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.
CHAPTER XXVII
THE PRINCIPLE OF GENERAL COVARIANCE, OR THE GENERAL PRINCIPLE OF RELATIVITY
WE have endeavoured to show that the great change in our views of the world brought about by the theory of relativity arises from the discovery that four-dimensional space-time and not the three-dimensional space and the independent time of classical science correspond to reality. It might seem but a natural generalisation to extend to space-time the conclusions we arrived at in the last chapter when discussing the covariance of the natural laws to a change of mesh-system in space. We should then expect the laws of nature to express relationships between vectors and tensors in space-time, hence to remain covariant to a change in our space-time mesh-system.