A Pictorial Introduction to Differential Geometry, Leading to Maxwell's

Home , Jan Arnoldus Schouten

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Scalar . . . . submanifold . of 2.4 . . Boundaries . . . . 2.3 . Submanifolds. . . 2.2 Manifolds. 2.1 onayo umnfl ...... submanifold . a of . boundary ...... untwisted . . . external and ternal ttt faclrtrScience. accelorator of stitute ie Geometry-ArXiv.tex File: jonathan-gratus 11 11 11 10 10 8 5 3 9 9 8 6 6 6 6 5 4.1 Multiplication of vectors and 4 A selection of submanifolds and forms by scalar fields...... 14 theirboundaries ...... 7 4.2 Addition of forms...... 14 5 Pathological examples which are 4.3 The wedge product also known as notsubmanifolds...... 7 the exterior product...... 14 6 A representation of a scalar field. . 8 4.4 Integration...... 15 7 A scalar field as a function of x 4.5 Conservation Laws ...... 16 and y...... 8 4.6 DeRham Cohomology...... 16 8 A scalar field depicted as contours. 8 4.7 Fourdimensions...... 17 9 A representation of a vector field. . 8 10 Internal orientations...... 8 5 Non-closed forms. 18 11 Equivalent orientations...... 9 5.1 The exterior differential operator. . 19 12 Creating a helix...... 10 5.2 Integration and Stokes’s theorem 13 Inheritance of internal orientation for non-closed forms...... 19 onaboundary...... 11 6 Other operations with scalars, vec- 14 Inheritance of external orientation tors and forms 20 onaboundary...... 11 15 Representation of a top-form. . . . 12 6.1 Combining a 1-form and a vector field to give a scalar field...... 20 16 An untwisted 2-form on a 3-dim 6.2 Vectors acting on a scalar field. . . 20 manifold...... 12 6.3 Internal contractions...... 20 17 An untwisted 2-form on a 3-dim manifold...... 12 7 The Metric 21 18 An untwisted 1-form on a 3-dim 7.1 Representing the metric: Unit el- manifold...... 12 lipsoids and hyperboloids . . . . . 21 19 Addition of 1-forms in 2-dim . . . 14 7.2 Length of a vector ...... 22 20 Wedge product of untwisted 1- 7.3 Orthogonal subspace ...... 22 formand2-form...... 14 7.4 Angles and γ-factors ...... 23 21 Wedge product of two untwisted 7.5 Magnitude of a form...... 23 2-form...... 14 7.6 Metric dual...... 23 22 Wedge product of three untwisted 7.7 Hodgedual...... 24 2-form...... 15 7.8 The measure from a metric . . . . 24 23 Wedge product of twisted forms . . 15 24 Wedge product of a twisted and 8 Smooth maps 24 anuntwistedform...... 15 8.1 Pullbacks of p-forms...... 25 25 Integration of a 2-form...... 15 8.2 Pullbacks of metrics...... 27 26 Integration of a 1-form...... 16 8.3 Pushforwards of vectors...... 27 27 A conservation law...... 16 28 Conservation law for all space. . . 17 9 Electromagnetism and Electrody- 29 DeRham Cohomology on an an- namics 27 nulus...... 17 9.1 Electrostatics and Magnetostatics. 29 30 DeRham Cohomology on a Torus. 17 9.2 Maxwell’s equations as four space- 31 Failed non-zero integrals...... 18 timepictures ...... 30 32 2-form in 4-dimensions: plane pic- 9.3 Lorentz force equation ...... 31 ture...... 18 9.4 Constant space slice of Maxwell’s 33 2-form in 4-dimensions: moving equations...... 32 linepicture...... 18 10 Conclusion 32 34 Example of a non-closed 1-form (black) in 2-dim...... 19 35 A non-closed, non-integrable 1-form. 19 List of Figures 36 More non-closed forms...... 19 37 Demonstration of Stokes’s theorem. 20 1 Zoominginonasphere...... 5 38 Combining vectors and 1-forms . . 20 2 Zoominginonacorner...... 5 39 Demonstration of internal con- 3 0, 1, 2 and 3-dimensional manifolds. 6 traction...... 20

2 40 Representations of the metric in 1 Introduction 3-dimspace...... 21 41 The spacetime metric...... 21 Differential geometry is a incredibly useful 42 The metric ellipsoid and length. . . 22 tool in both physics and mathematics. Due 43 The metric hyperboloid and length. 22 to the heavy technical mathematics need to 44 The metric ellipsoid and orthogo- define all the objects precisely it is not usually nality...... 23 introduced until the final years in undergrad- 45 The metric hyperboloid and or- uate study. Physics undergraduates usually thogonality...... 23 only see it in the context of general relativ- 46 The metric ellipsoid and angles. . . 23 ity, where spacetime is introduced. This is 47 The metric hyperboloid and γ- unfortunate as differential geometry can be factor...... 23 applied to so many objects in physics. These 48 The metric, magnitude of a 1-form 24 include: 49 Themetricdual...... 24 • The 3-dimensional space of Newtonian 50 TheHodgedual...... 25 physics. 51 The measure from the metric. . . . 25 52 Projection 3-dim to 2-dim...... 25 • The 4-dimensional spacetime of special 53 Projection 3-dim to 1-dim...... 26 and general relativity. 54 Pullback, 2-dim to 3-dim, 2-form . 26 • 2-dimensional shapes, like spheres and 55 Pullback, 2-dim to 3-dim, 1-form . 26 torii. 56 Pullback, 1-dim to 3-dim, 1-form . 26 • 6-dimensional phase space in Newto- 57 Pullback, tangential ...... 26 nian mechanics. 58 Pullback, 3-dim to 2-dim, 2-form . 27 • 7-dimensional phase-time space in gen- 59 Pullback, 3-dim to 2-dim, 1-form . 27 eral relativity. 60 Pullback, 3-dim to 1-dim, 1-form . 27 • The state space in thermodynamics. 61 The pullback of a metric onto an • The configuration space of physical sys- embedding...... 27 tem. 62 The Push forward of the vector field. 28 • The solution space in differential equa- 63 The magnetic 2-form in space. . . 29 tions. 64 Gauss’s Law in electrostatics. . . . 29 65 The B and H fields from a wire. . 30 There are many texts on differential geome- 66 The B and H fields from a magnet. 30 try, many of which use diagrams to illustrate 67 Conservation of charge ...... 30 the concepts they are trying to portray, for 68 Conservation of magnetic flux . . . 31 example [1–8]. They probably date back to 69 The excitation fields ...... 31 Schouten [8]. The novel approach adopted 70 The constitutive relations in the here is to present as much differential geom- vacuum ...... 31 etry as possible solely by using pictures. As 71 The Lorentz force law...... 32 such there are almost no equations in this ar- 72 Faraday’s law of induction on the ticle. Therefore this document may be used plane z =0...... 32 by first year undergraduates, or even keen 73 Maxwell Ampères law on the school students to gain some intuition of dif- plane z =0...... 33 ferential geometry. Students formally studying differential geometry may use this text in conjunction with a lecture course or standard List of Tables text book. 1 External orientations...... 9 In geometry there is always a tension be- 2 Types of orientation for all objects. 10 tween drawing pictures and manipulating al- 3 Concatenation of orientations. . . . 10 gebra. Whereas the former can give you in- 4 Examples of twisting and untwist- tuition and some simple results in low dimen- ing...... 11 sions, only by expressing geometry in terms 5 The fields in electromagnetism . . 28 of mathematical symbols and manipulating 6 The relativistic field in electro- them can one derive deeper results. In addi- magnetism...... 28 tion, symbols do not care whether one is in 1,

3 3, 4 dimensions or even 101 dimensions (the of 4-dimensional manifolds with a spacetime one with the black and white spots), whereas metric. It is the metric which defines curva- pictures can only clearly represent objects in ture and therefore gravity. General relativity 1, 2, or 3 dimensions. This is unfortunate therefore is the study of how the metric af- as we live in a 4-dimensional universe (space- fects particles and fields on the manifold and time) and one of the goals of this document how these fields and particles effect the met- is to represent electrodynamics and Maxwell’s ric. There are many popular science books equations. and programs which introduce general rela- As well as presenting differential geometry tivity by showing how light and other object without equations, this document is novel be- move in a curved space, usually by showing cause it concentrates on two aspects of dif- balls moving on a curved surface. Therefore ferential geometry which students often find one of the things we do not cover in this doc- difficult: Exterior differential forms and ori- ument are the spacetime diagrams in special entations. This is done even before introduc- relativity and the diagrams associated with ing the metric. curvature, general relativity and gravity. Exterior differential forms, which herein we Since the metric is so important one may will simply call forms, are important both ask why one should study manifolds without in physics and mathematics. In electromag- metrics. There are a number of reasons for netism they are the natural way to define the doing this. One is that many manifolds do electromagnetic fields. In mathematics they not possess a metric. To be more precise, are the principal objects that can be inte- since on every manifold one may construct grated. In this article we represent forms pic- a metric, one should say it has no single pre- torially using curves and surfaces. ferred metric, i.e. one with physical signif- Likewise orientations are important for icance. For example phase space does not both integration and electromagnetism. possess a metric, neither do the natural mani- There are two types of orientation, in- folds for studying differential equations. Even ternal and external, also called twisted in general relativity, one may consider varying and untwisted, and these are most clearly the metric in order to derive Einstein’s equa- demonstrated pictorially. Some physicists tions and the stress-energy-momentum ten- will be familiar with the statement that the sor. Transformation optics, by contrast, may magnetic field is a twisted vector field, also be considered as the study of spacetimes with known as a pseudo-vectors or an axial vector. two metrics, the real and the optical metric. Here the twistedness is extended to forms As a result knowing about objects and oper- and submanifolds. ations which do not require a metric is very The metric to late is one of the most im- useful. The other reason for introducing dif- portant objects in differential geometry. In ferential forms before the metric is that, if fact without it some authors say you are one has a metric, then one can pass between not in fact studying differential geometry 1-forms and vectors. Thus they both look but instead another subject called differential very similar and it is difficult for the student topology. Without a metric you can think of to get an intuition about differential forms as a manifold as made of a rubber, which is in- distinct from vectors. finity deformable, whereas with a metric the One of the goals of this article is to in- manifold is made of concrete. The metric troduce the equations of electromagnetism gives you the length of vectors and the an- and electrodynamics pictorially. Electromag- gle between two vectors. It can also tell you netism lends itself to this approach as the ob- the distance between two points. In higher jects of electromagnetism are exterior differ- dimensions the metric tells you the area, vol- ential forms, and most of electromagnetism ume or n-volume of your manifold and sub- does not need a metric. Indeed the four manifolds within it. It gives you a measure macroscopic Maxwell’s equations can be writ- so that you can integrate scalar fields. ten (and therefore drawn) without a metric. General relativity is principally the study It is only when we wish to give the addi-

4 tional equations which prescribe vacuum or (a) (b) the medium that we need the metric. This has lead to school of thought [9] which suggest that the metric may be a consequence of electromagnetism. The challenge when depicting Maxwell’s (c) equations pictorially is that they are equations on four dimensional spacetime and we lack 4-dimensional paper. Unfortunately, Figure 1: Zooming in on a sphere. As we go were we blessed with such 4-dimensional pa- from (a) to (b) to (c), the patch of the sphere per, we would have to live in a universe becomes ﬂatter. with 5 spatial dimensions and therefore 6- dimensional spacetime. Whatever ﬁeld the 5-dimensional equivalent of me would be at- tempting to represent in pictures he/she/it would presumably be bemoaning the lack of 6-dimensional paper. (a) (b)

This article is organised as follows. We Figure 2: Zooming in on a corner. As we go start in section 2 with basic differential ge- from (a) to (b), the patch does not get any ometry, discussing manifolds, submanifolds, flatter. This is not a manifold. scalar and vector fields. In section 3 we introduce both untwisted and twisted orienta- time and two spatial dimensions. We then tion, dealing first with orientations on sub- conclude in section 10 with a discussion of manifolds, since this is the easiest to trans- further pictures one may contemplate. fer to other objects like forms and vectors. All figures are available in colour online. We then introduce closed exterior differential forms, section 4, showing how one can add them and take the wedge product. We then talk about integration and hence conservation 2 Manifolds, scalar and laws. This leads to a discussion about man- vector fields ifolds with “holes” in them. Finally in this section we discuss the challenge of represent- 2.1 Manifolds. ing forms in four dimensions. In section 5 we discuss non-closed forms and how to vi- Manifolds are the fundamental object in dif- sualise and integrate them. In section 6 we ferential geometry on which all other object summarise a number of additional operation exist. These are particularly difficult to define one can do with vectors and forms. The met- rigorously and it usually take the best part of ric is introduced in section 7 where we show an undergraduate mathematics lecture course how we can use it to measure the length of a to define. This means that the useful tools vector, angles between vectors as well the the discussed here are relegated to the last few metric and Hodge dual. Section 8 is about lectures. smooth maps which shows how forms can be Intuitively we should think of a manifold pulled from one manifold to another. Finally as a shape, like a sphere or torus, which is we reach our goal in section 9 which shows smooth. That is, it does not have any cor- how to picture Maxwell’s equations and the ners. If we take a small patch of a sphere Lorentz force equation. We start with static and enlarge it, then the result looks flat, like electromagnetic fields as these are the easiest a small patch of the plane, figure 1. By con- to visualise and then attempt to draw the full trast if we take a small patch of a corner then, four dimensional pictures. We then return to no mater how large we made it, it would still three dimensional pictures but this time one never be flat, figure 2.

5 least in some direction. In ﬁgure 4 we see that examples 4a, 4c, 4f and 4g are all compact, whereas the others are not compact. A compact submanifold may either have a bound- (a) A 0-dimensional (b) A 1-dimensional ary, as in section 2.3 below, or close in on manifold: a point. manifold: a curve. itself like a sphere or torus.

2.3 Boundaries of submanifold Unlike the embedding manifold, it is also useful to consider submanifolds with boundaries. See figure 4. Both compact and non compact submanifolds can have boundaries. The importance of the boundary is when doing integration, in particular the use of Stokes’ theorem. The boundary of a submanifold is (c) A 2-dimensional (d) A 3-dimensional an intuitive concept. For example the bound- manifold: a surface. manifold: a volume. ary of a 3-dimensional ball is its surface, the Figure 3: 0, 1, 2 and 3-dimensional mani- 2-dimensional sphere, figure 4f. We assume folds. that the boundary of an n-dimensional submanifold is an (n − 1)-dimensional submanifold. We observe that the boundary of the In figure 3 are sketched manifolds with 0, boundary vanishes. 1, 2 and 3 dimensions. 0-dimensional manifolds are simply points, 1-dimensional manifolds are curves, 2-dimensional manifolds are 2.4 Scalar fields. surfaces and 3-dimensional manifolds are vol- Having defined manifolds, the simplest ob- umes. ject that one may define on a manifold is a Topologically, there are only two types of 1 scalar field. This is simply a real number dimensional manifolds, those which are lines associated with every point. See figure 6. and those which are loops. Higher dimen- The smoothness corresponds to nearby points sional manifolds are more interesting. having close values, also that these values can be differentiated infinitely many times. A 2.2 Submanifolds. scalar field on a 2-dimensional manifold may be pictured as function as in figure 7. Alter- A submanifold is a manifold contained in- natively one can depict a scalar field in terms side a larger manifold, called the embedding of its contours, figure 8. We will see below, manifold. The submanifold can be either the section 4, that these contours actually depict same dimension or a lower dimension than the the 1-form which is the the exterior derivative embedding manifold. Examples of embedded of the scalar field. submanifolds are given in figure 4 There are all kinds of bizarre pathologies 2.5 Vector fields. which can take place when one tries to em- bed one manifold into another, such as self A vector field is a concept most familiar to intersection, figure 5a, or the a manifold ap- people. A vector field consists of an arrow, proaching itself figure 5b. However we will with a base point (bottom of the stalk) at avoid these cases. each point on the manifold, see figure 9. Submanifolds may either be compact or Although we have only drawn a finite num- non compact. A compact submanifold is ber of vectors, there is a vector at each point. bounded. By contrast a non compact sub- The smoothness of the vector field corre- manifold extends all the way to “infinity” at sponds to nearby vectors not differing too

6 (b) The lower por- (c) The disc (green) is (d) The half plane is (a) The interval tion is a non com- a compact 2-dim sub- a non compact 2-dim (green) is a compact pact 2-dim submanifold manifold of a 2-dim submanifold (green) of 1-dim submanifold (green) of a 2-dim man- manifold. The bound- a 3-dim manifold. Its of a 1-dim manifold. ifold. The boundary ary of the disk is the boundary is the non The boundary of of this submanifold is 1-dim circle (blue), a compact 1-dim curve the submanifold are the non compact 1-dim compact submanifold. (blue). the two points, this submanifold (blue). is a compact 0-dim submanifold (blue).

(e) A non compact 2- (f) A compact 2-dim (g) A compact 2-dim sub- dim submanifold (blue) submanifold, the sur- manifold, the surface of a of a 3-dim manifold. face of the sphere, of torus. It is the boundary It is the boundary of a 3-dim manifold. It of the compact 3-dim solid the non compact 3-dim is the boundary of the torus. submanifold consisting compact 3-dim ball. of it and all points above it.

Figure 4: A selection of submanifolds and their boundaries. Observe that the boundaries is itself a submanifold, but it does not have a boundary. This implies ate the boundary must extend to inﬁnity or close in on itself.

much. To multiply a vector field by a scalar field, simply scale the length of each vector by the value of the scalar field at the base point of that vector. Of course if the scalar field is negative we reverse the direction of the ar- (a) A self intersecting (b) A curve approach- row. We are all familiar with the addition of curve. ing itself. vectors from school, via the use of a parallel- ogram. Figure 5: Pathological examples which are not submanifolds. The vector presented in figure 9 are untwisted vectors. We will see below in section 3.6 that there also exist twisted vectors.

7 4 4 3 3 5 2 2 2 2 2 2 2 23 1 1 1 2 1 1 − 1 -1 1 0 -1 1 2 0 2

(a) Some of the values (b) Intermediate val- of the scalar field are ues of the scalar field Figure 9: A representation of a vector field. shown. are also shown.

Figure 6: A representation of a scalar ﬁeld.

+ − ) (a) 0-dim (b) 1-dim (c) 2-dim mani- x, y (

f manifold. manifold. fold.

y x

Figure 7: A scalar ﬁeld as a function of x and y. Right handed Left handed (d) 3-dim manifold.

3 Orientation. Figure 10: Possible (internal) orientations on 0, 1, 2 and 3 dimensional manifold. We now discuss the tricky role that orientation has to play in differential geometry, es- pecially the difference between twisted and • Orientation is important for integration untwisted orientations. However the distinc- and Stokes’s theorem. tion is worth emphasising since the objects in • The orientation usually only affects the electromagnetism are necessarily twisted and overall sign. However it is important when others are necessarily untwisted. adding forms. The take home message is: • The pictures are pretty.

3.1 Orientation on Manifolds (internal) x We start by showing the possible orientations on manifolds. Not all manifolds can have an orientation. If a manifold can have an ori- y entation it is called orientable. Examples of orientable manifolds include the sphere and Figure 8: A scalar ﬁeld depicted as contours. torus. Non-orientable manifolds include the The arrows point up. M¨obius strip and the Kline bottle. Even if a

8 Embedding dimension 0 1 2 3 0 + −

1 + − Figure 11: Equivalent external orientations of 1 and 2 dimensional submanifold in 3-dim. The actual length of the arc or arrow leaving 2 is irrelevant. What matters is simply what the Submanifold dimension + − circulation around 1-dim submanifold and side of the 2-dim submanifold it points. 3 + − ternal orientation. Table 1: Possible external orientations on a The internal orientation of submanifolds 0, 1, 2 and 3-dim submanifold of a 0, 1, 2 and are exactly the same as those for manifolds, 3-dim embedding manifold. as shown in figure 10, with the type of the orientation: sign, arrow, arc or helix, depending manifold is orientable, one may choose not to on the dimension of the submanifold. put an orientation on the it. For manifolds By contrast external orientations depend of 0, 1, 2 or 3 dimensions the two possible not just on the dimension of the submanifold (internal) orientations are shown in figure 10. but also on the dimension of the embedding These are as follows: manifold. They are similar, plus or minus • An orientation for a 0-dimensional sub- sign, arrow, arc or helix, as the internal ori- manifold consists of either plus or minus, entations. However in this case it takes place figure 10a. However convention usually dic- in the embedding external to the submani- tates plus. fold. In table 1 we see examples of external • On a 1-dimensional manifold it is an orientations for 0, 1, 2, 3 dimensional sub- arrow pointing one way or another, figure manifolds embedded into 0, 1, 2, 3 dimen- 10b. sional manifolds. As we see, the type of the • On a 2-dimensional manifold it is an arc external orientation depends on the difference pointing clockwise or anticlockwise, figure in the dimension of the embedding manifold 10c. and submanifold. Thus the external orienta- • On a three dimensional manifold one tion is a plus or minus sign if the two mani- considers a helix pointing one way or an- folds have the same dimension. It is an arrow other, figure 10d. Interestingly, observe if they differ by 1 dimension, an arc if they that unlike in the 1-dim and 2-dim cases, differ by 2 dimensions and a helix if they dif- there is no need for an arrow on the helix. fer by 3 dimensions. We emphasise that when specifying an orientation, we can point the respective arrow or 3.2 Orientation on submani- arcs in a variety of directions without chang- ing the orientation. For example, in figure folds: Internal and external 11 we see how the same orientation can be Again submanifolds may be orientable or represented by several arcs or arrows. non-orientable. However with orientable manifold one now has a choice as to which 3.3 Internal and external, type of orientations to chose: internal or ex- twisted and untwisted ternal. With a manifold there is no concept of an external orientation because there is no For manifolds and submanifold it is useful to embedding manifold in which to put the ex- label internal orientations as untwisted and

9 Always z definable & Internal External Essential y z Manifold No Untwisted Undefined Submanifold No Untwisted Twisted Vectors Yes Untwisted Twisted Forms Yes Twisted Untwisted x Table 2: Objects which can have orientations (a) A curve created by x y and the correspondence between untwisted going a short distance versus twisted and internal versus external. in x (red), followed by (b) The same diagram a short distance in z as in figure 12a but now (blue) followed by go- followed which with the axes all point- becomes + by is ing a short distance in ing towards the reader. y (green) and then re- followed which peating. becomes by + is z

followed which becomes by is

followed which becomes x y by is (c) The same diagram as in figure 12b but followed which with a offset each loop to show the helix. becomes by is Figure 12: Demonstration that going a short distance in x then z then y and repeating cre- Table 3: Concatenation of orientations. ates a helix. The reader is encouraged to ex- periment with some pipe cleaners. external orientations as twisted. As we have stated, as well as manifolds and submani- of two orientations, as in rows three and four folds, vectors and forms also have orienta- of table 3. Likewise the helices, figure 10d tions. These may also be twisted or untwisted arise by concatenating an arc with a perpen- and may also be internal or external. The pic- dicular line as in rows five and six of table tures of the orientations depend on whether 3. they are internal or external, but the assign- Using pipe cleaners may help you to see ment to which are twisted or untwisted de- that that an arc followed by a perpendicular pend on the object, according to table 2. line is equivalent to a helix. Likewise a helix arises in 3 dimensions by tracing out a path, going in one direction, then a second, then 3.4 Concatenation of orienta- the third and repeating as in figure 12. tions We can combine two orientations together. 3.5 Inheritance of orientation Simply take the two orientations and join by the boundary of a sub- them together, the second onto the end of the manifold first. See table 3. You can see that the untwisted orientations If a submanifold has an internal orientation for two dimensions, figure 10c, i.e. clockwise and a boundary then the boundary inherits or anticlockwise arise from the concatenation an internal orientation, see figure 13. Like-

10 Submanifold Submanifold Embedding Embedding external internal + dimension orientation orientation orientation

- (a) (b) (c) Boundary of a 3 Boundary Boundary 3-dim submanifold. of a 1-dim of a 2-dim The helix is the Table 4: Examples of twisting and untwist- submani- submani- orientation of the ing. fold fold 3-dim ball.

Figure 13: Inheritance of an internal orien- nal orientations. They look like the external tation of a submanifold by its boundary for orientations of 1-dimensional submanifolds as 1,2,3-dimensional submanifold embedded into depicted in table 1. In the case of vectors in a 3-dimensional manifold. 3-dimension one can see why they are call axial.

+ 3.7 Twisting and untwisting If a manifold has an orientation then we can use it to change the type of orientation of a submanifold. Thus we can convert an exter- (a) (b) (c) Boundary of a nal orientation into an internal orientation. Boundary Boundary 3-dim submanifold, We can also convert an internal orientation of a 1-dim of a 2-dim The + is the external submani- submani- orientation of the into an external orientation. The convention fold fold 3-dim ball and the we use here is to say that if we concatenate arrow pointing in- the external orientation (ﬁrst) followed by the wards is the external internal orientation (second) one must arrive orientation of the at the orientation of the manifold. See exam- 2-dim sphere. ples in table 4. For example if the orientation of a 2-dim Figure 14: Inheritance of an external orien- manifold is clockwise and the untwisted ori- tation of a submanifold by its boundary for entation is up then the twisted orientation is 1,2,3-dimensional submanifold embedded into right. See fourth line of table 3. a 3-dimensional manifold. Starting with an untwisted form, then twisting and then untwisting it does not wise if a submanifold has an external orienta- change the orientation. tion then the boundary inherits the external orientation, ﬁgure 14. 4 Closed forms.

3.6 Orientations of vectors In this section we introduce exterior differential forms, which we call simply forms. On an In section 2.5 and figure 9 we saw the usual n type of vectors. These are untwisted vec- -dimensional manifold a form has a degree, which is an integer between 0 and n. We refer tors, since they have an untwisted orienta- for a form with degree p as a p-form. A closed tions which is the internal orientations. There form of degree p can be thought of as a col- exists another type of vector called a twisted lection of submanifolds1 of dimension n − p. vector. These are also called axial or pseudo- vectors. An example of such a vector is the 1Some authors [2, 8, 9] use a double sheet to in- magnetic field B. Twisted vectors have exter- dicate 1-forms and likewise they replace 1-dim form-

11 ⊖ ⊖ ⊖ ⊖ ⊖ ⊖⊖ ⊖ ⊖⊖⊖ ⊖ ⊖⊖⊖⊖⊖ ⊖ ⊖⊖⊖ ⊕⊖ ⊖⊖⊖ ⊖⊖⊖⊖⊖ ⊖ ⊖⊖⊖⊖⊖⊖⊖ ⊕ ⊖ ⊖ ⊕ ⊖⊖ ⊕⊕ ⊖ ⊖ ⊖ ⊕ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕⊕ ⊕ ⊕⊕⊕ ⊕ ⊕⊕⊕⊕⊕ ⊕ ⊕⊕⊕ ⊕ ⊕ ⊕ ⊕⊕⊕⊕ ⊕⊕⊕⊕⊕ ⊕ ⊕ ⊕ ⊕ ⊕⊕⊕⊕⊕⊕⊕ ⊕ ⊕ ⊕ ⊕ ⊕⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

(a) Low density. (b) Increased density, while reducing the value of each dot.

Figure 15: Representation of a twisted top- form, i.e. an n-form on an n-dimensional manifold.

Figure 17: An untwisted 2-form on a 3-dim manifold.

Figure 16: An untwisted 2-form on a 3-dim manifold. Figure 18: An untwisted 1-form on a 3-dim manifold. Note that the 1-form must vanish in the region between the two opposing orientations. A diagram of a twisted 1-form in 3 submanifolds with cylinders to indicate (n−1)-forms. It is true that for a single 1-form at a point it is dimensions is given in figure 26. necessary to have two (n − 1)-dim form-elements- submanifold in order indicate the magnitude of the form. Consider in figure 38 one needs at least 2-lines We call these form-submanifolds. Similar to indicate the combination of a 1-form with a vector to a vector field, there is a submanifold of the where both are defined only at one point. However in correct dimension passing though each point this document we only consider smooth form-fields. in the manifold. They must not have bound- I.e. a form defined at each point. Thus there is a form-submanifold passing through each point. Fur- aries, therefore they either continue off to “in- thermore since the form-field is smooth form subman- finity” or they close in on themselves, like ifolds for nearby points are nearly parallel. There- spheres. In section 5 we see that the form- fore there is sufficient information to indicate the submanifolds with boundaries correspond to magnitude of the form via the density of the form- non-closed forms. submanifolds. Looking at figure 38 again we see there is sufficient information to calculate the action of the Although the points, lines, surfaces corre- vector on the form. sponding to p-forms, for p ≥ 1, do not have

12 values, they do have an orientation and as with each points. As such the number of before these can be untwisted or twisted. As dots multiplied by their sign inside a par- stated in table 2 these are the opposite way ticular n-volume is approximately constant round than for submanifolds. Thus an un- during the limiting process. Compare this twisted form has an external orientation and to the scalar field, where as we stated, the a twisted form has an internal orientation. value of each point remains constant during The untwisted forms are mathematically sim- the limiting process. pler and much easier to take wedge products For untwisted top-forms each point car- of. See section 4.3 below. Indeed the stan- ries an external orientation, as in 0 dimen- dard method of prescribing the internal ori- sional submanifolds in table 1. entation of an n-dim manifold is in terms of • On an n-dimensional manifold the an untwisted n-form. closed (n − 1)-forms correspond to 1- dimensional form-submanifolds, i.e. curves We have the following: as depicted in figures 16 and 17. The curves • An untwisted 0-form (internal orienta- do not have start points or end points. tion) is simply a scalar field and is best Therefore they must either close on them- visualised as above in figures 6, 7 and 8. selves to form circles or they must go from Note that a closed 0-form is a constant and and to “infinity”. They also do not in- not very interesting. From the prescription tersect. Again there is really a curve go- above, a 0-form on an n-dim manifold is an ing though each point. The smoothness n-dim form-submanifold at each point. again requires that between a high density A twisted 0-form (external orientation) of curves and a low density there should be requires an external orientation for a 0-dim an intermediate density, and that between submanifold (point) as in table 1. curves of opposite sign there should be a • On an n-dimensional manifold, the gap. form-submanifolds for an n-form consists Figure 16 shows an untwisted 2-form (ex- of a collection of dots. For the rest of this ternal orientation) in 3 dimensions. report, we shall refer to n-form as top- Twisted (n − 1)-forms are lines with di- forms. rection. (Internal orientation). One may For twisted top forms (internal orienta- consider these as flow lines of a fluid or sim- tion) these dots carry a plus or minus sign. ilar. An example is shown in figure 17. See figure 15. The density of the dots cor- You can see from figures 9 and 17 that responds loosely to the magnitude of the untwisted vectors and twisted (n−1)-forms top-form. The smoothness of the field re- look similar, and indeed they are. With quires that between a high density of dots just a measure we can convert a vector into and a low density dots, there should be a a 1-form via the internal contraction. See region of intermediate density. section 6.3. A non-vanishing twisted top-form with • On an n-dimensional manifold the only plus dots is called a measure. We closed 1-forms correspond to (n − 1)- see below that these are very useful for dimensional form-submanifolds. integrating scalar fields (section 4.4) and Untwisted 1-forms are the submanifold for converting between vector fields and corresponding to the contours or level sur- (n − 1)-forms (section 6.3). We see in sec- faces of a scalar field, as in figure 8. The tion 7.8 that a metric automatically gives external orientation point in the direction rise to a measure. of increasing value of the scalar field. The Although, as with vector fields, there is faster the rate of increase of the scalar field, really a “dot at each point”, therefore the the closer the level surfaces. density of dots is actually infinite. However One usually takes the gradient of a scalar we can think of a limiting process. Start field which is the vector corresponding to with a set of discrete dots, then (approxi- the direction of steepest assent. This gra- mately) double the density of dots, while at dient is perpendicular to the level surfaces. the same time halving the value associated

13 (a) Addition of untwisted 1- (b) Addition forms of twisted 1-forms

Figure 19: Addition of untwisted (left two) and twisted (right) 1-forms in 2-dim. Figure 20: In 3-dimensions, the wedge prod- The solid 1-form-submanifold (blue and red) uct of an untwisted 1-form (blue) with a un- are added to produce the dashed 1-form- twisted 2-form (red) to give a untwisted 3- submanifold (green). form (green).

However to do this one needs a concept of orthogonality which comes with the metric deﬁned below in section 7. An example of an untwisted 1-form in 3- dimensions is given in ﬁgure 18.

4.1 Multiplication of vectors and forms by scalar fields. Figure 21: In 3-dimensions, the wedge product of an untwisted 1-form (vertical, blue) To multiply a p-form field by a scalar field with an untwisted 1-form (horizontal, red) to simply increase the density of dots, curves, give an untwisted 2-form (green). surfaces of the p-form by the value of the scalar field. 4.3 The wedge product also The action of multiplying a p-form by a known as the exterior prod- negative scalar field changes the orientation uct. of the p-form. It is worth noting that if you multiply a In order to take the wedge product of two closed p-form with p < n by a non constant forms simply requires taking their intersec- scalar the result will not be a closed form. tion, figures 20-24. In places where the two form-submanifolds are tangential then the resulting wedge product vanishes. The wedge product of two untwisted forms 4.2 Addition of forms. is an untwisted form. Simply concatenate the two orientations. See figures 20 and 21. The It is only possible to add forms of the same wedge product for three untwisted 1-forms is degree and twistedness. Thus two untwisted given by the intersection of three (n − 1)-dim p-forms sum to an untwisted p-form and two form-submanifolds, figure 22. twisted p-forms sum to a twisted p-form. In The wedge product of two twisted forms is contrast to multiplying by a scalar, adding an untwisted form (figure 23) and the wedge two closed forms gives rise to a closed form. product of a twisted form and an untwisted Although algebraically adding forms is form is a twisted form (figure 24). However trivial, the visualisation of the addition of two the rules of this is more complicated than p-forms is surprisingly complicated. Figure simply concatenation. The only guaranteed 19 show how to add 1-forms in 2-dimension. method is to choose an orientation for the

14 Figure 24: In 3-dimensions, the wedge product of the untwisted 1-form (blue) with the twisted 1-form (red) to give the green twisted Figure 22: In 3-dimensions, the wedge prod- 2-form. uct of three untwisted 1-forms. The ﬁrst vertical (blue) followed by the second facing reader (brown) then the third horizontal (red). This gives an untwisted 3-form (green).

Figure 25: Integration of a twisted 2-form on Figure 23: In 3-dimensions, the wedge prod- a 2-sphere embedded in a 3-dimensional man- uct of the twisted 1-form (vertical, blue) with ifold. Observe that every curve that enters the the twisted 1-form (horizontal, red) to give sphere also leaves. the green untwisted 2-form.

1 manifold, then convert the twisted forms into figure 15b each dot must have value 4 and the 1 untwisted forms, take the wedge product and integral is 6 2 . As the density of dots increases then apply a twisting if necessary. One can and their value tends to zero, this sum will check that the result is invariant under the converge. Contrast this to the case of the choice of orientation of the manifold. How- scalar field, figure 6, which does not converge ever it may depend on the twisting convention as the density increases. In order to integrate described in section 3.7. a scalar field it is necessary to multiply it by a measure. To integrate an untwisted top-form it is 4.4 Integration. necessary to use the orientation of the man- We can only integrate twisted top-forms, i.e. ifold to convert the untwisted top-form into n-forms on an n-dimensional manifold, figure a twisted top-form. If the manifold is non 15. To do this we simply add the number orientable, such as the Möbius strip then one of positive dots and subtract the number of can only integrate twisted top-form. negative dots. Then take the limit the the What about integrating forms of lower di- density tends to infinity and the value tends mension? To integrate a p-form we must in- to zero. Let figure 15 refer to a 2-dimensional tegrate it over a p-dimensional submanifold. manifold. Thus in figure 15a, assuming each In addition untwisted forms are integrated dot value has 1, then its integral equals 9. In over untwisted submanifolds and likewise for

15 ﬁnal time time

initial time

space space space

Figure 26: Integration of a twisted 1-form on Figure 27: The closed twisted 3-form (red) is a twisted circle embedded in a 3-dimensional integrated over the 3-dim surface of a cylinder manifold. Observe that every surface that with net result zero. Any current which en- crosses the circle must cross again in the op- ters at the initial time (lower blue disc) must posite direction. either leave at the ﬁnal time (upper blue disc) or through the curved surface of the cylinder (yellow). Note that the orientation of the sub- twisted forms and twisted submanifolds; with manifold is out (given by the yellow and blue plus if the orientations agree and minus oth- arrows). This is because the sum of the inte- erwise. We will see a generalisation of this in grals is zero. However the contribution from section 8.1 on pullbacks. initial time is negative.

4.5 Conservation Laws See section 4.6 below for a comment about holes. We can see from figures 25 and 26 that every In order to convert the integral of a closed curve or surface that intersects the subman- current into a conservation over all time it ifold also leaves. Therefore the integral over is necessary to expand the curved surface of the sphere in figure 25 and over the circle in the cylinder out to infinity. Thus the initial figure 26 are both zero. This gives rise to and final discs become all of space. One must conservation laws which are very important therefore guarantee that no current “escapes in physics. For example the conservation of to infinity”. See figure 28. charge or the conservation of energy. In figure 27 we see that the closed twisted 3-form is integrated over the surface of a 3 dimensional 4.6 DeRham Cohomology. cylinder. In this context we refer to the closed As mentioned in section 4.5 in order to guar- 3-form as a current. antee that the integral of a closed p-form over However there are three conditions on the a closed p-dimensional submanifold is zero we submanifold which together imply that the need that the larger manifold in which all the integral of a closed form is zero. submanifolds are embedded has no “holes”. • The submanifold (of dimension p) must As an example of a manifold with holes con- be the boundary of another submanifold (of sider the annulus, which is disc with a hole, dimension p + 1). given in figure 29. When integrating around • The (p + 1)-dimensional manifold must a circle which does not encircle the hole then not go off to infinity. the integral is zero. However when integrat- • The larger manifold in which all the ing around the hole then the integral is non- submanifolds are embedded must not have zero. any “holes”. The use of integration to identify holes in a

16 ﬁnal time

initial time

(a) A conserved ﬁeld. Figure 29: A annulus which is a 2-dim manifold with a hole. When the untwisted 1-form (red) is integrated in a circle (blue) which does not enclose the hole the result is zero. By con- ﬁnal time trast when one integrates over circle (green) which does enclose the hole the result is non zero.

initial time

(b) A non conserved ﬁeld.

Figure 28: In 4-dimensions, integration of the twisted 3-form (red), showing it is conserved as long as nothing “escapes to infinity”. The 3-dimensional submanifolds (blue) correspond to “equal time slice”. Note that, com- Figure 30: On a 2-dim torus, integrating the paring with figure 27 we have let the curved 1-form (red) over the circle (blue) is not zero, surface (yellow) pass to infinity. We have also despite the fact that the 1-form is closed. swapped the orientation of the initial timeslice so that it has the same orientation as the finial time slice. form is continuous, in 31b it is discontinuous. We will see physical examples of such discontinuous forms in electrostatics and magneto- manifold is known as DeRham Cohomology. statics, section 9.1 which correspond to the Note that the “hole” need not be removed fields generated by point and line sources such from the manifold like a hole is removed from as electrons and wires. a disc to create an annulus. In figure 30, one integrates over a circle which encircles the 4.7 Four dimensions. “hole” of a torus. If the manifold does not have any holes, There are significant yet obvious prob- then as stated the integral is zero. It is inter- lems when trying to visualise forms in 4- esting to see attempts to create a non zero dimensions. However there are important integral as in figure 31. In figure 31a the reasons to try. As we will see below, we of form-submanifolds terminate, whereas in fig- course live in a 4-dimensional universe and ure 31b the form-submanifolds all intersect the pictorial representation of electrodynam- at a point. In both cases these correspond ics would be particularly enlightening. Sec- to non closed forms as we will see below in ondly there are phenomena which only exist section 5. The difference is that in 31a the 1- in four or more dimensions. In lower dimen-

17 (a) t< 0 (b) t = 0 (c) t> 0

Figure 32: The plane representation of a 2-form-submanifold in 4-dimensions. The 2- form-submanifold is represented by both the line (blue), for all time, and the momentary plane (red) at the time t = 0. The self wedge (a) In this case the form-submanifolds termi- is represented the dot (green). nate. This corresponds to the 1-form (red) not being closed.

(a) t< 0 (b) t = 0 (c) t> 0

Figure 33: The moving line representation of a 2-form-submanifold in 4-dimensions. The 2- form-submanifold is represented by two lines (red and blue). The self wedge is represented by the dot (green).

(b) In this case the form-submanifolds all inter- dimensional form-submanifolds which intersect. This corresponds to the 1-form (red) not sect at that point. being continuous at this point. One way to attempt to visualise a 2-form- Figure 31: Two attempts to create a non submanifold in 4-dimensions, is to use 3- zero integral as in figure 29, but on a man- dimensional space and 1-dimension of time. ifold which does not have a hole. On the This is particularly relevant for visualising 2-dimensional disc, the 1-form (red) is inte- spacetime. There are two possibilities. Let’s grated over circle (green). assume that the two 2-dimensional form- submanifolds intersect at time t = 0. Then one possibility, shown in figure 32 is that sions any p-form, for p ≥ 0 wedged with it- for all t =6 0 the form-submanifold is a line, self is automatically zero. This is because ev- whereas at t = 0 it is that line and an inter- ery form-submanifold is tangential to itself. secting plane. The alternative representation This does not apply to scalar fields, viewed is given by two moving lines as seen in figure as 0-forms, which can be squared. How- 33. The result of wedging the 2-form with ever there are 2-forms in 4-dimensions which itself gives a 4-form. This top-form is repre- are none zero when wedged with themselves. sented by a dot, which is the intersection of In particular, this includes the electromag- the 2-form-submanifold with itself. netic 2-form which we write F . Thirdly, all closed form-submanifolds up to now have a single smooth submanifold passing through 5 Non-closed forms. each point. However 2-forms in 4-dimensions, which are non-zero when wedged with them- Non closed p-forms consist of a small sub- selves, correspond, at each point, to two 2- manifold of dimension (n − p) or surface at

18 Figure 34: Example of a non-closed 1-form (black) in 2-dim. If we integrate from bottom Figure 36: An integrable non-closed 1-form left to top right on the green square, we see in 3-dim: Two 1-form-submanifolds (green) we get a different value depending on the path each with a scalar field, represented by con- taken. tour lines (red). The exterior differential of this 1-form are the closed red contour lines. We assume that the scalar field is increasing towards the inner contours. Both possible twistedness orientations are shown on the same diagram. For untwisted orientations (lower sheet, blue) concatenate the inward arrow of the contours with the up arrow of the Figure 35: An Attempt to show a non-closed, submanifold sheets. The twisted orientation non-integrable 1-form in 3-dim (green). If we (upper sheet, grey) simply inherits the orien- try and close the elements up in to surface tation of the sheet. we see that we fail. Starting the red arc on the lightest green and going round anticlockwise we end up on the darkest green, we have The exterior derivative of a scalar field is helixed down. simply the 1-form-submanifolds corresponding to the contours, also known as level (n − each point. We call these form-elements. 1)-surfaces, of the scalar field. For untwisted The difference is that in non-closed forms, the scalar fields the orientation points in the di- form-elements do not connect up to form a rection the scalar field increases, as in figure submanifold. Examples are given in figures 34 8. and 25. We can see from figure 34 that the The exterior differential of an integrable integral now depends on path taken. Thus non-closed form are the contours of the scalar non-closed forms correspond to non conser- field on each form-submanifold. See figure 36. vative fields. Clearly if the p-form is closed then it has no In some (but not all) cases a non-closed boundary and the result is zero. In addition, form can be thought of as a collection of since a boundary has no boundary, the ac- closed form-submanifolds, but each of these tion of the exterior derivatives operator twice has a scalar field on it, see figure 36. Such vanishes. forms are known as integrable. For example the 1-form (in 2-dim) in figure 34 is integrable whereas the 1-form (in 3-dim) in figure 35 is 5.2 Integration and Stokes’s not. theorem for non-closed forms. 5.1 The exterior differential op- The integration of non-closed forms is no erator. longer necessarily zero, even if there are no holes in the manifold. Recall to integrate The exterior differential operator takes a p- a p-form we must integrate it over a p- form and gives a (p+1)-form. One may think dimensional submanifold. In addition un- of this as taking the boundary of the form- twisted forms are integrated over untwisted submanifolds as given by figures 13 and 14. submanifolds an likewise for twisted forms

19 +

+ − + − − + Figure 38: Combining an untwisted 1-form − − (black) and an untwisted vector ﬁeld (green) to give a scalar ﬁeld, in 2-dim. The scalar at − − the base point of the vector would have value + + −5.

+ +

Figure 37: Demonstration of Stokes’s theorem in 2-dimensions. The orientation of the twisted disk is + and the twisted circle is in- wards. The twisted 1-form is not closed and Figure 39: In a 3-dim manifold. The inter- its exterior derivative is the positive and neg- nal contraction of the untwisted 2-form (red) ative dots. The integration of these dots over with the untwisted vector (blue) to produce the disk is −7. Likewise the integration of the the untwisted 1-form (green). The orienta- 1-form over the circle is also −7. tion of the vector followed by the orientation of the 2-form gives the orientation of the 1- form. and twisted submanifolds; with plus if the orientations agree and minus otherwise. submanifolds the vector crosses. See figure In figure 37 we see a demonstration of 38. From this we see that if a 1-form con- Stokes’s theorem. That is, the integral of a tracted with a vector is zero, then the vector (p−1)-form over a (p−1)-dimensional bound- must lie within the 1-form-submanifold. ary of a submanifold equals the integral of the exterior derivative of the (p−1)-form over the p-dimensional submanifold. Again we have to 6.2 Vectors acting on a scalar assume the embedding manifold has no holes. field. With a bit of work defining the divergence and curl in terms of exterior differential oper- A vector acting on a scalar field gives a new ator, one can see that our version of Stokes’s scalar field. This is given by contracting the theorem is a generalisation of the divergence exterior derivative of the scalar field with the and Stokes’s theorem in 3-dimensional vector vector, as defined above. Thus it corresponds calculus. to the number of times a vector crosses the contour submanifolds of the scalar field. 6 Other operations with 6.3 Internal contractions. scalars, vectors and Internal contractions is an operation that forms takes a vector field and a p-form and gives a (p−1)-form. Pictorially it corresponds to the 6.1 Combining a 1-form and a (p − 1)-form-submanifold created by extend- vector field to give a scalar ing the p-form-submanifold in the direction of field. the vector. Thus increases the dimension of the form-submanifold by 1. See figure 39. If Given a vector field and a 1-form field, one the vector lies in the p-form-submanifold the can combine them together to create a scalar internal contractions vanishes. field. The scalar is given by the number form- This is a generalisation of the contracting of

20 a vector and a 1-form given in subsection 6.1. Since the vector lies in the resulting (p − 1)- form-submanifold, internally contracting by the same vector twice is zero.

7 The Metric

The metric gives you the length of vectors (a) The unit spheres in (b) The unit spheres in (figures 42 and 43) and the angle between two flat Riemannian space. Riemannian space. vectors (figures 44 and 45). It can also tell you how long a curve is between to points Figure 40: Representations of the metric in 3-dim space. and given two points it can tell you which is the shortest such curve, which is called a geodesic. Thus it can tell you the distance between two points. This is the length along space space space the geodesic between them. time space time space time space In higher dimensions the metric tells you the area, volume or n-volume of your manifold and submanifolds within it. It gives you a measure so that you can integrate scalar (a) The two- (b) The dou- (c) The one- fields, figure 51. It enables you to convert 1- sheet (upper ble cone. sheet hyper- forms into vectors and vectors into 1-forms and lower) boloid. (called the metric dual). It also enables you hyperboloid. to convert p-forms into (n − p)-forms (called the Hodge dual). It doesn’t, however, give Figure 41: The spacetime metric (in 3 di- you an orientation. mensions). The two-sheet hyperboloid (a) is used for measuring timelike vectors. The up- Metrics possess a signature. A list of n per hyperboloid is for future pointing vectors, positive or negative signs, where n is the and the lower for past pointing vectors. The dimension of the manifold. There are a one-sheet hyperboloid (c) is used for measur- number of important dimensions and signa- ing the length of spacelike vectors. Between tures: the two hyperboloids is the double cone, which • One dimensional Riemannian mani- all lightlike vectors lie on. folds. • Two dimensional Riemannian mani- nature (−, +, +), and two dimensional space- folds, i.e. signature (+, +). These are sur- time, signature (−, +). This is because they faces, which may be closed like spheres and are easier to draw. torii, with a number of “holes”. They may also be open like planes. They also included non orientable surfaces such a the 7.1 Representing the metric: Möbius strip. Unit ellipsoids and hyper- • Three dimensional Riemannian manifolds, i.e. signature (+, +, +). These in- boloids clude the three dimensional space that we Here we represent the metric is by the unit el- live in. lipsoid or hyperboloid. See also [6]. Around a • Four dimensional spacetime, i.e. sig- point consider all the vectors which have unit nature (−, +, +, +). Here the minus sign length. For a Riemannian manifold, these refers to the time direction and the three vectors form an ellipse in 2 dimensions and an pluses to space. ellipsoid in 3 dimensions. In figure 40 we show However for this document we will also be the ellipsoid for a Riemannian manifold. If looking at three dimensional spacetime, sig- the ellipsoids are the same at each point then

21 2 time 1.5 2 1 0.5 1

1 2 space

Figure 42: Using the metric ellipsoid to cal- Figure 43: Using the metric hyperboloid to culate the length of a vector. The unit ellip- measure the length of vectors. soid is scaled to various sizes. The vector has • The diagonal (green) lines are the path of length 2. light rays. Thus the diagonal vector (dark green) is a lightlike vector. • The left and right (red) curves are the hyper- the metric is flat as in figure 40a. By con- boloid for measuring the length of spatial vec- trast for a curved metric the ellipsoids must tors. (In two dimensions these two curves are be different at each point as in figure 40b. disjoint. However in three or more dimensions For spacetime metrics the ellipsoids is re- these two lines connect to create the one-sheet placed by a hyperboloid, as in figure 41. Since hyperboloid.) The right-pointing (black) vec- there is a minus in the signature, vectors are tor is spatial and has length 2. divided into three types: • The upper and lower (blue) lines are for mea- • Timelike vectors, these represent the 4- suring the length of timelike vectors. The up- velocity of massive particles. These vectors ward (orange) line is timelike and has length 1 point to one of the components of the two- in the forward direction. The downward (pur- ple) line is timelike and has length 2 in the sheet hyperboloid, as seen in figure 41a. backward direction. As the name suggest, the two-sheet hyperboloid has two components, the upper and lower. This distinguishes between future figure 42. point and past pointing vectors. We see in figure 43, we use the two-sheet • Lightlike vectors, the 4-velocity of light. (upper and lower) hyperboloid to measure These vector lie on the double cone. See fig- the length of forward and backward pointing ure 41b. Again one can distinguish between timelike vectors. We also use the one-sheet past and future pointing lightlike vectors. hyperboloid (left and right curves) to mea- • Spacelike vectors, These are not the 4- sure the spacelike vectors. velocity of anything. These vectors point to the one-sheet hyperboloid. One cannot distinguish between past and future space- 7.3 Orthogonal subspace like vectors. Given a vector then the metric can be used to specify all the directions which are orthogonal 7.2 Length of a vector to that vector. Orthogonal vectors are also known as perpendicular vectors or vectors at Using the metric ellipsoid or hyperboloid to right angles to each other. This is given by calculate the length of a vector is easy. For the line tangent to the ellipsoid (figure 44) or a Riemannian metric the ellipsoid represent hyperboloid (figure 45) at the point where the all vectors of unit length. Therefore simply vector crosses it. In 3 dimensions the set of compare length of the vector with the radius orthogonal directions form a plane and in 4 of ellipsoid in the direction of the vector as in dimensions they would form a 3-volume.

22 ℓ

Figure 46: Using the metric ellipsoid to measure the angle between two vectors. The Figure 44: Using the metric ellipsoid to pre- (red) lines are orthogonal to one of the vec- scribe the orthogonal space (red) to a vector tors (black). The length ℓ is the cosine of the (blue). The orthogonal space is tangential to angle between the two vectors. the ellipsoid. time time

space space

Figure 47: Using the metric hyperboloid to Figure 45: Using the metric hyperboloid to measure the angle between two vectors. The prescribe the orthogonal space (red) to a vec- (red) lines are orthogonal to one of the vectors tor (black). The orthogonal space is tangen- (black). The length γ is the γ-factor required tial to the hyperboloid. to boost one of the vectors to the other.

7.4 Angles and γ-factors is given by counting the (n − 1)-dimensional Given two vectors then a Riemannian metric form-submanifolds inside the ellipsoid and di- can be used to find the cosine of the angle viding by 2. In figure 48 we see how to between them as shown in figure 46. This use the metric ellipse to measure the magni- cosine will always be a value between −1 and tude to a 1-form in 2 dimensions. Measuring 1. the magnitude of a 2−form, requires count- In the case of a spacetime metric, then in ing the number of (n − 2)-dimensional form- figure 47 we get a value which is greater than submanifolds inside the ellipsoid and dividing 1. We call this value the γ-factor. It is one of by π. Similar formula exist for higher degrees the key values in special relativity. Assuming and for spacetime manifolds. that the two vectors are the 4-velocity of two particles, then the γ-factor is a function of the 7.6 Metric dual. relative velocity between the two particles. The metric can be used to convert a vec- 7.5 Magnitude of a form. tor into a 1-form and a 1-form into a vector. Given a vector then the 1-form is the (n − 1)- One can use the metric to measure the mag- form-submanifold which is perpendicular to nitude of forms. On an n−dimensional Rie- the vector, as given in figures 44 and 45. See mannian manifold the magnitude to a 1-form also figure 49.

23 (a) Spacelike vector (blue). The orientation (red) of the 1-form is unchanged.

Figure 48: Using the metric to measure the magnitude of a 1-form in 2 dimensions. Sim- ply count the number of form-submanifold inside the ellipse and divide by 2. In this case the 1-form has magnitude 3.5 (b) Timelike vector (blue). The orientation (red) of the 1-form is reversed. The twistedness of the orientation is unchanged. For spacelike vectors the orientation is preserved, whereas for timelike vectors the orientation is reversed. For lightlike vectors, the vector is orthogonal to itself. Therefore (c) Lightlike vector (blue). The orientation it lies in its own orthogonal compliment. For (red) of the 1-form points in the direction of these the orientation points in the direction the nearby spacelike vectors (green). of the nearby spacelike vectors. Figure 49: The metric dual converts a vector into a 1-form and visa versa. The untwisted 7.7 Hodge dual. vector (blue) is converted into the untwisted 1-form (red). The Hodge dual takes a p-form and gives the unique (n−p)-form which is orthogonal to it. It changes the twistedness of the form. In 8 Smooth maps Riemannian geometry, if one starts with an untwisted p-form it gives a twisted (n − p)- A smooth map is a function that maps one form with the same orientation. If one starts manifold into another. with a twisted p-form it gives the untwisted The simplest case of a smooth map is a form with either the same or the opposite ori- diffeomorphism where there is a one to one entation. This is given by double twisting as correspondence between points in the source follows: manifold and points in the target manifold. • If p is even or (n − p) is even then the As a result the source and target manifolds orientation is unchanged. must have the same dimension and in some • If both p and (n − p) are odd then the respects they look the same, that is all the orientation is reversed. concepts in sections 2-6. By contrast when a For non Riemannian geometry the formula for metric is included the two manifolds can look the resulting orientation is more complicated. very different. One example is in transformation optics where one maps between one 7.8 The measure from a metric manifold where light rays travel in straight lines and another manifold where light rays A metric naturally gives rise to a measure. travel around the hidden object. That is a twisted top-form. This is actu- One class of smooth maps that we have en- ally the Hodge dual of the untwisted constant countered is that of the embedding of sub- scalar field 1. It is given by packing together manifolds as seen in figure 4. In this case we the metric ellipsoids. See figure 51. We can map a lower dimensional (embedded) mani- integrate this measure to give the size of any fold into a higher dimensional (embedding) manifold even of it is not orientable. manifold. In this case every point in the em-

24 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ (a) In 2-dim. Hodge dual of the untwisted 1- ⊕ form (red), becomes the twisted 1-form (blue). Figure 51: The measure from the metric. In 2 dim, the twisted top form is given by packing together the metric ellipses.

(b) In 3-dim. Hodge dual of the untwisted 1- form (red), becomes the twisted 2-form (blue).

Figure 52: A projection mapping a 3-dim (c) In 3-dim. Hodge dual of the untwisted 2- manifold onto a 2-dim manifold (green). All form (red), becomes the twisted 1-form (blue). the points on each line (red and blue) are projected onto the respective dots. Figure 50: Hodge dual of untwisted forms (red) to twisted forms (blue) in 2 and 3- dimensional Riemannian geometry. For space- 8.1 Pullbacks of p-forms. time geometry the resulting orientation may be reversed. No matter how complicated the map is it is always possible to pullback the form. We will only consider here the pullback with respect to diffeomorphisms, embeddings and projections. Pullbacks preserve the degree of the bedded submanifold corresponds to a unique form and also the twistedness of the forms. point in the embedding manifold. However The pullback with respect to diffeomor- some points in the embedding manifold do phisms simply deforms the shape of the form- not correspond to a point in the embedded submanifolds. However unless there is a met- manifold. ric the question is: deforms with respect to what? This corresponds to our statement Another class are regular projections. that without a metric one cannot distinguish These map a higher dimensional manifold between manifolds which are diffeomorphic into a lower dimensional manifold. See fig- and therefore as we said in the introduction, ures 52 and 53. In this case every point in the one is dealing with rubber sheet geometry. image manifold corresponds to a submanifold For the pullback with respect to an em- of points in the domain. bedding, one simply takes the intersection of submanifold with the form-submanifolds Other maps however are none of the above as seen in figures 54, 55 and 56. This pre- and may include self intersections (figure 5), serves the degree of the form. If we pullback folding and a variety of other more compli- a p-form from an m-dimensional manifold to cated situations. an n-dimensional manifold then the p-form-

25 Figure 55: The pullback with respect to a 2- dim submanifold (red plane) embedded in a 3- Figure 53: A projection mapping a 3-dim dim manifold. The pullback of an untwisted 1- manifold onto a 1-dim manifold (green). All form (blue planes) gives an untwisted 1-form the points on the plane (blue) are projected (green lines). onto the dot.

Figure 54: The pullback with respect to a 2- dim submanifold (red plane) embedded in a 3- Figure 56: The pullback with respect to a dim manifold. The pullback of an untwisted 2- 1-dim submanifold (red curve) embedded in a form (blue curves) gives an untwisted 2-form 3-dim manifold. The pullback of an untwisted (green dot). 1-form (blue plane) gives an untwisted 1-form (green dots). submanifolds are of dimension (m − p), hence the intersection of the (m − p)-dim form- the orientations agree and minus otherwise, submanifolds with the n-dimensional mani- as was discussed in section 5.2. folds will have dimension (n − p) correspond- For the pullback with respect to projections ing correctly to a p-form. This assumes the the resulting form-submanifold is simply all form-submanifolds and the submanifold in- the points which mapped onto the original tersects transversely. However if the subman- form-submanifold. See ﬁgures 58 and 59. For ifold and the form-submanifold intersect tan- untwisted forms the orientation is obvious. gentially then the result is zero. See ﬁgure For twisted forms one can use orientation for 57. both the source and target manifolds, untwist For untwisted forms there is a natural pull- the form and convert it back. Alternatively back of the orientation. For twisted forms it is one can decide an orientation of the projec- necessary to pullback onto a twisted subman- tion and use that. We will not describe this ifold. To be guaranteed to get the orientation correct it is easiest to choose an orientation for the embedding manifold, untwist both the form and the submanifold and then twist the pullbacked form. This gives the result that: The orientation of the pullbacked form con- Figure 57: The pullback with respect to a catenated with the orientation of the sub- 1-dim submanifold (red curve) embedded in a manifold equals the orientation of the original 3-dim manifold. The 1-form (blue plane) is form. In particular the pullback of a twisted tangential to 1-dim submanifold. The result p-form onto a twisted submanifold is plus if is zero.

26 Figure 60: The pullback with respect to a Figure 58: The pullback with respect to a projection from a 3-dim manifold to a 1-dim projection from a 3-dim manifold to a 2-dim manifold (green curve). The pullback of the manifold (green plane). The pullback of the untwisted 1-form (blue dot) gives the 1-form untwisted 2-form (blue dots) gives the 2-form (red plane). (red lines).

Figure 61: The pullback of a Riemannian metric onto an embedding as represented by unit spheres. The unit spheres in 3-dim (green) are pulled back into unit circles (blue) on the 2-dim submanifold (red).

Figure 59: The pullback with respect to a projection from a 3-dim manifold to a 2-dim jection. manifold (green plane). The pullback of the untwisted 1-form (blue curve) gives the 1-form 8.3 Pushforwards of vectors. (red plane). We can always pushforward a vector field with respect to a diffeomorphism. However, here. in general the pushforward of a vector field with respect to any other map will not result 8.2 Pullbacks of metrics. in a vector field. As can be seen in figure 62 some points in the target manifold have no We can only pullback a metric with respect vector associated with them and others have to diffeomorphisms and embedding. With re- more than one. spect to diffeomorphisms it makes the two However the pushforward of vectors are manifolds have the same geometry. very useful. For example the velocity of a The pullback of a metric with respect to single particle is the tangent to its worldline. an embedding is called the induced metric. This is only defined on its worldline. This is the metric we are familiar with on the I will not discuss these further here. sphere or torus which is embedded into flat 3-space. Figure 61 shows the result of the induced metric. 9 Electromagnetism and We cannot pullback a metric with respect Electrodynamics to a projection as we don’t have enough information. In particular we do not know the Maxwell’s equations, when written in stan- length of vectors in the direction of the pro- dard Gibbs-Heaviside notation are four dif-

27 Vector Form Relation between names notation notation vector and form untwisted untwisted E Electric field. E Metric dual vector 1-form untwisted twisted Metric dual and D Displacement field. D vector 2-form Hodge dual twisted “B-field”, Magnetic field or untwisted Metric dual and B B vector Magnetic flux density. 2-form Hodge dual “H-field”, Magnetic field intensity, twisted untwisted H Magnetic field strength, H Metric dual vector 1-form Magnetic field or Magnetising field. untwisted twisted ρ charge density. ̺ Hodge dual scalar 3-form untwisted twisted Metric dual and J current density. J vector 2-form Hodge dual

Table 5: The ﬁelds in electromagnetism, in terms of the familiar vector notation, the corresponding form notation and the operations needed to pass between the two.

Spacetime Pull back Remaining form onto space component untwisted untwisted untwisted F B E 2-form 2-form 1-form twisted twisted twisted H D H 2-form 2-form 1-form twisted twisted twisted J ̺ J Figure 62: Push forward of the vector field 3-form 3-form 2-form (red) of the 1-dim manifold (blue) embedding into the 2-dim manifold (white) produces vec- Table 6: Combining the fields in electromag- tors (red). However these vectors do not form netism into the relativistic quantities. a vector field as they are not defined in some places and in others have multiple values. just stated the vacuum constitutive relations, namely that D = ǫ0E and B = µ0H. How- ferential equations of four vector fields E, B, ever in a medium, such as glass, water, a D and H. The source for the electromag- plasma, a magnet, a human body or what- netic fields is the charge density ρ and the ever else you may wish to study, these con- current J. In our notation it is clearer to stitutive relations are more complicated. The replace these with forms. In table 5 the rela- simplest constitutive relations consider a con- tionship between the vector notation and the stant permittivity ǫ and permeability µ. In form notation is given. this medium there is no dispersion, i.e. ra- As can be seen in table 5, both the fields diation of different frequency travel at the B and H may be referred to as the magnetic same speed. This would mean for example field. Here we will simply refer to them as the that rain drops would not give rise to rain- “B-field” and “H-field”. bows and may therefore be described as “an- As we said, Maxwell’s equations are differ- tediluvian”! They are however easy to work ential equations in E, B, D and H. However with. A more physically reasonable constitu- there is insufficient information to be able to tive relations is to allow the permittivity or solve these equations. They need to be re- permeability to depend on frequency. This is lated by additional information which relate a good model for water, wax or the human the four quantities E, B, D and H. These body. If the electric or magnet fields are very are called the constitutive relations. We have strong then the permittivity and permeability

28 depend on the electric or magnet ﬁelds. These are non linear materials. Exotic and meta- materials can have more complicated constitutive relations. The degree of complexity of these relations is limited only by the imagination of the scientists studying the material and their ability to build them. Indeed physicists are currently looking at models of the vacuum where the simple relations D = ǫ0E Figure 63: The magnetic 2-form B in space B µ0H and = break down in the presents as unbroken curves. Since the curves are of intense ﬁelds, either due to quantum ef- unbroken this corresponds to the non diver- fects or because Maxwell’s equations them- gence of B. The external orientation is given selves breakdown. (green). The circle (red) encloses some of the B-lines.

Just as in relativistic mechanics where we E combine energy and momentum into a single relativistic vector the 4-momentum, so D we combine the electric and magnetic B-field into a single spacetime untwisted 2-form F . Likewise we combine the displacement current and H-field into a single twisted 2-form H, and we combine ρ and J in a single twisted 3-form J , as summarised in table 6. As a result relativity mixes the E and B fields and Figure 64: Gauss’s Law in electrostatics. The separately mixes the D and H fields. The positive charge, is given by the twisted 3-form constitutive relations then relate F and H. ̺ is the dot (red) in the centre. This is the The simplest case is the vacuum where the re- exterior derivative of the twisted 2-form D (black) lines. In the vacuum or similar con- lationship is simply given by the Hodge dual. stitutive relations, then D is the Hodge dual of the untwisted 1-form E (green spheres). As stated, Maxwell’s equations give the electric and magnetic fields for a given cur- 9.1 Electrostatics and Magne- rent. It is also necessary to know how the tostatics. charges, that produce the current, respond to the electromagnetic fields. This is given by In this subsection we deal with electrostat- the Lorentz force equation. ics and magnetostatics as 3-dimensional pictures are easier to draw. One has that B is divergence-free, Gauss’s law which states that In this section we start, section 9.1, with the divergence of D equals the charge, Fara- electrostatics and magnetostatics as these are day’s law which states that E is curl-free and in three dimensions and therefore easier to Ampere’s law which states that the curl of depict pictorially. We then, in section 9.2 H is the current. In the language of forms to draw Maxwell’s equations as four space- this corresponds to the fact that B and E are time pictures. In section 9.3, we depict the closed, whereas the exterior derivative of D Lorentz force equation and a spacetime pic- and H are ρ and J respectively. ture. In section 9.4 we look at Maxwell’s In figure 63 we see that the B lines are un- equations in 3-dimensional spacetime. This is broken and therefore B is closed. In figure 64 the pullback of the 4-dimensional spacetime we see Gauss’s law, that a charge twisted 3- equations onto a fixed plane. This contrasts form ̺ is the exterior derivative of the twisted the the static cases when we pullback onto a 2-form D. For the vacuum or for any consti- fixed timeslice. tutive relation where the permittivity is con-

29 Time

Space Space Space

Figure 65: Ampere’s law: The constant line current, the twisted 2-form J (Black), is the Figure 67: Conservation of charge: There exterior derivative of the twisted 1-form H. In exists a closed twisted 3-form J called the the vacuum or similar constitutive relations, electric current (green). The green line may then H is the Hodge dual of the untwisted be interpreted as charged particles, distributed 2-form B (blue circles). over all of spacetime. stant or depends only on frequency, then the J in the wire, is the exterior derivative of the D is the permittivity times the Hodge dual twisted 1-form H, and H is the Hodge dual of the electric 1-form E. In figure 65 we see of B. Ampere’s law. The current, a twisted 2-form In figure 66 we show the fields for a permanent magnet. This is an example of a slightly more complicated constitutive relations. Outside of the magnet, the twisted 1- form H is the Hodge dual of the untwisted 2-form B. However in the magnet, the ori- N entations of H and B are in the opposite direction, due to the constitutive relations for a permanent magnet.

9.2 Maxwell’s equations as four spacetime pictures S Figures 67 to 70 represent conservation of charge and Maxwell’s equations in spacetime. Thus there are four axes, one of time and three of space. We start with conservation of charge, fig- Figure 66: A permanent magnet. The untwisted magnetic 2-form B (blue curves) and ure 67, which is not one of Maxwell’s equa- twisted 1-form H (red ellipsoids) generated by tions but a simple consequence of them. In a permanent magnet (grey rectangle). Ob- figure 67 we see the 1-dim form-submanifolds serve that outside of the magnet H fields are of J do not have a boundary and therefore perpendicular to the B fields. Thus the Hodge J is closed. One may think of the form- dual, figure 50, of H coincides with B. Inside submanifolds of J as worldlines of the indi- the magnet however the orientations of H and vidual point charges such as electrons or ions. B are in opposite directions. This is due to Look again at figure 63, but now consider the non vacuum constitutive relations of per- that the fields are not static. We may imagine manent magnets. that the magnetic field lines move in space.

30 aaa’ a fidcin nti aethe case this In of of form-submanifolds form induction. integral of the electro- law by an Faraday’s given cross generate as may will force lines This motive field circle. the red of the some result a As u ollnsi pctm.Ti steclosed the is 3-form This spacetime. in worldlines out eainhpbetween relationship n hrfr tcrepnst h untwisted 2-form the to closed boundaries corresponds it have therefore and not will These form-submanifold spacetime. 4-dim in form-submanifold lsd2-form closed httetitd2-form twisted the that 2-form the 2-form the for D form-submanifolds 1-dim the charges the If sidctdi section in indicated intersect as self can recall 4-dimensions should in 2-forms one that However spacetime. 4-dim futitd2-form untwisted of in.I h aumoehsthat has one vacuum rela- the constitutive In the tions. using augmented be to r h og ul of duals Hodge the are eni figures in seen aldteeetoantcfil bu) This fields two 2-form (blue). the field encodes untwisted electromagnetic closed the called a exists There 68: Figure eetn h aepoeuefrfigure for procedure same the Repeating ssae eoe awl’ qain need equations Maxwell’s before, stated As iete2dmfr-umnfls h clo- The of form-submanifolds. sure out- 2-dim loop a the is side orientation the 4-dimensions in awl qain hc contain which equations Maxwell

a u h -i omsbaiod for form-submanifolds 2-dim the out map Time J F si figure in as seuvln otetomacroscopic two the to equivalent is H hsgvsfigure gives This . F ̺ osraino antcflux: magnetic of Conservation F

64 s2dmwtotbudr in boundary without 2-dim as Space r o ttcte hymap they then static not are nﬁgure In .

Space Space and F F B E 65 67 nohrmdathe media other In . 4.7 ilmpota2-dim a out map will and H E and nfigure In . saconsequence a As . . steHdedual Hodge the is and 68 B H ic eare we Since . edpc the depict we B smr com- more is 69 E scnbe can as , . and D 70 and esee we B . F 64 H . 31 qain h oet oc a.A depicted As remaining figure law. in one force Lorentz is the equation, there electrodynamics equation In force Lorentz 9.3 plicated. ssatitd2-form twisted a ists 69: Figure aum h wse 2-form twisted The vacuum: 70: Figure D contain which equations Maxwell macroscopic deriva- exterior its and is tive closed not is it general ed(e) tecdstefields the encodes It (red). field eew tep oso that show to attempt we Here og ulo h nwse 2-form untwisted the of dual Hodge iesos hyhv h aeorientation. same the have They 4- dimensions. in form-submanifolds 2-dim orthogonal are Time Time and J 71 H gen.Ti seuvln otetwo the to equivalent is This (green). . hssae hti pctm the spacetime in that states This .

h osiuierltosi the in relations constitutive The h xiainﬁls hr ex- There ﬁelds: excitation The

Space Space Space Space Space Space H aldteexcitation the called H D rd sthe is (red) H and F and (blue). H In . F . osatsaesieof figures static the consider slice may We space Constant 9.4 2-form electromagnetic of the 4-velocity F and the particle both to the orthogonal is force iei otisifrainaotboth about information and contains it time this However form-submanifold. dimensional coor- time coordinates. one space with two diagram and dinate a is result The ie ewl alti ln the all plane for this plane call will this We onto time. back pulled look when fields and like space electromagnetic in the plane what a consider choose to is spacetime in spacetime. through slice dimensional Maxwell-Ampères law. plane the in loop a for duction evn h op hc steitga fthe of figure integral Similarly the force. is electromotive differ- which the loop, the is leaving which the bottom, in and ence top the at F orsodt h ulak sdsusdin discussed as pullback, section the to correspond . -eoiyo h particle. the of 4-velocity rhgnlt obt h 2-form the is force both The to (red). to arrow orthogonal the straight and the (green) is curve the force is particle the of world-line the (blue), form-submanifold 2-dim the h 2-form The natraiesieo awl’ equations Maxwell’s of slice alternative An oc a napril.Hr h 2-form the Here particle. a on law force 71: Figure scoe h ieec ewe h field the between difference the closed is E Figure . awl’ equations. Maxwell’s 8

ftefields the of , Time B natmtt rwteLorentz the draw to attempt An ed utcrepn ofield to correspond must field, 72 F hw aaa’ a fin- of law Faraday’s shows

sacoe nwse 1- untwisted closed a is Space F

Space Space , H and z z F .Since 0. = J plane. 0 = 63 n the and 73 na3- a on and F sthe is is 64 B 32 ilfrs nodrt riea figures at arrive to order in forms, tial dif- differen- of on concentrating proportion geometry, large ferential a presented have We Conclusion 10 osi lsa nti ae dal one ideally case, this paper. In dimensional 7 have plasma. would and a electrons of in flow the ions electromag- as well the as Aharonov–Bohm depict fields one netic the does on How light effect? shed to one best and can it potential, electromagnetism is the how the rep- draw example, pictorial For a from resentation. gain may which netism informa- different tion. or more give which natives ini hte o ie -om figure 2-form, given 2- a depict or for whether to is attempt tion section the dimensions, 4 in in is forms chal- One open some still lenges. are there geometry, tial electro- describe dynamics. that diagrams the are which n hrfr h ulakof pullback the therefore and h ubro ie evn h ie fthe of sides the equals leaving disc lines cylinder. top of the number leaving disc the bottom number the the entering and lines of number the z lelnsaecniuu,ie as i.e. continuous, are lines blue h lelnsaeplbc fteelectromag- the 2-form of netic pullback are lines blue The plane 72: Figure hr r aypeoeai electromag- in phenomena many are There ntrso itral rsnigdifferen- presenting pictorially of terms In 33 0 = scretadwehrteeaealter- are there whether and correct is z scoe.Tu h ieec between difference the Thus closed. is 0 = time nertdoe ieadspace. and time over integrated , aaa’ a fidcino the on induction of law Faraday’s F noteplane the onto F space noteplane the onto 4.7 z F 0 = n ques- One . sclosed is The . 67 - 32 71 [4] Enzo Tonti. Finite formulation of the electromagnetic field. Progress In Elec- tromagnetics Research, 32:1–44, 2001. + [5] Karl F Warnick, Richard H Selfridge, and David V Arnold. Electromagnetics − − made easy: differential forms as a teach- time ing tool. In Frontiers in Education Con- ference, 1996. FIE’96. 26th Annual Con- ference., Proceedings of, volume 3, pages space 1508–1512. IEEE, 1996.

Figure 73: Maxwell Ampères law on the [6] Charles W Misner and John A Wheeler. plane z = 0, integrated over time and space. Classical physics as geometry. Annals of The green dots are the intersections of the physics, 2(6):525–603, 1957. positive and negative charges as they cross z = 0. The red lines are the pullback of the [7] Friedrich W Hehl and Yuri N Obukhov. excitation 2-form H onto the plane z = 0. A gentle introduction to the foundations These terminate at charges. Thus the differ- of classical electrodynamics: The mean- ence between the number of lines entering the ing of the excitations (d, h) and the bottom disc and the number leaving the top field strengths (e, b). arXiv preprint and sides of the cylinder equals the charge in- physics/0005084, 2000. side the cylinder. [8] Jan Arnoldus Schouten. Tensor analy- Acknowledgments sis for physicists. Courier Corporation, 1954. I am grateful grateful for the support pro- vided by STFC (the Cockcroft Institute [9] Friedrich W Hehl and Yuri N Obukhov. ST/G008248/1 and ST/P002056/1), EPSRC Foundations of classical electrodynam- (the Alpha-X project EP/J018171/1 and ics: Charge, flux, and metric, vol- EP/N028694/1). I am grateful to the mas- ume 33. Springer Science & Business ters students who took my course Advanced Media, 2012. Electromagnetism and Gravity and 2017 and [10] Jonathan Gratus. Lancaster university read the document, making suggestions and internal report. 2017. to Taylor Boyd and Richard Dadhley who pointed out some inconsistencies. All figures are used with kind permission of Jonathan Gratus [10].

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