Toy Universes

Based on what we have stated up to now concerning tensors, you must be eager to learn how to make the connection between the index-free and full-index languages.  This post aims to serve as a first connection between both languages. You may find out, however, that the connection will come in a very natural way: basically from the linearity property of tensors. For now, our present exposition will be as simple as possible.

We are working with tensors living in “simple” spaces, namely, Euclidean tensors. Recall from previous posts that we have extended the definition of tensors to include multilinear functionals. So the picture we have up to now is that a tensor is like a machine with a certain number of slots to be filled with input vectors, and that, after “processing” these input vectors, the machine retrieves another tensor of a given rank (0, 1, 2,…), depending on the nature of the tensor.

For example, recall the 3D Euclidean metric , given by the Pythagorean formula:

.

The situation above is that you have two points,, in a space , with the metricattached to it, so that the metric function gives you a rule to calculate distances between these points. The coordinates (ordered triples)of each point can be compactly written as a vector, that is, a geometrical object which, under the Cartesian coordinatization, has components. We agree that all such “position” vectors are “attached” to the origin (their “tail” is at the origin, and their “head” is at the given point. This is just a useful image for vectors in Euclidean spaces. In general manifolds, we will have to give up on that vision occasionally).

Therefore, the metric is nothing but a linear machine with two slots that outputs a real number. It is a tensor of rank 2, because we have two input vectors and one output scalar, and a scalar counts as zero for the computation of the rank (as already noted in a previous post).

To emphazise our picture of tensors as machines, we shall temporarily use the following (unorthodox) notation for tensors: given a tensorof rank, it hasinput slots (operates onvectors) and outputs another tensor of rank. We write the tensoras:

.

For example, the metric tensor (rank 2) is written as:

,

where is a scalar.

Suppose now another tensor, also of rank 2, but now its nature is a bit different from the former:

.

That tensor receives as input the vectorand outputs another vector,. Let us examine the above expression carefully. You can change perspective and see that the vectoris in fact a vector-valued function of the vector:

.

But the vectorcan be written in terms of the three orthonormal basis vectors (i.e., mutually ortogonal and of unit length),:

,

where the triple of real numbers () are the coordinates of the “head” of the vector(i.e., its components). Insert that into the previous equation, and you have:

.

Because of the linearity property of tensors, the above equation is:

.

But each of the objectsis a vector, right? That means you can also expand them in terms of basis vectors, as you did with. Let us write each expansion as (*):

,

,

.

From the above expansions, you can see clearly that you need all these 9 numbers,, in order to express the set of objectsin terms of the basis. In fact, you can see now that the tensor T, if to be seen in component form under a given basis, needs a specification of 9 numbers, or its 9 components. That happens because you must specify how the tensor T acts on each basis vector. You must also realize that these 9 numbers do depend on the chosen basis. Also, it is evident that in a 2D space, you would need 4 of such numbers (); in a n-dimensional space, numbers.

Now, let us go back to the expression:

.

Clearly, it is also true that you can write the vectoras:

.

So, we have:

.

But we have already deduced above that:

.

Or:

.

Rearrange that in the following way:

.

Equate each component of at the left side with the corresponding expression in parenthesis in the left side. You get:

,

,

.

Evidently, you can write that system of equations compactly by collecting the components of T as a matrix operating on the given vector:

.

Now, let us change notation in order to avoid cumbersome writting. Just letbe labeled as numbers,. So it is easy to see that the system of equations above can be written compactly as:

,

with indicesrunning from 1 to 3. We shall often use the Einstein’s summation convention (see Notation page on the side bar). So that can be written simply as:

.

In summary, we see thatgets as input the vectorand outputs vector. In the chosen basisthe components of these vectors are related by the equation above.

It is left to you to show (see a little bit above, (*)) that:

.

But that is simply an expression for a new transformed basis:

.

We have above precisely a law for the transformation of the basis vectors by the action of the tensor. See the inversion now: for the transformation of the basis instead offor the transformation of vectors.

You may be wondering how all that came out right. Well, we simply made the right choice of notation when writing the components of T — see (*) above — because we wanted all to be consistent at the end.

Ref.: [Nea10], [MTW73]