The evolution of scientific thought from Newton to Einstein

by A. D'Abro

were not linked together by the stringent mathematical rules we have referred to, we should have to recognise that we could not have been contemplating the same vector when we passed from the first co-ordinate system to the second. Conversely, if the mathematical rules were found to be satisfied by the components, we should realise that we were indeed dealing with the same vector. In this way the vector would no longer be some indeterminate magnitude. It would be a definite and precise one to which we might attribute an objective existence transcending our choice of a mesh-system of reference. The components themselves have no absolute existence; they do not transcend our mesh-system. They represent but partial aspects of the vector, mere modes or shadows, varying, as they do, with our system of reference. But to the vector itself an objective existence can be conceded.

Now, mathematical investigation soon showed that there existed two different types of vectors, distinguished by two different rules of mathematical transformation when we passed from one co-ordinate system to another. As these two rules of mathematical transformation, known as the contravariant and the covariant rule, respectively, appeared to be the only consistent ones, it was recognised that there existed only these two different types of vectors, the so-called contravariant vector and the covariant vector. In a certain sense this distinction is somewhat artificial, for we may always regard a given vector as contravariant or covariant, as we see fit. All that we mean to imply by this terminology is that we intend to subject the components of the vector to the contravariant or the covariant rule of transformation when we change our co-ordinate system.

So far we have considered two types of magnitudes, invariants or scalars, and vectors. Mathematicians discovered, however, that there existed a vast array of other types of magnitudes named tensors,[88] whose components likewise obeyed rigid rules of mathematical transformation when we passed from one co-ordinate system to another, and which for this reason could be credited, just as truly as vectors and invariants, with an objective existence transcending our mesh-system. Tensors differed from vectors owing to the greater number of their components. Whereas a vector possessed only as many components as the space had dimensions, a tensor could have a much greater number. According to the number of their components in a space of a given number of dimensions, tensors were grouped into tensors of the second order, of the third order, of the

th order.

In a three-dimensional space a tensor of the second order has nine components instead of three, as has a vector. A tensor of the third order has twenty-seven, and so on. Just as in the case of vectors, the components of a tensor situated at a point in space vary in value with the choice of our mesh-system, and, again, just as with vectors, the values of the components in one mesh-system are connected with their values in another mesh-system by rigid mathematical rules of transformation. In the present case, however, these rules of transformation are more complex. They are seen to be generalisations of the rules regulating the transformations of vector components. As an immediate sequel, tensors appear to be mere generalisations of the older known vectors.

In particular, a vector was found to be nothing else than a tensor of the first order, while an invariant appeared to be a tensor of zero order. The general name “tensor” expresses, therefore, all those types of mathematical magnitudes which possess an objective significance transcending our system of co-ordinates, although it will often be convenient to reserve the name “tensor” for tensors of the second and higher orders, retaining thereby the classical appellations “vector” and “invariant” for tensors of lower order.

Now we shall see that the distinction we have made between the two different types of vectors (contravariant and covariant) will extend automatically to tensors. Let us consider, for instance, a tensor of the second order. When we change our co-ordinate system, the nine components of this tensor will vary, and to obtain their values in the new co-ordinate system we apply a rule of transformation which is, so to speak, the vector rule taken twice in succession. As there are two vector rules, one contravariant and one covariant, we get, in the case of tensors of the second order, a twice contravariant rule, a twice covariant one or a mixed rule both contravariant and covariant. In this way we are led to distinguish three different types of second-order tensors: the twice contravariant, the twice covariant and the mixed variety.

Here it must be stated that the two rules, the contravariant and the covariant, are in certain respects the antitheses of each other. What one rule does the other undoes. If, therefore, we combine any vector considered as contravariant with any vector considered as covariant, the result will be similar to what happens when a squirrel runs in a drum which is rotating backwards. The forward motion of the squirrel is counteracted by the backward motion of the drum, so that finally the squirrel does not move at all. The result of combining a contravariant and a covariant vector will be, therefore, to obtain a scalar or invariant. In the same way, a covariant tensor of the second order combined with two contravariant vectors, or a covariant tensor combined with a contravariant one of the same order, will each yield an invariant.