Toy Universes

If you have been following our previous posts on tensors, you have probably noticed that we have essentially signaled a few general and abstract remarks about tensors, and have not made any reference to more concrete techniques on how to compute with them (like, e.g., a reference to tensors as matrices).  That fact is certainly one of the first disturbances that specially bothers those who came from a physics-like background. It certainly disturbed me (and still does).

For that reason I have decided to write this short note, motivated by a reader with an engineer background (see a brief clarification in a subsequent comment). To relief that natural distress, at least for a moment, we must state at this point that matrices can represent tensors.

So let us have this straight, at least as much as we can from what we have already learned. What we have been saying here is that, more generally, tensors are multilinear functionals: machines (in the language of [MTW73]) with a certain number of slots that may be completed or partially filled with inputs (namely — up to this point — vectors), giving as output another tensor (either of rank 0, 1, etc). The result depends on the nature of what the machine “operates” on.

For example, we have the Riemann tensor, so much used in General Relativity: it can be thought of as a machine with three slots (so it operates on three vectors), giving as output a vector. We will go into what that means when time comes, with both coordinate (index) language and free-index language. (Notice that in our abstract setting, up to now, we are using a free-index language. No reference to coordinates.) Just hang on!

Now, matrices. Tensors can be represented by matrices. The set of numbers that constitutes a matrix depends on the coordinate system used. (A few tensors can be represented by matrices that do not depend on the coordinate system, e.g., the isotropic tensors. But, generally, matrices representing tensors depend on the basis used, i.e. the coordinate system).

But whatever the notion that you carry on tensors, it is important to realize that a tensor is a geometrical object that exists independently of the coordinate system. So when you represent a tensor by a matrix,  that is, by fixing a given basis, that matrix is a set of given numbers. When you represent the same tensor in another coordinate system, the associated matrix will in most cases be a set of completely different numbers. Because tensors — purely as they are (not when represented by matrices) — do not depend on the coordinate system used, it is clear that when you see tensors as matrices the transformation law that connects both coordinate bases will have an important role. That is why that, in index notation, tensors are usually defined as objects that transform in “such and such” way.

But, see, Nature does not care about coordinates that we humans use. A given phenomenon will happen regardless of the system we choose to make measurements. This seems a little dumb, but it is the essence of all mathematical tower to be constructed here. If an apple falls to the ground, we must express that phenomenon using a universal law, valid in any reference system. Such laws are generally expressed by tensor equations. Tensors are geometrical objects, independent of coordinates, so that when you say that a tensor gives zero (that is a tensor equation, right?) at some frame () , that must be true in any other frame () — they have the same form. (You should not come up with, say,…) Such equations with the same form in all equivalent reference systems are an expression of the universality of the law they represent. That’s the principle of general covariance. Specific numerical values of physical quantities may be different in different frames, but the tensor equation has the same form.

For that reason, an appropriate relation between frames must provide an expression that connects exactly the measured quantities related to the given phenomenon, since they do differ from coordinate frame to coordinate frame. That includes the matrix representation of tensors.

But do not suppose from that statement that such relations between frames are only relevant when we make specific computations!

For instance, in special relativity, quantities measured in one inertial frame are connected by quantities measured in another inertial frame by the Lorentz transformation. That transformation can be regarded as a matrix as well. But a matrix is just a way to see it. What it does express is a fundamental symmetry of space-time (manifested essentially by the constancy of the velocity of light as measured by different observers). It forms a group (and this is a huge and fundamental subject that will be able to address at some point, hopefully!). All that does not need a coordinate to have a meaning.

As we proceed, we will be able to return to what have been stated here, in much more detail. For now, in summary, tensors can be represented by matrices, which are a useful concept in calculations. But we want to see them not only as a set of numbers, as we could easily lose sight of the forest for the trees.