Appendix A Introduction to Tensors This appendix focuses on tensors. There is a deep enough relationship between differential forms and tensors that covering this material is important. However, this material is also distinct enough from differential forms that relegating it to an appendix is appropriate. Indeed, If we were to cover the material on tensors in the same spirit and with the same diligence as we covered differential forms this book would be twice the current size. Therefore this appendix is more of a quick overview of the essentials, though hopefully with enough detail that the relationship between differential forms and tensors is clear. But there is also another reason for including this appendix, our strong desire to include a full proof of the global formula for exterior differentiation given at the end of Sect. A.7. Strangely enough, the full global exterior derivative formula for differential forms uses, in its proof, the global Lie derivative formula for differential forms. The material presented in this appendix is necessary for understanding that proof. In section one we give a brief big-picture overview of tensors and why they play such an important role in mathematics and physics. After that we define tensors using the strategy of multilinear maps and then use this, along with what we already know, to introduce the coordinate transformation formulas. Section two looks at rank-one tensors, section three at rank-two tensors, and section four looks at general tensors. Section five introduces differential forms as a special kind of tensor and then section six gives the real definition of a metric tensor. We have already discussed metrics in an imprecise manor several times in this book. Finally, section seven introduces the Lie derivative, one of the fundamental ways of taking derivatives on manifolds, and then goes on to prove many important identities and formulas that are related to Lie derivatives and their relationship to exterior derivatives. With this material we are finally able to give a rigorous proof of the global formula for exterior differentiation. A.1 An Overview of Tensors This book, as the title clearly indicates, is a book about differential forms and calculus on manifolds. In particular, it is a book meant to help you develop a good geometrical understanding of, and intuition for, differential forms. Thus, we have chosen one particular pedagogical path among several possible pedagogical paths, what we believe is the most geometrically intuitive pedagogical path, to introduce the concepts of differential forms and basic ideas regarding calculus on manifolds. Now that we have developed some understanding of differential forms we will use the concept of differential forms to introduce tensors. It is, however, perfectly possible, and in fact very often done, to introduce tensors first and then introduce differential forms as a special case of tensors. In particular, differential forms are skew-symmetric covariant tensors. The word anti-symmetric is often used instead of skew-symmetric. Using this definition, formulas for the wedgeproduct, exterior derivatives, and all the rest can be developed. The problem with this approach is that students are often overwhelmed by the flurry of indices and absolutely mystified by what it all actually means geometrically. The topic quickly becomes simply a confusing mess of difficult to understand formulas and equations. Much of this book has been an attempt to provide an easier to understand, and more geometrically based, introduction to differential forms. In this appendix we will continue in this spirit as much as is reasonably possible. In this chapter we will define tensors and introduce the tensorial “index” notation that you will likely see in future physics or differential geometry classes. Our hope is that when you see tensors again you will be better positioned to understand them. Broadly speaking, there are three generally different strategies that are used to introduce tensors. They are as follows: © Springer Nature Switzerland AG 2018 395 J. P. Fortney, A Visual Introduction to Differential Forms and Calculus on Manifolds, https://doi.org/10.1007/978-3-319-96992-3 396 A Introduction to Tensors 1. Transformation rules are given. Here, the definition of tensors essentially boils down to the rather trite phrase that “tensors are objects that transform like tensors.” This strategy is certainly the one that is most often used in undergraduate physics classes. Its advantage is that it allows students to hit the ground running with computations. Its disadvantage is that there is generally little or no deeper understanding of what tensors are and it often leaves students feeling a bit mystified or confused. 2. As multi-linear maps. Here, tensors are introduced as a certain kind of multilinear map. This strategy is often used in more theoretical physics classes or in undergraduate math classes. This is also the strategy that we will employ in this chapter. In particular, we will make a concerted effort to link this approach to the transformation rules approach mentioned above. This way we can gain both a more abstract understanding of what tensors are as well as some facility with computations. 3. An abstract mathematical approach. This approach is generally used in theoretical mathematics and is quite beyond the scope of this book, though it is hinted at. A great deal of heavy mathematical machinery has too be well understood for this approach to introducing tensors. It is natural to ask why we should care about tensors and what they are good for. What is so important about them? The fundamental idea is that the laws of physics should not (and in fact do not) depend on the coordinate system that is used to write down the mathematical equations that describe those laws. For example, the gravitational interaction of two particles simply is what it is and does not depend on if we use cartesian coordinates or spherical coordinates or cylindrical coordinates or some other system of coordinates to describe the gravitational interaction of these two particles. What physically happens to these two particles out there in the real world is totally independent of the coordinate system we use to describe what happens. It is also independent of where we decide to place the origin of our coordinate system or the orientation of that coordinate system. This is absolutely fundamental to our idea of how the universe behaves. The equations we write down for a particular coordinate system maybe be complicated or simple, but once we solve them the actual behavior those equations describe is always the same. So, we are interested in finding ways to write down the equations for physical laws that 1. allow us to do computations, 2. enable us to switch between coordinate systems easily, 3. and are guaranteed to be independent of the coordinate system used. This means the actual results we get do not depend on, that is, are independent of, the coordinate system we used in doing the calculations. Actually, this is quite a tall order. Mathematicians like to write things down in coordinate-invariant ways, that is, using notations and ideas that are totally independent of any coordinate system that may be used. In general this makes proving things much easier but makes doing specific computations, like computing the time evolution of two particular gravitationally interacting particles, impossible. In order to do any sort of specific computations related to a particular situation requires one to define a coordinate system first. It turns out that tensors are exactly what is needed. While it is true that they can be quite complicated, there is a lot of power packed into them. They ensure that all three of the above requirements are met. It is, in fact, this ability to switch, or transform, between coordinate systems easily that gives the physics “definition” of tensors as “objects that transform like tensors.” A tensor, when transformed to another coordinate system, is still a tensor. Relationships that hold between tensors in one coordinate system automatically hold in all other coordinate systems because the transformations work the same way on all tensors. This is not meant to be an exhaustive or rigorous introduction to tensors. We are interested in understanding the big picture and the relationship between tensors and differential forms. We will employ the Einstein summation notation throughout, which means we must be very careful about upper and lower indices. There is a reason Elie Cartan, who did a great deal of work in mathematical physics and differential geometry, said that the tensor calculus is the “debauch of indices.” A.2 Rank One Tensors A tensor on a manifold is a multilinear map ∗ ∗ T : T M ×···× T M × TM×···× TM −→ R r s contravariant covariant degree degree T α1,...,αr ,v1,...,vs −→ R. A Introduction to Tensors 397 That is, the map T eats r one-forms and s vectors and produces a real number. We will discuss the multilinearity part later so will leave it for now. The number r, which is the number of one-forms that the tensor T eats, is called the contravariant degree and the number s, which is the number of vectors the tensor T eats, is called the covariant degree. We will also discuss this terminology later. The tensor T would be called a rank-r contravariant, rank-s covariant tensor, or a rank (r, s)- tensor for short. Before delving deeper into general tensors let us take a more detailed look at two specific tensors to see how they work and to start get a handle on the notation. Rank (0, 1)-Tensors (Rank-One Covariant Tensors) A (0, 1)-tensor, or a rank-one covariant tensor, is a linear mapping T : TM −→ R.