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A PEDAGOGICAL INVESTIGATION OF THE DEVELOPMENT OF USING DIFFERENTIAL FORMS

by

BENJAMIN DAVID SABREE

Submitted in partial fulfillment of the requirements for the degree of

Master of Science

Thesis Advisors: Prof. David A. Singer (CWRU )

and Prof. Kenneth L. Kowalski (CWRU )

Department of Mathematics

CASE WESTERN RESERVE UNIVERSITY

August 2008 CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the thesis/dissertation of

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candidate for the ______degree *.

(signed)______(chair of the committee)

______

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(date) ______

*We also certify that written approval has been obtained for any proprietary material contained therein. Contents

Acknowledgments iv

Abstract v

1 Introduction 1

2 Some General Advantages of Differential Forms 5

3 Differential Forms in General Relativity 12

4 Conclusions 20

References 23

iii Acknowledgments

There are many people without whom this thesis would not have been possible. I would like to extend my sincere gratitude to my department chair, Professor James

Alexander, my primary advisor, Professor David Singer, and my secondary advisor,

Professor Kenneth Kowalksi, for serving on my thesis committee. I especially appre- ciate the assistance I received from Professors Kowalski and Singer while researching and writing this thesis. I would like to thank Professor Thomas Ivey for his gener- ous input on the advantages of differential forms. Thanks is also due to Professor

Kowalski for helping greatly with my search for a thesis topic.

I owe a debt of gratitude to my parents, Anna and Steve Sabree, for their constant and encouragement. Without it, I would not be where I am today.

Most of all, I would like to extend a heartfelt thanks to my greatest source of unwavering support, my wife Emily.

iv A Pedagogical Investigation of the Development of General

Relativity Using Differential Forms

Abstract

by

Benjamin David Sabree

General relativity is widely applicable to many of physics research.

Some ‘math first’ treatments of the subject employ differential forms while others do not. This paper advocates those approaches that utilize differential forms by

first outlining some of the general mathematical advantages of forms and then by comparing developments of selected topics from the different treatments.

v 1 Introduction

General relativity, the modern theory of , was developed by Albert Ein- stein near the end of 1915. The theory attributes gravity to the effects of a curved four-dimensional , and it does so in the language of pseudo-Riemannian differential geometry. Exceptionally well-supported by numerous experiments and observations of the last century, general relativity is now understood to have conse- quences across a range of very diverse fields of physics. Some of the most obvious and immediate implications of the theory led to the beginning of modern cosmology, while in the last half of the twentieth century relativity was tied to the study of par- ticle physics. Despite having gone through an unfashionable period for a number of decades after its formulation, general relativity is now at the forefront of a great deal of current scientific research and experimentation.

The teaching of general relativity has a somewhat troubled history. For many years after it was first published, several factors contributed to the consideration of relativity as being on the fringes of mainstream physics study. Initially, relativity was perceived as being a mathematically difficult theory and its brilliant creator was still active in the field. Additionally, the theory was almost immediately experimentally reinforced as far as the limits of contemporary observational precision would allow, most notably by its agreement with the documented advance of the perihelion of

1 Mercury and its successful prediction of the aberration of starlight by the sun [1].

Due to the limited accuracy provided by instruments, extreme circumstances were required for relativistic effects to be detectable. As a consequence, the theory was seen as having very few applications. Being outside of the normal body of physics research, there was little need to teach relativity.

During the latter half of the twentieth century, technological advances brought general relativity back into the fold of mainstream physics. The increased accuracy of astronomical observations along with advances in experimental precision began to reveal a wealth of phenomena which exhibited general relativistic effects [2, 3, 4, 5, 6].

It was also during the late twentieth century that general relativity was first tied into the study of other fields, most notably particle physics [7].

This newfound prominence of general relativity in theoretical and experimental physics demands its adoption into the standard physics curriculum. However, due to the fact that the theory has only been actively taught for a relatively short period of time, the pedagogy is quite underdeveloped and no method has yet to achieve a

‘tried and true’ status.

When developing the subject of general relativity for introduction to advanced undergraduate and beginning graduate students, there are two standard approaches.

The deductive approach involves first developing the requisite mathematical machin-

2 ery, then motivating and deriving the field equations, and finally exploring solutions

to the field equations and considering modern topics. This approach is quite natu-

rally referred to as ‘math first’. There are many texts that follow this approach, not

the least of which is the exceptional and well-respected tome by Misner, Thorne, and

Wheeler, Gravitation [8]. A more appropriate and bite-sized treatment can be found

in Schutz’s A First Course in General Relativity, which is the primary text I used for this paper [9].

The second approach, aptly called ‘physics first’, involves beginning with solutions to the field equations and exploring their consequences and interesting phenomena, with the actual derivation of the field equations and the associated mathematics rel- egated to near-appendix status. Champions of the ‘physics first’ approach claim that time constraints and other limitations, such as student mathematical background, more often than not preclude the use of a ‘math first’ development [10]. The primary text leading this charge is Hartle’s Gravity, which quite nicely accomplishes its goal of motivating the study of relativity by immediately examining fascinating phenomena while evading much of its mathematical foundation [11].

Despite the fact that in some cases, time constraints or student interests may necessitate a ‘physics first’ approach, every teaching situation must be assessed and dealt with on an individual . Therefore, it is best to avoid discounting a ‘math

3 first’ approach unless the only offerings of general relativity are to be survey courses

where students possess no interest in, nor necessity of, pursuing the subject beyond

an introductory overview.

Among those developments which subscribe to the ‘math first’ approach, differ-

ent techniques can be observed. Classically, the subject is developed from the idea

of vectors and vector fields directly into the land of , in which the story of

relativity is set [12, 13, 14, 7]. In this case, the difficulty of the mathematics is exac-

erbated by a failure to use more modern mathematical techniques. Using the more

contemporary concept of differential forms allows for discussions not necessarily tied

to specific coordinates, which in turn simplifies many of the arguments along the road

to deriving the field equations [8, 9, 15]. Because the complexity of the mathematics

required is one of the major factors contributing to adoption of the ‘physics first’ ap-

proach, perhaps the benefits gained by initially investing time in the understanding

of differential forms will result in the deductive, more thorough treatment of general

relativity being seen as a more palatable approach.

Differential forms were first developed by Elie´ Cartan near the turn of the 20th century. Cartan’s innovation was to combine the one-forms of Pfaff with the of Grassman in an attempt to solve systems of differential equations arising from the study of thermodynamics [16]. While the utility of differential forms initially

4 went unrecognized for many years, they have since found their way into the study of subjects ranging from and geometric optics to quantum mechanics and [17]. The importance of differential forms to general relativity is perhaps best reflected by their role in Gravitation [8].

2 Some General Advantages of Differential Forms

The most obvious, and perhaps most trivial, advantage of using differential forms is economy of notation. The gain of differential forms over vector notation is similar to, but not quite so pronounced as, that achieved by vector notation over sets of equations. For instance, using the vector notation developed by Gibbs, Heaviside was able to express the twenty scalar equations of Maxwell as the four vector equations commonly known today [18, 19, 20]. One example of the economy of differential forms is seen in Stokes’ theorem. Classically, Stokes’ theorem is expressed in vector notation as: Z ZZ F · dr = F · dS , (1) C S where S must be oriented, piecewise-smooth, and bounded by C, a simple, piecewise- smooth, closed curve with positive orientation, and F is a vector field whose compo- nents have continuous partial on an open region containing S.

5 Alternatively, expressed in the language of differential forms, Stokes’ theorem is given as: Z I dω˜ = ω , (2) M ∂M where M is an oriented, piecewise-smooth of n, ∂M is its bound- ary, and ω is a compactly supported continuously differentiable n − 1 form on M.

Just as economy of notation is not the only advantage of a vector formulation, neither is it the only advantage of a differential forms formulation. This expression of Stokes’ theorem in differential forms notation is a more generalized theorem which usurps the theorem as well.

In addition to providing an economy of notation, differential forms are often more natural to work with than other objects. For example, consider the following:

Let σ be a curve, M and N be , TM and TN be their

(at some points p and f(p), respectively), and f be a map from M to N:

R σ  / M f N

TpM Tf(p)N

0 The σ = vp is a vector which lives in TpM. If we compose the map f with

6 0 σ and consider the derivative (f ◦ σ) , we have a vector living in Tf(p)N. We can

0 then define the map f∗ by f∗(vp) ≡ (f ◦ σ) , which takes vectors in TpM to vectors in

Tf(p)N:

R JJ JJ f◦σ σ JJ JJ JJ  J% / M f N

f∗ TpM / Tf(p)N

The map f∗ can push any vector forward, but it will not take vector fields in TM to vector fields in TN unless f is a diffeomorphism.

On the other hand, if ω is a one-form defined everywhere in the , we have:

R JJ JJ f◦σ σ JJ JJ JJ  J% / M f N

f∗ TpM / Tf(p)N

ω  R

7 ∗ Here we can define f ω ≡ ω ◦ f∗ to be the of ω:

R JJ JJ f◦σ σ JJ JJ JJ  J% / M f N

f∗ TpM / Tf(p)N I II II ω ∗ II f ω II I$  R

So we can always pull a differential form back, but we must have a diffeomorphism to push vector fields forward.

Another situation for which using differential forms is more natural arises when considering the . Although the gradient is introduced to multi-variable cal- culus students as a vector, it is actually a one-form.

Take a scalar field φ(t, x, y, z) defined at every spacetime point along the of some observer. Parameterizing the world line by the value of τ at each on the path gives us the curve

φ(τ) = φ [t(τ), x(τ), y(τ), z(τ)] . (3)

The four-velocity of our observer along the world line is given by

dt dx dy dz U~ = ~e + ~e + ~e + ~e . (4) dτ t dτ x dτ y dτ z 8 Examining the rate of change of φ along our curve, we find

dφ ∂φ dt ∂φ dx ∂φ dy ∂φ dz = + + + dτ ∂t dτ ∂x dτ ∂y dτ ∂z dτ (5) ∂φ ∂φ ∂φ ∂φ = U t + U x + U y + U z . ∂t ∂x ∂y ∂z

dφ ~ Notice that dτ , a number, is a of U, so we have defined a one-form,

the gradient:

dφe : TM −→ R . (6)

The usual definition of a gradient vector is the vector which points in the direction of greatest change per unit length. However, without a on our manifold, comparing lengths of vectors pointing in different directions is impossible. Because a metric is not required to describe the gradient one-form, it is a more natural object than the gradient vector [21, 9].

Similarly, a normal one-form is also a more natural object than a normal vector.

A normal vector is a vector orthogonal to some at a point, meaning it is orthogonal to all vectors tangent to the surface at that point. To define this we require a scalar product, which in turn requires a metric. On the other hand, defining a normal one-form to the surface does not require a metric. A normal one-form to a surface is simply a one-form whose value when contracted with every vector tangent to the surface at that point is zero [9].

9 In particular, for a surface defined as the solution to

φ(t, x, y, z) = constant , (7)

we see that the gradient, dφ˜ , is a normal one-form. We can therefore picture dφ˜ as a

set of surfaces of constant φ, which can be quite useful intuitively [9].

Finding transformation rules for objects is yet another situation in which differ-

ential forms can be quite valuable. Classically, vectors are defined as objects that

transform under an arbitrary coordinate transformation in the same way as displace- −→ ments transform. That is, a vector ∆r can be represented as a displacement: in

∆x  ∆r Cartesian coordinates as ∆y , in polar coordinates as ( ∆θ ) , or in arbitrary coor-

∆ξ  ∆x  dinates as ∆η . Then for small ∆y , we can transform to arbitrary coordinates

according to

      ∆ξ ∂ξ ∂ξ ∆x    ∂x ∂y      =     . (8)    ∂η ∂η    ∆η ∂x ∂y ∆y

We can then define the transformation for an arbitrary vector V~ as

0α α β V = Λ β V , (9)

10 where

  ∂ξ ∂ξ α  ∂x ∂y  Λ β =   . (10)  ∂η ∂η  ∂x ∂y

−→ ∆x  Here we have defined a single vector, ∆r = ∆y , and transformations of others have followed by analogy [9].

Consider instead the approach that can be taken using one-forms. Given a scalar

∂φ ∂φ field φ and a (ξ, η), we can find the derivatives ∂ξ and ∂η . We can

˜ ∂φ ∂φ then define the one-form dφ to be the object whose components are ( ∂ξ ∂η ) in our coordinate system. This gives us an entire class of one-forms, one for each different scalar field. The transformation of components follows from the

∂φ ∂φ ∂x ∂φ ∂y = + , (11) ∂ξ ∂x ∂ξ ∂y ∂ξ

so that

      ∂x ∂x  ∂ξ ∂η  ∂φ ∂φ = ∂φ ∂φ   . (12) ∂ξ ∂η ∂x ∂y  ∂y ∂y  ∂ξ ∂η

By first defining a class of one-forms, the transformation properties follow automati- cally from the chain rule rather than by analogy, as they did with vectors. Now using

11 the of vectors and one-forms, a vector can be defined as a linear function of

one-forms into R. That is, dφ = hdφ,˜ V~ i , (13) dτ

~ dξ dη where V , with components dτ and dτ , depends only on the curve on which φ is defined,

while dφ˜ depends only on φ. With our parametrization τ unaffected by a change of

coordinates, we see that V~ transforms according to

      ∂ξ ∂ξ ∂ξ ∂x  ∂τ   ∂x ∂y   ∂τ    =     , (14)  ∂η   ∂η ∂η   ∂y  ∂τ ∂x ∂y ∂τ

the same transformation law we derived earlier in (8), but which we have now found

by using more satisfying, if slightly lengthier, methods [9].

3 Differential Forms in General Relativity

After initially establishing the concept of differential forms, the first major departure

between the different ‘math first’ developments of general relativity appears with the

introduction of tensors. When tensors are introduced classically, they are typically

defined as “ things that transform like tensors,” meaning that tensors are objects

whose components have indices which obey certain transformation rules [12,

12 13, 14, 7]. It would be difficult to defend a claim that this definition provides any kind of clear understanding of tensors, much less any particularly useful machinery for further development of the subject.

Using knowledge of differential forms, however, the concept of a tensor can be

m explained in a much more illuminating fashion. The definition of an ( n ) tensor as a multi-linear function of m one-forms and n vectors into R gives a much clearer understanding of the nature of tensors. The subtle note that this definition makes no mention of the components of these m one-forms or n vectors places us in a position to use tensors as powerful tools rather than simply information holders [8, 9, 15].

This increased understanding of tensors provides what is perhaps the strongest support for a differential forms treatment over a more classical treatment of rela- tivity. Following a classical development, objects can only be verified as tensors by confirming that they obey certain transformation rules after they have been derived.

Alternatively, by using the idea that a tensor is a function on one-forms and vec- tors which is independent of their components (and therefore independent of any coordinate frame), general relationships can be established from specific cases.

For example, knowledge about vectors and one-forms in Cartesian coordinates can be used to derive a general relationship between the and covariant derivatives. In Cartesian coordinates, the components of a one-form Ve and its related

13 vector V~ are equal, so, because ∇ is just differentiation of components in Cartesian

coordinates, the components of the covariant derivatives of the one-form and vector

must also be equal. That is, if

Ve = g(V,~ ) (15) then

~ ∇βVe = g(∇βV, ) , (16) or, in component notation,

ν Vµ;β = gµν V ;β . (17)

But covariant differentiation of Ve yields

ν ν Vµ;β = gµν;β V + gµν V ;β , (18) so we must have that

gµν;β = 0 . (19)

Because (17) and (18) are tensor equations, we see that (19) must be valid in all coordinate frames [9].

Classically, the same result is obtained by differentiating the metric to find

∂g µν = Γκ g + Γκ g . (20) ∂xλ µλ κν νλ κµ

0 By then using the rule for covariant differentiation of a ( 2 ) tensor, we find that

∂g g = µν − Γγ g − Γγ g . (21) µν;λ ∂xλ µλ γν νλ γµ 14 ∂gµν Substituting (20) into (21) for ∂xλ yields

gµν;λ = 0 ,

the same result obtained in (19), although the former derivation was considerably

more intuitive [7].

Knowledge of differential forms also helps to make the role of the metric tensor

0 g much more transparent. As a ( 2 ) tensor which defines the scalar product between two vectors,

g[A,~ B~ ] = A~ · B,~ (22)

if given a single vector, the metric tensor becomes a one-form,

g[A,~ ] , (23)

a linear function of vectors into R. This verifies why we use the metric tensor to lower

indices on tensors:

β Vα = gαβ V , (24)

0 because we see that giving one vector to the metric tensor yields a ( 1 ) tensor, or

a one-form, which can then be contracted with another vector to obtain a number.

−1 2 Similarly, we see that the inverse metric g , a ( 0 ) tensor defined by

αβ α g gβλ = δλ , (25)

15 takes a single one-form to a vector:

α αβ V = g Vβ , (26)

and can therefore be used to raise indices [9].

This idea of giving a single vector to the metric tensor and obtaining a one-form

leads to an intuitively helpful way of representing inertial frames in Minkowski .

In , the four-velocity of a momentarily comoving reference frame for

some particle is given by 1   1        0    U~ =   . (27)    0        0

Feeding this four-velocity to the metric tensor yields the corresponding one-form U˜ with components

β Uα = gαβ U . (28)

Specifically, we find that   U˜ = −1 0 0 0 , (29)

1Here we use geometrized units where c = 1 and adopt the East Coast for the . That is, (ds)2 = −(dt)2 + (dx)2 + (dy)2 + (dz)2.

16 which is just −dt˜ . So for a particle in flat space, we can picture U˜ as a set of surfaces

of constant t. These are the surfaces of simultaneity for the particle and they define

its frame [9].

For further examples of the utility of differential forms in developing general rel-

γ ativity, we look to the affine , or Christoffel symbols. The symbols, Γαβ, provide a rule for of vectors in the pseudo- of general relativity. They can be defined by considering the derivative of a vector V~ , given by ∂V~ ∂V α ∂~e = ~e + V α α , (30) ∂xβ ∂xβ α ∂xβ where ∂~e α ≡ Γµ ~e , (31) ∂xβ αβ µ

∂~eα µ noting that ∂xβ can be written as a of basis vectors with Γαβ as

µ ∂~eα the coefficients. Under this interpretation, Γαβ is the µ-th component of ∂xβ . That is,

α denotes the basis vector being differentiated, β denotes the coordinate with respect

to which it is being differentiated, and µ denotes the component of the resulting

tensor [9].

λ Let us examine the justification of Γµν not being a tensor. Classically, the argu-

ment goes as follows:

17 Consider the tensor ∂V V = µ − Γλ V . (32) µ;ν ∂xν µν λ

For this to indeed be a tensor, it must transform like a tensor, so we must have that

α β 0 ∂x ∂x Vµ;ν = 0µ 0ν Vα;β ∂x ∂x (33) ∂V 0 = µ − Γ0λ V 0 . ∂x0ν µν λ Substituting (32) into the first part of (33), we find

∂xα ∂xβ ∂V  V 0 = α − Γγ V µ;ν ∂x0µ ∂x0ν ∂xβ αβ γ (34) ∂xα ∂xβ ∂V ∂xα ∂xβ = α − Γγ V . ∂x0µ ∂x0ν ∂xβ ∂x0µ ∂x0ν αβ γ

Because Vν is a tensor, we know that it transforms according to

∂xν V 0 = V . (35) µ ∂x0µ ν

Differentiating with respect to x0λ gives us

∂V 0 ∂2xν ∂xν ∂xα ∂V µ = V + ν . (36) ∂x0λ ∂x0µ ∂x0λ ν ∂x0µ ∂x0λ ∂xα

We can substitute this into (34) to obtain

∂V 0 ∂2xα ∂xα ∂xβ V 0 = µ − V − Γγ V . (37) µ;ν ∂x0ν ∂x0µ ∂x0ν α ∂x0µ ∂x0ν αβ γ

Using the second part of (33) gives

0 0 2 α α β ∂V 0 ∂V ∂ x ∂x ∂x µ − Γ λ V 0 = µ − V − Γγ V , (38) ∂x0ν µν λ ∂x0ν ∂x0µ ∂x0ν α ∂x0µ ∂x0ν αβ γ 18 so we see that 2 α α β 0 ∂ x ∂x ∂x Γ λ V 0 = V + Γγ V . (39) µν λ ∂x0µ ∂x0ν α ∂x0µ ∂x0ν αβ γ

Substituting (35) into (39) gives us

κ 2 α α β 0 ∂x ∂ x ∂x ∂x Γ λ V = V + Γγ V . (40) µν ∂x0λ κ ∂x0µ ∂x0ν α ∂x0µ ∂x0ν αβ γ

∂x0λ Multiplying everything in sight by the appropriate ∂xγ yields

2 α 0λ α β 0λ 0 ∂ x ∂x ∂x ∂x ∂x Γ λ V = V + Γγ V , (41) µν κ ∂x0µ ∂x0ν ∂xα α ∂x0µ ∂x0ν ∂xγ αβ γ

γ so we find the transformation law for Γαβ:

2 α 0λ α β 0λ 0 ∂ x ∂x ∂x ∂x ∂x Γ λ = + Γγ , (42) µν ∂x0µ ∂x0ν ∂xα ∂x0µ ∂x0ν ∂xγ αβ

γ and we see that Γαβ is not a tensor because of the double derivative term. The fact

that the Christoffel symbols fail to transform like a tensor is the only indication we

have that they are not a tensor [12].

Using our better understanding of tensors, the justification is quite a bit more

µ intuitive. Recall equation (31), repeated here, and our interpretation of Γαβ:

∂~e α ≡ Γµ ~e . ∂xβ αβ µ

1 Because ~eα is a set of vectors, ∇~eα is a set of ( 1 ) tensors whose components are

µ Γαβ. Changing µ or β only changes the component under discussion, but changing α

19 changes the set of tensors ∇~eα. A change of coordinates changes the the basis ~eα to

~eλ, so ∇~eλ is a new set of tensors, not different components of the same set of tensors.

µ µ Therefore Γαβ is not a tensor (or the components of a tensor), and so Γαβ in one frame

µ0 is not obtained by a tensor transformation of Γα0β0 in another frame. For instance, in

µ Cartesian coordinates, Γαβ = 0, but this is not true in polar coordinates [9].

Beginning with the derivation of the Riemann and Ricci tensors, the different

‘math first’ treatments converge and are nearly identical in their application of the

Bianchi identities leading to the derivation of the Einstein field equations.

4 Conclusions

We have seen that the use of differential forms can be of great advantage when

following a deductive, ‘math first’ development of general relativity. In addition to

often being more natural to work with, differential forms can provide clearer pictures

of physical situations and help guide us to better insights. The primary benefit of

using differential forms in general relativity, however, is in establishing a much deeper

understanding of the nature of tensors. Being that general relativity is a tensor theory,

this benefit is a significant one. With this better understanding of tensors, much of

the development of the subject follows considerably more smoothly than it would

20 otherwise.

Developing general relativity using differential forms does entail additional math-

ematical overhead in order to introduce the concepts and machinery needed. If the

only gain from this added overhead was a conceptual one, leading to better intuition

with the subject, it would be difficult to argue that this development is always the best practice. Similarly, the same argument would be unconvincing if the only gain was utilitarian, leading to much simpler derivations of some of the necessary mate- rial. However, considering that using differential forms in the treatment of general relativity is not only conceptually and intuitively helpful, but also utilitarian, it is difficult to argue that the more classical approach has any place whatsoever in the subject’s development.

Biased as I may be toward a solid mathematical foundation, I believe that the im- proved intuition and greater understanding of mathematical concepts underlying the theory make the differential forms approach to teaching general relativity an excellent choice. Perhaps the optimal course of action would be similar to that espoused by

Moore, in which the presentation of the mathematical underpinnings of the theory is interwoven with interesting physical consequences throughout [22]. By following this strategy using differential forms, not only would students be given a very clear and precise mathematical foundation for the theory, they would also be kept motivated

21 throughout their study by being able to examine some of the fascinating phenomena which general relativity can explain.

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