We now come to our concepts and judgments of space. It is essential here also to pay strict attention to the relation of experience to our concepts. It seems to me that Poincaré clearly recognized the truth in the account he gave in his book, “La Science et l’Hypothèse.” Among all the changes which we can perceive in a rigid body those which can be cancelled by a voluntary motion of our body are marked by their simplicity; Poincaré calls these, changes in position. By means of simple changes in position we can bring two bodies into contact. The theorems of congruence, fundamental in geometry, have to do with the laws that govern such changes in position. For the concept of space the following seems essential. We can form new bodies by bringing bodies B, C, . . . up to body A; we say that we continue body A. We can continue body A in such a way that it comes into contact with any other body, X. The ensemble of all continuations of body A we can designate as the “space of the body A.” Then it is true that all bodies are in the “space of the (arbitrarily chosen) body A.” In this sense we cannot speak of space in the abstract, but only of the “space belonging to a body A.” The earth’s crust plays such a dominant rôle in our daily life in judging the relative positions of bodies that it has led to an abstract conception of space which certainly cannot be defended. In order to free ourselves from this fatal error we shall speak only of “bodies of reference,” or “space of reference.” It was only through the theory of general relativity that refinement of these concepts became necessary, as we shall see later.
I shall not go into detail concerning those properties of the space of reference which lead to our conceiving points as elements of space, and space as a continuum. Nor shall I attempt to analyse further the properties of space which justify the conception of continuous series of points, or lines. If these concepts are assumed, together with their relation to the solid bodies of experience, then it is easy to say what we mean by the three-dimensionality of space; to each point three numbers, x1, x2, x3 (co-ordinates), may be associated, in such a way that this association is uniquely reciprocal, and that x1, x2, and x3 vary continuously when the point describes a continuous series of points (a line).
It is assumed in pre-relativity physics that the laws of the configuration of ideal rigid bodies are consistent with Euclidean geometry. What this means may be expressed as follows: Two points marked on a rigid body form an interval. Such an interval can be oriented at rest, relatively to our space of reference, in a multiplicity of ways. If, now, the points of this space can be referred to co-ordinates x1, x2, x3, in such a way that the differences of the co-ordinates, Δx1, Δx2, Δx3, of the two ends of the interval furnish the same sum of squares,
for every orientation of the interval, then the space of reference is called Euclidean, and the co-ordinates Cartesian.* It is sufficient, indeed, to make this assumption in the limit for an infinitely small interval. Involved in this assumption there are some which are rather less special, to which we must call attention on account of their fundamental significance. In the first place, it is assumed that one can move an ideal rigid body in an arbitrary manner. In the second place, it is assumed that the behaviour of ideal rigid bodies towards orientation is independent of the material of the bodies and their changes of position, in the sense that if two intervals can once be brought into coincidence, they can always and everywhere be brought into coincidence. Both of these assumptions, which are of fundamental importance for geometry and especially for physical measurements, naturally arise from experience; in the theory of general relativity their validity needs to be assumed only for bodies and spaces of reference which are infinitely small compared to astronomical dimensions.
The quantity s we call the length of the interval. In order that this may be uniquely determined it is necessary to fix arbitrarily the length of a definite interval; for example, we can put it equal to 1 (unit of length). Then the lengths of all other intervals may be determined. If we make the xv linearly dependent upon a parameter λ,
we obtain a line which has all the properties of the straight lines of the Euclidean geometry. In particular, it easily follows that by laying off n times the interval s upon a straight line, an interval of length n · s is obtained. A length, therefore, means the result of a measurement carried out along a straight line by means of a unit measuring rod. It has a significance which is as independent of the system of co-ordinates as that of a straight line, as will appear in the sequel.
We come now to a train of thought which plays an analogous rôle in the theories of special and general relativity. We ask the question: besides the Cartesian co-ordinates which we have used are there other equivalent co-ordinates? An interval has a physical meaning which is independent of the choice of co-ordinates; and so has the spherical surface which we obtain as the locus of the end points of all equal intervals that we lay off from an arbitrary point of our space of reference. If xv as well as (v from 1 to 3) are Cartesian co-ordinates of our space of reference, then the spherical surface will be expressed in our two systems of co-ordinates by the equations
How must the be expressed in terms of the xv in order that equations (2) and (2a) may be equivalent to each other? Regarding the expressed as functions of the xv, we can write, by Taylor’s theorem, for small values of the Δxv,
If we substitute (2a) in this equation and compare with (1), we see that the must be linear functions of the xv. If we therefore put
or
then the equivalence of equations (2) and (2a) is expressed in the form
It therefore follows that λ must be a constant. If we put λ = 1, (2b) and (3a) furnish the conditions
in which δαβ = 1, or δαβ = 0, according as α = β or α ≠ β. The conditions (4) are called the conditions of orthogonality, and the transformations (3), (4), linear orthogonal transformations. If we stipulate that s2 = ΣΔxv2 shall be equal to the square of the length in every system of co-ordinates, and if we always measure with the same unit scale, then λ must be equal to 1. Therefore the linear orthogonal transformations are the only ones by means of which we can pass from one Cartesian system of co-ordinates in our space of reference to another. We see that in applying such transformations the equations of a straight line become equations of a straight line. Reversing equations (3a) by multiplying both sides by bvβ and summing for all the v’s, we obtain
The same coefficients, b, also determine the inverse substitution of Δxv. Geometrically, bvα is the cosine of the angle between the axis and the xα axis.
To sum up, we can say that in the Euclidean geometry there are (in a given space of reference) preferred systems of co-ordinates, the Cartesian systems, which transform into each other by linear orthogonal transformations. The distance s between two points of our space of reference, measured by a measuring rod, is expressed in such co-ordinates in a particularly simple manner. The whole of geometry may be founded upon this conception of distance. In the present treatment, geometry is related to actual things (rigid bodies), and its theorems are statements concerning the behaviour of these things, which may prove to be true or false.
One is ordinarily accustomed to study geometry divorced from any relation between its concepts and experience. There are advantages in isolating that which is purely logical and independent of what is, in principle, incomplete empiricism. This is satisfactory to the pure mathematician. He is satisfied if he can deduce his theorems from axioms correctly, that is, without errors of logic. The questions as to whether Euclidean geometry is true or not does not concern him. But for our purpose it is necessary to associate the fundamental concepts of geometry with natural objects; without such an association geometry is worthless for the physicist. The physicist is concerned with the question as to whether the theorems of geometry are true or not. That Euclidean geometry, from this point of view, affirms something more than the mere deductions derived logically from definitions may be seen from the following simple consideration.
Between n points of space there are distances, sμv; between these and the 3n co-ord
inates we have the relations
From these equations the 3n co-ordinates may be eliminated, and from this elimination at least equations in the sμv will result.* Since the sμv are measurable quantities, and by definition are independent of each other, these relations between the sμv are not necessary a priori.
From the foregoing it is evident that the equations of transformation (3), (4) have a fundamental significance in Euclidean geometry, in that they govern the transformation from one Cartesian system of co-ordinates to another. The Cartesian systems of co-ordinates are characterized by the property that in them the measurable distance between two points, s, is expressed by the equation
If K(xv) and K′(xv) are two Cartesian systems of co-ordinates, then
The right-hand side is identically equal to the left-hand side on account of the equations of the linear orthogonal transformation, and the right-hand side differs from the left-hand side only in that the xv are replaced by the . This is expressed by the statement that is an invariant with respect to linear orthogonal transformations. It is evident that in the Euclidean geometry only such, and all such, quantities have an objective significance, independent of the particular choice of the Cartesian co-ordinates, as can be expressed by an invariant with respect to linear orthogonal transformations. This is the reason that the theory of invariants, which has to do with the laws that govern the form of invariants, is so important for analytical geometry.
As a second example of a geometrical invariant, consider a volume. This is expressed by
By means of Jacobi’s theorem we may write
where the integrand in the last integral is the functional determinant of the with respect to the xv, and this by (3) is equal to the determinant of the coefficients of substitution, bvα. If we form the determinant of the δμα from equation (4), we obtain, by means of the theorem of multiplication of determinants,
If we limit ourselves to those transformations which have the determinant +1* (and only these arise from continuous variations of the systems of co-ordinates) then V is an invariant.
Invariants, however, are not the only forms by means of which we can give expression to the independence of the particular choice of the Cartesian co-ordinates. Vectors and tensors are other forms of expression. Let us express the fact that the point with the current co-ordinates xv lies upon a straight line. We have
Without limiting the generality we can put
If we multiply the equations by bβv (compare (3a) and (5)) and sum for all the v’s, we get
where we have written
These are the equations of straight lines with respect to a second Cartesian system of co-ordinates K′. They have the same form as the equations with respect to the original system of co-ordinates. It is therefore evident that straight lines have a significance which is independent of the system of co-ordinates. Formally, this depends upon the fact that the quantities (xv – Av) – λBv are transformed as the components of an interval, Δxv. The ensemble of three quantities, defined for every system of Cartesian co-ordinates, and which transform as the components of an interval, is called a vector. If the three components of a vector vanish for one system of Cartesian co-ordinates, they vanish for all systems, because the equations of transformation are homogeneous. We can thus get the meaning of the concept of a vector without referring to a geometrical representation. This behaviour of the equations of a straight line can be expressed by saying that the equation of a straight line is co-variant with respect to linear orthogonal transformations.
We shall now show briefly that there are geometrical entities which lead to the concept of tensors. Let P0 be the centre of a surface of the second degree, P any point on the surface, and ξv the projections of the interval P0 P upon the co-ordinate axes. Then the equation of the surface is
In this, and in analogous cases, we shall omit the sign of summation, and understand that the summation is to be carried out for those indices that appear twice. We thus write the equation of the surface
The quantities αμv determine the surface completely, for a given position of the centre, with respect to the chosen system of Cartesian coordinates. From the known law of transformation for the ξv (3a) for linear orthogonal transformations, we easily find the law of transformation for the αμv*:
This transformation is homogeneous and of the first degree in the aμv. On account of this transformation, the aμv are called components of a tensor of the second rank (the latter on account of the double index). If all the components, aμv, of a tensor with respect to any system of Cartesian co-ordinates vanish, they vanish with respect to every other Cartesian system. The form and the position of the surface of the second degree is described by this tensor (a).
Tensors of higher rank (number of indices) may be defined analytically. It is possible and advantageous to regard vectors as tensors of rank 1, and invariants (scalars) as tensors of rank 0. In this respect, the problem of the theory of invariants may be so formulated: according to what laws may new tensors be formed from given tensors? We shall consider these laws now, in order to be able to apply them later. We shall deal first only with the properties of tensors with respect to the transformation from one Cartesian system to another in the same space of reference, by means of linear orthogonal transformations. As the laws are wholly independent of the number of dimensions, we shall leave this number, n, indefinite at first.
Definition. If an object is defined with respect to every system of Cartesian co-ordinates in a space of reference of n dimensions by the nα numbers Aμvρ . . . (α = number of indices), then these numbers are the components of a tensor of rank α if the transformation law is
Remark. From this definition it follows that
is an invariant, provided that (B), (C), (D) . . . are vectors. Conversely, the tensor character of (A) may be inferred, if it is known that the expression (8) leads to an invariant for an arbitrary choice of the vectors (B), (C), etc.
Addition and Subtraction. By addition and subtraction of the corresponding components of tensors of the same rank, a tensor of equal rank results:
The proof follows from the definition of a tensor given above.
Multiplication. From a tensor of rank α and a tensor of rank β we may obtain a tensor of rank α + β by multiplying all the components of the first tensor by all the components of the second tensor:
Contraction. A tensor of rank α – 2 may be obtained from one of rank α by putting two definite indices equal to each other and then summing for this single index:
The proof is
In addition to these elementary rules of operation there is also the formation of tensors by differentiation (“Erweiterung”):
New tensors, in respect to linear orthogonal transformations, may be formed from tensors according to these rules of operation.
Symmetry Properties of Tensors. Tensors are called symmetrical or skew-symmetrical in respect to two of their indices, μ and v, if both the components which result from interchanging the indices μ and v are equal to each other or equal with opposite signs.
Condition for symmetry: Aμvρ = Avμρ.
Condition for skew-symmetry: Aμvρ = –Avμρ.
Theorem. The character of symmetry or skew-symmetry exists independently of the choice of co-ordinates, and in this lies its importance. The proof follows from the equation defining tensors.
Special Tensors.
I. The quantities δρσ (4) are tensor components (fundamental tensor).
Proof. If in the right-hand side of the equation of transformation we substitute for Aαβ the quantities δαβ (which are equal to 1 or 0 according as α = β or α ≠ β), we get
The justification for the last sign of equality becomes evident if one applies (4) to the inverse substitution (5).
II. There is a tensor (δμvρ . . .) skew-symmetrical with respect to all pairs of indices, whose rank is equal to the number of dimensions, n, and whose components are equal to +1 or –1 according as μvρ . . . is an even or odd permutation
of 123 . . .
The proof follows with the aid of the theorem proved above .
These few simple theorems form the apparatus from the theory of invariants for building the equations of pre-relativity physics and the theory of special relativity.
We have seen that in pre-relativity physics, in order to specify relations in space, a body of reference, or a space of reference, is required, and, in addition, a Cartesian system of co-ordinates. We can fuse both these concepts into a single one by thinking of a Cartesian system of co-ordinates as a cubical frame-work formed of rods each of unit length. The co-ordinates of the lattice points of this frame are integral numbers. It follows from the fundamental relation
that the members of such a space-lattice are all of unit length. To specify relations in time, we require in addition a standard clock placed, say, at the origin of our Cartesian system of co-ordinates or frame of reference. If an event takes place anywhere we can assign to it three coordinates, xv, and a time t, as soon as we have specified the time of the clock at the origin which is simultaneous with the event. We therefore give (hypothetically) an objective significance to the statement of the simultaneity of distant events, while previously we have been concerned only with the simultaneity of two experiences of an individual. The time so specified is at all events independent of the position of the system of co-ordinates in our space of reference, and is therefore an invariant with respect to the transformation (3).
A Stubbornly Persistent Illusion Page 25
