector field.
It is clear that U(1,1)〈f, g〉 is Leibniz in both of its arguments, therefore v w ∈ ΓT 2M bydefinition 10. Next consider a similar calculation using indices and with f ∈ ΓΛ0M.
U〈f〉 = (v w)〈f〉 = v〈w〈f〉〉
= v
⟨wa
∂f
∂xa
⟩= vb
∂
∂xb
(wa
∂f
∂xa
)= vb
∂wa
∂xb∂f
∂xa+ vbwa
∂2f
∂xb∂xa
=
(vb∂wa
∂xb∂
∂xa+ vbwa
∂2
∂xb∂xa
)f =
(vb∂wa
∂xb∂
∂xa+
(vbwa + vawb
)2
∂2
∂xa∂xb
)f
The final step exploits the natural symmetry in the definition of a second order vector, namelyUab = U ba. This is true for all f therefore after relabelling, the final line is exactly equation(58).
The notation (vw) : ∇ used in definition 21 is therefore perfectly logical. The choices of Ua andUab made in this definition correspond exactly to the calculated first and second componentsof v w respectively.
There are two other observations which further justify the suitability of this definition. First ofall, it is clear to see that any first order vectors v, w ∈ ΓTM must satisfy by definition of theLie bracket
v w − w v − [v, w] = 0 (60)
Immediately then, the following is also true.
(v w − w v − [v, w]) : ∇ = (v w) : ∇− (w v) : ∇− [v, w] : ∇ = 0 (61)
Definition 21 must be consistent with this equation.
20
5.2 Third Order Vectors & the Connection
Lemma 24. Given a general connection ∇ on M, (u v) : ∇ ∈ ΓTM satisfies
(v w) : ∇− (w v) : ∇− [v, w] : ∇ = 0 (62)
Proof. Since the Lie bracket of two vector fields is itself a vector field, definition 18 of a firstorder vector field combining with the connection will be required.
(v w) : ∇− (w v) : ∇− [v, w] : ∇ = ∇vw −1
2T (v, w)−
(∇wv −
1
2T (w, v)
)− [v, w]
= (∇vw −∇wv − [v, w])− 1
2T (v, w) +
1
2T (w, v)
= T (v, w)− T (v, w)
= 0
The second observation is that the coordinate expression for (U : ∇)e is very nearly the exactcomponent expansion of ∇vw. It would therefore be expected that any additional terms in thedefinition of (vw) : ∇ would involve first order covariant derivatives only. The torsion betweenv and w, as stated in definition 9 is T (v, w) = ∇vw−∇wv− [v, w], namely a sum of first ordercovariant derivatives. Furthermore, the definition involves specifically two vectors, v and w.A regular vector component must have only one free contravariant (upstairs) index. Simplyby considering the number of upstairs and downstairs indices in the coordinate expression, theproduct vawb can only be multiplied by a tensor of the form Qcab. Torsion is the only candidate.
5.2 Third Order Vectors & the Connection
During initial research, the motivation for definition 19 originally came from considering thecoordinate expansion of ∇vw. If the idea of combining a second order vector and the connectionis to be extended to third order, a natural consideration would be the coordinate expansion of∇u∇vw.
Lemma 25. Given u, v, w ∈ ΓTM, then
(∇u∇vw)e = ucvawb(
ΓeabΓdcd +
∂Γeab∂xc
)+(uavc∂cw
b + ucwb∂cva + ucva∂cw
b)
Γeab (63)
+uc∂cvb∂bw
e + ucvb∂2bcwe
Proof.
∇u(∇vw)e = ∇u(vawbΓeab + va
∂we
∂xa
)= uc
(vawbΓfab + va
∂wf
∂xa
)Γecf + uc
∂
∂xc
(vawbΓeab + va
∂we
∂xa
)= ucvawbΓfabΓ
ecf + ucvaΓecf
∂wf
∂xa+ ucvawb
∂Γeab∂xc
+ ucvaΓeab∂wb
∂xc+ ucwbΓeab
∂va
∂xc
+ uc∂va
∂xc∂we
∂xa+ ucva
∂2we
∂xc∂xa
= ucvawbΓfabΓecf + ucvawb
∂Γeab∂xc
+ Γeab
(uavc
∂wb
∂xc+ ucva
∂wb
∂xc+ ucwb
∂va
∂xc
)+
(uc∂va
∂xc∂we
∂xa+ ucva
∂2we
∂xc∂xa
)
At first sight this, equation (63) may not seem too enlightening. It is however straightforwardto show, using an identical method to that used to prove lemma 23, a similar lemma. Here itis stated without proof.
21
5.2 Third Order Vectors & the Connection
Lemma 26. Given u, v, w ∈ ΓTM, then V ∈ ΓT 3M if
V = u v w (64)
Furthermore in index notation this may be written
V =
(ua∂vb
∂xa∂wc
∂xb+ uavb
∂2wc
∂xa∂xb
)∂
∂xc+
(ubvc
∂wa
∂xc+ ucwa
∂vb
∂xc+ ucvb
∂wa
∂xc
)∂2
∂xa∂xb(65)
+uavbwc∂3
∂xa∂xb∂xc
Proof. Simply apply ua∂a to both sides of equation (58).
It is clear by comparing equations (63) and (65), that the coefficients of the coordinate expansionof ∇u∇vw are exactly those of the third order vector u v w. As with the second order vectorcase, the fully symmetrised coefficients of a third order vector V are now considered. By doingthis, the following third order analogue of definition 19 is constructed.
Definition 27. Given a third order vector field V ∈ ΓT 3M such that V = V a∂a+ V ab
2 ∂2ab+V abc
6 ∂3abc and a general connection ∇ on M,
(V : ∇)e =V abc
6
(ΓdabΓ
ecd + ∂cΓ
eab
)+V ab
2Γeab + V e (66)
It should be recognised that the same notation has been used for this combination of a thirdorder vector with the connection, (V : ∇)e as with the combination of a second order vectorand the connection (U : ∇)c. The reason for this is as with (U : ∇)c introduced in definition 19,the right hand side of equation (66) transforms as a vector. This result will now be proven bymeans of a rather long calculation. For clarity, colours have been used to distinguish the originsof each term. This is also useful for tracking terms down the page as the proof continues.
Lemma 28. Given a third order vector V ∈ ΓT 3M, the object (V : ∇)e is a vector quantity.That is to say (
V : ∇)ε
=∂yε
∂xe(V : ∇)e (67)
Proof. This proof requires many of the results already shown in the report. The expansion ofterms at the beginning requires almost all of the results in section 4. To simplify the resultingexpressions toward the end, the identities relating partial derivatives shown in lemmas 1 and 2will be needed.(V : ∇
)ε=V αβγ
6
(ΓδαβΓεγδ + ∂γΓεαβ
)+V αβ
2Γεαβ + V ε
=V abc
6
∂yα
∂xa∂yβ
∂xb∂yγ
∂xc
[(∂yδ
∂xh∂xf
∂yα∂xg
∂yβΓhfg +
∂yδ
∂xf∂2xf
∂yα∂yβ
)(∂yε
∂xe∂xi
∂yγ∂xj
∂yδΓeij +
∂yε
∂xj∂2xj
∂yγ∂yδ
)+
∂
∂yγ
(∂yε
∂xe∂xf
∂yα∂xg
∂yβΓefg +
∂yε
∂xf∂2xf
∂yα∂yβ
)]
22
5.2 Third Order Vectors & the Connection
+1
2
[V ab∂y
α
∂xa∂yβ
∂xb+V abc
3
(∂yα
∂xa∂2yβ
∂xb∂xc+∂yα
∂xb∂2yβ
∂xa∂xc+∂yβ
∂xc∂2yα
∂xa∂xb
)](∂yε
∂xe∂xf
∂yα∂xg
∂yβΓefg
+∂yε
∂xf∂2xf
∂yα∂yβ
)+ V a ∂y
ε
∂xa+V ab
2
∂2yε
∂xa∂xb+V abc
6
∂3yε
∂xa∂xb∂xc
=V abc
6
∂yα
∂xa∂yβ
∂xb∂yγ
∂xc
[(∂yδ
∂xh∂xf
∂yα∂xg
∂yβΓhfg
∂yε
∂xe∂xi
∂yγ∂xj
∂yδΓeij +
∂yδ
∂xf∂2xf
∂yα∂yβ∂yε
∂xe∂xi
∂yγ∂xj
∂yδΓeij
+∂yδ
∂xh∂xf
∂yα∂xg
∂yβΓhfg
∂yε
∂xj∂2xj
∂yγ∂yδ+∂yδ
∂xf∂2xf
∂yα∂yβ∂yε
∂xj∂2xj
∂yγ∂yδ
)+
(∂2yε
∂xe∂xd∂xf
∂yα∂xg
∂yβ∂xd
∂yγΓefg
+∂yε
∂xe∂2xf
∂yα∂yγ∂xg
∂yβΓefg +
∂yε
∂xe∂xf
∂yα∂2xg
∂yβ∂yγΓefg +
∂yε
∂xe∂xf
∂yα∂xg
∂yβ∂xd
∂yγ∂Γefg∂xd
+∂xd
∂yγ∂2yε
∂xd∂xf∂2xf
∂yα∂yβ
+∂yε
∂xf∂3xf
∂yα∂yβ∂yγ
)]+
1
2
[V ab∂y
α
∂xa∂yβ
∂xb∂yε
∂xe∂xf
∂yα∂xg
∂yβΓefg + V ab∂y
α
∂xa∂yβ
∂xb∂yε
∂xf∂2xf
∂yα∂yβ
+V abc
3
∂yα
∂xa∂2yβ
∂xb∂xc∂yε
∂xe∂xf
∂yα∂xg
∂yβΓefg +
V abc
3
∂yα
∂xb∂2yβ
∂xa∂xc∂yε
∂xe∂xf
∂yα∂xg
∂yβΓefg
+V abc
3
∂yβ
∂xc∂2yα
∂xa∂xb∂yε
∂xe∂xf
∂yα∂xg
∂yβΓefg+
V abc
3
∂yα
∂xa∂2yβ
∂xb∂xc∂yε
∂xf∂2xf
∂yα∂yβ
+V abc
3
∂yα
∂xb∂2yβ
∂xa∂xc∂yε
∂xf∂2xf
∂yα∂yβ+V abc
3
∂yβ
∂xc∂2yα
∂xa∂xb∂yε
∂xf∂2xf
∂yα∂yβ
]+V a ∂y
ε
∂xa+V ab
2
∂2yε
∂xa∂xb+V abc
6
∂3yε
∂xa∂xb∂xc
=V abc
6
∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂yδ
∂xh∂xf
∂yα∂xg
∂yβ∂yε
∂xe∂xi
∂yγ∂xj
∂yδΓhfgΓ
eij
+V abc
6
∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂yδ
∂xf∂2xf
∂yα∂yβ∂yε
∂xe∂xi
∂yγ∂xj
∂yδΓeij+
V abc
6
∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂yδ
∂xh∂xf
∂yα∂xg
∂yβ∂yε
∂xj∂2xj
∂yγ∂yδΓhfg
+V abc
6
∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂yδ
∂xf∂2xf
∂yα∂yβ∂yε
∂xj∂2xj
∂yγ∂yδ
+V abc
6
∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂2yε
∂xe∂xd∂xf
∂yα∂xg
∂yβ∂xd
∂yγΓefg +
V abc
6
∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂yε
∂xe∂2xf
∂yα∂yγ∂xg
∂yβΓefg
+V abc
6
∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂yε
∂xe∂xf
∂yα∂2xg
∂yβ∂yγΓefg +
V abc
6
∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂yε
∂xe∂xf
∂yα∂xg
∂yβ∂xd
∂yγ∂Γefg∂xd
+V abc
6
∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂xc
∂yγ∂2yε
∂xc∂xf∂2xf
∂yα∂yβ+V abc
6
∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂yε
∂xf∂3xf
∂yα∂yβ∂yγ
+V ab
2
∂yα
∂xa∂yβ
∂xb∂yε
∂xe∂xf
∂yα∂xg
∂yβΓefg +
V ab
2
∂yα
∂xa∂yβ
∂xb∂yε
∂xf∂2xf
∂yα∂yβ+V abc
6
∂yα
∂xa∂2yβ
∂xb∂xc∂yε
∂xe∂xf
∂yα∂xg
∂yβΓefg
+V abc
6
∂yα
∂xb∂2yβ
∂xa∂xc∂yε
∂xe∂xf
∂yα∂xg
∂yβΓefg +
V abc
6
∂yβ
∂xc∂2yα
∂xa∂xb∂yε
∂xe∂xf
∂yα∂xg
∂yβΓefg
+V abc
6
∂yα
∂xa∂2yβ
∂xb∂xc∂yε
∂xf∂2xf
∂yα∂yβ+V abc
6
∂yα
∂xb∂2yβ
∂xa∂xc∂yε
∂xf∂2xf
∂yα∂yβ
+V abc
6
∂yβ
∂xc∂2yα
∂xa∂xb∂yε
∂xf∂2xf
∂yα∂yβ
+V a ∂yε
∂xa+V ab
2
∂2yε
∂xa∂xb+V abc
6
∂3yε
∂xa∂xb∂xc
=V abc
6δfaδ
gb δicδjh
∂yε
∂xeΓhfgΓ
eij +
V abc
6δicδ
jf
∂yα
∂xa∂yβ
∂xb∂yε
∂xe∂2xf
∂yα∂yβΓeij
+V abc
6δfaδ
gb
∂yγ
∂xc∂yδ
∂xh∂yε
∂xj∂2xj
∂yγ∂yδΓhfg +
V abc
6
∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂yδ
∂xf∂yε
∂xj∂2xf
∂yα∂yβ∂2xj
∂yγ∂yδ
23
5.2 Third Order Vectors & the Connection
+V abc
6
∂2yε
∂xe∂xdδfaδ
gb δdcΓefg +
V abc
6
∂yγ
∂xc∂yα
∂xa∂yε
∂xeδgb
∂2xf
∂yα∂yγΓefg +
V abc
6
∂yβ
∂xb∂yγ
∂xc∂yε
∂xeδfa
∂2xg
∂yβ∂yγΓefg
+V abc
6
∂yε
∂xe∂Γeab∂xc
+V abc
6
∂yα
∂xa∂yβ
∂xb∂2yε
∂xc∂xf∂2xf
∂yα∂yβ+V abc
6
∂yε
∂xf∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂3xf
∂yα∂yβ∂yγ
+V ab
2δfaδ
gb
∂yε
∂xeΓefg +
V ab
2
∂yε
∂xf∂yα
∂xa∂yβ
∂xb∂2xf
∂yα∂yβ+V abc
6δfa∂yε
∂xe∂2yβ
∂xb∂xc∂xg
∂yβΓefg
+V abc
6δfb
∂2yβ
∂xa∂xc∂yε
∂xe∂xg
∂yβΓefg +
V abc
6δgc
∂2yα
∂xa∂xb∂yε
∂xe∂xf
∂yαΓefg +
V abc
6
∂yα
∂xa∂2yβ
∂xb∂xc∂yε
∂xf∂2xf
∂yα∂yβ
+V abc
6
∂yα
∂xb∂2yβ
∂xa∂xc∂yε
∂xf∂2xf
∂yα∂yβ+V abc
6
∂yβ
∂xc∂2yα
∂xa∂xb∂yε
∂xf∂2xf
∂yα∂yβ
+V a ∂yε
∂xa+V ab
2
∂2yε
∂xa∂xb+V abc
6
∂3yε
∂xa∂xb∂xc
=V abc
6
∂yε
∂xeΓdabΓ
ecd +
V abc
6
∂yα
∂xa∂yβ
∂xb∂yε
∂xe∂2xd
∂yα∂yβΓecd
+V abc
6
∂yγ
∂xc∂yδ
∂xd∂yε
∂xj∂2xj
∂yγ∂yδΓdab +
V abc
6
∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂yδ
∂xf∂yε
∂xj∂2xf
∂yα∂yβ∂2xj
∂yγ∂yδ
+V abc
6
∂2yε
∂xe∂xcΓeab +
V abc
6
∂yα
∂xa∂yγ
∂xc∂yε
∂xe∂2xf
∂yα∂yγΓefb +
V abc
6
∂yβ
∂xb∂yγ
∂xc∂yε
∂xe∂2xg
∂yβ∂yγΓeag
+V abc
6
∂yε
∂xe∂Γeab∂xc
+V abc
6
∂yα
∂xa∂yβ
∂xb∂2yε
∂xc∂xf∂2xf
∂yα∂yβ+V abc
6
∂yε
∂xf∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂3xf
∂yα∂yβ∂yγ
+V ab
2
∂yε
∂xeΓeab +
V ab
2
∂yε
∂xf∂yα
∂xa∂yβ
∂xb∂2xf
∂yα∂yβ+V abc
6
∂yε
∂xe∂2yβ
∂xb∂xc∂xg
∂yβΓeag
+V abc
6
∂2yβ
∂xa∂xc∂yε
∂xe∂xg
∂yβΓebg +
V abc
6
∂2yα
∂xa∂xb∂yε
∂xe∂xf
∂yαΓefc +
V abc
6
∂yα
∂xa∂2yβ
∂xb∂xc∂yε
∂xf∂2xf
∂yα∂yβ
+V abc
6
∂yα
∂xb∂2yβ
∂xa∂xc∂yε
∂xf∂2xf
∂yα∂yβ+V abc
6
∂yβ
∂xc∂2yα
∂xa∂xb∂yε
∂xf∂2xf
∂yα∂yβ
+V a ∂yε
∂xa+V ab
2
∂2yε
∂xa∂xb+V abc
6
∂3yε
∂xa∂xb∂xc
=∂yε
∂xe
[V abc
6
(ΓdabΓ
ecd+
∂Γeab∂xc
)+V ab
2Γeab+V
e
]+V ab
2
(∂yα
∂xa∂yβ
∂xb∂yε
∂xf∂2xf
∂yα∂yβ+
∂2yε
∂xa∂xb
)+V abc
6
(∂yα
∂xa∂yβ
∂xb∂yε
∂xe∂2xd
∂yα∂yβΓecd +
∂yγ
∂xc∂yδ
∂xd∂yε
∂xj∂2xj
∂yγ∂yδΓdab
+∂2yε
∂xe∂xcΓeab +
∂yα
∂xa∂yγ
∂xc∂yε
∂xe∂2xf
∂yα∂yγΓefb+
∂yβ
∂xb∂yγ
∂xc∂yε
∂xe∂2xg
∂yβ∂yγΓeag
+∂yε
∂xe∂2yβ
∂xb∂xc∂xg
∂yβΓeag +
∂2yβ
∂xa∂xc∂yε
∂xe∂xg
∂yβΓebg +
∂2yα
∂xa∂xb∂yε
∂xe∂xf
∂yαΓefc
)+V abc
6
(∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂yδ
∂xf∂yε
∂xj∂2xf
∂yα∂yβ∂2xj
∂yγ∂yδ+∂yα
∂xa∂yβ
∂xb∂2yε
∂xc∂xf∂2xf
∂yα∂yβ+
∂3yε
∂xa∂xb∂xc
+∂yε
∂xf∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂3xf
∂yα∂yβ∂yγ+∂yα
∂xa∂2yβ
∂xb∂xc∂yε
∂xf∂2xf
∂yα∂yβ+∂yα
∂xb∂2yβ
∂xa∂xc∂yε
∂xf∂2xf
∂yα∂yβ
+∂yβ
∂xc∂2yα
∂xa∂xb∂yε
∂xf∂2xf
∂yα∂yβ
)=∂yε
∂xe(V : ∇)e +
V abc
6
(∂yα
∂xa∂yβ
∂xb∂yε
∂xe∂2xd
∂yα∂yβΓecd +
∂yγ
∂xc∂yδ
∂xd∂yε
∂xj∂2xj
∂yγ∂yδΓdab
+∂2yε
∂xe∂xcΓeab +
∂yα
∂xa∂yγ
∂xc∂yε
∂xe∂2xf
∂yα∂yγΓefb+
∂yβ
∂xb∂yγ
∂xc∂yε
∂xe∂2xg
∂yβ∂yγΓeag
24
5.2 Third Order Vectors & the Connection
+∂yε
∂xe∂2yβ
∂xb∂xc∂xg
∂yβΓeag +
∂2yβ
∂xa∂xc∂yε
∂xe∂xg
∂yβΓebg +
∂2yα
∂xa∂xb∂yε
∂xe∂xf
∂yαΓefc
)+V abc
6
(∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂yδ
∂xf∂yε
∂xj∂2xf
∂yα∂yβ∂2xj
∂yγ∂yδ+∂yα
∂xa∂yβ
∂xb∂2yε
∂xc∂xf∂2xf
∂yα∂yβ
)+V abc
6
(∂3yε
∂xa∂xb∂xc+∂yε
∂xf∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂3xf
∂yα∂yβ∂yγ
+∂yα
∂xa∂2yβ
∂xb∂xc∂yε
∂xf∂2xf
∂yα∂yβ+∂yα
∂xb∂2yβ
∂xa∂xc∂yε
∂xf∂2xf
∂yα∂yβ+∂yβ
∂xc∂2yα
∂xa∂xb∂yε
∂xf∂2xf
∂yα∂yβ
)+V ab
2
(∂yα
∂xa∂yβ
∂xb∂yε
∂xf∂2xf
∂yα∂yβ+
∂2yε
∂xa∂xb
)=∂yε
∂xe(V : ∇)e +
V abc
6
(∂yα
∂xa∂yβ
∂xb∂yε
∂xe∂2xd
∂yα∂yβΓecd +
∂yγ
∂xc∂yδ
∂xd∂yε
∂xj∂2xj
∂yγ∂yδΓdab
+∂2yε
∂xe∂xcΓeab +
∂yα
∂xa∂yγ
∂xc∂yε
∂xe∂2xf
∂yα∂yγΓefb+
∂yβ
∂xb∂yγ
∂xc∂yε
∂xe∂2xg
∂yβ∂yγΓeag
+∂yε
∂xe∂2yβ
∂xb∂xc∂xg
∂yβΓeag +
∂2yβ
∂xa∂xc∂yε
∂xe∂xg
∂yβΓebg +
∂2yα
∂xa∂xb∂yε
∂xe∂xf
∂yαΓefc
)+V abc
6
(∂yα
∂xa∂yβ
∂xb∂yγ
∂xc∂yδ
∂xf∂yε
∂xj∂2xf
∂yα∂yβ∂2xj
∂yγ∂yδ− ∂yα
∂xa∂yβ
∂xb∂yε
∂xj∂yγ
∂xf∂yδ
∂xc∂2xj
∂yγ∂yδ∂2xf
∂yα∂yβ
)=∂yε
∂xe(V : ∇)e +
V abc
6
[(∂yα
∂xa∂yβ
∂xb∂yε
∂xe∂2xd
∂yα∂yβΓecd +
∂2yβ
∂xc∂xa∂yε
∂xe∂xg
∂yβΓebg
)+
(∂yγ
∂xc∂yδ
∂xd∂yε
∂xj∂2xj
∂yγ∂yδΓdab +
∂2yε
∂xe∂xcΓeab
)+
(∂yβ
∂xb∂yγ
∂xc∂yε
∂xe∂2xg
∂yβ∂yγΓeag +
∂yε
∂xe∂2yβ
∂xb∂xc∂xg
∂yβΓeag
)+
(∂2yα
∂xa∂xb∂yε
∂xe∂xf
∂yαΓefc +
∂yα
∂xc∂yγ
∂xa∂yε
∂xe∂2xf
∂yα∂yγΓefb
)]=∂yε
∂xe(V : ∇)e
+V abc
6
[(∂yα
∂xa∂yβ
∂xb∂2xd
∂yα∂yβ+
∂2yβ
∂xa∂xb∂xd
∂yβ
)∂yε
∂xeΓecd +
(∂yγ
∂xc∂yδ
∂xd∂yε
∂xf∂2xf
∂yγ∂yδ+
∂2yε
∂xd∂xc
)Γdab
+
(∂yβ
∂xb∂yγ
∂xc∂2xg
∂yβ∂yγ+
∂2yβ
∂xb∂xc∂xg
∂yβ
)∂yε
∂xeΓeag +
(∂yα
∂xa∂yγ
∂xc∂2xf
∂yα∂yγ+
∂2yα
∂xa∂xc∂xf
∂yα
)∂yε
∂xeΓefb
]=∂yε
∂xe(V : ∇)e
Where to get to the penultimate line, the symmetry V abc = V cab has been used on two occasions.
Now that it has been shown that equation (66) represents a bona fide vector, it is reasonableto assume there is a coordinate free interpretation. This was the case with equation (52), thesecond order expression, which was found to be linear in torsion. Such a relationship was to beexpected since the object came about by considering ∇vw. This third order expression howevercomes about by investigating ∇u∇vw, which has been shown to involve both derivatives andproducts of Christoffel symbols. There is therefore a far wider variety of terms that could appear.For example, some kind of curvature dependence would be expected, or indeed derivatives orsquares of torsion. Using a similar yet less forceful approach to that of the last section, the casewhen V ∈ ΓT 3M is such that V = u v w is considered. The starting point is with a newtype of method which exploits the f-linearity and Leibniz properties of our object. Two thirdorder analogues of equation (60) are then used. It is clear that any first order vectors u, v and
25
5.2 Third Order Vectors & the Connection
w must satisfy both of the following identities.
u v w − v u w − [u, v] w = 0 (68)
u v w − u w v − u [v, w] = 0 (69)
In section 5.2.2, it is shown how these two equations alone can be used to justify a coordinatefree definition of (V : ∇)e given a torsion free connection.
A Brief Aside
Looking back to section 5.1, the coordinate free definition of (v w) : ∇ was ‘derived’ by writingthe coordinate expression in terms of vectors which have a specific coordinate free definition.A significant amount of time was spent attempting to use the same method to get from thecoordinate to coordinate free definition of (V : ∇)e. The main issue was finding the correctinterpretation for the third order vector component V abc. By definition of a higher order vector,V abc is completely symmetric. That is to say, with the result of lemma 26, it would be expectedthat V abc would take the following form for V = u v w.
This way, V abc = V cba = V bca = · · · . On the other hand, in order to show that (V : ∇)e
transforms as a vector (lemma 28), the only symmetry which is used is V abc = V cab. This is infact a cyclic permutation of abc and would imply that V abc could look something like
V abc ∝ A(uavbwc + ucvawb + ubvcwa
)+B
(ubvawc + uavcwb + ucvbwa
)(71)
For constants A and B. It is straightforward to show that this form of the component satisfiesV abc = V cab. Due to the length of each calculation that such a method involves, this turned outto be a highly inefficient way of dealing with the problem and all attempts were unsuccessful.For this reason a number of new, more indirect methods were developed. These were largelymore successful in directing the research toward a firm definition.
5.2.1 A General Connection
As has already been discussed, it is expected that a coordinate free (V : ∇)e will involvecurvature terms and those which are derivatives of, or are second order in the torsion. Oneother possibility are terms of the form T (∇−−,−). These arise due to the appearance ofV ab ∝ ucwa∂cv
b in the coordinate definition. To see how each of these terms feature, thefull coordinate free definition of (V : ∇)e with a general connection will now be given. Afull justification will follow. As explained, due to lack of time and methods available, theexact coefficients of each and every term were not calculated, however the overall form of theexpression is clear.
Definition 29. Given u, v, w ∈ ΓTM and a general connection ∇ on M,(u v w) : ∇ ∈ ΓTM is such that
(u v w) : ∇ = ∇u∇vw −1
3R(u, v)w − 1
3R(u,w)v + T3 (72)
Where
T3 = −1
2T (u,∇vw)− 1
2T (∇uv, w)− 1
2T (v,∇uw) +A(∇uT )(v, w) +B(∇vT )(u,w) (73)
+C(∇wT )(u, v) +DT (T (u, v), w) + ET (T (v, u), w) + FT (T (w, u), v)
A,B,C,D,E and F are constants yet to be determined.
26
5.2 Third Order Vectors & the Connection
It will first be shown by using nothing more than specific f-linear and Leibniz requirements of(u v w) : ∇, that terms of the form T (−,∇−−) not only must appear, but can also onlyhave coefficients ±1
2 or 0.
Lemma 30. Given f ∈ ΓΛ0M, u, v, w ∈ ΓTM and a general connection ∇ on M. Thef -linearity and Leibniz requirements of (u v w) : ∇ ∈ ΓTM force it’s coordinate freeexpression to be of the form
(u v w) : ∇ = −1
2T (u,∇vw)− 1
2T (∇uv, w)− 1
2T (v,∇uw) + “f -linear terms” (74)
Proof. Investigation of (fu) is trivial and yields nothing new. Consider then (fv).
(u (fv) w) : ∇ = u〈f〉v w : ∇+ f(u v w) : ∇
= u〈f〉(∇vw −
1
2T (v, w)
)+ f(u v w) : ∇
= u〈f〉∇vw −1
2
(T (∇u(fv), w)− fT (∇uv, w)
)+ f(u v w) : ∇
= ∇u(f∇vw)− f∇u∇vw −1
2
(T (∇u(fv), w)− fT (∇uv, w)
)+ f(u v w) : ∇
= ∇u∇(fv)w −1
2T (∇u(fv), w) + f
((u v w) : ∇−∇u∇vw +
1
2T (∇uv, w)
)Hence
(u (fv) w) : ∇−∇u∇(fv)w +1
2T (∇u(fv), w) = f
((u v w) : ∇−∇u∇vw +
1
2T (∇uv, w)
)
=⇒ (u v w) : ∇ = ∇u∇vw −1
2T (∇uv, w) + “other terms f -linear in u and v” (75)
This method clearly only highlights terms which must appear in the definition in order tocompensate for the Leibniz structure of the left hand side. There can therefore be any numberof other terms which are f-linear in u and v, hence the additional “+ other terms f-linear in uand v.” Next consider (fw).
(u v (fw)) : ∇ = (u (v〈f〉w)) : ∇+ (u (fv w)) : ∇= u〈v〈f〉〉w : ∇+ v〈f〉u w : ∇+ u〈f〉v w : ∇+ f(u v w) : ∇
= ∇u(v〈f〉w
)− v〈f〉∇uw + v〈f〉∇uw −
1
2v〈f〉T (u,w) + u〈f〉∇vw
− 1
2u〈f〉T (v, w) + f(u v w) : ∇
= ∇u(∇v(fw)− f∇vw
)− 1
2T (u,∇v(fw)− f∇vw) +∇u
(f∇vw
)− f∇u∇vw
− 1
2T (v,∇u(fw)− f∇uw) + f(u v w) : ∇
= ∇u∇v(fw)− f∇u∇vw −1
2T (u,∇v(fw)) +
f
2T (u,∇vw)
− 1
2T (v,∇u(fw)) +
f
2T (v,∇uw) + f(u v w) : ∇
27
5.2 Third Order Vectors & the Connection
Hence
(u v (fw)) : ∇−∇u∇v(fw) +1
2T (u,∇v(fw)) +
1
2T (v,∇u(fw)) = f
((u v w) : ∇ (76)
−∇u∇vw +1
2T (u,∇vw) +
1
2T (v,∇uw)
)Combining this result with (75)
=⇒ (u v w) : ∇ = ∇u∇vw −1
2T (∇uv, w)− 1
2T (u,∇vw)− 1
2T (v,∇uw) (77)
+ “other terms f -linear in u, v and w”
As before, f-linear terms must be accounted for. This is exactly equation (74).
This method uses nothing more than the Leibniz property of first order vectors and our def-inition of second order vectors combining with the connection. With no other assumptions,every term which is not f-linear in all of u, v and w has been attained. These terms just hap-pen to look like a nice extension of the second order result, it may be there is an underlyingpattern. Roughly speaking, going from (v w) : ∇ to (u v w) : ∇, ∇vw → ∇u∇vw andT (v, w) → T (∇uv, w) + T (u,∇vw) + T (v,∇uw). Even at this early stage, the pattern pointsto a possible generalisation to nth order combination with the connection.
By definition of the connection and torsion tensors, all terms involving derivatives and squaresof torsion are f-linear in all of their arguments. This is the reason definition 29 includes cyclicsums of both, with the unknown coefficients A through to F . Only these six terms are necessarydue to the antisymmetry of the torsion tensor. Unfortunately, the exact coefficients of these sixterms were never found due to lack of constraints. This problem will be addressed in detail insections 6 and 7. However, by considering (u v w) : ∇ in a torsion free regime, the exactform of (u v w) : ∇0 can be fully justified. Assuming that lemma 28 holds, it may be thatat higher orders, object such as (V : ∇)e only exist in the absence of torsion. If this is the caseit could point to something more fundamental, which at the current level of understanding isbeing overlooked.
5.2.2 A Torsion Free Connection
Considering a torsion free connection makes for a greatly simplified problem, in this case it ispossible to take T3 = 0 in equation (72). It will now be shown that by enforcing equations (68)and (69), one arrives at the following definition.
Definition 31. Given u, v, w ∈ ΓTM and a torsion free connection ∇0 on M,(u v w) : ∇0 ∈ ΓTM is such that
(u v w) : ∇0 = ∇0u∇0
vw −1
3R(u, v)w − 1
3R(u,w)v (78)
It has been argued by looking at the coordinate form of (V : ∇)e that (uvw) : ∇0, in additionto ∇u∇vw, will only involve curvature terms. It is not obvious however that these two particularcurvature terms should be in the definition at all, let alone be sure that they are indeed theonly curvature terms that can feature. This said, Bianchi’s first identity, equation (20) requires
28
5.2 Third Order Vectors & the Connection
that the cyclic sum of three curvature tensors be zero. This means no more than two termsof a cyclic sum can appear. Using Bianchi again, these two cyclic terms can be written as thenegative of the third term in the cyclic sum, hence no two terms which are cyclic permutationsof each other can appear simultaneously. Furthermore there are only two ways to arrange u, vand w such that their cyclic sums are independent. That is to say, the cyclic sum of (u,w, v) isonly different to (u, v, w), not for example (w, v, u). This is easily verified by writing down allof the permutations. With these arguments alone, the following hypothesis can be made.
(u v w) : ∇0 = ∇0u∇0
vw +AR(u, v)w +BR(u,w)v (79)
Where A and B are constants. These constants can then be found using equations (68) and(69). Instead of doing this calculation explicitly, it will simply be shown that (u v w) : ∇0
given by definition 31, does indeed satisfy both equations.
Lemma 32. Consider a torsion free connection ∇0 on M and (u v w) : ∇0 ∈ ΓTM suchthat (u v w) : ∇0 = ∇0
u∇0vw − 1
3R(u, v)w − 13R(u,w)v. Then
u v w : ∇0 − v u w : ∇0 − [u, v] w : ∇0 = 0 (80)
u v w : ∇0 − u w v : ∇0 − u [v, w] : ∇0 = 0 (81)
Proof. Beginning with the left hand side of (80).
u v w : ∇0 − v u w : ∇0 − [u, v] w : ∇0 = ∇0u∇0
vw −1
3R(u, v)w − 1
3R(u,w)v
−(∇0v∇0
uw −1
3R(v, u)w − 1
3R(v, w)u
)−∇0
[u,v]w
=(∇0u∇0
vw −∇0v∇0
uw −∇0[u,v]w
)− 1
3R(u, v)w − 1
3R(u,w)v
+1
3R(v, u)w +
1
3R(v, w)u
= R(u, v)w − 2
3R(u, v)w +
1
3R(v, w)u+
1
3R(w, u)v
=1
3
(R(u, v)w +R(v, w)u+R(w, u)v
)= 0
By Bianchi’s first identity, equation (20). Now onto the left hand side of (81).
u v w : ∇0 − u w v : ∇0 − u [v, w] : ∇0 = ∇0u∇0
vw −1
3R(u, v)w − 1
3R(u,w)v
−(∇0u∇0
wv −1
3R(u,w)v − 1
3R(u, v)w
)−∇0
u[v, w]
= ∇0u
(∇0vw −∇0
wv − [v, w])− 1
3R(u, v)w − 1
3R(u,w)v
+1
3R(u, v)w +
1
3R(u,w)v
= ∇0u (T (v, w)) = 0
Since the connection is torsion free.
It would seem therefore that definition 31 correctly reflects how a third order vector combineswith a torsion free connection. For many applications in physics, a torsion free connection is allthat is required for a solid theory. As has already been mentioned, general relativity is based onthe idea of a torsion free connection[12]. There is also the Fundamental Theorem of Riemannian
29
5.3 Third Order Vectors & the Connection, a Scalar
geometry, revisited later in section 6.
The existence of these vectorial objects, constructed using the non-tensorial Christoffel symbolsalong with second and third order vector components, is quite astonishing. A natural stepforward would be to look at whether vectors of this form exist when dealing with vectors of nth
order. The work done at these low orders strongly suggests the possibility of such a definition.This topic will be discussed in greater detail in section 6.
5.3 Third Order Vectors & the Connection, a Scalar
It has now been explicitly shown that it is possible to combine higher order vectors with theconnection and construct a first order vector. During research it was found that it is alsopossible to build a scalar quantity from higher order vector components and the Christoffelsymbols. There is no obvious way to do this with a first order vector, but at second order theresult can be written down almost trivially.
Definition 33. Given a second order vector field U ∈ ΓT 2M such that U = Ua∂a+ Uab
2 ∂2ab,a metric g ∈ Γ
⊗M and a general connection ∇ on M,
U... ∇ =
Uab
2gab (82)
Here a triple colon is used to distinguish this expression from the object (U : ∇)e, which is of
course a vector. The claim is that U... ∇ transforms as a scalar quantity. It was shown in section
4 that for a second order vector, the component Uab is a symmetric (2, 0) tensor. The metricis by definition a symmetric (0, 2) tensor, hence when the two tensors are combined the indicescan be contracted and the result is a scalar. The right hand side of equation (82) also leads toa nice coordinate free definition. Using the same approach as with the vectorial objects, thespecific case of U = v w is considered.
Definition 34. Given v, w ∈ ΓTM, a metric g ∈ Γ⊗M and a general connection ∇ on
M, (v w)... ∇ ∈ ΓΛ0M is such that
v w... ∇ = g(v : ∇, w : ∇) (83)
This definition is easily justified by expanding the right hand side of equation (82) into is fullysymmetric form introduced in lemma 23.
Lemma 35. Let a second order vector field U ∈ ΓT 2M have components Uab = vawb+vbwa
and arbitrary Ua. ThenUab
2gab = g(v : ∇, w : ∇) (84)
30
5.3 Third Order Vectors & the Connection, a Scalar
Proof.
Uab
2gab =
1
2(vawb + vbwa)gab
=1
2(gabv
awb + gabvbwa)
=1
2(gabv
awb + gabvawb)
=1
2(2gabv
awb)
= g(v, w) = g(v : ∇, w : ∇)
It would appear that definition 34 suggests a relationship between first and second order vectorscombining with the connection. Although this does imply the existence of some sort of inductivedefinition for the combination of arbitrary order vectors and the connection, a first order vectorcombining with the connection is a trivial result. Recall that u : ∇ = u. To show that there isindeed a strong link between subsequent orders, higher orders must be investigated. A possibledefinition is now given for a scalar constructed from a third order vector and the connection.
Definition 36. Given a third order vector field W ∈ ΓT 3M such that W = W a∂a +Wab
2 ∂2ab + Wabc
6 ∂3abc and a general connection ∇ on M,
W... ∇ = W abcΓcab +W abgab (85)
Where Γabc = gadΓdbc are the Christoffel symbols of the first kind.
The existence of such a definition was a great surprise since the right hand side involves onlytwo of three possible third order components. For this reason, it would be expected that anysymmetry required to show the object’s invariance under coordinate transform, to be broken.With the following lemma it is clear that this assumption is incorrect.
Lemma 37. Given a third order vector field W ∈ ΓT 3M, the object W... ∇ transforms as
a scalar quantity. That is to sayW
... ∇ = W... ∇ (86)
Proof. Note the use of equation (45) for the Christoffel symbol of the first kind transformationlaw.
W
... ∇ = WαβγΓγαβ + Wαβ gαβ
= W abc∂yα
∂xa∂yβ
∂xb∂yγ
∂xc
(∂xd
∂yα∂xe
∂yβ∂xf
∂yγΓfde + gde
∂2xd
∂yα∂yβ
∂xe
∂yγ
)+ gde
∂xd
∂yα∂xe
∂yβ
(W ab∂y
α
∂xa∂yβ
∂xb+W abc 1
3
(∂yα
∂xa∂2yβ
∂xb∂xc+∂yα
∂xb∂2yβ
∂xa∂xc+∂yβ
∂xc∂2yα
∂xa∂xb
))
31
5.3 Third Order Vectors & the Connection, a Scalar
= W abcδdaδebδfc Γfde +W abcgdeδ
ec
∂yα
∂xa∂yβ
∂xb∂2xd
∂yα∂yβ+W abgdeδ
daδeb
+W abcgde1
3
(δda∂xe
∂yβ∂2yβ
∂xb∂xc+ δdb
∂xe
∂yβ∂2yβ
∂xa∂xc+ δec
∂xd
∂yα∂2yα
∂xa∂xb
)= W abcΓcab +W abgab +W abcgdc
∂yα
∂xa∂yβ
∂xb∂2xd
∂yα∂yβ
+W abcgae1
3
∂xe
∂yβ∂2yβ
∂xb∂xc+W abcgbe
1
3
∂xe
∂yβ∂2yβ
∂xa∂xc+W abcgdc
1
3
∂xd
∂yα∂2yα
∂xa∂xb
= W... ∇−W abcgdc
∂xd
∂yα∂2yα
∂xa∂xb+
1
3
(W abcgae
∂xe
∂yβ∂2yβ
∂xb∂xc+W abcgbe
∂xe
∂yβ∂2yβ
∂xc∂xa
+W abcgdc∂xd
∂yα∂2yα
∂xa∂xb
)= W
... ∇−W abcgcd∂xd
∂yα∂2yα
∂xa∂xb+
1
3
(W cabgcd
∂xd
∂yβ∂2yβ
∂xa∂xb+W bcagcd
∂xd
∂yβ∂2yβ
∂xa∂xb
+W abcgcd∂xd
∂yβ∂2yβ
∂xa∂xb
)= W
... ∇−W abcgcd∂xd
∂yβ∂2yβ
∂xa∂xb+
1
3W abcgcd
(∂xd
∂yβ∂2yβ
∂xa∂xb+∂xd
∂yβ∂2yβ
∂xa∂xb
+∂xd
∂yβ∂2yβ
∂xa∂xb
)= W
... ∇−W abcgcd∂xd
∂yα∂2yα
∂xa∂xb+W abcgcd
∂xd
∂yβ∂2yβ
∂xa∂xb
= W... ∇
So what would be a coordinate free interpretation of this result, that is to say (u v w)... ∇?
Unfortunately the same problem is encountered as was had when defining (V : ∇)e. Thecoordinate definition includes the fully symmetric component W abc, an object which has proven
very difficult to interpret. In order for W... ∇ to transform in the correct way, it is demonstrated
in the proof that the only symmetries required are W abc = W cab = W bca. Notice that onceagain, these are the cyclic permutations of abc. As before, it is easy to verify that the conditionW abc = W cab = W bca is satisfied if
W abc = A(uavbwc + ucvawb + ubvcwa
)+B
(ubvawc + uavcwb + ucvbwa
)(87)
Where A and B are arbitrary constants. The component W ab should also be fully symmetric.Referring back to the result of lemma 26, a suitable form for this component to take is
W ab = C
((ubvc
∂wa
∂xc+ ucwa
∂vb
∂xc+ ucvb
∂wa
∂xc
)+
(uavc
∂wb
∂xc+ ucwb
∂va
∂xc+ ucva
∂wb
∂xc
))(88)
Where C is another constant. With this component interpretation, the following coordinatefree version of definition 36 can be proposed.
Definition 38. Given u, v, w ∈ ΓTM, a metric g ∈ Γ⊗M and a general connection ∇ on
M, (u v w)... ∇ ∈ ΓΛ0M is such that
u v w... ∇ = 2
(g(u,∇vw
)+ g(v,∇uw
)+ g(w,∇uv
))(89)
−(g(u, T (v, w)
)+ g(v, T (u,w)
)+ g(w, T (u, v)
))32
5.3 Third Order Vectors & the Connection, a Scalar
This definition is justified by considering the coordinate expression of W... ∇ and is the starting
point of the next lemma. The right hand side has been written in this way so that the torsion
and torsion free parts of W... ∇ are clear. Later, this definition will be rewritten in a simpler
form.
Lemma 39. Take a metric g ∈ Γ⊗M and let a third order vector field W ∈ ΓT 3M have
components given by (87), (88) and arbitrary W a. Then
5.3 Third Order Vectors & the Connection, a Scalar
Since during the proof it is taken that A = B = C = 1, notice that the coefficient W abc originallyassumed to be cyclicly symmetric, actually turns out to be fully symmetric. That is to say,invariant under all permutations of abc. Now that this coordinate free result has been shown, itwas previously mentioned that definition 38 can be rewritten in a more elegant fashion. Simplyby rearranging the right hand side of (89) and dividing by 2 one has the following.
Definition 40. Given u, v, w ∈ ΓTM, a metric g ∈ Γ⊗M and a general connection ∇
on M, (u v w)... ∇ ∈ ΓΛ0M is such that
1
2
(u v w
... ∇)
= g(u, v w : ∇
)+ g(v, u w : ∇
)+ g(w, u v : ∇
)(91)
This result shows that there is indeed some sort of link between third order vectors combiningwith the connection, and second order vectors combining with the connection. The existenceof such a coordinate free relationship only strengthens the claim that an inductive definitionfor combining arbitrary order vectors and the connection is possible. Using coordinates aloneit would be almost impossible to spot this relationship.
In this section, suggestions have been made for coordinate and coordinate free definitions whichdemonstrate how first, second and third order vectors can be combined with the connection
to form both scalar, U... ∇ and vector quantities, (U : ∇)e. Working with these low orders, a
relationship between subsequent orders has been found by expressing U... ∇ in terms of (U : ∇)e
for various U ∈ ΓT 2M. Next, the most important results of the research are discussed andtheir possible applications to physics considered.
34
6 Analysis & Discussion
This section will act as an overall review of the main results presented in the report so far, whichare believed to be original. There will also be a more in depth discussion about the possiblephysical applications of the work.
6.1 Discussion of Results
Beginning first of all with the vectorial quantity (U : ∇)c for U ∈ ΓT 2M.
Result 1. (Lemma). Given a second order vector field U ∈ ΓT 2M such that U = Ua∂a +Uab
2 ∂2ab and a general connection ∇ on M,
(U : ∇)c =Uab
2Γcab + U c (92)
is a vector quantity.
Result 2. (Definition). Given v, w ∈ ΓTM and a general connection ∇ on manifold M,(v w) : ∇ ∈ ΓTM is such that
(v w) : ∇ = ∇vw −1
2T (v, w) (93)
Going from the coordinate, to the coordinate free definition in the second order case was straight-forward after noticing the link between the symmetry of the coefficient Uab and the torsion.The fact that Uab must equal U ba meant that Uab could not just be proportional to vawb, buthad to be proportional to vawb + vbwa. In the coordinate expression (result 1), this term ismultiplied by a Christoffel symbol Γcab. For a general connection of course, Γcab 6= Γcba hencethe definition of (v w) : ∇ is forced to contain a torsion term. Notice that for a torsion freeconnection, (v w) : ∇ reduces to the covariant derivative of w in the direction of v.
Now (V : ∇)e for V ∈ ΓT 3M is considered.
Result 3. (Lemma). Given a third order vector field V ∈ ΓT 3M such that V = V a∂a +V ab
2 ∂2ab + V abc
6 ∂3abc and a general connection ∇ on M,
(V : ∇)e =V abc
6
(ΓdabΓ
ecd + ∂cΓ
eab
)+V ab
2Γeab + V e (94)
is a vector quantity.
Result 4. (Definition). Given u, v, w ∈ ΓTM and a torsion free connection ∇0 on M,(u v w) : ∇0 ∈ ΓTM is such that
(u v w) : ∇0 = ∇0u∇0
vw −1
3R(u, v)w − 1
3R(u,w)v (95)
A different set of methods were required to extract a coordinate free definition from (V : ∇)e,due to the complexity of the symmetric expansion of V abc. From f-linearity requirements alone,
35
6.1 Discussion of Results
it was shown that (u v w) : ∇ must consist of terms f-linear in all arguments, along withthree terms of the form T (∇−−,−). Using the first Bianchi identity and equations (68) and(69), it was found that two of the f-linear terms must be curvature tensors. This was enough tofully define (u v w) : ∇ in the case of a torsion free connection, result 4. This expression tiestogether the concepts of curvature and third order vectors, a relationship which is believed hasnot been recognised before. Due to lack of time and methods, the exact form of (u v w) : ∇for a general connection was not found. All known identities that (u v w) : ∇ should satisfywere exhausted calculating the first six terms. Looking back to definition 29, this left 6 unknowncoefficients after symmetry considerations.
As well as combining higher order vectors with the connection to form vectorial quantities, itwas also found that it is possible to construct scalars. For this kind of object, the notation
W... ∇ was introduced. The case where W ∈ ΓT 2M was found trivially.
Result 5. (Lemma). Given a second order vector field W ∈ ΓT 2M such that W = W a∂a+Wab
2 ∂2ab, a metric g ∈ Γ⊗M and a general connection ∇ on M,
W... ∇ =
W ab
2gab (96)
is a scalar quantity.
Result 6. (Definition). Given v, w ∈ ΓTM, a metric g ∈ Γ⊗M and a general connection
∇ on M, (v w)... ∇ ∈ ΓΛ0M is such that
(v w)... ∇ = g(v : ∇, w : ∇) (97)
This was the first expression to be found relating vectors of subsequent orders combining withthe connection. By brute force calculation directly from the coordinate definition, third orderanalogues of results 5 and 6 were found.
Result 7. (Lemma). Given a third order vector field W ∈ ΓT 3M such that W = W a∂a +Wab
2 ∂2ab + Wabc
6 ∂3abc and a general connection ∇ on M,
W... ∇ = W abcΓcab +W abgab (98)
is a scalar quantity.
Result 8. (Definition). Given u, v, w ∈ ΓTM, a metric g ∈ Γ⊗M and a general connec-
tion ∇ on M, (u v w)... ∇ ∈ ΓΛ0M is such that
1
2
(u v w
... ∇)
= g(u, v w : ∇
)+ g(v, u w : ∇
)+ g(w, u v : ∇
)(99)
Results 6 and 8 are perhaps the most important to come out of the research. Both not only show
that it is possible to move between W : ∇ and W... ∇, but more importantly relate first/second
36
6.2 Physical Applications
order vectors combining with the connection and second/third order vectors combining withthe connection respectively. As has already been highlighted, this points to a possible inductivedefinition which combines arbitrary order vectors and the connection.
All eight of these results have arisen from a natural relationship between the connection andthe higher order vector components. It has been shown that both the Christoffel symbolsand higher order vector components are in general non-tensorial. Take the specific example ofU ∈ ΓT 2M with first component, Ua. Under a change of coordinate frame, both transformwith a piece which is tensorial and an additional non-linear piece, dependant on second orderderivatives of each coordinate function. Combining the two together in the right way has theeffect of cancelling out the additional, non-tensorial term. It was explained in section 3.3 thatthe fundamental reason for this cancellation is their dual jet space relationship.
6.2 Physical Applications
The study of higher order vectors is fairly abstract, yet it has been shown that combining themwith the connection leads to relationships between them and useful, measurable geometricquantities. Covariant derivatives, curvature and torsion lend themselves well to the study ofgravity, where the nature of the space in question has direct consequence in the theory. Generalrelativity for example has the geodesic deviation equation. This equation states that the onlyway gravity can be ‘measured’ is to look at the curvature of the manifold in which a test particlemoves[12]. It is natural then to expect, that it may be possible to express some equations fromEinstein’s theory, in terms of these new coordinate free objects. In lemma 41, the conditionwhich a vector must satisfy in order for it to be Killing is rewritten. A vector u ∈ ΓTM isKilling if Lug = 0, that is to say that the Lie derivative of the metric in the direction of u iszero[12]. Every Killing vector corresponds to a conserved quantity in the spacetime describedby g, energy or momentum for example[12]. It is straightforward to show assuming metriccompatibility and using the Leibniz rule that
Lug = 0 =⇒ u〈g(v, w)〉 = g([u, v], w
)+ g(v, [u,w]
)(100)
For all v and w.
Lemma 41. Consider first order vectors u, v, w ∈ ΓTM, a metric g ∈ Γ⊗M and a metric
compatible connection ∇ on M. u is a Killing vector if
u〈g(v, w)〉 =1
2
([u, v] w
) ... ∇+1
2
([u,w] v
) ... ∇ (101)
Proof. Beginning with equation (100). u is a Killing vector if for all v and w it satisfies
u〈g(v, w)〉 = g([u, v], w
)+ g(v, [u,w]
)(102)
Now consider the following.
1
2
(u v w
... ∇)− 1
2
(v u w
... ∇)
= g(u, v w : ∇
)+ g(v, u w : ∇
)+ g(w, u v : ∇
)− g(v, u w : ∇
)− g(u, v w : ∇
)− g(w, v u : ∇
)= g(w, u v : ∇
)− g(w, v u : ∇
)= g(w, [u, v] : ∇
)= g(w, [u, v]
)37
6.2 Physical Applications
Then immediately by relabelling.
1
2
(u w v
... ∇)− 1
2
(w u v
... ∇)
= g(v, [u,w]
)Substituting these two expressions directly into equation (102) gives
u〈g(v, w)〉 =1
2
(u v w
... ∇)− 1
2
(v u w
... ∇)
+1
2
(u w v
... ∇)− 1
2
(w u v
... ∇)
=1
2
(u v w − v u w
) ... ∇+1
2
(u w v − w u v
) ... ∇
=1
2
([u, v] w
) ... ∇+1
2
([u,w] v
) ... ∇
This is a nice result which involves both of the new second and third order scalar objects.
A possible application of the vectorial objects (U : ∇)c in a similar area of physics, are to newcosmological models. The method for doing such modelling usually begins with the constructionof a Lagrangian, which is then integrated to obtain the action. The equations which define thephysical laws of the universe in question, are obtained by finding the stationary points of theaction. In theory, the Lagrangian contains all of the necessary information for a completedescription of the physical system. For a given universe, it is sensible to require that theLagrangian be invariant under Lorentz group transformations. This assures that any equationsof motion respect special relativity. The requirement is satisfied by the following Lagrangianwhich yields Maxwell’s equations in a vacuum[16].
LMaxwell = −1
2dA ∧ ?dA+A ∧ ?J (103)
Where A is the electromagnetic potential 1-form and J is the 4-current 1-form. The advantageof using coordinate free language to write down this Lagrangian is that Lorentz invariance isautomatically built in. With this in mind, the following cosmological Lagrangian featuringU ∈ ΓT 2M such that U = v w for v, w ∈ ΓTM, can be suggested.
The first term is dynamical and the second corresponds to the field mass, each have a couplingof κ1 and κ2 respectively. This is in complete analogy with the Lagrangian for a massive scalarfield given in (118). In accordance with equation (103), wedging each of the two forms must
give an overall 4-form. This can be achieved by setting ˜(U : ∇) to be a 1-form on M. The
manifoldM is 4-dimensional, which means that ? ˜(U : ∇) is in fact a 3-form onM. The degreestherefore add correctly when the two forms are wedged together. It is straightforward to check
that having ˜(U : ∇) as a 1-form ensures that the dynamical term is also an overall 4-form. Amore detailed discussion of exterior calculus can be found in appendix section A.
By writing down this Lagrangian, second order vectors are being viewed as possible new sourcesof matter. Looking back to the coordinate free result, result 2, this could be seen as a fairlyreasonable suggestion. The expression is written in terms of curvature and torsion, both ofwhich are quantities which play a central role in general relativity and Einstein-Cartan theoryrespectively. The Einstein-Cartan model of gravity is similar to general relativity but with non-zero torsion. It is believed that torsion may feature in a theory of gravity in order to capture
38
the effects of matter with spin[4]. It was suggested in a 2010 paper by Poplawski that torsioncan not only remove the big bang singularity, but also explain cosmic inflation by relaxing thetorsion free condition in the Friedman equations[13]. It has been shown in this report that evenwhen combining just second order vectors with the connection, a linear torsion is introducednaturally. It is possible that the universe described by equation (104) has no big bang singular-ity, but preserves all of the observed properties of general relativity. It is not just gravitationalmodels which make use of torsion. Another example is in the modelling of crystal defects in thecontinuum, more specifically dislocations and disclinations[3]. The properties of such a spacelend themselves well to a description through torsion[3].
By the same justification as was used to write down LT 2M, a second Lagrangian involving athird order vector V ∈ ΓT 3M such that V = u v w for u, v, w ∈ ΓTM, can be suggested.
Due to the definition of a third order vector combining with a general connection being incom-plete, this Lagrangian would correspond to a torsion free theory. The Fundamental Theoremof Riemannian geometry states however that given a metric, there is a unique connection on itwhich is metric compatible and torsion free[11]. There is no reason to believe therefore that aLagrangian of this form, would not predict anything new or of consequence.
7 Conclusion
It has been shown that it is possible to combine higher order vectors and the connection insuch a way, that the resulting objects are expressible in terms of useful geometric quantities.These results were formed on the assumption that such objects must exist, given the naturalrelationship between the connection and higher order vectors, which becomes evident when therespective transformation laws are compared. The coordinate definitions of these new objectswere obtained by writing down expressions involving products of Christoffel symbols and higherorder vector components, while ensuring the correct number of free indices to indicate vector andscalar quantities. Once these expression were explicitly proven to be tensorial, the coordinatefree definitions were obtained by considering the special cases of second and third order vectors,vw ∈ ΓT 2M and uvw ∈ ΓT 3M. In all but one case, complete definitions were obtained forgeneral connections by simply respecting the symmetries of the higher order components. Thisapproach was unsuccessful for the third order vectorial object, where new methods to decipherthe exact form had to be found. With 2 equations involving the Lie bracket, f-linearity andsymmetry considerations, and the Bianchi identities, the problem was reduced from 13 to 6unknowns. From this a complete torsion free definition could be extracted. As explained, itmay be that the elusiveness of a definition fully inclusive of torsion, despite the result of lemma28, implies some deeper problem which is currently being overlooked. The final outcome is aset of coordinate free definitions showing how second and third order vectors can be combinedwith the connection to obtain 2 vector quantities and 2 scalar quantities.
The physical implications of these definitions were discussed at length in section 6.2, highlightingpossible applications to gravitational and cosmological theories. It is the natural occurrence oftorsion in the definitions, a frequently overlooked quantity, which could lead to new predictionsin these fields. In order to draw something physical from a Lagrangian however, it must first beintegrated and varied. To extract anything meaningful from equations (104) and (105) would
39
therefore require a method of computing the functional derivative of U : ∇. Such mathematicshas not yet been developed. Finally, notice that a significant portion of the work features aconnection and no metric. Questions can therefore be asked about the possibility of building amanifold abstractly, with a connection and no metric.
If this work were to be taken further, the ultimate goal would be an inductive definition whichdescribes how an nth order vector can be combined with the connection in a coordinate freeway. Results 6 and 8 which relate higher order vectors of subsequent order combining with theconnection, only support the existence of such a definition. Looking at the first, second andthird order vectorial combinations with the connection, there is a clear pattern emerging. Annth order definition is likely to be of the following form.
(u1 · · · un) : ∇ = ∇u1 · · · ∇un−1un + Sn (106)
Where u1, · · · , un ∈ ΓTM and Sn : [ΓTM]n → ΓTM. That is to say Sn is a function whichtakes n first order vectors and gives a first order vector. It is also reasonable to assume thatSn will be made up completely of curvature and torsion tensors, along with their higher ordercovariant derivatives and products. Sn may contain for example
Indeed, any combination of torsions, curvatures and del operators which can accommodate nfirst order vectors are a possibility. It is clear from the rate of increase in complexity of Sn, thatworking with higher orders would require a computer program. For example, the next logicalstep would be to investigate (u1 u2 u3 u4) : ∇, S4 could contain any of the following.
∇−R(−,−)−(∇−∇−T )(−,−)
T (∇−−,∇−−)
R(T (−,−),−)−
R(∇−−,−)−(∇−T )(∇−−,−)
T (T (−,−), T (−,−))
R(−,−)T (−,−)
R(−,−)∇−−(∇−T )(T (−,−),−)
T (T (T (−,−),−),−)
T (R(−,−)−,−)
Before taking into account any symmetries in the arguments of the vectors, there are 4! waysin which 4 first order vectors can be placed into each of the slots. That makes for a grandtotal of 288 unknowns. Furthermore, even with the aid of a computer program, solving such anexpression for the exact definition would require 288 conditions. Recall that for the third ordercase there were still 6 unknowns, with no known method to reduce this number any further. Apossible solution which, due to lack of time was never developed far enough to contribute, isobserving that a general connection ∇ can be written in the following form.
∇ = ∇′ + αQ , α ∈ R, Q ∈ Γ⊗M (108)
This is a one parameter family of diffeomorphisms. If for example it is chosen that the connection∇′ be completely torsion free, it is straightforward to show that taking α = 1/2 and Q = Tsatisfies this choice.
∇ = ∇′ + 1
2T (109)
The connection ∇ is still any general connection. Equation (108) could be used to substitutefor ∇ in the incomplete coordinate free definition of (u v w) : ∇. By then carefully choosingdifferent values of α, it may be that the remaining 6 unknowns could be extracted.
This masters project has been successful in defining two ways in which first, second and thirdorder vectors can be combined with the connection to form tensorial quantities. It is fair tosay that if equipped with the correct techniques, there are many ways in which the researchcould be taken forward and continued. However, what more can efficiently be achieved withoutthe development of appropriate computational methods or a completely different approach, islimited.
40
8 Glossary of Notation
This section acts as a quick reference for allnotation used in this report, that is to say norigorous definitions are given.
Multi-Index Notation
Given I = [i1, · · · , iq], then unless otherwisestated.
|I| = i1 + · · ·+ iq , ||I|| = lenI (110)
I! = i1! · · · iq! , xI = xi11 · · ·xiqq (111)
The multi-index partial derivative.
DI =∂
∂xi11· · · ∂
∂xiqq
(112)
Basic Latin and Greek ScriptThis excludes all types of vector and other ten-sor spaces.
Notation Explanation
M A manifold.
m Dimension of M.
p A point on M.
n Order of a vector.
k Degree of a form.
a, b, c, · · ·α, β, γ, · · · Free/dummy indices.
q, r Natural numbers.
I, J Multi-indices.
i1, · · · , iq Indices contained in I.
κq Coupling constants.
Vectors and Vector/Tensor Spaces
1st Order Vector nth Order Vector Scalar k-Form
At a Point, p u,v,w ∈ TpM U,V,W ∈ TnpM f |p, g|p, h|p n/a
At all Points u,v,w ∈ TM U,V,W ∈ TnM n/a n/a
Field u, v, w ∈ ΓTM U, V,W ∈ ΓT 2M f, g, h, λ ∈ ΓΛ0M µ, ν, η ∈ ΓΛkM‘At all points’ refers to the following disjointunion, the set of all vectors at all points.
TM =⋃p∈M
TpM (113)
The set Γ⊗M denotes the space of all tensor
fields on M.
Other Spaces, Objects & Operations
Coordinate Coordinate Free Explanation
ua∂af u〈f〉 An arbitrary vector acting upon a scalar.
Not required. µ : v A arbitrary 1-form acting upon a vector.
Γcab, Γcab Not required. 1st and 2nd kind Christoffel symbols.
Not required. ∇/∇0 A general/torsion free connection.
T cab / uavbT cab T / T (u, v) Torsion tensor.
Rdabc / ubvcwaRdabc R / R(u, v)w Curvature tensor.
gab g(u, v) The metric tensor.
Not required. Jrf/(Jrf)∗ rth order jet/dual of jet of scalar f.
Not required. rϕ Element of rth order jet of scalar f.
ua∂a(vb∂b) u v Vector u operating/acting on vector v.
(U : ∇)e U : ∇Higher order vector combining with the
connection to form a vector.
W... ∇ W
... ∇Higher order vector combining with the
connection to form a scalar.
Not required. Lu The Lie derivative in direction of u.
Not required. g The metric dual.
Not required. ? The Hodge star operator.
Not required. d The exterior derivative.
Not required. ∧ The wedge product.
Not required. A/J Electromagnetic potential/4-current 1-forms.
41
Appendices
A Exterior Calculus
In section 6.2, possible applications of the work are discussed. In one example, two Lagrangiansare written down and analysed using aspects of differential geometry which are not requiredanywhere else during the main research phase. These are the exterior derivative ‘d’, the metricdual ‘g,’ the wedge product ‘∧’ and the Hodge star operator ‘?’. For the purposes of this report,that is to say in order to understand the Lagrangian application, only a basic knowledge ofthese ideas is necessary. If a formal definition is not essential, it has not been included.
The wedge product, ∧. This operation allows higher degree differential forms to be con-structed from 1-forms (as introduced in section 2.2). To build a 4-form for example, the typerequired for Lagrangians (104) and (105), two 1-forms are first wedged together to give a 2-form. Next, two of these 2-forms can be wedged to give an overall 4-form. In general the wedgeproduct can be seen as the following function[11].
Definition 42. Given µ ∈ ΓΛkM and ν ∈ ΓΛqM, the wedge product is a function ∧ : ΓΛkM×ΓΛqM→ ΓΛk+qM, with (µ, ν) 7→ µ∧ν such that it is associative and has graded commutativity.
µ ∧ ν = (−1)kqν ∧ µ (114)
It is also plus and f-linear in all of its arguments.
Higher degree differential forms are a far more well established tool in physics and mathematicsthan higher order vectors. It has already been mentioned that it is possible to reduce Maxwell’sequations down to just two expressions. To do this the electric and magnetic fields are combinedinto a single ‘electromagnetic’ 2-form[12].
Equipped with the wedge product and keeping in mind the 1-form basis introduced in section2.2, the coordinate expression for a general k-form can be written down[12].
Lemma 43. Given an m-dimensional manifold M with coordinates (x1, · · · , xm) and multi-indexed scalar fields fI ∈ ΓΛ0M, a general k-form on M, µ ∈ ΓΛkM can be expressed
µ =1
k!fIdx
I , dxI = dxi1 ∧ · · · ∧ dxim (115)
The factor of 1k! is to account for the symmetry in the wedge product due to its graded com-
mutativity. This coordinate expression will be used when talking about the Hodge star.
Hodge star, ?. Although this operator is used in Lagrangians (104) and (105) which arecoordinate free, for the purposes of the project it is best to define the Hodge star using indexnotation. A succinct coordinate free definition by induction does exist, however it requires theconcept of internal contraction which does not feature in the report. The action of the Hodgestar on a general k-form is calculated in the following way[2].
Lemma 44. Given an m-dimensional manifold M with metric g ∈ Γ⊗M, multi-indexed
scalar fields fI ∈ ΓΛ0M and a general k-form µ ∈ ΓΛkM such that µ = 1k!fIdx
Where εj1···jkjk+1···jm is the Levi-Civita symbol.
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The Hodge star is therefore a function ? : ΓΛkM→ ΓΛm−kM, taking a k-form and producingan (m−k)-form. Most notably its definition is dependant on the choice of metric and dimensionof the manifold. Taking the wedge product of a form with its own Hodge dual results in a formof maximum degree in that particular space. This is the property which has been used in thediscussion section to construct the two Lagrangians.
Exterior derivative, d. The exterior derivative is an operator which allows the degree of aform to be increased by 1. As with most coordinate free objects, d can be defined as a functionwhich obeys a set of rules. Here it is sufficient to understand how the exterior derivative of adifferential form can be calculated using a coordinate basis[6][12].
Lemma 45. Given an m-dimensional manifoldM with coordinates (x1, · · · , xm), multi-indexedscalar fields fI ∈ ΓΛ0M and a general k-form µ ∈ ΓΛkM such that µ = 1
k!fIdxI ,
dµ =1
k!
∂fI∂xj
dxj ∧ dxI (117)
The original k-form has become a (k + 1)-form. It is straightforward to show that d2 = 0 dueto the equality of mixed partial derivatives[6]. If classical vectors in R3 are viewed as 1-forms,this property can be used to demonstrate the well known result ∇ × ∇g = 0, where g is anywell behaved scalar field[6]. The exterior derivative is used in Lagrangians (104) and (105) toconstruct a kinetic term. This is in complete analogy with how kinetic and mass terms are builtinto Lagrangians in quantum field theory. The Lagrangian for a free scalar field ψ with mass λis given by[17]
L =1
2(∂aψ)(∂aψ)− 1
2λ2ψ2 (118)
Using the differential geometric approach, partial derivative ∂a has been replaced by exteriorderivative d.
Metric dual, g. The metric dual provides a way to transition between differential 1-forms andfirst order vectors and vice-versa. The dual of a 1-form field µ ∈ ΓΛ1M for example, is denotedµ and is a vector field. Formally, it is best understood through its coordinate free definition[7].
Definition 46. Given µ ∈ ΓΛ1M and ν ∈ ΓΛ1M and metric g ∈ Γ⊗M, the metric dual is
a function g : ΓΛ1M× ΓΛ1M→ ΓΛ0M, with (µ, ν) 7→ g(µ, ν) such that
g(µ, ν) = g(µ, ν) (119)
Such an operation therefore makes it possible to apply the work done with vectors in the researchphase of the project, to a covariant Lagrangian formalism.
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REFERENCES
References
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