Basics of Tensor Calculus and General Relativity-Vectors and Introduction to Tensors (Part I: Vectors)

SOURCE FOR CONTENT: Neuenschwander D.E.,2015. Tensor Calculus for Physics. Johns Hopkins University Press. 

At some level, we all are aware of scalars and vectors, but typically we don’t think of aspects of everyday experience as being a scalar or a vector. A scalar is something that has only magnitude, that is it only has a numeric value. A typical example of a scalar would be temperature. A vector, on the other hand, is something that has both a magnitude and direction. This could be something as simple as displacement. If we wish to move a certain distance with respect to our current location we must specify how far to go and in which direction to move. Other examples, (and there are a lot of them), including velocity, force, momentum, etc. Now, tensors are something else entirely. In Neuenschwander’s “Tensor Calculus for Physics”, he recites the rather unsatisfying definition of a tensor as being

” ‘A set of quantities T_{s}^{r} associated with a point P are said to be components of a second-order tensor if, under a change of coordinates, from a set of coordinates x^{s} to x^{\prime s}, they transform according to

\displaystyle T^{\prime r}_{s}=\frac{\partial x^{\prime r}}{\partial x^{i}}\frac{\partial x^{j}}{\partial x^{\prime s}}T_{j}^{i}. (1)

where the derivatives are evaluated at the aforementioned point.’ “

Neuenschwander describes his frustration when encountered this so-called definition of a tensor. Like him, I found I had similar frustrations, and as a result, I had even more questions.

We shall start with a discussion of vectors, for an understanding of these quantities are an integral part of tensor analysis.

We define a vector as a quantity that has 3 distinct components and an angle that indicates orientation or direction. There are two types of vectors; those with coordinates and those without. I will be discussing the latter first, but I will consider Neuenschwander’s description in the context of the definition of a vector space. Then I shall move on to the former. Consider a number of arbitrary vectors \displaystyle \textbf{U}, \textbf{V}, \textbf{W},etc. If we consider further more and more vectors, we can therefore imagine a space constituted by these vectors; a vector space. To make it formal, here is the definition:

Def. Vector Space:

A vector space \mathcal{S} is the nonempty set of elements (vectors) that satisfy the following axioms

\displaystyle 1.\textbf{U}+\textbf{V}=\textbf{V}+\textbf{U};  \forall \textbf{U},\textbf{V}\in \mathcal{S},

\displaystyle 2. (\textbf{U}+\textbf{V})+\textbf{W}= \textbf{U}+(\textbf{V}+\textbf{W}); \forall \textbf{U},\textbf{V},\textbf{W}\in \mathcal{S},

\displaystyle 3. \exists  0 \in \mathcal{S}|\textbf{U}+0=\textbf{U},

\displaystyle 4. \forall \textbf{U}\in \mathcal{S}, \exists -\textbf{U}\in \mathcal{S}|\textbf{U}+(-\textbf{U})=0,

\displaystyle 5. \alpha(\textbf{U}+\textbf{V})=\alpha\textbf{U}+\alpha\textbf{V}, \forall \alpha \in \mathbb{R},  \forall \textbf{U},\textbf{V}\in \mathcal{S}.

\displaystyle 6. (\alpha+\beta) \textbf{U}= \alpha\textbf{U}+\beta\textbf{U}, \forall \alpha,\beta \in \mathbb{R},

\displaystyle 7. (\alpha\beta)\textbf{U}= \alpha(\beta\textbf{U}), \forall \alpha,\beta \in \mathbb{R},

\displaystyle 8. 1\textbf{U}=\textbf{U}, \forall \textbf{U}\in \mathcal{S},

and satisfies the following closure properties:

 1. If \textbf{U}\in \mathcal{S}, and \alpha is a scalar, then \alpha\textbf{U}\in \mathcal{S}.

2. If \textbf{U},\textbf{V}\in \mathcal{S}, then \textbf{U}+\textbf{V}\in \mathcal{S}.

The first closure property ensures closure under scalar multiplication while the second ensures closure under addition.

In rectangular coordinates, our arbitrary vector may be represented by basis vectors \hat{i}, \hat{j},\hat{k} in the following manner

\displaystyle \textbf{U}=u\hat{i}+v\hat{j}+w\hat{k}, (1)

where the basis vectors have the properties

\displaystyle \hat{i}\cdot \hat{i}=\hat{j}\cdot \hat{j}=\hat{k}\cdot \hat{k}=1, (2.1)

\displaystyle \hat{i}\cdot \hat{j}=\hat{j}\cdot \hat{k}=\hat{i}\cdot \hat{k}=0.(2.2)

The latter of which implies that these basis vectors are mutually orthogonal. We can therefore write these in a more succinct way via

\displaystyle \hat{e}_{i}\cdot \hat{e}_{j}=\delta_{ij},  (2.3)

where \displaystyle \delta_{ij} denotes the Kronecker delta:

\displaystyle \delta_{ij}=\begin{cases} 1, i=j\\ 0, i \neq j \end{cases}. (2.4)

We may redefine the scalar product by the following argument given in Neuenschwander

\displaystyle \textbf{U}\cdot  \textbf{V}=\sum_{i,j=1}^{3}(U^{i} \hat{e}_{i}) \cdot (V^{j} \hat{e}_{j})=\sum_{i,j=1}^{3}U^{i}V^{j}(\delta_{ij})=\sum_{l=1}^{3}U^{l}V^{l}. (3)

Similarly we may define the cross product to be

\displaystyle \textbf{U}\times \textbf{V}=\det\begin{pmatrix} \hat{i}  & \hat{j} & \hat{k} \\ U^{x} & U^{y} & U^{z} \\ V^{x} & V^{y} & V^{z}  \end{pmatrix}, (4)

whose i-th component is

\displaystyle (\textbf{U}\times \textbf{V})^{i}=\textbf{U}\times \textbf{V}=\sum_{i,j=1}^{3}\epsilon^{ijk}U^{j}V^{k}, (5)

where \epsilon^{ijk} denotes the Levi-Civita symbol. If these indices form an odd permutation \epsilon^{ijk}=-1, if the indices form an even permutation \epsilon^{ijk}=+1, and if any of these indices are equal \epsilon^{ijk}=0.

As a final point, we may relate vectors to relativity by means of defining the four-vector. If we consider the four coordinates x,y,z,t, they collectively describe what is referred to as an event. Formally, an event in spacetime is described by three spatial coordinates and one time coordinate. We may replace these coordinates by x^{\mu}, where \mu \in \mathbb{Z}^{\pm} in which I am defining \mathbb{Z}^{\pm}=\bigg\{x\in \mathbb{Z}|x\geq 0\bigg\}. In words, this means that the index \mu is an integer that is greater than or equal to 0.

Furthermore, the quantity x^{0} corresponds to time, x^{1,2,3} correspond to the x,y, and z coordinates respectively.  Therefore, x^{\mu}=(ct,x,y,z) is called the four-position.

In the next post, I will complete the discussion on vectors and discuss in more detail the definition of a tensor (following Neuenschwander’s approach). I will also introduce a few examples of tensors that physics students will typically encounter.






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