Part 3 — CHAPTER XXVI TENSORS AND THE LAWS OF NATURE

WE have now reached a point where further progress would appear to require at least a few elementary notions on tensors and the absolute calculus. We shall endeavour to present these matters in as clear a way as possible by considering, first of all, tensors from the standpoint of pure mathematics, without any reference to their applications in Einstein’s theory.

Let us suppose that we are dealing with the ordinary three-dimensional Euclidean space of classical science. We remember that it is often convenient to refer the positions of objects situated in space to some definite system of reference or system of co-ordinates. This procedure is equivalent to drawing an imaginary three-dimensional mesh-system with respect to which we locate the points of space. The precise shape of this mesh-system is arbitrary. It may split up space into equal cubes (in which case it is called Cartesian), or again it may be curvilinear and split up space into irregularly shaped volumes. In all cases, however, we shall assume that these mesh-systems remain fixed and do not vary in shape or orientation once they have been drawn.

Now, though we have stated that the precise shape of the co-ordinate system, or mesh-system, is arbitrary, there are mathematical reasons which render the use of Cartesian systems especially convenient. So long as we are dealing with Euclidean space, in which, as we know, Cartesian mesh-systems are always permissible, we need not be concerned with Gaussian co-ordinate systems. It was for this reason that classical science, when dealing with physical problems, confined itself to Cartesian co-ordinates; and by a change of co-ordinate system it meant a rotation or a displacement of the Cartesian frame of reference from one position to another.

Nevertheless, we always had to view the possibility of having to investigate problems in which Cartesian co-ordinate systems would be impracticable, owing to the curvature or non-Euclideanism of the continuum. Accordingly, even before the advent of the theory of relativity, mathematicians had endeavoured to perfect a form of calculus which would free them from all dependence on any particular type of co-ordinate system. This task was accomplished chiefly by Levi-Civita and Ricci, and the mathematical instrument evolved was called the absolute calculus. And so, when in the general theory Einstein was led to his conception of a curved or non-Euclidean space-time, in which Cartesian co-ordinate systems were impossible, the mathematical instrument he required was at hand. All he had to do was to put it to practical use.

Although, in the Euclidean space of classical physics which we now propose to discuss, there was no need to consider Gaussian or curvilinear co-ordinate systems, yet, in view of the greater generality of the method, we shall treat the subject matter of this chapter from the standpoint of all systems of co-ordinates, whether Gaussian or Cartesian. The conclusions we shall reach must, however, be construed as holding for all manner of co-ordinate systems. If these preliminary points are understood, we may proceed to a more detailed discussion of the problem.

Let us consider a straight rod situated in this space. If we agree on a unit of measurement, we can assign some definite length to the rod. Obviously, this length, whatever it may be, will remain unaffected by a change in our co-ordinate system in space. Magnitudes of this sort, which transcend in every respect our choice of a co-ordinate system, are known as scalars, numbers or invariants.

But suppose now that we wish to determine the exact orientation of our rod. Orientation, in contrast to length, has no meaning until some mesh-system has been prescribed; for we can define a direction in empty space only by referring it to some accepted system of standard directions at each point. The meshes of our mesh-system at each point will, of course, define such standard directions for us. When, therefore, we attempt to determine the orientation of the rod, our procedure will be to project its length upon the three meshes that pass through one of its extremities. We thereby obtain what are known as the three components of the rod; and when these three components are known, the orientation and the length of the rod are fully determined.[86]

From this example we see that a considerable difference exists between the directed magnitude we are here considering and the invariant one which was given by the mere length of the rod irrespective of its orientation. Whereas the invariant magnitude possessed no components, the directed one has as many components as the space has dimensions, three in the present case. A directed magnitude of this sort is termed a vector.

Suppose now that we change our system of reference, rotate it slightly, for instance. We shall thereby obtain three new components for our vector in the new co-ordinate system, and these new components will differ in magnitude from our former ones. However, the change in the value of the components for a definite change of frame of reference is not arbitrary, but will be submitted to well-defined mathematical rules.[87] This fact enables us to discover a criterion of objectivity.

For, suppose that in a first co-ordinate system certain components, , , , are revealed by measurement, and that in a second co-ordinate system new components , , are obtained. If under these circumstances we were to find that the two trios of components , , and , , 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.

Let us now pass to one of the most important characteristics of tensors and vectors. The fact that the components of all contravariant vectors are submitted to the same rules of change when we pass from one co-ordinate system to another, proves that if two contravariant vectors situated at the same point of space are equal, that is, if the components of the one are equal to the components of the other in our co-ordinate system, this equality will inevitably endure in any other co-ordinate system. In other words, the equality of two vectors at a point of space constitutes an equality which a change in our co-ordinate system can never destroy; the equality is thus an absolute. Similar conclusions apply to the equality of two tensors at the same point of space.

It must be noted, however, that this equality endures only because the components of either vector or of either tensor on the two sides of the equation are submitted to the same rules of transformation when our co-ordinate system is changed. This implies that we are dealing with vectors of the same nature (covariant or contravariant), or with tensors of the same nature and order. If these conditions are satisfied, we see that equations between vectors or tensors, often called vector equations and tensor equations, exhibit the remarkable property of remaining unaffected by a change of mesh-system.

In a certain sense, then, vector equations and tensor equations constitute invariant or absolute equations. This last appellation would, in fact, be perfectly justified were we to argue in terms of the absolute vectors and tensors which exist independently of our mesh-system. But in practice it is very difficult to dispense with a mesh-system. As a result, what we come into contact with—what we measure in reality—is not the vector or the tensor itself, which is absolute, but its components in the mesh-system that we have selected. These components are relative, since they change in value when we change our mesh-system. True, these changes being the same on either side of the equation, the equality endures regardless of our choice of mesh-system. But, on the other hand, a definite numerical change has taken place in the value of either side of our equation. The form of the equation as a whole has remained unaltered, but the contents of the two sides has changed.

Let us explain this point more fully. Consider two vectors which are situated at the same point and which are equal to each other. In three-dimensional space, this equality is expressed by the respective equality of the three components of the first vector with the three components of the second vector in our co-ordinate system; hence we have three equations, such as

If, now, we change our mesh-system, the components of either vector change in exactly the same way, so that our equations subsist. In the new mesh-system, however, becomes and becomes , etc., so we now have

Obviously, although the equations subsist, yet the expressions on either side of our three equations have varied; for is different from and so on. Thus we see that it would not be quite correct to state that our equations remained completely unchanged when we passed from one mesh-system to another.

For these reasons it is customary to speak of the covariance rather than of the invariance of vector or tensor equations. The word “covariance” expresses the fact that both sides have varied in exactly the same way. The appellation “invariance” would tend to make us believe that neither side had varied at all, which would, of course, be incorrect. In short, we must remember that it is the relationship of equality between the two sides, and not the individual terms themselves, that remain unaffected by a change of mesh-system.[89]

Now, there exists still another type of vector and of tensor equation. If, for instance, a vector or a tensor vanishes, this means that all the components vanish individually. A simple glance at the rules of transformation shows that in this case all the components will continue to vanish in any co-ordinate system, so that the equating of a tensor, a vector or an invariant to zero at a point of space will endure in all co-ordinate systems if it endures in any particular system.

Summarising, we find that there exist two types of covariant equations:

The first type is given by the equality of two vectors (the vectors being both covariant or both contravariant) or, more generally, by the equality of two tensors, these tensors being of the same order and same nature (i.e., covariant, mixed or contravariant). A limiting case would be given by the equality of two invariants; we should then have an invariant equation.

The second type is given by the vanishing of a vector or a tensor at a point in space. A limiting case would be given by the vanishing of an invariant.

All these deductions obtained mathematically are in a certain sense self-evident physically when we recall that a tensor, a vector and an invariant represent magnitudes to which an objective existence may be conceded. It is obvious, indeed, that if two such magnitudes are equal to each other at a point in space, a mere change of our co-ordinate system can never disturb this intrinsic equality. In the same way, if a concrete entity, such as a tensor, has vanished at a point in space, a mere change of our co-ordinate system is incapable of bringing it into existence. Once again, we realise the strict dependence which exists between physical representations of this sort and the fact that invariants, vectors and tensors must be recognised to have an objective existence transcending the particular point of view of our co-ordinate system.

If these mathematical considerations have been grasped, we may pass to physical applications of these purely mathematical discoveries. We will restrict ourselves for the present to the three-dimensional Euclidean space of classical science; hence by a change of co-ordinate system we again mean a mere change in shape, position and orientation of our mesh-system. We do not mean the passage from one system to another in relative motion thereto.

In classical science we often come upon invariants and vectors. Examples of invariants are given by the mass and the density of mass of a body. Vectors are illustrated by a velocity, an acceleration, a force, a displacement; and we may say that in a general way a vector can be represented by an arrow. The simplest physical illustration of a tensor at each point of space is afforded by the state of compression and tension at each point of an elastic medium which has been distorted.

It is impossible to express this state of tension and compression at a point by means of three components in our three-dimensional space. Owing to lateral wrenching effects at each point, nine components are necessary; and we realise that we can no longer appeal to a vector. We are now in the presence of a tensor. The physical origin of the appellation “tensor,” coined by Willard Gibbs, thus becomes apparent. In the present case the tensor is a second-order tensor, and, furthermore, it is what is known as a symmetrical tensor. The characteristics of a symmetrical tensor are defined by the identity of certain of its components two by two; this reduces the number of independent components of a second-order tensor in three-dimensional space to six in place of nine.

It can be proved that the numbers which we have often mentioned constitute symmetrical second-order covariant tensors. The tensor nature of the ’s means nothing else than that the components of the ’s in our co-ordinate system vary according to the rules of tensors when we pass from one co-ordinate system to another. The ’s being second-order covariant tensors, we may infer that by combining them with two displacements, or vectors, of opposite nature (i.e., contravariant vectors), we shall obtain an invariant. It is for this reason that yields an invariant,[90] which, as we know, is nothing but the square of the length .

In three-dimensional space there are nine of the components, but, the tensor being symmetrical, this number is reduced to six. Thus . For this reason we may in the case of three-dimensional space write out as follows: instead of [91]

When we come to examine the concrete significance of the ’s, we find them to be expressive of the angles between our co-ordinate meshes, and of the general shape of our mesh-system from place to place. Since, however, the shape of our mesh-system, though arbitrary in large measure, is nevertheless restricted to certain types compatible with the structure of the space in which we trace them, these ’s are also representative of the structure or geometry of our space. For this reason they are called structural or fundamental tensors.

Let us recall once more that we are not yet dealing with Einstein’s discoveries or with space-time; we are discussing three-dimensional space, and all we have said refers to pre-Einsteinian science.

Other illustrations of tensors are afforded by an angular velocity and by a magnetic force. The tensor nature of these two magnitudes is less obvious because it so happens that these second-order tensors are of a special type known as anti-symmetrical. In tensors of this type, certain components are always zero in all systems of co-ordinates, and the remaining ones are of the same magnitude but of opposite sign. The result is that in the case of three-dimensional space anti-symmetrical tensors of the second order possess only three independent components just like vectors, and for this reason are often represented by arrows and confused with vectors. As a matter of fact, this confusion need not arise; for it is obvious that a rotation, for example, is better represented by a corkscrew curve than by an arrow, and the same applies to a magnetic force when we realise that it is generated by a whirling of electrons. At all events, had it not been for the accidental fact that space had three dimensions, the confusion could never have occurred at all; for, in a four-dimensional space, anti-symmetrical tensors of the second order have a number of components greater than four (which is the number of components of a vector in a four-dimensional space). Still another illustration of an anti-symmetrical tensor of the second order is furnished by an element of a surface.

In short, we see that the magnitudes in which classical science was chiefly interested were either invariants, vectors or tensors. At this we need not be surprised, seeing that the magnitudes to which physical science pays greatest attention are always those to which we may concede an objective existence transcending our choice of a mesh-system. Thus, an object itself appears to be of greater significance than the shape of its shadow; for the shadow will change as we alter the orientation and shape of the surface on which it falls, whereas the existence of the object will transcend any such changes.

If now we consider the laws of nature, we can anticipate, for two different reasons, that they must be vector or tensor equations. First, because all those physical magnitudes to which we attribute an objective existence and which enter into the expression of the laws of nature are invariants, vectors and tensors; and, secondly, because a law of nature must obviously remain unaffected by a mere change of orientation or shape of our frame of reference.

It is, indeed, evident that a mesh-system is a mere artifice of calculation. It can have no intrinsic significance, and, as a matter of fact, we might dispense with it altogether. Such an absolute condition of the world as the equality of two vectors at a point or the vanishing of a vector or of a tensor would endure even had mesh-systems never been invented by Descartes and Gauss. Suppose, then, that a law appeared to hold when we selected some particular mesh-system, but no longer when our meshes were squeezed together or rotated to some new position. Obviously our law would possess no generality; it could refer to nothing objective and would be no law at all.

We may summarise these discoveries by saying that all the laws of classical science were vector or tensor laws. They remained covariant or unchanged in form when we changed the orientation or shape of our mesh-system while maintaining it fixed in space. This last point is important; for in classical science, time being absolute and separate from space, it was solely from the point of view of a spatial change in the shape and orientation of our mesh-system that the laws remained covariant.

We may also mention that in addition to the covariance of all natural laws to a spatial change of mesh-system, there existed a second requirement which all natural laws had to obey. We refer to the test of dimensionality. A brief discussion will make this point clear.

The numerical values of all physical quantities depend on our choice of standard units for the measurement of distance, duration and mass. If, therefore, we change our units, if we adopt the minute instead of the second, or the foot instead of the metre, or the pound instead of the kilogramme, the numerical values of both sides of the equation representing the law will be affected accordingly. For the equation to endure in spite of the change in value of its two terms, it is necessary that a change in our unit of time or of length or of mass should affect the numerical values of both sides of the equation in exactly the same way. This condition is expressed by saying that both sides of the equation must possess the same dimensionality. It is perfectly obvious that a law of nature must constitute an equation of this type. It would be ridiculous to suppose that a change in our units from the avoirdupois to the metric system, for example, could ever upset the validity of a law of nature. This test of dimensionality was often useful in determining the second side of the equation representing some unknown law. We might ignore the exact form of the second side, but we could always tell in advance that its dimensionality would have to be the same as that of the first side. In this way a certain amount of a priori information was often obtained.