Einstein’s Results
Thus we approach the absolute character of the surface through the relative nature of the observer’s reference system. There is a relationship common to all values of the g’s that belong to the same curvature. This relationship is expressed by a differential equation. It is this equation of curvature that the airship’s observer must find. Einstein’s problem was similar, but he was concerned with four dimensions, which entailed a general formula with ten g’s, and he had to find a set of differential equations of the second order to determine the law of Fabric curvature. He divided the Fabric into regions: I. World-Frame—beyond influence of energy. II. Empty region—free of energy, but under its influence. III. Region containing free energy only. Each region has a characteristic curvature. By means of an absolute differential calculus—a wonderful mathematical scaffolding erected by Riemann, Christoffel and others—involving the theory of tensors, he succeeded in finding such a set of equations. He kept the following points in view: (1) The equations must not only give the character of region II, but must satisfy the special case of region I; (2) They must be independent of any partitioning system, because the General Theory of Relativity demands that a law of nature be in a form appropriate for all observers whatever their position and motion; (3) They must be concerned with energy which is conserved, not mass which the Special Theory showed dependent on velocity. This set of differential equations which shows how the curvature of the Fabric at any point links to the curvature at neighboring points is the law of gravitation, a law which has been severely tested by the practical observation of the solar eclipse already referred to. At a first approximation these equations degenerate into Newton’s Law. At a second approximation they account for the motion of the perihelion of Mercury, which had hitherto baffled astronomers. All the laws of mechanics are deducible from this law of World-Fabric curvature, i.e. conservation of energy (which includes conservation of mass since we re-define mass as energy) and conservation of momentum (re-defined by a relativist). It must be noted that this law and the General Theory show that the velocity of light is not absolutely constant, but, like everything else, a light-pulse is affected by the Fabric curvature in a gravitational field. In conclusion we will contrast some conspicuous differences in the old world view of classical mechanics and the new view presented by Einstein.
1. A three-dimensional ether medium with variously conceived properties which communicated the supposed inherent attractive force in matter in some unexplained way, and transmitted electromagnetic waves, has been replaced by a four-dimensional external World-Fabric, the geometrical character of which controls the motion of matter (energy) and accounts for all mechanical laws.
2. After separating the observer’s subjective share in definitions from nature’s share in the things defined, space, time, and force, hitherto regarded as absolute, have been shown to be purely relative and dependent on the observer’s track. Mass has also proved to be relative to velocity unless re-defined as energy. As classical mechanics bases all definitions on space, time, and mass units, the relativity of such defined quantities is now apparent.
3. Newton’s laws of motion, his law of gravitation, and the laws of conservation, hitherto regarded as unrelated, are now synthesised in a basic law of mechanics.
Einstein has not disturbed the electric theory of matter, and both the old and new physics have in common the “Principle of Least Action.” We obtain a glimpse of this principle in the unique tracks pursued by freely moving bodies, which may be regarded as tracks of least effort, force only being manifested as an expression of the Fabric’s resentment when bodies depart from these natural tracks. Einstein has approached nearer to the truth in regard to the laws underlying nature, and, as always, this means a simplification. His theory, which entails a readjustment of such fundamental conceptions as space and time, opens up fresh fields to scientific investigation and to philosophic thought. It reveals a bridge uniting the domains of physics and philosophy, and it heralds a new era in the history of science.
[1] Commander McHardy uses the term “event” in a sense somewhat different from that seen in a majority of the essays. He reserves for the four-dimensional element—the instant of time at a point in space—the name “point-event”; and the term “event” he applies to a collection of these forming, together, an observable whole. An actual physical happening, like a railroad wreck or a laboratory experiment, it will be realized is of the latter sort, occupying an appreciable region of space rather than a single point, and an appreciable interval of time rather than a single second. To the element, the “point-event” of Commander McHardy’s essay, this bears the same relation that the geometer’s solid bears to his point. This comment is in no sense to be taken as criticism of Commander McHardy’s terminology, which rather appeals to us; we make it merely to guard against confusion in the reader’s mind.—Editor. [↑]
[2] This paragraph is the result of an editorial revision of the author’s text, designed to retain the substance of his presentation, while tying up what he has to say more definitely with the preceding essays, and eliminating the distinction between finite and infinitesimal intervals, which we believe to be out of place in an essay of this character. We will not apologize to our mathematical readers for having used finite and differential notation in the same equation, in violation of mathematical convention.—Editor. [↑]
[3] Although gravitational force in a small region can be imitated or annulled by accelerating motion, there remains the disturbing influence of gravitational matter already referred to and expressed in the fabric curvature. It is this that defines how unique tracks run, or rather, how bodies progress.—Author. [↑]
XV
THE NEW WORLD
A Universe in Which Geometry Takes the Place of Physics, and Curvature That of Force
BY GEORGE FREDERICK HEMENS, M.C., B.SC., LONDON
It is familiar knowledge that the line, the surface and ordinary Euclidean space are to be regarded as spaces of one, two and three dimensions respectively and readers of this journal are aware that a hypothetical space of four dimensions has been closely investigated. The most convenient space to study is the surface or two-space, since we can regard it as embedded in a three-space. If a surface is curved it is generally impossible to draw a straight line on it, for as we see clearly, the “straightest” line is changing its direction at every point. To describe this property accurately it is necessary to ascribe to each point a magnitude which expresses what happens to the direction of a short line in the region when displaced a short distance parallel to itself. This is called the direction-defining magnitude. Different sets of values of this magnitude relate to surfaces of different curvatures.
A second fundamental property has recently been pointed out. There is inherent in every part of a space a measure of length peculiar to that particular region and which in general varies from region to region. To describe this variation accurately it is necessary to ascribe to each point another magnitude called the length-defining magnitude, which expresses the change from each point to the next of the unit of length. These two magnitudes define the surface completely.
Similarly, a space of any number of dimensions is defined completely by a similar pair of magnitudes. A space is the “field” of such a magnitude-pair and the nature of these magnitudes defines the dimensions of the space. The four-space usually described is the Euclidean member of an infinity of four-spaces.
When we look into a mirror we see a space differing from ordinary space in that right and left are interchanged and this is described mathematically by saying that if we locate points as usual by specifying three distances
,
,
of the point from three mutually perpendicular planes, then a point
,
,
, in actual space corresponds with a point
,
,
in the mirrored space: in other words the mirrored space is derived from the real space by multiplying the
coordinates by
. If we were to multiply by
instead of
we should derive a different space; in this case, however, we have no mirror to show us what it looks like. Such a space is said to have one negative dimension and it has the peculiar property that in the figure derived from the right triangle of ordinary space the square of the “hypotenuse” equals the difference and not the sum of the squares of the other two sides, so that the length of a line may sometimes have to be represented by the square-root of a negative number, a “complex” number.
In considering what at first sight may appear to be fantastic statements made by this theory, it must be borne in mind that all our knowledge of the external universe comes through our sense-impressions, and our most confident statements about external things are really of the nature of inferences from these sense-impressions and, being inferences, liable to be wrong. So that if the theory says that a stone lying on the ground is not a simple three-dimensional object, and that its substance is not the same as its substance a moment before, the matter is one for due consideration and not immediate disbelief.
The idea that the universe extends in time as well as in space is not new, and fiction-writers have familiarized us with wonderful machines in which travellers journey in time and are present at various stages of the world’s history. This conception of the universe, to which the name “space-time” is usually applied, is adopted by the new theory and assigned the status of a physical reality.