## Tensor Calculus II

In the last post I described a contravariant tensor of rank 1 and a covariant tensor of rank 1. In this post we will consider generalizations of these. We will introduce tensors of arbitrary rank $(k,l)$ where $k$ is the number of contravariant indices and $l$ is the number of covariant indices. How many numbers does such a tensor represent? It is easy to see that if the tensor is defined in $n$ dimensional space, it defines $n^{(k+l)}$ real numbers for each point of space, and each coordinate system. The notation for such a tensor is

$A^{i_1 i_2 \dots i_k}_{j_1 j_2 \dots j_l}$

Now before we go ahead, I would like to clarify that the above represents just 1 single component of the tensor , out of the $n^{(k+l)}$ components. The reason why I clarified this is that with tensors and the Einstein summation convention introduced in the last post, indices can sometimes be used as a summand. Specifically we said that if an index repeats in a tensor expression in a contravariant postion and a covariant position it stands for summation. You need to sum over that index for all possible values (1 to $n$). Also, multiple indices can repeat in a tensor expression. In that case one needs to sum over all such indices varying from 1 to $n$. Therefore a tensor expression in which 1 index is to be summed upon expands into $n$ quantities and a tensor equation in which 2 indicies are to be summed upon expands into a sum of $n^2$ quantities. Guess what? armed with this understanding we are already ready to understand the general transformation law of tensors. Despair not my friends looking at this expression. If you do not understand this – dont worry. We will probably never see such complicated mixed tensors of general rank $(k,l)$. However, it is necessary to understand how to compute with tensors and their transformation law. So after reading the general law of transformation and understanding what it means, I woul like you to forget it temporarily instead of getting bogged down by the weird equation! I will follow the definition with an example. This will be restricted to 3 dimensions and to our familiar spherical polar and cartesian coordinate systems. If you can understand the stuff in there, I would like you to remember the general computational method there. The general law of tensor transformation for a tensor of rank $(k,l)$ is

$\bar{A}^{i_1 i_2 \dots i_k}_{j_1 j_2 \dots j_l} = A^{u_1 u_2 \dots u_k}_{v_1 v_2 \dots v_l}\frac{\partial{\bar{x}^{i_1}}}{\partial{x^{u_1}}} \frac{\partial{\bar{x}^{i_2}}}{\partial{x^{u_2}}} \dots \frac{\partial{\bar{x}^{i_k}}}{\partial{x^{u_k}}} \cdot \frac{\partial{x^{v_1}}}{\partial{\bar{x}^{j_1}}} \frac{\partial{x^{v_2}}}{\partial{\bar{x}^{j_2}}} \dots \frac{\partial{x^{v_l}}}{\partial{\bar{x}^{j_l}}}$

In this equation, the left hand side represents the value of 1 component out of the $n^{(k+l)}$ components in the “bar” coordinate system [which is why we denote the $A$ with a bar as the tensor]. The right hand side on the other hand is a huge sum of $n^{(k+l)}$ quantities because of the repeating indices $u_1, u_2, \dots, u_k, v_1, v_2, \dots, v_l$. Each term of the summation is a product of a component of the tensor with the appropriate partial derivatives – that is, each term is a particular instantiation of the $u_1, u_2, \dots, u_k, v_1, v_2, \dots, v_l$. The transformation law for the general rank tensor is a direct generalization of the transformation law for rank 1 tensors. The tensors value is needed for the $i_1,i_2,\dots,i_k,j_1,j_2,\dots,j_l$ index and the corresponding terms for the primed coordinate system occur in the numerator or denominator according to its for the $i$‘s – the contraviant indices or the $j$‘s – the covariant indices. The rest of the new introduced $u_s, v_t$ have been summed upon since they are repeating indices.

Wow! We have stated the general transformation law and now we will proceed to an example. Before we do that, I would like to state our general roadmap in the future posts. Remember that our goal is to understand the equations of General Relativity. We are studying tensors just because we want to understand the weird symbols and equations in the profound equations laid down by Einstein. So in the next post, we will study general operations on tensors like – addition, subtraction, inner products etc. Along with this we will state some rules to recognize tensors. We will state a general rule by which we can ascertain that a set of number form the components of a tensor. We will use this general rule to prove that some collection of numbers are tensors in later posts. Here we will not however talk of the derivative of a tensor because that involves some more machinery – but hey .. without derivatives there is no calculus so get back we will! The next few posts will talk about the fundamental tensor which is related to measuring distance in space between two points. That will define define distance for us. In the next few posts we will then discuss derivatives of tensors. In doing so we will introduce some tensors in terms of the fundamental tensor. These will be helpful for defining derivatives. Armed with all this machinery about distances and derivatives, we will then state equations for geodesics which are shortest paths between 2 points in space. In ordinary space this is the straight line. But in spaces, where the fundamental tensor is more complicated the geodesics are not “straight” lines [Well by then we will probably wonder what straight means anyway!]. Finally we will have the notions of distance, derivatives and shortest paths in our bag so we will talk about what we really need to understand the Einstein equations – curvature. Specifically we will introduce quantities which represent how curved a space is at a point. After this we will state the Einstein equations and if I can manage I will show some simple consequences of these equations. The Einstein equations somehow relate the curvature of our playground [which is the “space” of 4 dimensional spacetime], to the distribution of mass and energy.

So much for the future posts – all that is probably going to take several posts! But for the time being lets get back to an example using familiar 3 space. In familiar 3 space, we are going to work with 2 coordinate systems – one : cartesian $x^1=x, x^2=y, x^3=z$ and the other spherical polar $\bar{x}^1=r, \bar{x}^2=\theta, \bar{x}^3 = \phi$. Recall that for a point $P$, $r$ is the distance from the origin $O$, $0 \leq \phi \leq \pi$ is the angle made by $OP$ with the $z$ axis and $0 \leq \theta < 2\pi$ is the angle made by the projection of $OP$ onto the $xy$-plane with the $x$ axis. Then it is easy to see that

$x^1 = \bar{x}^1 \sin{\bar{x}^3} \cos{\bar{x}^2}$

$x^2=\bar{x}^1 \sin{\bar{x}^3} \sin{\bar{x}^2}$

$x^3=\bar{x}^1 \cos{\bar{x}^3}$.

The inverse transformations equations are

$\bar{x}^1 = \sqrt{ {x^1}^2 + {x^2}^2 + {x^3}^2}$

$\bar{x}^2=\tan^{-1}\left( \frac{x^2}{x^1}\right)$

$\bar{x}^3=\cos^{-1}\left( \frac{x^3}{\sqrt{ {x^1}^2 + {x^2}^2 + {x^3}^2}}\right)$

See here for an illustration. Assume we are working in a small region of space where all the $x^1,x^2,x^3$ are nonzero so that $\bar{x}^2,\bar{x}^3$ are well defined. We also need to verify that the Jacobian matrix of the partial derivatives is non-singular but that is left as an exercise!

We will take a simple case here.

Example

Suppose there is a contravariant tensor that has components $\bar{A}^1, \bar{A}^2, \bar{A}^3$. What are the components in the first coordinate system? For illustration we only show $A^2$. Now by the tensor transformation law

$A^2 = \bar{A}^i \frac{\partial{x^2}}{\partial{\bar{x}^i}}$

and so evaluating the partial derivatives and summing them up $A^2$ evaluates to

$A^2=\bar{A}^1\sin{\bar{x}^2}\sin{\bar{x}^3} + \bar{A}^2\bar{x}^1\cos{\bar{x}^2}\sin{\bar{x}^3}+\bar{A}^3\sin{\bar{x}^2}\cos{\bar{x}^3}$

Of course the above represents the value at a specific point in space. A contravariant tensor of rank 2 will have 9 such terms to sum up. We encourage the reader to evaluate the components of such a tensor.

Before the next post do think of some of the things you can do with tensors. Specifically think about

1. When can two tensors be added/subtracted to produce new tensors?
2. Can two tensors be multiplied to produce a new tensor?
3. Consider a function $f$ defined for each point of space. Suppose for each coordinate system $(x^1,x^2,\dots, x^n)$ we define $n$ components at a point by $A_i= \frac{\partial{f}}{\partial{x^i}}$. Prove that these numbers define a covariant tensor of rank 1.