It is postulated that the system of equations expressing the laws of pre-relativity physics is co-variant with respect to the transformation (3), as are the relations of Euclidean geometry. The isotropy and homogeneity of space is expressed in this way.* We shall now consider some of the more important equations of physics from this point of view.
The equations of motion of a material particle are
(dxv) is a vector; dt, and therefore also , an invariant; thus is a vector; in the same way it may be shown that is a vector. In general, the operation of differentiation with respect to time does not alter the tensor character. Since m is an invariant (tensor of rank 0), is a vector, or tensor of rank 1 (by the theorem of the multiplication of tensors). If the force (XN) has a vector character, the same holds for the difference . These equations of motion are therefore valid in every other system of Cartesian co-ordinates in the space of reference. In the case where the forces are conservative we can easily recognize the vector character of (Xv). For a potential energy, Φ, exists, which depends only upon the mutual distances of the particles, and is therefore an invariant. The vector character of the force, is then a consequence of our general theorem about the derivative of a tensor of rank 0.
Multiplying by the velocity, a tensor of rank 1, we obtain the tensor equation
By contraction and multiplication by the scalar dt we obtain the equation of kinetic energy
If ξv denotes the difference of the co-ordinates of the material particle and a point fixed in space, then the jn have vector character. We evidently have so that the equations of motion of the particle may be written
Multiplying this equation by ξμ we obtain a tensor equation
Contracting the tensor on the left and taking the time average we obtain the virial theorem, which we shall not consider further. By interchanging the indices and subsequent subtraction, we obtain, after a simple transformation, the theorem of moments,
It is evident in this way that the moment of a vector is not a vector but a tensor. On account of their skew-symmetrical character there are not nine, but only three independent equations of this system. The possibility of replacing skew-symmetrical tensors of the second rank in space of three dimensions by vectors depends upon the formation of the vector
If we multiply the skew-symmetrical tensor of rank 2 by the special skew-symmetrical tensor δ introduced above, and contract twice, a vector results whose components are numerically equal to those of the tensor. These are the so-called axial vectors which transform differently, from a right-handed system to a left-handed system, from the Δxv. There is a gain in picturesqueness in regarding a skew-symmetrical tensor of rank 2 as a vector in space of three dimensions, but it does not represent the exact nature of the corresponding quantity so well as considering it a tensor.
We consider next the equations of motion of a continuous medium. Let ρ be the density, uv the velocity components considered as functions of the co-ordinates and the time, Xv the volume forces per unit of mass, and pvσ the stresses upon a surface perpendicular to the σ-axis in the direction of increasing xv. Then the equations of motion area, by Newton’s law,
in which is the acceleration of the particle which at time t has the co-ordinates xv. If we express this acceleration by partial differential coefficients, we obtain, after dividing by ρ,
We must show that this equation holds independently of the special choice of the Cartesian system of co-ordinates. (uv) is a vector, and therefore is also a vector. is a tensor of rank 2, is a tensor of rank 3. The second term on the left results from contraction in the indices σ, τ. The vector character of the second term on the right is obvious. In order that the first term on the right may also be a vector it is necessary for pvσ to be a tensor. Then by differentiation and contraction results, and is therefore a vector, as it also is after multiplication by the reciprocal scalar . That pvσ is a tensor, and therefore transforms according to the equation
is proved in mechanics by integrating this equation over an infinitely small tetrahedron. It is also proved there, by application of the theorem of moments to an infinitely small parallelepipedon, that pvσ = pσv, and hence that the tensor of the stress is a symmetrical tensor. From what has been said it follows that, with the aid of the rules given above, the equation is co-variant with respect to orthogonal transformations in space (rotational transformations); and the rules according to which the quantities in the equation must be transformed in order that the equation may be co-variant also become evident.
The co-variance of the equation of continuity,
requires, from the foregoing, no particular discussion.
We shall also test for co-variance the equations which express the dependence of the stress components upon the properties of the matter, and set up these equations for the case of a compressible viscous fluid with the aid of the conditions of co-variance. If we neglect the viscosity, the pressure, p, will be a scalar, and will depend only upon the density and the temperature of the fluid. The contribution to the stress tensor is then evidently
in which δμv is the special symmetrical tensor. This term will also be present in the case of a viscous fluid. But in this case there will also be pressure terms, which depend upon the space derivatives of the uv. We shall assume that this dependence is a linear one. Since these terms must be symmetrical tensors, the only ones which enter will be
For physical reasons (no slipping) it is assumed that for symmetrical dilatations in all directions, i.e. when
there are no frictional forces present, from which it follows that If only is different from zero, let by which α is determined. We then obtain for the complete stress tensor,
The heuristic value of the theory of invariants, which arises from the isotropy of space (equivalence of all directions), becomes evident from this example.
We consider, finally, Maxwell’s equations in the form which are the foundation of the electron theory of Lorentz.
i is a vector, because the current density is defined as the density of electricity multiplied by the vector velocity of the electricity. According to the first three equations it is evident that e is also to be regarded as a vector. Then h cannot be regarded as a vector.* The equations may, however, easily be interpreted if h is regarded as a skew-symmetrical tensor of the second rank. Accordingly, we write h23, h31, h12, in place of h1, h2, h3 respectively. Paying attention to the skew-symmetry of hμv, the first three equations of (19) and (20) may be written in the form
In contrast to e, h appears as a quantity which has the same type of symmetry as an angular velocity. The divergence equations then take the form
The last equation is a skew-symmetrical tensor equation of the third rank (the skew-symmetry of the left-hand side with respect to every pair of indices may easily be proved, if attention is paid to the skew-symmetry of hμv). This notation is more natural than the usual one, because, in contrast to the latter, it is applicable to Cartesian left-handed systems as well as to right-handed systems without change of sign.
Reprinted courtesy of Princeton University Press
* This relation must hold for an arbitrary choice of the origin and of the direction (ratios Δx1 : x2 : x3 of the interval.
* In reality there are equations.
* There are thus two kinds of Cartesian systems which are designated as “right-handed” and “left-handed” systems. The difference between these is familiar to every physicist and engineer. It is interesting to note that these two kinds of systems cannot be defined geometrically, but only the contrast between them.
* The equation may, by (5), be replaced by ,from which the result stated immediately follows.
* The laws of physics could be expressed, even in case there were a preferred direction in space, in such a way as to be co-variant with respect to the transformation (3); but such an expression would in this case be unsuitable. If there were a preferred direction in space it would simplify the description of natural phenomena to orient the system of co-ordinates in a de
finite way with respect to this direction. But if, on the other hand, there is no unique direction in space it is not logical to formulate the laws of nature in such a way as to conceal the equivalence of systems of co-ordinates that are oriented differently. We shall meet with this point of view again in the theories of special and general relativity.
* These considerations will make the reader familiar with tensor operations without the special difficulties of the four-dimensional treatment; corresponding considerations in the theory of special relativity (Minkowski’s interpretation of the field) will then offer fewer difficulties.
Selections from
The Evolution of Physics
During the first half of the twentieth century, quantum theory was transforming the landscape of physics, much as the theory of electromagnetism did a century earlier. In The Evolution of Physics, Albert Einstein and Leopold Infeld describe this revolution from the eye of the storm. Today we have grown so accustomed to the idea of nanotechnology and microelectronics, technologies which could not exist without quantum mechanics, that it is easy to forget what a monumental shift in our understanding is required to think in quantum terms.
In the continuous picture, a piece of iron, for example, may have any mass whatsoever. In the quantum picture, this is shown to be an illusion. Each lump of iron has a certain number of atoms in it, and each atom has a fixed mass. Another lump can only differ by an integer number of atoms and thus, by “quantized” masses. Atoms themselves are made of still smaller quantized elements, protons and neutrons. And that’s not even the end of it! About two decades after the publication of The Evolution of Physics, Murray Gell-Mann and Kazuhiko Nishijima proposed that protons and neutrons were made of yet smaller quantized particles known as quarks.
The idea that particles may be only divided a finite number of times before reaching the atomic scale was not a new one; it had its origins as far back as Democritus and the early Greek atomists. The strength of modern quantum theory came, rather, from the properties ascribed to microscopic particles. While on the human scale, we normally say that a particle has a well-defined position and speed, we can make no such statements on the quantum scale. Instead, particles are defined by their probability waves. One of the strangest examples of quantum weirdness comes from the idea that prior to observation by an experiment, an electron does not have a well-defined position, but that by observing it, we “force” it into a particular state. Let’s be clear that quantum theory doesn’t say that we don’t know the position prior to observations, but that such a thing as definite position really doesn’t exist!
The amazing thing is that while the microscopic world is governed by statistics, the macroscopic world seems governed by Newton’s laws, which are themselves deterministic. How can that be, since, in the end, macroscopic objects are made of protons, neutrons, and electrons? We see the same effect when we think about the air in a room. While the individual molecules fly around in a haphazard manner, on the human scale, they normally seem much steadier. In a sense, the distinction between the wave properties and particle properties of matter are really just a function of physical scale. The quantum theory shows that on the smallest scales, particles look more and more wavelike, and are governed more and more by statistics.
This wave-particle duality doesn’t just exist for objects like electrons and protons, however. Isaac Newton originally proposed that light must have particle properties, a theory which was rejected in the nineteenth century when light was observed to exhibit interference patterns, a property of waves. Ultimately, light was understood to have both wave properties, as with radio waves, and quantized particle properties, which came to be known as photons. Modesty must have prevented Einstein from noting that it was his own interpretation of the photoelectric effect that ultimately gave rise to the modern particle picture of light. In this experiment, an ultraviolet beam is shined on metal, and electrons are ejected, a very particle-like behavior. His 1905 paper describing this effect earned him the Nobel Prize in 1921.
Einstein’s Evolution of Physics gives us an insight into the state of science in the early twentieth century, including glimpses into his own considerable contributions. Nearly seventy years later, though they have refined their models considerably, physicists are still dealing with the fallout of the weirdness born from the quantum picture of the universe.
FIELD, RELATIVITY
The field as representation . . . The two pillars of the field theory . . . The reality of the field . . . Field and ether . . . The mechanical scaffold. . . Ether and motion . . . Time, distance, relativity . . . Relativity and mechanics . . . The time-space continuum . . . General relativity . . . Outside and inside the elevator . . . Geometry and experiment. . . General relativity and its verification . . . Field and matter
THE FIELD AS REPRESENTATION
DURING the second half of the nineteenth century new and revolutionary ideas were introduced into physics; they opened the way to a new philosophical view, differing from the mechanical one. The results of the work of Faraday, Maxwell, and Hertz led to the development of modern physics, to the creation of new concepts, forming a new picture of reality.
Our task now is to describe the break brought about in science by these new concepts and to show how they gradually gained clarity and strength. We shall try to reconstruct the line of progress logically, without bothering too much about chronological order.
The new concepts originated in connection with the phenomena of electricity, but it is simpler to introduce them, for the first time, through mechanics. We know that two particles attract each other and that this force of attraction decreases with the square of the distance. We can represent this fact in a new way, and shall do so even though it is difficult to understand the advantage of this. The small circle in our drawing on page 286 represents an attracting body, say, the sun. Actually, our diagram should be imagined as a model in space and not as a drawing on a plane. Our small circle, then, stands for a sphere in space, say, the sun. A body, the so-called test body, brought somewhere within the vicinity of the sun will be attracted along the line connecting the centers of the two bodies. Thus the lines in our drawing indicate the direction of the attracting force of the sun for different positions of the test body. The arrow on each line shows that the force is directed toward the sun; this means the force is an attraction. These are the lines of force of the gravitational field. For the moment, this is merely a name and there is no reason for stressing it further. There is one characteristic feature of our drawing which will be emphasized later. The lines of force are constructed in space, where no matter is present. For the moment, all the lines of force, or briefly speaking, the field, indicate only how a test body would behave if brought into the vicinity of the sphere for which the field is constructed.
FIG. 1.
The lines in our space model are always perpendicular to the surface of the sphere. Since they diverge from one point, they are dense near the sphere and become less and less so farther away. If we increase the distance from the sphere twice or three times, then the density of the lines, in our space-model, though not in the drawing, will be four or nine times less. Thus the lines serve a double purpose. On the one hand they show the direction of the force acting on a body brought into the neighborhood of the sphere-sun. On the other hand the density of the lines of force in space shows how the force varies with the distance. The drawing of the field, correctly interpreted, represents the direction of the gravitational force and its dependence on distance. One can read the law of gravitation from such a drawing just as well as from a description of the action in words, or in the precise and economical language of mathematics. This field representation, as we shall call it, may appear clear and interesting but there is no reason to believe that it marks any real advance. It would be quite difficult to prove its usefulness in the case of gravitation. Some may, perhaps, find it helpful to regard these lines as something more than drawings, and to imagine the real actions of force passing thr
ough them. This may be done, but then the speed of the actions along the lines of force must be assumed as infinitely great! The force between two bodies, according to Newton’s law, depends only on distance; time does not enter the picture. The force has to pass from one body to another in no time! But, as motion with infinite speed cannot mean much to any reasonable person, an attempt to make our drawing something more than a model leads nowhere.
We do not intend, however, to discuss the gravitational problem just now. It served only as an introduction, simplifying the explanation of similar methods of reasoning in the theory of electricity.
We shall begin with a discussion of the experiment which created serious difficulties in our mechanical interpretation. We had a current flowing through a wire circuit in the form of a circle. In the middle of the circuit was a magnetic needle. The moment the current began to flow a new force appeared, acting on the magnetic pole, and perpendicular to any line connecting the wire and the pole. This force, if caused by a circulating charge, depended, as shown by Rowland’s experiment, on the velocity of the charge. These experimental facts contradicted the philosophical view that all forces must act on the line connecting the particles and can depend only upon distance.
A Stubbornly Persistent Illusion Page 26